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A G A R D

SECOND GUIDED MISSILES SEMINAR

GUIDANCE AND CONTROL

VENICE, ITALY

SEPTEMBER 1956

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North At lant ic Treaty Organizat ionAdvisory Group for Aeronaut icalResearch and Development

SECOND GUIDED M ISSILES SEMINARGUIDANCE AND CONTROL24-28 Septemter 1956, Venice, Italy

Twenty-four pape rs by di fferent author s ar eincluded. The main topic dealt with is theguidance and co nt ro l of m iss i les . Specificp a p e r s c o n s i d e r t h e r a n g e o f p r o b l e m s f r o mthe selection of a missile guidance and contro l sy stem to i t s final testing in actual fl ight.


SECOND GUIDED MISSILES SEMINARGUIDANCE AND CONTROL

24-28 Septemter 1956, Venice, Italy

Twenty-four pape rs by di fferent author s ar eincluded. The main topic dealt with is theguidance and co ntro l of m iss ile s. Specificpapers consider the range of problems f romthe selection of a missile guidance and contro l sy stem to i ts final testing in actual fl ight.


SECOND GUIDED MISSILES SEMINARGUIDANCE AND CONTROL24-28 September 1956, Venice, Italy

Twenty-four pape rs by di fferent autho rs ar eincluded. The main topic dealt with is theguidance and con t rol of m iss i les . Specificp a p e r s c o n s i d e r t h e r a n g e o f p r o b l e m s f r o mthe selection of a missile guidance and contro l sy stem to i ts final testing in actua l flight.


SECOND GUIDED MISSILES SEMINARGUIDANCE AND CONTROL

24-28 September 1956, Venice, Italy

Twenty-four pape rs by di fferent autho rs ar eincluded. The main topic dealt with is theguidance and con t ro l of m iss i les . Specif icp a p e r s c o n s i d e r t h e r a n g e o f p r o b l e m s f r o mthe selection of a missile guidance and contr o l sy stem to i ts final testing in actua l fl ight.

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NORTH ATLANTIC TREATY ORGANIZATION

ADVISORY GROUP FOR i\ERONAUTICAL R E S E J \ R C H i \ND DEVELOPMENT

(ORGANISATION DU TRA ITE DE L'ATL AN TIQU E NORD)

A G A R D

SECOND GUIDED MISSILES SEMINAR

GUIDANCE AND CONTROL

0 0 0 1 0 9 3 8

P a p e r s p r e s e n t e d a t a n A G J \ R D GuidedM i ss i l es S emi n ar , h el d 2 4 -2 8 S ep t emb er 1 9 5 6 ,

in Venice, Italy

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The pa pe rs i nc l ude d i n t h i s A G A R D o gra ph w e re o bt a i ne dt hro ugh t he c o ur t e s y o f t he U ni te d St a t e s A i r F o r c e , t he U ni te dSt a t e s A r m y, t he U nit e d St a t e s N avy, a nd the A i rc ra f t I nd us t r i e sA s s o c i a t i o n . C o l o ne l C . E. Eva ns o f t he A RD C o rga ni z e d a c o m m i t t e e c o n s i s t i n g o f himself, L i e u t e n a n t C o l o n e l R o l a n d V . T i e d e ,C o m m a n d e r W i l li am M . R o m b e r g e r , a nd M a j o r G e n e r a l J . F .P h i l l i p s , w h o o b t a i n e d t h e p a p e r s f r o m t h e i r r e s p e c t i v e o r g a n i z a t i o n s , a nd m a d e a r r a n g e m e n t s f o r t h e p r e s e n t a t i o n in V e n i c e .

Th i s publ i c a t i o n s up po r t e d by t he U ni te d St a t e s A i r F o r c eunde r C o nt ra c t N o . A F 18(603) - 133, m o ni t o re d by t he A i r F o r c eO f f i c e o f Sc i e nt i f i c R e s e a rc h o f t he A i r R e s e a rc h a nd D e ve l o pm e ntC o m m a n d .

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INTRODUCTION

T h e p ap ers p resen t ed i n t h i s v o l u me su rv ey on e of t h e maj or

d i v i s i on s of g u i d ed mi ss i l es d es i g n , an d p rov i d e i n fo rmat i on wh i chwil l be useful to tho se engaged in the design of m is si le s and m is si l es y s t e m s .

In ad d i t i on , t h e S emi n ar i l l u s t ra t ed t h e v al u e of d i scu ss i n gm a t t e r s p e r t a i n i n g t o g u i de d m i s s i l e s i n i n t e r n a t i o n a l m e e t i n g sand pointed the way towards future act ivi t ies in this f ield.

THEODORE VON KARMANCh ai rman , AGARD

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1.

2.

3 .

4 .

5.

6.

7 .

T ABL E OF CONT E NT S

In t rod u ct i on

W eap on s S y st em P h i l osop h y , Georg e H. Cl emen t

0 R0 W eap on S y st em P h i l osop h y , Har l an C. M eal

New P r i n c i p l e s i n t h e Desi g n of S u p er i o r Co mm u n i cat i o n s , Nav i g ati on ,an d M i ss i l e Gu i d an ce S y st ems, W . P . L ear , S r .

G u i da n c e T e c h n i q u e s , W a l t e r L . W e b s t e r , J r .

Con si d era t i on s i n t h e Ch oi ce of a M i ss i l e Gu i d an ce an d Con t ro l S y st em,R o b e r t W . M a y e r

I n e r t i a l N a v ig a ti o n , N o r m a n F . P a r k e r a nd C h a r l e s P . G r e e n i n g

Aiding the Inert ial Navigat ion System, Will iam F. Ballhaus andF r e d e r i c k S t e v e n s , J r .

P a g e

i

1

11

29

39

57

67

87

8 . L i n ea r Ho mi n g Nav i g ati on , Ro b er t K. Ron ey 1 01

9 . P i t fa l l s i n M i ss i l e Co n t ro l , Ro b er t L . Joh n so n 1 15

10 . T h e E f fe c ts o f A i r f r a m e C h a r a c t e r i s t i c s o n C o n t r o l S y s te m D e s ig n ,

F . E . P e r r y 13 9

11. Geomet r i cal S t ab i l i zat i on Based on S erv od r i v en Gi mb al s an d In t eg rat i n gG y r o U n i t s , C h a r l e s S . D r a p e r a nd R o g e r B . W o o d bu r y 1 53

12 . S am p l ed -Dat a S y st em s, Joh n R. Rag azzi n i 1 83

13. Di g i ta l T ech n i q u es in M i ss i l e Gu i d an ce S y s t em s, S id n ey Dar l i n g to n 2 29

14. T h e Use of Di g i t a l Comp u t er T ech n i q u es i n M i ss i l e Desi g n an d Con t ro l ,

D. H. Gr idley 249

15 . T h e A p p l i c a t i o n o f N o i s e a n d F i l t e r T h e o r i e s t o G u i d a n c e P r o b l e m s ,R . J . P a r k s a nd R o b e r t M . S t e w a r t 2 65

i i i

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TA B LE O F C O N TEN TS( C o nt i nue d)

P a g e

16 . R e c e nt D e ve l o pm e n t s in Fi xe d a nd A da pt ive Fi l t e r i ng , A . G . C a r l t o na nd J . W. Fo l l i n , J r . 285

17. P r a c t i c a l P r o b l e m s E n c o u n t e r e d i n M i s s i l e G u i d a n c e a n d C o n t r o l D e s i g n ,R. E . Whiffen 301

18 . A ppl i c a t i o n o f M e t ho ds o f Sc i e nc e t o t he Pro bl e m o f R e l i a bi l i t y ,

C. Raym o nd Knight 315

19 . R e l i a b i l i t y o f G ui ded M i s s i l e s , Edw i n A . Spe a km a n 331

20 . L a b o r a t o r y v s . F l i g h t E v a l u a t i o n o f A i r b o r n e G u i d a n c e C o m p o n e n t s ,

W. H . C l o he s s y 341

2 1 . Tr e n ds in Fi e l d Te s t i ng o f G ui ded M i s s i l e s , Er ns t A . Ste inho ff 347

2 2 . Lo w Si gna l Le v e l M i s s i l e I ns t ru m e nt a t i o n , L. G . de B e y 359

2 3 . O n t h e W a y t o A u t o m a t e d P r o c e s s i n g o f F l i g h t M e a s u r e m e n t ,W . E . K l e m p e r e r 3 75

2 4 . P a p e r o n t he G ui da nc e a nd C o nt ro l o f M i s s i l e s , S t e phe n Wa l dro n 387

iv

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WEAPONS SYSTEM PHILOSOPHYG e o r g e H . C l e m e n t *

SUMMARY

In achieving an effect ive defense and an increased weapons capabil i ty , there has beena n e v e r l a r g e r e x p e n d i tu r e o f o u r n a ti o n a l r e s o u r c e s . T h e d if fi cu lt p r o b l e m e x i s t s inmai n t a i n i n g an ad eq u at e l ev el o f d efen se w i th a mi n i m al d r a i n on n at i on al r es o u r ce s . T h eweap on s sy s t em p h i l osop h y i s essen t i a l l y a p o i n t o f v i ew emb raci n g a t ech n i q u e fo r o rd er i n g ,c l a s s if y i n g, a nd a n a l yz i n g a t e c h n o l o g i c al ly c o m p l e x m e c h a n i s m , o r g a n i z a ti o n o r p r o c e s sso t h at each e l em en t an d each p r o b l em can b e co n si d er ed in i t s p r o p er con t ex t and t re at edin i t s p r o p er p e rs p e ct i v e . T h e ob j ect i v e of su ch an ap p r o ach i s t o a t t a i n t h e o b j ect i v es oft h e sy s t em as a wh ol e wi th a mi n i m u m ex p en d i t u r e of r es o u r ce s . In t e ll i g ent l y ap p l ied , th i sap p ro ach can b e a p owerfu l t oo l a i d i ng t h e d es i g n , d ev el o p men t , and man ag e men t of mi l i t a ryw e a p o n s s y s t e m s .

SOMMAIRE

Dan s l a re al i sa t i o n d 'u n e d efen se ef fi cace e t d 'u n e cap ac i t e acc ru e d es a r m es ,i l y a eu t ou j o u rs un e d ep en se p l u s g ra n d e d e n os re ss o u r ce s n at i o n al es . II ex i s t e u n p ro b l em ediffici le, celui de m aint enir un niveau de defense adequat en depe nsant au minim um lesr e s s o u r c e s n a t i o n a l e s . L a p h i l o s o p h i e d u s y s t e m e d ' a r m e s e s t e s s e n t i e l l e m e n t un p o i ntd e v u e e m b r a s s a n t u n e te c h n iq u e q u i v i s e a o r d o n n e r , c l a s s i f i e r , e t a n a l y s e r u n m e c a n i s m ed on t l a t ech n ol o g i e es t co mp l ex e, u n e o r g an i sa t i on ou u n p reced e^ d e t e l l e m an i e re q u ec h a qu e e l e m e n t e t q ue c ha q u e p r o b l e m e p u i s s e e t r e c o n s i d e r e d a n s s o n p r o p r e d o m a i n e e tt r a i t e d a n s s a p r o p r e p e r s p e c t i v e . L ' o b j e t d ' u ne t e l l e a p p r o c h e e s t d ' a t t e i n d r e l e s b u tsd u s y s t e m e c o n s i d e r e d a n s s o n e n s e m b l e a v e c u ne d e p e n s e m i n i m u m d e s r e s s o u r c e s .

1. INTRODUCTION

Men have long been concerned with thed e f e n s e o f t h e i r n a t i o n a l r e s o u r c e s a n d t h e i rnat ional , way of l i fe against the possiblee n c r o a c h m e n t o f a p o t e n t ia l a t t a c k e r . M i l i t a ry fo rces h av e b een con t i n u al l y d es i g n ed ,d ev el op ed , o rg an i zed , an d t h en mod i f i ed t oc o u n t e r t h e i n c r e a s i n g t h r e a t i m p o s e d b ytechno log ical adva nce and by the changingso ci a l s t r u c t u r e of t h e wo r l d in wh i ch wel i v e . T h e fu n cti on of th ese m i l i t a r y fo rc esh as b een t o d et e r an y p ot en t i a l a t t ac k er f ro m

c a r r y i n g o u t a n a g g r e s s i v e a c t i o n b y i n c r e a s i n g t h e p r o b a b i l i t y o f h i s f a i l u r e .

In t h e re al m of mo d er n mi l i t a r y af fa i rsw e c o n t i n u e t o d e s i r e a n e v e r i n c r e a s i n gcap ab i l i t y t o sa t i s fy o u r d efen se n eed s , andn o o n e, I b el i ev e, wou ld a rg u e ag ai n s t t h efu n damen t al wi sd om of seek i n g t h i s i n cre asedcap ab i l i t y . T h e m arg i n of su p er i o r i t y i n t h ep e r f o r m a n c e o f o u r w e a p o n s , f o r e x a m p l e ,t h e ran g e, sp eed , a l t i t u d e, an d p ay l oad cap ab i l i t i es of o u r a i r p l an es co mp are d wi th t h oseof a potential enemy may spel l the difference

T h e RAND Co rp o ra t i o n , S an t a M o n i ca , Cal i fo r n i a .

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be t w e e n v i c t o ry a nd de f e a t i n a po s s i b l ef u t ure c o nf l i c t. Wha t w e o f t en fa i l t o a p pr e c i a t e i s t ha t t he i nc r e a s e d c a pa bi l i t y w e ne e da nd s t r i v e t o a c h i e v e i s p u r c h a s e d a t t h e c o s to f a n i nc re a s e d c o m pl e xi t y o f t he e qui pm e nta n d o r g a n i z a t i o n t h a t r e s u l t s .

T h e c o n s e q u e n c e o f t h i s i n c r e a s e d c o m p l e x it y i s t o r e q u i r e an e v e r l a r g e r e x p e n d i t u r e o f o u r n a t i o n a l r e s o u r c e s t o a c h i e v ea de qua t e l e ve l s o f de f e ns e e f f e c t i ve ne s s .T h e r e s o u r c e s t h a t w e h a v e a t o u r d i s p o s a la r e a l w a y s l i m i t e d a n d h a v e v e r y r e a la l t e r n a t i v e u s e s . F i s s i o n a b l e m a t e r i a l e x pe nde d i n the c o ns t r uc t i o n o f bo m bs c a nno t beu s e d f o r t h e g e n e r a t i o n o f e l e c t r i c p o w e r .M a n p o w e r c o m m i t te d t o t h e m a i n t e n a n c e o f

t he m i l i t a r y f o rc e s c a nno t be us e d in a nyp r o d u c t i v e c a p a c i t y i n t h e n a t io n a l e c o n o m y .A di ff ic ul t p ro bl e m e x i s t s in a c hi e vi ng a ndm a i nt a i n i ng a n a de qua t e le ve l o f de f e ns ee f f e c t iv e n e s s w i th a m i n i m a l d r a i n o n o u rn a t i o n a l r e s o u r c e s .

T h e m o d e r n i m p l e m e n t s o f w a r f a r e a r e a no r d e r o f m a g n it u d e m o r e c o m p l e x a nd m o r ee x p e n s i v e t h an t h e i r p r e d e c e s s o r s o f a fewy e a r s a g o . T h e i n c r e a s e d r a n g e , g r e a t e r

s p e e d , a n d h i g h e r a l t it u d e o f o u r b o m b e r s h a sre s ul t e d i n t he ne e d f o r be t t e r na vi ga t i o na la i d s , b o m b i n g an d f i r e c o n t r o l m e c h a n i s m s ,a n d c o m m u n i c a t io n e q u i p m e n t ; a l s o l o n g e rr u n w a y s , m o r e h ig h ly t r a i n e d a i r c r e w s , a ndm o r e e x t e n s i v e m a i n t e n a n c e a n d r e p a i r o r g a n i z a t i o n s . T h e a i r p l a n e a s a w e a p o n h a sb e c o m e m o r e t h a n a s i m p l e a s s e m b l a g e o fp o w e r p l a n t , f u e l, a i r f r a m e a nd c o n t r o l s . I ti s no w a c o m pl e x a nd h i ghl y i n t e gr a t e dm e c h a n i s m c o n s i s t i n g o f g r o u n d a n d a i r bo rne c o m po ne nt s w hi c h w e c a l l a w e a po n

sy ste m . I would l ike to spend the next fewm i n u t e s e x p l o r i n g t h e p o t e n t ia l o f a p a r t i c u l a rpoint of view, which we cal l the weaponss ys t e m phi l o s o phy, a s i t a pp l i e s t o t hed e v e l o p m e n t o f m i l i t a r y e q u i p m e n t .

2 . TH E SY STEM S C O N C EPT

When we speak of weapons we usual lyt h i nk o f t he i ndi v i dua l e l e m e nt s , t he t o o l so f w a r f a r e . F i r e p o w e r o n t a r g e t i s t ho u g h to f i n t e r m s o f b u l l e t s , b o m b s , a n d g r e n a d e s .M e c h a n i s m s f o r t r a n s p o r t i n g t h is f ir e p o w e rt o t he v i c in i t y o f t he t a r ge t b r i ngs t o m i nds u c h t h in g s a s g u n s , t a n k s , t r u c k s , a i r p l a n e s ,a nd g ui de d m i s s i l e s . T h e p e o p l e w ho o p e r a t et he e qui pm e nt a re t ho ught o f a s i nf a nt rym e n,p i l o t s , n a v i g a t o r s , r a d i o o p e r a t o r s , a ndg u n n e r s . T h e d e v i c e s o f c o m m u n i c a t i o n a r et h o u g h t o f a s t e l e p h o n e s , r a d i o s , a n d r a d a rse ts . In a sen se eac h of the se things taken byi tself i s a weapo n. But I think i t i s re ad i lyeviden t that e ach of the se taken by i tsel f i sva l ue l e s s . A bul l e t i s o f l i t t l e m i l i t a r y us e

w i t ho ut a gun, w i t ho ut a s o l d i e r t o f i re t heg u n , w i t h o u t s o m e m e a n s o f c o m m u n i c a t i o n t od i re c t t he e f f o r t o f t he s o l d i e r , a nd s o o n .

I s t he n a w e a po ns s ys t e m a n a ggre ga t i o no f t he s e c o m po ne nt s ? In a s e n s e ye s , but i ti m p l i e s m o r e t h a n t h i s . T h e w e a p o n s s y s t e mphi l o s o p hy i s i n t i m a t e l y c o nc e r ne d no t o n l yw i t h t he c o m po ne nt s o f w a rf a re s i ngl y a ndi n a g g r e g a t e b u t a l s o w i t h t h e i n t e r a c t i o n sa n d i n t e r r e l a t i o n s h i p s a m o n g t h e s e c o m p o n e n t s , w i th t h e e x p e n d i t u r e of r e s o u r c e s

n e c e s s a r y t o b r i n g t h e m i l i t a r y fo r c e in t obe i ng a nd ke e p i t o pe ra t i ng a t a n a de qua t el e ve l o f e f f e c t i ve n e s s , a nd w it h t he p r o c e s so f us e o f t he i n t e gra t e d w ho l e a s a m e a ns o fa c h i e v i n g s o m e d e s i r e d m i l i t a r y p u r p o s e .

What then would be an adequate def ini t iono f a w e a po ns s y s t e m ? I t h ink t ha t w e ha vet o re c o gni z e t ha t t he c o nc e pt o f a w e a po nss ys t e m c a n m e a n d i f f e re nt t h i ngs t o d i f f e re nti ndi vi dua l s , de pe ndi ng upo n t he i r po i n t o fv i e w . T h e p a r t i c u l a r c o n c e p t o f a w e a p o n s

s y s t e m c a n r e p r e s e n t d i ff e r e n t l e v e l s o fa gg re g a t i o n de pe ndi ng upo n the i n t e re s t a ndc o nc e rn o f t he i ndiv i dua l v i e w e r . Thus t hehe a d o f a na t i o na l go ve rn m e nt m a y be c o nc e r n e d w i th a s o c i a l s y s t e m , a p o l i t i c a l

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s ys te m , a n e c o no mic s ys te m , a nd a mi l i ta r ys ys te m a s the ins t r um e nts o f na t io na l po l ic y .The ma n in c ha r ge o f the mi l i ta r y s ys te mviews i t in turn as being composed of an airfo r c e , a n a r m y, a nd a na vy; in o the r wo r d s ,

a n a i r , a g r o und, a nd a s e a s ys te m . Thechief of the air fo rce thinks in te r m s of ana i r d e fe n s e s y s t e m , a s t r a t e g i c a i r s y s t e m ,a t a c ti c a l a i r s y s t e m , an a i r t r a n s p o r t s y s t e m ,and so on.

Skipping a few ste ps , o ne can view the a ir p la ne itself, the in te r c e pt o r in the a i r de fe ns ea r m , a s a s y s t e m . T h e i n t e r c e p t o r in t u r nc o nta ins s uc h th ings a s a p r o puls io n s ys te m,a na viga tio n s ys te m, a c o m munic a t io ns y s t e m , a f i r e c o n t r o l s y s t e m , a n d s o o n .

Fo r the pur po s e s o f th is d is c u s s io n , Ith ink we c a n l imi t o ur s e lve s to c o ns ide r ingsuch things as the air defense system, thes t r a t e g i c a i r s y s t e m , o r t h e t a c t i c a l a i rs ys te m a s be ing we a po ns s ys te ms a t thehighest level of aggregation with which wewill have to dea l . Accepting this conc ept,the c o mple te we a po ns s y s te m inc lude s the nno t o nly the ve hic le to ge the r wi th i t s a r ma me nt , c o m mun ic a t io ns , na viga tio n , f i r e c o nt r o l , a nd o the r s y s te m s ; but a ls o , a nd a s aninte gr a l pa r t o f the s ys te m, the s uppo r t ingfa c i l i t ie s , the ba s e s , the c o mmunic a t io nsnet, the maintenance and test equipment, thelo gis t ic a l s upply a nd t r a ns po r t , the t r a in ingins ta l la t io ns , the m e c ha n ic s , a nd o the r s p e c ia l iz e d pe r s o nne l . All o f the s e c o mpo ne ntsa r e jus t a s muc h a pa r t o f the we a po nssystem as the vehicle itself.

It is useful when thinking of weaponss y s t e m s a n d t h e i r c o m p o n e n t s t o r e g a r d t h e ma s being d i f fe r e nt o r de r s o f . the s a m e kindof thing, al l members of a continuum ofs y s t e m s , s u p e r s y s t e m s , a n d s u b s y s t e m s . T othe p r o puls io n e ngine e r a n e ngine is a s ys te m .At the same t ime i t is a member of a highero r d e r s ys te m , the a i r c r a f t in whic h i t i sins ta l le d . Fu r t he r , the e ngine i s c o m po s e d

o f m a n y l o w e r o r d e r s y s t e m s o r s u b s y s t e m ssuch as an ignit ion system, a fuel system, ane xha us t s ys te m , a nd s t i l l o the r s . W e s e ethe n tha t the s ys te ms a ppr o a c h, the we a po nss ys te m s phi lo s o phy, i s in pa r t a te c hnique

o f c la s s i fy ing a nd o r d e r ing a c o m ple xm e c h a n i s m o r p r o c e s s s o th a t e a c h e l e m e n to r p r o b l e m c a n be c o n s i d e r e d in i t s p r o p e rp e r s p e c t i v e .

The s ys te m s a ppr o a c h is no t a new c o nc e ptno r did it find its o ri gin in dealing withm i l i t a r y p r o b l e m s . S y s t e m s a n a l y s i s ,s ys te ms e ngine e r ing, the s ys te ms c o nc e pthas been applied for many years by thee ngine e r s a nd ma na ge r s o f s uc h indus t r ia le n t e r p r i s e s a s l a r g e t e le p h o n e c o m p a n i es a nd

t h e p r o d u c e r s a n d d i s t r i b u t o r s o f e l e c t r i cpo we r . As we ha ve s e e n, the s ys t e m sa p p r o a c h a r i s e s o u t o f c o m p l e x it y a nd r e p r e s e n t s a m e t h o d o f o r d e r i n g an d c o n t r o l l in gfor effective use a technologically complexs t r u c t u r e .

3. A TYP ICAL AIR DEFE NSE SYSTEM

At this point it will be useful to examinea typic a l we a po ns s y s te m s uc h a s a n a i rd e fe n s e s y s t e m . A s w e d i s c u s s e d e a r l i e r ,

the pur po s e o f a de fe nse s ys te m is to de te ra ny po te nt ia l a t ta c ke r f r o m c a r r y ing o ut a na g g r e s s i v e a c ti o n by i n c r e a s i n g t he p r o b abil i ty of his fai lur e o ver that which he mightexpect in the absence of our defensive effor t .In fact we seek to ma xim ize the pro babil i tyof his fai lur e by co nstr ucti ng as effective ad e fe n se s y s t e m a s o u r l i m i te d r e s o u r c e s w i l lp e r m i t .

A c o nve nie nt me a s u r e o f the e ffe c t ive ne s sof an ai r d efense sy ste m is a quanti ty we cancall the "k il l po ten tial ." Kill po tential isdefined a s the expected n um ber of bo m ber sde s t r o ye d by the de fe ns e s ys te m unde r s o mestand ar d co ndit ion. In cho o sing between twoai r d efense sy ste m s that can be developed ando pe r a te d fo r a n e quiva le nt e xpe ndi tur e o f

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r e s o u r c e s , t h e s y s t e m h a v i n g t h e h i g h e r k i l lp o t e n t ia l w o u ld b e p r e f e r r e d . S i m i l a r l y ,act i on s wh i ch t en d t o i n crease t h e k i l l p otent ia l of the sy ste m a s a who le wo uld bec o n s i d e r e d d e s i r a b l e , w h i le t h o s e t h a t t en d t o

red u ce t h e k i l l p o t en t i a l wou l d b e con si d eredu n d esi r ab l e . Now th e o p er a t i o n al fu n cti on s ofa d e p l o y e d a i r d e f e n s e f o r c e a r e c o n c e r n e dwi th t h e d et ect i o n , i d en t i fi cat i on , i n t e r ce p t io n , a nd d e s t r u c t i o n o f h o s t i l e a i r c r a f t .

In o r d e r t o d e te c t h o s t i l e a i r c r a f t , s o m emet h od of su rv ei l l an ce of t h e a i r t raf f i c i nt h e r eg i o n t o b e d efen d ed mu st b e p ro v i d ed .T h i s i s c o m m o n l y a c c o m p l i s h e d b y m e a n s o fa r a d a r s u r v e i l l a n c e n e t d e p l o y e d o v e r t h eg eo g ra p h i c a r ea t o b e d efen d ed, an d fu r t h e r ,

i n o r d e r t o m a x i m i z e w a r n i n g t i m e , r e a c h i n gas fa r b eyon d t h e p e r i m et er o f t h e d efen d eda r e a a s i s p r a c t i c a l . T h e s e r a d a r i n s t a l lat io ns may be lo cated at f ixed si te s on landand augmented by picket ships and picketa i r c r a f t a lo n g c o a s t a l f r o n t i e r s . T h e r a d a rn et may fu r t h er b e su p p l eme n t ed by aco u st i ca la n d o p t i c a l d e t e c t i o n n e t s s u c h a s o p e r a t e din the U nited Sta tes by the Gro und O bs er ve rC o r p s . T h i s d e t e c t io n n e t i s r e s p o n s i b l e f o ra c q u i r i n g r a w d a t a p e r t a i n i n g to t h e c u r r e n ta i r t raf f i c s i t u at i on , map p i n g i n d i v i d u al a i r

c r a f t o r g r o u p s o f a i r c r a f t , an d d e t e r m i n i n gt h ei r a l t i t u d e an d t h ei r n u m b er . T h e i n d i v id u al d et ect i o n i n s t a l l a t i o n s mu st b e t i edt o g e t h e r b y m e a n s o f a n e x t e n s i v e c o m m u n i c a t io n s y s t e m o v e r w h ic h t h e r a w s u r v e i l l a n c ed at a can b e t ran smi t t ed t o i d en t i f i cat i onf a c i l i t i e s .

At the identif icat ion faci l i ty , the raws u r v e i l l a n c e d a ta a r e r e c e i v e d , p r o c e s s e d ,and eva luate d. Individual piec es of ai rm o v e m e n t s i n f o r m a t io n a r e c o r r e l a t e d toe s t a b l is h a i r c r a f t t r a c k s . In t h is p r o c e s s ,su p p l em en t ar y d at a may b e re q u es t ed andre cei v e d f ro m t h e d et ect i o n s i t e s . Af t er th et rack s a re es t ab l i sh ed , t h ey mu st b e i d en t i f i edas f r i en d l y o r h o s t i l e . Un kn own s mu s t b e

su b j ect ed t o su ff i ci en t i n v est i g at io n t o d e t e r m i n e t h e i r s t a t u s . T h e i nd iv i du a l t r a c k s a r es u m m a r i z e d t o d e s c r i b e t h e a i r s i tu a t i o n int h e g eo g ra p h i c a r ea fo r wh ich t h e i d en t i f i cati on faci l i ty i s r esp o n si b l e . T h i s a i r

s i t u at i on i n fo r mat i o n t h en mu st b e co mm u n i cat ed t o i d en t i f i cat i on faci l i t i es resp on si b l efo r ad j acen t g eog rap h i c a reas an d t o t h ea p p r o p r i a t e i n t e r c e p t i o n f a c i l i t i e s .

At th e i n t e rc ep t i o n faci li t y th e a i r s i t u at i ond at a a r e re cei v ed an d th e t h r eat i s ev al u at ed .On t h e b as i s o f t h i s ev al u at i on an ap p rop r i a t ef ract i on of t h e av ai l ab l e d efen si v e fo rce i sc o m m i t te d t o t h e a i r b a t t l e . O r d e r s a r ei s s u e d t o s c r a m b l e i n t e r c e p t o r s a nd t o b r i n gs u r f a c e - t o - a i r g ui de d m i s s i l e a nd a n t i a i r

c r a f t a r t i l l e r y u n i t s i n to a c t i o n . I n fo r m a t io non th e p o s i t i on of t h e h o s t i l e a i rc ra f t i sco mm u n i cat ed in t h e fo r m of g u i d an ce i n s t r u c t i o n s t o t h e a p p r o p r i a t e w e a p o n s u n it s o ft h e a i r d e f e n s e f o r c e .

At t h e weap o n s u n i t s , t h e g u i d an ce i n fo rm a t i o n r e c e i v e d f r o m t h e i n t e r c e p t i o n f a c i l it yi s emp l oy ed t o b r i n g f i re p o wer t o b ea r ont h e h o s t i le a i r c r a f t . I n t e r c e p t o r s a nds u r f a c e - t o - a i r g uid ed m i s s i l e s a r e v e c t o r e dt o t h e v ic i n i ty of t h e h o s t i l e a i r c r af t w h er e

t h e i r a i r b o r n e s e a r c h , d e t e c ti o n , a nd t r a c k in geq u i p men t en ab l es t h em t o c l o se on t h een emy an d b r i n g t h e h os t i l e a i rc raf t u n d erf i r e . A n t i a i r c r a f t a r t i l l e r y u n i t s r e c e i v ep o s i t i o n i n fo r m a t i o n o n h o s t i le a I rc raf t so t h att h e i r f i r e p o w e r m a y b e p r o p e r l y e m p l o y e d .

T e c h n o l o g i c a l a d v a n c e s i n t h e a i r c r a f t a r ta re con t i n u al l y i n creas i n g t h e ran g e, a l t i t u d e,an d sp eed of b o mb ard m en t v eh i c l es . T h e sep e r f o r m a n c e i n c r e a s e s , p a r t i c u l a r l y t he i n c r e a s e in s p e e d , m e a n s th a t o v e r th e y e a r s ,

l es s and l es s t im e wil l b e av ai l ab l e to b r i n gt h e a i r d efen se fo r ce i n to act i o n . T o co u n t ert h e con t i n u al l y sh r i n k i n g t i me av ai l ab l eb et ween i n i t i a l d et ect i on an d t h e b omb re l easet i m e , d ev el op men t of eq u i p men t t o p erfo rm

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many of the detection, identification, interception and destruction functions of the airde fe ns e fo r c e by a uto ma t ic o r s e mia uto ma t icme ans is und er way. Since the detectio n,identification, and inter ceptio n facil i t ie s of an

a i r d e f e n s e f o r c e a r e c o m m o n l y r e f e r r e d t oa s the "gr o und e nvi r o nm e nt" th is new e quipment is being called the SemiautomaticGr o und Envi r o nme nt o r SAGE Sys te m.

At this point one might well say, "Howdo es this affect m e ? " Th is is a l l well andgood but how do es this apply to pr o ble m s thata r is e in the de ve lo pme nt o f mis s i le guida nc ea nd c o nt r o l s ys te m s ? How do e s th is affe ctme ? To i l lus t r a te th is le t us fo r the mo me nta s s ume tha t the guida nc e a nd c o nt r o l s ys te m

we a r e c o nc e r ne d wi th i s to be c o me a pa r to f a s u r f a c e - to -a i r m is s i le whic h in tu r n i sneeded as a compo nent of an ai r defensesyst em such as the one we have just d esc r i be d.

4. THE DEVELO PMENT O F A SYSTEM

As we a l l r e c o gniz e , the p r o c e s s o f de s igna nd de ve lo pme nt i s o ne o f c o mpr o mis e .De c is io ns ha ve to be ma de th r o ugho ut th isp r o c e s s a m o n g a l t e r n a t i v e m e t h o d s o f

a c h ie v in g p a r t i c u l a r g o a l s . F o r e x a m p l e ,s ho uld the mis s i le guida nc e s ys te m c o nta ins o me me a ns o f ho ming o n the ta r ge t? Ifs o , s h o u l d t h e m i s s i l e c a r r y a n a c t i v e r a d a rfo r te r mina l guida nc e o r s ho uld i t ma ke us eo f e ne r gy r e f le c te d f r o m the ta r g e t a i r c r a f twhen i l luminated by a gro und in stallat io n?Or s ho uld the g r o und ins ta l la t io n t r a c k thepresent posit ion and velocity of both theta r g e t a i r c r a f t a nd the m is s i l e and o n theba s is o f th is info r ma t io n c o mpute the p r o pe rma n e uve r to be pe r fo r m e d by the m is s i le a nd

is s ue ins t r uc t io ns to i t t o do s o ?

The r e s o lu t io n o f the s e que s t io ns c a nno tpr o pe r ly be m a de o n the ba s is o f e xa miningthe guidance and control function in isolation,nor even on the basis of examining the

p e r f o r m a n c e o f t h e m i s s i l e b y itself; r a t h e ri t must include a consideration of the influenceof such questions on the operation of the airde fe ns e s ys te m a s a who le .

So me m ight a r gue that we a r e c o nc e r ne dhe r e wi th the a c c ur a c y o f the m is s i le ; thati t is obvious that if the missile can be mademo r e a c c ur a te the n i t wi l l c e r ta in ly r e s ul ti n m o r e k i l l s .

It is not a t a l l c lear , however , that makingt h e m i s s i l e a s " a c c u r a t e a s p o s s i b l e " w i llne c e s s a r i ly r e s ul t in ma xim iz ing the e ffe c t ive ne ss o r kill po tential of the defense sy stem. In this case, the numerical value of thekil l potential is given as the product of three

q u a n t i t i e s . F i r s t , t h e n u m b e r o f m i s s i l e sthat can be launched during the engagement;s e c o n d, the p r o ba bi l i ty tha t a g iven mis s i lewi l l no t ma lfunc tio n o r r e s ul t in a g r o s se r r o r ; a nd th i r d , the p r o ba bi l i ty tha t a no n-ma lfunc t io ning m is s i le k i l l s the bo mbe r . Ino ur e x a mple a n inc r e a s e in the a c c ur a c y o fthe mis s i le wo uld r e s ul t in inc r e a s ing thevalue of the th ird facto r in the kil l potentialequation , the pr o babil i ty that a no n-ma lfunc t io ning m is s i l e wo uld k i l l the bo mb e r .This wo uld r e s ul t in a n inc r e a s e d k i l l po te nt ia lonly if the value of the fi rs t two fact o rs inthe e qua t io n a r e no t ma te r ia l ly de gr a de d.

If in the c o ur s e o f a c hie ving the g r e a te ra c c ur a c y, a ddi t io na l guida nc e a nd c o nt r o le quipme nt w e r e a dde d to the m is s i le , ther e s u l t ing inc r e a s e in we ight might r e d uc ethe r a nge o r s pe e d o f the m is s i le which inturn could r esu lt in reducing the num ber ofmissiles that could be launched during thee nga ge m e nt . Fu r t he r , the a dde d e quipme ntmight ad ver sely affect the r el iab il i ty of them i s s i l e . E i t h e r o r b o th o f t h e s e p o s s i b i l i ti e sthen might well reduce the effectiveness of thesys tem as a whole in spite of the enhancedacc ur acy of the unit . I think i t is c le ar , then,that the manifold dec is io ns and cho ices amonga l te r na t iv e s tha t mu s t be ma de dur ing the

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cou rse of t h e d es i g n an d d ev el op men t of am i s s i l e m u s t i n c lu d e a c o n s i d e r a t i o n o f t h einfluence and feedback on the weapo ns sy ste mas a who le as well as the effect on the p a r t i cu l ar comp on en t i n q u est i on .

F r o m t h e p oi n t o f v i ew of t h e m i ss i l ed es i g n e r , t h en , i t i s i mp o r t a n t t o h av e ana p p r e c i a t i o n , a n u n d e r s t a n d i n g o f t he m i l i t a r yen v i ro n m en t i n wh i ch t h e d ev i ce wi l l o p e ra t e .It is n e c e s s a r y t o a p p r e c i a t e t h e r e l e v a n tt e c h n i c a l , o p e r a t i o n a l , a nd l o g i s t i c a l f a c t o r so f t h i s e n v i r o n m e n t t o u n d e r s t a n d t h e i n t e r a c t i o n s a n d i n t e r r e l a t i o n s h i p s a m o n g t h e s efact o rs an d h ow t h ey i n fl u ence th e op e ra t i o nof the s ys tem as a who le. I l ike to think oft h e s e t e c h n i c a l , o p e r a t i o n a l , a nd l o g i s t i c a lfact o rs as b ei n g t h e an at o my , th at i s , th es t r u c t u r e o f t h e w e a p o n s s y s t e m . T h et e c h n i c a l f a c t o r s i n c l u d e s u c h a r e a s a s p r o p u l s i o n , f u e l s , g u i d a n c e , c o n t r o l , c o m m u n i cat ion, and the l ike.

E n g i n e e r s a r e a c c u s t o m e d t o t hi nk o f t h ep e r f o r m a n c e o f m i l i t a r y a i r v e h ic l e s in t e r m sof su ch q u an t i t i es as ra n g e , sp ee d , a l t i tu d e,r a t e o f c l i m b , m a n e u v e r a b i l i t y , a nd a c c u r a c y .I t is gen er al ly held that an exce llent vehic le ison e i n wh i ch t h e v al u es of t h ese p erfo rman cep a r a m e t e r s a r e m a d e a s l a r g e a s p o s s i b l e .

T h e l e v e l o f p e r f o r m a n c e a c t u a l l y a c h i e v e dw i th a p a r t i c u l a r v e h i c l e , h o w e v e r , d e p en d sn ot on l y u p on t h e cu r ren t l ev el o f t ech n ol og y ,b ut a l s o u pon t h e d es i g n ch o i ces t h at a r ema d e in t h e c o u r se of t h e d ev el o p men t oft h e v eh i cl e .

I t hi nk w e a l l r e c o g n i z e f r o m l o n g e x p e r i en ce t h at t h e d es i g n an d d ev el op men t ch oi ceswe ma ke a r e l ikely to affect al l , not justo n e o f th e p e r f o r m a n c e p a r a m e t e r s . F o rex amp l e , t h e ch oi ce of t h e p owerp l an t can

influence not only the speed of the missi leb u t a l so i t s ran g e, a l t i t u d e, ra t e of c l i mb ,an d man e u v er ab i l i t y as we l l . I t fo l l ows t h ent h at q u i t e i n d ep en d en t f rom an y con si d era t i ono f ho w th e d e s i r e d p e r f o r m a n c e p a r a m e t e r s

were sp eci f i ed , t h e f i n al resu l t wi l l , i n a l ll i k e l i h o o d , r e p r e s e n t a c o m p r o m i s e i n w h i c hm a n y o f th e d e s i r e d p e r f o r m a n c e l e v e l s a r eon l y p ar t i a l l y sa t i s f i e d . T h e q u esti on i s t h en" W h at a r e go o d c o m p r o m i s e s ? " T o a n sw e rt h i s q u est i on we mu st ex ami n e t h e res t o ft h e w e a p o n s s y s t e m a n a t o m y , t h e o p e r a t i o n a la n d l o g i s t i c a l f a c t o r s .

T h e op era t i on al fact o rs wh i ch ap p l y i n c l u d e su ch c o n si d er a t i o n s as t h e mo b i l it y oft h e s y s t e m , i t s s t a t e o f r e a d i n e s s , i t s r e l i ab i l i t y , i t s su sce p t i b i l i t y t o co u n t e r-m e a s u r e s , a nd i t s d a t a r e q u i r e m e n t s . T h es p e c i f i c m e a n i n g a t t a c h e d t o t h e s e c o n s i d e r a t i on s an d t h e i r i n f lu en ce on t h e ef fect i v en essof t h e en d eav o r d ep en d s u p on t h e ch ar ac t er

of t h e w eap on s sy s t e m wi t h wh ich we ar edealing, that is , whether i t be an ai r defenses y s t e m , a s t r a t e g i c a i r s y s t e m , o r a t a c t i c a la i r s y s t e m .

M o b i li ty of t h e sy s t em en ab l es t h e fo rceto be rapidly deployed and moved as theo c c a s i o n d e m a n d s . It e n a b l e s t h e c o n c e n t r a t i o n o f f i r e p o w e r a t t i m e s a nd p l a c e swh er e i t i s n eed ed . As ap p l i ed t o m i ss i l e s ,t h e o p er a t i o n al co n cep t of mo b i l i ty m ean s

that the effect ive ra nge and spe ed of thes y s t e m s h o u l d b e a s g r e a t a s p o s s i b l e . T h i sran g e an d sp eed can b e su p p l i ed e i t h er b yt h e i n h eren t ra n g e an d sp eed of t h e m i ss i l eor by the capabil i ty of the weapon systemfor ra p i d d ep l oy m en t an d mo v eme n t t o n ewb a s e s o f o p e r a t i o n .

T h e s t a l e of read i n ess of t h e sy s t em i si ts abi l i ty to re spo nd quickly when cal le du p on . In an a i r d efen se sy s t em o n ly t h o sem i ss i l e s t h at a r e av ai l ab l e can b e of an y

use when the at tac k c o m es . If the guidanceo r c o n t r o l e q u i pm e n t in t h e m i s s i l e r e q u i r e sa n e x c e s s i v e w a r m u p o r p r e p a r a t i o n t i m e ,the mi ss i le wil l be of l i t t le us e to the defens eo p e r a t i o n .

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R e l ia bil i ty r e f e r s to the p r o ba bi l i ty tha tthe system will function within its designl imi ts whe ne ve r i t i s c a l le d upo n to pe r fo r m.Again, for ex am ple , if the guidan ce andc o nt r o l s ys te m is me c ha niz e d by e quipme nt

of such a delicate nature that i t cannot bem a i n t ai n e d u n d er n o r m a l o p e r a t i n g c o n d i t i o n s ; o r if in an effor t to sque eze out thela s t b i t o f pe r fo r ma nc e the e quipme nt i spushed into operation at levels beyond i tscapac ity; o r i f the equipment is co ntinuallys ubjec te d to unkno wn o r po o r ly unde r s to o de nvi r o nme nta l c o ndi t io ns , a h igh fa i lu r e r a tew i ll r e s u l t . T h e g r o s s e r r o r s , m a lf u nc ti o n s,and excessive maintenance that fo l low do notcontr ibute to the effectiveness of the defenses y s t e m .

The da ta r e qui r e me nts o f the s ys te m ha veto do wi th the qua nt i ty , c ha r a c te r , a nd r a teof flow of the information that is essentialto the o pe r a t io n . B e fo r e a de fe ns e s ys te mc a n br ing f i r e po w e r to be a r o n a n a t ta c ke r ,the bo mb e r s mus t be de te c te d , t r a c ke d , a ndidentified. Again, if the data r eq uir em en tsof any co mpo nent, such as the guidance andc o nt r o l e quipme nt , a r e e xc e s s ive o r unus ua lo r if they incr ea se the susc eptib il i ty of thes y s t e m t o c o u n t e r m e a s u r e s , t h e n t h e p e r

formance of the system as a whole wills uffe r .

Tur ning no w to the lo gis t ic a l fa c to r s weinc lude c o ns ide r a t io n o f s uc h i te m s a sfa c i l i t ie s , e quipme nt , s upply , ma in te na nc e ,p e r s o n n e l , an d t r a i n i n g . L o g i s t ic s p r o v i d e sthe me a ns fo r the c o nduct o f a i r o pe r a t io nsa nd is a n impo r ta nt fa c to r in de te r mining theeffectiveness of the weapo ns sys tem . Duringthe c o u r s e o f the de ve lo pme nt o f o u r s u r f a c e -t o - a i r m i s s i l e t he d e s i g n e r , e i t h e r e x p l ic i t lyo r impl ic i t ly , i s de te r mining the lo gis t ic a lr e qui r e me nts o f the s ys te m. B e fo r e de c idingto use that unusual compound as a fuel , onthe basis of, say enhancing the range perfo r ma nc e o f the mis s i le , the lo g is t ic a l

implications of the decis ion should be investi gate d. Can i t be ma nufactu red andt r a n s po r te d to the po int o f us e in the r e q ui r e dqua nt i t ie s ? Amo ng i t s p r o pe r t ie s i s thepr o po s e d fuel s o to xic a nd c o r r o s iv e tha t i t

r e q u i r e s i n o r d i n a t e l y c o m p l e x s u p p l y , t r a n s po r t , s to r a ge , and ha ndling pr o c e du r e s ? Ifs o , t h e o v e r a l l p e r f o r m a n c e o f t h e s y s t e mmay be deg ra ded in sp ite , of the enhancedr a nge tha t wo uld r e s ul t .

Now that we have exam ined the anatomy ofa weapon system and found that i t containsma ny e le me nts whic h p la c e s e e mingly c o nf l ic t ing r e qu i r e me nts o n the s ys te m, 1 thinki t i s fa i r to a s k a ga in "Ho w c a n o ne de te r m inew h at i s a g o o d c o m p r o m i s e ? " Ho w c a n o n e

d e t e r m i n e a p r e f e r r e d c o u r s e o f a c ti o n i n t hemid st of this w elte r of conflict ing detail?To investigate this question let us againr e s ta te o ur unde r s ta nding o f the we a po nssystem philosophy and then examine the ro leo f o p e r a t i o n s r e s e a r c h a s a p p l i e d t o t h epr o ble m o f we a po ns s ys te m de ve lo pm e nt .

The weapons system philosophy is a pointo f v iew, e m br a c ing a te c hnique fo r o r d e r in gand classifying a techno log ically com plexm e c h a n i s m , o r g a n i z a t i o n , o r p r o c e s s s o t h a t

e a c h e l e m e n t o r p r o b l e m c a n b e c o n s i d e r e din i ts proper context to the end that theobjective of the system as a whole may ber e a l iz e d wi th a minima l e xpe ndi tur e o f r e s o u r c e s .

As we no te d e a r l ie r , the s ys te m a ppr o a c his not a new concept but has been applied formany years as an aid in the design, developme nt , a nd m a na ge me nt o f la r ge a nd c o m ple xi n d u s t r i a l e n t e r p r i s e s . T h e f i r s t f o r m a la ppl ic a tio n o f th is te c hnique to mi l i ta r ypr o ble ms a ppe a r s to ha ve be e n ma de by theBr it ish dur ing Wo rld War 11, when theydeveloped and employed a management devicetha t ha s c o me to be kno wn a s o p e r a t io nsa na lys is . At tha t t ime the B r i t i s h we r ec o nc e r ne d wi th de te r mining ho w be s t to

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emp l oy t h e weap on s , eq u i p men t , an d p erson n el t h ey t h en h ad av ai l ab l e t o mostef fect i v el y accomp l i sh an ass i g n ed mi l i t a rym i ss i o n . T h e su c ce ss of t h i s t ech n i q u e inc o n s e r v i n g r e s o u r c e s o r i n m a x i m i z in g t he

mi l i t a ry ef fect i v en ess of a g i v en fo rce wass o o n a p p a r e n t . O t h e r n at i o n s q u i ck l y r e c o g nized the benefi ts of this technique, andap p l i ed i t t o t h ei r own p rob l ems of h ow t oob t a i n t h e max i mu m effect i v en ess f romex i s t i n g weap on s sy s t e m s. As we n ot ede a r l i e r , t he t r e n d to w a r d g r e a t e r te c h n o l og i cal comp l ex i t y of mi l i t a ry weap on ssy st e m s h as co n t i nu ed an d i s co n t in u i n g a t anev er acce l er a t i n g p ac e. I t hi nk i t n at u r a l ,t h en , t h at t h e sy s t e m s an al y s i s t ech n i q u e b eapplied not only to the use of exist ing

weap on s sy s t e m s b u t sh ou l d be ex t en d ed anda p p l i e d t o s o l v i n g p r o b l e m s t h a t a r i s e d u r i n gt h e c o u r se of t h e d es i g n and d ev el op m en t ofw e a p o n s y s t e m s a s w e l l .

T h e ap p l i cat i on of t h e sy s t ems an al y s i st e c h n i q u e , o r o p e r a t i o n s r e s e a r c h a s i t i ss o m e t i m e s c a l l e d , to p r o b l e m s a r i s i n g int h e d es i g n an d d ev el o p men t of weap o n ss y s t e m s a f fo r d s u s t he o p p o r t u n i t y o f a r r i v i n ga t d e c i s i o n s t h a t a r e i n th e p r o p e r c o n t e xt ,t h at a r e co n si s t en t wi t h t h e o v er a l l ob j ect i v esof t h e weap o n s sy s t e m . It p ro v i d es t h ef r a m e w o r k f o r f o r m i n g j u d g m e n t s a n d h e l p su s to p h r a s e a n d s e l e c t t h e c r i t e r i a o n t h eb as i s o f wh i ch we can d e t er m i n e an an s we rt o o u r e a r l i e r q u e s t i o n " W h a t i s a g o o dc o m p r o m i s e ? "

5. PRINCIPLES FOR ANALYSIS OFWEAPONS SYSTEMS

D r , C h a r l e s H i t c h , o f t h e R A N D C o r p o r a t i o n , h a s d e v e lo p e d p r i n c i p l e s t ha t a r e o fsu ch i m p o r t a n ce i n an al y zi n g weap o n s sy s t emd ev el op m en t p r o b l em s t h at I wou ld li k e t ore p ea t two of t h em h e r e . T h e f i rs t p o i n t Iwi sh t o d raw f rom Dr . Hi t ch 's wo rk i s t h at inan al y zi n g a weap on s sy s t em p rob l em, t h e

effects o f a p ro p o sed act i on mu s t b e as se ss edat least at the next higher level of aggregation,a nd f u r t h e r , m u s t b e a s s e s s e d in t e r m s o ft h e op era t i on o r fu n ct i on t h at t h e weap on ss y s t e m i s p e r f o r m i n g .

R e c a l l i n g o u r e a r l i e r e x a m p l e w h en w ewe re d i scu ss i n g t h e i nf lu en ce of a t t emp t i n gt o i n c r e a s e t he a c c u r a c y of t he s u r f a c e - t o - a i rm i s s i l e g u i d a n c e a n d c o n t r o l s y s t e m , w el e a r n e d t h a t a n i n c r e a s e i n m i s s i l e a c c u r a c ywou ld n ot n ec ess ar i l y b e co n si s t en t wit h t h eob j ect i v es of t h e weap on s sy s t em at i t s n ex th i g h er l ev el o f ag g r eg at i o n . T h at i s , ani n c r e a s e i n m i s s i l e a c c u r a c y w o u l d n o tn ec es sar i l y l ead t o an i n c re as e in t h e k i llp o t en t i a l o f t h e a i r d efen se sy s t em ; in fact ,

i t mi g h t ev en m at e r i a l l y re d u ce t h e o v er a l lef fect i v en ess of t h e sy s t e m . I t was n ot g ooden ou g h t o assess t h e i n f l u en ce of t h e p rop osedactio n o n the bas is of i ts effect on the pe r fo rm an ce of th e mi ss i l e . T h e ef fects o f t h ep r o p o s e d a c t i o n m u s t b e a s s e s s e d a t l e a s t a tthe next higher level of aggr ega tio n, and mu stb e a s s e s s e d in t e r m s o f t h e o p e r a t i o n o rfun cti on t h e weap on sy s t e m i s p er fo rm i n g .

The seco nd point that I wish to dr aw fro mD r . H i t c h ' s w o r k a l s o c o n c e r n s t h e s e l e c t i o no f c r i t e r i a in t e r m s of w h ic h t he c o n s e q u e n c e so f s o m e p r o p o s e d c o u r s e o f a c t i o n a r eas se ss e d. In analyzing the influence ofp r o p o s e d d e v e l o p m e n t a l t e r n a t i v e s o n t h eeffect i v en ess of a weap on s sy s t em , i t i s acom mo n an d of ten fa t a l mi s t ak e t o se l e ctc r i t e r i a t h at c o n c e n t r a t e a t t e n ti o n o n a s i n g l ei n p ut ; t o ma x i m i ze so me o b j ect i v e 's fu nct ion ,say b o m b er s k i l l ed , fo r a g iv en q u an ti t y ofinput , say number of picket ai rcraft in thed et ect i o n n et , o r t o mi n i mi z e t h e n u m b er ofp i c ke t a i r c r a f t r e q u i r e d t o a c h i e v e a c e r t a i nn u m b e r o f b o m b e r s k i l l e d .

T o d o t h i s c o m p l e t e ly o v e r l o o k s t h er e s o u r c e s u b s t i tu t i o n p o s s i b i l i t i e s th a t m a yex ist . An o bject ive can usually be achievedby v a r i o u s c o m b i n a t i o n s o f r e s o u r c e s . A n a i r

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defense sys tem of any specified level ofeffectiveness may be achieved with ver ydiffe r e nt mix e s o f g r o und e nvi r o nme nt , in te r c e p t o r s , a n d s u r f a c e - t o - a i r g u i d e d m i s s i l e s .S o m e r e s o u r c e s u b s t i t u t i o n p o s s i b i l i t i e s a r e

di r e c t a nd o bvio us ; ma ny o the r s a r e indi r e c tand subtle . Judic io us application of theo p e r a t i o n s r e s e a r c h t e c h n i q u e t o w e a p o n ss ys te m de ve lo pme nt p r o ble ms wi l l unc o ve rthe fact that the po ssib il i t ie s of substi tutio nin c a r r y ing o ut a ny o pe r a t io n a r e typic a l lymuc h gr e a te r tha n a r e a t f i r s t a ppa r e nt .

At this point I would like to inter ject ano te o f c a ut io n . The we a po ns s ys te m philoso phy like any useful too l can not only beproperly and usefully applied, but i t can also

be wantonly m isapplie d with deva stating r e s u l t s . The weapons system philosophy and i tsr e la te d c o nc e pts a nd te c hnique s , s ys te m sa n a l y s i s o r o p e r a t i o n s r e s e a r c h , a r e p o w e r ful to o ls . Pr o pe r ly us e d they fur nis h ab a s i s , a fram ew o rk , that helps us thinkthr o ugh a nd unde r s ta nd a c o m ple x pr o ble m .It r e pr e s e nts in no wa y, ho w e ve r , a s ubs t i tu tefor thought. If the techniqu es we have beendis c us s ing a r e b l indly , unima gina t ive ly , o runthinkingly applied o nly r i go r m o r t is canr e s u l t .

Pr o pe r a ppl ic a t io n o f the we a po ns s ys te ma ppr o a c h to d e s ign a nd de ve lo pme nt p r o ble m se na ble s us to c ho o s e f r o m a mo ng a s e t o fa l te r na t ive s tho s e tha t a r e go o d a s o ppo s e dto tho s e tha t a r e po o r . It ma y e ve n pe r mitus to r a nk the c ho ic e s a c c o r ding to s o m es c h e m e o f p r e f e r e n c e f r o m b e s t t o w o r s t .H o w e v e r , s e a r c h e s f o r t h e " o p t i m u m " o r" b e s t p o s s i b l e " a l t e r n a t i v e s a r e l ik e ly t obe no t o nly t ime -c o ns um ing, but a ls o f r u i t l e s s . The funda me nta l r e a s o n fo r th is , I

believe, is that the analytical t reatment ofa ny pr o ble m r e q ui r e s the c o ns t r uc t io n o f a na na lyt ic mo de l whic h a ppr o xima te s the r e a lwo r ld s i tua tio n . The de gr e e o f a ppr o xima t io nc a n r a nge f r o m qui te c r ude to ve r y s o phi s t ic a te d , but i t s t i l l r e ma ins a n a ppr o xima t io n .

While ver y useful infer ence s can be madefro m a s tudy of the mo del, it should be keptin mind that it is a model, an idealizeda ppr o xima t io n , a nd o ne s ho uld gua r d a ga ins to ve r -a na lyz ing o r us ing the mo de l to o bta in

r e s u l ts in a r e a s fo r whic h i t i s uns ui te d .

A s e c o nd po s s ib le mis a ppl ic a t io n o f thewe a po ns s ys te m phi lo s o phy c a n a r is e in thef ie ld o f ba s ic r e s e a r c h . W e ha ve be e n d i s cussing today the weapons system philosophyand how it applies to the design and development of m ili ta ry equipm ent. I think we havet o r e c o g n i z e h e r e t h a t t h e r e i s a c h a r a c t e r is t ic difference between develop ment andfundame nta l r e s e a r c h . De ve lo pm e nt a c t ivi t ies ar e aimed at a specific goal , a new or

impr o ve d wea po n, ma c hine .o r p r o c e s s . B a s icr e s e a r c h ha s no s uc h s pe c i f ic go a l , i t do e s ,ho w e ve r , ha ve a pur po s e , unde r s ta ndingnature in al l i ts bewilder ing and infinitev a r i a t i o n s . F u n d a m e n ta l s c i e n c e s e a r c h i n gthro ugh th is w ild er nes s ad ds piece by pieceto o ur s to r e of know ledge. As each new factis bro ught to l ight so methin g that was unknowno r u n c e r t a i n b e c o m e s kn o w n, b e c o m e s m e a s ur ed, d eter mi ned , and ver ified. I think i t isa ls o unive r s a l ly a ppr e c ia te d tha t in th iste c hno lo gic a l a ge a s t r o ng mil i ta r y , in fa c t as t r o ng s o c ie ty , mus t r e s t o n a f i r m ba s e o ffundamental scie nce . In fact , a l l technologyha s i t s r o o ts in a nd gr o ws o ut o f ba s icscientific und ersta ndin g. I t is in this factand in the fact that basic research is essential ly an exploration of the unknown that Ithink the dang er l ie s . In m o st applicatio ns ofthe we a po ns s ys te m phi lo s o phy we a r e in te r e s t e d i n d e t e r m i n i n g a p r e f e r r e d c o u r s e o fa c t io n f r o m a n a na lys is of the p r e dic ta bleconsequences that would result i f any of anum be r o f a l te r na t ive a c t io ns we r e ta ke n.How can one predict the consequences of the

unknown? It mu st f i r s t be found, disc o ver ed,m ea sur ed , and ver ified before i t can beevalua ted. If the patt er n of thinking r ep r e sented by the weapons system philosophy wereto be m isemp lo yed, say in an effor t to

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ORO WEAPON SYSTEM PHILOSOPHYHa r la n C . M e a l*

SUMMARY

The func tio n o f o pe r a t io ns r e s e a r c h i s to s e le c t a nd o r ga n iz e info r m a t io n whic h wi l la s s is t mi l i ta r y p la n ne r s to a r r ive a t de c is io n s le a ding to a n o r d e r ly s uc c e s s io n o f we a po nss y s te m s . The ta s k i s c o mpl ic a te d by the r a pid te c hno lo gic a l a dva nc e s whic h o f te n r e nde ra weapon o bso lete b efo re i t is even placed in pro ductio n. All the facto rs such as s tudy, r e s e a r c h a nd de ve lo pm e nt , c o ns t r uc t io n o f p r o to ty pe s , te s t ing , p r e pr o duc t io n , p r o duc t io n ,t r a in in g, de ve lo pme nt o f ta c t ic a l do c t r i ne , a nd f inal ly de plo yme nt a nd o pe r a t io n o f we a po nsmus t be take n in to a c c o unt be fo r e s e t t l ing o n the a do pt io n o r d is c o nt inua nc e o f a p r o gr a m .Selection of weapon p ro gr am alte rn ativ es mu st be made with the effectivene ss of the wholede fe ns e s ys te m in mind. A s o und we a po ns p r o gr a m mus t p r o vide c o nt inuing e ffe c tive ne s so ve r a lo ng t im e . Fo r th is r e a s o n, a c o nt inuing r e ne wa l of the a na ly s is o f we a po ns s ys te m s ,and their effectivene ss in the pr ese nc e of changing thr ea ts is a task that is under taken byt h o s e e ng ag ed in o p e r a t i o n s r e s e a r c h .

SOMMAIRE

Le but de la r e c he r c he e n o pe ' r a tio n e s t de s e le c t io nne r e t d ' o r g a n is e r 1 'info r m a tio nq ui a s s i s t e r a l e s p r o j e t t e u r s m i l i t a i r e s , a a r r i v e r a d e s d e c i s i o n s c o n c e r n a n t u n e s u c c e s s i o no r d o n n e e d e s y s t e m e s a r m e s . L a t a c h e e s t c o m p l i q ue e p a r l e s p r o g r e s t e ch n o l o g i q u esr a pide s qui s o uve nt r e nde nt une a r me de s ue te a va nt me me qu ' e l le ne s o i t mis e e n p r o duc t io n .To us le s fa c te u r s te ls que 1'etude, la r e c he r c he e t le de ve lo ppe me nt , la c o ns t r uc t io n depr o to type s , le s e s s a is , la p r e p r o duc t io n e t la p r o duc t io n , la mi s e e n r o ute , le deve lo ppe me ntd ' une do c t r ine de ta c t ique e l f ina l le me nt le de plo ie me nt e t [ ' o pe r a t io n de s a r me s do ive nt e t r epris en compte avant la mise sur pied et 1'adoption o u l e r e j e t d 'u n p r o g r a m m e . Un p r o g r a m m e d ' a r m e s s o l i d e m e n t e t a b li d o i t p o u v o i r p r o c u r e r d u r a n t u ne l o ng ue p e r i o d e , u nee ff ic a c i te c o n t inue . Po ur c e la , une a na lys e c o nt inue l le me nt r e no uve l le e de s s ys te m e sd ' a r m e s e t de le ur e ff ic a c i te e n p r e s e nc e de me n a c e s c ha nge a nte s a e te e n t r e p r i s e pa r le sp e r s o n n e s t r a v a i l l a n t d a n s l a r e c h e r c h e e n o p e r a t i o n s .

T h e g o a l o f p r e s e n t A r m y p r o g r a m s i s t opro vide a high level of effectivene ss o ver along per io d of t im e. The o bject of the sep r o g r a m s i s to d e t e r , t h e r e f o r e t h e r e i s n ope a k ye a r , no r a c e to bui ld a pa r t ic u la r we a po ns ys te m to ma ximum e ffe c t ive ne ss a t a ny pa r t i c u l a r t i m e . T h e s e c o n s i d e r a t i o n s , w h ic happly to guided weapon sy ste m s a s they do toa l l Ar my pr o gr a m s , c a n be s a id to e mbo dy the

c u r r e c t Ar m y we apo n s ys te m phi lo s o phy. Iwi l l no t e la bo r a te fu r the r o n th is phi lo s o phybut instead i l lu str ate how that philosophy ispr act ice d. Specifically, I want to show howthe Ope r a t io ns R e s e a r c h Off ic e o f The J o hnsHo pkins Unive r s i ty i s a t te mpt ing to de ve lo patechnique for supplying the Army with better

infor matio n on which to b ase i ts planningd e c i s i o n s .

"Ope r a t io ns R e s e a r c h Off ic e , The J o hns Ho pkins Unive r s i ty .

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A co ntinuing high level of effec t iven ess isach i ev ed b y p l ann i n g re se ar ch an d d ev e l o p m e n t a n d p r o c u r e m e n t p r o g r a m s w h i c h l e a dt o a n o r d e r l y s u c c e s s i o n o f w e a p o n s s y s t e m s .T e c h n o l o g i c a l a d v a n c e s o c c u r a t s u c h a r a t e

t h at , i n o rd er t o car ry ou t t h i s p l an n i n g t ask ,a g r e a t d e a l o f i n f o r m a t i o n i s r e q u i r e d . T h i sm u s t b e s e l e c t e d a nd o r d e r e d in f o r m a t i o n , n o tj u s t a l l t he i n f o r m a t i o n . T h e p r o c e s s o fsel ect i n g an d o rd er i n g t h i s i n fo r mat i o n wi l li l l u s t r a t e ho w t h e A r m y p h il o s o p h y i s p r a c t iced , s ince the info rm atio n used in anyp l an n i n g p rocess wi l l i n f l u en ce, i f n o t d et e rmi n e, t h e resu l t i n g p l an .

In t h e an al y s i s o f weap on s sy s t ems, e i t h erof fen si v e o r d efen si v e, we s t a r t wi th a g iv en

cap ab i l i t y of a weap o n . T h i s co m es f rom t h eweap on d es i g n e r . As a resu l t o f a re q u es t byt h e m i l i t a r y , u su al l y i n t h e fo r m of m i l i t a r yc h a r a c t e r i s t i c s , t he w e ap o n d e s i g n e r s p r o d u ce a d es i g n p r o p o sal fo r a weap on wh i chwi l l shoo t a cer t a i n d i s t a n ce , wi th a sp eci fi eda c c u r a c y , c a r r y i n g a l i s t e d p a y l o a d , a n d s ofo r t h . Al th ou gh a g r eat d eal o f i mp r o v em en tha s o cc ur r ed , i t is of ten in the specifying ofm i l i t a r y c h a r a c t e r i s t i c s t h a t p l a n n i n g d i f f i c u lt y f i r s t o c c u r s . O fte n t he m i l i t a r y p l a n n e r sa r e a l l t o o f a m i l i a r w i t h t h e t a c t i c a l p r o b l e m

and the kind of weapon which would solve it ,if i t co uld be buil t , but a r e not fam il ia r enoughwith the sta te of the weapon ar t to ask for af e a s i b l e w e a po n o r w e a po n s y s t e m . F o re x a m p l e , a s e t o f m i l i t a r y c h a r a c t e r i s t i c smi g h t cal l fo r an an t i a i rc ra f t mi s s i l e wh ichh as a r an g e of 2 0 mi l e s , an a l t i t u d e cap ab i l i t yf rom zero t o 6 0 , 0 0 0 feet , p rob ab i l i t y of k i l lag ai n s t a s i n g l e a i rc ra f t (e i t h er a l on e o r i nfo r mat i o n ) of 0 . 8 , a m i ss i l e weig h t of n o t mo r ethan 600 lbs. , and with the ent i re unit havingt h e mob i l i t y of a s t an d ar d m i l i t a r y t ru ck .

I am su r e t h at t h e ab su r d i t y of su ch a co m b i n at io n o f c h a r a c t e r i s t i c s i s o b v i o u s , b ut i tmay n ot b e so ob v i ou s t o t h e mi l i t a ry manwh o h as b een man y t i m es am azed by t h e

a c c o m p l i s h m e n t s of t he r e s e a r c h a n d d e v e l o p m e n t o r g a n i z a t i o n s , a n d h a s g r a d u a l l ybeen co nvinced that they can do anything.O p e r a t i o n s r e s e a r c h c a n a s s i s t bo t h t he m i l i t a r y p l a n n er an d th e weap on d es i g n er by

t ra n sl a t i n g ca p ab i l i t i es i n to ef fect i v en ess andb y t r a n s l a t i n g t a c t i c a l e f f e c t i v e n e s s i n t o r e q u i r e m e n t s t o d e v e lo p an i n c r e a s e d c a p ab i l i t y .

Th is is ju st o ne exa m ple of how diff icul t iesin s e l e c t i n g a nd o r d e r i n g in f o r m a t i o n o c c u r .T h e r e a r e m a n y s u c h e x a m p l e s . T o g e t t ot h e r e s e a r c h a n d d e v e l o p m e n t g r o u p s t h einfo rm atio n which they need to design goo dw e a p o n s , t o ke e p t he m i l i t a r y o r g a n i z a t i o n si n fo rm ed as t o t h e cu r r en t s t a t e of t h eweap on a r t , t o r e l a t e bo t h of t h ese to t h eat t ack cap ab i l i t y an d t h e co s t o f p ro p o sedp r o g r a m s , a nd t o d o t h i s i n m i l i t a r y r a t h e rt ha n t e c h n ic a l t e r m s , r e q u i r e s m u c h h a n dl in gof info rm atio n. The way in which this is donei l l u s t ra t es h ow t h e p h i l osop h y i s p u t i n t op r a c t i c e .

L et u s now look a t how th i s p r o ce ss act u al l yt ak es p l ac e . S t a r t i n g i n i t i a l ly wi th p ro p o sednew we apo ns co ming up to f i l l a need, howd o es t h e Army d eci d e wh i ch o n e, o r wh atm i x t u r e , t o a d o p t? T h e o p e r a t i o n s r e s e a r c ht ea m s ex am i n e t h e p r o b l em in t h i s fash i on .Re fer ri ng to Fig . 1 , in taking an ini t iale s t i m a t e o f th e p r o b l e m , t h e o p e r a t i o n sa n a l y s i s i s c e r t a i n t o b e v e r y u n s o p h i s t ic a t e d .T h e weap on d es i g n ers es t i mat e t h at t h ey canhave the we apo ns by 1958 which will havec e r t a i n c a p a b i l i t i e s . I n te l li g e n ce e s t i m a t e sthat in 1958 the ai r thr ea t is , say , 8-1200b o m b e r s , s i m i l a r t o U . S . B - 5 0 ' s , f ly in g a tmed i u m al t i t u d e. T h e ef fect i v en ess s t u d yc o m b i n e s t h e s e t w o w it h s o m e s i m p l e m a t h e m at i c s an d e s t i m at e s t h at i t wi l l t ak e 50 0

units of the type in quest io n to l imit thed a m a g e t h e e n e m y b o m b e r s c a n d o t o t h ed e f e n d e d t a r g e t s y s t e m t o s o m e s p e c i f i e da m o u n t . F u r t h e r m o r e , t h e s e u n i ts w i ll h a v ea c e r t a i n c o s t .

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In do ing th is f i r s t a na ly s is o f the p r o ble m ,a check is m ade that the design pro po sed istechnically feasible and that reasonable leadt im e s fo r de ve lo pme n t , te s t ing , to o l ing,t r a in ing, p r o duc t io n , a nd de plo yme nt ha ve

been allow ed fo r . No ver y detailed s tudy oftac tics is m ade, but at leas t a par t of the to tals pe c t r um o f e ne my c o ur s e s o f a c t io n wi l l beexamined to see what the enemy can do thath u r t s m o s t . M o s t c o m m o n a m o n g s u chsimp lificatio ns is that the effect of co un ter -measures will be ignored and that low alt i tudea t ta c ks ne e d no t be c o ns ide r e d.

The a na lys is o f e ffe c t ive ne s s the n be c o me squi te s imple , o r a t le a s t s o i t s e e ms a f te rt h e f a c t. C o m m o n l y , a s s u m p t i o n s a r e m a d e

to avo id c o n s ide r ing the m o r e c o m ple x of theo pe r a t io na l que s t io ns . An e xa mp le i s thes ta te me n t that f i r e wi l l be d is t r ib ute d e i th e runifo r mly o r a t r a ndo m o ve r the a t ta c ke r s ina n y p a r t i c u l a r e n g a g e m e n t . A m o r e s w e e p i n gkind of assumption is that one can look ato nly pa r t o f the we a po n s ys te m, fo r e xa mple ,the weapon itself, igno r ing the info r ma t io ng a t h e r i n g a n d c o n t r o l s y s t e m .

The pr o ble m o f c o s t i s no t ve r y s o phi s t icated ei the r . I t may s imply s tat e that the

c o s t o f the r e qui r e d numbe r o f we a po ns i s abil l ion do llar s and i t wil l cost a hundr edm i l l io n d o l l a r s a y e a r to o p e r a t e t h e m .C o mpa r is o n o f the r e s ul ts o f s uc h a na lys e s fo rs e v e r a l s y s t e m s a l lo w s t h e c o m p a r i s o n o fc o s t -e ffe c t ive n e s s r a t io s , the we a po n g ivingthe minimum c o s t o v e r e ffe c t ive ne s s be ing theb e s t .

Actually, I have already skipped the realf i r s t , o r m o s t e l e m e n t a r y a n a l y s i s w h i c hdevelops a figure of merit fo r a unit and the

co st of that unit. Fo r exam ple, i t has beensaid in the past that the kil l po tential o rexpected number of kil ls per battal ion, dividedby the effective annual cost , gave an effectivene s s -c o s t r a t io which c o uld be us e d to c o mpa r e weapo ns. Effective annual co st is the

a nnua l c o s t o f the s ys t e m , p lus an a m o r t i zation of the init ial investm ent o ver theexpected useful l ife of the system, usuallyf o u r y e a r s .

What is wrong with such an analysis?Fun dam entally i t igno re s the effect of t im eand that things change with tim e. Since it isthe c a s e tha t no ta c t ic a l o r s t r a te gic s o lu t io nwill remain per t inent for long, i t is a lso thecase that analysis of a weapon or weaponss y s t e m p r o g r a m , e v en if d i r e c t e d t o w a r d as pe c i f ic t ime , mus t r e f le c t the s e c ha nge s .

In the pa s t , po l ic ie s we r e wo r ke d o uts lo wly o v e r a pe r io d o f ye a r s a nd mo dif ie ds lo wly. A we a po n wo uld be de s igne d, a p r o to

type constructed and extensively tested, a fewmo dels fabr icated and put into the hands ofthe m i l i ta r y us e r to t r a in wi th a nd e s ta bl is hdo c t r ine , lo ng be fo r e a ny tho ught o r que s t io nof quanti ty pro ductio n came up. In fact ,usually no s ignificant pro ductio n would takepla c e unt i l ho s t i l i t ie s we r e immine nt o r ha da l r e a dy br o ke n o ut . The t ime s c a le wa s s olong that it could be taken to be infinite, thati s , the effects of changes with t ime ignored.

The t im e s c a le no w is no t muc h s h o r te r inye a r s tha n it wa s twe nty ye a r s a go . Thedifferen ce is that we mo ve thro ugh i t muchfa s te r . W e a po ns a nd we a po ns s ys te m s a r edesigned, buil t , tes ted, and produced inqua nt i ty , a nd d is c a r de d a s o bs o le te , wi tho utev er being used in co mb at. In addit ion , thetechnology mo ves so fast that even befo re aweapon is in pro ductio n the ideas which ma kei t o bs o le s c e nt e xis t ; be fo r e the t r o o ps ha vebecome familiar with a new weapon, thes upe r s e ding we a po n is we l l in to de ve lo pme nt .

It must be so , s ince offense and defense bothhave rapidly moving technologies which theyboth must exploit if they are not to fallbehind. It is a littl e like the legen dar y RedQueen who had to run as fast as she could ino r d e r t o s t a y w h e r e s h e w a s .

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I n o rde r t o run e ve n a l i t t l e f a s t e r , a no r g a n i z a t io n a l o r p la n n in g s c h e m e , a p h i l o so phy, mu st ex ist which al lo ws , in factr e q ui r e s , t he e f f e c t s o f t i m e c ha n ge s t o bes ho w n i n t he a na l ys i s o f o pe ra t i o na l va l ue o f

a ny w e a po n o r s uc c e s s i o n o f w e a po ns . In ara pi d l y c ha ngi ng te c hno l o gy a s t a t i c a na l y s i s ,which does not show the effects of changesw i th t i m e , i s w o rs e t ha n va l ue l e s s , i t i sd a n g e r o u s .

Let us look now at how the ana lys iss c h e m e s ho w n i n t he f i r s t c ha r t m u s t bem o di fi ed i n o r de r t o s uppl y p l a nn e r s w i thi nf o rm a t i o n re l a t i ve t o t he e f f e c t i ve ne s s o fa p re di c t e d de f e ns i ve c a pa bi l i t y i n m e e t i nga p r e di c t e d o f f e nsi ve c a pa b i l i t y . M o re i m po r t a n t l y , t he s c he m e m us t be a bl e t o s ho who w a s u c c e s s i o n o f o f f e ns ive c a pa bi l i t i e s c a nb e m e t by a n o r d e r l y d e v e l o p m e n t an d d e p l o y m e n t o f m o r e a n d m o r e a d v a n c e d w e a p o n s .

o f f e ns i ve c a pa bi l i t y , t he s a m e a va i l a bi l i t ya na l ys i s a s f o r t he de f e ns i ve c a pa bi l i t i e s m us tbe do ne . It s ho ul d s pe c i fy , i ns o f a r a s po s s i b l e ,r a n g e s , a c c u r a c i e s , e t c . , a n d t h e d a t e s t h e s ec a n be e xpe c t e d t o be o pe r a t i o na l l y a va i l a bl e .

I n a s i m i l a r f a s h i o n , t h e s p e c t r u m o f d e f e ns i ve c a p a bi l i t i e s i s p r e di c t e d . Th i s i se a s i e r , h o w e v e r , f o r w e c a n g o a n d c o n f e rw i th t he de s i g ne r a nd de t e r m i ne w ha t d i f fi c u l t i e s h e e x p e c t s t o e n c o u n t e r in t h e p r o c e s so f de ve l o pi ng t he de vi c e i n que s t i o n . A l s o ,a nd v e r y i m p o r t a n t , w e c a n f ind o ut ho w hep r o p o s e s t o m o d i fy t h e p r e s e n t d e s i g n w i thd e v e l o p m e n t s e x p e c te d t o m a t e r i a l i z e f r o mi de a s a l re a dy e xi s t e nt . A go o d e xa m pl e o ft h i s i s t he us e o f t r a n s i t o r s . G i ven a gui ded

m i s s i l e w h i c h i s c o n s t r u c t e d u s i n g v a c u u mt ube s , w he n w i l l t ra ns i t o rs be us e d a nd w ha ti m p r o v e m e n t s c a n b e e x p e c t e d i n r e l i a b i l i t y ,w e ig h t r e d u c t i o n , l o w e r n o i s e f i g u r e r e c e i v e r s , a nd o t h e r f a c t o r s .

F i r s t , a de t a i l e d p ro j e c t i o n o f t he o f f e ns ivecapa bi l i ty is needed. Th is can be o btained toa c e r t a i n e x t e n t f r o m i n t e ll i g e n c e s o u r c e s ,but it i s a l s o b a s e d o n p r o j e c t i o ns fro m t heknown sta te of the a r t . Tha t is , if the enemyha s kno w l e dge o f c e r t a i n k i nds o f m i c ro w a ve

po w e r s o urc e s , t he n i t i s c e r t a i n t ha t he c a nha ve a c e r t a i n k i nd o f ra da r , a f t e r a l e a dt i m e w hi c h i s p r e di c t a bl e w i t hi n f a i r l y na r ro wlim its . In this way, the ava i labi l i ty of a wideva r i e t y o f o f f e ns i ve c a pa bi l i t i e s c a n be p re dicte d. No te that he r e i t i s not enough top r e d i c t a s u c c e s s i o n o f w e a p o n s o f t h e s a m et ype a s , f o r e xa m p l e , the re p l a c e m e nt o f ap i s t o n - e n g i n e d b o m b e r f o r c e b y a j e t b o m b e rf o rc e . A l s o , new w e a po ns s y s t e m s a nd t hec a pa bi l i t y o f us i ng ne w t a c t i c s m us t beincluded . Th e pha sing of di fferen t type s of

e l e c t r o n i c c o u n t e r m e a s u r e s i s a n e x a m p l e .T h e a r r i v a l o f b a l l is t i c a nd c r u i s e m i s s i l e si s a no t he r o bvi o us k i nd o f t h re a t w hi c h m us tbe include d. But , to r ep ea t , it i s not enoughs i m pl y t o i nc l ude i t . Fo r a ny pa r t i c u l a r

I t i s not enough to confer only with thede s i gn e r , but t he e ngi n e e r i n g a nd p ro duc t i o nstaffs m ust es t im ate how long i t wi l l take toge t t he m o di f ic a t io n i n t o p ro duc t i o n , p ro duc ei t , and modify the exis t ing e quip me nt . Only ifthis is do ne in a deta i led and pain stakin g

m a n n e r , h o w e v e r , c a n t h e a n a l y s i s p r o d u c er e s ul t s w hi c h a r e o f po s i t i ve va l ue t o t hep l a n n e r s .

Fig. 2 shows what the resul t of such a s tudymight be. Th is is the sam e type of lead t im ea na l ys i s a s i s do ne in t he s t a t i c c a s e m e nt io ne d be f o re . Th i s i s , ho w e ve r , o n l y t hebe gi nni ng, f o r a t s o m e po i n t i n t he p ro c e s s anew idea , an invent ion , may a pp ear and wed es i r e to know what effect , if any, this canh a v e o n t h e p r o g r a m . F o r e x a m p l e , a d v a n c e s

in r a d a r t e ch n o l o g y m a y m a k e d e s i r a b l e t h eaddi t ion of a se ek er to the m is si le , if i t i s ana i r de f e ns e m i s s i l e , o r if i t i s a n o f f e ns i vew e a po n p ro gra m , a ne w na vi ga t i o na l t e c hni quem a y a p p e a r t o b e f e a s i b l e .

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Of cou rse , t h e same so r t o f t h i n g mu st b eacc o m plishe d if the in te r es t is in an offensiveweap on sy s t e m . T h at i s , t o p ro j ect o f fensi veweap on cap ab i l i t i es we mu st con si d er t h ed e f e n si v e m e a n s a t t h e d e f e n d e r ' s d i s p o s a l .

T h e Army 's g u i d ed weap on p h i l osop h y i s n o td i f fe ren t i n t h i s case , a l t h ou g h t h e a rea ofo p e r a t i o n s m a y b e . In t h e f ie ld a r m y ' so p e r a t i o n s t h e s u r f a c e s u p p o r t m i s s i l esy s t em wi l l b e a t tack i n g a som ewh at d i f fe ren tkind of target than the Army is defending inthe co ntinenta l United State s. If the defensiveo b j ect i v e i s t h e a i r d efen se of a f ie l d a r m y , wemu st look a t t h e sa m e p ro b l em fro m b ot hs i d e s of t h e l i n e . W e mu st work t h e sa m ep r o b l e m i n o p p o s i te d i r e c t i o n s .

S ome of th e fact o rs f ro m b ot h s i d es of t h e

p r o b l e m m a y b e t h e s a m e . F o r e x a m p l e ,a n a i r d e f e n s e m i s s i l e s y s t e m m a y h a v e as u r f a c e s u p p o r t c a p a b i l i ty . T h e p r e s e n c e o fan a i r d efen se sy s t em may l i mi t t h e t act i calmo bil i ty of the gro und fo rc e and in that wayi n fl u ence i t s n at u r e as a t a r g et , and so on .

What is th e differ ence be tween the effect i v e n e s s a n a l y s i s th a t p r e d i c t s e f f e c ti v e n e s so v er t i me an d t h e on e wh i ch was men t i on edb efo re , t h e s t a t i c an a l y s i s? T h i s p ar t o f t h etechniqu e is no t yet so far advance d as ar e the

t wo i n pu t an a l y se s men t i o n ed . T o d at e , a l lwe h av e b een ab l e to d o is t o p r ed i c t ef fect i v en ess a t su ccess i v e p oi n t s i n t i me an d d raw asmoot h cu rv e b et ween t h e p oi n t s , wh i ch i s , Ithink, ade qua te. The thing that is needed mo sti s a t ech n i q u e fo r an a l y s i s o f t h e t o t a lsy s t em ef fect i v en ess a t each p oi n t i n t i m e. S ofar , we a re mat ch i n g cap ab i l i t i es , b al an ci n gan offensive weapon system with a defensivew e a po n s y s t e m . W e a r e n o t , i n m a n y c a s e s ,ev en co n si d er i n g t h e co mp l et e weaponsy s t em , ma t ch i n g on l y weap on cap a b i l i t i es i n

s t ead of d et e rmi n i n g weap on sy s t em ef fect i v e n e s s .

T h i s s i t u a ti o n i s i l l u s t r a t e d by c o n s i d e r i n gh ow ef fect i v en ess an al y ses of a i r d efen seweap o n s sy s t e m s d o o r do n ot i n cl ud e th e

w e a po n c o n t r o l s y s t e m . T h e s i m p l e s t e f f ec t i v en e ss s t u d y i g n o re s t h e en v i ro n me n t oft h e weap on . S uch an an a l y s i s i s u n sa t i s fact o r y fo r an y t h in g b ut co m p ar i so n ofweapons, but the statement is made that a

co mp ar i so n can b e mad e wi th ou t s i g n if i cante r r o r . T h i s i s n o t c o r r e c t . Such a s ta t e m e n tcan be sub stan tiate d only by studying theweapon in i ts environment and f inding out thatthe effect i s , in fact , the sam e for the twos y s t e m s . In a s u r f a c e - t o - a i r m i s s i l e de f en s estudy, the ki l l potential per unit may be determi n ed u n d er t h e assu m p t i on of p er fect , u n i f o r m , o r r a n d o m d i s t r i b u t i o n o f f i r e o v e r t h eat t ack i n g a i r c r af t . Non e of t h ese a r e ach i e v able in the r ea l ca se . Also , i t is eas y tod emon st ra t e t h at i t mak es a d i f fe ren ce wh i ch

of t h ese i s u sed , s i n ce t h e comp ar i son b et weenweap on s o rd i n ar i l y i s d i f fe ren t fo r eachd i s t r i b u t i o n o f f i r e .

I f t h e ef fect i v en ess of i n t e rcep t o rs i s b ei n gst u d i ed , t h e assu mp t i on of c l ose con t ro l o rb road cast con t ro l i s mad e, wi t h d i f fe ren tn u m b e r s o f e x p e c t e d s u c c e s s e s p e r s o r t i e di n t e r c e p t o r r e s u l t i n g f r o m t he tw o a s s u m p t i o n s . In this case, as in the guided missi lean a l y s i s , t h i s i s n o t h i ng mo r e o r l ess t h an t h esubs t i tut io n of intui t io n for fact . We havei g n or ed t h e en v i ro n m en t , b ut we es t i m at eintui t ively that the engagement wil l be thisp a r t i c u l a r w a y a nd b r o a d c a s t c o n t r o l o ru n i fo rm d i s t r i b u t i o n of f i re wi l l g i v e a g oodap p ro x i mat i o n to t h e r es u l t . T h i s may b e anaccu rat e es t i mat e , b u t i t i s a n on v er i f i ab l ees t i m at e an d h as a s i n g u l ar d isad v an t ag e inthat i t has subs t i tuted the intuit ion of theana lyst fo r the intui t ion of the executiv e, ar i sk y s i t u at i on a t b es t .

A so m e w h a t b e t t e r a p p r o a c h t o w e a po nan al y s i s i s t o assu me an en v i ron men t fo r t h ew e a p o n . T h i s ap p r o a c h a t t e m p t s t o s t a t er e q u i r e m e n t s f o r t h e w e a p o n c o n t r o l s y s t e m .We have done analyses in which we say thatt h e ef fect i v en ess of a mi ss i l e d efen se , u s i n gg i v e n f i r i n g d o c t r i n e s , h a s b e e n d e t e r m i n e d .

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The dis t r ibut io n o f f i r e o ve r a t ta c king a i r craf t is not specified , but the r ul es fore ngaging ta r ge ts a r e s pe c i f ie d . An e xa mp leo f s uc h a s e t o f r u le s i s the "ne a r e s t une nga g e d" do c t r in e . In th is do c t r ine the f i r s t

p r io r i ty ta r g e t i s tha t ta r ge t ne a r e s t the bo mbr el ea se point which is not engaged by o therba t te r ie s ; if the r e a r e no une nga ge d ta r ge ts ,then the ne a r e s t ta r ge t i s s e le c te d . Othe rd o c t r i n e s , s o m e m o r e c o m p l i c at ed , s o m el e s s , ha ve a ls o be e n us e d.

Now the n, wha t a r e the r e qu i r e me nts whic hthe c o nt r o l s ys t e m m us t s a t isfy if the f i r ingdo ctr ine used is to be feasible ? What info rmation, what kinds and how much, must bea v a i la b l e t o t h e b a t t e r y c o m m a n d e r in o r d e r

to c a r r y o ut the f i r ing do c t r in e ? This inc lude s the Info r ma t io n whic h mus t be t r a ns mit te d f r o m ba t te r y to ba t te r y , f r o m ba t te r yto the co m ma nd o r c o nt r o l c e n te r , the da tar a te s r e q ui r e d , the t r a c k c a pa c i ty which thec o nt r o l c e nte r mu s t ha ve , the a c c u r a c y wi thwhic h ta r ge t po s i t io n mus t be me a s ur e d, a ndthe acc ur acy with which these data mu st bet r a n s m i t t e d . A ll t h e s e , a nd m o r e , c o m p r i s ethe li s t o f c h a r a c te r is t ic s whic h the a s s um e de n v i r o n m e n t h a s .

This l is t can be an input to another s tudywhic h t r ie s to de te r m ine if s uc h a n e nvi r o nment could be rea lize d, and if so , when. Th isstudy might tur n o ut to show that the as su m ption was invalid, and h er ein l ies o ne of theprimary difficult ies of an analysis of weapone ffe c t ive ne s s whic h a s s um e s a pa r t i c u la rc o nt r o l s ys te m . If the s tudy a s s um e s a ne nvi r o nm e nt , i t i s apt to g ive the im pr e s s io nthat the a s s um e d e nvi r o nme nt c o uld be r e a le n v i r o n m e n t , a nd t h e r e f o r e n o p r o b l e me x i s t s . I t i s s o me wha t be t te r tha n igno r ingthe environment, fo r the at tention of thee xe c ut ive c a n be d i r e c te d to the a r e a o func er tainty . He can mak e his decis io n withgr e a te r c o nf idenc e in the e ffe c t ive ne s sa na lys is . The a na lys is ma y pr e s e nt a s e t o f

r e s ul ts which wo uld o bta in fo r e a c h o f s e v eral different environments and he can applyhis e xpe r ie n c e a nd in tu it io n to a c le a r lys ta te d pr o ble m . If no ne o f the c o nt r o ls y s te m s indic a te d c a n be r e a l iz e d, the n the

r e s u l t s o f t h e a n a l y s i s a r e , o f c o u r s e ,m e a nin gle s s . B ut the e xe c ut ive a nd p la nn e r sa t le a s t kno w tha t the r e s ul ts a r e me a ningl e s s , a fact which they might not have knownif the environment had been ignored.

The kind of effectiven ess ana lysis that isneeded, the an aly sis that is r equ ir ed (if wea r e r e a l ly go ing to be a ble to le a ve thed e c i si o n m a k e r s f r e e t o m a k e d e c i s i o n sbe twe e n we a po n pr o gr a m a l te r na t ive s , wi thco nfidence that th ey hav e a good idea of the

co nsequ ence s of the decis io n) is one whichc o ns id e r s the e ffe c t ive ne s s o f the who led e f e n s e s y s t e m .

To continue with the example of a weapona nd i t s c o nt r o l s y s te m , the s e m us t be c o ns ide r e d to ge the r , a s o ne s ys te m . Only in tha twa y c a n we a ns we r que s t io ns s uc h a s , "W ha tf i r ing do c t r ine s a r e fe a s ib le us ing th iss u r f a c e - t o - a i r m i s si l e s y s t e m ? " o r " Ho wma ny in te r c e p to r s c a n be use d wi th wha teffect , given the technical cap abi l i t ies of the

SAGE system and the expected t raffic dens i t i e s ? " When we have appr o ached defensesys tem effec tiveness from this point of view,we wi l l be a ble to p r e s e nt a l te r na t ive s whic hwil l ma ke muc h s i m ple r the p r o c e s s o fc ho o s ing wha t mix tur e o f we a po ns i s de s i r a blein the defense system and also we can beginto see how to decide which weapons should bephased out and at what t imes.

The mo s t im po r ta nt a dva nta ge which

a c c r u e s to s uc h a n a na l ys is i s tha t thee ffe c t ive ne s s a na lys is c a n s e ns ibly fe e dba c kr e q u i r e m e n t s t o t h e r e s e a r c h an d d e v e l o p me nt pe o ple , r e q ui r e m e n ts which a r e ba s e do n o pe r a t io na l e ffe c t ive ne s s r a the r tha n s o meidea that this new de vice wo uld be "n ic e to

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h a v e . " I t can sh ow wh en a mi smat ch occu rsb et ween weap on an d en v i r o n m en t . A m i ss i l emay b e d es i g n ed wi t h mu ch l on g er ran g e t h ani t can u se , g i v en t h e l i m i t a t i o n s of t h e sy s t emt o s u pp ly t a r g e t d a t a o n d i s ta n t t a r g e t s .

S i mi l a r l y , t h e an al y s i s may sh ow t h at d at aacc u r acy sh ou l d b e sacr i f i ced i n fav or o fh i g h e r d a t a r a t e s , o r t h e c o n v e r s e m i g h t b et r u e . T h e r e a r e m a n y o t h e r s u c h q u e s t i o n s ,including, no do ubt , m any that we a r e notc u r r e n t l y a w a r e o f. It i s c l e a r , h o w e v e r , t ha tt h e o p t i mu m weap on sy s t em can n ot b ed esi g n ed u n t i l t h e weapon an d i t s en v i ro n m en tar e s t u d i ed an d d es i g n ed t o g et h er . As l on ga s e i t h e r s y s t e m a t t e m p t s t o d i c t a te r e q u i r e m en t s fo r t h e o t h er , as i s now t h e ca se , n on -o p t im u m p e r f o r m a n c e i s a l m o s t c e r t a i n to

r e s u l t .

T h i s d i scu ss i o n of t h e i n fl u en ce of t h ei n fo r m a t i o n g a t h e r i n g a nd w e ap o n c o n t r o lp a r t s o f a weap on sy s t em ap p l i es eq u al l y wel lt o t h e s u r f a c e s u p p o r t m i s s i l e c a s e . S uc ha s y s t e m h a s t h e s a m e s o r t o f d e p e n d e n c e sa s i t s e n v ir o n m e n t a nd t h e w e a po n c h a r a c t e r ist ics should be influenced by the kinds ofi n fo r m a t i o n a v a i l a b l e t o t h e w e a po n s y s t e m .A ta c t i c a l s u p p o r t w e a po n d e s i g n e d t o d e s t r o yt ar g e t s of o p p o r t u n i t y can n ot b e su cc essfu l int h at r o l e u n l e s s t h e i n fo r m a t i o n s y s t e m c a nl o c a t e t a r g e t s a nd t h e s y s t e m c a n r e a c t f a ste n o u gh t o p r o d u c e t h e m i s s i l e b u r s t p r i o r t od i sa p p e ar an c e of th e t a r g e t . T h i s k in d ofc o n s i d e r a t i o n m a y i n f lu e n ce t h e d e s i r e d m a x i mu m ra n g e of t h e sy s t em . A weap on of5 0 -m i l e ran g e may b e n i cel y ma t ch ed wit h t h ed e si r ed mo b i l i ty of t h e weap on , t h e d i s t an cebehind l ine of co ntact that this mo bil i ty r e qu ir es i t to be pla ced, and the quali ty ofi n t e l l i g en ce av ai l ab l e on t a rg et s j u s t b eh i n dt h e b at t l e zo n e. Of co u r se , t h e Arm y al way sw i ll h a ve u s e f o r l o n g e r r a n g e m i s s i l e s f o ru se ag a i n s t re l a t i v e l y f ix ed t a r g e t s , su ch assu p p l y d u mp t ran sp or t a t i on faci l i t i es an d t h el i k e . B u t f o r t a c t i c a l s u p p o r t , th e m i s s i l eu s e d a n d i t s i n f o r m a t i o n g a t h e r i n g a nd p r o ces s i n g s y s t e m sh ou l d i n fl u en ce each o t h er .

T h e c o n s i d e r a t i o n o f t he i n fo r m a t i o ng a t h e r i n g a n d p r o c e s s i n g s y s t e m , a s w e l l a st h e weap on , i s o n ly o n e of sev er a l t h i n g swhich need to be added to the analyses whichar e cu r re n t l y d on e t o g i v e t h em th e k in d of

g e n e r a l i t y c o m m e n s u r a t e w i th th e s c o p e o f t h ep rob l ems wh i ch h av e t o b e so l v ed . I m e n t i o nthis o ne in detai l sinc e i t is o ne which m is si lede sig ne r s have not taken sufficient acco unt ofi n t h e p as t . T o p ro d u ce an o rd er l y weap ons y s t e m s u c c e s s i o n , t h e A r m y m u s t c o n s i d e rc o m b i n a t io n s o f w e a p o n s a n d s y s t e m s to m e e tal l t h e p a r t s o f a comp l ex t i me p h ased t h re at .

Let us look now at an exa m ple of howsu ch a g en eral ef fect i v en ess an al y s i s mi g h tc o m e o u t . R e f e r r i n g to F i g . 6 , f o r th e t i m e

p er i od i n q u est i on , t h e n u mb ers of b ombc a r r i e r s m i gh t lo o k l ik e t h e u p p e r g r a p h( th e sa m e as F i g . 5 ) . T h e d efen si v e cap ab i l it i e s in t e r m s of s u r f a c e - t o - a i r m i s s i l e smi g h t l ead t o n u m b er s of u n i t s av ai l ab l e asshown in the midd le gr aph . Both of the set wo g r a p h s a r e m u c h o v e r s i m p l i f ie d . T h ec h a r t d o e s no t s ho w c o u n t e r m e a s u r e s o ra l t i t u d e s o r t a c t i c s o r m a n y o f t h e o t h e rt h r e a t c h a r a c t e r i s t i c s t ha t n e e d t o b e a d de dt o t h e n u mb er of b omb ers i n t h e a t t ack .S i m i l ar l y , t h e d efen se cap a b i l i t i es shown ar e

on l y weap o n s , no t weap on sy s t e m s. On ly on eof the infini te possible number of differentmi x t u res an d p h asi n g sch ed u l es i s sh own .

In an y case , as an ex amp l e , t h e resu l t o ft h e ef fect i v en e ss an a l y s i s mi g h t b e t h e b ot t omg ra p h wh i ch g i v es t h e n u mb er of b o mb s ont a r g e t o r f r a c t io n o f th e t a r g e t s y s t e m d e st r o yed as a function of the date of the at tac k.T h e d ot t ed l i n e mi g h t b e a t o l e rab l e l ev el o fd am ag e , t h at amo u n t of d amag e we cansus tain and have a high pro babil i ty of beingab l e t o p u l l ou rs el v es t o g et h er ag ai n an d f ig h tto win. Th is kind of plot for a var iety of dif

fe ren t mi x es of weap on s an d sch ed u l es wi l la l low t h e Army t o d eci d e wh i ch p r o g r am i tw a n t s t o p r e s e n t .

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I have not yet said anything about co st s .This i s no t be c a us e the y a r e no t impo r ta nt ,be c a us e the y ma y ha ve o ve r r id in g imp o r tanc e. In fact , bec ause of this , they mayo p e r a t e a s b o u n d ar y c o n d i t i o n s . T h e r e a r e a

wide var iety of boundary condit ions which canbe in effect in an atta ck -de fen se situ atio n ofthis kind. I will mentio n only two such co nstraints and show how costs work in each oft h e s e .

One po ssible framew o rk in which such ananal ysis might be acco mp lished is that ofs ta te d e ffe c t ive ne s s . Tha t i s , the s ys te m sc o ns ide r e d mus t l imi t the da ma ge to theta r ge t s ys te m to x pe r c e nt , o r no m o r e tha nn bo m bs a r e to de to na te o n ta r ge t . In s uc h af r a me wo r k the e ffe c t ive ne s s s tudy de te r min es what quanti t ies of the feasible s ys te m sa r e r e q u i r e d t o s a ti sf y t he r e q u i r e m e n t .C o s ts o f a p r o gr a m to do th is might a ppe a ras depicted in Fig . 7 . Th is tel ls the plannerhow much it will co st him to achi eve thede sir ed effectiveness and in what ye ar s hewill spend the mo ney. Co sts of sm al le r andla r ge r p r o gr a ms wi l l no t , in ge ne r a l , be al inea r function of the se, s ince effectivene ss isnot o rd inar i ly a l in ear function of cost anda ls o be c a us e d i f fe r e nt pha s ings o f p r o gr a msa r e r e q u i r e d . A n o t h e r , an d m u c h m o r e

c o mmo n to the Ar my, i s the r e qui r e me nt topr o duc e the ma ximu m e ffe c t ive ne s s fo r al imi te d budge t. This i s c o ns ide r a bly m o r ec o mpl ic a te d s inc e c o s ts c a nno t be de te r mine dunt il a f te r qua nt i t ie s ha ve be en pr o gr a m me do ver t im e. The sam e techniques apply, i t isjust that the usual way is by successivea p p r o x i m a t i o n .

Let us look now at a diagram of how allthis an alysis is put to gethe r in a sch em ewhic h a s s is ts the Ar my by pr o viding i tcontinuously updated information needed toput into practice i ts philosophy of providingcontinuing effectiveness at minimum cost asshown in Fig. 8. The defense capabil i tyanalysis must be done in detail fo r a wide

va r ie ty o f c a pa bi l i t ie s ; i t mus t r e f le c t theease of modification of this system and whatnew capabil i t ies are obtained by this modificatio n. Th is is the gro wth po tential of thds ys te m . The e ffe c t ive ne s s a na ly s is wi l l

weigh whether that potential is great o rs m a l l , whe the r it i s va lua ble o r no t . Theo ffe nsive c a p a bi l i t ie s a r e t r e a te d in the s a m eway and with the same detail , insofar as ispo s s ib le . B o th o f the s e mus t r e f le c t a l lpo s s ib le c o ns t r a in ts . An e xa mple we ha vee nc o unte r e d i s : "Yo u c a n ha ve s ys te m A th isyea r o r sys tem B next ye ar . If you want themboth, but as soon as possible, then system Awill be next year and sys tem B the ye ara f te r . " This wa s s imply a c a s e o f a s ho r ta g eo f r e s e a r c h a nd de ve lo pme nt c a pa bi l i ty . Ama xim um e ffo r t o n e i the r p r o gr a m wo uld

de la y the o the r . Tr a in ing ca n be a fa c to rwhich l im its the availa bil i ty. The t rainin gp r o g r a m r e q u i r e s m u c h p r e p a r a t i o n a nd i t i ss low to produce the kind of specialis t calledfo r in la r ge num be r s in guide d m is s i l es y s t e m s .

The e ffe c t ive ne s s m a tc he s a va i la bles ys te m s a ga ins t the a va i la ble th r e a t . In do ingt h i s , i t mus t ta lk a bo ut c o mple te s ys te ms ,both offensive and defe nsiv e, and if not thee nt i r e de fe ns e s ys te m, a t le a s t a c o mple tewe apo n s ys t e m . It i s no be t te r , ho w e ve r , tom a t c h a c o m p l e t e w e a p o n s y s t e m , e n v i r o n me nt , m ixtu r e s a nd a l l , a ga ins t a pa r t ia lthreat such as manned bombers , than i t is toma tc h a co m ple x th r e a t a ga ins t a pa r t ia lwe a po ns s ys te m . In e i the r c a s e , the r e s ul tmay be quite valuable, but i t leaves much tobe de s i r e d be c a us e the una s ke d que s t io nsm a y h a v e t h e m o s t i m p o r t a n t a n s w e r s .

The effectiveness an aly sis feeds back tothe a va i la bi l i ty s tudy r e qu i r e m e nts o f two

k i n d s . The fi rs t is the knowledge of whatt ime the qua nt i t ie s r e qui r e d wi l l be a va i la ble .The s tudy will f i rs t have been done on thebasis of f i rs t unit availabil i ty and a nominalde plo yme nt r a te . If the qua nt i t ie s a r e la r ge .

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n ew con d i t i o n s may ap p e ar . It may b en e c e s s a r y t o p la n a d d i ti o n a l f a c i l i t i e s ; m a n p ow er av ai l ab i l i t y m ay b eco me a l i mi t on t h ed e p l o y m e n t r a t e . T h e e f f e c t iv e n e s s a n a l y s i sa l so p rov i d es t h e s t a t e of u rg en cy u n d er wh i ch

the ava ilabi l i ty is to be pr ed icte d. It mighti n d i cat e t h at an a l l -ou t ef fo r t i s req u i red t ome et a p r ed i c t e d t h r ea t . It may i n d i cat ep a r a l l e l i n g o f d e v e l o p m e n t o p e r a t i o n s i no r d e r t o h a v e h i g h a s s u r a n c e t h a t t h e p r o j e c twi l l b e read y a t t h e sch ed u l ed t i me.

A seco nd kind of feedback fr o m theeffect i v en es s s t u d y t o t h e av ai l ab i l i t y a n al y s i si s t h e re q u i re m en t t o d et e r m i n e wh en newd ev e l o p m en t s , n o t p re v i o u sl y an al y zed , can b ema d e rea d y t o f i ll a n eed d i sc o v er ed in t h eeffe ct ive nes s study . When can a low al t i tudeso l u t i on b e d ep l oy e d ? W h en wi l l t h e an t i j a m m i n g f e a t u r e s b e i n c o r p o r a t e d i n t her a d a r s ? T h e r e a r e f e ed b ac k s f r o m th eeffect i v en ess an a l y s i s t o t h e t h r eat an al y s i sa l s o . An al y si s m ay sh ow t h at so m e of t h eo f f e ns iv e c a p a b i l i t i e s d o n o t p o s e r e a l t h r e a t s .T h e a t t a c k e r c a n d i s c o v e r t h i s , t o o , a n d w i l ll ik ely red i s t r i b u t e h i s ef fo r t . T h e mo d i fi edt h r eat n u m b er s an d t y p es g o i n t o t h e ef fect i v en es s an al y s i s ag ai n . S ome at tack t act i c s wi l lp r o v e t o b e s u p e r i o r . R e c o g n i ti o n o f t h i s m a yi n d i cat e d ev el o p m en t s to cap i t a l i ze on t h i s

s u p e r i o r i t y .

F rom t h e ef fect i v en ess an al y s i s an d t h et ime of deployment indicated by the avai lab i l i t y an al y s i s , t h e an n u al cos t s o f t h ed e f e n s e p r o g r a m m a y b e d e t e r m i n e d . In t hec a s e o f a p r o g r a m w h ic h i s t o p r o d u c e a s t a t e deffect i v en ess , if p os s i b l e , t h e an n u al ex p en d i t u r e s m a y v a r y w i d e l y a s n e w s y s t e m s p u r c h a s e d a r e m o r e o r l e s s e x p e n s i v e . N o t et h a t t h e s e c o s t s a r e n o t j u s t o p e r a t i n g c o s t s ,n o t j u s t p r o c u r e m e n t c o s t s , b u t b o th o f

t h e s e , a s w e l l a s r e s e a r c h an d d e v e lo p m e n tc o s t s , t oo l i n g , t es t i n g , t ra i n i n g , an d a l l o t h erc o s t s n e c e s s a r y to p a y f o r t h e d e f e n se s y s t e m .C o s t s a r e u s u a l l y e x p r e s s e d i n t e r m s o f m a n p o w e r , d o l l a r s , a nd s o m e t i m e s a m o u n t o f

n u cl e ar ma t er i a l . T h e eq u at i o n s wh ich a ll owt r a n s f o r m a t i o n b e t w e e n t h e s e a r e n o t y e tes tab lish ed and m ust be left to the higherex ecu t i v e an d p ol i cy mak i n g b od i es t o d et e rm i n e .

The cost analysis feeds back to theeffect i v en ess an al y s i s an d t h e av ai l ab i l i t yanalysis whether the specif ied budget l imith as b een ex ceed ed an d wh et h er t h e d ep l oy m en tsch ed u l es sh o u l d b e s t r e t c h ed ou t o r t h e t o t a ln u m b e r s m o d i fi e d. T h e p r o g r a m m a y h a ve t ob e r e p r o g r a m m e d s o m e w h a t in o r d e r t o p r o v i d e con t i n u i n g ef fect i v en ess o v er t i me wi t h ou t ex ceed i n g b u d g et l i mi t s . A feedback to thet h r eat may a p p ea r i n t h e case of a n on -b u dg et l i mi t ed p r o g r am . If ef fect i v en ess i ssta ted , then i t may be quite expe nsive toa c h i e v e . S o m e t h r e a t s m a y b e e a s y t o m e e tt ech n i cal l y , b ut t he p r o g r am s t o d o so may b em o r e e x p e n s iv e t ha n t he p r o g r a m s w h ic h p o s et h e m o r e d i ff icu lt tech n i cal p r o b l e m s. Anex am p l e wou ld b e t he p ro d u ct i on of v ery l a r g en u mb ers of b omb ers wh i ch a re easy en ou g h t osh oot d own b u t wh i ch wou l d req u i re v ery l a rg en u m b er s of m i ss i l e s to d o it . If t h e a t t ack erhad a s a go al the long r ang e bankrupting of theeco n o my , h e mi g h t p i ck su ch a co u r se . T h i sk in d of con si d er a t i o n i s no t n o rm al l y ap ar t o ft h e an al y s i s , b u t fact o rs b ear i n g on i t can b e

p oi n t ed ou t .

T h i s , t h en , i s h ow t h e an al y s i s ap p ears .Only by est abl ishin g such an ana lysiso rg an i zat i on an d k eep i n g i t op era t i n g con t inuou sly can we hope to be able to make ther i g h t d e ci s i o n s . On e can n ot ex p ect t o g eta l l t h e a n s w er s a t o n ce. By t h e t i me t h ean al y s i s i s d on e on ce, i t i s ou t o f d at e .H o w e v e r , e a c h s u c c e s s i v e r e v i e w i s e a s i e ran d can an d sh ou l d b e m o re g en e ra l . T h ef i r s t a p p r o x i m a t i o n a n a l y s i s t h a t 1 men t i on ed

at the beginning could be cal led the f i rstro u n d t h rou g h su ch a l oo p . S u cce ss i v e t r i p st h ro u g h t h i s k in d of l oop a r e n o t d i s cr e t e ;the feedback should be co ntinuo us and res ul tin a continuous renewal and modificat ion of the

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p r o g r a m . N ew t h r e a t p a r a m e t e r s m a y g e n e r a t e new r e s e a r c h a nd d e v e lo p m e n t r e qu ir em ent s . We mu st know when thes e can bemet and whether the modification or newweapon can be in t im e to meet the new th r ea t .

We must know what this costs and if we canafford i t in the desired amount.

Afte r th is p r o c e s s ha s p r o c e e de d fo r at i m e , all new weapons and sy ste m s becom ejust m od ifications of the defense sy ste m . Thea r r iva l o f ma r ke dly d i ffe r e nt we a po ns wi l lhave been an ticip ated long enough that thesyste m can have react ed to the point wh er e themo dific at io n i s no t d is r upt ive . T he h igh o bs o le s c e nc e r a te o f we a po ns s ys te m s , o c c a s io ne dby the r a pid te c hnic a l p r o gr e s s in bo tho ffense and defen se cap abi lity, and the longlead t im es for new equipment caused by thegr e a t c o mple xi ty o f mo de r n we a po n s ys te ms ,r e q u i r e s t h at p r o g r a m s b e c o n t in u a ll y r e evaluated . In so doing, we mu st be carefu lto leave open all the al te rn ativ es w hich wemight wish to fo llow up next yea r . Th is m ean s

t h a t m a n y m o r e r e s e a r c h a n d d e v e l o p m e n tp r o g r a m s m u s t b e p u r s u e d t ha n b e f o r e . M o s tof the se will not be fol lowed thro ugh to de ployed weapons but will be ca r r ie d along unti li t i s r e la t ive ly c e r ta in tha t the r e s e a r c h a nddevelo pmen t is not leading to a weapon syste mthat we may need. Alt er nativ es mu st be lefto pe n s o tha t ne xt ye a r ' s s o lu t io n i s de te r m ine dby ne xt ye a r ' s kno wle dge , no t p r e de te r mine dby the les se r knowledge we have this ye ar .

The o bje c t o f the we apo ns p r o gr a m is toprovide a continuing effectiveness over a longt i m e . W e de s i r e the e ffec t o f de te r r ingpo s s ib le a t ta c k a nd a r e the r e fo r e no t po in t ingfo r a ny pa r t ic ula r pe a k e ffe c t ive ne s s ye a r .It might be said that the object is to produceweapo ns sy ste m s s tr o ng enough that they willnot have to be used. With thes e ideas in m ind,i t is c le ar that we mu st o btain effectiven essby the ye a r , fo r s e ve r a l ye a r s , a nd c o nt inua l lyr e n e w o u r a n a l y s i s o f t h e s e s y s t e m s in o r d e rto ge l a s ne a r the o pt imum s ys te m , de fe ns e o ro ffe ns e , a s po s s ib le .

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C a p a b i l i t y

Ef fe c t ! vene»s

Study

IA v a i l a b i l i t y

ICos t

Threa t

F i g . 1 . S i mp l if i ed an a l y s i s sch em e.

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Study

R & D

P r o t o t y p e t a n d T e s t

P r e - p r o d u c t i o n

P r o d u c t i o n

T r a i n i n g a n d D e v e l o p m e n t

o f T a c t i c a l D o c t r i n e

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O p e r a t i o n

Fig . 2 . Le a d t ime pha s ing.

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S t u d y M o d i f i e d o r N e w W e o p o n

O r i g i n a l W e a p o n

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L

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F i g . 3 . L ead t i me p h asi n g .

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1 M o d i f i e d o r N e w W e o p o n

StudyO r i g i n a l W e a p o n

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P r o t o t y p e s a n d T e s t i n g


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Fig . 4 . Le a d t ime pha s ing.

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c'>

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. ^ • " • " " "p rop s /

S ub l a u n c h e d m i s s i l e s

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b o m b e r s /

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F i g . 5. Availabil i ty of c o m p l e x t h r e a t .

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S ub l a u n c h e d m i s s i l e s

T i m e

E f f e c t i v e n e s s

Fig. 6 . To p, a va i la bi l i ty o f c o m ple x th r e a t .Middle, availabil i ty of defense capabil i ty.B o t to m, e ffe c t ive ne s s .

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F i g . 7 . P r o g r a m c o s t s .

A v a i l a b i l i t y o f

C a p a b i l i t y

P

R e q u i r e m e n t s

( R & D , q u a n t i t i e s )

S t a t e o f u r g e n c y

D e f e n s e S y s t e m

E f f e c t i v e n e s s

O v e r T i m e

B u d g e t L i m i t ?

1 r

C o s t O v e r

T i m e

A v a i l a b i l i t y o f

C o m p l e x T h r e a t

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U p d a t e t h r e a t

E c o n o m i c B a l a n c e

F i g . 8 . An al y si s sch em e.

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NEW PR INC IPLE S IN THE DESIGN OF SU PERIORCOMM UNICATIONS, NAVIGATION, AND MISSILE GUIDANCE SYSTEMS

W . P . L e a r , S r . *

SUMMARY

This pa pe r de a ls wi th the a ppl ic a t io n o f s ta t i s t ic a l c o nc e pts to the de s ign o f s ys t e m s .Pa r t ic u la r a t te ntio n i s de vo te d to the o pe r a t io na l p r inc ip le s o f the SCAN s ys t e m . Thepo s s ib i l i ty o f a pplying the s e p r inc iple s to c o mpute r s i s po in te d o ut .

SOMMA1RE

C e t te no te t r a i t e de 1 ' appl ic a tio n de c o nc e pts s ta t i s t iqu e s a 1'etude d e s y s t e m e s .El le t r a i te e n pa r t ic u l ie r de s p r inc ipe s o pe r a t io nne ls du s ys te m e SCiVN e t indique la po s s i b i l i t e d ' a p p l i q u e r c e s p r i n c i p e s a u x c a l c u l a t r i c e s .

1. INTRODUCTION

C o mm unic a tio n i s ge ne r a l ly c o nc e ive d a sthe c o nve ying o f ide a s o r s ymbo ls f r o m o nepe r s o n to a no the r . In th is d is c u s s io n , we wi l ld e fi ne c o m m u n i c a ti o n a s t h e p r o c e s s of t r a n s f e r r i n g m e s s a g e s o v e r s u b s t a n t ia l d i s t a n c e sf r o m mind to mind, mind to ma c h ine , ma c hineto mind, o r m a c hine to m a c hine . W ith th isdefinit io n, navigation and m iss ile guidances y s t e m s w h ic h a c c e p t e x t e r n a l l y - g e n e r a t e dda ta be c o m e s pe c ia l c a s e s o f the ge n e r a l

c o m m u n i c a t i o n p r o b l e m .U n d e r a d v e r s e c o n d i t i o n s , t h e p r o c e s s o f

c o m munic a t io n i s a ga m e pla ye d a ga ins tNa tu r e . B y e xplo i t ing the unc e r ta in ty tha tmus t e xis t a t the r e c e iving e nd o f a c o mmunic a tio n s ys te m Na tur e p la ys a ga me o fa ugme nt ing o r s upp r e s s ing the in tende dm e s s a g e . In r a d i o s y s t e m s t he m e c h a n i s mfo r a ugme nt ing the in te nde d me s s a ge i s thege ne r a t io n o f r a ndo m s igna ls (o r no is e ) inthe a tmo s phe r e a nd in the r e c e iving e quipm e n t . S u p p r e s s i o n o f t h e m e s s a g e o c c u r swhen the r e a r e o bs ta c le s a ffe c ting pr o pa g a tion of a s igna l between dis tan t po ints ; in

t h i s c a s e f o r w a r d - s c a t t e r i n g , s k i p , m u l t i p at h ,s e le c t ive fa ding, a nd o the r a no ma l ie s p r o duc eu n c e r t a i n t y in t h e m o m e n t - b y - m o m e n t r e c e pt io n o f the r a dia te d s igna l .

I c la im tha t the va s t ma jo r i ty o f to da y ' sc o mm unic a t io n , na viga tio n , a nd m is s i le guida nc e s ys te ms a r e g r o s s ly ine ff ic ie nt a ndineffectual co m par ed to what might bea c hie ve d. In mo s t c a s e s the e quipme ntl imi ta t io ns a r e inhe r e nt in and a r is e f r o m theini t ia l s ta te me nt o f o bje c t iv e s . M a ny s ys te m sar e inefficient with gr ea t po wer input ande xt r a va ga nt us e o f the s pe c t r um be c a us e thes ys te m ha s no t be e n a da pte d to i t s c o mple tee nvi r o nm e nt . W e ha ve o f ten p la ye d the ga mea ga ins t Na tur e wi th unna tur a l r u le s ; fo rins ta nc e , we o f te n de s ign s ys te ms whic h wi l lw o r k o n ly w h en f a v o r a b l e s i g n a l - t o - n o i s ec o ndi t io ns e xis t a nd we the n s e t a bo ut ge ne r a t ing the r e qui r e d a mo unt o f po we r , ho we ve runr e a s o na ble tha t ma y be . I wo uld s ugge s tthat i t is more logical to recognize fully anda c c e pt the na tur a l p r o c e s s e s a nd the n s e t

a bo ut the de ve lo pm e nt o f s ys te m s whiche xplo i t the r u le s o f the ga me the r e by impo s e d.

*L e a r , Inc ., Sa nta M o nic a, C a l i fo r n ia .

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T h e f i e l d of i n fo rmat i on t h eory , b ased u p ont h e c l a ss i c w o rk s of W i en er an d S h an n on , h asfo rm al i zed a q u an t i t a t i v e ap p r o ach t oc o m m u n i c a t io n . Of g r e a t s i g n if i c a nc e a r e t h et h e o r e m s r e l a t i v e t o t h e i n t e r c h a n g e a b i l i t y o ft i m e , b a n d w i d t h , a n d s i g n a l - t o - n o i s e r a t i o sin s u ch s y s t e m s . M o r e o v e r , i n fo r m a t io nt h e o r y h a s d e fi ne d s u ch m e s s a g e c h a r a c t e r i s t i cs as red u n d an cy an d coh eren ce, an d h asi n d i cat ed t h ei r v al u e i n i n creas i n g t h e l i k el i h ood of m ess ag e r ecep t i o n i n a sy s t em wh ichi s d e v i s e d t o u t i l i z e s u c h p r o p e r t i e s . T h es u b s t a n t i a l v a l u e o f i n t e g r a t i o n i n s i g n a l - t o -n o i s e i m p r o v e m e n t , w h e r e t h e a p p l i c a t i o np e r m i t s i t s u s e , h a s b e en d e m o n s t r a t e d .O t h e r p o w e r f u l t o o l s f r o m i n f o r m a t i o n t h e o r ya r e a u t o c o r r e l a ti o n a nd c r o s s - c o r r e l a t i o n .E x p er i men t s h av e sh own t h e ab i l i t y of t h ec o r r e l a t i o n m e th o d s , p a r t i c u l a r l y c r o s s -c o r r e l a t i o n , t o r e v e a l t he p r e s e n c e o f s i g n a l sman y d eci b el s b el ow t h e av erag e n oi se l ev el ;h e r e , as wi th i n t eg ra t i o n , o n e t r ad es t i me fo rs i g n a l - t o - n o i s e i m p r o v e m e n t . C e r t a i n l yt h e s e t e c h n i q u e s f r o m i n f o r m a t i o n t h e o r y a r ei m p o r t a n t a i d s i n c o m m u n i c a t i o n s y s t e md esi g n . ' Ho wev er , t o t h i s t i m e t h ey h av e b eenu sed mai n l y i n t h e measu remen t of t h eme t h o d s w e h av e, r a t h er t h an t o p o i n t t h e wayt o s u p e r i o r m e t h o d s .

Ad d i t ion al t o o l s a r e av ai l ab l e f rom t h ef ie ld o f s t a t i s t i c s . M e a s u r e m e n t s m a d e i np r a c t i c a l s y s t e m s a lw a y s co n t a i n s o m e d e g r ee of u n c er t a i n t y . T h e ex t en t o f t h e u n cer t a i n t y i s i n p ar t a fu nct ion of t h e me asu r i n gu n i t . F o r i n s t a n ce, if on e mad e re p ea t edm e a s u r e m e n t s f r o m h e r e to th e N o r t h P o l ew i th a o n e - m e t e r r o d , a v a r i e t y o f a n s w e r ss e p a r a t e d a t o n e - m e t e r i n t e r v a l s co u ld b eex p e ct ed . In su ch a s i m p l e sy s t em t h em e a s u r e m e n t s w o uld l i e s y m m e t r i c a l l y a bo u tt h e t r u e v al u e i n acc o rd an c e wi th t h e Gau ssi an

d i s t r i b u t i o n . T h e wi d th of t h e d i s t r i b u t i o nwi l l v ary wi t h t h e p reci s i on of t h e measu r i n gs y s t e m a nd t h e e x t r a n e o u s d i s t u r b a n c e s , o rn o i s e , a d d e d t o t h e p r o c e s s . T h e s i g n i f i c a n c eof an y s i n g l e m ea su r em en t can on l y b e

e x p r e s s e d in t e r m s o f p r o b a b i l i t y . H o w e v e r ,in a l i n ear sy s t em , t h e av e ra g e of a v as tn u m b e r o f s u c h m e a s u r e m e n t s w o u l d y i e l da n a n s w e r o f g r e a t l y e n h an c e d a c c u r a c y .F u r t h e r m o r e , f r o m t h e f ie ld o f r a n d o mp r o c e s s e s , i t f o ll o w s t h at a ny s y s t e m e r r o rr e s u l t i n g fr o m a v e r y l a r g e n u m b e r o fr a n do m , u n c o r r e c t e d e r r o r s o u r c e s w illhave a pro bab le va lue which again fol lows theGau ssi an fu n ct ion . T h ese s t a t e m en t s su g g estm e a n s o f im p r o v i n g p e r f o r m a n c e in t h o s esy st ems wh ere t h ey can b e u t i l i zed .

O u r p r e s e n t n e ed i s t o e v o l v e s y s t e m swhich wil l function under the widest possiblera n g e of co n d i t i o n s . F u n d am en t al i s . t h en eed t o d ev i s e a met h od wh i ch u t i l i zes aUof t h e av ai l ab l e d at a . T o of fse t th e ran d o m

n es s of N at u r e , i t wi ll b e n e ces sar y to p l ayt h e g ame ag ai n s t Nat u re by p l aci n g p r o b ab i l i s t i c b et s , b ased on sou n d l o g i c , as t o t h emo st l i k el y v alu e of each d at a sam p l e . F o rex am p l e , i n n av ig at io n an d g u i d an ce p ro b l em sone is deal ing with physical bodies havingb ou n d ed v a l u es of o r i g i n a l p o s i t i on , v el oci t ya nd a c c e l e r a t i o n . T h e r e f o r e , a t a ny m o m e n tal l o f t h e p o ss i b l e m es sa g e s in su ch a sy s t emdo not have equal pro bab le value . Why nota t t emp t t o fin d t h e p ro b ab l e v al u e t o eachd a t a s a m p l e ?

2. THE SCAN SYSTEM

M y comp an y i s cu r ren t l y d ev el op i n g an av i g ati on sy s t em wh i ch ex p l o i t s so me of t h ep r eced i n g p r i n ci p l es an d wi l l se r v e as ani l l u s t r a t i v e e x a m p l e o f t h e t e c h n i q u e s d i s cu s sed . T h i s sy s t em i s ca l l ed t h e Self-

C o r r e c t i n g A u t o m a t i c N a v i g a t o r , o r S CA Nf o r s h o r t .

Navigat io n s ys te m s can be divided into

t wo c l as se s : t h e se l f -co n t a i n ed an d t h ee x t e r n a l l y - r e f e r e n c e d . T h e s e lf - c o n t a in e dcl a ss of n av i g at ion eq u ip men t i n cl u d es i n e r t i a l , d o p p l e r - s u p e r v i s e d , an d d ea d r e c k o n i ng t y p e s . T h e e x t e r n a l l y - r e f e r e n c e d c l a s s

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inc lude s all of the r a d i o , r a d a r , o p t i c a l , and

c e le s t ia l type s . The a c c u r a c y of self-

c o nta ine d s ys te ms is t i m e - d e p e n d e n t . The

a c c u r a c y of the e x t e r n a l l y - r e f e r e n c e d c l a s sis l imited by wavelength and l i n e - o f - s i g h t or

o t h e r p r o p a g a t io n p r o b l e m s , i nc lu d in g s i g n a l -t o - n o i s e r a t i o s . The e r r o r s o u r c e s for e a c hc l a s s of s y s t e m are different in c h a r a c t e r .

The SCAN technique is a ppl ic a ble whe r enavigational data from both self-contained ande x t e r n a l l y - r e f e r e n c e d s y s t e m s are a v a i l a b l e .Any given self-contained navigational equipm e n t , for ins ta nc e a d e a d - r e c k o n i n g c o m p u t er , w i ll h av e m e a s u r a b l e e r r o r c h a r a c t e r i s t i c s . A t y p ic a l d e a d - r e c k o n i n g c o m p u t e r is

shown in Fig. 1. Air c r a f t he a ding and t r u e

a i r s p e e d m e a s u r e m e n t s are supplied to th ise q u i p m e n t. H e a di ng a nd a i r s p e e d c o r r e c t i o n sdue to wind must be i n s e r t e d m a n u a l l y .Or dina r i ly s uc h da ta are not a c c u r a t e l yknown, thereby reducing the e quipme nt re

l iabil i ty. The SC AN s ys te m pr o vide s the

m e a n s for a u t o m a t i c a l l y and a c c u r a t e l yle a r ning the s e wind va lue s . The c o r r e c t e dve lo c i ty ve c to r may now be c o n v e r t e d f r o mi t s p o l a r f o r m i n t o n o r t h - s o u t h and e a s t -w e s t c o m p o n e n t s . By in te gr a t ing e a c h ve lo c i ty com ponent with re sp ect to l i m e , one

o b t a i n s the d i s t a n c e t r a v e r s e d in thato r d i n a t e . By adding the o r i g i n a l c o o r d i n a t eposit ion, which is the c o ns ta nt of i n t e g r a t i o n ,o ne o b ta ins p r e s e nt po s i t io n .

A t yp ic al e r r o r c h a r a c t e r i s t i c for such an

e quipme nt is shown by the c u r v e in Fig. 2.

T h e c u r v e d e p i c t s an e r r o r v e r s u s t i m ec o nto ur whic h wi l l c o nta in a lmo s t all s t a t i s t i c a l l y o b s e r v e d e r r o r s ; say 99.5% of all

o bs e r va t io ns fal l wi th in th is c o nto ur , whic h wes ha l l c a l l the c o nfide nc e l imi ts of the self-

c o n t ai n ed s y s t e m . T h e r e f o r e , a f te r o p e r a t i n go v e r an i n t e r v a l tx w i t h o u t r e c a l i b r a t i o n ,e r r o r s as g r e a t as plus or m i n u s «x are

acknowledged as p o s s i b l e . H o w e v e r , s i n c et h e t o t a l e r r o r r e s u l t s f r o m t h e s u m m a t i o n of

s e v e r a l u n c o r r e l a t e d s o u r c e s (in the d e a d -r e c k o n e r t h e s e are the r a n d o m e r r o r s in

t r o d u c e d by the d i r e c t i o n a l r e f e r e n c e , the

a i r s p e e d t r a n s d u c e r , and the c o o r d i n a t e c o n v e r t e r s and i n t e g r a t o r s ) the pr o ba bi l i ty of

v a r i o u s e r r o r m a g n i t u d es f o ll o w s theGa us s ia n funct io n . Th is indic a te s g r e a te rl ikelihood of the o c c u r r e n c e of s m a l l e r r o r sthan of l a r g e o n e s . The c r o s s s e c t i o n of

the confidence l imit can t h e r e f o r e be said to

have a G a u s s i a n p r o f i l e . The e xis te nc e of a

c o nfidenc e l imi t , ba s e d upo n phys ic a l m e a s u r e m e n t s m a d e on a p a r t i c u l a r set of h a r d w a r e , is n o r m a l l y not r e c o g n i z e d as m e a n ingful data and is not util ized in s y s t e md e s i g n .

If one e x a m i n e s the m a n n e r in whiche x t e r n a l l y - r e f e r e n c e d n a v i g a t i o n s y s t e m sa r e u s e d at p r e s e n t , for e x a m p l e a r a d i osystem like VOR-IME or T a c a n , we find a

m o s t p e c u l i a r f o r m of d a t a p r o c e s s i n g .F i r s t of all, the o b s e r v e r m u s t e i t h e r c o m pletely believe or c o mple te ly d is be l ie ve wha tthe e quipme nt indic a te s . If no o t h e r n a v igation data are available, belief in the one

se t of data is a p u r e act of faith. The

o b s e r v e r , h o w e v e r , is e n c o u r a g e d in suchfaith when the dia ls or i n d i c a t o r s are r e l a t ively s table and mo ving ata r e a s o n a b l e r a t e .This s ta bi l i ty is a m e a s u r e of s i g n a l - t o -n o i s e . T h e r e f o r e , w he n f a v o r a b l e s i g n a l - t o -no is e r a t io s e xis t , c o nf ide nc e is high; but

w h e n p o o r s i g n a l - t o - n o i s e r a t i o s are en

c o u n t e r e d and dia ls s ho w e xc e s s ive wa nde r ,al l confidence is los t , even though the e q u i p me nt may on f r e que nt o c c a s io ns be indicatingvalid data.

It may now be app ar ent that the confidencel i m i t s and the a s s o c i a t e d G a u s s i a n p r o b ability function which applies to the self-

c o nta ine d s ys te m can be used in evaluatinge a c h s a m p l e of da ta f r o m the e x t e r n a l l y -r e f e r e n c e d s y s t e m . By weighting each datas a mple wi th the Gaussian function all s a m p l epo ints o uts ide the confidence l imits are given

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a w e ig h t of z e r o a nd a r e a u t o m a t i c a l ly d i s car d ed . M l p o i n t s i n s i d e t h e con f i d en cel i m i t s , wh e t h er n o i se o r v al id d at a , a r e g i v ena Gau ssi an wei g h t an d a re t h en i n t eg ra t ed .S i n c e n o i s e h a s n o p r e f e r e n t i a l p o s i t i o n , o v e r

an i n t e rv al i t wi l l i n t eg ra t e t oward zerov a l u e . H o w e v e r , i f a v a l id e r r o r e x i s t s a ndev en i f on l y occas i on al d at a samp l es a rere cei v ed , su ch i n fo r mat i o n i s no t a ran d o mp r o ce ss and wi l l r esu l t in a d ef i n it e i n t e g r a t o r o u t p u t. T h e e x i s t e n c e o f s u c h p o s i t io n a l e r r o r s , t h a t i s d i f f er e n c e s b e t w e ent h e se l f -co n t a i n ed an d t h e ex t er n a l l y -re fer en c ed i n d i cat i o n s , i s u sed in a m an n e rwh i ch cau se s t h e two sy s t e m s t o co me in t ob e t t e r a n d b e t t e r l o n g - t e r m a g r e e m e n t .

F i g . 3 s h o w s t h e m a n n e r o f c o n v e r t i n g

t h e e x a m p l e d e a d - r e c k o n e r i n t o a s y s t e mwh ich ac cep t s an d wei g h s o u t s i d e d at a . Ob serv e t h at f rom t h e i n i t i a l i n p u t s , h ead i n g ,a nd a i r s p e e d , d o w n t o p r e s e n t - p o s i t i o n d at at h i s i s s t i l l t h e same u n i t p rev i ou sl y sh own .P r o v i s i o n i s m a d e f o r a c c e p t i n g d a t a f r o mt h e r a d i o s y s t e m a n d s u b t r a c t i n g t h e s e f r o mt h e l ocal d at a . Ob v i ou sl y , i d en t i cal co o rd i n at e s and u n i ts mu s t b e u sed . T h e" d at a f i l t e r " sh own in t h e f i g u re i s afunctio nal unit which em bo dies the co nfidencel i mi t s an d ap p l i es t h e Ga u ssi an wei g h t i n g .

T h e o u t pu t f ro m t h i s u n i t , wh i ch i s p r o b a b i l it y - w e i g h te d e r r o r , i s t h en i n t e g r a t e d .T h i s i n t eg r a l v al u e can b e sh own t o b e t h ev e lo c i t y c o r r e c t i o n r e q u i r e d in o r d e r toc a u s e t h e l o c a l l y m e a s u r e d v e l o c i t y t o c o m ei nt o l o n g - t e r m a g r e e m e n t w i th t h e e x t e r n a ld a t a . F u r t h e r m o r e , s i nc e t he e x t e r n a l d a taa r e m e a s u r e d a g a in s t a g r o u n d r e f e r e n c et h i s v e l o c i t y c o r r e c t i o n c o n v e r t s t h e l o c a lm e a s u r e m e n t f r o m a i r s p e e d a nd h e a di n g t og r o u n d s p e e d a nd g r o u n d - t r a c k . T h i s c o r r ect io n is th er efo r e the wind effect (i f al l

l o c a l m e a s u r e m e n t s w e r e e r r o r - f r e e ) a n d i si n j ected a t t h e d e ad - r ec k o n er wi nd i n p u t s .

F i g . 4 i l l u s t r a t e s t he u n d er l y i n g p h i lo sop h yof t h e SCAN sy st e m . He re d at a f ro m a self-

c o n t a in e d a n d a n e x t e r n a l l y - r e f e r e n c e d

s y s t e m a r e c o m p a r e d o r s u b t r a c t e d o n e f r o mt h e o t h e r . T h e d i f fe ren ce wi l l co n t a i n t h en o i s e o f b o t h s y s t e m s p l u s a ny v al id e r r o rda ta. If o ne had an o pt imu m fi l ter ( in thisca se a f i l t e r cap ab l e of accep t i n g val i d d at a

and r eject in g- no ise) the output of the f i l terw o u l d c o n s i s t o f o n l y s u c h c o r r e c t i v e d a t aa s r e q u i r e d t o m a k e t h e tw o s y s t e m s a g r e e .T h e p r a c t i c a l f i l t e r , w hi ch a p p r o a c h e s t h eo p t i m u m in p r a c t i c e , c o n s i s t s o f t he s t a t i s t i cal p r o ces s i n g of t h e d at a f i l t e r an d t h efu n ct ion i n g of t h e d ou b l e i n t eg ra t i n g c l o sed -l oop s sh own i n F i g . 3 .

In p r a c t i c e , t h e m e a s u r e d s t a t i s t i c a l d r i f tc h a r a c t e r i s t i c s o f t h e s e l f - c o n t a i n e d s y s t e ma r e u s e d t o e s t a b l i s h a c o n f i d e n c e -l i m i tc i rc l e ab ou t t h e i n d i cat ed p os i t i on of t h e

v eh i c l e . T h e rad i u s of t h e co n f i d en ce- l i mi tc i r c l e d ep e n d s u p on t h e i n t e rv a l o f t i me s i n cea c c e p t a bl e c o r r e c t i v e d at a w e r e r e c e i v e da nd t h e s y s t e m d r i f t r a t e s . R a d i o p o s i t i o nd at a s am p l e s a r e i n d iv i d u all y g iv en a p r o b ab i l i t y o r c red i b i l i t y coef f i c i en t d et e rmi n edb y t h e re l a t i o n sh i p of t h e i n d i cated p os i t i onan d t h e con f i d en ce l i mi t . T h i s co n st i t u t esa m u l t i p l i c a t i o n o f t h e r a d i o s y s t e m d i s t r i b u t i o n b y t h e s e l f - c o n t a i n e d s y s t e m e r r o rp r o b ab i l i t y . T h i s p ro d u ct , wh ich i s t h ep r o b a b i l i ty - w e i g h t e d e r r o r , is i n t e g r a t e d t o

p r o v i d e t r u e c r o s s - c o r r e l a t i o n . L o n g - t er mi n t eg r at i o n fu r t h e r co n st i t u t e s t h e t ak in g ofa l a r g e s t a t i s t i c a l s a m p l e ; n o c o n c l u s io n sa r e e v e r d r a w n o n o n e , o r a f e w , s a m p l e sof d at a .

T h e s p e c i a l p u r p o s e m a c h i n e t h us d e v i s e d" k n o w s " ho w t o m a x i m i z e th e c o r r e l a t i o n a ndi n so d o i n g l earn s t h e wi n d s a l of t an d cau sest h e t wo se t s o f n av i g at i on al d at a t o comei nt o c l o s e r a nd c l o s e r a g r e e m e n t o v e r t h el o n g - t e r m . F u r t h e r m o r e , it c an o b t ai n a dd edc o r r e c t i o n d a t a e v e n w i th s p o r a d i c r a d i os i g n a l s o r a d v e r s e s i g n a l - t o - n o i s e r a t i o s .I t c a n p r o v i d e v e l o c i t y m e m o r y f o r a c c u r a t ee x t r a p o l a t i o n d u r i n g i n t e r v a l s w h e n e x t e r n a ld at a a r e u n av ai l ab l e an d k nows h ow t o p a ssj u d gm e n t o n d a t a w h en s i g n a l s a r e r e s t o r e d .

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B e c a us e o f the s ys te m' s s o phis t ic a te d da taf i l te r ing , i t i s r e la t ive ly immune f r o m in te r f e r e n c e e f fe c ts f r o m e i t h e r r a n d o m o r s p e c i f ic a l ly ge ne r a te d s igna ls .

The SCAN s ys te m jus t de s c r ibe d ma ys e r v e a s a n e x a m p l e o f m y b r o a d e r t h e s i s ;t h e r e a r e , h o w e v e r , n o s e t r u l e s f o r th ed e v el o p m e n t o f o p t i m a l s y s t e m s . E v e r ya s pe c t o f the p r o ble m mus t be e xplo r e d;e ve r y b i t o f kno wn o r impl ie d info r m a t io nm ust be uti l ized. Often the sett ing of thepr o ble m mus t be a djus te d in o r d e r to pe r mi tthe us e o f be s t te c hnique s . Fo r ins ta nc e ,in the SCAN s ys te m lo ng-te r m in te gr a t io nb e c a m e p o s s i b l e o n l y w h en t he t e r r e s t r i a lc o o r din a te s w e r e c ha nge d in to a mo ving s e t

o f c o o r dina te s a t ta c he d to the ve hic le .

M o s t info r m a t io n s o ur c e s a r e h igh inr e dunda nc y o f da ta ; few s ys te m s ma keeffective use of this added info rm atio n. Inge ne r a l , a ny s ys te m whic h s imply d is pla ysc ur r e nt ly r e c e ive d s igna ls , whic h do e s no tus e s to r a ge o r do e s no t a t le a s t kno w there cen t his to ry of the data , can be said to beno no pt imum. Th is a ppl ie s to a l l a pp l i c a t io n s e xce pt tho s e whic h t r a ns m it c o m p l e t e l y r a n d o m m e s s a g e s o r n u m b e r s .

3. C OM PUTER S

As o ne fur the r , a nd mo r e s pe c ula t iv e ,e xa mp le o f th is a ppr o a c h to s ys te m de s ign,I wo uld l ike to d is c us s s o me uno r tho do xt h o u g h ts a bo u t c o m p u t e r s . T h e c o m p u t e rma y be r e g a r de d a s a c o m munic a t io n p r o ble mwh erein only the eleme nt of dis ta nce has beene l i m i n a t e d . T h e o r d i n a r y a p p r o a c h t o g e n e r a l - p u r p o s e d i g i t a l c o m p u t e r d e s i g n i s c h a r

acter ized by the s izable tonnage and kilowattra t in gs of the end pro duc t. Such equipmentr e s ul ts f r o m a de s ign o bje c t ive which s e e ksequipmen t infall ibil i ty. Th is o bjective en f o r c e s p r o d i g i o u s s i g n a l - t o - n o i s e r a t i o s u n dmult ip l ic a t io n o f c o mple xi t ie s due to the

a ddi t io na l s e l f -c he c king a nd o the r s a fe g u a r d s . E a c h o f t h e s e s t e p s f u r t h e r i n c r e a s e sinput po wer and si ze . Infallibility itself isoften an elus ive go al , s inc e the s te ps takento a c hie ve i t o f te n , o f the m s e lv e s , r e duc e

r e l i a b i l i t y .

No w, fo l lo wing the p r inc iple s p r e vio us lydis c u s s e d, le t us c o ns ide r the a ppl ic a tio n o fs t a t i s t i c a l p r o c e s s e s t o s u ch a c o m p u t e r .Fi r s t o f a l l , le t us s a y we wi l l us e t r a ns is to r sand ex tr em ely low power input. In fact , le tus wo rk with s ignals at o r nea r the no isele ve l , s o tha t the r e s u l ta nt p r o ba bi l i ty o fg e t ti n g a c o r r e c t a n s w e r t o a p r o b l e m m a ybe o nly te n pe r c e n t . W hat go o d is a c o mp ute r

with ten per cen t re l iab il i ty ? Well , le t usacknowledge that if we knew how to designsuch a co m pu ter i t might be mad e vastlys m a l l e r t ha n i t s b i g, r e l i a b l e c o u n t e r p a r t .Pe r h a p s i t co uld be ma de to o c c upy o nly o ne -third of a desk- to p and be pow ered by as ingle d r y ba t te r y .

No w le t us a t ta c k i t s va lue a s a c o mpute r .W hat if we c ho s e to us e th r e e s uc h c o m pute r s(s t i l l ve r y , ve r y m uc h s m a l le r than i t s

c o unte r pa r t ) a nd c o nne c t the m s o that p r o ble ms we r e a ppl ie d to a l l th r e e in pa r a l le l .Le t the a ns w e r s o ut of e a c h c o mpu te r the n got o a s m a l l p r o g r a m m e r w h ic h m a k e s ac o inc ide nc e c he c k o n the s e v e r a l num e r ic a lr e s ul ts . Only when a l l th r e e ma c h ine sa gr e e wi l l the a ns w e r be a c c e pte d; unt i lthe y a l l a gr e e o n a c o mmo n a ns we r thep r o g r a m m e r c o n t i n u al ly r e i n i t i a t e s th e s a m epr o ble m . Eve ntua l ly s uc h a n a gr e e m e nt wi llo ccu r and the ans we r will co me o ut. Ifs i m u l t a n e o u s l y c o r r e c t a n s w e r s a r e r e

qui r e d , the p r o ba ble numbe r o f c o mputa t io ns(fo r te n- pe r c e nt r e l ia bi l i ty) i s o ne tho us a nd.Ho we ve r , i f s o me me mo r y c a pa bi l i ty i sincluded in the design and nonsimultaneousa gr e e m e nts a c c e pte d, the p r o b a ble numbe r o fc o mputa t io ns c a n be c o ns ide r a b ly r e du c e d.

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If the computer in quest ion had as l i t t leas t en t h ou san d ou t p u t n u mb ers , each ofwh i ch w er e eq u a l l y l i k el y t o b e n o i se i n d u ced , t h e p rob ab i l i t y of a mu t u al l y ag reedu pon w ro n g a n sw er i s l es s t h an ab ou t

1 0 '

9

* . T h i s i s a v e r y r e s p e c t a b l e r e l i ab i l i t y , ev en fo r t h e so-cal l ed i n fal l i b l em a c h i n e .

You may n ow b e p re p ar ed t o ask wh yt h r e e c o m p u t e r s , w hy n o t t e n o r t wo o r o n e ?T en o r t wo wi l l on l y ch an g e t h e s t a t i s t i csof re l i ab i l i t y . On e mac h i n e wi t h t h r ee

* T h i s c o m e s f r o m ( 1 0 4 ) 3 o r 1 0IZ f o r o n ec a l c u l a t io n ; h o w e v e r , s i n c e o n t h e a v e r a g eat most 1 0 3 c a l c u l a t i o n s a r e r e q u i r e d , t h i sfigure must be divided by 10 3 .

co mp u t at i on s m ay see m t h e eq u i v al en t , , b uti s n o t . T h e s t a t i s t i c a l p r o c e s s w i ll p r o v i d es i g n a l - t o - n o i s e i m p r o v e m e n t o n ly if t he e r r o ip r o c e s s e s a r e r a n d o m . T h i s c an n o t b ea s s u r e d in a s i n g le s y s t e m ; c o n s i d e r s p e

ci f i cal l y a co mp o n en t fa i l u r e w h ich wou l dc o n s i s t e n t ly t en d t o r e p e a t a g i ve n e r r o r .

I f t h e r e a r e a n y c o m p u t e r d e s i g n e r sr e a d i n g t h i s p a p e r , I o f f er m y s i n c e r ea p o l o g i e s f o r a n y a p p a r e n t l y d i s p a r a g i n gco m me n t s I may h av e ma d e. In fact , Is t an d i n t h e u t most awe of t h ei r mon u men t albut cap able ma ch ine r y. It is my hope thatt h i s c o m p u t e r s p e c u l a ti o n d o e s s e r v e t od r a m a t i z e t h e p o s s i b i l i t i e s a n d m e t h o d s o fs t a t i s t i c al d e s i g n . T o o l on g , I t h in k , h av e wetr ied to l ive in a black and white wo rld,p o p u l a te d o n l y w it h o n e s o r z e r o s f o r p r o b ab i l i t y .

REFERENCES

1. S h an n on , C. E . , an d W eav er , W .o f I l l i n o i s P r e s s , 1 9 4 9 .

' T h e M a t h e m a t i c a l T h e o r y o f C o m m u n i c a t i o n , " U n i v e r s i t y

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Fig . 1 . B lo c k d ia g r a m o f s imp le de a d r e c ko ne r (o pe n lo o p) .

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o

+ t .

t i m e •

F i g . 2 . T y p i c a l m a x i m u m e r r o r a s a f un ct io n o f t i m e w i th o u t c o r r e c t i o n d a t a .

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H e a d i n g

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with po la r r a dio da ta .

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S e l f - Con t a ined

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GUIDANCE TECHNIQUESW a l t e r L , W e b s t e r , J r ,

SUMMARY

T h er e a r e a n u m b er of p o ss i b l e g u i d an ce t ech n i q u es wh i ch can b e u sed t o g u i d e am i s s i l e t o w a r d a c o l l is i o n w i th i t s t a r g e t . T h e s e a r e c o m m a n d , b e a m r i d e r , r a d i o n a v i g at i on al , h om i n g , an d com b i n ed t ech n i q u e s , each of wh i ch h as i t s ad v an t ag es an d d i sa d v a n t a g e s . T h e c o m m a n d s y s t e m r e q u i r e s c o m p l e x g r o u n d e q u i p m e n t s a nd g u i d e s t h em i ss i l e by co n t i n u ou sl y m o n i t o r i n g t h e p o s i t io n of b o t h t h e m i ss i l e an d t h e t a r g e t . T h eb e a m r i d e r f o ll o w s a b e a m w h ic h i r r a d i a t e s an d l o c k s o n t h e t a r g e t f r o m t h e m i s s i l e s o u r c e .Rad i o n av i g at i on al sc h em es a r e v er y ac cu ra t e in p o s i t i on al p l acem en t of t h e m i ss i l e b u tr e q u i r e a gr o u n d b a s e l i n e i n s t a l l a t i o n . H o m i n g t e c h n i q u e s r e l y o n r e f l e c t i o n s o r r a d i a t i o n sfro m the tar get and have the advantage of having high er and highe r signa l leve ls ava i lablea s t h e m i s s i l e a p p r o a c h e s i t s t a r g e t . T h e c h o i c e o f s y s t e m d e p e n ds o n a n u m b e r o f fa c t o r ssu ch as t h e accu racy , t h e med i u m f rom wh i ch t h e mi ss i l e i s l au n ch ed , t h e n u mb er of s i mu l t an eou s t a r g e t s and m i ss i l e s w h ich m u st b e h an d l ed , and t h e ra n g e o v er wh ich t h e m i ss i l e s

must be effect ive.

SOMMAIRE

II ex i s t e u n cer t a i n n o mb r e d e t ech n i q u es d e g o u v ern e p o ss i b l e s q u i p eu v en t e t r eu t i l i see s d an s l a g ou v ern e d 'un m i ss i l e v er s la co l l i s i on av ec sa c i b l e . Ceu x so n t : l es t ech n i q u e s d e c o m m a n d e m e n t , d e s u i v e u r d e r a y o n , d e r a d i o n a v i g a t i o n n e l l e , d e " r e n t r e e " e t l e st ech n i q u es c o m b i n ee s , d o n t ch acu n e a se s av an t ag es e t se s d es av a n t ag es . L e sy s t em e d eco mm an d emen t ex i g e d es eq u i p em en t s au so l co mp l ex es e t il g u id e l e m i ss i l e en re l a t an tcon t i n u men t l es p o s i t i on s re sp e ct i v es d u m i ss i l e e t d e l a c i b l e . L e su i v eu r d e r ay on su i t u nray o n q u i i r r ad i e e t se fi xe su r l a c i b l e a p a r t i r d e l a so u r ce d u m i ss i l e . L e s d i sp o s i t i f s d en a v ig a ti o n p a r r a d i o s o n t t r e s p r e c i s e s d a n s l e p o s i t i o n n e m e n t du m i s s i l e , m a i s i l s e x ig e n t

u n e i n st a l la t i o n b ie n d e t e r m i n e e a u s o l . L e s t e c h n iq u e s d e " r e n t r e e " s ' a p p ui e n t s u r l e sref l ex i o n s e t l e s r ad i a t i o n s eman an t d e l a c i b l e e t e l l e s on t 1'avantage d 'ob t en i r d es s i g n au xd 'en t re e d on t l e n i veau es t d e p l u s en p l u s e l ev e a m es u r e q u e l e m i ss i l e s 'ap p r o ch e d e saci b l e . L e ch oi x d 'u n sy s t em e d ep en d d 'u n ce r t a i n n o mb re d e fact e u r s t e l s q u e, l a p r eci s i o n ,l e v eh i cu l e d uq u el l e m i ss i l e es t l an ce, l e n o mb re d e c i b l e s e t d e m i ss i l e s qu i d o i v en t e t r es i m u l ta n d m e n t m a n i a s e t l e s d i s t a n c e s p o u r l e s q u e l l e s l e s m i s s i l e s d o i v e n t e t r e e f f i c a c e s ;

1. INTRODUCTION

A m i ss i l e g u i d an ce sy s t em h as b eend efi n ed as a g ro u p of co mp o n en t s wh ichm e a s u r e s t he p o s i ti o n of a g ui de d m i s s i l ew i t h r e s p e c t t o i t s t a r g e t a n d c a u s e s c h a n g e sin the f light path a s r eq uir ed . A subject a s

b r o a d a s t h i s , o n e w h ic h e n c o m p a s s e s s o m a n yof t h e en g i n eer i n g an d b as i c sc i en ces , can n ot

be t r ea ted in deta i l in the length of t im ea l l o c a t e d t o a ny o n e s p e a k e r . I w i ll t r y ,t h e r e f o r e , t o c o v e r w i th a r a t h e r b r o a d b r u s ht h e v ar i o u s t y p es of g u i d an ce sy s t em s wi t hwh i ch I am fami l i a r , t h ei r ad v an t ag es an d

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d i s a d v a n t a g e s , a n d s o m e o f t h e a s s o c i a t e de ngi ne e r i ng p ro bl e m s w hi c h w e i n t he U ni t e dSt a t e s ha ve e x pe r i e nc e d i n t he de s i gn andt e s t i n g o f s u c h s y s t e m s d u r i n g th e p a s td e c a d e .

In t he us ua l c a s e , t he gui da nc e s ys t e m w i l li n cl u d e s e n s i n g , c o m p u t in g , d i r e c t i n g , s t a b i l i z i ng , a nd s e rv o c o m po ne nt s . N o t a l l o ft h e s e c o m p o n e n t s n e e d be i n c o r p o r a t e d in t h em is si le and in fact i t wi l l be see n that in am a j o r i t y o f th e p r a c t i c a l c a s e s a p o r t i o n o ft h e s e c o m p o n e n t s i s e x t e r n a l t o t h e m i s s i l eand lo cated at the launching si te . Thes t a b i l i z i n g a n d s e r v o c o m p o n e n t s a r e a l w a y sl o c a t e d in t he m i s s i l e a n d a r e o fte n r e f e r r e dt o a s t h e c o n t r o l s y s t e m , a lt h o u g h t he y a r ec o n s i d e r e d t o b e c o m p o n e n t p a r t s of th eo v e r a l l g u i d a n c e s y s t e m .

The o bj e c t i ve o f a gui da nc e s ys t e m i s t ode l i ve r a m i s s i l e a t t he t a rge t w i t h s uf f i c i e nta c c u r a c y t o a c c o m p l i s h i t s m i s s i o n . In t h em i l i t a r y s e n s e t h e m i s s i o n i s u s u a l ly t a r g e td e s t r u c t i o n . M i s s i l e t a r g e t s a r e g e n e r a l l yc l a s s i f i e d i n to t w o b r o a d g r o u p s , a i r t a r g e t sa nd s u r f a c e t a r g e t s . S u r f a c e t a r g e t s m a y b ef u r t h e r b r o k e n d o w n i n t o s t a t i o n a r y t a r g e t sa nd m o vi ng t a r ge t s ; 1 t hi nk yo u w i l l a g re et h a t a l l a i r t a r g e t s a r e m o v i n g t a r g e t s . Itw i l l be f o und t ha t t he a bo ve c a t e go r i e s o f

t a r ge t s w i l l ha ve a g r e a t i nf lue nc e o n thet ype o f gui da nc e s ys t e m w hi c h i s c ho s e n f o ra n y g i ve n m i s s i l e s y s t e m .

At this t im e I would l ike to o ut l ine f iveb r o a d c a t e g o r i e s o f g u i d a n c e in t o w h ic h m o s tkno w n s ys t e m s w i l l f a l l :

( a ) C o m m a nd G ui da nc e

(b ) B e a m - R i d e r G u i d a nc e

( c) R a di o N a vi ga t i o na l G ui da nc e

( d) H o m i ng G ui da nc e

( e ) C o m bi ne d G ui da nc e Sys t e m s

2 . COMMAND GUIDANCE SYSTEM

A c o m m a nd gui da nc e s ys t e m i s de f ine d a so ne in w hi ch gui da nc e i n t e l l i ge nc e t ra ns m i t t e dt o t h e m i s s i l e f r o m a r e m o t e s o u r c e c a u s e st he m i s s i l e t o f o llo w a d i re c t e d fl ight pa t h .

I n h e r e n t l y , c o m m a n d s y s t e m s a r e t h e s i m pl e s t o f a l l the t e c hni que s c o ns i d e r e d .C o m m a nd gui da nc e s y s t e m s ha ve lo ng be e nu s e d f o r t h e r e m o t e c o n t r o l o f b o a t s , t a n k s ,m o d e l a i r c r a f t , an d t a r g e t a i r c r a f t . Ing e n e r a l , c o m m a n d s y s t e m s r e q u i r e t h at t h ebe ha v i o r o f t he m i s s i l e a nd t he t a r ge t bem o n i t o r e d , t h at a p r e s c r i b e d c o l l is io n c o u r s ebe c o m put e d, a nd t ha t i nf o rm a t i o n be t ra ns m i t t e d t o th e m i s s i l e c o n t r o l s y s t e m s o a s t oal ign the missi le f l ight path toward ani n t e r c e p t w i th t h e t a r g e t . I ts m o s t s e r i o u s

g e n e r a l l i m i t a ti o n f o r m i l i t a r y m i s s i l e u s e i sl o w t ra f f i c - ha ndl i ng c a pa c i t y , f o r t he t ypi c a lc o m m a nd s ys t e m w i l l s o l ve o nl y o ne i n t e rc e p tpr o ble m at a t im e. Owing to the fact that thebe ha v i o r o f t he m i s s i l e a nd t he t a rg e t m us t bem o ni t o re d a nd t ha t t h i s i s m o s t o f t e n a c c o m pl i s he d by t ra c ki ng de vi c e s l o c a t e d a t t hel a unc h s i t e , a no t he r s e r i o us l i m i t a t i o n i s i t sd e c r e a s i n g a c c u r a c y w ith i n c r e a s e in r a n g e .

The p r i nc i pa l f unc t i o ns o f a c o m m a ndgui da nc e s ys t e m a r e t he a c qui s i t i o n a nd

t ra c ki ng m e a ns , t he c o m put a t i o n o f f l i ghtp a th e r r o r , c o m m u n i c a t io n o f t h e i n t e ll i g e n c et o t h e m i s s i l e , a nd a s e n s i n g a nd c o n t r o lm e c h a n i s m in t h e m i s s i l e to a c c o m p l i s hf l i g h t p a t h c o r r e c t i o n .

A t ypi c a l c o m m a nd s ys t e m w o ul d lo o kfunct ional ly so mew hat as shown in Fig . 1.T h i s s y s t e m p r o v i d e s f o r s e p a r a t e t r a c k i n go f t he m i s s i l e a nd t a rge t i n be a r i ng a ndr a n g e , a n d a s e p a r a t e c o m m a n d l in k . T h e r ea r e , o f c o u r s e , m a n y v a r i a t i o n s o f t h i s t yp e

o f s y s t e m a nd it m a y b e u s e d f o r a i r - t o - a i r ,a i r - t o - g r o u n d , a nd e v e n s u r f a c e - t o - s u r f a c em i s s i l e s a s w e ll a s f o r t he s u r f a c e - t o - a i rc a s e s ho w n h e r e . T h e o b v i o u s a d v a n ta g e o fs uc h a s ys t e m i s de r i ve d f ro m t he f a ct t ha t

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most of t h e comp l ex eq u i p men t i s l ocat ed a tthe launch si t e instea d of in the m is si le . Them i s s i l e c a r r i e s o n l y a b a r e m i n i m u m o fe l e c t r o n i c s - a c o m m a n d r e c e i v e r a n d s t a b i l i zat i on an d con t ro l eq u i p men t , t h ereb y i n

c r e a s i n g i t s r e l i a b i l i t y a n d d e c r e a s i n gm i s s i l e c o s t .

The design of the launching si te equipm entfo r t h i s sy s t em i s con v en t i on al , i t s comp l ex i t ydepending upon the specif ic ap plicat io n. Int h e c a s e s ho w n , t h e t r a c k i n g r e q u i r e m e n t s f o rm i s s i l e a nd t a r g e t a r e p r a c t i c a l l y i d e n t ic a l ,w h i l e i n t h e c a s e o f t h e s u r f a c e - t o - s u r f a c em i ss i l e w h ere t h e t a rg e t i s f ix ed t h e re wi llb e n o r e q u i r e m e n t f o r t a r g e t t r a c k i n g , a n dt h e t a rg et co o rd i n at es can b e se t in t o t h ec o m p u t e r . T h e c o m p u t e r d e s ig n a g a in w i llb e s t ra i g h t fo rw ard b ut wi l l v ar y in co m plexity with type of flight path solutionre q u i r ed and th e n u mb e r of ad d i t i on alfu n ct ion s it i s ca l l ed on t o p er fo rm , su ch asp ara l l ax comp u t at i on , p o i n t i n g of t h e l au n ch i ng p l a t fo r m, e t c .

S i n ce commu n i cat i on of t h e g u i d an ce i n t e l l i g en ce t o t h e mi ss i l e i s most o f t enacco m plishe d by ra dio l ink, i t wil l be foundt h at in th e n o n m i l i t a ry ca se t h i s com p on en tis the least diff icul t to design of al l theel em en t s of t h e g u i d an ce sy s t e m . In g en er al ,UHF and VHF fr equ enc ies wil l be used to takead v an t ag e of t h e red u ced s i ze of comp on en t sreq u i red i n t h e mi ss i l e , a l t h ou g h t h e con t ro l l i n g fact o r wi l l b e t h e ran g e a t wh i ch t h em i s s i l e s y s t e m w i ll b e r e q u i r e d to o p e r a t e .In t h e mi l i t a ry ap p l i cat i on , comman d l i n ksec u r i t y w i ll b e a d es i g n co n si d er a t i o n ofu t mo st i mp o r t a n ce . Id eal ly , t h e l in k mu st b ei m m u n e t o i n t e r f e r e n c e fr o m i d e n t i c a ls y s t e m s o p e r a t i n g i n t h e s a m e o r a d j a c e n ta r e a s , i n t e r f e r e n c e f r o m c o m m u n i c a t i o n a n d

n av i g ati on al s y s t em s of f r i en d ly fo rc es , andf rom d et ect i on , an a l y s i s , and j amm i n g b y t h ee n e m y . S uc h c o n s i d e r a t i o n s g r e a t l y c o m plica te the pr o ble m s involved in the desig n ofa comman d l i n k fo r mi l i t a ry ap p l i cat i on .

3 . BEAM-RIDER GUIDANCE SYSTEM

A b e a m - r i d e r g u i d a n c e s y s t e m i s d e f i n e das a sy s t em fo r g u id i n g m i ss i l e s wh i chu t i l i zes a b eam d i rect ed i n t o sp ace, su ch t h at

t h e cen t e r o f t h e b eam ax i s fo rm s a l i n eal on g wh i ch i t i s d es i red t o d i rect a mi ss i l e .(T h e most p ract i cal t y p e of su ch a sy s t em i st h at u s i n g ra d a r t o fo r m th e b eam . ) T h em i s s i l e c o n t a i n s e q u i p m e n t t h a t d e t e r m i n e st h e d i re ct i o n an d mag n i t u d e of t h e e r r o r wh ent h e m i ss i l e d ev i at es f rom t h e cen t er o f t h eb e a m .

F i g . 2 s h o w s a s i m p l e d i a g r a m o f a b e a m -r i d i n g m i s s i l e g u i d a n c e s y s t e m w h e n a r a d a rbeam is used as the guiding medium.

A r a d a r a n t e n n a i s d i r e c t e d s o t h a t t h ecen t er o f t h e b eam i s on t h e t a r g et . In fo rm a t i o n i s s u p e r i m p o s e d o n t h i s r a d a r b e a mw h ic h p e r m i t s t h e m i s s i l e t o m e a s u r e i t sp o s i t i on wi t h re sp e ct t o t h e cen t er o f t h erad ar b eam an d t o mov e i n a d i rect i on wh i chw i ll r e d u c e th i s e r r o r t o w a r d z e r o . T h ee x a m p l e s ho w n i s an a i r - t o - a i r a p p l i c a t io n ;s i m i l a r s y s t e m s c an b e d e v is e d fo r s u r f a c e -t o - a i r an d a i r - t o - s u r f a c e a p p l i c a ti o n . It w i llb e seen t h at in a l l ap p l i cat i o n s of t h e b ea m -

r i d e r p r i n c i p l e , t h e r e m u s t b e a l in e - o f - s i g h tp at h b et ween t h e g u i d an ce rad ar an d t h et a r g e t a t a l l t i m e s .

On e r eq u i re m en t of t h e sy s t em i s a h ig ho r d e r o f a c c u r a c y i n p l a c in g t he r a d a r b e a mon th e t a r g e t . In mo st ca se s t h e r ad ar wi l l bea n a u to m a t i c a l l y t r a c k i n g r a d a r in w h ic h c a s ea p r e c i s i o n t r a c k i n g r a d a r w i t h h i g h a n g u l a ra c c u r a c y , s u c h a s i s u s ed f o r a i r b o r n ei n t e r c e p t o r s h i p bo a r d f i r e c o n t r o l , w i llm e e t t h e r e q u i r e m e n t s .

M en t i on h as b een mad e of t h e n ecess i t yfo r t h e b eam t o co n t a in i n fo rm at i on wh i ch wi l lp e r m i t t h e m i s s i l e t o o r i e n t i t se l f w it h r e s p e c tt o th e cen t e r o f t h e b eam . T h e g u i d an ce

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i n fo r mat i o n mu st co n t a i n a t l eas t t wo fact s :t he a m p l i tu d e o f t he e r r o r a n d t h e d i r e c t i o nof t h e e r r o r . W i th t h i s i n t e l l i g en ce t h em i s s i l e c a n c o r r e c t i t s e r r o r in p o s i ti o nm e a s u r e d f r o m t h e c e n t e r o f t h e b e a m .

In t h e con v en t i on al scan n i n g r ad ar t h ecen t er o f t h e rad ar b eam i s p o i n t ed a t anangle f ro m the scan axis and the bea m isro t a t ed so t h at t h e ce n t e r o f t h e b eam fo l l owst h e scan c i rc l e as sh own i n F i g . 3 .

I f a recei v er i s p l aced an y wh ere on t h es c a n a x i s , th e s i g na l s t r e n g t h m e a s u r e d byt h e r ec ei v er wi l l b e co n st an t as th e b eamc i r c l e s a r o u n d t he a x i s . If t h e r e c e i v e r i sd i sp l ac ed f ro m t h e ax i s t h e am p l i t u d e wi l l b emo d u l at ed a t t h e scan f req u en cy . F i g . 4sh ow s t h e v ary i n g amp l i t u d e mod u l at i on wh i chr e s u l t s f r o m p l a c in g t h e r e c e i v e r a t t h r e ed i f fe ren t p o i n t s wit h res p ec t t o t h e scan ax i s .

W h en t h e recei v er an t en n a i s d i sp l acedf ro m t h e s can ax i s at (a) , a s i n e wav emod u l at i on ap p ears i n t h e amp l i t u d e of t h ere cei v ed s i g n al wi t h t h e p h ase of t h e mo d u l a t i on , su ch t h at t h e max i mu m amp l i t u d ea p p e a r s a t 9 0 d e g r e e s . W h en t h e r e c e i v e ris at point (b) , the modulat ion is st i l l p resentbut reduc ed in am pli tude and the phas e hasch an g ed so t h at t h e max i mu m ap p ears a t

1 80 d e g r ee s . At p o i n t (c), t h e re cei v ed s i g n ali s c o n s t a n t .

It w il l be found that as long as the r e cei v er i s mai n t a i n ed n ear t h e scan ax i s oft h e rad ar , t h e s i g n al amp l i t u d e mod u l at i onw i ll b e d i r e c t l y p r o p o r t i o n a l to t h e d i s p l a c e m e n t o f t h e r e c e i v e r f r o m t h e c e n t e ro f t h e b e a m . A b e a m r a d a r r e c e i v e r ,t h e r e f o r e , a l r e a d y h a s a m e a s u r e o f t hea m o u n t o f e r r o r o f t he m i s s i l e ' s d i s p l a c e m ent . I t has al r ea dy been pointed out thatt h e amo u n t of e r r o r i s n o t enou g h , b ut t h at

an addit ional fact must be supplied to them i s s i l e ; n a m e l y, t he d i r e c t i o n o f t he e r r o r .T h e p h ase of t h e f req u en cy mod u l at i on i sav ai l ab l e i n t h e mi ss i l e recei v er an d n eed so n ly to be c o m p a r e d w i th a r e f e r e n c e m o d u

l a t io n a t th e ra d a r scan f req u en cy t o d i sc l o set he d i r e c t i o n o f th e e r r o r . T h e r e f e r e n c emo d u l at io n may b e o b t a i n ed f ro m t h e ref

e r e n c e g e n e r a t o r a l r e a d y e x is t in g in t he r a d a ra nd m a y be t r a n s m i t t e d t o t h e m i s s i l e r e

c e i v e r i n an y m a n n e r th e d e s i g n e r m a ysel e ct . If a su i t ab l y s t ab l e o sc i l l a t o r can b ed e s i g n e d , th e r e f e r e n c e g e n e r a t o r m i g h t e v enb e c a r r i e d i n t h e m i s s i l e a n d s y n c h r o n i z e dwi t h t h e r ad ar j u s t p r i o r to l au n ch . Af te rt h e s i g n al s a re recei v ed i n t h e mi ss i l e , t h eya r e s e p a r a t e d i nt o a v o l t a ge p r o p o r t i o n a l t ot h e d i sp l ac em en t of th e mi ss i l e f rom t h e scana x i s o f t h e r a d a r a nd a v o l t a g e p r o p o r t i o n a lt o t h e a n g u l ar p o s i t io n of t h e m i ss i l e in t h er a d a r b e a m a s m e a s u r e d f r o m s o m e a r b i t r a r yz e r o r e f e r e n c e . T h e c o m b i n at io n o f t he tw os i g n a l s r e p r e s e n t s i n p o l a r c o o r d i n a t e s t h ee r r o r o f th e m i s s i l e f r o m t h e c e n t e r o f t h eb e a m . T h i s i n fo r m a t i o n m a y b e c o n v e r t e di nt o r e c t a n g u l a r c o o r d i n a t e s to c o n t r o l t h emi ss i l e i n p i t ch an d y aw.

In a b e a m - r i d e r s y s t e m t h e e q u ip m e n tc a r r i e d b y t h e m i s s i l e i s m o r e c o m p l e x t h a nin t h e s i m p l e c o m m a n d s y s t e m p r e v i o u s l yd escr i b ed , b u t l ess comp l ex t h an most o f t h es y s t e m s t o fo l lo w . T h e o v e r a l l s y s t e m i s l e s scomp l ex t h an t h e t y p i cal comman d sy st em,s i n c e o n ly o n e tr a c k i n g r a d a r i s r e q u i r e d an d

c o m p u t e r r e q u i r e m e n t s a r e m u c h l e s ss e v e r e . A lth o u gh a b e a m - r i d e r g u id a n c er a d a r i s r e q u i r e d f o r e a c h t a r g e t b e in ga t t a c k e d , m o r e t h an o n e m i s s i l e m a y b eg u id ed s i mu l t an eo u sl y b y t h e b eam . T h ep r o b l e m o f m u t u a l i n t e r f e r e n c e b e tw e e nsi m i l a r g u i d an ce sy s t e m s b ei n g u sed in t h esame area ex i s t s i n t h i s t y p e of g u i d an ce b u ti s fa r l e ss se v e r e th an i n t h e p re v i o u s cas e .

4. RADIO NAVIGATIONAL GUIDANCE

Bot h c i r cu l ar an d h y p er b o l i c n av i g at i on

t e c h n iq u e s a r e a p p l i c a b l e t o m i s s i l e c o n t r o lan d v ar i ou s fo rms of each h av e b een u sedf o r y e a r s i n t h e c o n t r o l o f p i lo t e d a i r c r a f t .T h ey a r e p r i m a r i l y a p p l ic a b le to s u r f a c e - t o -s u r f a c e a nd s u r f a c e - t o - a i r m i s s i l e s y s t e m s .

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Hy p erb ol i c n av i g at i on i s a g en eral met h odfor d et e rmi n i n g l i n es of p os i t i on b y measu r i n g t h e d i f fe ren ce i n d i s t an ce of t h e a i r c r a f t o r m i s s i l e f r o m tw o o r m o r e s t a ti o n sw h o s e p o s i t i o n s a r e k no w n . B y m e a s u r i n gt h e d i f fe ren ce of t i me of a r r i v al o f s i g n al st r a n s m i t t e d f r o m t h e s t a t i o n s , t h e d i s t a n c e sc a n b e d e t e r m i n e d .

A t y p ical h y p erb o l i c g u id an ce sy s t e m i ss ho w n in F i g . 5 . A m a s t e r t r a n s m i t t e r i sc a r r i e d in th e m i s s i l e a nd p e r i o d i c a l ly t r a n s m its a signal which is re ce ived by each of theg r o u n d s t a t i o n s . E a c h s t a t io n t he n t r a n s m i t sa s i g n al b ack wh i ch i s recei v e d by t h e mi ss i l e .A t i me co mp ar i so n of t h e two s i g n al s i s mad ein a c o m p a r a t o r a n d f r o m t h i s t h e n e c e s s a r yt i m e o r d i s t a n c e d i ff e r e n c e i s d e r i v e d . T h e

m i ss i l e can t h en b e mad e t o fly on a h y p er b o l i cp a th w h ic h r e p r e s e n t s a c o n s t a n t d i s t a n c ed i f fe ren ce an d wh i ch p asses t h rou g h t h et ar g et . Al t it u de i s co n t ro l l ed by an a l t i m et e r .Di s t an ce f rom ei t h er o f th e g ro u n d s t a t i o n scan be m easu re d i n d ep en d en tl y by m eas u r i n ga c t u a l r o u n d t r i p t i m e o f t r a n s m i s s i o n o f t h esignal and when this distance equals thed i s t an ce f rom t h e s t a t i on t o t h e t a rg et , t h et e rmi n al d i v e can b e i n i t i a t ed .

Hy p erb o l i c sy s t em s can b e d es i g n ed t o gi v eex cel l en t acc u r acy . T h ei r t raf f ic h an d li n gcap aci t y i s g ood an d man y wi d el y -sp acedlaunching si t es co uld be used to launchm i ss i l e s i n to a s i n g l e h y p erb o l i c g u i d an cep at t e rn . T h e m i ss i l e d o es n ot h av e t o b et ra ck ed , as i n t h e com man d t y p e sy s t em ,al though the tact ical si tuat ion may make thisd e s i r a b l e f o r i n f o r m a t i o n p u r p o s e s .

A v e r y s i m i l a r r a d i o n a v ig a t io n a l t yp es y s t e m , d if fe r i ng p r i m a r i l y in g e o m e t r y i s t he

s o - c a l l e d c i r c u l a r n a v ig a t io n a l s y s t e m s ho w nin Fig . 6 . In this sys tem the m is si le f l iesal on g a p at h of con st an t d i s t an ce f rom ag r o u n d t r a n s m i t t e r d e s c r i b i n g , of c o u r s e , t h ear c of a c i r c l e . If t h i s co n st an t d i s t an c e i s

eq u al t o t h e ran g e of t h e g rou n d s t a t i on f romt h e t a rg et , t h en t h e a rc a l on g wh i ch t h em i s s i l e f l ie s w i l l p a s s o v e r t h e t a r g e t .

T h e ad v an t ag es of su ch a sy s t em areidentical with those of a hyperbol ic types y s t e m a n d t h e s a m e t a c t i c a l c o n s i d e r a t i o n sap p l y ; n am el y , t h e t a r g et mu st b e a s t a t i o n ar yon e an d t h e d i s t an ce of t h e t a r g et f ro m t h etwo sla ve sta t io ns must be known. I t should ben ot ed th at i n b o t h t h ese sy s t e m s, on l y d i st a n c e s a r e r e q u i r e d a n d t h a t t h e d i s t a n c ebetween the two slave stat ions need not bek n own ex act l y .

5. HOMING GUIDANCE SYSTEMS

A h om i n g g u id an ce s y s t e m may b e d efi n edas a g u i d an ce s y s t e m b y wh i ch a mi s s i l es t ee r s i t se l f t o ward a t a rg et b y m ean s of asel f -con t a i n ed mech an i sm wh i ch i s ac t i v at edb y s o m e d i s t i n g u i s h i n g c h a r a c t e r i s t i c o f t h et a r g e t . T h e r e a r e t h r e e g e n e r a l t y p e s o fh o m i n g s y s t e m s ; i . e . , a c t i v e h o m i n g , s e m i -act i v e h omi n g , an d p ass i v e h omi n g .

A p a s s i v e h o m i n g g u i d an c e s y s t e m m a y bed efi ned as a h om i n g sy s t e m wh e rei n t h em i s s i l e m a k e s u s e o f e n e r g y e m a n a t i n g f r o m

t h e t a r g e t . In s u c h a s y s t e m n o t r a n s m i t t e ro r i l l u mi n at i n g so u r ce of any k in d i s re q u i r edi n t h e mi ss i l e , b u t m er el y a r ec ei v er wh ich i scap ab l e of d et ect i n g an d t rack i n g t h e p art i cu l ar t y p e of en er g y b ein g ra d i a t ed . T h i sene rg y may be in the fo rm of l ight , hea t ,s o u n d , o r e l e c t r o m a g n e t i c r a d i a t i o n .

O f t h e t h r e e b a s i c h o m i n g s y s t e m s , t h ep a s s i v e s y s t e m r e q u i r e s t h e l e a s t e q u i p m e n ti n t h e mi ss i l e an d , t h er efo re , h o l d s ou t t h ep r o m i s e o f g r e a t e s t r e l i a b il i ty . T h e g e n e r a l

g u i d an ce t ech n i q u e fo r v i su al li gh t o r i n f ra r edhoming guidance is the same, differing only ind et a i l s o f t h e sen s i t i v e e l emen t an d t h e sy s t emo f o p t i c s . A s e n s i t i v e e le m e n t , a n o p t i c a lsy s t e m t o focu s t h e en er g y on t h e e l em en t ,

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and a method of scanning such as is used inr a d a r s y s t e m s i s r e q u i r e d . By c o m m u t a ti o n ,t h e amp l i t u d e of t h e i n f ra red s i g n al s f rom fou rq u a d r a n t s m a y b e c o m p a r e d a n d u p - d o w n ,l e f t - r i g h t s i g n a l s g e n e r a t e d f o r m i s s i l e c o n t r o l . An o b v i o u s dr a w b a c k o f t h e i n f r a r e d o r

o p t i c a l g u i d a n c e s y s t e m i s i t s p o o r p e r f o r m a n c e u n d e r a d v e r s e w e a t h e r c o n d i t i o n s ;o b v i ou sl y t h e o p er a t i n g ran g e of su ch as y s t e m w i l l b e r e d u c e d c o n s i d e r a b l y b y h a z ean d ra i n an d wi l l ap p roach zero u n d erco n d i t i on s of h eav y fog an d r a i n . F u r t h er m o r e , s i n c e t h e s e n s i t i v i t y o f t h e s y s t e m sd ep en d s u pon t h e co n t r as t o f t h e t a r g et wi t hi t s b ack g rou n d , t h ose t a rg et s wh i ch d o n otem it l ight o r h eat sufficiently different ini n t en si t y f ro m t h ei r b ack g ro u n d wi l l mak et h e s y s t e m u s e l e s s .

R a d i o f r e q ue n c y p a s s i v e h o m i n g s y s t e m sare b es t ex emp l i f i ed b y t h ei r re l a t i v el ys i m p l e c o u n t e r p a r t in s h i p s a nd a i r c r a f tn av i g at ion , t h e ra d i o d i r ect i o n f i n d er . T h er e q u i r e m e n t s a r e f o r a r e c e i v e r w h ic h c a n b et un e d a u t o m a t i c a l ly t o t h e t a r g e t t r a n s m i t t i n gfrequen cy and an antenna and signa l co m p a r i s o n m e t h o d t o g i v e t h e r e q u i r e d d i r e c t io n a l i n f o r m a t i o n . B o t h p h a s e c o m p a r i s o nan d amp l i t u d e comp ar i son t ech n i q u es can b eu s e d t o d e r i v e t h i s d i r e c t i o n a l i n f o r m a t i o n .

T h e m aj o r p ro b l em i nv ol v ed i n t h e d es i g no f s u c h a s y s t e m i s t h e r e q u i r e m e n t f o rt r e m e n d o u s d y n a m i c r a n g e i n t h e r e c e i v e r ,s i n ce i t mu st op era t e on a s i g n al wh i ch i sre l a t i v el y weak a t lon g ran g e an d v er ys t r o n g a s t he m i s s i l e a p p r o a c h e s c l o s e tot h e t a r g e t .

An act i v e h omi n g sy s t em i s on e i n wh i chb o t h t h e t r a n s m i t t e r o f e n e r g y r e q u i r e d t oi l l u mi n at e t h e t a rg et an d t h e recei v er o f t h eref l ect e d en er g y a r e l oca t ed wi t h in t h e

mi ss i l e i t se l f as sh own i n F i g . 7 .

T h e en er g y u se d t o i l l u mi n at e t h e t a r g etmay b e i n th e fo rm of l i g h t, h eat , rad i o , o rsou n d , t h e mo st co mm o n fo r m b ei n g t h e

r a d a r s e e k e r . W h il e t h e o r e t i c a l l y e i t h e rC W r a d a r s u s i n g v e l o c i t y m e a s u r i n g t e c h n i q ue s o r p u l s e r a d a r s u s in g r a n g e m e a s u r i n gtechn iques can be used equally w ell in thisapplic at io n, i t wil l be found that ce r ta inf u n d a m e n t a l c h a r a c t e r i s t i c s o f t h e C W r a d a r

m ake i t ver y diff icul t to eng inee r into ana c t i v e m i s s i l e s e e k e r . S in c e t he r e c e i v e r an dt h e t r a n s m i t t e r o f t h e CW r a d a r m u s t b e o na t th e s a m e t i m e , th e r e c e i v i n g an d t r a n s mi t t i n g an t en n as mu st b e p h y si cal l y o r e l ect r i c a l l y s e p a r a t e d . In t he p r a c t i c a l c a s e t hea n t e n n a s a r e p h y s i c a l l y s e p a r a t e d b y a g r e a ten ou g h d i s t an ce t o e l i m i n at e th e p ro b l em off e e d - t h r o u g h o f t r a n s m i t t e r e n e r g y d i r e c t l yi n to t h e r ec ei v er . In a sm al l mi ss i l e i t i sp r a c t i c a l l y i m p o s s i b l e t o a c h i e v e t h is p h y s i c a l s e p a r a t i o n a n d e l e c t r i c a l s e p a r a t i o n i s

co mp l ex an d d i ff icu lt t o ach i e v e. F o r t h eser ea so ns i t wil l be found that for ac t iver a d a r s e e k e r s t h e p u l s e s y s t e m i s t he m o s tp r a c t i c a l .

T h e m o s t s e v e r e p r o b l e m w h ic h t h ed e s i g n e r o f a n y a c t i v e r a d a r ho m i ng s y s t e mwill have to face is the l imited space availa b l e i n t h e m i s s i l e . A m o n g o t h e r v a r i a b l e s ,t h e m a x i m u m r a n g e o f a r a d a r s y s t e m i sd ep en d en t u p on t r an s m i t t ed p o wer an d an t en n ac r o s s s e c t i o n . S in c e a tw o fo l d i n c r e a s e i n

ran g e can on l y b e ach i ev ed b y i n creas i n g t h et r a n s m i t t e r p e a k p o w e r s i x t e e n t i m e s , it i su n d e r s t a n d a b l e w hy d e s i g n e r s o f a i r b o r n er a d a r s u s u a l l y d e p e n d u p o n i n c r e a s e i nan t en n a s i z e t o ach i ev e ad d i t io n al ran g ecap ab i l i t y . In t h e ca se of t h e g u id ed m i ss i l e ,p a r t i c u l a r l y in t he a i r - l a u n c h e d m i s s i l e , al i mi t i s soo n re ach e d on t h e s i ze of t h ea n t e nn a t ha t c a n b e u s e d . F o r t h e s e r e a s o n st h e a c t i v e r a d a r g u i d a n c e s y s t e m i s i n h e r e n t l y a s h o r t r a n g e d e v i c e .

T h e m o st ob v i o u s ap p l i cat i on of th e act i v eh omi n g g u i d an ce sy s t em i s ag ai n s t a i rc raf tt a r g e t s a nd i n b o th t h e a i r - t o - a i r a n d s u r f a c e -t o-a i r ap p l i cat i on i t h as an ad v an t ag e ov ero t h e r s y s t e m s w h ic h ca n n o t be i g n o r e d ;

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n am el y , on ce t h e g u i d an ce sy s t e m h asob t ai n ed a l oc k -o n , t h e m i ss i l e sy s t em i scapab le of independently solving the f i r ec o n t r o l p r o b l e m . T h i s m e a n s t h at t h et raf f i c-h an d l i n g cap ab i l i t y of t h e sy s t em i sfa r b et t e r t h an t h e comman d t y p e g u i d an ces y s t e m a n d s o m e w h a t i m p r o v e d o v e r t h eb eam r i d e r . In e i t h er ap p l i cati on t h e t ra f f i c-h an d li n g cap aci ty i s , o f co u r se , n o t u n l i mi t edb e c a u s e a c e r t a i n s e q u e n c e o f e v e n t s m u s tt ak e p l ace p r i o r t o l au nch of t h e m i ss i l e . Int he a i r - t o - a i r c a s e t h e ta r g e t m u s t f i r s t b eacq u i red b y t h e p i l o t op t i cal l y o r b y mean so f t h e a i r b o r n e i n te r c e p t r a d a r . T h e p i lo tm u s t t h e n fly t h e i n t e r c e p t o r to t h e c o r r e c th e a d i n g a n d r a n g e t o a s s u r e m i s s i l e r a d a rl o ck -o n . Upon l au n chi n g t h e m i ss i l e , t h epilot is then f re e to br eak off the at tac k o r

l au n ch ad d i t i on al mi ss i l es .

.Althou gh act i v e se ek er sy s t e m s h av ep o s s i b l e a p p l ic a t io n t o a i r - t o - s u r f a c e a n ds u r f a c e - t o - s u r f a c e s y s t e m s , i t s u s e w i ll b el i m i te d t o t h o s e c a s e s w h e r e t h e r e i s s o m ed i s ti n g u is h i n g c h a r a c t e r i s t i c o f t he t a r g e tw i th r e s p e c t to i t s b a c k g r o u n d . P r i m a r i l y ,i t s u se wi l l b e l i mi t ed to i so l a t ed t a r g e t s , su chas sh i p s a t sea .

A sem i act i v e h om i n g gu i d an ce sy s t e m i s

o n e w h e r e i n t h e r e c e i v e r in t h e m i s s i l er e c e i v e s r e f l e c t e d e n e r g y f r o m t h e t a r g e t ,t h e en erg y h av i n g b een t ran smi t t ed f rom as o u r c e o t h e r t h an t he m i s s i l e , a s s ho w n i nFig. 8 .

i n t e ll i g e n c e i s d e r i v e d f r o m e n e r g y t r a n s m i t t e d f r o m a p o i nt e x t e r n a l to t he m i s s i l e .

In t h e case of t h e act i v e rad ar h omi n gs y s t e m , i t w a s s t a t e d th a t b e c a u s e o f r e s t r i c t i o n s i m p o s e d b y t h e s i z e o f t he m i s s i l e ,su ch a g u id an ce d ev i ce was re l a t i v el y s h o r tr a n g e . It i s f o r t h i s r e a s o n , a m o n g o t h e r s ,t ha t s e m i a c t i v e r a d a r g u id a n c e s y s t e m s b e c o m e o p e r a t i o n a l l y a t t r a c t i v e . T a k i n g f o re x a m p l e a g r o u n d - t o - a i r m i s s i l e s y s t e m , i tsh ou l d b e ap p ar en t t h at s i n ce t h e t ra n sm i t t i n gp o r t i on of t h e sy s t em wi l l b e on t h e g ro u n d ,t h e t r a n s m i t t e r p o w e r a n d a n te n n a c r o s ss e c t i o n c a n b e i n c r e a s e d c o n s i d e r a b l y o v e rthat p o ssib le in a m is si le . If, in fact , thep o w e r o u t p u t c a n t h e r e b y b e i n c r e a s e d f o u r fo ld an d t h e t ra n sm i t t i n g an ten n a a r ea t en fo l d ,

t hen t h e t h eo re t i cal max i mu m ran g e wi l l b ei n c r e a s e d 2 . 5 t i m e s o v e r t h e a c t i v e s e e k e rca se as sh own b y t h e s i mp l i f i ed c al cu l a t i o n .

P S G T

, P o G R

= 1.4 x | .7.

(1 )

An ot h er ad v an t ag e of semi act i v e h omi n gf rom t h e ov era l l sy s t em s t an d p oi n t sh ou l d n ot

g o u n n ot i ced an d t h at i s i t s u se i n co m b i na t io n w i th o t h e r m i d c o u r s e g u i d a nc e m e a n sf o r t e r m i n a l g u id a n c e t o a c h i e v e c o m p a r a t i v el y l o n g r an g es wit h a l l t h e accu ra cyi n h e r e n t in h o m i n g s y s t e m s .

T h e t ra n s m i t t e r may b e lo cat ed a t t h el au n ch i n g s t a t i on o r a t a p o i n t sep ar a t ed f romt h e m i s s i l e l a u n ch in g s i t e . T h e t r a n s m i t t e rmay b e l ocat ed o n t h e su rfac e of t h e ea r t h , ass ho w n in th e i l l u s t r a t i o n o r m a y b e c a r r i e d i nan a i r c r af t . As i n t h e ca se of ac t i v e h om i n g

g u i d an ce, t h e t ran s m i t t ed en erg y may b e i nt h e fo r m o f l ig h t , h e a t, r a d i o , o r s o u n d w a v e s .T h e mai n d i f fe ren ce b etween t h e t wo sy s t e m si s t h at t h e semi act i v e sy s t em i s n o t i n d ep en d en t of ou t s i d e sou rces , s i n ce i t s g u i d an ce

S i n ce i n man y sy st ems t h e t a rg et mu st b et r a c k e d b y a f i r e c o n t r o l t y p e r a d a r i n o r d e rto launch a missi le of any kind, then thisi l l u mi n at i on can b e u sed b y t h e mi ss i l e t oh o m e s e m i a c t i v e l y o n t h e t a r g e t . H o w e v e r ,t h e mi ss i l e d o es n ot n eed t o h o me al l t h e way

from launching point to target , but may bec o n t r o l l e d t h r o u g h a m i d c o u r s e p h a s e b yc o m m a n d g u i d a n c e o r b e a m - r i d e r g u i d a n c eu n t i l it r ea ch es a p o i n t c l o se en ou g h t o t h et a r g e t t o s t a r t h o m i n g o n i t .

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As p oi n t ed ou t ear l i e r i n t h i s p ap er , t h ea c c u r a c y o f a c o m m a n d s y s t e m o r a b e a m -r i d e r s y s t e m d e t e r i o r a t e s w ith r a n g e , b ut ah o m i n g - a l l - t h e - w a y s y s t e m , w h e t h e r a c t i v eo r s e m i a c t i v e i s e s s e n t i a l l y s h o r t r a n g e a sco mp are d wi t h t h em. T h e ad v an t ag e of t h i sco mb i n at i on g u i d an ce sc h em e i s t h at i t p r o v i d es re l a t i v el y l on g ran g e wh i le s t i l l r e t a i n in g t h e a ccu ra cy i n h eren t i n a h om i n gsy s t em . Of co u r se , as a l way s, we mu st p ay ap r i ce fo r t h i s g ai n an d i n t h i s case t h e p r i cewe mu st p ay i s t h at t h e i l l u mi n at i n g rad armu s t r em ai n t ra i n ed o n t h e t a r g et un t ili n t e r cep t h as b een ach i ev ed , a l t h ou g h an yn u m b er of mi ss i l e s can b e l au n ched ag ai n s ta s i n g l e t a r g et , on l y on e t a r g et can b e h an d ledat a t i me.

W h en semi act i v e h omi n g i s u sed fo r ana i r - t o - a i r t yp e m i s s i l e , t h e m i s s i l e i sl au n c he d by a f i g ht e r o r i n t e r c e p t o r a i r c r a f tan d t h e t a rg e t i s il l u mi n at ed by t h e i n t e r c e p t o r t yp e r a d a r c a r r i e d b y t h e a t t a ck i n ga i r c r a f t . T h e m e t h o d o f a c q u i r i n g t h e t a r g e tand launching the missi le wil l be identicalw i t h t h o s e w h e r e a n a c t i v e m i s s i l e h o m i n gsy s t em i s u sed . Ho wev er , wi th t h e act i v esy s t em t h e a i r c r af t i s f ree t o b re ak away t oa t t a ck a n o t h e r t a r g e t a s s o o n a s t h e m i s s i l ei s l au n ch ed . T h i s i s n o t t r u e fo r t h e se m i -

act i v e ca se . T h e i l l u mi n at i n g r ad ar , as i n t h es u r f a c e - t o - a i r c a s e , m u s t b e p o i n te d a t t h et a rg et con t i n u ou sl y u n t i l mi ss i l e i n t e rcep t i sach i ev e d . T h i s i s g en e ra l l y accep t ed as al i ab i l it y o f t h e s e m i a c t i v e s y s t e m ; h o w e v e r ,a s t u d y of t h e p aramet ers i n v ol v ed wi l l sh owt h i s t o be a m i n o r c o n s i d e r a t i o n . D u e t o th esh o r t m i ss i l e f li gh t t i m e i n v ol v ed , t h e l i mi t edm a n e u v e r a b i l i t y o f h i g h - s p e e d a i r c r a f t , a ndt h e a b i l i t y o f m o d e r n t r a c k i n g r a d a r s t oma i n t a i n t ra ck i n g l ock o v er wi d e an g l e s an dan g l e ra t es , i t i s p oss i b l e wi t h semi act i v es y s t e m s t o e x e c u t e t h e b r e a k a w a y m a n e u v e rwi th p ra ct i ca l l y t h e sa m e sp eed as wi t h ana c t i v e s y s t e m .

An ot h er ad v an t ag e of t h e semi act i v esy s t em n ot ap p are n t a t f i rs t g l an ce i s t h at we

now h av e a sy s t em wh i ch au t o ma t i cal l y o v e r comes t h e b as i c d i f f i cu l t y p rev i ou sl y en co u n t er ed wh ich i s i n h ere n t to a CW t y pes y s t e m , t h e a n t en n a f e e d -t h r o u g h p r o b l e m .As I sa i d b efo re , s i n ce t h e rec ei v er an dt r an sm i t t e r o f a CW sy st e m mu st b e on a t t h es a m e t i m e , t h e r e c e i v i n g a nd t r a n s m i t t i n gan t en n as mu st b e sep ara t ed p h y si cal l y o re l e ct r i ca l l y . In t h e ca se of th e act i v e h om i n gsy ste m , we found th is difficult to ach ieve , buti n t h e se m i act i v e sy s t em we h av e a b u i l t - i np h y si cal se p ar a t i o n of t h e an t en n as . T h i s ,t h e r e f o r e , m a k e s i t f e a s i b le t o u s e CW r a d a rt ech n i q u es fo r a semi a ct i v e h om i n g sy s t emw her e i t is exceedingly diff icul t to do so fora n a c t i v e h o m i n g s y s t e m .

6. COMBINATION GUIDANCE SYSTEM

I h av e j u s t men t i on ed t h e p oss i b l e co m b i n at i on of a semi act i v e h omi n g sy s t em wi t ha b eam r i d er an d h av e sh own t h e ad v an t ag esof su ch a sy s t e m . In t h e p r act i cal ca se i twill be found m o r e o ften than not that am i s s i l e s y s t e m w i ll r e q u i r e a co m b i na t io n o fg u i d an ce t ech n i q u es i n o r d e r t o b es t meet th et a c t i c a l r e q u i r e m e n t .

T h e m i s s i l e g u i d a n c e p r o b l e m m a y b e

co n si d e re d in t h r ee p a r t s : l au nch i ng g u i d a n c e , m i d c o u r s e g u i d a n ce , a nd t e r m i n a l g u i d a n c e . It i s in t h e s u r f a c e - t o - s u r f a c e t yp em is si le that we will be m o st apt to f ind i tn e c e s s a r y t o u s e a c o m b i n a ti o n o f a l l p a r t sb eca u se of th e l on g r an g es i n v ol v ed . W h en,f o r e x a m p l e , th e m i d c o u r s e p h a s e is ah y p e r b o l i c o r c i r c u l a r r a d i o n a v i g a t i o n s y s t e m , s u c h a s d e s c r i b e d e a r l i e r in t h e t e x t,t h e mi ss i l e may wel l b e l au n ch ed a t somed i s t an ce fro m t h e g u i d an ce sy s t e m . It wi l lb e n e c e s s a r y u n d e r t h e s e c i r c u m s t a n c e s t o

p r o v i d e s o m e f o r m o f s i m p l e i n e r t i a l g u i d a nc eo r c o m m a n d g ui d a n c e t o p l a c e t h e m i s s i l ewi t h in t h e m i d co u r se g u i d an ce p at t e r n . If t h ewarh ead l e t h al i t y p at t e rn i s su ch as t o req u i ree x t r e m e a c c u r a c y i n t h e t e r m i n a l p h a s e o f

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such a m iss i le f l ight , i t wil l pro bably ben e c e s s a r y t o a dd s o m e fo r m o f t e r m i n a lg u i d an ce t o su ch a sy s t e m . T h i s t e r m i n alg u i d an ce co u ld t ak e t h e fo r m of a p ass i v eh o m e r o r in t he c a s e of d i s c r e t e r a d a r

t a r g e t s , i t m i g h t b e a n a c t i v e r a d a r h o m e r .

In th e c a s e o f s u r f a c e - t o - a i r m i s s i l e s , o n em i g h t u s e a h o m i n g - a l l - t h e - w a y s y s t e m , ab e a m - r i d e r s y s t e m , o r a c o m m a n d s y s t e m ift h e r a n g e s r e q u i r e d w e r e r e l a t i v e l y s h o r t ;h o w e v e r , lo n g r a n g e , s u r f a c e - t o - a i r m i s s i l e sw i l l, in g e n e r a l , r e q u i r e c o m b i n e d m i d c o u r s ea nd t e r m i n a l s y s t e m s . A s p o i n te d o u te a r l i e r , t he c o m m a n d a nd b e a m - r i d e r s y s t e mw i l l l a c k t h e n e c e s s a r y a c c u r a c y a t l o n g

range and wil l have to be augmented by somefor m of t e r m i n al h om i n g wh i ch may b e act i v e .

sem i ac t i v e , o r p ass i v e to f it t h e tact i ca ls i t u a ti o n . T h e h o m i n g - a l l - t h e - w a y s y s t e m i si n h eren t l y sh o r t r an g e an d wi l l h av e t o b ec o m b in e d w i th a m i d c o u r s e s y s t e m t o a c h i e v et he n e c e s s a r y r a n g e .

In t he a i r - t o - s u r f a c e t y p e m i s s i l e , w e h a v ea s i m i l a r s i t u a t io n . S h o r t r a n g e m i s s i l e sc a n p r o b a b l y a c h i e v e t h e n e c e s s a r y r a n g ea nd a c c u r a c y w it h a c o m m a n d , b e a m - r i d e r , o rh om i n g sy s t em , b ut wh en a l on g ran g e m i ss i l ei s ca l l ed fo r we mu st g o t o a comb i n eds y s t e m t o a c h i e v e t h e r a n g e a n d a c c u r a c yr e q u i r e d . In t he a i r - t o - a i r a p p l ic a t io n , t h em i s s i l e s w i l l g e n e r a l l y b e i n t h e s h o r t r a n g ecl a ss an d a s i n g l e fo r m of g u i d an ce may b eu s e d . A s s t a te d e a r l i e r , b e a m - r i d e r , a c t i v e ,

semi act i v e , an d p ass i v e h omi n g may b e u sedi n t h i s ap p l i cat i on .

R E F E R E N C E S

1 . L o ck e, Art h u r S . , " G u i d an ce , " D . Van No st ran d and Co mp an y . In c . , New Yo rk , N. Y. , 1 95 5 .

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CONSIDERATIONS IN THE CHOICE OFA MISSILE GUIDANCE AND CONTROL SYSTEM

Rob er t W i l l i am M ay er*

SUMMARY

T h i s p ap er p r ese n t s so me of t h e maj o r fact o rs wh ich sh ou ld b e co n si d er ed in mak i n ga p r o p e r c h o i c e o f a m i s s i l e g u i da n c e a n d c o n t r o l s y s t e m . T h e f i r s t p a r t d i s c u s s e s t h ei m p o r t a n c e o f t h o r o u g h l y e s t a b l i s h i n g r e q u i r e m e n t s b e f o r e m a k in g a c h o i c e . T h e s e c o n dp a r t e n u m e r a t e s t he v a r i o u s c r i t e r i a f o r ju d gi ng a s y s t e m a g a i n s t s u c h r e q u i r e m e n t s . Insu m m ar y , t h e p ap er p o i n t s ou t t h at t h e re ex i s t s a g r eat n eed fo r d ev elo p i n g fu r t h er t ech n i q u est o p rov i d e b et t e r mean s of wei g h t i n g t h e c r i t e r i a i n mak i n g a sy s t em ch oi ce .

SOMMAIRE

Cet t e n o t e p re 'sen t e q u el q u es u n s d es p r i n ci p au x fact eu rs q u i d ev rai en t e t re con si d ered

d a n s u n c h o ix c o r r e c t du s y s t e m e d e c o n t r o l e e t d e l a g o u v e r n e d 'u n m i s s i l e . L a p r e m i e r ep a r t i e t r a i t e d e 1 ' im p o r t a nc e d ' e t a b l i r c o m p l e t e m e n t le s p e r f o r m a n c e s d e m a n d ^ e s a v an t d ef a i r e u n c h o i x . L a d e u x i e m e p a r t i e e n u m e r e l e s d i f fe r e n t s c r i t e r i u m s p e r m e t t a n t d e j u g e run s y s t e m e s u r d e t e l l e s p e r f o r m a n c e s . E n r e s u m e , c e t t e n o t e m o n t r e q u ' i l e x i s t e d a n sce d o mai n e, u n g ran d b esoi n d e d ev e l op p er d es t ech n i q u es p l u s p o u sse es en v ue d e fou r n i rd e s m e i l l e u r s p r e c e d e s , e n f a is a n t le c h o i x d u n s y s t e m e , d ' e v a l u e r l e s c r i t e r i u m s .

1. INTRODUCTION

T h e ob j ect i v e of t h i s p ap er i s t o p resen tt h e maj o r fact o rs wh i ch sh ou ld b e co n si d er ed

in making a choice of a missi le guidance andc o n t r o l s y s t e m . I c h o s e t h is o b j e c t iv e b e cau se (1 ) i n man y ca se s , t h e p ro b l em ofm a k i n g t h e p r o p e r c h o i c e o f s y s t e m f o r agiven application is equally as difficult as thep rob l ems of d et a i l ed sy n t h esi s an d an al y s i s o fa s ys tem , and (2) t ime spent in ma king thep rop er d eci s i on s on ch oi ce of t h e sy s t em, i nthe beginning of a development, can resul t insavings in t ime and money in later phaseso f t h e p r o g r a m .

To me et this ob ject ive , I should l ike tos t r e s s t h e i m p o r t a n c e o f t h o r o u g h l y e s t a b l i sh in g t he re q u i re m en t s of t h e sy s t em b efore

mak i n g a ch o i ce . I sh al l t hen d i s cu ss so meo f t h e m o r e i m p o r t a n t c r i t e r i a w h i c h s h o u l db e con si d ered i n mak i n g su ch a sy s t em ch oi ce ,a nd i l l u s t r a t e s o m e o f t h e s e w i th e x a m p l e s .

2. ESTABLISHING THE REQUIREMENTSFOR THE GUIDANCE AND CONTROLSYSTEM

It i s e x t r e m e l y i m p o r t a n t t h at t he r e q u i r e men t s of a sy s t em b e c l ear l y es t ab l i sh ed an du n d ers t o o d b efo re a t t em p t i n g t o mak e ach o i ce . T h e b es t d es i g n ed sy s t e m wil l b e ap oor on e i f i t d oes n ot sa t i s fy t h e req u i reme n t s of t h e ap p l i cat i o n . T o i n su re th at t h ec o r r e c t s y s t e m i s ch o s e n f o r t h e j o b r e q u i r e sc o n s i d e r a t i o n o f f a c t o r s d e s c r i b e d in t h ef o l l o w i n g p a r a g r a p h s .

" Gen eral E l ect r i c Comp an y , P h i l ad el p h i a , P en n sy l v an i a .

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a. T h e M i ss i on of t h e Co mp l et e W eaponSystem

In d es i g n i n g an y sy s t e m o r su b s y st emwh i ch i s a p ar t o f a comp l et e weap on sy s t em

it is impossible to specify in detai l al l ther e q u i r e m e n t s o f th e s u b s y s t e m ; t h e r e f o r e th eg u i da n c e a n d c o n t r o l s y s t e m d e s i g n e r s m u s thave sufficient know ledge of the o ver al lsy s t e m . T h i s k n owl edg e a l l ow s t h em t oex er ci s e j u d g men t in mak i n g a ch o i ce wh eres p e c if i c r e q u i r e m e n t s a r e no t , o r c a n n o t . b edefined. Fo r th is re aso n they need to knowt h e ov era l l ob j ect i v es of t h e weap on sy s t em,su ch as ;

(1) The natu re of the tar get and

t a c t i c a l o r s t r a t e g i c c o n d i t i o n sunder which it will be engaged.

(2) The type of payload to be ca r r ie da nd i ts c h a r a c t e r i s t i c s . ( I n s o m ec a s e s , payload choice should beco n cu r r en t wi th g u i d an ce ch o i ce . )

(3) The vehicle f ro m which the m is sile will be launched and itsmot i on s , i f an y .

(4 ) G e n e r a l e n v i r o n m e n t a l c o n d i t i o n s , i . e . , wh ere d o es t h eact i on t ak e p l ace?

b. M i ssi o n of t h e Gu i d an ce an d Co n t ro lS y st em

If the des ign er has the knowledge of them i ss i o n of t h e o v er a l l s y s t em , h e can t henc o n s i d e r t h e m o r e s p e c i f i c m i s s i o n o f t h eg u i d an ce and co n t ro l sy s t em . T h i s i s d e fined by such specific factors as:

(1) Ran ge. How far mus t the m iss i lebe guided?

(2) Altitu de. At what height is thew a r h e a d a c t i v a t e d ?

(3) Accu ra cy. How clo se must i tc o m e t o t h e t a r g e t ?

(4) Re liabi l i ty . What per cen t fai l ures can be al lowed due to the

g u i d an ce an d con t ro l sy s t em?

T h e se fou r fact o rs d ef in e th e b as i c ob ject ive of the system, i .e. , what i t mustacc o m p l i sh . T h ey d o n ot , h o wev er , show howt h e s y s t e m a c c o m p l i s h e s t h e s e o b j e c t i v e swi t hi n t h e f ramew o rk of t h e o v er a l l sy s t em .The definit ion of " ho w " is co ntained in as e r i e s o f d e t a i le d s p e c if ic r e q u i r e m e n t s .

c . S p ecif ic Re q u i r em en t s

Def in it ion of t h ese sp eci f i c r eq u i r em en t sr eq u i r es a k nowl ed g e of th e o v er a l l sy s t emscheme and the interfaces which exist betweent h e g u i d an ce an d con t ro l sy s t em an d o t h ers u b s y s t e m s .

S uc h r e q u i r e m e n t s c a n , of c o u r s e , b ec o m equite detai led but , in general , should coverareas su ch as t h e fo l l owi n g :

(1) T h e t ra j ect o ry sch em e; d efi n in gwh ere i mp or t an t fu n ct i on s t ak ep l ace , su ch as b u rn o u t , a r mi n gan d fu zi n g , man eu v ers , e t c .

(2) P o wer su p p l i es av ai l ab l e andt h e i r c h a r a c t e r i s t i c s .

(3) P r o p u l si o n c h a r a c t e r i s t i c s .

(4) Space and weight al lo cat io ns, both

in a i r f r am e and g rou n d eq u i p m e n t .

(5 ) E n v i ro n me n t al co n d i t io n s b efo reand during flight.

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d. A re as of Flex ibi l i ty

Not a l l r eq u i re m en t s of t h e g u i d an ce andco ntro l sys tem should be fixed ear ly in thed e s ig n s t a g e . I n s te a d , s o m e o f t he r e q u i r e

m en t s of t he g u i d an ce and co n t ro l sy s t emsh ou l d remai n f l ex i b l e ov er a g i v en d es i g nra n g e . T h i s wi ll p ro v i d e m o re la t i t u d e i nf ix in g d e s ig n p a r a m e t e r s in c r i t i c a l p o r t i o n sof o t h er su b sy st ems an d a t t h e same t i mep r o v i d e t h e g r e a t e s t p o s s i b il i ty of a m o r eo p t i m u m o v e r a l l m i s s i l e w e ap o n s y s t e m .In o r d e r t o r e a l i z e m a x i m u m g a i ns fr o m t h i sflexibil i ty i t is ne ce ssa r y that the guidancea n d c o n t r o l s y s t e m d e s i g n e r s u n d e r s t a n d t h el i m i t a t i o n s a n d p r o b l e m s o f t h e o t h e r s u b s y s t e m s . T h e y m u s t u n d e r s t a n d , t o o , t h ei n t er re l a t i on sh i p s wh i ch ex i s t b et ween t h esesu b sy st ems an d t h e g u i d an ce an d con t ro lsy s t e m . T h i s th or o u g h u n d ers t an d i n g can a i dt h e comp l et e i n t eg ra t i on of t h e g u i d an ce an dc o n t r o l s y s t e m w i th t h e o v e r a l l s y s t e m .

S ome of th e l i mi t a t i o n s t o b e co n si d er edi n mai n t a i n i n g f l ex i b i l i t y of req u i remen t sa r e :

(1 ) L i mi t a t i on s on t h e p rop u l s i onsy st e m . Very d ef in i t e l i mi t a t i o n s ex i s t in t h erepeatability of cutoff and difficulty in maintaining thrust and total impulse within closet o l e r a n c e s . F o r e x a m p l e , t he g u i da n c e a ndcon t ro l d es i g n er sh ou l d con si d er i f i t i sw o r t h w h i le , f r o m a n o v e r a l l s y s t e m s t a n d p oi n t , t o h av e a co mp l ex t h ru s t co n t ro l ino rd er t o s i mp l ify t h e g u i d ance sch em e.

P e r h a p s e l i m i n a t i o n o f t h r u s t c o n t r o lw o u ld no t c a u s e a p r o p o r t i o n a t e i n c r e a s e ing u i d an ce sy s t em comp l ex i t y .

(2) Lim itat io ns of pow er supply.P o w e r s u p p l y r e q u i r e m e n t s s h o u l d r e m a i nflexible until i t has been established definitelywhether or not i t is best to build complexr e g u l a t i n g s u p p l i e s o r p r o v i d e i n c r e a s e dcom p l ex it y in t h e g u i d an ce c i r cu i t ry t o accep tw i d e r p o w e r s u p p l y v a r i a t i o n s .

(3) L i m i t a t i o n s in a i r f r a m e . A c e r -t a i n amou n t of f l ex i b i l i t y i n req u i remen t ssho uld exis t to al low for the best balanc ebetween guidance and control complexity anda i r f r a m e c o m p l e x it y , in s u ch a r e a s a s ;

( a) B a c k l a s h t o l e r a n c e s o n c o n t r o l s u r f a c e s .

(b) T o l e r a n c e s o n m i s a l i g n m e n to f a i r f r a m e s u r f a c e s a nds e c t i o n s .

(c ) T o l e r a n c e s o n t h r u s t m i s a l i g n m e n t .

(d) T o l e r a n c e s i n d e t e r m i n a t i o no f a e r o d y n a m i c c o e f f i c i e n t s .

T h e r eq u i r em en t s fo r t h e g u i d ance an dco n t ro l sy s t e m d efi n e t h e d es i g n g o al s . It i si m p o r t a n t , e s p e c i a l l y d u r i n g t h e p r e l i m i n a r yd esi g n s t a g e, t h at t h ese b e co n si d er ed asg o a l s a nd n o t a s r i g i d r e q u i r e m e n t s s i n c ed e v i a ti o n s fr o m t h e s e r e q u i r e m e n t s m a yr e s u l t in a m o r e o p ti m u m o v e r a l l s y s t e m .As su ch , t h es e d es i g n g oal s b eco me a p ar tof t h e c r i t e r i a by wh ich a ch o i ce of sy s t emsh ou l d b e mad e.

3 . CRITER IA FOR CHOOSING A SYSTEM

In a d d it io n t o t he a b o v e r e q u i r e m e n t s t h e r ea r e o t h e r f a c t o r s o f e qu a l i m p o r t a n c e t o b econ si d ered i n t h e ch oi ce of a g u i d an ce an dc o n t r o l s y s t e m . 1 should l ike briefly to talkab ou t t h em and t h e p rev i o u sl y d i sc u sse dr e q u i r e m e n t s a s c r i t e r i a , a nd i nd i ca te s o m e

of th e co n si d e ra t i o n s wh i ch shou l d t ak e p l acet o i n s u r e a c o n s t a n t b a l a n c e o f o n e c r i t e r i o nag ai n s t t h e o t h er wh en mak i n g a sy s t emch o i ce . F i g . 1 i s h elp fu l i n i l l u s t r a t i n g t h es ef a c t o r s .

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a . A c c ura c y

When • c o n s i d e r i n g t he r e q u i r e m e n t o fa c c u r a c y a s a c r i t e r i o n , i t i s i m p o r t a n t n o t10 o ve r e m ph a s i z e i t at t he e xpe n s e o f o t he r

c r i t e r i a . It i s s o m e t i m e s e a s y f o r t h is t oha ppe n s i nc e a c c ur a c y c a n be e xpl i c i t l yde f i ne d a s a num be r , s uc h a s c i rc ul a rp r o b a b l e e r r o r , w h i l e , o n t he o t h e r ha n d , i ti s m o r e d i ff ic ult t o a t t a c h a num be r t o s o m eo f t he o t he r l e s s t a ngi bl e c r i t e r i a , w hi ch w i l lb e d i s c u s s e d s h o r t l y .

So m e f l e xib i l it y s ho ul d be pe r m i t t e d in t hede f in i ti o n o f a c c u ra c y . R a t he r t ha n e xpl i c i t l yd e fi n e a c c u r a c y a s t h e p r o b a b l e e r r o r i n, l e tus s a y , t he X a nd Y c o o rd i na t e s , it m a y be

a d v a n t a g eo u s t o d e fi n e i t a s c i r c u l a r p r o b a b l ee r r o r , t h u s g i v in g m o r e f l e x ib i li ty t o t h ed i s t r i b u ti o n o f e r r o r b e t w ee n c o o r d i n a t e s .

b . R e l i a bi l i t y

T h e r e l i a b i l i t y r e q u i r e m e n t c a n b e e a s i l ye x p r e s s e d a s a n u m b e r f o r u s e a s a c r i t e r i o n .H o w e ve r , go o d m e t ho ds a re s t i l l no t a va i l a bl ef o r r e a d i l y p r e d i c t i n g w h e t h e r o r n o t a g i v e nre l i a bi l i t y w i l l , in f a c t, be a c hi e ve d. T h e r e f o re , t o o b t a i n a n e xpl i c i t m e a s u r e of ho w w e l lt he re l i a bi l i t y c r i t e r i o n m a y be m e t byd i f f e r e n t c h o i c e s o f s y s t e m s i s p r e s e n t l ya d i ff ic ul t jo b . Lo gi c a l l y t he n , t he c o ns i de r a t i o n o f re l i a bi l i t y in m a ki ng a c ho i c e be tw e e ns y s t e m s o f te n m u s t b e c a r r i e d o u t o n ar e l a t i v e b a s i s . T h r e e m a j o r t o o l s a r e a v a i l a b l e f o r p r e d i c t i n g r e l i a b i l i t y : m a t h e m a t i c s ,e xp e r i e nc e , and i nfo rm a t i o n o n t he qua nt i t ya nd t ype s o f e qui pm e nt s .

( 1) M a t he m a t i c s . A f i gure f o r t he

re l i a bi l i t y o f a g i ve n s ys t e m m a y be de r i ve dby pre di c t i ng t he re l i a bi l i t y o f t he c o m po ne nt sw i th i n t he s y s t e m , b a s e d o n e i t h e r e x p e r i e n c eo r t e s t , a nd t he n c o m bi ni ng t he re s ul t i ngv a l u e s i n t he p r o p e r f as h io n t o g i v e a n o v e r a l l

re l i a bi l i t y f i gur e . Th i s f i gure m a y t he n bec o m p a r e d w i th o n e o b t a i n ed f o r a s i m i l a rs y s t e m , t he c o m po ne nt re l i a bi l i t i e s o f w hi chw e r e a r r i v e d a t in t h e s a m e m a n n e r . R e l i a bi l i t y f i gure s o bt a i ne d i n t h i s m a nne r a re

r e l a t i v e , r a t h e r t h a n e x p l i c i t , b e c a u s e o fa s s u m p t i o n s u s e d f o r p r e d i c t i n g r e l i a b i l i t y o fc o m p o ne n t s . H o w e ve r , t hey a r e suf f ic i e ntl yg o o d f o r c o m p a r i s o n s if li k e a s s u m p t i o n s a r eu s e d f o r c o m p o n e n t r e l i a b i l i t i e s .

(2 ) E x p e r i e n c e . E x p e r i e n c e p la y s al a r g e p a r t in a s s e s s m e n t of r e l a t i v e r e l i a bi l i t y . If a s ys t e m i s m a de up o f c o m po ne nt sw hi ch ha ve a l r e a d y s ho w n a h igh de gr e e o fr e l i a b i l i t y w h i le o p e r a t i n g in o t h e r s y s t e m s ,

i t can be expected that such a system wil l bet h e m o r e r e l i a b l e t h a n o n e c o m p r i s e d o fu n t r i e d c o m p o n e n t s . Of c o u r s e , t h i s i s t r u eo nl y i f t he c o m po ne nt s a re o pe ra t e d i n t hes a m e e n v i r o n m e n t . W h en a s s e s s i n g r e l a t i v ere l i a bi l i t y i n t h i s w a y, i t i s s o m e t i m e s he l pf u l t o l i s t t he c o m po ne nt s , i ndi c a t i ng t he i rp r e v i o u s u s e s an d e n v i r o n m e n t s . By d o i n gt h i s , t h e s y s t e m w i t h t h e m o s t " p r o v e n "c o m p o n e n t s c a n b e d e t e r m i n e d .

(3) Info rm at ion o n quant i ty and type so f e q u i p m e n t s . I n fo r m a t io n c o n c e r n i n gt he re l a t i ve c o m pl e xi t y o f a s ys t e m c a n beo b t a i ne d by c o n s i de r i ng t he num be r s o f l i kec o m p o n e n t p a r t s i n v o l v e d , s u c h a s e l e c t r o n i ct u b e s , r e s i s t o r s , c a p a c i t o r s , s o l e n o i d v a l v e sa nd r e l a ys . When do i ng s o , ho w e ve r , t ho ughtm ust be given to the types of equ ipm entsinvolved and to the type of fai lur e that maybe i nc ur re d . In a ddi t i o n , c o ns i de r a t i o n s ho ul dbe g i ve n t o t he p ro ba bi l i t y o f p re ve nt i ng t h i sf a i l u r e by a s y s t e m a t i c m a i n t en a n c e i n s p e c

t io n r o u t i n e . F a i l u r e in m e c h a n i c a l c o m ponents of ten can be predicted by visuali n s p e c ti o n a nd g r a d u a l d e g r a d a t i o n o f p e r f o r m a n c e , w h e r e a s so m e e l e c t r i c c o m p o n e n t sm a y f a i l c o m pl e t e l y w i t ho ut w a rni ng.

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c. Accu racy v s . Rel i ab i l it y

A h ig h a c c u r a c y s y s t e m w i th p o o r r e l iabi l i ty wil l r es ul t in few hi t s. In the sam eway , a low acc u r acy sy s t em wi th hi gh r e

l i ab i li t y a l so res u l t s in few h i t s . S i nceg r e a t e r a c c u r a c y g e n e r a l l y c a l l s f o r g r e a t e rco mp l ex i ty an d l e ss re l i ab i l i t y , it i s i mp o rt an t th at b o t h acc u r acy an d re l i ab i l i t y fact o rsb e b al an ced , as mu ch as p oss i b l e , t o p rov i d ea sy s t em wh i ch wi l l b es t i n su re t h e d el i v eryof the payload to the target .

T h i s b a l a n c e , o r c o m p r o m i s e , c o u l d b emad e if we wer e ab l e t o d raw cu r v es fo raccu ra cy and re l i a b i l i t y as a fu ncti on ofcomplexity and then combine the two types

of cu r v es t o g i v e a com p o si t e a s in F i g . 1.Unfo rtunately, i t is ex tr em ely difficult to dot h i s , e s p e c i a ll y f o r th e r e l i a b i l i ty c r i t e r i o n ,s i n c e m o s t o f t h e r e l i a b i l i t y e s t i m a t e s a r eo n l y r e l a t i v e , a s w a s p r e v i o u s l y d i s c u s s e d .Keep i n g su ch a comp osi t e cu rv e i n mi n dh ow ev er , i s a h el p wh en mak i n g a sy s t emc o m p a r i s o n .

d . O p e r a t io n a l C h a r a c t e r i s t i c s

The abil i ty of the system to meet the

o p e r a t i o n a l c h a r a c t e r i s t i c s i s e q u al ly a si m p o r t a n t a s i t s a c h i e v e m e n t o f o t h e r p e r f o r m a n c e c h a r a c t e r i s t i c s . O p e r a t io n a l c h a r act e r i s t i cs may b e sp eci f i ed fo r g u i d an cec o n t r o l d e s i g n e r s a s r e q u i r e m e n t s o r , i no t h er ca ses , t h ey may b e p res en t ed on ly ina b ro ad s en s e . W h en t h e l a t t e r i s t h e ca se ,t h e d es i g n e rs mu st i n su re t h at th e eq u i p men twi l l meet t h ei r b es t es t i m at e s of t h e re q u i r edo p e r a t i o n a l c h a r a c t e r i s t i c s . S pe ci fic a r e a swhich sho uld be co nsid er ed in ma king ach o i ce of sy s t e m ar e d i sc u sse d in t h e fo l l ow

i n g p a r a g r a p h s .

(1) E as e of handling. T he si ze s andwei g h t s of mi ss i l e an d g rou n d eq u i p men tp i eces mu st b e su ff ic i en tl y sm al l t o i n su re

easy e mp l acem en t an d h an d li n g i n t h e p r e p a r a t i o n f o r l a u nc h i ng . M o r e o v e r , t h eequipment must be designed for use in al l oft h e e n v i r o n m e n t s t h a t a r e t o b e e n c o u n t e r e d .

T h e an sw er t o t h e q u est i on of wh et h er t op u t more eq u i p men t i n t h e mi ss i l e o r on t h eg r o u n d in t he c a s e o f a s u r f a c e - t o - s u r f a c eweapon is of ten influenced st rongly by ad e s i r e f o r e a s y h a n d li n g. F o r e x a m p l e , at act i ca l s i t u at i on re q u i r i n g t h e u t mo st ea se ofh an d l i n g wou l d cau se on e t o ch oose a sy s t emhaving m o r e of the guidance equipm ent in them i ss i l e , as i n an i n er t i a l s y s t e m , t h an on t h eg r o u n d a s i n a r a d i o s y s t e m .

(2) Al i gn men t p ro ced u r e . T h e ea se

of making ini t ial s et t ing s and al ign m ents isa n o t h e r i m p o r t a n t c o n s i d e r a t i o n s i n c e i t h a san effect on the type and quan tity of equ ipm enta nd m a n p o w e r r e q u i r e d to c a r r y o u t t h ep r o ce ss . In ad d i t io n , t h e r a t e of f i re i sinfluenced by the ra pidi ty with which suchad j u st men t s can b e mad e.

T o i n s u r e t ha t t h e b e s t o v e r a l l s y s t e m i ssel ec t ed , i t i s i mp o r t an t t o b al an ce co mp l ex i t yo f o n - b o a r d g u id a n c e e q ui p m e n t a g a i n s t c o m plexity of gro und al ignm ent equipm ent. Th isi s n e c e s s a r y b e c a u s e a n y a l i g n m e n t p r o

c e d u r e s r e q u i r i n g c o m p l e x g r o u n d e q u i p m e n t s m a y r e s u l t in l o w e r o v e r a l l s y s t e mr e l i a b i l i t y .

(3) M ainten anc e. M aintenance is anarea wh i ch was n eg l ect ed d u r i n g some of t h ee a r l y p h a s e s o f m i s s i l e d e v e l o p m e n t . Ach o i ce b et ween sy s t e m s sh o u l d l ean h eav i l yt oward t h e on e t h at d oes n ot req u i re t h es e r v i c e s o f s y s t e m s d e s i g n e r s t o m a i n t a i np r o p e r o p e r a t i o n . S k il le d t e c h n i c ia n s a l o n esh ou l d b e ab l e t o m ai n t a i n t h e se l ec t ed

s y s t e m .Of co u r se , a t t en t i on sh ou l d b e gi v en t o t h e

l ocat i on of eq u i p men t s fo r ease of ad j u s t m e n t u n d e r t h e e n v i r o n m e n t a l c o n d i t i o n swh i ch wi l l b e ex p er i en ced .

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(4 ) S e t - u p t i m e . S e t - u p t i m e , w h i c hi n c l u d e s a l l o p e r a t i o n s n e c e s s a r y t o l o c a t e ,e m pl a c e , e r e c t , a nd tune up e qui pm e nt ne ede dt o l a unc h t he m i s s i l e , i s t i e d i n c l o s e l y w i t he a s e o f ha ndl i ng . Whe n a t a c t i c a l s i t ua t i o n

i n v o l v e s s e t - u p t i m e , i t c a n b e a n i m p o r t a n tf a c t o r i n t h e c h o i c e o f s y s t e m .

(5 ) R a t e o f f i r e . T h e r e q u i r e m e n t sf o r r a t e o f f i re a r e s t r o ngl y i nf l ue nc e d by t hem i s s i o n of t h e o v e r a l l s y s t e m . F o r e x a m p l e ,a de f endi ng m i s s i l e s ys t e m f o r us e a ga i n s ts a t u r a t i o n a t t a c k s n e c e s s i t a t e s a m a x i m u mr a t e o f f i re . O n t he o t he r ha nd, ra t e o f f i rer e q u i r e m e n t s f o r s t r a t e g i c - t y p e m i s s i l e sm a y n o t b e a s s t r i n g e n t .

( c) D o t he re qu i re m e n t s of t heg u i d a n c e a n d c o n t r o l s y s t e mc a u s e t he m i s s i l e to b e u n n e c e s s a r i l y lo n g a nd t h e r e f o r e e a s i l y s p o t t e d f r o m t h e

a i r ?

( d) A r e t he a l i gnm e nt p r o c e d ur e s s o l e ngthy t ha t t hee n e m y h a s t i m e t o d e s t r o yt h e m i s s i l e a n d a s s o c i a t e de q u i p m e n t ?

( e ) D o e s t h e s y s t e m r e q u i r eg r o u n d c o m m u n i c a t i o n l i n k swhich could easi ly bej a m m e d o r d e s t r o y e d ?

R a t e o f f ir e d e p e n d s h e a v i ly o n e r e c t i o nt i m e a t t he l a unc hi ng pa d, a l i gnm e nt t i m e ,o u t f o r m a i n t e n a n c e , e t c . F o r e x a m p l e , if ar a d a r g r o u n d e q u i p m e n t i s u s e d f o r g u i d a n c e ,ra t e o f f i re m a y be l i m i t e d by t he a m o unt o ft i m e t h e r a d a r m u s t t r a c k e a c h m i s s i l e d u r i n gt h e c o u r s e of i t s t r a j e c t o r y . T h i s c o u l d b e as t r o ng f a c t o r i n m a ki ng a c ho i c e be t w e e n ar a d i o a n d a n a l l - i n e r t i a l t y p e s u r f a c e - t o -s u r f a c e m i s s i l e s y s t e m .

(6 ) S e c u r i t y f r o m c o u n t e r m e a s u r e s .S e c u r i ty f r o m c o u n t e r m e a s u r e s i s o n e o f t hem o s t i m p o r t a n t c r i t e r i a o n w h ic h a s y s t e mcho ice sho uld be ma de . In the ca se of as u r f a c e - t o - s u r f a c e m i s s i l e , a c o n s i d e r a t io no f t h i s f a c t o r i nc l ude s s uc h que s t i o n s a s :

( a) C a n t he ra di o gui da nc es y s t e m b e j a m m e d ?

( b) D o e s t he s ys t e m i nvo l ve al a r g e g r o u n d g u i d a n c e s u b s y s t e m i n s t a l l a t io n , w h i chc o ul d e a s i l y be s po t t e d f ro mt h e a i r ?

e . R e q u i r e m e n t s f o r O t h e r S u b s y s t e m s

The gui da nc e a nd c o nt ro l s ys t e m s ho ul d bec ho s e n s uc h t ha t t he be s t po s s i b l e s ys t e m i so bt a i ne d w i t h i n t he t i m e s c a l e a l l o t t e d f o rd e v e l o p m e n t . T o i n s u r e t h a t t h e t i m e s c a l ewil l no t be exceede d, ca r e mu st be taken tom a ke c e r t a i n t ha t t he gui da nc e a nd c o nt ro ls y s t e m d o e s n o t o v e r c o m p l i c a t e o t h e r s u b s y s t e m s . T h i s s u b j ec t a l r e a d y h a s b e e ndi s c u s s e d s o m e w ha t unde r the s ubj e c t o ff l e xib i l it y o f r e q ui r e m e n t s , but I w o uld l i ket o s t r e s s it f u r t h e r n o w .

I n m a ki ng a c ho i c e o f s ys t e m t h i s c r i t e r i o ns ho ul d be j udge d by a s ki ng s uc h que s t i o ns a s :

(1) Do es the cho ice of guidance andc o n t r o l s y s t e m h i n d e r o r h e lp t he a r m i n g a ndf uzi ng f unc ti o ns ? Fo r e xa m pl e , a s u r f a c e -t o - a i r s e e k e r s y s t e m h av in g no r a n g e i n f o r m a t i o n m a y de pe nd o n ra t e o f c ha nge o f

l i ne - o f - s i gh t o r c ha nge in s i gn o f ra nger a t e f o r f uz ing. When c o m p a re d w i th as y s t e m w h i c h s u p p l i e s r a n g e i n f o r m a t i o nf o r f uzi ng t h i s m a y i m po s e a ha r ds h i p o nt he a rm i ng a nd f uz i ng s ys t e m .

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(2) Do es the cho ice of guidance andc o n t r o l s y s t e m i m p o s e d es i gn h a r d s h i p s o nt he a i r f r a m e in t e r m s o f s p a c e , a r r a n g e m e n t ,a n d a e r o d y n a m i c c o n t r o l c h a r a c t e r i s t i c s ?F o r e x a m p le , a s u r f a c e - t o - s u r f a c e s y s t e m

wh ere g u i d an ce eq u i p men t mu st b e car r i edin the no se may re su l t in diff icul t al ignm entp r o c e d u r e s if t h e m i s s i l e i s e r e c t e d v e r t i c a l l ybefore launching.

S p l i t t i n g of eq u i p men t i n t o v ar i ou s comp a r t m e n t s c a n a l s o b e t r o u b l e s o m e by r e q u i r i n g m o r e w i r e s f o r i n t e r c o n n e c t i o n s a n dp o ss i b l y r i g i d a l i g n men t sp eci f i cat i on s b e t w e e n c o m p a r t m e n t s .

(3) Do es the cho ice of guidance and

con t ro l sy s t em cau se comp l ex i t y i n t h ep r o p u l s i o n s y s t e m b y r e q u i r i n g t h r u s t c o n t r o l , i n t e r m e d i a t e t h r u s t s t a g e s , a u x i l i a r yc o n t r o l j e t s , e t c . ?

(2) W i ll i mp ro v em en t s in co m p o n e n t s m a k e it p o s s i b l e t o m e e t p e r f o r m a n c esp eci f i cat i on s wh i ch can n ot p resen t l y b each i ev ed wi th ou t fu r t h er sy s t e m co mp l i cat i o n ? If t h e an sw er i s " y e s , " th en i t may

b e wi se t o ch oose t h e s i mp l e sy s t em an dd ep en d on comp on en t d ev el op men t t o p rov i d et h e i m p r o v e m e n t , r a t h e r t h a n c h o o s e a m o r eco mp l ex , l es s re l i ab l e sy s t e m wh i ch wi l l b eb u r d e n s o m e i n t h e f u t u r e .

(3) Will the guidance and controlsy s t e m b e q u ick l y ou t mo d ed b y i mp ro v em en t si n o t h er s u b sy st em s? It may be u n wi se t och o o se an o v er l y co mp l ex g u i d an ce sy s t e m toc o p e w i th d e f i c ie n c i e s in o t h e r s u b s y s t e m s ,i f met h od s fo r e l i mi n at i n g t h ese d ef i c i en ci esar e on t h e h o r i zo n . And if su ch met h o d s a r e

fo r t h com i n g , co n si d er a t i o n sh ou l d be g i v en t och oos i n g a g u i d an ce an d con t ro l sy s t em t h atcan ea s i l y b e s i mp l i f i ed as su ch d ev el op m en t sb e c o m e a v a i l a b l e .

f. Gro wt h P o t en t i a l

T h e ch o i ce of g u i d an ce and co n t ro l sy s t emsh ou ld a l so i n clu d e a co n si d er a t i o n of t h eg rowt h p ot en t i a l o f t h e g u i d an ce an d con t ro ls y s t e m a n d t h e o t h e r m i s s i l e s u b s y s t e m s . O n

examining the abi l i ty of systems to meet thiscr i t e r i o n t h e d es i g n e r sh ou ld ask h i msel f su chquest ions as the fol lowing:

(1) Do es th e sy s t e m p ro v i d e fo rf u t u r e i n c o r p o r a t i o n o f n e w c o m p o n e n t s ,p resen t l y u n d er d ev el op men t , wi t h ou t ch an g ein b as i c sy s t em con cep t ? F o r ex am p l e , ag u i d an ce sy s t em mi g h t emp l oy n u mer o u sc o m p o n e n t s o r c o r r e c t i o n d e v i c e s in i t s e a r l ys t a g e s o f d e v el o p m e n t in o r d e r t o m e e ta c c u r a c y r e q u i r e m e n t s . H o w e v er , by p r o p e rd esi g n , t h e sy s t em may b e sy n t h esi zed in sucha way that , as the accuracy of basic guidancec o m p o n e n ts i n c r e a s e s , t he c o r r e c t i o n d e v i c e smay b e merel y e l i mi n at ed wi t h ou t ch an g i n gs y s t e m c o n c e p t .

g . T i mi n g

T h e b es t sy s t em i s u sel ess i f i t i s n o tav ai l ab l e i n t i m e. T h u s , i t i s o f ex t r em ei m p o r t a n c e t o b e s u r e t h a t t h e s e l e c t e dsy ste m is sufficiently sim ple so that i t can bed ev elop ed i n acco rd an ce wi th a sch ed u l e . Ina d d i t i o n , c a r e m u s t b e e x e r c i s e d t o i n s u r et h at c r i t i c a l co mp o n en t s wi l l b e av ai l ab l e i ns u ff ic ie n t t i m e . If t h e p e r f o r m a n c e - i m p r o v e m e n t f a c to r v e r s u s t i m e i s s m a l l , t he s y s t e mh a v i n g t h e l e s s e r p e r f o r m a n c e a n d e a r l i e rav ai l ab i l i t y sh ou l d b e fav ored .

h . Cost

T h e cos t o f a g u i d an ce an d con t ro l sy s t emi s of i m p o r t an ce n ot on l y f rom t h e d o l l a rs t an d p oi n t , b u t a l so b ecau se i t i n d i cat es t h eimpact on the avai labi l i ty of manpower andfac i l i t ies of design ing, pro ducing, ando p e r a t i n g s u c h a s y s t e m . W hen c o n s i d e r i n gcost s , t h e fo l l owi n g a rea sh ou l d b e ex p l o redt o i n su r e t h at th e l owest co s t , co n si s t en t wi tho t h e r r e q u i r e m e n t s , i s o b t a i n e d :

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(1 ) M a n p o w e r t o d e v e l o p c r i t i c a li t e m s . If th e s y s t e m c a n b e c h o s e n t o r e q u i r ea m i ni m um o f s uc h i t e m s , c o s t w i ll be l o w e r .

(2) E as e of deve lo pm ent test in g. If a

c ho i c e e xi s t s , i t w o ul d be w i s e t o c ho o s e t hes y s t e m m o s t e a s i l y t e s te d t o d e t e r m i n e p e r f o r m a n c e . C o n s i d e r a t i o n sh o u l d b e g iv e n t o( a) t h e n u m b e r o f d e v e l o p m e n t m o d e l s r e qui r e d a nd (b) t he t ype o f f a c i l i t i e s r e q ui r e d .A c h o i c e o f a m a r g i n a l e q u ip m e n t m a y r e q u i r ee x t e n s i v e e n v i r o n m e n t a l f a c i l i t ie s f o r p r o o ftest ing, whi le a choice of equipment that isa l r e a d y p r o v e n by o t h e r a p p l i c a t i o n s , h av i n gt h e s a m e e n v i r o n m e n t , w o u ld r e d u c e su c hr e q u i r e m e n t s .

m a y n o t o c c u r . It t h e r e f o r e b e c o m e s t heduty o f t he gui da nc e a nd c o nt ro l s ys t e md e s i g n e r t o s e t u p s u ch c r i t e r i a f o r j u d g in gt h e s y s t e m .

T o i n s u r e p r o p e r c h o i c e i t i s i m p o r t a n tt h a t t h e p r o c e d u r e o u t l i n e d , o r o n e s i m i l a rto i t , be ut i l iz ed. Often, b eca use of a t ights c he dul e a s ys t e m m a y be s e l e c t e d w i t ho utbenefi t of such a na ly sis . If the cho ice isb a s e d o n p a s t e x p e r i e n c e w i th s i m i l a rs y s t e m s a n d r e q u i r e m e n t s , i t w i ll p r o b a b l ybe a go o d o n e . H o w e ve r , if t he a ppl i c a t i o nis new, involving the use of newly developedt e c hni que s a nd c o m po ne nt s , a n a na l ys i s s uc ha s t ha t de s c r i be d s ho ul d be m a de .

( 3) Pr o duc t i o n f a c i l i t i e s . I s t hed e s i g n a d o p t a b le t o m a s s p r o d u c t i o n m e t h o d s ,s u c h a s p r i n t e d c i r c u i t t e c h n i q u e s ?

( 4) C o s t t o m a i nt a i n a nd o pe ra t e . I san exte nsiv e s taff of highly ski l le d tech nicia nsre qu i re d to m a i nt a i n t he e qui pm e nt ? I se x p e n s i v e s p e c i a l t e s t e q u i p m e n t r e q u i r e d ?Wi ll it s t a nd up unde r l o ng s t o r a g e pe r i o dsa n d w i ll i t w i t h st a n d t r a n s p o r t a t i o n e n v i r o n m e n t s w i th o u t n e e d f o r e x t e n s i v e r e p l a c e m e n t s ?

4 . CONCLUSION

If r e q u i r e m e n t s w e r e p e r f e c t l y w r i t t e n ,c r i t e r i a s u c h a s th o s e p r e s e n t e d , a n d o t h e r s ,w o ul d be a n i n t e gra l pa r t o f t he s t a t e m e nt o fr e q u i r e m e n t s ; h o w e v e r , in m a n y c a s e s t h i s

A n a ppro a c h s uc h a s ha s be e n o ut l i ne dhe re i s no t a n a na l y t i c a l o n e , it i s m o r e am a t t e r o f w r i t i ng do w n t he f a c t o rs a nd o fusing judgment in the appl icat ion of weight ingf a c t o r s to th e v a r i o u s c r i t e r i a . A s s u c h ,s o m e f a c t o r s s u c h a s r e l i a b i l i t y m a y b em o re he a vi l y w e i ght e d t ha n i s j us t i f i e d .

T h u s , t he r e e x i s t s to da y a g r e a t ne e d fo ra be t t e r m e a ns of w e i ght ing t he va r i o usc r i t e r i a . T h e i d e a l a p p r o a c h w o u ld b e t oa pply a w e i ght i ng f a c t o r t o e a c h c r i t e r i o n

and to add up in so m e fashion the to tal nu mb ero f po i n t s f o r e a c h s ys t e m , t he be s t c ho i c ebe i ng t he o ne w i th t he l a r ge s t num be r . Thebi g p r o bl e m , ho w e ve r , l i e s i n de t e r m i n i ng t hew e i g ht in g f a c t o r s . P e r h a p s , a s we a l l g a ine x p e r i e n c e in t h i s a r e a o f s y s t e m e n g i n e e r i n gw e c a n do be t t e r a t a ppl yi ng s uc h f a c t o rs .

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100

ACCURACY% OF HITS

WITHIN GIVEN CIRCLE

5 0

RELIABILITY% OF MISS ILES

SUPPLIED WHICHFUNCTION PROPERLY

EFFECTIVENESS% OF M,SS1LES

SUPPLIED WHICH

WHEN FIRED

LAND WITHIN

GIVEN CIRCLE

10 0

5 0

10 0

5 0

COMPLEXITY

COMPLEXITY

COMPLEXITY - —

F i g . 1. Effects of complexity on sys tem f actors .

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INERTIAL NAVIGATIONN o r m a n F . P a r k e r a n d C h a r l e s P . G r e e n i n g *

SUMMARY

Iner t ial navigat io n is ideal for high-s peed f l ight bec ause of i ts inhe re nt ac cur acy andf reed om f rom t h e o p er a t i o n al l i mi t a t i o n s of o t h er n av i g at ion me t h o d s . It i s an ap p l i cat i onof Newt o n 's secon d l aw of mo t i on , wh i ch i mp l i es t h at acc el er a t i o n re l a t i v e t o In er t i a l sp acecan b e d et ect ed wi th ou t r efe r en c e to ex t e rn al i n fo r mat i o n . T h e b as i c e l em en t s of a p h y si cals y s t e m f o r i n e r t i a l n a vi ga ti o n a r e d e s c r i b e d . A c c e l e r o m e t e r s m e a s u r e i nd e pe n de n t c o m p o n e n t s of h o r i z o n t a l a c c e l e r a t i o n , w h il e g y r o s c o p e s s t a b i l i z e t h e s e a c c e l e r o m e t e r s in ad es i re d o r i en t a t i o n . A co mp u t er f in d s p os i t i on and v el oci t y b y i n t eg ra t i n g t h e acc el er a t i o nc o m p o n e n t s s e n s e d in t h e v e h i c l e , a n d a l s o c a l c u l a t e s o r i e n t a t i o n c o r r e c t i o n s f o r m o t io no v e r t he e a r t h , r o t a t i o n o f t h e e a r t h , a nd o t h e r f a c t o r s . T h e s o u r c e a nd p r o p a g a t io n o fc e r t a i n e r r o r s in t h e s y s t e m a r e m e n t io n e d , w i th e m p h a s i s o n t h e i n i ti a l m i s a l i g n m e n t o ft h e a c c e l e r o m e t e r s a s a n e x a m p l e . T h e n a vi ga ti o n e r r o r s o f t he s y s t e m w h ic h r e s u l t f r o m

e r r o r s in t he a c c e l e r o m e t e r s a nd g y r o s c o p e s a r e s t a t e d .

SOMMAIRE

L a n av ig at ion p ar i n er t i e es t i d eal e p o u r l e v o l a g ra n d e v i t e sse p ar so n i n d ep en d an ceet son ex act i t u d e i n h eren t e au x l i mi t a t i o n s op er a t i o n n el l es d es au t r es m^ t h od es d e n av i g at io n .C 'es t une applicat io n de la seco nd loi de NEWTON qui implique que 1 'acce lera t io n r ela t ive pa rr a p p o r t a I ' e s p a c e f ix e , p e u t - e t r e m e s u r e e s a n s a u c un e r e f e r e n c e a un e i n fo r m a t i o n e x t e r i e u r e .L e s e l em en t s d e b ase d 'u n sy s t em e p h y si q u e d e n av i g ati on p a r i n er t i e son t d e sc r i t s . L e sa c c e l e r o m e t r e s m e s u r e n t l e s c o m p o s a n t s i n de p e nd a n ts d e 1 ' a c ce l e r a t io n h o r i z o n t a l e , p e nd a n tq ue l e s g y r o s c o p e s s t a b i l is e n t c e s a c c e l e r o m e t r e s d a n s u ne o r i e n ta t io n d e s i r e e . U ne c a l c u l a t r i ce t ro u v e l a p o s i t ion e t l a v i t es se en i n t eg ran t l es co mp o san t s d e I'acceleVat i on en r eg i s i r ee sd a n s l e m o b i le et c a l c u le e g a le m e n t l e s c o r r e c t i o n s d ' o r i e n t a t i o n d u e s a u d e p l a c e m e n t a u t o u rd e l a t e r r e , a la r o t a t i o n d e l a t e r r e et a ux a u t r e s f a c t e u r s . L a s o u r c e e t l a p r o p a g a t io n d ec e r t a i n e s e r r e u r s d a n s le s y s t e m e s o n t m e n t i o n n e e s , p a r e x e m p l e , l e c a s d 'u n a l ig n e m e n ti n it i a l i n c o r r e c t d e s a c c e l e r o m e t r e s . L e s e r r e u r s d e n a v ig a ti o n du s y s t e m e qu i r e "s ul te n t d e se r r e u r s d a n s l e s a c c e l e r o m e t r e s e t d a n s l e s g y r o s c o p e s s o n t e v a l u e e s .

1. INT RODUCT ION of v eh i cl e acc el e r a t i o n s , h as b eco me a p r a c t i cal rea l i t y an d i s assu mi n g an i mp or t an t

P r ev i o u s p ap er s h av e i n d icat ed t h at a ro l e i n t h e n av ig at ion of h i g h -sp eed a i r c r af t ,n u mb e r of i mp o r t an t ch an g es a r e now t ak i n g At t h e p r ese n t t i m e, t h e u se of i n er t i a lp l ace i n t h e a r t an d sc i en ce of n av i g at io n . n av i g ati on eq u i p men t i s l i mi t ed a l mo st en -On e su ch ch an g e i s t h at i n er t i a l n av i g ati on , t i re l y to m i l i t a r y ap p l i ca t i o n s . As a r es u l t ,t h e d et er mi n a t i on of v eh i cl e p o s i t io n an d sp eci f i c sy s t em d et a i l s an d p er fo rm an c ev e lo c i t y by t h e m e a s u r e m e n t a nd i n t e g r a t i o n a r e c l o a k e d by s e c u r i t y r e s t r i c t i o n s a nd a r e

*North .American Aviat ion, Inc. , Downey, California.

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no t ge n e r a l l y kno w n. H o w e ve r , e v i de nc e o ft he a dva nc e m e nt w hi c h ha s t a ke n p l a c e in th i sf ie ld i s p r o v i d e d b y th e s e r i o u s c o n s i d e r a t i o nc ur re nt l y be i ng g i ve n t o t he us e o f i ne r t i a ln a v i ga t io n e q u i p m e n t i n c o m m e r c i a l j e t

t r a n s p o r t a i r c r a f t .

T h e b a s i c p r i n c i p l e s o f i n e r t i a l n a v i g a t io n ha ve be e n kno w n f o r m a ny y e a r s . Inf a c t, t he phys i c a l l a w s w e re f o rm ul a t e d i n t het i m e o f G a l i l e o a nd N e w t o n, a nd s o m e o f t hem o r e s u b t l e e f f e c t s w e r e r e c o g n i z e d b y J e a nF o u c a u l t , M ax S c h u l e r , a n d o t h e r s . H o w e v e r ,o nl y r e c e nt l y , w i t h t he a dve nt o f h i gh- s pe e da i r c r a f t , h a v e t h e s h o r t c o m i n g s o f c o n v e n t i o na l na vi ga t i o n s c h e m e s be c o m e s uf f ic i e ntl yi m p o r t a n t t o j us t if y the i n t e ns e de ve l o p m e nt

e ff o r t r e q u i r e d to m a k e i n e r t i a l n a v i g at io np r a c t i c a l .

D e a d r e c k o n i n g t e c h n i q u e s , b a s e d o n a i r s p e e d m e a s u r e m e n t s a nd c o m p a s s d ir e c t i o ni n fo r m a t io n , a r e m a r g i n a l a s t o a c c u r a c y .T h i s i s p a r t i c u l a r l y t r u e a t h ig h a l t i tu d e sw h e r e j et s t r e a m s m a y be e n c o u n t e r e d , n e a rt h e e a r t h ' s p o l e s w h e r e m a g n e t i c c o m p a s sa c c ur a c y i s po o r , a nd in f l i ght s i n w hi c hc o n s i d e r a b l e d i s t a n c e m u s t b e t r a v e l e d b e t w e en kn o w n l a n d m a r k s o r o t h e r c h e c k p o i n t s .

R a d i o a n d r a d a r n a v i g a t i o n t e c h n i q u e s a r ev e r y w e l l s u i t e d f o r m a n y a p p l i c a t i o n s ,e s p e c i a l l y f o r gui da nc e ne a r t he o r i g i n a nddes t ina t io n of the f l ight . They depend on ther e c e i p t o f s i g n a l s f r o m t h e g r o u n d a nd , h e n c e ,a r e o f l i m i t e d u s e i n m i l i t a r y a p p l i c a t i o n s .T h e y a r e a l s o o f r e s t r i c t e d u s e o v e r l a r g eo c e a n s , an d a r e a ff ec te d b y m a g n e t i c s t o r m s .

C e l e s t i a l n a v i g a t i o n , b a s e d o n m e a s u r e m e n t s o f t h e a n g u l a r p o s i t io n s o f s t a r sr e l a t i v e t o t h e l o c a l g r a v i t a t i o n a l v e r t i c a l , i sa de qua t e f o r na vi ga t i o n o f s h i ps a nd l o w - s pe e da i r c r a f t . H o w e v e r , a s a i r c r a f t s p e e d i n c r e a s e s , t h i s m et ho d b e c o m e s u n r e l i a b le d u et o t he i na c c ura c y i nvo l ve d i n e s t a bl i s h i ng t hel o c a l v e r t i c a l . T h e d i r e c t i o n o f t h e l o c a l

v e r t i c a l c a n no t b e d e t e r m i n e d a c c u r a t e l y b yc o n v e n t i o n a l t e c h n i q u e s b e c a u s e a c c e l e r a t i o ns o f t he ve hi c l e c a nno t be d i s t i ngui s he df r o m t h e a c c e l e r a t i o n d u e t o g r a v i t y .

A n i ndi c a ti o n o f t he m a gni t ude of e r r o rinvolved is given by the fol lowing example:C o n s i d e r a n a i r c r a f t o n a c o u r s e w h i ch ap i lo t c o n s i d e r s t o b e s t r a i g h t b u t w h ic h i sa c t ua l l y c urve d w i t h a ra di us o f c urva t ure o f1000 na ut i c a l m i l e s ( 1830 k i l o m e t e rs ) . AtM a c h 1 , t he c e nt r i f uga l f o rc e a s s o c i a t e d w it ht h i s ve r y s l i ght c u rv e w o uld i n t ro duc e a ne r r o r o f a p p r o x i m a t e l y 1 7 n a u t i ca l m i l e s( 3 1 k i l o m e t e r s ) .

I ne r t i a l na v i ga t i o n , o n t he o t h e r ha nd, i s

i de a l l y s u i t e d f o r h i gh- s pe e d f l ight . Thes i g n a l t o b e m e a s u r e d , n a m e l y a c c e l e r a t i o n ,is lar ge and the effect of the t im e buildup ofe r r o r w h ic h o c c u r s in i n e r t i a l s y s t e m s i sre d uc e d by t he s ho r t f li ght t i m e s a s s o c i a t e dw i th h i gh- s p e e d a i r c r a f t . A l tho ugh i ne r t i a lna vi ga ti o n i s a l s o w e l l s u i t e d f o r m a nya ppl i c a t i o ns i n l o w - s pe e d ve h i c l e s , i t w a s t hes e v e r e n a v i g a t i o n a l r e q u i r e m e n t s a s s o c i a t e dwith high-speed f l ight that provided thei m p e t u s n e c e s s a r y t o b r i n g i n e r t i a l n a v i g at io nt o i t s p r e s e n t s t a t e o f d e v e l o p m e n t .

2 . PHYSICAL BASIS OF INERTIALNAVIGATION

I ne r t i a l na vi ga t i o n a s us e d i n t h i s pa pe rm e a n s na vi ga ti o n w it h re s pe c t t o s p a c e . It i sa c c o m p l i s h e d b y m e c h a n i z a t i o n o f N e w t o n ' ss e c o nd l a w o f m o t i o n w hi c h s t a t e s t ha t f o rc ei s e qua l t o t he t i m e r a t e of c ha nge o fm o m e nt um . Whi le po s i t io n a nd uni f o rmve l o c i t y a re no t d i re c t l y de t e c t a bl e w i t ho utr e f e r e n c e t o e x t e r n a l i n fo r m a t i o n , i t I s

a p p a r e n t f r o m a c o n s i d e r a t i o n o f N e w t o n ' ss e c o n d la w t h at a c c e l e r a t i o n i s d i r e c t l yde t e c t a b l e . If a m a s s i s s us pe nde d ( Fi g . 1)in suc h a way tha t i t wi l l be def lected f ro ma n e ut r a l p o s i t io n by a ny f o rc e a c t i ng upo n it .

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and so th at a res t o r i n g fo r ce a r i s es in t h ep r ese n ce of t h e d ef l ecti on , t h i s fo rce o rd ef l ect i on can b e u sed as a measu re ofa c c e l e r a t i o n w i t h r e s p e c t t o s p a c e .

If the initial position and velocity of the

a c c e l e r o m e t e r a r e k n o w n, p o s i t io n a nd v e l o c i ty at any la te r t im e can be co mputed byd et ect i n g a l l accel era t i on s an d i n t eg ra t i n gwi t h resp ect t o t i me - on ce t o ob t a i n v el oci t yan d t wi ce t o ob t a i n p os i t i on .

3. MECHANIZATION

T h e s i m p l e a c c e l e r o m e l e r ( F i g . 1) i s s oar ran g ed t h at i t i s sen si t i v e on l y t o accel era t i on s a l on g on e ax i s , ca l l ed t h e sen si t i v eax i s of t h e i n s t r u m en t . If two su ch ac ce l e r

o m e t e r s a r e m o u n t e d w i t h t h e i r s e n s i t i v eax es a t r i g h t an g l es t o each o t h er (F i g . 2 ) ,t h e y c a n b e u s e d t o m e a s u r e a n y a r b i t r a r yacc ele r at io n in the plane defined by the twoa x e s . I f w e c o n s i d e r a n a c c e l e r a t i o n a sindicated by the heavy ar ro w in a di r ec t io ns o m e w h a t e a s t o f n o r t h , t h e n o r t h - s o u t ha c c e l e r o m e t e r w i ll d e t e c t o n e c o m p o n e n t o ft he a c c e l e r a t i o n a n d t he e a s t - w e s t a c c e l e r o m e t e r t he o r t h o g o n a l c o m p o n e n t . T h e s ea c c e l e r a t i o n c o m p o n e n t s c a n b e i n t e g r a t e ds e p a r a t e l y a nd i n t e r p r e t e d a s d i s t a n c e s n o r t h

or sou t h an d eas t o r west o f t h e assu medst ar t i n g p oi n t .

I t is appar ent that th er e is a need fo rs t a b i l i t y o f a c c e l e r o m e t e r o r i e n t a t i o n a b o u tal l axe s. If the platfo rm upon which thea c c e l e r o m e t e r s a r e m o u n te d i s m i s a l i g ne d inazi mu t h , as i n d i cat ed (F i g . 3 ) , an acc el er a t i o ni n th e n o r t h e r l y d i re ct i o n wh ich sh ou l d p r o d u ce a read i n g on l y on t h e n or t h -sou t ha c c e l e r o m e t e r w i l l a l s o b e d e t e c t e d t o as l ig h t d e g r e e by t he e a s t - w e s t a c c e l e r o m e t e r .The integr ated output in this ca se wil l bes l i g h tl y in e r r o r in b o t h d i r ect i o n s s i n ce t h et ru e n o r t h er l y d i s t an c e wi ll b e s l i g h t lyg reat er t h an t h at i n d i cat ed an d t h e t ru e eas t -west d i s t an ce wi l l ac t u al l y b e zero , a l t h ou g ha value wil l be indicated.

It i s a l s o n e c e s s a r y t o p r e v e n t th e a c c e l e r o m e t e r p l a t f o r m f r o m t i l t i n g a b o u t a h o r i zo ntal axis (Fig . 3) bec ause ther e is no wayf o r a n y a c c e l e r a t i o n s e n s i n g de v i c e to d i s t i n g u ish b etween t h e acc el er a t i o n d u e t o

g ra v i t y an d an acc el er a t i o n d u e t o ch an g e i nv e l o c i t y .

T h e a c c e l e r o m e t e r p l a tf o r m c a n b e s t a b i l ized, even in a moving vehicle, by mountingt h e p l a t fo rm in a g i mb al sy s t em and p rov i d i n gg y r o s c o p e s w h i c h h a v e t h e p r o p e r t y o f m a i n t ai ni ng a f i x e d d i r e c t i o n i n s p a c e ( F i g . 4 ) . O nep o s s i b l e a r r a n g e m e n t i n v o l v e s t h e u s e o ft h r e e s e p a r a t e g y r o s c o p e s , e a c h o f w h ic h h a sa s i n g l e ax i s of f ree mot i on p erp en d i cu l ar t oi t s sp i n ax i s . E ach g y r o wi ll t h en p ro v i d est ab i l i t y ab ou t an ax i s p er p en d i c u l ar t o t h efre e axi s and the spin axi s (Fig . 5) . Atorque act ing about a stabi l ized axis wil l beop p osed b y t h e g y ro , wh i ch wi l l p recessab ou t i t s f ree ax i s . T h e an g l e of p r ece ss i o ncan b e u sed as a s i g n al to co u n t er act t h et o r q u e b y e n e r g i z i n g s e r v o m o t o r s in t h es t a b l e e l e m e n t g i m b a l s y s t e m .

If we expand the picture to include theeffect s o f cu rv at u re of t h e ear t h an d ro t a t i onof t h e ea r t h , we fin d t h at co r r ec t i o n s wi l l h av e

t o b e m a d e t o t h e p l a tf o r m o r i e n t a t i o n . L e tu s f i rs t co n si d e r t h e ef fect s o f mo t ion o v ert h e su rfac e of a sp h er i cal ea r t h (F i g . 6 ) .Beca u se t h e s t ab i l i z i n g g y r o s mai n t a i n t h ei rd i r e c t i o n i n a s p a c e c o o r d i n a t e s y s t e m ,mot i on of t h e sy s t em ov er t h e su rface of t h eearth wil l resul t in an apparent t i l t of thep l at fo rm re l a t i v e to t h e ea r t h . T h i s ti l t wi l li n t ro d u ce a com p on en t of g r av i t y i n to on e o rb o th o f t h e a c c e l e r o m e t e r s , u n l e s s t h ea c c e l e r o m e t e r p l a t fo r m i s d r i v e n t h r o u g h a na p p r o p r i a t e a n g l e .

E v en th ou g h t h e sy s t em may b e s t a t i o n ar yon the sur fac e of the ear th , i t wil l develo p ana p p a r e n t m i s o r i e n t a t i o n a f t e r a s h o r t t i m e o fo p e r a t i o n d u e t o t h e r o t a t i o n o f t h e e a r t h .

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Tha t i s , t he p l a t f o rm w i l l a ppe a r t o ro t a t ef ro m e a s t t o w e s t ( l i ke t he s t a r s ) a bo ut a na xi s pa ra l l e l t o t he a xi s o f ro t a t i o n o f t hee a r t h .

T o p r e v e n t t h e i n t r o d u c t i o n o f g r a v i t ye f fe c ts i n to t h e a c c e l e r o m e t e r s , it i s n e c e s s a r y t o c o r r e c t t h e o r i e n t a t i o n o f t h epl a t f o rm f o r bo t h o f t he a bo ve e f f e c t s c o nt i n u o u s l y .

T h e c o r r e c t i o n f o r m o t io n o v e r t he e a r t hr e q u i r e s k n o w l ed g e o f t h e a n g u l a r m o t i o n o ft he s y s t e m . The ra di us o f the e a r t h i s kno w na nd t he d i s t a nc e t ra ve l e d i s a va i l a bl e f ro mt h e a c c e l e r o m e t e r s t h e m s e l v e s . If t h e d i s t a nc e t ra ve l e d i s d i v i de d by t he ra di us o f t hee a r t h p l us t he a l t i t ude a bo ve t he s ur f a c e , t her e s u l t i n g a n g l e i s p r e c i s e l y t h e a n g l e t h r o u g hw hic h t he p l a tf o rm m u s t be ti l t e d t o m a i nt a i nt he p r o p e r o r i e n t a t i o n . T h e o u t p ut o f t h ec o m p u t e r w h i c h p e r f o r m s t h i s d i v i s i o n i sa ppl i e d t o t he p l a t f o r m c o nt r o l s i n s uc h aw ay a s t o k e e p t h e a c c e l e r o m e t e r s a l w a y sp r o p e r l y o r i e n t e d .

S i m i l a r l y , t h e o r i e n t a t i o n c a n b e c o r r e c t e df o r e a r t h ro t a t i o n by s upplyi ng a s i gna l w hi chis a funct ion of t ime and lat i tude of thes ys t e m ( Fi g . 7 ) . It i s a pp a re nt t ha t a ne a r t h - s t a b i l i z e d p l a t f o r m w i l l e x p e r i e n c e ac o m p o n e n t o f t h e e a r t h ' s r o t a t i o n a b o u t i t sno r t h - s o ut h a x i s , g i ve n by the c o s i ne o f t hel a t i t ude , a nd a c o m po ne nt a bo ut i t s a z i m ut hax is , given by the s in e of the lat i tu de. Thec o m p u t e r w i l l c o m p u t e t h e p r e s e n t l a t i t u d efro m the ini t ial la t i tud e and the seco ndi n t e g r a l o f t h e n o r t h - s o u t h a c c e l e r o m e t e ro u t put . It w i l l t he n re s o l ve t he c o ns t a nt e a r t hr a t e i n t o n o r t h - s o u t h a n d a z i m u t h c o m p o n e n t s , a n d s e n d t h e p o r t i o n s t o t h e a p p r o p r i a t e p l a t f o r m d r i v e s .

I n a ny but t he c rude s t a ppl i c a t i o ns , a c c ur a c y r e q u i r e m e n t s m a k e it n e c e s s a r y t o t a k ei n t o a c c o unt s e c o nda ry e f f e c ts due t o t heo bl a t e ne s s o f t he e a r t h a nd due to t he a ngu l a r

m o t io n o f a ny e a r t h - f i x e d c o o r d i n a t e s y s t e m( the w e l l - kno w n C o r i o l i s e f fe c t ). Th e s ee f f e c ts re qu i r e a ddi t io n a l e l e m e n t s in t hec o m p u t e r b ut i n t r o d u c e no n ew p r i n c i p l e s .

4 . SYSTEM OPERATION

The b l o c k d i a gra m ( Fi g . 8) s ho w s t hee s s e nt i a l e l e m e nt s o f a n i ne r t i a l na vi ga t i o ns y s t e m . F o r s i m p l i c i t y , a s i n g l e - a x i s s y s t e mi s s ho w n. I t s o pe r a t i o n c a n be s t be i l l us t r a t e dby c o ns i de r i n g t he ef fe c t o f a c c e l e r a t i o n o fthe vehicle in which the system is located.An a c c e l e r a t i o n t o t h e r i g h t , f o r e x a m p l e ,w i l l b e d e t e c t e d b y t he a c c e l e r o m e t e r . T h eo ut put o f t he a c c e l e ro m e t e r w i l l be do ubl yi n t e gr a t e d , a nd t he re s ul t i ng d i s t a nc e i nf o r

m a t i o n d i s p l ay e d f o r th e o p e r a t o r . In a ddi t i o n ,t h i s i nf o r m a t i o n m a y be us e d, a f t e r a nyn e c e s s a r y c o m p u t a t io n , t o p r o v i d e a n g u l a rc o r r e c t i o n i n f o r m a t io n t o t h e s t a b i li z e d p l a t f o r m . T h e s t a b i li z i n g g y r o w i ll r e s i s t a n yi na dv e r t e n t t o rqu e t e ndi ng to t i l t t he p l a t f o r ma nd w i l l p ro vi de s i gna l s t o t he g i m ba l s e rvo m o t o r t o c o u n t e r a c t t he d i s t u r b a n c e t o r q u ea nd m a i nt a i n t he p l a t f o rm i n i t s p r o pe ro r i e n t a t i o n .

W h en t h e a c c e l e r a t i o n c e a s e s , th e a c c e l e r

o m e t e r o u tp ut d r o p s t o z e r o . T h e f i r s ti n t e g r a l r e m a i n s c o n s t a n t , an d t h e s e c o n di n t e g r a l c o n t i n u e s t o i n c r e a s e a t a u n i f o r mr a t e , p r o v i d in g c o r r e c t d i s t a n c e i n f o r m a t i o na nd p l a t f o r m ro t a t i ng s i gna l s . If t he c a r r y i n gv e h i c l e s l o w s a n d s t o p s , t h e a c c e l e r o m e t e rs e n s e s t h e n e g a t iv e a c c e l e r a t i o n , c a u s i n g t h edi s t a n c e i ndi c a t o r t o c e a s e t u rn i ng. A nyi n c r e a s e o r d e c r e a s e in s p e e d d u e t o w i n dso r c o n t r o l f o r c e s w i ll b e d e t e c t e d a nd i n t e gra t e d i n t he s a m e w a y.

5 . ER R O R S

A n y m e c h a n i c a l o r e l e c t r i c a l s y s t e m c a nb e e x p e c t e d to d e p a r t s o m e w h a t f r o m p e r f e c tr e s p o ns e t o i nput s i g na l s . At f i r s t g l a nc e i t

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a p p e a r s t ha t t h e p e r f o r m a n c e o f t h e in e r t i a li n s t r u m e n t s ( t h a t i s , t h e a c c e l e r o m e t e r s a n dg y ros) wi l l h av e t o b e su rp ass i n g l y g ood i no r d e r t o p r o v i d e a c c u r a t e po s i ti o n i n fo r mat i on o v er a p er i o d of h o u r s . It mi g h t b e

expected, for exa m ple, that a f ixed platfo rmt i l t wou ld cau se an e r r o r t o b u il d u p as t h esq u ar e of t h e t i m e. (E • 1/2sin i s t i l t , g i s g r av i t a t i o n alacce l era t i o n , an d t i s t i me. ) S i m i l a r l y , g y r odrif t r at e might be expected to re su l t in ane r r o r t ha t i n c r e a s e s a s t h e c u b e o f t h e t i m e .Howev er , wh i l e i t i s t ru e t h at t h e req u i rem e n t s o n t h e s e i n s t r u m e n t s a r e s e v e r e , t h ec u r v a t u r e o f t he e a r t h , t o g e t h e r w it h p r o p e rd esi g n of t h e sy s t em, resu l t s i n a l i mi t i n ge ff ec t o n s y s t e m e r r o r s . T h i s e ff ec t c a nb est b e i l l u s t r a t ed b y co n si d er i n g an ex a mp l e .

A t y pi c al s o u r c e o f e r r o r i n a n i n e r t i a ln av i g at i on sy s t em i s an i n i t i a l mi sal i g n men to r t i l t of the pla tfo rm . If we exa m inequali tat ively the effects of a t i l t on a singleax i s p l a t fo rm wh i ch i s s t a t i o n ar y , as in al a b o r a t o r y o r a s t a n d i n g a i r c r a f t , w e c a n g e ta p i c t u r e of e r r o r p r o p a g a ti o n u n c o m p l i c a te db y a i r c r a f t a c c e l e r a t i o n s a n d b y i n t e r a c t i o n sb et ween ax es of s t ab i l i zat i on .

Let us as su m e (Fig. 9) that the sys tem iss t a r t ed an d a l i g n ed , wi t h an i n i t i a l i n cor rectt i l t , 4> • T h e a c c e l e r o m e t e r r e g i s t e r s a naccel era t i on of g s i n <p , which is integ ra tedand i n t e r p re t ed as a mo t ion to t h e l ef t .Hen ce, t h e comp u t er sen d s a s i g n al t o t h eplatfo rm t i l t ing i t to the left thro ugh an anglec o r r e s p o n d i n g t o t h e s p u r i o u s t r a v e l in t ha td i r e c t i o n o v e r t h e c u r v e d e a r t h .

At a late r t im e (F ig. 10), the t i l t to theleft ha s equaled the m isal ignm ent ang le, d> ,and the platfo rm is now level . Th er e is nown o in pu t t o t h e a c c e l e r o m e t e r . H o w e v e r , th ef i r s t i n t eg r a l i s s t i l l p os i t i v e an d t h e secon di n t eg ral i s s t i l l i n creas i n g , cau si n g a fu r t h erro t a t i on i n t h e same d i rect i on , resu l t i n g i n a

sen sed g rav i t y comp on en t op p osi t e t o t h eo rig inal one (Fig. 11). After a t im e thef i rs t i n t eg r a l i s b rou g h t t o ze r o an d t h e p l a t f o r m c o m e s t o r e s t , t h i s t i m e w i th a t i l to f - 4 > .

T h e rev erse of t h e o r i g i n al s i t u at i on n owex i s t s , and t h e p l a t fo r m i s t i lt ed b ack t ow ardt h e o r i g i n al o r i en t a t i on (F i g . 12) an d co n t i n u es t o osci l l a t e i n t h i s fash i on .

T h i s i l l u s t r a t i o n ( F i g . 1 3) s h o w s g r a p h i cal l y t h e accel era t i on , v el oci t y , p os i t i on , an dp l a tf o r m a n g le a s c o m p a r e d w i th t he p r o p e rv al u es fo r t h ese n u m b er s as a fu ncti on oft i me i n t h e s i t u at i on we h av e j u s t d escr i b ed .W e h a v e s e e n ho w th e a c c e l e r o m e t e r d e t e c t sa g c o m p o n e nt wh ic h d e c r e a s e s t h r o u g h z e r ot o t he op p o si t e p o l a r i t y an d th en r e t u r n s tothe o rig inal valu e. The effect upon the f i rs tand second integrals is shown in the veloci tyand d i s t an ce cu r v e s . T h e p l a t fo r m t i l t ,which is di rect ly dependent upon sensedd i s t a n c e , i s a l s o s h o w n .

T h e r esu l t o f t h i s b eh av i o r i s t o p ro v i d e abo u nd t o t h e p o s i ti o n e r r o r s r e s u l t i n g f r o m a ni n it ia l e r r o r . If t he t il t e r r o r w e r e p r o pagated in the usual fashion, as 1/2 (g sin<£) t 2 , an u nb ou n d ed p os i t i on e r r o r wou ldr e s u l t . H o w e v e r , b e c a u s e t h e e a r t h i s n o tf l a t a n d b e c a u s e t e r m s r e l a t e d t o t h e c u r v at u re a re fed b ack i n t o t h e p l a t fo rm con t r o l s , t he e r r o r i s l i m i t e d t o a v a lu e w h ic ho sci l l a t e s wi th i n n ar ro w, f ix ed b o u n d s . T h ep e r i o d o f t h i s o s c i l l a t i o n , f o r a p r o p e r l yd e s i g n e d s y s t e m i s d e t e r m i n e d b y t h e v a l u e so f th e e a r t h ' s r a d i u s a n d " g , " a nd i s 84m i n u t e s . It i s , i n o t h er w o rd s , a S ch u l ertuned system. I t is this phenomenon whichm a k e s i n e r t i a l n a v i g a t i o n p r a c t i c a b l e .

A n ul l e r r o r ( th a t i s , an e r r o r in t h el o c a t i o n o f t h e z e r o p o i n t ) , o r a s c a l e f a c t o re r r o r in th e a c c e l e r o m e t e r , w il l r e s u l t ins o m e w h a t s i m i l a r o s c i ll a t in g e r r o r c u r v e s(F i g . 1 4 ). T h e p h ase re l a t i o n sh i p s and t h e

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s i z e o f t he m e a n e r r o r v a r i e s fr o m o n e c a s et o a no t he r , but t he f re que nc y a nd ge ne ra lf o rm o f t he c u rv e s i s s i m i l a r . A dr i f t i n t hel e v el li n g g y r o s in t r o d u c e s a n e r r o r t e r mw hi ch i n c r e a s e s l i ne a r l y w i th t i m e , i n a ddi t i o n

t o a n o s c i l la t o r y c o m p o n e n t.

F o r l o n g n a v i g a t i o n t i m e s , d y n a m i c a lc o u p l i ng a m o n g p a r t s of t h e s y s t e m m u s t b ec o n s i d e r e d . A l s o , e r r o r s o u r c e s c a n u s u al lyb e b e t t e r d e s c r i b e d s t a t i s t i c a l l y t h a n a sc o ns t a nt s . The ne t r e s ul t o f t he s e t woc o n s i d e r a t i o n s i s t o r e d u c e t he l o n g t e r mbui ldup o f e r r o r f ro m t ha t w hi ch m i ght bee xpe c t e d o n t he ba s i s o f c o ns t a nt i nde pe nde nte r r o r s o u r c e s .

6 . C O M PO N EN T R EQ U I R EM EN TS

T h e c r i t i c a l e l e m e n t s of i n e r t i a l n a v i g a ti o ns y s t e m s a r e t h e a c c e l e r o m e t e r s a n d g y r o s c o p e s . T h e p r e s e n t s t a t e o f t h e c o m p u t e ra r t , bo t h a na l o g a nd d i g i t a l , m a ke s itp o s s i b l e t o m e e t t h e a c c u r a c y a n d c a p a c i t yr e q u i r e m e n t s b y c a r e f u l d e s i g n .

7. CONCLUSION

We ha ve s e e n t ha t i t i s t he o re t i c a l l ypo s s i b l e t o de ve l o p a na vi ga t io n a l de vi c ewhich depends only upon i ts ini t ial condi t ion

( po s i t i o n a nd ve l o c i t y) , a nd i t s a c c e l e ra t i o nsr e l a t i v e t o s p a c e in o r d e r t o p r o v i d e c o n t i n uo us po s i t io n i nf o r m a t i o n . The a dva n t a ge s o fs u c h a s y s t e m o f n a v ig a ti o n a r e n u m e r o u s . I tis independent of wind veloci ty and visibi l i tyc o n d i t i o n s w h i c h o f t e n r e n d e r m o r e c o n v e n t io n a l n a v i g a ti o n m e t h o d s i n a c c u r a t e o r u n usa ble . It i s equa l ly effect ive o ver land o rw a t e r , dur i ng n i ght o r da y , a nd p l a c e s nor e s t r i c t i o n s o n t h e c o u r s e o f th e v e h i c l e inw h ic h i t i s c a r r i e d . F u r t h e r m o r e , s i n c e n oe x t e r n a l i n f o r m a t i o n i s r e q u i r e d a f t e r s y s t e ma c t i va t i o n , t he s ys t e m c a nno t be p re ve nt e df r o m o p e r a t i n g b y e l e c t r i c a l o r m a g n e t icf ie l d s of a r t i f i c i a l o r n a t u r a l o r i g i n .

I t i s a pp a re nt t ha t a s ys t e m ha vi ng t he s ec h a r a c t e r i s t i c s w i ll fin d w i d e s p r e a d a p p l i c at io ns and wil l have a pro found inf luence ont h e m i l i t a r y a n d c o m m e r c i a l t r a n s p o r t a t i o np r o b l e m s o f t h e f u t u r e .

R E F E R E N C E S

1. Sl a t e r , J . M . , "C ho i c e o f C o o rd i n a t e Sys t e m s in I ne r t i a l N a v i ga t i o n , " N a vi ga ti o n , V o l um e5 , N o . 2 , J o urna l o f t he I ns t i t u t e o f N a vi ga t i o n , J une 1956.

2 . Sl a t e r , J . M . , a nd D unc a n, A . B . , " I n e r t i a l N a v i ga t i o n , " A e ro na ut i c a l Engi ne e r i ng R e vi e w ,V o l um e 15, N o . 1 , J a n ua r y , 1956, I ns ti t u t e o f t he A e ro na ut i c a l Sc i e n c e s .

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AIDING THE INERTIAL NAVIGATION SYSTEMW i l l i a m F . B a l l h a u s a n d F r e d e r i c k S t e v e n s , J r . *

SUMMARY

A m o ng t h e s e v e r a l s o u r c e s o f e r r o r i n a n in e r t i a l n a v ig a ti o n s y s t e m a r e g y r o d r i f t ,c o m p o n e nt t o l e r a n c e s , m e c h a n i z a t i o n a p p r o x i m a t i o n s , d ef l e c ti o n s o f th e v e r t i c a l , i n i ti a lsy s t em m i sal i g n m en t s , an d su r v ey i n acc u r acy of t h e lau n ch s i t e . W h i le man y of t h esee r r o r s w i ll u s u a ll y b e s m a l l , o t h e r s w i ll b e a p p r e c i a b l e . T h e c u m u l a ti v e e ff ec t o f e r r o r s i n al o n g - r an g e g u i d an ce sy s t e m i s su ch t h at a i d i n g t ech n i q u es wi l l b e re q u i r ed i n o r d e r t oa c h ie v e t h e t e r m i n a l a c c u r a c i e s g e n e r a l l y r e q u i r e d o f s u c h s y s t e m s . T h e r e a r e s e v e r a la i d i ng t ech n i q u es av ai l a b l e . Amon g t h ese a r e d ie u se of r ad ar o r v i su al ch eck p o i n t s , t h et rac k i n g of a r ad ar b eaco n , s t a r t ra ck i n g , an d com p u t at i on al d amp i n g t h ro u g h t h e u se of ane x t e r n a l v e l o c i t y m e a s u r e m e n t . T h e u s e o f t wo o r m o r e o f t h e s e p o s s i b i l i t i e s s i m u l t a n e o u s l yi s fou n d t o b e of g reat v al u e.

SOMMAIRE

P a r m i l e s m u l t i p le s s o u r c e s d ' e r r e u r s d a n s un s y s t e m e d e n a v ig a ti o n p a r i n e r t i e s et r o u v e n t , l a d e r i v e du g y r o s c o p e , l e s t o l e r a n c e s d e s o r g a n e s , le s a p p r o x i m a t i o n s m e c a n i q u e s ,l e s d e f le c t io n s d u v e r t i c a l , l e s a l l g n e m e n t s i n c o r r e c t s i n it ia u x du s y s t e m e e t 1'erreur d ed e t e r m i n a t i o n d e l a p o s i t i o n du la n c e m e n t . T a n d i s q ue b e a uc o u p d e c e s e r r e u r s s o n t g e n -£ r a l e m e n t p e t i t e s , d ' a u t r e s s o n t a p p r e c i a b l e s . L ' ef fe t c u m u l a ti f d e s e r r e u r s d a n s u n s y s t d m ed e g ou v er n e a l on g u e d i s t an ce e s t t e l q ue d es t ech n i q u es d 'a i d es ser o n t n e ce ssa i r es en v u ed ' o b t e n i r 1'exactitude f in a le g e n e r a l e m e n t r e q u i s e d a n s d e t e l s s y s t e m e s . P l u s i e u r s t e c h n iq u e sd ' a i d e s o n t d i s p o n s i b l e s . P a r m i c e l l e s - c i s e t r o u v e n t 1'utilisation d u r a d a r o u d e p o i n t s d ' o -r i e n t a t i o n v i s i b l e s , l a lo c a l i s a ti o n d ' u n fe u d e r a d a r , l a t r a c e d e s e t o i l e s e l p a r a m o r t i s s e m e n tcal cu l e , b ase ' su r 1'utilisation d e m e s u r e s d e v i t e s s e e x t e X e u r e . L ' u t i l i s a t io n s im u l t a n e e d e

d e u x o u d e p l u s i e u r s d e c e s p o s s i b i l i t e s s e t r o u v e n l e t r e d e g r a n d e v a l e u r .

INTRODUCTION

If Ch r i s t o p h er Co l u mb u s had b een wr o n g i nthe la t te r pa r t of the f i fteenth ce ntur y, and theear t h real l y h ad b een f l a t i n s t ead of rou n d , wew o u l d n o t h a v e t o d a y ' s s o l u t i o n s t o p r o b l e m si n g eo n av i g at io n . S i n ce Co l u mb u s was co r r ec t , and t h e ear t h i s n o t fl a t, man y of o u rp r o b l e m s o f n a v i ga t io n o n t h e e a r t h a r es i mp l i f i ed .

In Co l u m b u s 's d ay t h e r e w as n o sex t an t ,t h e re w as n o n av ig at io n a l m an ac , an d t h e rew a s n o s p e e d m e a s u r i n g d e v i c e . T h a t h ef in al ly r each ed No r t h Am er i ca i s a t r i b u t e t oh i s g e n i u s , h i s b u r n i n g d e s i r e , a n d h i s p r a c t i cal d ead reck on i n g wh i ch was , wi t h h i m,a l m o s t a s i x t h s e n s e .

T od ay , wi t h ou r cap ab i l i t y t o p rod u cep r e c i s i o n m e c h a n i s m , w ith o u r s u p e r s p e e de l e c t r o n i c c o m p u t e r s , a n d w i th o u r k no w l e d ge

* N o r t h r o p . A i r cr a f t, I n c ., H a w t h o r n e , C a l i f o r n i a .

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o f th e e a r t h ' s g e o m e t r y , it i s p o s s i b l e t on a v i g at e p r e c i s e l y o v e r l o ng d i s t a n c e s w i th o u tr e f e r e n c e t o a ny l a n d m a r k s , to a n y m a g n e t i cf i e l ds , o r t o a ny ra d i a t i o n f i e l ds . I ne r t i a ln a v i g a t i o n m a k e s t h i s p o s s i b l e .

L e t u s c o n s i d e r s o m e o f t h e p r o p e r t i e s o fi n e r t i a l s y s t e m s . O n e d i s t in g u i s h i n g p r o p e r t yo f s uc h s y s t e m s i s t he i r m e t ho d o f s e ns i ngp r e s e n t p o s i t io n . H e r e t h e s p h e r o i d e a r t hp e r m i t s t h e e s ta b l i s h m e n t o f a o n e - t o - o n ec o r r e s p o n d e n c e o f e a c h p o i n t o n t h e s u r f a c e oft he e a r t h w i t h t he d i re c t i o n o f a f o rc e ve c t o ra t t ha t po i n t . Fo r i l l us t ra t i o n , t he f o rc ev e c t o r c o u l d b e t h e g r a v i t a t i o n a l v e c t o r a tt ha t po i n t . F i g . 1 s ho w s ho w e a c h g r a v i t a t i o na l ve c t o r i s uni que ly a s s o c i a t e d w it h ap a r t i c u l a r p o s i t io n o n t h e s u r f a c e o f t h e

e a r t h . C o n v e r s e l y , e v e r y p o i nt o n t he e a r t h ' ss u r f a c e m a y b e a s s o c i a t e d w ith t h e a n g u l a r o rd i r e c t i o n a l c o o r d i n a t e s o f t h e v e c t o r .

I t w i l l be re c o gni z e d t ha t i n p re c i s e l y t h i sm a n n e r , t h r o u g h t h e m e c h a n i s m of t he s p h e r i c a l s t r e t c h m a p , p o i n t s o n th e e a r t h ' ss ur f a c e a r e i de nti f ie d by t he c l a s s i c a l na v i ga t o r . It s ho ul d be no t e d t ha t t he s e ge o de t i cc o o rd i na t e s o f l a t i t ude a nd l o ngi t ude a r ei n d e p e n d e n t o f l i n e a r m e a s u r e m e n t s c o n duc t e d o n t he s ur f a c e o f t he e a r t h a nd, f u r t he r ,

t ha t t he c o o rd i na t e s , ( i . e . , l a t i t ude a nd l o ngi t ude ) o f a ny po i n t m a y be de t e r m i ne d s o l e l yb y a n a n g u l a r m e a s u r e m e n t , v i z : t h e d i r e c t i o n o f t h e g r a v i t a t i o n a l v e c t o r o b s e r v e dat that point .

2 . THE INERTIAL SYSTEM

T h e h e a r t o f to d a y ' s i n e r t i a l s y s t e m l i e sin i t s a c c e l e r o m e t e r s , s u p p o r t e d by a s t a b i l i z e d p l a t f o r m w h o s e s p a c e o r i e n t a t i o n i ss l a v e d t o t h a t o f i t s r e f e r e n c e g y r o s c o p e s .

T h e s t a b l e p l a t fo r m i s i s o l a t e d f r o m c r a f ta ng ul a r m o t i o ns by m o unt ing i t o n g i m ba l s ,i n v o l v i n g a v a r y i n g n u m b e r o f r o t a t i o n a la x e s , e a c h o f w h ic h r e p r e s e n t s a d e g r e e o ff r e e d o m .

A c c e l e r a t i o n s a r e s e n s e d a nd do u b ly i n t e g r a t e d t o d e r i v e p o s i t io n a l i n f o r m a t io n , w i ths ubs e que nt f e e dba c k t o t he i ne r t i a l re f e re nc ei n o rde r t o i s o l a t e t he e f f e c t o f g ra vi t y f ro ml i n e a r a c c e l e r a t i o n s .

A di a gr a m o f the ba s i c bui ld i ng b lo c ks o fan ine rt ial sy ste m is shown in Fi g. 2. In theo r d i n a r y a n d m o r e e l e m e n t a r y d e s i g n , g y r o s c o p e s s e r v e a l o n e a s t he s t a b i l iz a t i o n r e f e r e nc e . I n i ti a ll y a l i gne d t o s o m e kno w n r e f e r e nc e , t he y a re c o nt i nuo us l y ro t a t e d i n i ne r t i a ls pa c e i n a c c o rd w i th t he a ngu l a r ve l o c i t y o ft h e a d o p t e d c o o r d i n a t e s y s t e m .

Any c o o r d i n a t e s y s t e m e r r o r w o u ld i n t r o d u c e c o r r e s p o n d i n g i n a c c u r a c y in p o s i t io ni ndi c a t i o n . A br i e f d i s c u s s i o n o f s pe c i f i ce r r o r s w h ic h o c c u r in r e a l s y s t e m s a nd t h e i re f fe c ts u po n s y s t e m a c c u r a c y i s n e c e s s a r y t oo r i e n t o u r f r a m e o f r e f e r e n c e .

In a n y r e a l i n e r t i a l s y s t e m , im p e r f e c t i o n sin g y r o s , a c c e l e r o m e t e r s , an d c o m p u t e rr e s p o n s e c a n r e s u l t in m e a s u r a b l e e r r o r sb e in g i n tr o d u c e d i n to t he s y s t e m . E r r o r sr e s u l t i n g fr o m s u c h i m p e r f e c t i o n s a r er e s o l v e d a s p itc h e r r o r s , r o l l e r r o r s , o r

h e a d i n g e r r o r s .

A p i t ch o r ro l l m i s a l i gnm e nt o f o ne m i nut eo f a r c c o r r e s p o n d s t o a d i r e c t p o s i t i o n a le r r o r o f o n e n a u ti c al m i l e o n t he e a r t h ' ss u r f a c e . A h e a d in g m i s a l i g n m e n t p r o d u c e s ac o n s i s t e n tl y i n c r e a s i n g p o s i ti o n a l e r r o rw hic h i nc r e a s e s w i th t he d i s t a nc e t ra v e l e df ro m t he p o in t w h e r e t he e r r o r o c c u r r e d . Ahe a di ng m i s a l i gnm e nt o f o ne m i nut e o f a rcc o r r e s p o n d s t o an e r r o r r a t e of 0 .6 6 m i l e sp e r t h o u s a n d m i l e s t r a v e l e d .

In a ddi t io n t o e i t h e r p i t c h , r o l l , o r he a dinge r r o r , t h e r e is a f u r th e r c h a r a c t e r i s t i c o fi n e r t i a l s y s t e m s w h i c h m u s t b e c l e a r l yu n d e r s t o o d .

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In t h e s i mp l e p erfect u n d amp ed i n er t i a ls y s t e m , if no e r r o r i s in t r o d u c e d , t h e s y s t e mwi l l a l way s y ie l d p erfec t r es u l t s . Ho wev er ,if a s p e c if i c p o s i t i o n o r v e l o c i ty e r r o r i si n tr o d u c e d i n to t h e p e r f e c t s y s t e m , t h i s e r r o r

wi l l be p er p et u at ed in t h e fo r m of an o sc i l l a ti n g e r r o r in v e l o c i ty o r p o s i t io n a s i l l u s t rated in the system in Fig. 3 .

If i n i ti a l e r r o r s i n p o s i t io n ep and veloci tye v e x i s t , t h e s y s t e m r e s p o n s e i s d e s c r i b e d b yt h e L a p l a c e t r a n s f o r m

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sin u e t .

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T h i s o s c i l l a t i o n i s s i m i l a r t o t h e o s c i l lat ion of a pendulum whose length equals thee a r t h ' s r a d i u s ; i t s c h a r a c t e r i s t i c n a t u r a lp er i o d , d et e r mi n ed b y g rav i t y an d t h e ef fect ive pendulum length, is 84.4 minutes.

I t can be readily seen too, in Fig. 4 , howg y r o s c o p e s a nd r e f e r e n c e m i s a l i g n m e n t se n t e r s y s t e m d y n a m i c s a s m a j o r o f f e n d e r sa m o n g t h e m a n y p o s s i b l e s o u r c e s o f s y s t e me r r o r .

F o r e x a m p l e , g y r o s c o p e s w h o s e d r i f t r a t ei s as low as . 0 1 d eg re e p e r h ou r can p er m i ta n e r r o r g r o w t h a t a r a t e o f a l m o s t o n e m i l ep e r h o u r , q u i t e a p a r t f r o m a n y m i s a l i g n m e n t s

wh ich wer e p er m i t t ed t o ex i s t p r i o r t o i n i t ia ls y s t e m o p e r a t i o n . E v e n t h is p r e c i s i o n i si n su ffi c ien t t o meet accu ra cy r eq u i re m en t s ins o m e l o n g - r a n g e o p e r a t i o n s r e g a r d l e s s o fp erfect i on i n t h e remai n d er of t h e sy s t em.

In sofar as i n i t i a l comp u t er se t t i n g s an di n i t i a l a l ig n men t of t h e s t ab l e refer en ce a r eco nce rn ed, l i t t le diff icul ty is enco unter ed ing r o u n d - b a s e d s y s t e m s , w h e r e e x t e n s i v e s u r v ey s an d f ix ed re fer en c es may b e emp l oy ed

t o y ie l d a l m o s t a n y p r a c t i c a l d e s i r e d a c c u ra cy . Ho wev er , su ch faci li t y i s n o t p r ese n tin s y s t e m s o p e r a t e d f r o m m o v i ng b a s e s s u c ha s s h i p s a t s e a o r a i r b o r n e a i r c r a f t , an dacc u r acy sp eci f i cat i on s may in t h i s ev en tp ro v e t oo d eman d i n g t o be met by con v en t i on alcon cep t s of s t ab l e p l a t fo rm mech an i zat i on an dsy st e m al i g n me n t . In ad d i ti on , so me g ro u n d -based sy ste m s which must be placed intoact i on o n sh o r t n o t i ce do n ot p er mi t su f fi c ien tt i m e f o r s a t i s f a c t o r y p r e f l ig h t o p e r a t i o n s .

A c c e l e r o m e t e r s o f d e m o n s t r a t e d r a n g e ,sen si t iv i ty , and l ine ar i ty sufficient to just ifyt h e u se of a s i mp l e u n d amp ed sy st em h av ebeen unavailable in t ime to be used for somen a vi g at io n s y s t e m s . T h e r e q u i r e d s e n s in gr a n g e p r e s e n t s a p a r t i c u l a r l y ac u t e p r o b l e minasmuch as i t affects achievement of theo t h e r d e s i r e d c h a r a c t e r i s t i c s . It i s c l e a rt h at sy s t e m s of t h e t y pe u n d er d i scu ss i o n maye x p e r i e n c e l a r g e a c c e l e r a t i o n s b o t h d u r i n gtakeoff and in f l ight m an eu ve r s. In navali n s t a l l a t i o n s , l a r g e t r a n s i e n t s m a y a l s o b e

cau sed by th e sh i p 's n at u r a l p i t ch an d r o l l .

I n t eg r a t o r r e s p o n s e p r e s e n t s s t i ll a n o t h e rp rob l em i n t h e comp u t at i on al asp ect o f i n ert i a l n av i g at io n . He re , an y i n accu r acy i si mm ed i at e l y ref l ect ed i n an eq u i v al en t e r r o rof i n t eg r a t i o n , wh ich co n t i n u es t o b e p r o p agated within the undamped system in i tsc h a r a c t e r i s t i c 84 - m i n u te o s c i l l a ti o n . T h i scan b e p ar t i cu l a r l y o f fen siv e i n a t o t a lc o o r d i n a t e c o m p u t a t i o n , i n w h i c h i n t e g r a t o rra n g e mu st b e sp rea d o v er t h at o f t h e

n e c e s s a r y fl ig ht r a n g e . M o s t a p p r o a c h e s a r eco nfro nted with diff icul ty w hich is enco unt e r e d in a c c e l e r o m e t e r d e s ig n - t h a t o fachieving both wide range and high sensi t ivi tyin an analog device.

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I t i s evident that the unaided inert ials y s t e m m a y e a s i l y p r o v e u n a b le t o p e r f o r ms a t i s f a c t o r i l y w i th in o p e r a t i o n a l r e s t r i c t i o n sa nd r a n ge s a s s o c i a t e d w i th a num be r o fa ppl i c a t i o ns i n w hi c h i t c o ul d o t he rw i s e

s e r v e t o r e a l a d v a n t a g e . T o t h e d e s i g n e rc o nf ro nt e d w i t h t h i s s i t ua t i o n , w ho de s i r e st o r e t a i n t he a dva nt a g e s o f i ne r t i a l na vi ga t i o n ,s e v e r a l a l t e r n a t i v e s a r e a t o n c e a p p a r e n t .The f i r s t i s t o p ro c e e d w i t h a s i m pl e i ne r t i a lsystem in the hopeful bel ief that eventuald e v e lo p m e n t o f a c c e p t a b l e a c c u r a t e p r o d u c t i o n c o m po ne nt s w i l l be po s s i b l e w i t h i n t hea l l o t t e d t i m e s p a n . A s e c o n d , an d m o r ec o n s e r v a t i v e c o u r s e i n v o l v e s a u g m e n t i n g t h eba s i c s y s t e m i n s uc h a m a nne r a s t o e l i m i n a t ei t s s i gni fi c a nt l i m i t a t i o ns a s a re s ul t o fc o m p o n e n t e r r o r a nd a t t h e s a m e t i m e t o

e s c a pe t he r i s ks i nvo l ved i n e xpe c t i ng re s ul t st o o s o o n in d i ff ic ult a nd unc e r t a i n de v e l o pm e nta r e a s .

3 . AIDED INERTIAL SYSTEM

O f t he s e ve ra l po s s i b l e m e t ho ds o f a i d i nga n i n e r t i a l s y s t e m , i n c o r p o r a t i o n of c h e c k po i n t i nf o r m a t i o n us u a l l y , a nd un de r s t a nd a bl y , i s g i ve n f i r s t p r i o r i t y . Th i s i s e s pe c i a l l yt ru e i n p i l o t e d a i rc ra f t a ppl i c a t i o ns , i n w hi ch

t he c re w c a n be o f a s s i s t a nc e , be c a us e o f i t so b v i o u s d i r e c t r e l a t i o n s h i p t o t h e p r i m ena vi ga t io n a l o u t put . Se a r c h r a d a r a nd v i s ua ls ys t e m s ha ve be e n e m pl o ye d i n t h i s re s pe c t ,t h e f o r m e r f o r i t s a d v a n ta g e o f a r e a c o v e r a g e ,t he l a t t e r f o r re l a t i ve de f i n i t i o n a nd p re c i s i o n .O f c o u r s e , s o m e r e a s o n a b l e d i s t r i b u t i o n o fc he c kpo i nt s m u s t e xi s t i n o r ne a r t he a r e aw he re a c c ura c y i s o f i n t e re s t i f us e f uli n fo r m a t i o n i s to b e d e r i v e d .

I t i s po s s i b l e t o a u t o m a t i z e t he c he c kpo i nt

o pe r a t i o n . The f a shi o n i n w hi c h c he c kpo i nti nf o r m a t i o n u s ua l l y i s i n t ro duc e d t o t heba s i c i ne r t i a l c o m p ut e r t o e ff e ct t he ne c e s s a r y c o r r e c t i o n i s q u i te e l e m e n t a r y , a nd i t seffect i s i l l us t r ate d in Fi g. 5. As show n,

r e s i d u a l v e l o c i ty e r r o r , u n af fe c te d b y c o r r e c t io n o f o b s e r v e d p o s i t i o n e r r o r , is l ef t t op r o p a g a t e a ne w , b u t n o t l a r g e r , e r r o r . T h u s ,a s e r i e s o f c h e c kp o i n t o b s e r v a t i o n s a t s u i t a b l ei n t e rv a l s c a n l e a d a t e a c h s t e p t o a f u r t he r

re duc t i o n o f bo t h po s i t i o n a nd ve l o c i t y unc e r t a i n t y , a s s ho w n i n Fi g . 6 .

T h e s e c o r r e c t i o n s , w h e n a p p l i e d specif

i c a l l y t o t h e c o m p u t e r , p e r m i t a n y r e f e r e n c em i s a l i gn m e nt t o co nt i nue a s a ' s o ur c e o fp o s i ti o n e r r o r . O t h e r a nd s o m e w h a t m o r ec o m p l e x a p p r o a c h e s t o c h e ck p o i n t c o r r e c t i o nw hi ch a ff ec t re f e r e nc e a s w e l l a s c o m p ut e ra l i g n m e n t a r e f e a s i b l e .

A i r l x )r n e r a d a r , t r a c k i n g a b e a co n w h o s epo s i t io n i s kno w n a c c ur a t e l y , c a n p ro vi de ar a p i d r e d u c t i o n o f e r r o r s in b o th p o s i t io na nd ve l o c i t y ; t h i s e s s e nt i a l l y co nt i nuo us f o r mo f ch e c k po i n t i n c o r p o r a t i o n c a n b e p e r f o r m e de qua l l y w e l l w it h i n t e rc ha n ge d l o c a t i o n o ft r a c ke r a nd be a c o n, if a c o m m uni c a t i o n l i nkis added to the sy ste m . Both velo ci ty andp o s i t io n c o r r e c t i o n s a r e e f fe c te d , w i th t i m ec o n s t a n t s c h o s e n o n t h e b a s i s o f a c o m p r o m i s e b e t w e e n t h e d e s i r e f o r r a p i d d e c a y o ft h e e r r o r a nd th e n e c e s s i t y t o av o id u n p l e a s ant effects of t ra ckin g no ise. When thebe a c o n c a n be t ra c ke d a t s ho r t ra n ge a n

a p p r e c i a b l e i n c r e a s e i n p r e c i s i o n c a n b ea c h i e v e d .

A f ina l m e t ho d o f c he c kpo i nt i nc o rpo ra t i o ni n v o l v e s th e t r a c k i n g o f s t a r s , a p r o c e d u r ef a m i l i a r t o a ny a i r f o r c e o r na vy n a v i g a t o r .I f t he d i re c t i o n o f t he ve r t i c a l i s a s s um e d t obe kno w n, a s w e l l a s a n a c c ur a t e i ndi c a t i o no f t im e , t he " s h o o t i n g " o f s t a r s c a n e s t a b l i s hl o c a l po s i t i o n . Thi s p r o c e du re l e nds i t se l ft o a u t o m a t i c m e t h o d s , w i t h t h e r e f e r e n c ep l a t f o r m s e r v i n g t o i n d i c a t e t h e v e r t i c a l .

P e r h a p s t h e m o s t i m p o r t a n t c o n s i d e r a t i o ni n c o nne c t i o n w i t h re l i a nc e upo n c he c kpo i nt si s t he d i s t a nc e t o be t ra ve l e d f ro m t he l a s tc he c kpo i nt t o t he t a rge t , s i nc e t he i ne r t i a l

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r e f e r e n c e ( a n d c o m p u t e r ) c a n e a s i l y b ed r i f t i n g d u r i n g t h e t i m e r e q u i r e d t o t r a v e r s ethis f inal leg of the f l ight . The pr o blemwh i ch t h i s p r es en t s i s , o f co u r se , a fu ncti onof craft speed as well as of the dis tan ce

involved.

W h e n o p e r a t i o n a l c o n d i t i o n s d o n o t p e r m i ta c c u r a t e a l i g n m e n t p r i o r t o takeoff, and whenav ai l ab l e g y r o sco p es can b e ex p ect ed t o a ll owex c ess i v e d r i f t o f t h e s t ab l e p l a t fo rm , ad d i t io n a l e q u ip m e n t t o c o r r e c t t h e s t a b l e p l a t f o r m i s n e c e s s a r y .

T o m e e t t h i s p r o b l e m , t h e s t a r - t r a c k i n geq u i p men t men t i on ed p rev i ou sl y can b e u sedwi th a d i f fe ren t ob j ec t i v e . On e o r m o r e

a u t o m a t i c s t a r t r a c k e r s c a n b e u s e d i ncon n ect i on wi t h t h e g y r o -s t ab i l i zed p l a t fo r mt o e s t a b l i s h t h e p l a t f o r m s p a c e o r i e n t a t i o n .

In op er a t i o n , t h i s u t i l iza t i on of s t a rt r a c k i n g p r o v i d e s s i g n a l s t o t he g y r o s c o p e sw h i c h s e r v e t o c o r r e c t t h e i r a n g u l a r v e l o c i t yas wel l as t h ei r an g u l ar p os i t i on i n i n er t i a ls p a c e . T h e s t a b i l i z e d p l a tf o r m c o n t i n u e s t oo p er a t e j u s t a s i t d o es i n t h e b as i c i n er t i a lsy s t em , wi t h t h e ex cep t i on th at i t s p er fo r m an ce i s en h an ced ; g y r o sco p e d r i f t i s e l i m i n at ed , i n ef fect , a s a r e m i sal i g n m en t s d u e t os e v e r e l au n ch s h o c k s a n d t o p o o r i n i ti a lco ndit ion s such a s ine xact kno wledge of thel au n c h s i t e c o o r d i n a t e s . D u r i n g a n y a b s e n c eo f s t a r - t r a c k i n g i nf o r m a t io n , t he g y r o -s t ab i l i zed p l a t fo rm p e r fo r m s j u s t as it wou l dh av e in an u n mo n i t o r ed sy s t em . S t ar tra ck i n gcan be ini t iated anew afte r per io ds of cloudc o v e r , w i th in l i m i t s o f t h e d e s ig n p a r a m e t e r sof t h e sy s t em su ch as t e l escop e f i e l d s i zeand exp ect ed d r i f t r a t es of t h e u n mo n i t o r edp l a t f o r m .

Au x i l i a ry t e l escop e p oi n t i n g , s t e l l a r d et ect io n , and s t e l l a r d at a reso l u t i o n eq u i p m en tc o m p l e te s t he s t a r - t r a c k i n g s y s t e m . An u mb er of ap p r o ach e s can b e fo l lowed i n i t s

d e s i g n , w h o s e b a s i s m a y r a n g e f r o m ag e n e r a l p u r p o s e c o m p u t e r t o a p l a y b a c k o fp r e c o m p u t e d te l e s c o p e o r i e n t a t i o n d a t a . T h el a t t e r a p p r o a c h i s f a v o r e d f o r i t s r e l i a b i l i t y ,t h e fo rm er fo r i t s f l ex i b i l it y .

T h e u t i l i za t io n of an i n d ep en d en t so u r ce ofv e l o c i t y i n f o r m a t i o n f o r i m p r o v i n g i n e r t i a ln a v i g a t i o n s y s t e m p e r f o r m a n c e t h r o u g h t h ed a m p i ng o f d y n a m i c e r r o r s i s e v i d e n t. L e tu s con si d er t h ree mod es of u t i l i z i n g v el oci t yi n f o r m a t i o n .

W e may p rocess t h e i n d ep en d en t v el oci t yi n fo r mat i o n i n su ch a m an n e r t h at i t may b ei n se r t e d in t o t h e co mp u t er as (1 ) a d i r ec tc o r r e c t i o n t o t h e c o m p u t e d l i n e a r v e l o c i t y ,(2) a d e r i v e d c o r r e c t i o n t o t he m e a s u r e o fl i n e a r a c c e l e r a t io n , o r ( 3) a d e r i v e d c o r r e c t i on t o t h e comp u t ed p os i t i on .

T h e m a t h e m a t i c a l e q u i v a l e n c e o f t h e s et h r e e a p p r o a c h e s h a s b e e n e s t a b l i s h e d f o rl i n ea r f i l t e r d es i g n . Bec au se of t h i s eq u i v al en ce an d th e d i r ec t n e ss of t h e mo d e wh er ei nt h e au x i l i a ry v el oci t y i n fo rm at i on i s u sed t oc o r r e c t t h e c o m p u t e d v e l o c i t y , t h i s m o d e w i l lb e u sed as a b a s i s o f fu r t h er d escr i p t i o n . Agen er al functio nal d iag ra m is given in Fig . 7.T h e con d i t i on s wh i ch d et ermi n e t h e f i n al i zedf i l te r d e s i g n f o r a ny p a r t i c u l a r s y s t e m a r e :

a . T h e s i mp l i c i t y of mec h an i zat i on d e s i r e d .

b . T h e c h a r a c t e r o f t h e n o i s e a s s o c i a t e dw i t h t h e m e a s u r e m e n t o f v e l o c i t y .

c . T h e d e s i r e d p e r f o r m a n c e .

G e n e r a l l y t h e d e s i r e d p e r f o r m a n c e i sd amp i n g of t h e 8 4 -mi n u t e osci l l a t i on associ

ated with the ini t ial condit ions, and at tenuationo f t h e e f f e c t s o f n o i s e i n t h e a c c e l e r o m e t e r -co mp u t er l oo p . T h e com p l ex i ty of t h ev e lo c i t y f i l t e r i s u s u a ll y r e s t r i c t e d to f i r s t -o r d e r o p e r a t i o n s a nd n o t m o r e th a n tw o

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p a r a m e t e r s . The s e l e c t i o n of t he o pt i m umm a gni t ude of t h e s e p a r a m e t e r s is uniquelyd e t e r m i n e d by t he de s i re d ra pi d i t y o f da m pi ngan d the n a t u r e of t h e v e l o c i ty m e a s u r e m e n t .T h e c u r r e n t m e th o d s of v e lo c i ty m e a s u r e m e n ta r e by E)oppler shift in r a d a r e c h o e s a n d byt r u e a i r s p e e d . In t h e l a t t e r a p p r o a c h , w i n dva r i a t i o ns m u s t be lum pe d w i th o t he r ve l o c i t yn o i s e for p u r p o s e s of d e t e r m i n i n g t h e f i l t e rp a r a m e t e r s .

V e lo c i ty d a m p in g r e m o v e s o n ly e r r o r s inpo s i t io n a nd ve l o c i t y a s s o c i a t e d w i t h t he 84 . 4 -m i nut e o s c i l l a t i o ns . Thi s i s s ho w n by de r i v i ngt h e L a p l a c e t r a n s f o r m of t h e p o s i ti o n a l e r r o rf o r t h e s y s t e m in F i g . 7. Thi s c a n be s ho w nto be

E p ( s ) e p ( 0 ) s + ev(0) + Ks

2

N(s )2 2 2

s + Ks «e + <i.e

(3 )

w h e r e K is a f i l t e r p a r a m e t e r a n d N ( s ) is theve l o c i t y no i s e f unc t i o n .

I n v e r t i n g t h i s t r a n s f o r m g i v e s the t i m e

r e s p o n s eeP(t) • e"

at|ep(o)[coS/St - | s in£t]

+ — s,n/3t| + eN ( 4 )

w h e r e eN is the n o i s e - f o r c e d e r r o r , a isK <J| / 2 an d 0 i s / o j | - a

2

F i g . 8 f u r t h e r i l l u s t r a t e s the fact thatv e l o c i t y d a m p i n g d o e s not c o r r e c t e r r o r sdue to p l a t f o r m m i s a l i g n m e n t .

To t h i s po i n t , the d i s c u s s i o n has beenc o n c e r n e d w i t h the effects of individualt e c h n i q u e s for a i d i n g t h e i n e r t i a l s y s t e m . Itha s be e n s ho w n t ha t s o m e of t h e s e s c h e m e sc o r r e c t the c o m p u t e r , w h il e o t h e r s s e c u r e

p r o p e r p l a t f o r m o r i e n t a t i o n ; in s o m e c a s e s ,e i t h e r the c o m p u t e r or t h e p l a t f o r m m a y beaided, depending upon the m a n n e r in whicht h e a u x i l i a r y i n f o r m a t i o n is i n t ro duc e d i n t ot h e b a s i c s y s t e m .

T h i s is a c c o m p l i s h e d t h r o u g h a c o m bi na t i o n of two or m o r e of t h e p o s s i b i l i t i e sa l r e a d y e n u m e r a t e d . F o r e x a m p l e , t he p l a t f o r m o r i e n t a t i o n c o u l d be c o r r e c t e d or i m p r o v e d by r a d a r c h e ck p o i n t w it h t h e c o m p u t e re m pl o yi ng ve l o c i t y da m p i ng. The c o m bi na t i o no f s t a r t r a c k i n g for m o r e e x a c t p l a t f o r mo r i e n t a t i o n w i t h r a d a r c h e c k p o i n t c o r r e c t i o n sfo r the c o m p u t e r is a l s o p o s s i b l e . H ig h lys a t i s f a c t o r y r e s u l t s c o u l d be o b t a i n e d c o m b i n i n g s t a r t r a c k i n g and v e l o c i t y d a m p i n g .The effect of t h e t r a c k i n g in t h i s c a s e w o ul dbe to i n s u r e t h at t h e c o m p u t e r is d a m p e d tot h e p r o p e r r e f e r e n c e , s o t h a t t h e c o m p u t a ti o n swould not be m e r e l y n o n o s c i l l a t i n g , buta c t u a l l y c o r r e c t .

4. CONCLUSION

It is i m p o r t a n t to no t e t ha t i ne r t i a l na vi g a t i o n s y s t e m s ar e no m e r e m a t h e m a t i c a lc o n c e p t s . T h e y a r e li v in g , b r e a t h i n g , w o r k i n ge l e m e n t s . T h e y c o m e in a l l s i z e s a nd s h a p e s ,w i t h a c c u r a c i e s to m a t c h . E v e r y t yp e ofa i d in g s c h e m e d i s c u s s e d h e r e i s e i t h e r a p a r tof an e x i s t i n g s y s t e m or has at least beenf l i g h t - d e m o n s t r a t e d . S t a r t r a c k i n g is asf e a s i b l e in dayl ight and t w i l i ght ho urs as int he da rk of ni ght; c he c kpo i nt s c a n be us e d

f o r c o m p u t e r c o r r e c t i o n ; v e l o c i ty d a m p i n gw o r k s as t h e o r y i n d i c a t e s it s h o u l d . S o m es y s t e m s use s i n g l e a i d s , s o m e a p p l y c o m b i n a t i o n s , and o t h e r s get along with no a i ds ata l l . The s t a t e of the art is such that ani n e r t i a l n a v i g a ti o n s y s t e m e i t h e r has be e nbuilt or c o u l d b e d e s i g n e d to m e e t an y r e a s o n a bl e set of r e q u i r e m e n t s if s i z e is notr e s t r i c t e d . D e v e lo p m e n t e f fo r t is nowd i r e c t e d to the r e d u c t i o n of s i z e a nd w e i ghto f i n e r t i a l s y s t e m s .

H o w s i m p l e C h r i s t o p h e r C o l u m b u s ' s jobo f d i s c o v e r y of N o r t h A m e r i c a a n d r e t u r n toSpain would have been if he could have ins t a l l e d an a u t o m a t i c i n e r t i a l g u i da n c e s y s t e min his N i na , P i n t a , or Sa nt a M a r i a I

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COGREENWICHMERIDIAN

POLARAXIS

EQUATORIALPLANE

GRAVITY AND

GEODETIC VERTICAL

GEOCENTRIC VECTOR

ELLIPSOID OF

REVOLUTION

F i g. 1 . R el a t i onshi p of grav i t a t i onal vect o r and poi nt s on surf ac e of t he ear t h .

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STABILIZED INERTIAL

REFERENCE

ACCELEROMETERSPLATFORM

CONTROL

GYROSCOPES

i 1

GYRO- RATE

COMPUTER

I M C C 5 T I A 1

INERTIAL uu iviru i en

VELOCITY AND

POSITION

INTEGRATORS

F i g. 2 . B asic i ner t i a l sys t e m.

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• O

FIRST SECONDINTEGRATOR INTEGRATOR

ACCELEROMETER

x~€>COMR

P'COMP.

SYSTEM RESPONSE TO INITIAL CONDITION ERRORS

TRANSFER FUNCTION

, . eD(o)s + cv(o)

cn\S i ' — -

S2 + wg2

«p(o) = INITIAL POSITION ERROR

*¥ (o) = INITIAL VELOCITY ERROR

«p(t) • €p(o)cos «Et • — sin o»Et

F i g. 3. U n d a m p e d i n e r t i a l c o m p u t e r .

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vO

POSITION

AMPLITUDE OF PITCH OR^ROLL MISALIGNMENT

POSITION

ERROR

MISALIGNMENT

8 4 MINUTES

T I M E

F i g . 4 . U n d am p e d s y s t e m r e s p o n s e t o r e f e r e n c e m i s a l i g n m e n t .

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5

POSITIONERROR

VELOCITYERROR

POINT OFCORRECTION

TIME

F i g. 5 . E f fect of s i ngl e checkpoi nt i ncorp or a t i on.

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-ooo

POSITIONERROR

VELOCITY

ERROR

FIRST SECONDCORRECTION CORRECTION

TIME

F i g. 6 . E f fect of repe at ed checkpoi nt inco rpo ra t i o n.

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VELOCITY

INFORMATION

FIRSTINTEGRATOR

SO

ACCELEROMETER »<g>

SECONDNTEGRATOR

PCOMP.

BASIC COMPUTER

F i g. 7 . Vel o ci t y- damped i ner t i a l sys t em .

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LINE4AR HOMING NAVIGATIONRob er t K. Ron ey *

SUMM4ARY

T h i s p ap er d ea l s wi t h th e an al y s i s o f p ro p o r t i o n al n av ig at i on . T h e eq u at i on s of n av i g at i on a r e d ev el op ed an d t h e effect o f sy s t em d y n am i cs an d smo o t h i n g t i me on t h e resu l t i n gt r a j e c t o r i e s a nd m i s s d i s t a n c e s i s b r i e fl y d i s c u s s e d .

SOMMAIRE

Cet l e n o t e es t u n e an al y s e d e la n av i g at i on p r o p o r t i o n n e l e . L es eq u at i o n s de n av i g at io nson t e cr i t e s e t l e s ef fe t s d es e l em en t s d y n ami q u es d u sy s t em e e t du t em p s d 'a t t en u at i o n s u rl e s t r a j e c t o i r e s e t l e s d i s t a n c e s m a n q u e e s s o n t b r i e v e m e n t d i s c u t e X

1. INTRODUCTIONL i n e ar h om i n g n av ig at io n i s a sp eci a l case

of l i n e-of -s i g h t n av i g at i on . By l i n e-of -s i g h tn av i g at i on i s mean t n av i g at i on u s i n g i n fo rmation on the bearing only of the intendedt ar g et , i . e . , wit h ou t u se of t h e ra n g e . Afami l i a r ex amp l e of su ch n av i g at i on i s t h e u seof a r ad i o d i re ct i o n f i n d er i n an a i r p l an e t oh o m e o n a r a d i o b e a c o n . T h i s d i s c u s s i o n w i llbe l im ited to guidance sy ste m s in which ther a t e of r o t a t i on of t h e v el oci t y v ect o r o f t h eg u id ed v eh i cl e , h er e i n af t e r ca l l ed t h e m i ss i l e ,

i s m a d e l i n e a r l y p r o p o r t i o n a l t o t he o b s e r v e dr at e of ro t a t i on of t h e l i n e-o f -s i g h t t o t h et a r g et , in su ch a way as to red u c e t h e line-of-

s i g h t ro t a t i on . W h en t h e ro t a t i on of t h e l i n e -o f - s i g h t i s z e r o , t h e m i s s i l e i s o n a l i n e a r l yp r e d i c t e d c o l l is i o n c o u r s e w i th i t s t a r g e t .S u ch g u i d an ce sch emes a re g en eral l y cal l edp r o p o r t i o n a l n a v i g a t i o n .

C o n s i d e r t h e c o o r d i n a t e s y s t e m o f F i g , 1 ,P e r f e c t p r o p o r t i o n a l n a vi ga ti o n i s d e s c r i b e dby the equation

X • Xci-

w h e r e y r e p r e s e n t s t h e a n g le o f d i r e c t i o n o ft h e mi ss i l e , o - rep resen t s t h e an g l e of t h el i n e - o f - s i g h t f r o m m i s s i l e t o t a r g e t , b o thm e a s u r e d w i th r e s p e c t t o s o m e f ix ed r e f e r en ce ; t h e d ot i n d i cat es d i ffe ren t i a t i on wit hr e s p e c t t o t i m e , a n d X is a co nstant cal led thenavigat ion gain.

It wi ll b e reco g n i zed t hat Y r ep r es en t s al a t e r a l a c c e l e r a t i o n a m , of the missi le andt h at 6- a r i s es f rom a com p on en t of t h er e l a t i v e ve l o c i t y n o r m a l to th e l i n e - o f - s i g h t ,

i . e . , a v el oci t y d ev i at i on f rom a co l l i s i onc o u r s e , t h u s

V Y

wh ere V i s t h e mi ss i l e sp eed an d

(2 )

(1)

(3)

w h e r e V i s a v e c t o r r e p r e s e n t i n g m i s s i l eveloci ty and Vc t h e v el oci t y req u i red fo r ac o l l i s i o n c o u r s e .

•Hu g h es Ai r c ra f t Co . , Cu l v er Ci t y , Cal i fo rn i a .

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W i th t h i s i n t e r p r e t a t i o n f irm l y in mi n d , th econ cep t i on i s eas i l y ex t en d ed t o t h r ee d i men s i on s wh erei n , fo r t h e most ef f i c i en t n av i g a t io n , t h e c o m p o n e n t o f t h e m i s s i l e l a t e r a la c c e l e r a t i o n n o r m a l t o t h e l i n e - o f - s i g h t i s

mad e i n th e n eg at i v e d i re ct i o n of t h e v el oci t ye r r o r . (S ee Ref. 1.) In o t h er w o rd s , t h emi ss i l e accel era t i on i s i n t h e p l an e ofr o t a t io n o f t he l i n e - o f - s i g h t . T h i s r e q u i r e sthat the navigation gain, as defined by Eq.(1), in the two prin cipa l plane s be the r at ioof the cosine of the lead angles.

An e l eg an t n o t a t i on of t h e g en er al t h r ee -d i men si on al l aw i s t h e v ect o r eq u at i on

Xtu ,

(4 )

in w h ic h w ^ r e p r e s e n t s t he r o t a t io n a l v e l o c i t y v ect o r o f t h e l i n e-of -s i g h t .

W i th t h i s g en e ra l i zat i o n , i t can b e sh ownt h at t h e resu l t an t t ra j ect o ry eq u at i on s can ,w i t h a p p r o p r i a t e l i n e a r i z a t i o n , b e s e p a r a t e di n t o t wo i d en t i cal i n d ep en d en t scal a r eq u at i o n s . T o s i mp l ify t h e d i sc u ss i o n , h en cefo r t hw e m a y c o n s i d e r o n l y t h e t w o - d i m e n s i o n a l

c a s e , o r on e of t h e i n d ep en d en t eq u at i o n s .

Re tur ning to the bas ic equatio n, i t sho uldbe pointed out that any actual mechanizat ionmu st r es ul t in mo dificat ion of the equatio n toaccou n t fo r t h e d y n ami cs of t h e s i t u at i o n .Ge n era l l y sp eak i n g , any me as u r e of t h e l i n e-o f -s i g h t d i rec t i o n o- mad e wit h in t h e m i ss i l ew i ll in c lu de r a nd o m e r r o r s , o r n o i s e . U n d e rm o s t p r a c t i c a l c o n d i t io n s , e l i m i n a t io n o rred u ct i on of t h i s n o i se wi l l b e req u i red b yf i l t e r i n g , o r smo o t h i n g , wit h a re su l t an t d el ayin s i g n al d et e r mi n ed b y t h e smo o t h i n g o rav er ag i n g t i m e. In ad d i ti on t h er e wi l l b eu n av oi d ab l e d y n ami c d el ay in t h e b as i c co n t ro leq u i p men t wi th whi ch th e re q u i r ed ac ce l e r a t i o n s a r e d e v e l o p e d .

In cl u d i n g t h ese d y n ami c t e rms an d n oi se ,t h e c o n t r o l e q u a t i o n b e c o m e s

Z ( P ) r -- Xo- + X!

(5 )

wh e re Z (p ) i s a p o l y n om i al (o r r a t i o o fp o l y n om i al s) i n th e d i f fe ren t i a l o p er a t o rp = d / d t , wi th s t a t i c v al u e u n i ty , ch ar ac t e r i z i ng t h e s y s t e m d y n a m i c s ; -) r e p r e s e n t se r r o r s i n t he m e a s u r e m e n t o f 6- .

2 . B4ASIC TR AJE CT OR Y SOLUTIONS

P e r h a p s t h e b e s t p i c t u r e o f t h e g e n e r a lt r a j e c t o r y r e l a t i o n s a r i s i n g f ro m p r o p o r t i o n al n av i g at i on i s rev eal ed b y d i rect so l u t i onof t h e p rop or t i on al n av i g at i on eq u at i on wi t h ou t sy s t em d y n ami cs , i . e . , Z (p ) : 1. This canbe do ne by not ing the re lat io n between Y and0 - i n h eren t i n t h e mi ss i l e- t a rg et k i n emat i cs y s t e m ( F i g . 1 ) , f r o m w h ic h o n e o b s e r v e sthat

Ra- - U sin (<f> - a ) - V s in ( Y - a )

R = U cos [<p - t r ) - Vc os ( Y - c r)

(6 )

wh er e R i s th e d i s t an c e b etween mi s s i l e andt ar g e t . If V an d U ar e co n st an t , i t fo l lowsfrom Eq. (6) ,

& = - 2o-R

Rm

Rr + R

(7)

w h e r e a t p i s t h e t a r g e t a c c e l e r a t i o n n o r m a lt o t h e l i n e - o f - s i g h t .

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W h en co mb i n ed wi th t h e co n t ro l eq uat i on(5), a f i r st o r de r differ ential equatio n in 6- iso btained w hich can be so lved di r ec t ly yielding

Xa - X - X^ = X a 0 ( J - )

A - 2 -{£)A - 2 ) t

( ^ - * )

(8)

w h e r e A = X R m /R, the effect ive navigat iongain, and Ye - \-r) , t h e m e a s u r e m e n t e r r o ri n t e r m s o f Y .

T h e i mp o r t an t feat u r e wh i ch sh ou l d b eo bse rv ed in this equation is the effect o n the

r a t e o f e r r o r d e ca y an d r e q u i r e d m i s s i l ea c c e l e r a t i o n b y t h e n a v i g a t i o n p a r a m e t e r A .N o t e in p a r t i c u l a r t ha t a s A a p p r o a c h e s 2 ,t h e a c c e l e r a t i o n r e q u i r e d to c o r r e c t i n i ti a le r r o r s d o e s no t d e c a y , a nd t he t e r m i n a la c c e l e r a t i o n r e q u i r e d b y e v a s i v e t a r g e tm a n e u v e r s b e c o m e s u n bo u n de d . T h e tw ot e r m s o f t he tr a j e c t o r y e q u at io n a r e s ho w ni n F i g s . 2 and 3 fo r v ar i o u s v al u es of A .

In o r d e r t o m i n i m i z e t h e t o t a l t r a j e c t o r yc u r v a t u r e , i t is c l e a r l y d e s i r a b l e t o m a k e t h e

con st an t X as h i g h as a l l owed b y o t h erc o n s i d e r a t i o n s . O n t h e o t h e r h a nd , a s m a yb e d i s c e r n e d f r o m t h e b a s ic c o n t r o l e q u a t io n ,the magnitude of X is l imited by the al loweda c c e l e r a t i o n s r e s u l t i n g fr o m n o i se o r r a n d o mi n p u t s , and by s t ab i l i t y co n si d e ra t i o n s a r i s i n gou t of sp u r i o u s feed b ack m ech an i s m s i n t h em i s s i l e s y s t e m .

F i g s , 4 a nd 5 i l l u s t r a t e t h e t r a j e c t o r i e sd esc r i b ed b y E q . (8) fo r v ar i o u s A . F o rc l a r i t y , t h e s e a r e d r a w n f o r a s t a t i o n a r ytar ge t . No ting that dr if t o r unbalance in them i ss i l e co n t ro l eq u at i on i s eq u i v al en t t o at a r g e t a c c e l e r a t i o n a l w a y s n o r m a l t o t h el i n e - o f - s i g h t , t h e t r a j e c t o r i e s f o r b o t h i n it i a le r r o r a nd s t e a d y m a n e u v e r c a n b e s ho w nwithout motion of the target point .

3 . MISS EQUATIONS AND WEIGHTINGFUNCTION

W h en t h e con t ro l d y n ami cs , i . e . , t h e t i med e l a y s , a r e i n cl u de d , t h e t r a j e c t o r y e q u a t io n s

can n ot in g en er al b e so l v ed ex p l i c i t ly . T h eycan , o f c o u r se , b e p l o t t ed by s imu l at i ont ech n i q u es , o r i n t eg ra t ed n u meri cal l y wi t hd i g it a l c o m p u t e r s . F o r t u n a t e l y , d e t a i l s o f t hea c t u a l t r a j e c t o r i e s a r e s e l d o m r e q u i r e d if t heg e n e r a l c h a r a c t e r i s t i c s o f t he t r a j e c t o r i e sar e k nown . T h e mai n p oi nt o f i n t e r es t i s t h ec l o s e s t a p p r o a c h , o r m i s s d i s t a n c e , t o t h eta r ge t . By applicat io n of the ini t ial valuet h e o r e m t o t h e L a p l a c e t r a n s f o r m o f t h et ra j ect o ry d i f fe ren t i a l eq u at i on , i t can b esh own t h at t h e re i s a fo r mal an al y t i cal

so l u t i on of t h i s mi ss d i s t an ce i n p erfectg en eral i t y fo r an y t ran sfer fu n ct i on fo r t h eco n t ro l d y n a m i cs . (S ee Ref, 2 .) T h i sso lut ion can be applied to any ini t ial co ndit i on s and an y d i s t u rb an c es su ch as t a r g etm a n e u v e r o r n o i s e .

T h e g e n e r a l s o l u ti o n , h o w e v e r , i s r a t h e rfo rmi d ab l e , as i n d i cat ed b y t h e fo rmal eq u at i o n f o r t h e m i s s :

m(o) = y t ( o ) - X y $ g (s ) Y t(s) ds

(9 )

o r

> £ [ g ( - s )Y t X - s ) ] T = o - j f , (o) .

(10)

T h e i n t eg ra l wi l l b e recog n i zed as t h er es idu e at infini ty of the integr and . X" 1 i st h e i n v e r s e L a p l a c e t r a n s f o r m . Y t (s) is justt he L a p l a c e t r a n s f o r m o f t h e t a r g e t e v a s i o nand no ise dr iving functions, and Yt is thesam e t r an s fo rm wi th th e co n t r i b u t i on of t h edisco ntinuity at T = 0 o m it ted; g(s) isan al og ou s t o t h e sy s t em fu n ct i on i n o rd i n aryc o n s t a n t- c o e f f i c ie n t l i n e a r s y s t e m a n a l y s i s .

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and is read i ly c alcu lated f ro m the fol lowinge qua t i o n i f t he c o nt ro l t ra ns f e r f unc t i o nZ( p) i s kno w n i n f a c t o re d f o rm :

g(s) = exp [ A / dp /pZ (P )l •L

-0 0

(11)

A s y m p t o t i c s o l u t i o n s o f t h e f o r m a l m i s se q u a ti o n a r e f a i r l y r e a d i l y c a l c u l a t e d , h o w e v e r , f o r p a r t i c u l a r c a s e s of i n t e r e s t . T h ef o rm i n w hi c h t he s o l u t i o ns a re o f m o s ti n t e r e s t i s t he s o - c a l l e d w e i ght ing func ti o n ,i . e . , t he m i s s re s ul t i ng f ro m a s pe c i f i e dm o m e n t a r y d i s t u r b a n c e a s a f u nc ti o n o f t h et i m e o r ra n ge a t w hi ch t he d i s t u rb a nc e i sa p p l i e d . F r o m s u i t a b l e l i n e a r c o m b i n a t io n so f t he s e w e i ght ing f unc t io ns , t he m i s s a r i s i n g

o ut o f a ny c o nc e i va bl e t a rge t e va s i ve m o t i o n ,m e a s u r e m e n t e r r o r ( n o i s e ), o r in i ti a l c o n d i t i o n s m a y b e d e t e r m i n e d .

While the weight ing funct ions can bec o m put e d in t he f a s hi o n d i s c us s e d , it i sge n e r a l l y f a r m o re e f f ic i e nt t o p l o t t he mdi re c t l y w i t h a s i m ul a t o r us i ng t he a dj o i n tt e ch n i qu e o n t h e t r a j e c t o r y e q u a t i o n s . Afew such weight ing funct ions are shown inFi g . 6 .

I t i s c o nve ni e nt t o no rm a l i z e t he w e i ght i ngf un c ti o n s i n t e r m s o f s o m e c h a r a c t e r i s t i ct i m e c o ns t a nt o f t he m i s s i l e s ys t e m , us ua l l yt h e t o t a l sm o o t h i n g t i m e . P a r t i c u l a r a t t e n t io nis cal led to the weight ing funct ions for veloci tye r r o r ( l au n ch in g e r r o r ) an d t a r g e t a c c e l e r a t i o n . N o t e pa r t i c u l a r l y t ha t a t s uf fi c ie nt lylong f l ight t ime, for a navigat ion constant atl e a s t a s g r e a t a s t h r e e , t h e m i s s r e s u l t i n gf r o m i n it ia l e r r o r s o r s te a d y t a r g e t m a n e u v e r s ( a n d h e n c e f r o m i n t e r n a l s y s t e m d r i f t s ,o r z e r o e r r o r s ) v an i sh . A n o t h er w ay tos t a t e t h i s o b s e r v a t i o n i s t h at f o r a ny s p e c i f ie d f l ight t i m e , t he m i s s re s u l t i ng f ro m t he s e

e r r o r s v a n i s h e s f o r s u ff ic i en t ly s m a l ls m o o t h i n g t i m e . T h e t i m e r e q u i r e d to b e c o m e i n s e n s i t i v e t o in i t ia l o r s t e a d y d i s t u r b a nc e i s re l a t i ve l y i ns e n s i t i ve t o the s ys t e mga i n , o r na vi ga t i o n c o ns t a nt .

4 . OPTIMIZATION OF DESIGNP A R A M E T E R S

T h e r e a r e tw o m a in p a r a m e t e r s c h a r a c t e r i z i n g a l i n e a r h o m i n g s y s t e m , t h e n a v i ga t io n c o ns t a nt a nd t he sm o o t hi ng t i m e .S i m i l a r l y th e r e a r e tw o p e r f o r m a n c e c h a r a c t e r i s t i c s o f p r i m e i n t e r e s t t h at a r e a f fe c te dby t h e d e s i g n p a r a m e t e r s : t h e m i s s d i s t a n c e ,a nd th e m a x i m u m a c c e l e r a t i o n e x p e r i e n c e d .W e a r e g e n e r a l l y i n t e r e s t e d in t h e s e c h a r a c t e r i s t i c s a s s t e m m i n g f r o m f o u r s o u r c e s :i n i t i a l c o ndi t i o ns , t a rge t e va s i o n , no i s e , a nds y s t e m e r r o r s , e . g ., d r i f ts o r z e r o s h i f t s.We s ha l l now s e e ho w t he s e f a c t o rs a r ere l a t e d in a g e ne ra l w a y in t he o pt i m i z a t i o no f s u c h a s y s t e m .

T h e a c c e l e r a t i o n r e q u i r e d t o c o u n t e rs t ea d y e v a s i v e m a n e u v e r s o r s y s t e m z e r oe r r o r s , p r o v i d e d t h e flig ht t i m e i s l o ngc o m p a r e d w i t h t h e s m o o t h i n g t i m e , v a r i e s a sA / ( A - 2)(c.f. Eq. (8)) , and thus diminishes

a s t he na vi ga ti o n ga in i n c r e a s e s . O n t heo t he r ha nd, a s w e c a n re a d i l y s e e f ro m t heb a s i c c o n t r o l e q u at io n , t h e a c c e l e r a t i o n r e s u lt in g f r o m r a n d o m , s h o r t p e r i o d , e r r o rs i g n a l s o n l i n e - o f - s i g h t m e a s u r e m e n t g i v e sa c c e l e r a t i o n s , a t l e a s t a t lo n g r a n g e , p r o p o r t ional to the navigat ion gain, but reduced byt he a t t e nua t i o n o f t he s m o o t hi ng f i l t e rs w hi c hdepends on the exact nature of the f i l ter andt he s pe c t r a l f re que nc y o f t he ra ndo m no i s e .

In g e n e r a l , t h e r e s u l t i n g a c c e l e r a t i o n w i llb e i n v e r s e l y p r o p o r t i o n a l t o t h e s q u a r e r o o to f the s m o o t hi ng t i m e . Lo o ki ng a t t he ga i ne f f e c ts f i r s t . F i g . 7 i l l u s t r a t e s t he t ra de - o f fb e tw e e n a c c e l e r a t i o n r e q u i r e m e n t s f o rc o u n t e r i n g t a r g e t a c c e l e r a t i o n a n d n o i se a s afunction of A . No te that fo r a spe cifiedt a r ge t a c c e l e r a t i o n a nd a s pe c i f ie d m a gni t udeo f no i s e po w e r de ns i t y a nd s m o o t hi ng t i m e ,t h e r e i s a l w a y s a m i n i m u m a c c e l e r a t i o nr e q u i r e m e n t a t s o m e f in i te A g r e a t e r t ha n

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t w o . I t has been found that a l inear combina t i o n o f t h e t w o r e q u i r e m e n t s c a n b e c o n s i d e r e d a c r i t i c a l m a n e u v e r a b il it y r e q u i r e men t , wi t h t h e resu l t i n g mi ss r i s i n g sh arp l yi f t h e accel era t i on ab i l i t y of t h e mi ss i l e

fa l l s b el ow t h i s c r i t i ca l v al u e.

The bro ken l ine on Fig. 8 sho ws theag g rav at i n g ef fect o f an y i n ci d en t al o r sp u rious feedback in the m iss i le sy ste m whichmay final ly l im it the maximu m X o btain able .

A s i m i l a r c o m p r o m i s e g o v e r n s t he s e l e c t i o n o f t h e o t h e r m a j o r n a v i g a t i o n p a r a m e t e r ,t h e smoo t h in g t i m e. W e h ave o b ser v ed t hatif the flight time is sufficiently great inc o m p a r i s o n t o th e s m o o t hi ng t i m e , t he e r r o r s

in the initial flight co nditio n w ill have vanished, and insignificant m is s wil l r es ul t f ro mst ead y ev asi v e man eu v ers o r i n ci d en t al u n b al an ces in th e sy s t em . On t h e o t h er h an d .

t he d i s p e r s i o n o f t r a j e c t o r i e s d ue to n o i s e ,t e n d s to i n c r e a s e w i th d e c r e a s e d s m o o t h in gt i m e , i n creas i n g v ery sh arp l y wh en t h ec r i t i c a l m a n e u v e r a b i l it y e x c e e d s t h e a b i li t yof t h e m i ss i l e . S i n ce in an y p r act i c al mi s s i l e

we are l imited to some fini te maximum fl ightt i m e , a c o m p r o m i s e i n f i l t e r t i m e m u s t b em a d e .

The t radeo ff inher ent in this si tuat io n isindicated in Fig . 9 . The mi ss es due tor a n d o m n o i s e d i s t u r b a n c e s a r e v i r t u a l l yindependent of the f l ight t ime, but graduallyd e c r e a s e w i th i n c r e a s e d f i l te r i n g . T h e m i s sd u e to i n i ti a l e r r o r s i s d ep e n de n t o n t h e r a t i oof f li g ht t i me t o smo o t h i n g t i m e, r i s i n gsh arp l y as t h e smoot h i n g t i me ex ceed s ab ou t

o ne-te nth the fl ight t im e. Since the two typeso f m i s s e s a r e g e n e r a l l y u n c o r r e l a t e d , a m e a ns q u a r e s u m m i n g g i v e s a r a t i o n a l c r i t e r i o n f o ro p t i m i z a t i o n .

REFERENCES

1 . A d l e r , F . P . , " M i s s i l e G u id a n c e by T h r e e - D i m e n s i o n a l P r o p o r t i o n a l N a v i g a t io n , "Jou rn al Ap p l i ed P h y si cs , 2 7, 1 9 5 6 , 5 0 0 -5 0 7.

2. Ben n et t , R. R „ an d M at h ews, W . E . , " A n al y t i cal D et erm i n at i o n of M i ss Di s t an ce fo rL i n e a r H o m i n g N a v ig a ti o n S y s t e m s , " H u gh e s A i r c r a f t C o . , T e c h n i c a l M e m o r a n d u mNo . 260, March 31, 1952.

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F i g . 4. T y p i c a l t r a j e c t o r i e s w i t h i n i t i a l e r r o r .

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P I T F A L L S IN M I S S IL E C O N T R O L

R o b e r t L . J o h n s o n *

S U M M A R Y

U n kn o w n e f f e c ts o r p r o b l e m s a r e u s u a l l y t h e o n e s w h ic h c a u s e t h e m o s t t r o u b l ei n t h e r e a l i z a t i o n o f a n a d e q u a t e l y c o n t r o l l e d m i s s i l e . In a n a t t e m p t t o r e d u c e t h i s a r e a o fu n k no w n s f o r o t h e r s , t h i s p a p e r p r e s e n t s a n u m b e r o f p r o b l e m s w h i c h h a v e o e e n e n c o u n t e r e din th e d e s i g n o f p a s t m i s s i l e s . T h e s e i n c l u d e : s t a t i c a n d d y n a m i c a e r o e l a s t i c i t y ; s t r u c t u r a lf e e d b a c k s; c r o s s - t a l k ; a e r o d y n a m i c , h y d r a u l i c , m e c h a n i c a l , a n d g e o m e t r i c a l n o n l i n e a r i t i e s ;

o v e r l o a d s o f v a r i o u s k i n d s ; t h e a d v e r s e e f f ec t s o f n o i s e , d r i f t , a nd t o l e r a n c e s ; a n d a m e n t i o no f s u c h o p e r a t i o n a l p r o b l e m s a s e n v i r o n m e n t a l an d s a f e ty c o n s i d e r a t i o n s . S i n c e m a t h e m a t i c a lf o r m u l a t i o n u s u a l l y f o l lo w s d i r e c t l y f r o m a n u n d e r s t a n d i n g o f th e p h y s i c a l p r o b l e m , t h et r e a t m e n t i s r e s t r i c t e d t o a n e x p l a n a t i o n o f t h e p h y s i c s o f t y p i c a l e x a m p l e s in e a c h c a s e .

S O M M A I R E

L e s p h e ' n o m e n e s i n c o n n u s s o n t g e n e r a l e m e n t c e u x q u i c a u s e n t l e p l u s d ' e n n u i s d a n s lar e a l i s a t i o n d ' u n m i s s i l e c o n t r o l e d ' u n e m a n i e r e a d e q u a t e . D a n s l e b u t d e r e d u i r e Ic n o m b r ed e c e s p r o b l e m e s p o u r l e s a u t r e s , c e t t e n o t e p r e s e n t e u n c e r t a i n n o m b r e d e p r o b l e m e s q u io n t e t e r e n c o n t r e s d a n s le p a s s e d a n s 1'etudedes m i s s i l e s . U s c o m p r e n n e n t : 1 ' a e r o e l a s t ic i t es t a t i q u e e t d y n a m i q u e , l e s c o n t r e - r e a c t i o n s ,d e s t r u c t u r e ; l e m e l a n g e d ' i n f o r m a t i o n s ; l e sn o n - l i n e a r i t e s a e r o d y n a m i q u e s , m ^ c a n i q u e s , h y d r a u l i q u e s e t g e o m e t r i q u e s ; l e s s u r c h a r g e sd e n a t u r e d i f f e r e n t e ; l e s e ff e t s d e f a v o r a b l e s d u b r u i t , d e l a d e r i v e e t d e s t o l e r a n c e s ; e t u n em e n t i o n d e s p r o b l e m e s o p e r a t i o n n e l s t e l s q u e l e s c o n s i d e r a t i o n s d e s d c u r i t e e t d ' e n -v i r o n n e m e n t . B i e n q u e l a m i s e s o u s f o r m e m a t h e m a t i q u e s u i t h a b i t u e l l e m e n t d i r e c t e m e n t l ac o m p r e h e n s i o n d u p r o b l e m e p h y s i q u e c e t t e n o t e s e r e d u i l u n i q u e m e n t d a n s c h a q u e c a s a u n ee x p l i c a t i o n d e s p h e n o m e n e s p h y s i q u e s r e g i s s a n t d e s e x e m p l e s t y p i q u e s .

1 . I N T R O D U C T I O N

«An i n d i v i d u a l o r a n o r g a n i z a t i o n , i n g o i n gf r o m t h e d e s i r e t o c o n t r o l a m i s s i l e t o t h e

a c t u a l i ty o f a c o n t r o l l e d m i s s i l e , p a s s e st h r o u g h tw o g e n e r a l s t a g e s . T h e f i r s t isc o n c e r n e d w it h w h y c o n t r o l s h o u l d w o r k .I t c o n s i s t s o f l e a r n i n g t h e b a s i c l i n e a rt h e o r i e s o f s e r v o m e c h a n i s m s an d d y n a m i c sa n d t h e i r a p p l i c a t i o n in t h e f o r m o f b a s i c

c o m p o n e n t s . T h e s e c o n d s t a g e r e v o l v e sa r o u n d t h e p r o b l e m s of w hy c o n t r o l d o e s n o tw o r k , a t l e a s t i n i t i a l l y . It c o n s i s t s o f t h e

g r o u n d a n d f l ig h t t e s t s w h i c h u n c o v e r t h e

u n k n o w n s o r p i t f a l l s w h i c h w e r e n o t i n

c l u d e d i n t h e i n i t i a l e f f o r t s .

T h e s e u n k n o w n s a r e n o t u s u a l l y in t h e m s e l v e s d if fi cu lt t o c o r r e c t , o n c e o n e r e a l i z e st h e i r e x i s t e n c e . W h i l e t h ey m a y c o m p l i c a t et h e d e s i g n , o r a t l e a s t t h e e f f o r t in o b t a i n i n g

i t , t h e r e a r e u s u a l l y w a y s o f h a n d l i n g t h e

p r o b l e m s . T h i s m e a n s t h a t f o r e w a r n i n g o ft y p i c a l p r o b l e m s s h o u l d m a t e r i a l l y e x p e d i t et h e o b t a i n i n g o f a d e q u a t e c o n t r o l . P r i o r

* D o u g l a s A i r c r a f t C o m p a n y , S a n t a M o n i c a , C a l i f o r n i a .

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c o n s i d e r a t i o n o f k no w n p r o b l e m s r e d u c e s t h ediff icul t ies on a given missi le to those whicha r e p e c u l i a r t o t h at m i s s i l e .

The subjec t of this p ap er is a brief

c o v e r a g e o f t h e s e k no w n p r o b l e m s . G e n e r a lc l a s s e s o f p r o b l e m s c o v e r e d a r e : l i n e a rp r o b l e m s , n o n l i ne a r p r o b l e m s , r a n d o me f f e c ts , a nd o p e r a t i o n a l p r o b l e m s . T h et r e a t m e n t i s r e s t r i c t e d t o an e x p l a na t io n o fthe ph ysic s involved in typical ca se s so thatm o r e e x a m p l e s m ay b e g i v en . M a t h e m a t ic a lf o r m u l a t i o n us ua l l y f o l lo w s d i r e c t l y f ro ms u c h a p h y s i c a l u n d e r s t a n d i n g . No a t t e m pt t ob e c o m p r e h e n s i v e h a s b e e n m a d e ; o n l y i t e m swhich have caused specif ic diff icul t ies int h e a u t h o r ' s e x p e r i e n c e a r e c o v e r e d .

The a dve nt o f l a rge s c a l e a na l o g a ndd i g i t a l c o m p u t i n g m a c h i n e s m a k e s t h e d i s t in c t io n b e tw e e n l i n e a r a nd n o n l i n e a r p r o b l e m s so m e w h a t une s s e nt i a l . It i s ba s e d upo nt he " p r e m a c h i n e " fa ct t ha t l i n e a r p r o b l e m sc o ul d us ua l l y be s o l ve d a nd unde rs t o o d ,w hi l e no nl i ne a r p r o bl e m s c o ul d no t . A tpr e s e nt , t he a v a i l a bi l i t y o f c o m put i ngm a c h i n e s f o r c o n t r o l a n a l y s i s r e s u l t s in a nunde rs t a ndi ng o f t he phys i c a l phe no m e na a st he c o nt ro l l i ng f a c t o r . A bi l it y t o s o l ve t he

m a t h e m a t i c a l e q u a t i o n s d e s c r i b i n g t h e p h y s i c s f o r m e r l y w a s th e c o n t r o l l in g f a c t o r .

Th i s do e s no t m e a n t hat o ne c a n ne gl e c tl e a r n i n g t h e t i m e - h o n o r e d a nd b a s i c t e c h n i q u e s o f l in e a r a n a l y s i s . T r a n s f e r f u n c ti o n s ,f r e q u e n c y c h a r a c t e r i s t i c s , N y q u i s t d i a g r a m s ,B o de p l o t s , and ro o t - l o c us t e c hn i que s f o r ma n a l m o s t i ndi s pe ns a bl e ba c kgro und f o r t hec o n t r o l s d e s i g n e r . N e i t h e r d o e s i t m e a n t h a ta n a l y s i s o f t h e p r o b l e m , n o m a t t e r h o w e a s yby m a c hi ne t e c hni qu e s , i s t he e nd r e s ul t .

M a ny t i m e s it i s po s s i b l e to e l i m i na t e t hes o ur c e o f t he p ro bl e m r a t h e r tha n l ive w i thi t thro ugh an al ys is . Again, kno wledge of thee x i s t e n c e o f p r o b l e m s i s p a r a m o u n t , t o w h ic hkno w l e dge l e t us p ro c e e d.

2 . LINE4AR PROBLEMS

a . A e r o e l a s t i c i t y

A e r o e l a s t i c i ty i s a s s o c i a t e d w ith t h e e l a s

t i c d e f o r m a t io n o f t h e a i r f r a m e u n d e r a e r o dyna m i c l o a ds . B o t h s t a t i c a nd dyna m i ce f f e c t s m us t be c o ns i de re d .

An example of the s tat ic effect would bethe def lect ion of a tai l surface under theloa ds which i t pick s up at ang les of at ta ck .Such deflect ions change the loads which wouldo t he r w i s e be e xpe c t e d a nd l e a d t o d i ff e re nc e 'sb e t w e e n a c t u a l a n d t h e o r e t i c a l a e r o d y n a m i cde r i va t i v e s . In t h i s c a s e , t he m a j o r e ff e ctw o ul d be o n t he li ft ing c h a r a c t e r i s t i c s . F i g . 1shows how the l ift would be affected for al o a d a f t o f t he s ur f a c e t o rs i o na l a x i s . Thes i tu a t io n w o u ld b e r e v e r s e d f o r a l o ad f o r w a r do f t he t o r s i o na l a x i s . Th e s e c ha ng e s i n l if tde r i va t i v e s s ho w up a l s o a s c ha nge s i nm o m e nt d e r i va t i ve s . B o t h a ff ec t t he dyna m i c so f t h e a i r f r a m e t o b e c o n t r o l l e d .

Si m i l a r e f f e c ts o c c u r if a c t ua t i ng t o r quero ds o r t he m i s s i l e bo dy de f l e c t . D e pe ndingu p o n t h e p a r t i c u l a r a i r f r a m e c o n f i g u r a t i o ni nvo l ve d, o ne m us t c o n s i de r t he e f f e c ts o fo t he r s t r uc t ur a l de f l e c t i o ns . It m a y w e l l bet ha t s o m e pa r t s w i l l be c r i t i c a l in t h i sr e g a r d , r a t h e r t h an in s t r e n g t h . In a nye v e n t , t h e r e v i s e d a e r o d y n a m i c d e r i v a t i v e sm us t be us e d i n t he a ut o m a t i c c o nt ro l a na l ys i s .

S u c h a e r o e l a s t i c i t y , h o w e v e r , i s n o t n e c e s sa r i ly bad. One may emp loy i t to modifya e r o d y n a m i c c h a r a c t e r i s t i c s a s a f un cti o n o flo a d s o a s t o m i n i m i z e v a r i a t i o n s w it h a i r spee d and al t i tu de. The effect upon f lut terc h a r a c t e r i s t i c s a l s o m u s t a lw a y s b e

c o n s i d e r e d .

D y n a m i c a c r o e l a s t i c i t y a s u s e d h e r e i nr e f e r s t o th e d y n am i c c o u p l in g of an a e r o e l a s t i c phe no m e no n w i t h t he a ut o m a t i c c o nt ro l

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sy st e m . M o re sp eci f i cal l y , t h e en d re su l t o fa n a u to m a t i c c o n t r o l s y s t e m i s t h e e x e r t i o nof a con t ro l fo rce of so m e k i n d . T h i s fo r cen o t o n l y a c t s u p o n t h e a i r c r a f t c o n s i d e r e d a sa r ig id body but as an ela st ic body. If the

el as t i c b od y d ef l ect s u n d er t h i s con t ro l fo rcein a manner which can affect that force, acou p l i n g o r feed b ack ex i s t s wh i ch mu st b eco n si d er ed . Not on l y s t a t i c b ut t r an s i en t ,i . e . , d y n ami c, ef fect s mu st b e con si d ered .

As an ex am p l e , co n si d e r F i g . 2 wh i chs h o w s a b lo c k d i a g r a m o f a c o n t r o l s y s t e mc o n t a i n i n g a r a t e g y r o a n d a n a c c e l e r o m e t e r .A co n t ro l su rf ace d ef lect i on wi l l n o t on l yexcite the missi le as a r igid body but i t wil li mp ose fo rces on t h e mi ss i l e wh i ch d ef l ect

i t , i .e . , exc i te i t , a s an ela st ic body. Themi ss i l e i n d ef l ect i n g wi l l mov e t h e ra t e g y roa nd th e a c c e l e r o m e t e r a nd t he r e s u l t i n gs i g n a l s w i l l, t h r o u g h th e p o s i t io n s e r v o , r e s u l tin f u r t h e r c o n t r o l s u r f a c e d e f l e c t io n s . T h ee l e m e n t " s t r u c t u r a l d y n a m i c s " i s n o t as i m p l e o n e b e c a u s e a l l m i s s i l e s h a v e s e v e r a lmodes of vibrat ion in which they can deflect .Usu al l y on e f in d s t h at t h e mo st i mp o r t an tmo d es a r e t h e f i r s t , seco n d , e t c . , b en d in gm o d es . F i g . 3 i n d i cat es t h e sh ap e of t h ef i r s t t h r e e o f t h e s e m o d e s . F i g . 4 s h o w s h o w ,o n th e f i r s t m o d e , th e a c c e l e r o m e t e r a nd r a t eg y r o in t h e ex am p l e i n t e r p r e t t h i s b en di n gm o t i o n .

As so ciated with each mo de is not o nly ashape but a natural f requency and a dampingr a t i o . T h erefo re , on e can p l o t t h e f req u en cyb eh av i o r o f a m i ss i l e as a t ra n sf er fu nct ionr e l a t i n g o u t pu t m o t io n t o i np ut f o r c e o r c o n t r o ldeflect ion at a given point . The mo tion in ou rex am p l e may be e i t h er t h e v er t i cal m ot i on a tt h e a c c e l e r o m e t e r l o c a t io n o r t h e a n g u l a r

mo t io n a t t h e r a t e g y r o l ocat i o n . F i g . 5sh ow s a t y p ical p l o t w h er e , i n t h e i n t e r es t so f s i m p l i c i t y , th e p h a s e c h a r a c t e r i s t i c s a r en ot sh own . T h ey , as wel l as t h e am p l i t u d ec h a r a c t e r i s t i c s , m a y b e d e t e r m i n e d e i t h e re x p e r i m e n t a l l y o r t h e o r e t i c a l l y .

W i t h t h e s t ru ct u ra l d y n ami cs k n own , i t i sa s i m p l e m at t e r t o i n cl u d e t h e ex t r a feed backl oop i n t h e eq u at i on s g o v ern i n g t h e sy s t em .S t ab i l it y may b e d et e rm i n ed as b efo r e an d , ifn e ce ssa r y , s t ep s t ak en t o o b t a i n i t . T h e se

s t e p s m a y c o n s i s t o f e l e c t r i c a l o r m e c h a n i c a lf i l t e r s o r p l aci n g th e i n s t ru m en t s so t h at t heyare i n sen si t i v e t o t h e b en d i n g mot i on s .

b . S t r u c t u r a l F e e d b a c k s

T h er e may b e o t h er feed b ack p at h s t hro u g ht h e s t r u c t u r e n o t a s s o c i a t e d d i r e c t l y w i thai r l o ad s and h en ce n ot c l a sse d wi th t h ea e r o e l a s t i c p h e n o m e n a . A g o o d e x a m p l e i st h e v i b ra t i o n i nd u ced b y th e i n er t i a l oad si m p o s e d o n t he s t r u c t u r e by h ig h p e r f o r m a n c e

s e r v o s y s t e m s . T h e r a p id a c c e l e r a t i o n s o fw h i c h t h e s e s y s t e m s a r e c a p a b l e r e s u l t i nb ot h t ra n s l a t i o n al an d ro t a t i on a l ex ci t a t i onb ei n g ap p l ied to th e a i r f r am e. If t h e re a r el o c a l r e s o n a n c e s in a n y o f t h e c o n t r o l i n s t r u m e n t s o r t h e i r m o u n t s , in c o n t r o l v a l v e s o ri n s e r v o a m p l i f i e r s , i n s t a b il i ty m a y r e s u l t .T h e p rob l em of i n s t ru men t mou n t s sh ou l d b emen t i on ed se p ar at e l y . T o o of t en t h e r i g i d i t yof t h e mou n t i n g s t r u c t u r e i s n o t g i v en ad eq u at eat ten t io n. It is qui te cl ea r that if the natur alf requency of the mo unt ( including the weight

of th e i n s t r u m en t ) i s b elow t h at o f t h e i n s t r u me n t , t h e mo u n t freq u en cy wi l l b e t h e co n t ro l l i n g d y n am i c p h en om en on . T h e ad d i t i on alu n su sp ect ed p h ase l ag s wh i ch re su l t may l eadt o d y n ami c i n s t ab i l i t y a rou n d o r ab ov e t h emou n t n at u ra l f req u en cy .

An o t h er re so n an c e wh i ch mig h t b e ex ci t edb y w in g i n e r t i a l o a d s , m o t o r r o u g h n e s s , o ro t h er ex ci t a t i on feed i n g t h rou g h a i rc raf ts t r u c t u r e i s i n te r n a l i n s tr u m e n t r e s o n a n c e .E x a m p l e s a r e p o t e n ti o m e t e r b r u s h e s , p r e s

su re g ag e d i ap h r ag m s, e t c . All o f t h eseeffects can be handled in a l inear analysisb y t h e same t ech n i q u e as p resen t ed fo r t h ea e r o e l a s t i c p h e n o m e n a . H o w e v e r , a b e t t e ra p p r o a c h i s to e l i m i n a t e t he r e s o n a n c e r a t h e rthan design around i t .

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c . C r o s s - t a l k

C r o s s - t a l k i s t he a p p e a r a n c e in o n e c o n t r o lax i s of s i g n al s wh i ch a r e i n ten d ed fo r o rr e s u l t f r o m a c t io n s i n a n o t h e r c o n t r o l a x i s .

C o n s i d e r a v e r t i c a l s i g n a l v e c t o r i n e a r t hc o o r d i n a t e s b e i n g r e s o l v e d i n t o c r u c i f o r mm i s s i l e c o o r d i n a t e s w i th o n e s e t o f w i n g slev el. With the wings in fact level , ther e s o l u t i o n r e s u l t s i n o n l y a n u p m a n e u v e r .H o w e v e r , if t h e m i s s i l e i s r o l l e d s o m e w h a t , ah o r i z o n t a l m a n u e v e r a l s o r e s u l t s , a s s ho w ni n F i g . 6 . T h e o n e s i g n a l h a s " c r o s s - t a l k e d "i n t o t h e o t h er a x i s . T h e ch an g e i n t h e i n ten d edd i r ect i o n i s u su al l y sm al l . In c l o sed l oops y s t e m s , c r o s s - t a l k e r r o r s w il l t en d t o b ec o r r e c t e d b u t t h e r e a r e l i m i t s t o ho w l a r g e

this coupling can be.

C r o s s - t a l k a r i s e s f r o m a n u m b e r o f d i ff e r e n t e f f e c t s . E x a m p l e s a r e :

(1) In s t r u m en t s a r e n o t mo u n t ed i nt h e p r o p e r d i r e c t i o n s o t h a t t h e ya r e s e n s i t i v e t o m o t i o n s i n o t h e ra x e s .

(2 ) S e r v o v a lv e s p o o l s a r e s e n s i t i v et o a c c e l e r a t i o n s i n a n o t h e r a x i s

f ro m t hat wh ich t h ey co n t r o l .

( 3) R a t e g y r o s a r e b e c o m i n g s e n s i t ive in another axis as theyd e f l e c t w h i l e m e a s u r i n g r a t e sabout the intended axis.

(4) Rat e g y ro s a r e act i n g l i k e an g u l ara c c e l e r o m e t e r s a b o u t t h e g i m b a la x i s .

( 5) T h r e e - d i m e n s i o n a l c o u p li n g d u et o t h e n on or t h og on al i t y of f reeg y r o g i m b a l s .

(6 ) Aerod y n ami c cou p l i n g resu l t i n gf ro m s t ee r i n g an g l es of a t t acka n d c o n t r o l s u r f a c e a n g l e s .

W h e n l i n e a r o r l i n e a r i z e d , t h e s e c r o s s talk effects may be handled by analyt ic techn i q u es . Usu al l y t h ey a re n on l i n ear an d mu stb e c o n s i d e r e d b y o t h e r m e t h o d s .

It should be noted that the above list onlyp r o v i d e s e x a m p l e s an d i s no t n e a r l y c o m p l e t e .Nor can i t b e comp l et e b ecau se each sy s t emwill have i ts own pr o ble m s which can only beu n c o v e r e d by th e d e s i g n e r a s ' h e c o n s i d e r st he e f f e c ts o f d e p a r t u r e s fr o m t he d e s i r e dst a t e of af fa i rs .

3. NONLINEAR PROBLEMS

A s p r e v i o u s l y n o t e d , n o n l i n e a r p r o b l e m s

h a v e b e e n c o n s i d e r e d s e p a r a t e l y o n l y b e c a u s et h e t ech n i q u es fo r h an d l i ng t h em ar e d i f fe ren t .T h e eq u at i on s of mot i on a re s t i l l wr i t t en i nt h e same way a l t h ou g h man y t i mes t h e n on -l i n e a r i t i e s a r e s u ch t h at th e e q u a t io n s a r ed i ff icu l t t o fo rm u l at e . Ho wev er , t hey may n ol on g er b e h an d l ed t h rou g h u su al op era t i on alcal cu l u s met h od s an d p rev i ou sl y men t i on edg ra p h i cal t ech n i q u es . S ol u ti on of t h ese n on l i n e a r p r o b l e m s u s u a ll y r e q u i r e s a n al o g o rd i g i t a l co mp u t i n g ma ch i n e t ech n i q u e s . L i n e a r i z a t i o n a b o u t s o m e q u i e s c e n t o p e r a t i n g

point is of ten em ploy ed with good r es ul ts ifi n t e rp re t ed i n t h e l i g h t o f a res t r i c t eds o l u t i o n .

a . A e r o d y n a m i c N o n l i n e a r i t i e s

T h e a e r o d y n a m i c c h a r a c t e r i s t i c s o f t hea i r f r a m e to b e c o n t r o l le d a r e c l e a r l y o fm a j o r i m p o r t a n c e t o th e c o n t r o l s y s t e mde sig ne r . In fact , as indicated in the pap erby P e r r y , th e a i r f r a m e c h a r a c t e r i s t i c s a r eso i mp o r t an t t h at t h e con f i g u rat i on mu st no tb e d es ig n ed an d p re sen t ed to t h e co n t r o l s mana s a " f a i t a c c o m p l i . " C o n s id e r a t io n o fc o n t r o l r e q u i r e m e n t s m u s t h e l p e s t a b l i s h t h eco nfigurat io n if a balanc ed design is to beo b t ai n ed . F o r ex am p l e , so me of t h e b en d in g

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sy st e m . M o re sp eci f i cal l y , t h e en d r esu l t o fa n a u to m a t ic c o n t r o l s y s t e m i s t h e e x e r t i o nof a con t r o l fo r ce of so m e k i n d . T h i s fo rcen ot on l y act s u p on t h e a i rc raf t con si d ered asa rig id body but as an ela st ic bo dy. If the

e l a s t i c b o d y d e f l e c t s u n d e r t h i s c o n t r o l f o r c ein a manner which can affect that force, acou p l i n g o r feed b ack ex i s t s wh i ch mu st b eco n si d er ed . Not on l y s t a t i c b ut t r an s i en t ,i . e . , d y n ami c, ef fect s mu st b e con si d ered .

As an ex am p l e , co n si d e r F i g . 2 wh i chs h o w s a b lo c k d i a g r a m o f a c o n t r o l s y s t e mc o n t a i n i n g a r a t e g y r o a n d a n a c c e l e r o m e t e r .A co n t ro l su rfa ce d ef lect io n wi l l n o t on l yexcite the missi le as a r igid body but i t wil li mp ose fo rces on t h e mi ss i l e wh i ch d ef l ect

i t , i .e . , exc i te i t , as an el as t ic body. Themi ss i l e i n d ef l ect i n g wi l l mov e t h e ra t e g y roa nd th e a c c e l e r o m e t e r a nd t h e r e s u l t i n gsi g n al s wi l l , t h rou g h t h e p os i t i on serv o , resu l tin f u r t h e r c o n t r o l s u r f a c e d e f le c t i o n s . T h ee l em e n t " s t r u c t u r a l d y n a m i c s " i s no t as i m p l e o n e b e c a u s e a l l m i s s i l e s h a v e s e v e r a lmodes of vibrat ion in which they can deflect .Usu al l y on e f in d s t h at t h e mo st i mp o r t an tmo d es a r e t h e f i r s t , secon d , e t c . , b en d in gm o d es . F i g . 3 i n d i cat es t h e sh ap e of t h ef i r s t t h r e e o f t h e s e m o d e s . F i g , 4 s h o w s h o w ,o n th e f i r s t m o d e , th e a c c e l e r o m e t e r a nd r a t eg y r o in t h e ex am p l e i n t e r p r e t t h i s b en di n gm o t i o n .

Ass o ciated with each mo de is not only ashape but a natural f requency and a dampingr a t i o . T h e r e f o r e , o n e c a n p l o t t h e f r e q u e n c yb eh av i o r o f a m i ss i l e as a t r an s fer fu ncti onr e l a t i n g o u tp ut m o t i o n to i np ut f o r c e o r c o n t r o ldeflect ion at a given point . The mo tion in o urex am p l e may b e e i t h er t h e v er t i ca l mo t io n a tt h e a c c e l e r o m e t e r l o c a t io n o r t h e a n g u l a r

m o t io n a t t he r a t e g y r o l o c a t i o n . F i g . 5sh ow s a t y p i cal p lo t w h er e , i n t h e i n t e r es t so f s i m p l i c i t y , th e p h a s e c h a r a c t e r i s t i c s a r enot show n. They , as well as the am pli tudec h a r a c t e r i s t i c s , m a y b e d e t e r m i n e d e i th e re x p e r i m e n t a l l y o r t h e o r e t i c a l l y .

W i t h t h e s t ru ct u ra l d y n ami cs k n own , i t i sa s i m p l e m at t e r t o i n cl u d e t h e ex t r a feed backloo p in t h e eq u at io n s g o v ern i n g t h e sy s t em .Stabil i ty may be de ter m ine d a s befo re and, ifn e ces sar y , s t ep s t ak en t o o b t a i n i t . T h e se

s t e p s m a y c o n s i s t o f e l e c t r i c a l o r m e c h a n i c a lf i l t e r s o r p l aci n g th e i n s t r u m en t s so t h at t h eya r e i n s e n s i t i v e t o t h e b e n d i n g m o t i o n s .

b. S t r u c t u r a l F e e d b a c k s

T h er e may b e o t h er feed b ack p at h s th ro u g ht he s t r u c t u r e n o t a s s o c i a t e d d i r e c t l y w i thai r l o ad s an d h en ce n ot c l a ss ed wi th t h ea e r o e l a s t i c p h e n o m e n a . A g o o d e x a m p l e i st h e v i b ra t i o n i nd u ced b y th e i n er t i a l oad si m p o s e d o n t h e s t r u c t u r e b y h ig h p e r f o r m a n c e

s e r v o s y s t e m s . T h e r a p id a c c e l e r a t i o n s o fw h i c h t h e s e s y s t e m s a r e c a p a b l e r e s u l t i nb ot h t ra n s l a t i o n al an d ro t a t i o n al ex ci t a t i onb ei n g ap p l ied to th e a i r f r am e. If t h e re a r el o c a l r e s o n a n c e s in a n y o f th e c o n t r o l i n s t r u m e n t s o r t h e i r m o u n t s , i n c o n t r o l v a l v e s o rin s e r v o a m p l i f i e r s , i n s ta b i li t y m a y r e s u l t .T h e p rob l em of i n s t ru men t mou n t s sh ou l d b emen t i on ed sep ar at e l y . T oo of t en t h e r i g i d i t yof t h e mo u n t in g s t r u c t u r e i s n o t g i ven ad eq u at eat ten t io n. I t is qui te cl ea r that if the natur alf requency of the mo unt ( including the weight

of th e i n s t r u m en t ) i s b elow t h at o f t h e i n s t r u me nt, the mo unt f requency wil l be the co nt ro l l i n g d y n ami c p h en omen o n . T h e ad d i t io n alu n su sp ect ed p h ase l ag s wh i ch re su l t may leadt o d y n ami c i n s t ab i l i t y a rou n d o r ab ov e t h emou n t n at u ra l f req u en cy .

An o t h er re so n an c e wh i ch mi g h t b e ex ci t edb y w in g i n e r t i a l o a d s , m o t o r r o u g h n e s s , o ro t h e r e x c i t a t i o n f e e d i n g t h r o u g h a i r c r a f ts t r u c t u r e i s i n te r n a l i n s tr u m e n t r e s o n a n c e .E x a m p l e s a r e p o t e n t i o m e t e r b r u s h e s , p r e s s u r e g ag e d i a p h r a g m s , e t c . A ll o f t h e s eeffects can be handled in a l inear analysisb y t h e same t ech n i q u e as p resen t ed fo r t h ea e r o e l a s t i c p h e n o m e n a . H o w e v e r , a b e t t e ra p p r o a c h i s t o e l im i n a t e t he r e s o n a n c e r a t h e rt h an d es i g n a rou n d i t .

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c . C r o s s - t a l k

C r o s s - t a l k i s t he a p p e a r a n c e i n o n e c o n t r o la xi s o f s i gn a l s w hi ch a r e i n te nde d f o r o rr e s u l t f r o m a c t i o n s i n a n o t h e r c o n t r o l a x i s .

C o n s i d e r a v e r t i c a l s i g n a l v e c t o r i n e a r t hc o o r d i n a t e s b e i n g r e s o l v e d i n t o c r u c i f o r mm i s s i l e c o o r d i n a t e s w i th o n e s e t o f w i n g slev el . With the wings in fact level , ther e s o l u t i o n r e s u l t s i n o n l y a n u p m a n e u v e r .H o w e v e r , i f th e m i s s i l e i s r o l l e d s o m e w h a t , ah o r i z o n t a l m a n u e v e r a l s o r e s u l t s , a s s ho w nin F i g . 6 . T h e o n e s ig n a l h a s " c r o s s - t a l k e d "i n t o t he o t he r a x i s . T he c ha nge i n t he i n t e nde ddi r e c t i o n i s us ua l l y s m a l l . In c l o s e d l o o ps y s t e m s , c r o s s - t a l k e r r o r s w il l t en d t o b ec o r r e c t e d b u t t h e r e a r e l i m i t s t o ho w l a r g e

t h i s c o upl i ng c a n be .

C r o s s - t a l k a r i s e s f r o m a n u m b e r o f d if f er e n t e f f e c t s. E x a m p l e s a r e :

( 1) I ns t r um e nt s a r e no t m o unt e d i nt h e p r o p e r d i r e c t i o n s o t h a t t h e ya r e s e n s i t i v e t o m o t i o n s i n o t h e ra x e s .

(2 ) S e r v o v a l v e s p o o l s a r e s e n s i t i v et o a c c e l e r a t i o n s i n a n o t h e r a x i sf ro m t hat w hic h t he y c o nt r o l .

( 3) R a t e g y r o s a r e b e c o m i n g s e n s i t i ve i n a no t he r a x i s a s t he yd e f l e c t w h i l e m e a s u r i n g r a t e sa bo ut t he i n t e nde d a xi s .

( 4) R a t e gy ro s a r e a c t i ng l i ke a ngu l a ra c c e l e r o m e t e r s a b o u t t h e g i m b a la x i s .

( 5) T h r e e - d i m e n s i o n a l co u p l i n g d u et o t he no no r t ho go na l i t y o f f re eg y r o g i m b a l s .

( 6) A e ro dyna m i c c o upl i ng re s ul t i ngf r o m s t e e r i n g a n g l e s o f a t ta c ka n d c o n t r o l s u r f a c e a n g l e s .

W h e n l i n e a r o r l i n e a r i z e d , t h e s e c r o s s talk effects may be handled by analyt ic techni qu e s . U s ua l l y t he y a r e no nl i ne a r and m us tb e c o n s i d e r e d b y o t h e r m e t h o d s .

I t should be noted that the above l is t onlyp r o v i d e s e x a m p l e s an d i s n o t n e a r l y c o m p l e t e .N o r c a n i t be c o m pl e t e be c a us e e a c h s ys t e mwil l have i ts own pr o ble m s which can only beu n c o v e r e d b y t h e d e s i g n e r a s ' h e c o n s i d e r st h e e f fe c ts o f d e p a r t u r e s fr o m t h e d e s i r e ds t a t e o f a f f a i r s .

3 . NONLINEAR PROBLEMS

A s p r e v i o u s l y n o t e d , n o n l i n e a r p r o b l e m s

h a v e b e e n c o n s i d e r e d s e p a r a t e l y o n l y b e c a u s et he t e c hni que s f o r ha ndl i ng t he m a re d i f f e re nt .The e q ua t i o ns o f m o t io n a r e s t i l l w r i t t e n i nt he s a m e w a y a l t ho ugh m a ny t i m e s t he no n-l i n e a r i t i e s a r e s uc h th a t t h e e q u a t io n s a r edifficult to fo rm ula te. Ho weve r, they may nol o n g e r b e h a n d le d t h r o u g h u s u a l o p e r a t i o n a lc a l c u l u s m e t h o d s a n d p r e v i o u s l y m e n t i o n e dgr a ph i c a l t e c hn i que s . So l u t io n o f t he s e no nl i n e a r p r o b l e m s u s u a ll y r e q u i r e s a n al o g o rd i g i t a l c o m put i ng m a c hi ne t e c hni q ue s . Li n e a r i z a t i o n a b o u t s o m e q u i e s c e n t o p e r a t i n gpo int is of ten em plo yed with good r es u l ts ifi n t e rp re t e d i n t he l i ght o f a re s t r i c t e ds o l u t i o n .

a . A e r o d y n a m i c N o n l i n e a r i t i e s

T h e a e r o d y n a m i c c h a r a c t e r i s t i c s o f t hea i r f r a m e to b e c o n t r o l le d a r e c l e a r l y o fm a j o r i m p o r t a n c e t o t he c o n t r o l s y s t e mde s i g ne r . In f a c t , a s i ndi c a t e d in t he pa p e rb y P e r r y , t h e a i r f r a m e c h a r a c t e r i s t i c s a r es o i m po r t a n t t ha t t he c o n f i gura t io n m us t no tb e d e s i g n e d a n d p r e s e n t e d t o t h e c o n t r o l s m a na s a " f a i t a c c o m p l i . " C o n s id e r a t io n o fc o n t r o l r e q u i r e m e n t s m u s t h e lp e s t a b l i s h t h eco nfigur at io n if a balanc ed design is to beo bt a i ne d. Fo r e xa m p l e , s o m e o f t he be nding

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diff ic ul t ie s me nt io ne d e a r l ie r c a n be e a s e d byp r o p e r c h o i c e o f l en g th t o d i a m e t e r r a t i o .A l s o , ma ny o f the a e r o dyna m ic no nl ine a r i t ie sme nt io ne d be lo w c a n be e l imina te d by pr o pe rc ho ic e o f c o nf igur a t io n . Suc h no nl ine a r i t ie s

a r e no t ne c e s s a r i ly fa ta l but the i r e ffe ctsm u s t c e r t a i n l y b e c o n s i d e r e d .

One type of nonlinear i ty is that shown inFig . 7 whic h o c c u r s ne a r z e r o angle o f a t ta c k.This type is an actual s tat ic instabil i ty atsm all an gles of at tac k. Depending upon thetype of control feedback employed, this canlead to low amplitude oscil lat ions wasteful inc o nt r o l e ne r gy a nd in k ine t ic e ne r gy, i .e . ,d r a g . Ano the r type , s ho wn in Fig . 8, r e p r e s e n ts a lo s s o f r e s to r in g mo me nt a t h igh

ang les of at tac k. Th is can lead to tumblingthe m is s i le if it o ve r s ho o ts o r i s c o m ma nde dinto the po o r c o nt r o l r e gio n . Sta bi l i ty c ha nge sl ike th is a r e us ua l ly a s s o c ia te d wi th v is c o uscro ssflo w effe cts . Hence , they may be ase v er e function of bank angle which can ber u i n o u s t o a n a g i l e m i s s i l e .

Ano the r type o f a e r o dyna mic no nl ine a r i ty ,which could also be mentioned under thec r o s s - t a l k h e ad i ng , i s t h at o f p i t c h / y a w / r o l li n t e r a c t i o n s . T h e s e r e s u l t in r o l l in g m o m e n t s

due to s te e r ing, a nd in s te e r ing mo me nts , dueto ro l l ing. Lo ss of s tabil i ty at high ang les ofattack at differ ent bank ang les as indic ated inFig. 8 is an exam ple of the la t te r . Fig . 9s ho ws typic a l r o l l ing mo me nt c ur ve s fo rv a r i o u s s t e e r i n g c o n d i t io n s . C a r e m u s t b eta ke n tha t the s e r o l l ing m o me n ts do no t o ve r p o w e r t h e r o l l c o n t r o l s y s t e m , e i t h e r s t a t ically or dyn am ically. Fig . 10 sho ws thedyna mic r e s po ns e o f a mis s i le a t te mpt ing toho ld a z e r o r o l l po s i t io n unde r the r o l l ingm o m e n t s i m p o s e d b y n o i sy s t e e r i n g c o m

ma n ds . The da s he d c ur ve s ho ws the r e s po ns ebefore the real m agnitu des of the ro l l ingm o m e n t s w e r e a p p r e c i a t e d . T h e s o l i d c u r v es ho ws the g r e a t ly impr o ve d a nd e nt i r e lya d e q u a t e b e h a v i o r a f t e r d y n a m i c p r e s s u r es e ns i t ive ga in c ha nging c i r c ui ts we r e a dde d.

b. H y d r a u l i c N o n l i n e a r i t i e s

Pe r h a p s the mo s t a nno ying no nl inc a r i tyasso ciat ed w ith hyd ra ulic s is the tendency forbr o ke n l ine s to s pr a y o i l o n e ve r yo ne e xc e pt

hydr a u l ic e ngi ne e r s . Th is tende nc y no t be ingsubject to an aly sis , le t us pas s on to typicalh y d r a u l i c n o n l i n e a r i t i e s .

Flo w ve r s u s p lunge r d is pla c e me nt in c o ntro l valves is apt to be nonlinear because ofla nd o ve r la p a nd ve lo c i ty s qua r e d pr e s s ur ed r o p a c r o s s o r i f i c e s . F i g . 11 i l l u s t r a t e s t h eb a s i c c h a r a c t e r i s t i c . A l so i n d ic a te d o n F i g .11 by the do t te d l ine s a r e the qua s i - l ine a rgain change effects which result fo r differenta mp l i tude s o f mo t io n . Th e s e p r o vide ar e a s o na ble wa y o f de te r mining whe the r s uc hga in c ha nge s wo uld be t r o uble s o me . Fo re xa m ple , in te r pr e t ing the Nyquis t d ia gr a m o fFig . 12 sho ws that a gain reductio n of 3.5would pro bably r esu lt in instabil i ty at thefrequency indicated as f2. Th is would be al imi te d lo w-a mpl i tude o s c i l la t io n be c a us ehighe r a mp l i tude s wo uld inc r e a s e the ga in andpr o vide s ta bi l i ty . Suc h o s c i l la t io ns c a n bet r o u b l e s o m e i n t e r m s o f l o s s o f h y d r a u l i co i l , i n c r e a s e in d r a g , d e c r e a s e d a c c u r a c y ,e t c .

Ac tua lly , the c o nt r o l va lve c h a r a c t e r is t i co f f ina l impo r ta nc e i s no t f r o m plunge r d is pla c e me nt to flo w, but f r o m c o nt r o l c u r r e n t(o r vo l tage ) to f lo w. Th is in t r o duc e s o the rn o n l i n e a r i t i e s s u ch a s c o n t r o l c u r r e n t t oplunge r fo r c e a nd p lunge r po s i tio n to p lunge rfo r c e . This la t te r i s a s s o c ia te d wi th the a xia lfo r c e s impo s e d o n s l id ing va lve p lunge r s a sthey d ive r t the f lo w. Exp e r im e nta l de te r minatio n of both the com ponent and o ver all

no nlin ear i t ies is quite feasible if one isa le r te d to lo o k fo r the m. C o mp r e s s ib i l i ty o fo i l , e i the r f r o m i ts o wn bulk mo dule s o r f r o me n t r a p p e d a i r , i n t r o d u c e s n o n l i n e a r i t i e swhic h a r e de pe nde nt upo n the va r io us vo lume sthr o ugho ut the s ys te m . C lo s e ly a s s o c ia te d

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w i th t h i s i s t h e a p p a r e n t c o m p r e s s i b i l i t yi n t ro duc e d by e xpa ns i o n o f a c t ua t i ng c y l i nde rsa nd l i n e s u n d e r p r e s s u r e l o a d s . In g e n e r a l ,t he s e e f f e c t s a pp e a r a s pha s e l a gs a nd unw a n t e d r e s o n a n c e s .

c . M e c h a n i c a l N o n l i n e a r i t i e s

T h e u s u a l m e c h a n i c a l n o n l i n e a r i t i e s o fd e a d s p o t , d r y f r i c t i o n , p r e l o a d s , e t c . , e n t e rm o s t e n g i n e e r s ' t h i n k i n g a t o n c e . O n e w h ic h i sl e s s o b vi o us i s t he s t e p by s t e p o ut put vo l t a geo f a w i r e w o u n d p o t e n t i o m e t e r a s t h e b r u s hi s m o ve d s m o o t hl y f ro m o ne e nd t o t he o t he r .I n h i g h g a i n s e r v o s y s t e m s , t h e s e s t e pf unc ti o ns c a n e xc i t e r e s o na nc e s o r le a d t o lo wa m p l i t u d e o s c i l l a t i o n s a s t h e s y s t e m o s c i l l a t e s f ro m t u rn t o t u rn . F i g' . 13 s ho w s t her e s u l t s o f a n a n a l o g s t ud y o n c o n t r o l s y s t e m sh a vi ng s u c c e s s i v e l y c o a r s e r r e s o l u t io n . T h ec o n t i n u e d o s c i l l a t i o n s w e r e a l s o e n c o u n t e r e di n f li ght o c c a s i o na l l y .

d . G e o m e t r i c a l N o n l i n e a r it i e s

M a n y n o n l i n e a r i t i e s a r i s e p u r e l y f r o mg e o m e t r i c a l c o n s i d e r a t i o n s w h i l e o p e r a t i n go v e r l a r g e a n g l e s . F o r e x a m p l e , t he c o n v e n t io n a l a e r o d y n a m i c e q u a t io n s a s s u m i n g s m a l la ngl e s , s o t ha t t he c o s i ne e qua l s o ne a nd t hesine e qu als the an gle, m ay not be val id form i s s i l e s o p e r a t i n g a t h ig h a n g l e s o f a t t a c k .I n h a n d l i n g t h r e e - d i m e n s i o n a l p r o b l e m s , t h ea c t u a l d i r e c ti o n c o s i n e s o r E u l e r a n g le t r a n s f o r m a t i o n s m u s t b e u s e d w h ic h a r e n o n l i n e a r .T h e f r e e g y r o g im b a l n o n o r t h o g o n a l it y m e n t io ne d e a r l i e r i s a c a s e in po i n t . M a nym i s s i l e t r a j e c t o r i e s e n t a i l l a r g e c h a n g e s int he ve lo c i t y ve c t o r w hi c h i nvo l ve t h r e e -d i m e n s i o n a l p r o b l e m s .

A l e s s o b v i o u s g e o m e t r i c a l n o n l i n e a rpr o bl e m i s t ha t invo l ve d i n t he d i f f e re nt i a t i o no f p e r i o d i c f u n c ti o n s . F o r e x a m p l e , r a t h e rt ha n i ns t a l l a ra t e g yr o t o p ro vi de da m pi ng in

a c o n t r o l s y s t e m , o n e s o m e t i m e s d i ff e r e n t i a t e s t he po s i t io n s i gn a l . U s ua l ly o net a c i t l y a s s u m e s t ha t t he po s i t io n o ut put i sl i ne a r , a nd i t i s us ua l l y f o r t he w o rki ngr a n g e . H o w e v e r , f o r l a r g e r a n g l e s , t h e

po s i t i o n p i c ko f f i s us ua l l y pe r i o di c , put t i ngo u t e i t h e r a s i n e w a v e o r a t r i a n g u l a r w a v e(Fig. 14) .

Wi t hi n the ra n ge ± 45 de gr e e s , bo t h t hea ppr o xi m a t e m a gn i t ude a nd t he s i gn o fd ( A n g le ) /d t = d ( V o l ta g e ) / d t a r e c o r r e c t . B e t w e e n ± 90 de gr e e s , t he s i gn i s a t l e a s tc o r r e c t . B ut b ey o n d 9 0 d e g r e e s , th e s i gn o fd ( V o l t a g e ) / d t h a s r e v e r s e d , e v e n t h o u g hd( A ngl e ) / d t i s t he s a m e . Thi s m e a ns t ha t int h i s r e g i o n t h e d a m p i n g v o l t a g e h a s r e v e r s e da nd i s no w unda m pi ng t he s y s t e m . A c yc l i c

i ns t a bi l i t y , a l t ho ugh no t s i nus o i da l , i s us ua l l yt he re s ul t w he n o ne g e t s pa s t t he 90 de g re ep o i n t s . F o r t h e t r i a n g u l a r w a v e , t h e s y s t e mb e h a v e s l i n e a r l y o u t t o 9 0 d e g r e e s . F o r t hes i ne w a ve , t he da m pi ng d r o ps o ff a s o nea p p r o a c h e s 9 0 d e g r e e s b e c a u s e o f t h e r e d u c e ds l o p e .

e . Sa t ura t i ng

A f re que nt t ype o f s ys t e m no nl i ne a r i t y i st ha t a s s o c i a t e d w i th t he s a t u ra t i ng , i . e . ,

re ach ing the l im it of a given quan t i ty, event ho ugh w i t h i n t he o pe ra t i ng ra nge t he qua nt i t ym a y b e l i n e a r . T y p i c a l e x a m p l e s a r e :

( 1 ) T h e s a t u r a t i n g o f i n s t r u m e n t ss u c h a s a c c e l e r o m e t e r s a n d r a t eg y r o s w h e n t h e i r o p e r a t i n g r a n g ei s e xc e e de d. S t o ps a r e us ua l l ypre s e nt t o a vo i d da m a gi ng t hei n s t r u m e n t s u n d e r s u c h c o n d i t i o n s .

(2) C o nt ro l s u r f a c e l i m i t s in e i t h e r

p o s i t io n , v e lo c i t y, o r a c c e l e r a t i o n . The f o rm e r i s a f unc ti o no f t he a i r f ra m e de s i gn, w hi l e t hel a t t e r tw o a r e a c t u a to r c o n s i d e r a t i o n s .

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(3 ) In t en ti on al l i m i t e r s a r e of t enp l a c e d in c o n t r o l s y s t e m s s o th a td a m a g i n g m a n e u v e r s w i l l n o t b er e q u e s t e d . T h e s e m a y b e c o m man d l i m i t e r s on i np u t v o l t ag e o r

t h ey may b e t o rq u e l i mi t a t i on s ont h e c o n t r o l s u r f a c e a c t u a t o r s .

f. G e n e r a l C o m m e n t s

T h e a b o v e r e p r e s e n t t y p i c a l p h y s i c a l p h e nomena, the effects of which must be cons i d e re d in an y g i v en d es i g n . Ho wev er , on eneed not live with all of the m . If th ei r i llef fects a r e r eal i zed , so me may b e av oi d ed bysu i t ab l e d es i g n se l ect i o n . T h i s i s cer t a i n l y

t r u e o f o v e r l o a d s i n i n s t r u m e n t r a n g e s ,p r o p e r c h o i c e of p o t e n t io m e t e r r e s o l u t io n ,e t c . S o me t i mes t h e i ll e ffect s o f se v e r a l maycan c el . F o r ex a m p l e , a d ead sp ot i n t h esy st e m may s t op an o sci l l a t i o n wh ich wou l d b ep r e s e n t du e t o p o t e n t i o m e t e r r e s o l u t i o n . Ifthe adve r se effect of the dead spo t ona c c u r a c y c a n b e t o l e r a t e d , t h i s m a y b e a na c c e p t a b l e s o l u t i o n . In g e n e r a l , h o w e v e r ,t h i s i s p rob ab l y r i sk y b u si n ess , esp eci a l l y i ft h e n on l i n ear i t i es a re n o t s t ab l e wi t h t i mean d com p on en t ch an g es . T h e p oi n t o f t h es e

c o m m e n t s i s t h a t r e a l i z i n g a p r o b l e m ' se x i s t e n c e e i t h e r p e r m i t s a v o i d a n c e , o r th eb e s t c o m p r o m i s e w i t h i t .

4. RANDOM E F F E CT S

P r e v i o u s l y m e n t io n e d p r o b l e m s a r e o f af ix ed n a t u r e , t h at i s , t h e p ro b l em can b ed es cr i b ed o n ce an d fo r a l l and a so l u t io no b t a in e d. H o w e v er , t h e r e a r e p r o b l e m sar i s i n g f rom ran d o m effect s wh i ch mu st b e

c o n s i d e r e d o n a s t a t i s t i c a l b a s i s . T h e t h r e et o b e c o n s i d e r e d h e r e a r e n o i s e , d r i f t , a n dt o l e r a n c e s . O n e c a nn o t e l i m i n a t e t h e s ep r o b l e m s b y s p e c i a l d e s i g n b e c a u s e t h e y a r ea l w a y s p r e s e n t . R a t h e r , o n e d e s i g n s s o th a tt h ei r ef fect i s t o l e rab l e .

a . Noi se

N o i s e i s th e p r e s e n c e s o m e w h e r e i n t h el oo p , u su al l y on t h e i n t e l l i g en ce s i g n al s , o fu n w an te d r a n d o m d i s t u r b a n c e s . E x a m p l e s

a r e t h e s c i n t i ll a t io n o f r a d a r s i g n a l s , t h efad i n g an d s t a t i c on rad i o s i g n al s , p owers u pp l y n o i s e d ue to v i b r a t o r n o i s e o r s o m eg o v e r n o r a c t i o n , p o t e n t i o m e t e r s c r a t c h , e t c .All t h ese af fect co n t ro l in t h e sam e g en e r alw a y s :

(1 ) I n tr o d u c ti o n o f e r r o r s d u e t os a t u r a t i o n a n d r e c t i f i c a t i o n a s i l l u s t r a t e d i nF i g . 1 5 . T h e p re sen ce i n t h e sy s t e m of l i mi to f s o m e k i nd , s t r u c t u r a l , e l e c t r i c a l , e t c . ,cl ip s the to p off the no ise as shown by the

sh ad ed p o r t i o n s . S i n ce t h e b ot t om i s n o tc l i p p ed , t h e re su l t i s an e r r o r b et ween t h er e p o r t e d a p p a r e n t a v e r a g e a n d t h e a c t u a la v e r a g e .

(2 ) T h e ex ces s i v e u se of so me fo rmo f s y s t e m p o w e r , e i t h e r e l e c t r i c a l , h y d r a u l i c ,p r o p u l s iv e , o r a e r o d y n a m i c (d r a g ) . T h e r ereal ly is no need to fol low al l this noise witht he m i s s i l e b e c a u s e t h e t a r g e t o r d e s i r e dr e f e r e n c e c e r t a i n l y i s n ' t d o i n g i t . T r y i n g t od o so i s wast efu l o f en erg y .

Th e usu al m ethod of red ucing the effectsof n o i se wh en t h e sou rce can n ot b e e l i mi n at edi s t o p ro v i d e f i l t e r i n g of so me k in d . Va r i o u sk i n d s of n o t ch o r l ow -p ass f i l t e r s may b ee m p l o y e d .

b. Dri f t

Dri f t i s t h e i n ab i li t y of c i r cu i t s t o ma i n t a i n

cal i b r a t i o n ab o u t t h ei r DC l ev el . It may b eco n si d e re d t o b e a v er y low f req u en cy n o i seb ut i s men t i o n ed s ep a ra t e l y b ecau se i t a f fect sd esi g n s i n a d i f fe ren t m an n er . On e d o es n otf i l t e r d r i f t b ut op er a t e s a t DC l ev el s wh i cha r e such that drif t i s a sm all value of the

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c o n t r o l l e d q u a n ti ty . F o r e x a m p l e , a r e a s o n able value for dr if t on the f i r st g rid of a DCam p l i f i e r i s 5 0 m i l l i v o l t s . Cl e ar l y t h en , on emu st n o t op era t e a t i n p u t s i g n al l ev el swh e re 5 0 m i l l i v o l t s i s a s i g n if i can t p ar t o f

the input to the am plif ier . Such things ast e m p e r a t u r e , t i m e , v i b r a t i o n , t ub e a g in g , e t c . ,a l l ef fect d r i f t . T h er e a r e way s of s t ab i l i z i n gam p l i f i e r s wi th r esp ect to d r i f t su ch asc h o p p e r s t a b i l i z a t i o n . H o w e v e r , th e y a r e n o ta s s u i t a b l e f o r a i r b o r n e u s e a s f o r g r o u n di n s t a l l a t i o n s .

c . T o l e r a n c e s

W h e n o n e h a s s e l e c t e d c e r t a i n c i r c u i t ,m ech a n i cal , and h y d r au l i c v a l u es fo r u se i na c er t a i n d es i g n , i t i s a g r ea t t emp t at i o n t od eman d t h at t h e eq u i p men t b e man u fact u redt o t h ose v al u es . Un fo r t u n at e l y , n o man u fact u r i n g o rg an i zat i o n can fu n cti on i n t h i sma n n er b eca u se t h e re j ect i o n r a t e wou l d b ep r o h i b i t iv e . T h e r e m u s t b e s o m e t o l e r a n c eo n e v e r y d i m e n s i o n , c i r c u i t v a l u e , g a i n , e t c .Cl e ar l y , t h e f i n al d es i g n mu st b e cap ab l e ofma n u fact u r e an d , h en c e, th e a r t i c l e can n ot b econ si d ered d es i g n ed u n t i l i t can b e man u fact u red wi th i n t he l i mi t a t i o n s of ex i s t i n gm e t h o d s . T h e r e f o r e , in e s t a b li s h in g c i r c u i t s ,

o n e m u s t a ll o w f o r 1 p e r c e n t r e s i s t o r s t ob eco me 5 p er cen t wi t h ag e an d fo r am p l i f i e rg ai n t o ch an g e ± 3 d b . S t r u ct u ra l a l i g n m en t smu st b e wi t hi n t oo l i ng l i m i t s . T h ei r ef fect so n a e r o d y n a m i c d e r i v a t i v e s m u s t b e c o n s i d e re d an d , if u n acce p t ab l e , e i t h e r t h e b as i cd esi g n ch an g ed o r b et t e r man u fact u r i n gm e t h o d s c o n c e i v e d . T h e d e s i g n - m a n u f a c t u r i n g cy cl e i s a c l osed l oop , t oo , an d on epa rt m ust be al lo wed to feed back to theo t h e r .

5 . OP E RAT IONAL P ROBL E M S

T h er e a r e s t i l l i t em s wh ich mu st b ec o n s i d e r e d b e f o r e a d e si g n c a n b e c o n s i d e r e dco m p l et e . On ce ma n u fact u r ed , i t wi l l b e i n

op erat i on al u se an d i t mu st op era t e i ne n v i r o n m e n t s w h ic h a r e f o r e i g n t o t h e e n g i n e e r i n g d e p a r t m e n t . F u r t h e r m o r e , o n e m u s tc o n s i d e r t h e c o n s e q u e n c e s o f i t s n o t w o r k i n g .

a . E n v i r o n m e n t

T h e en v i ron men t wi t h i n wh i ch t h e sy s t emm u s t o p e r a t e m a y b e c o n s i d e r e d t o c o n s i s tof c l i mat i c , d y n ami c, an d ed u cat i on al en v i ron m e n t s . B y c l i m a t i c e n v i r o n m e n t s w e m e a nt e m p e r at u r e , h u mi d i t y , san d an d d u st , fun g u s,ra i n , sn ow, mu d , e t c . T o h ow man y of t h esei t e m s w i l l t h e s y s t e m c o m p o n e n t s b e s u b j ect ed ? If su b j ect ed t o cer t a i n co n d i t i on s ,t he y m u s t e i t h e r o p e r a t e o r b e p r o t e c t e d s ot h e con d i t i on s won ' t af fect t h em.

T h e d y n a m i c e n v i r o n m e n t m a y b e c o n s i d ered t o b e t h e sh ock , v i b ra t i on , an ds t e a d y - s t a t e a c c e l e r a t i o n s to w h ic h s y s t e mc o m p o n e n t s a r e s u b je c t e d . T h i s m u s t i n c l u d e sh i p men t , s t o rag e, an d f i n al u se con d i t i o n s .

T h e ed u cat i o n al en v i ro n men t i s t h e i n tel l ige nce and t rai ning level of the peo plewh o wi l l b e op era t i n g an d mai n t a i n i n g t h eequip m ent. It do es l i t t le goo d to put out theb est p oss i b l e sy s t em i f i t req u i res a t eam

o f d o c t o r s o f e n g in e e r i n g t o o p e r a t e it .Sim plici ty and rel iab i l i ty co mbined with al i t t l e h u man en g i n eer i n g a re t h e k ey n ot es .He re ag ai n , i t i s i mp o ss i b l e t o p ut d owng e n e r a l n u m b e r s o r e q u a t i o n s . T h e a b o v ee n v i r o n m e n t s m u s t b e c o n s i d e r e d an d a p p li e dwi th su i t ab l e m ar g i n s of safe ty t o eachs p e c i f i c c a s e .

b . Safety

All of this pa per thus far has been devo ted

t o d es i g n i n g th e sy s t em s o i t wi l l wo rk . Bu twh at if i t d o esn ' t wo rk o ccas i o n al l y on ce i tg et s in t o se r v i c e? T h e co n seq u en ces of su chfai l u re mu st b e con si d ered an d , p erh ap s , t h ed esi g n af fected acco rd i n g l y . F o r ex am p l e ,wh at i f t h e serv oamp l l f i e r o f an a i r l au n ch ed

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g u id ed m i ss i l e fa i l s a t lau n ch an d d r i v e s t h ec o n t r o l s u r f a c e s h a r d o v e r . C l e a r l y t hem i ss i l e sh o u l d not s t r i k e t h e l au n ch in g a i r c r a f t . P e r h a p s t h i s r e q u i r e s a t i m e d e la yb e f o r e t h e c o n t r o l s u r f a c e s c a n m o v e o r

b reak l i n k s t o a l l ow t h e con t ro l su rface t oc e n t e r u n d e r e x c e s s i v e lo a d . O t h e r p o i n t swh ere safe t y con si d era t i on s sh ou l d af fect t h edesign wil l undo ubtedly ar is e in any givenm i s s i l e s y s t e m .

6. CONCLUSION

In a sen se , t h i s p a p er h as h ad an o p p o rt u n it y t o p r ese n t l i t t l e mo r e t han a l is t o fp a s t d i f f ic u l t i e s. P r o p e r e m p l o y m e n t sh o u l d

p rev en t t h ei r b ecomi n g fu t u re d i f f i cu l t i es ,a t l eas t wi th ou t war n i n g . Un fo r t u n at e ly , t h er eis no co m plete l ist of the unknowns ofm i s s i l e c o n t r o l . F u t u r e e f f o r t s w o uld t he nb e s o e a s y .

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REFERENCEANGLE OF / * * ^

ATTACK

ACTUALLIFT 4

fWw^n^mmn^m^m^mmi^^

EXPECTEDLIFT

j ^ y x m ^ . ' . ^ :

ELASTICAXIS

L I F T

EXPECTED

^ACTUAL

iMd&ft&i&'iSt•^S'^-v^A*"S £fe

ANGLE OF ATTACK

Fig. 1. Static a e r o e l a s t i c i t y .

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CONTROLDEFLECTION

INPUT

*f"*t

t oC/i

POSITION

SERVO

RIGID

BODYDYNAMICS

RATE

GYRO

STRUCTURAL

DYNAMICS

< S H — ^

ACCELER

OMETER-<8H-

OUTPUT

F i g. 2 . Dynamic aeroelasticity feedback.

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FIRST MODE

SECOND MODE

THIRD MODE

F i g , 3 . Ben d in g mo d e sh a p e s .

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APPARENT MISSILE

TRANSLATION

APPARENT MISSILEROTATION

F i g. 4 . I ns t ru men t i nt e rp re t a t i o n of bendi ng.

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oo

AMPLITUDEPER CONTROL

DEFLECTION

FIRST MODE

SECOND MODE

THIRD MODE

- J ' ^ J J L W i ' '

FREQUENCY

F i g . 5 . T y p i c al s t r u c t u r a l r e s o n a n c e s .

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LARGE PERCENTUP MOTION

STILL ACHIEVED i

t o

COMMAND ASSUMEDWINGS LEVEL

I MOTION TO RIGHTRESULTS

F i g. 6 . C r o ss- t a l k example*

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PITCHINGMOMENT

isa s

II

III|

I

o

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15 —10 —5

ANGLE OFATTACK

^.^miWiW^w^iWSi^^mtm15

ACTUALCHARACTERISTIC

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F i g. 7 . P i t chi ng mome"ht no nl i near i t y a t sma l l angl es of a t t ack.

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CO

IUMBLINGPITCHINGMOMENT .

5 10 15 X 2 0

dE

RESTORINGPITCHINGMOMENT

CHARACTERISTICS

BANK

F i g. 8 . P i t chi ng mo ment no nl i near i t y a t bank angl e and hi gh angl e of a t t ack.

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coK5

2O

oz

oO k .

I

P.

BANK = 22.5°6 P = 6 Y = o °

^ * w * < y * * * , b L ^ w ; y w < . y ^

10 15 \ 20 25

PITCH ANGLE OF ATTACK ^BANK = 0°8Y = + 15°O p — ( O p ) TRIM

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COCO

i , ROLL POSITIONDEGREES

ORIGINAL

+ 10x1

rr

If i :TIME, SECONDS

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F i g. 10. I nduced ro l l t ra ns i en t s .

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IFLOW

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ACTUAL FLOWCHARACTERISTIC

EQUIVALENT LINEARG A I N - H I G HAMPLITUDES

LOW AMPLITUDES^ i ? * ^ ^ ^ ^ ^ ^ ^ ^

PLUNGER POSITION

F i g. 11. Hyd raul i c val ve f low nonl i nea r i t y .

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HIGH AMPLITUDEG A I N

LOW AMPLITUDEG A I N

F i g . 1 2 . Ny q u is t d i a g r am effect o f v al v e n o n l i n ear i t y .

135

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AILERONPOSITION I

NORMAL NUMBER TURNS

ONEDEGREE

ROLL RATE

£.NORMAL

1— NORMAL h

,f

250DEG/SEC

0.2SECOND

F i g . 1 3 . E ff ec t o f a i l e r o n p o t e n t i o m e t e r r e s o l u t i o n .

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OUTPUT

VOLTAGEm

CO *S#^^gSS&«l

ASSUMED LINEAR

OUTPUT

ACTUAL TYPICALOUTPUTS

INPUTANGLE

F i g. 14. T ypi cal pos i t i on pi ckof f out put s .

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CO00

UNACHIEVEDC O M M AN D S

TIME

Fig. 15. Effect of l imit ing on no ise.

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T HE E F F E CT S OF AIRF RAM E CHARACT E RIS T ICSON CONTROL SYSTEM DESIGN

F . E . P e r r y *

SUMMARY

T h i s p a p e r d e a l s w i th t ho s e a s p e c t s o f a i r f r a m e c h a r a c t e r i s t i c s w hi ch a ff ec t c o n t r o lsy s t e m d esi g n . Det ai l ed co n si d er a t i o n i s g i ven t o s t a t i c s t ab i l i t y , sh if t o f aer o d y n am i c p ro p e r t i e s w i th t i m e - o f - f li g h t o f t he m i s s i l e , n o n l i n e a r i t i e s , v a r i o u s ty p e s o f i n t e r a x i s c o u p l in g ,an d p aras i t i c feed b ack d u e t o s t ru ct u ra l e l as t i c i t y .

SOMMAIRE

C e t te n o t e t r a i t e d e c e s a s p e c t s d e s c a r a c t e r i s t i q u e s d e s t r u c t u r e e n v o l q ui a f f ec te n tI 'e tu d e d u sy s t e m e d e co n t ro l e . E l l e co n si d e re en d et a i l l a s t ab i l i t e s t a t i q u e , l e ch an g emen td e s p r o p r i e t e ' s a e ' r o d y n a m i q u e s a v e c l e n o m b r e d ' h e u r e s d e v o l d u m i s s i l e , l e s n o n - l i n e a r i t e ' s .

l e s d if f e r e n ts t y p e s d e c o u p l ag e i n t e r a x e s , e t la c o n t r e - r e a c t i o n p a r a s i t e d ue a 1'elasticited e l a s t r u c t u r e .

1. INTRODUCTION

Hi st o r i ca l l y , t h e p i l o t ed a i rp l an e h as b eend esi g n ed wi t h ch aract er i s t i cs wh i ch mak e i teas y for the human pi lot to con tro l and whichma ke i t inher ently stab le in the abse nce of ana u to m a t ic c o n tr o l s y s t e m . C o n t r o l s y s t e m sw e r e f i r s t ad d ed a s a d e s i r a b l e a c c e s s o r yfor the prevention of pi lot fat igue and, inl a t e r y e a r s , t o i m p r o v e t h e a c c u r a c y of f li gh tan d sp eed of resp on se u n d er cer t a i n t act i calco n d i t i on s . Ho wev er , ev en wi th au t o ma t i cc o n t r o l s y s t e m s in u s e , t h e d e s i r e f o r go o dst a t i c s t ab i l i t y remai n ed , b ecau se i t wasd esi rab l e t o f l y t h e a i rp l an e man u al l y a tt i m e s , an d a l so b ecau se of p oss i b l e fa i l u reso f t h e c o n t r o l s y s t e m .

In d e v el o p i ng a i r f r a m e s f o r m i s s i l e s w en eed t o rev i se ou r t h i n k i n g a b i t , recog n i zi n gcer t a i n fu n d amen t al d i f fe ren ces wh i ch ex i s tb et ween t h e mi ss i l e an d t h e p i l o t ed a i rp l an e,

a nd c h an g in g th e a i r f r a m e ' s c h a r a c t e r i s t i c sa c c o r d i n g l y . P r o p e r t i e s w h ic h w o u ld b e q u it ed e si r ab l e in a pi l o t ed a i r p l an e may wel l mak et h e j o b o f t he m i s s i l e c o n t r o l s y s t e m d e s i g n e rmore d i f f i cu l t .

P e r h a p s t he m o s t i m p o r t a n t d i ff e r e n c e

between the pi loted ai r pla ne and the guidedmi ss i l e i s t h at ( i n most cases) t h e mi ss i l e i sn o t r e q u i r e d t o o p e r a t e w i t h o u t a u t o m a t i cc o n t r o l . T o de s i g n a m i s s i l e w hi ch i s i n h e r en t ly s t ab l e wi th ou t co n t ro l wou ld b e u s el es s ,s i n ce a con t ro l sy s t em fa i l u re wi l l g en eral l yre su l t i n an ab o r t ed mi ss i o n an y way . If t h i si s rec o g n i zed i n t h e ear l y s t a g es of a m i ss i l ed e ve lo p m e n t p r o g r a m , t he o v e r a l l p e r f o r m an ce of t h e mi ss i l e can b e i mp rov ed , an d t h ej ob of con t ro l sy s t e m d esi g n can b e mad e l es sdifficult. The ability to fly stab ly witho uta u t o m a t i c c o n t r o l m a y p r o v e t o b e a li a b i li t y ,

r a t h e r t ha n a n a s s e t , f o r r e a s o n s w hi ch w i llb e d i s c u s s e d l a t e r .

•Boei n g Ai rp l an e Comp an y , S eat t l e , W ash i n g t on .

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An ot h er d i f fe ren ce i s t h at p i l o t ed a i rc raf ta r e g en e ra l l y d es i g n ed fo r p o s i t i v e l oadf a c t o r s d u r i n g m a n e u v e r s , w h i le t h e t e r m i n a lg u i d an ce ac cu r acy of a m i ss i l e may wel l b ei mp ai red i f t h e same ap p roach i s u sed .

A t h i rd maj o r d i f fe ren ce b et ween t h ep i l o t ed a i r p l a n e an d t h e gu i ded m i ss i l e l i esi n s t r u c t u r a l r e q u i r e m e n t s . T h e p e r f o r m a n c er e q u i r e m e n t s o f t h e m i s s i l e m a y b e m u c hm o r e s e v e r e t ha n t h o s e o f t h e a i r p l a n e ,req u i r i n g h i g h g ai n s i n t h e con t ro l sy s t em.Th is can lead to difficulty if st r uc tu r alreson an ces fa l l wi t h i n t h e p assb an d s ofc o n t r o l - l o o p e l e m e n t s .

It is the pur po se of this pap er to o ut l ineb r ie f ly s o m e o f t he a i r f r a m e c h a r a c t e r i s t i c s

wh i ch i n f l uen ce co n t ro l sy s t em d esi g n , an d t od e s c r i b e t h e i r e f f e c ts . S uc h a d i s c u s s i o nwou ld n ot b e com p l et e , o f co u r se , wi t hou tmen t i o n of so me of t h e t ech n i q u es fo r h an d li n gu n d e s i r a b l e c h a r a c t e r i s t i c s w h en t he y e x i s t ,b e c a u s e a n o p t i m u m d e s i g n f r o m a c o n t r o lsy s t em v i ewp oi n t wi l l g en er al l y n ot b e anop t i mu m d esi g n f rom t h e v i ewp oi n t s of t h ea e r o d y n a m i c s e n g i n e e r , t he p r o p u l s io ne x p e r t , a n d t h e s t r u c t u r a l d e s i g n e r .

Bec au se of t h e se co n f l i c t s , i t i s ab so l u t e l y

ess en t i a l t h at t h e v i ewp oi n t o f t h e co n t ro ls y s t e m d e s i g n e r b e r e f l e c t e d i n t h e p r e l i m i n a r y d e s i g n o f a m i s s i l e , s o th a t a r e a s o n a b l eb al an ce can b e ob t a i n ed bet ween t h e v ar i o u sfact o rs wh i ch g ov ern i t s ov era l l d es i g n . I fthe r e is any single idea which should beemp h asi zed , i t i s t h at con t ro l sy s t em con s i d er a t i o n s sh ou l d b e i n t eg r a t ed in t o t h edesign f rom the beginning, and not ignoredu n ti l th e a i r f r a m e d e s ig n i s c o m p l e t e . F a i l u r et o d o so may r esu l t in t h e ad d i ti on of i n nu m e r a b l e e l e c t r o n i c d e v i c e s , o r " b l a c kb o x e s , ' ' a t s o m e l a t e r d a t e , s o th a t t h e a d v e r s ec h a r a c t e r i s t i c s o f th e a i r f r a m e m a y b e o v e r co me . T h i s ad d s t o sy s t e m com p l ex i t y , wi thc o r r e s p o n d i n g d e c r e a s e in r e l i a b i l i t y , w h ic hi s on e of t h e most d i f f i cu l t p rob l ems t h em i s s i l e d e s i g n e r m u s t f a c e .

2 . AERODYNAMIC CONSIDERATIONS

T h i s s e c t i o n d e a l s w i t h s o m e o f t h e a e r o d y n a m i c p r o p e r t i e s o f a n a i r f r a m e w h i c hinfluence and l im it the desig n of a m is si le 's

a u t o m a t i c c o n t r o l s y s t e m . T h e m e t h o d s u s e di n o b t a i n i n g d e s i r a b l e c o n t r o l c h a r a c t e r i s t i c sa r e p r o p e r l y t he d o m a i n o f t h e a e r o d y n am i ci s t ; h en ce we wi ll n o t co n si d er i nd et a i l th e m ean s of ob t a i n i n g t h em. T h eintent is sim ply to indicate what the de si r ab lec h a r a c t e r i s t i c s a r e s o t h a t t h e y m a y a s s u m et h e i r p r o p e r p l a c e in t he p r e l i m i n a r y d e s i g np r o c e s s in w h ic h p r o p u l s io n , a e r o d y n a m i c ,a nd c o n t r o l f a c t o r s m u s t a l l b e c o n s i d e r e d .Thu s the d esign may evolve as a who le witha r e a l i s t i c b a l a n c e o f f a c t o r s .

a. Sta tic Stability

P o s i t i v e s t a t i c s t a b i l i t y h a s b e e n r e c o g n i zed fo r so me t i me as an esse n t i a l q u al it yof t h e p i lo t ed a i r c r af t . Ot h erw i se , ev en ift h e ra t e of d i v erg en ce were l ow en ou g h t op e r m i t h u m a n c o n t r o l , a m o m e n t o f i n a t t e n t ion on the part of the pi lot might well resul tin t h e a i r c r a f t a t t e m p t i n g t o t u r n e n d - f o r - e n d ,w i th d i s a s t r o u s r e s u l t s . O n t h e o t h e r h a nd ,

t h e mi s s i l e , wi th an au t o mat i c co n t ro l sy s t emi n con t i n u ou s op era t i on , i s n o t su b j ect t osu ch r es t r i c t i o n s . It can b e o p er a t ed wi thn eu t r a l o r ev en n eg at i v e s t ab i l i t y , s i n ce t h es t a t i c s t ab i l i t y of t h e a i r f r am e i s su p p lan t edby the dynamic stabi l i ty of the overal ls y s t e m . In t h e p r o c e s s , s o m e b e n ef ic i alr e s u l t s m a y a r i s e .

T h e d eg ree of s t a t i c s t ab i l i t y of an a i r f rame i s d et e rmi n ed b y t h e d i s t an ce b et weeni t s a e r o d y n a m i c c e n t e r a n d i t s c e n t e r o fg rav i t y (c . g . ) . L o cat i on of t h e a i r f r am ec. g . ah ead of t h e aer o d y n am i c cen t er r es u l t si n a r e s t o r i n g m o m e n t w h i c h t e n d s t o r e t u r nt h e a i r f r am e t o a n eu t r a l p o s i t io n wh en i t i sd i sp l aced f rom i t s t r i mmed con d i t i on , i . e . ,when a change in angle of at tack oc cu r s.

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Or stated differe ntly, the s lope of the pitchingmo ment vs . l if t coefficient cu rv es ar e suchtha t a no s e -do wn mo me nt e xis ts fo r no s e -upc ha nge s in a ngle o f a t ta c k f r o m the t r imme dcondition, and thus a negative s lope is

r e qui r e d , us ing the us ua l no ta t io n .

La r ge a mo unts o f s ta t ic s ta bi l i ty a r ege ne r a l ly u nde s i r a ble in a n a i r f r a m e in te ndeds o le ly fo r a uto ma t ic c o nt r o l , fo r the fo l lo wi n g r e a s o n s :

To s e e tha t a c o nt r o l le r i s c a pa ble o fpro viding s tab il i ty and damping which ar einter chan geab le with the s tabi l i ty and dampinginhe r e nt in the a i r f r a m e , le t us c o ns ide rf i r s t the s o -c a l le d f ixe d-s t ic k a i r f r a m e e qua

t io n fo r a ngula r m o t io n abo ut the p i tch a xi s ,a nd, s e c o nd, the e quat io n whic h de s c r ibe so n e p a r t i c u l a r t yp e o f c o n t r o l l e r w he nins ta l le d in the s a m e a i r f r a m e . Summa tio nof to rques about the pitch axis gives, for thea i r f r a m e o n l y ,

(1) Ea s e o f c o nt r o l a nd a la r gede gr e e o f s ta t ic s ta bi l i ty a r e mutua l ly inc o mp a t i b l e . R e l a ti v e ly l a r g e c o n t r o l - s u r f a c ea ngle s , whic h inc r e a s e the t r im dr a g, wi l lb e n e c e s s a r y t o t r i m t h e a i r f r a m e t o a g i v e n

l if t c o ndi t io n . Als o , no nl ine a r i t ie s ma y beint r o duc e d in the p i tc hing mo me nt vs . s u r fa c ede f le c t io n r e la t io ns hip , be c a us e o f la r ges u r f a c e a n g l e s . If l a r g e s u r f a c e s a r e r e q u i r e d ,f l u t t e r p r o b l e m s m a y r e s u l t .

i -s£i +yd t

2

dgdt

+ ce

( i )

w h e r e I y , B , C , a r e c o ns ta nts which r e p r e s e n tthe a i r f r a m e pi tc h mo me nt o f ine r t ia , it sdamp ing coefficient, and i ts s ta t ic s tab il i ty,r e s p e c t i v e l y .

(2) La r g e h inge mo me n ts , a nd c o ns e que nt ly la r ge a mo unts o f ins ta l le d a c tua to rt o r q u e m a y b e n e c e s s a r y t o p r o v i d e th en e c e s s a r y a i r f r a m e t r i m a n g l e s .

(3) La r g e r to ta l s ur fa c e a r e a wi l l be

r e q ui r e d to p r o vide a g ive n de gr e e o f s ta t icstabil i ty than would be re qu ire d if the sam ede gr e e o f dyna mic s ta bi l i ty we r e p r o vide dby a c o nt r o l le r ; c o ns e que nt ly the d r a g a nds t r uc tur a l we ight c a n be le s s if the c o nt r o l le rs uppl ie s the s ta bi l i ty .

(4 ) In a n e u t r a l l y - s t a b l e , t a i l - c o n t r o la i r f r a m e , the ta i l ma y pr o vide us eful l if t.If a la r g e d e gr e e o f s ta t ic s ta bi l i ty i s p r e s e nt ,th is i s no t po s s ib le fo r a l l t r i m c o ndi t io ns .

(5) In co nfigur ation s wh ere i t isde s i r a ble to us e the s a m e s ur fa c e s fo r . bo thpi tc h (o r yaw) a nd r o l l c o nt r o l , the la r g es ur fa c e t r im a ngle s r e qui r e d fo r s o me f l ightco ndit ion s could r esu lt in insufficient dif

fe r e n t ia l s ur fa c e de f le c tio n fo r r o l l c o nt r o l .

At sufficiently low freq uenc ies we mayi g n o r e s u c h th i ng s a s p o w e r - s e r v o , f i l t e r ,a nd c o n t r o l - i n s t r u m e n t p h a s e l a g s , a nda s s u m e the e xis te nc e o f a n ide a l c o nt r o l le rwho s e r e s po ns e i s s uc h tha t i t c a n ins ta nta ne o us ly e xe r t a c o nt r o l mo me nt , D, (due

to s ur fa c e de f le c t io n) p r o po r t io na l to thed i s p l a c e m e n t Q , an d a m o m e n t E , p r o p o r t io na l to the bo dy a ngula r r a te df l /d t . W enow have.

I & • +y d t 2

( B + E ) £ + (C + D)0 = 0.

(2)

The damping has been m odified by thera tio (B + E) /B and the s tat ic s tabil i ty bythe facto r (C + D) /C. It wil l be reco gnize dtha t the fa c to r s D a nd E a r e s ubje c t to c o nt r o lby adjustmen t of gain co nstan ts within thec o nt r o l lo o ps . Thu s , e ve n wi th ne ga t ive s ta bi l ity in the b a s ic a i r f r a m e , we c a n pr o videpo s i t ive s ta bi l i ty by me a ns o f the c o nt r o l

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s y s t e m . A l s o , in a p r a c t i c a l c a s e , t he d a m p i n g p rov i d ed b y t h e con t ro l l e r may b e an o rd erof mag n i t u d e o r m o re g r ea t e r t h an t h at wh i chi s i n h ere n t in t h e a i r f r am e a l o n e. T h e ab ov eca se cou l d b e ex t en d ed t o i n cl u d e s i g n a l s

p r o p o r t i o n a l t o b o dy a n g u l a r a c c e l e r a t i o n a swe ll . App are ntly , then, the sta t ic stabi l i tywhich is designe d into co nventio nal pi lotedai rp l an es sh ou l d b e ap p l i ed t o mi ss i l e d es i g nonly when we wish to f ly without automaticc o n t r o l . N e u t r a l s t a b il i ty i s a b e t t e r c h o i c e .

b . Cen t er o f Grav i t y T rav el an d S h i f t si n A e r o d y n a m i c C e n t e r

R e g a r d l e s s o f t h e c o m m e n t s a b o v e , t h e

d y n am i cs d es i g n of a co n t ro l sy s t em can b et a i l o re d t o an y on e of a wi d e ra n g e of p o s i t i v ea nd n e g a t iv e s t a b i l i ty v a l u e s . T r o u b l e a r i s e s ,h owev er , wh en l a rg e ch an g es i n s t ab i l i t yo ccu r d u r i n g f li g h t. T h ese ch an g es may s t emf r o m s u c h t h i ng s a s m o v e m e n t o f t h e a i r f r a m ec. g . as i t s d i sp o sab l e l oad i s u sed u p , o rf rom sh i f t s in t h e aer o d y n am i c cen t e r as th eM a ch n u m b e r v a r i e s . A c o n t r o l s y s t e m d e s i g nf o r s u c h a n a i r f r a m e m a y s u f f e r i n c r e a s e dc o m p l e x it y o r a c o m p r o m i s e in p e r f o r m a n c ef o r s o m e fl ig ht c o n d i t io n s . T h e r e f o r e , c o n s i d er a t i o n sh ou l d b e g iv en t o ch o o s i n g ana i r f r a m e c o n f i g u r a ti o n w h i ch a vo i d s t h e s ep r o b l em s b y h av in g t h e d i s t an c e b et ween t h ea e r o d y n a m i c c e n t e r a n d t h e e . g . r e m a i n a sn e a r l y c o n s t a n t a s p o s s i b l e .

T h i s i d eal a i r f r am e can on l y b e ap p ro ach edin v a r y i n g d e g r e e s in a c t u a l p r a c t i c e . T h u si t may b e n ecessary , as v el oci t y an d fu ell o a d i n g c h a n g e , t o p r o g r a m c o n t r o l s y s t e mco n st an t s as a fun cti on of t i me, o r t o co m p en s at e t h em a cco rd i n g t o so me fu ncti on ofs t a t i c a nd to t a l p r e s s u r e s . ( T h e l a t t e r m a y

b e n ec es sar y an yway b ecau se of ch an g es ina e r o d y n a m i c c o e f f i c ie n t s .) T h i s o f c o u r s efal l s i n t h e cat eg ory of a cu re ra t h er t h ant h e p rev en t i on of t h e i l l n ess itself, but it mayavoid diff icul ty els ew he re in the des ign. So

we ad d a comp en sat i n g d ev i ce t o t h e sy s t em,an d sy s t em comp l ex i t y an d p rob ab i l i t y off a i l u r e i n c r e a s e . A b e t t e r c h o i c e i s t o g iv et h e p r o b l e m s of e . g . t rav el and sh i ft i n ae r o d y n ami c cen t er carefu l co n si d er a t i o n i n t h e

p r e l i m i n a r y d e s i g n o f t h e m i s s i l e .

c . S y s te m N o n l i n e a r i t i e s

A n u m b e r o f n o n l i n e a r i t i e s n e c e s s a r i l yex i s t i n an y a i r f r am e and i t s co n t r o l l e r , int h e fo rm of su ch t h in g s as t h e s t r u c t u r a ll i mi t o f t h e a i r f rame, wh i ch can n ot b eexce ede d, and the deflect io n and veloc i tyl i m i t s o f c o n t r o l s u r f a c e s . In t he p a s t s e v e r a ly e a r s , m e t h o d s f o r d e l i b e r a t e l y i n t r o d u c i n gn o n l i n e a r i t i e s f o r o p t im u m s e r v o s y s t e mp e r f o r m a n c e h a v e r e c e i v e d c o n s i d e r a b l eat t en t i on . As y et we l l -p ro v en n o n l i n eard esi g n t ech n i q u es a r e n o t g en er al l y av ai l ab l e ,h o w e v e r , a n d s y s t e m p e r f o r m a n c e m a yr e q u i r e c o m p r o m i s e s w h e r e v e r n o n l i n e a r i t i es ap p e ar . An al og s may b e u sed t o s t u dyt h e ef fect s o f n o n l i n e ar i t i es , bu t t h e sy s t emd e s i g n u s u a l l y b e c o m e s m o r e c u m b e r s o m e .

As an a i r f r am e i s o p er a t ed a t h i g h er an dh i g h er a l t i t u d es , t h e p rob l em of d ev el op i n ge no u g h l ift t o p e r f o r m m a n e u v e r s d i c t a t e si n cr eas i n g l y l a r g e an g l es of a t t ack . Co n s e q u e n t ly , u n l e s s th e a e r o d y n a m i c c h a r a c t e r i s t i c s o f t h e a i r f r a m e a r e c a r e f u ll y in v e s t i g at ed t h rou g h o u t t h e en t i r e op era t i n ge n v e lo p e o f t h e m i s s i l e , s o m e r e g i o n s m a yb e e n c o u n t e r e d w h e r e t h e c o n t r o l s y s t e md esi g n j ob i s m o r e d i f fi cu lt . F i g . 1 i l l u st r a t e s o n e p o s s i b l e t y p e o f n o n l i n e a r r e l a t ionship between angle of at tack and pi tchingmoment which may exist at h igh pi tch anglesof a t t ac k . S i n ce t h e i n cr em en t al ch an g e ofpi tching mo me nt with angle of at tac k

d e cr ea se s a t t h e l a r g e r lif t v al u e s , an eq u i v al en t c h a ng e i n c o n t r o l - l o o p g a i n o c c u r s . S uc ha ch an g e may resu l t i n i n s t ab i l i t y u n l essc o n t r o l - s y s t e m p e r f o r m a n c e i s a ll o w e d t od e t e r i o r a t e b y u s i ng l o w e r g a i n s , o r u n l e s s

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c o m p e n s a ti n g d e v i c e s a r e a d d ed . A c o m pensat ing device might take the form of at r an s d u c er wh i ch se n s es an g le of a t t ack an dp rov i d es i n p u t s i g n al s t o a mu l t i p l i e r , i n t h ef o r m o f a s m a l l s e r v o o r a n e l e c t r o n i c

ci r cu i t . This devic e would then mo difysy ste m gain co nsta nts as a function of angleo f a t t a c k . A n o t h e r " b l a c k b o x " w o u ld t h e r e fo re b e ad d ed , an d sy s t em re l i ab i l i t y wou l dd e c r e a s e s t i l l f u r t h e r .

St i l l a thi rd opt ion is apparent , that ofl i mi t i n g th e a l lo wab l e m an eu v er s i n t h eq u e s ti o n a b l e a r e a s . T h i s , h o w e v e r , i s h a r d l ya d es i ra b l e so l u t i on . S t ep s sh ou l d b e t ak en ,if p o ss i b l e , d u r i n g t h e a i r f r am e d esi g n t oi n s u r e l i n e a r b e h a v i o r o v e r t h e e n t i r eo p e r a t i n g r a n g e .

d. Inter axis Co upling

T h e h ig h v e l o c i t i e s a nd r e s p o n s e r a t e swh i ch a re t y p i cal o f h ig h p er fo rm an ce m i ss i l ed es i g n s , cou p l ed wit h o p era t i on a t l a r g ean g l es of a t tack a t t h e h i g h er a l t i t u d es , h asc a u se d a i r f r a m e b e h a v io r to b e co m e i n c r e a s i n g l y c r i t i ca l t o i n t e ract i on b et ween t h e a i r f r a m e a x e s . T h u s i t b e c o m e s m o r e a nd m o r ed es i r ab l e t h at con s i d er a t i o n b e g iv en d u r i n gt h e p r e l i m i n a r y d e s i g n o f a n a i r f r a m e t o t h e

p a r a m e t e r s w h i c h c a u s e c r o s s c o u p l i n g .N e g l e c t i n g e l e c t r i c a l c r o s s c o u p l i n g , s i n c ew e a r e c o n c e r n e d h e r e o n ly w i th a i r f r a m ec h a r a c t e r i s t i c s , t h e i n t e r a c t i o n s b e t w e e n t h ev a r i o u s a i r f r a m e a x e s a r e o f t w o t y p e s :aer o d y n am i c an d i n e r t i a l . S i n ce b ot h t y p esc o n t r i b u t e t o u n w a n t e d a i r f r a m e m o t i o n s , i ti s a p p r o p r i a t e to c o n s i d e r e a ch m a j o r c r o s s -cou p l i n g t e rm an d t h e av ai l ab l e mean s fo re l i mi n at i n g i t o r red u ci n g i t s ef fect .

T ech n i q u es fo r ev al u at i n g t h e ef fect s o fc r o s s c o u p l i n g o n t h e c o n t r o l l e d o r u n c o n t r o l l e d a i r f r a m e i n c l u d e t h e a n a l y t ic a l s o l u t i on of t h e eq u at i on s of mot i on an d comp l et et h r e e - d i m e n s i o n a l a n a lo g s i m u l a t o r s t u d i e sw h ic h i n c lu d e a l l i m p o r t a n t c r o s s - c o u p l i n gt e r m s .

A c o m p a r i s o n o f t h e r e l a t i v e v a l u e s o f a l lt h e in e r t i a l a nd a e r o d y n a m i c c r o s s - c o u p l i n gt e r m s w i l l g en er al l y i n d i cat e t h at a n u mb erof them m ay be negle cted. Such a studys ho u l d c o n s i d e r t he f o ll o w in g t e r m s :

e . G y r o s c o p i c C r o s s - C o u p l i n g

Highs l e n d e rr e s u l tyawingof yawiing anda n g u l a r

ro l l v el oci t i es , cou p l ed wi t h l on gb o d i es fo r t h e red u ct i o n of d r ag ,

in an in cr ea se in the pi tching andmomen t s i n d u ced b y comb i n at i on s

ng and ro l l ing veloc i ty , and of pi tc h-r o l l in g v e l o c i t y , r e s p e c t i v e l y . T h ea c c e l e r a t i o n s d u e t o t h i s a c t i o n a r e :

Ix- lyI i

pq

(3)

an d

I z - I x Pr = q

(4 )

wh ere p , q , an d r a r e r o l l , p i t ch , an d y awa n g u l a r v e l o c i t i e s r e s p e c t i v e l y ; a n d I x , I y ,and I z are t h e momen t s of i n er t i a ab ou t t h emi ss i l e ro l l , p i t ch , an d y aw ax es .

Since any increase in body length inc r e a s e s Iy and I z , w h i l e s m a l l e r b o d y c r o s ss e c t i o n s d e c r e a s e I x , t h e n u m e r a t o r s o fEqs. (3) and (4) tend to increase in thes u p e r s o n i c m i s s i l e d e s i g n . T h u s , t h e e q u a t i o n s o f m o t i o n s h o u l d i n c l u d e t h e s e t e r m swh en an al y zi n g t h e p erfo rman ce of su chm i s s i l e s . It i s p r o b a b l e t h at o t h e r d e s ig n

con si d era t i on s wi l l p ermi t l i t t l e f l ex i b i l i t yi n c o n t r o l l i n g t h e s i z e o f t h e s e t e r m s , b u tt h e d es i rab i l i t y of k eep i n g ro l l ra t es as l owa s p o s s i b l e c o n s i s t e n t w i th t he r e q u i r e dm i s s i l e p e r f o r m a n c e i s a p p a r e n t .

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f. Ro ll Mo me nts Due to Side Velo ci ty

Rol l momen t s d u e t o s i d e v el oci t y may b ei n t rod u ced b y wi n g d i h ed ral , o r b y u n eq u all a t e ra l d rag ab ov e an d b el ow t h e l on g i t u d i n al

c e n t e r l i n e o f t he a i r f r a m e . T h i s c h a r a c t e r i s t i c i s u s u a l l y d e l i b e r a t e l y i n t r o d u c e di n a con v en t i on al a i rp l an e, so t h at t h e a i r plane wil l automatical ly bank such that i tss i d e v el oci t y i s re d u ced b y ro l l i n g th e v el o ci ty v ec t o r i n to t h e p i t ch p l an e. T h i s i sa p p r o p r i a t e , s i n c e l o a d f a c t o r s a r e g e n e r a l l yap p l i ed i n t h e p o s i t i v e d i r ect i o n i n t h e p i l o t eda i r p l a n e .

T h e t e r m i n a l p h a s e a c c u r a c y r e q u i r e me n t s of a m i ss i l e a r e of t en su ch t h at i t i sne ce ss ar y to develo p high val ues of l if t in

b ot h t h e p os i t i v e an d n eg at i v e d i r ec t i o n s . I tmay be shown by so lut ion of the f ixe d-st icke q u a ti o n s o f m o t i o n fo r a n a i r f r a m e w i th r o l ldue to yaw that i t wil l be unstab le for so m el if t co n d i t i on s an d n ot fo r o t h e r s . As a co n seq u e n ce, a h ig h d eg r ee of s t ab i l i t y i n ro l lf o r p o s i t i v e l o a d s i s n o t d e s i r a b l e , s i n c e t h i swi l l g en eral l y d et ract f rom t h e s t ab i l i t y oft h e a i r f r a m e u n d e r n e g a t i v e l o a d c o n d i t i o n s ;a nd t h e c o n t r o l s y s t e m m u s t o f c o u r s e becap ab l e of co n t ro l l i n g t h e m i ss i l e i n t h i sl a t t e r m o d e o f o p e r a t i o n .

Un l ess we recog n i ze ro l l d u e t o s i d ev el o ci t y as a p o t en t i a l so u r ce of t ro u b l e i nt h e d e s i g n o f t h e m i s s i l e ' s c o n t r o l s y s t e ma nd e x e r c i s e th e n e c e s s a r y c a r e , r o l l -y a wcou p l ed sy s t em i n s t ab i l i t y may resu l t d u r i n gfl ight. At be st , the co mp lexity of the sy ste mm a y i n c r e a s e a s a r e s u l t o f t h e c r o s sc o u p l i n g , r e q u i r i n g s u c h m e a s u r e s a s t h ec r o s s - f e e d o f e l e c t r i c a l s i g n a l s b e t w e e n t h er o l l a nd ya w s y s t e m e l e c t r o n i c s . A s i n t h ecase of negative longitudinal stabi l i ty , theco n t ro l sy s t em may b e ab l e t o ma i n t a i n s t ab l eflight o ver the de si r ed r ang e of negative l if tc o n d i ti o n s ; h o w e v e r , a m o r e s t r a i g h t fo r w a r da p p r o a c h i s to u s e a s y m m e t r i c a l a i r f r a m ewhen both posi t ive and negative load condit i o n s a r e t o b e e n c o u n t e r e d .

g. Ro ll Mo ment Due to Rudd erDeflect ion

In a i r f r a m e s h av in g r u d d e r a r r a n g e m e n t swh ich a r e n o t sy m m et r i ca l ab ou t t h e l on g i

t u d in al ax i s , r o l l i n g mo me n t s wi ll b e i n t r o d u ced b y ru d d er d ef l ect i on . T h u s , a sp u r i ou sro l l m o men t i s g en er at ed by t h e y aw -ax i sco n t r o l l e r ; an d th i s can ad v e rs el y affectsy s t e m s t ab i l i t y i f n o t t ak en i n to co n si d e r at i on i n t h e d es i g n of t h e con t ro l sy s t em.

S e v e r a l t e c h n i q u e s a r e a v a i l a b l e i n o v e r co mi n g t h e ef fect s o f ro l l -y aw cou p l i n g .One of these is to use relat ively high gainsi n t h e ro l l sy s t em, so t h at t h e a i l e ron sr e a c t r a p i d l y t o c o u n t e r a c t r o l l m o m e n t s .

Un fo r t u n at e l y , t h e s t r u c t u r a l s t if fn ess of t h ewi n g s wi l l g en er al l y i m p o se a p r act i c alu p p er g ai n l i mi t , d u e t o t h e f i rs t assy met -r i cal b en d i n g mod e of t h e wi n g s . E l i mi n at i on of ro l l d u e t o ru d d er d ef l ect i on can a l sob e accomp l i sh ed b y a i l e ron act i on i n an ot h erw a y . If t h e a i l e r o n s e r v o s a r e p r o p e r l ys i g n a l e d , th r o u g h th e r o l l c o n t r o l s y s t e m ,b y ru d d er mot i on , a i l e ron act i on may b eu s e d t o c o u n t e r a c t t h e r o l l t o r q u e g e n e r a t e dby the r ud de r . Me chanica l coupling wouldp r o b a b l y n o t b e d e s i r a b l e , s i n c e r u d d e rm o t io n w o u ld c o u n t e r a c t n o r m a l a i l e r o nmo t io n u n l ess a u n i l a t e r a l cou p l in g d ev i cecou l d b e d ev i sed . Re g ar d l e ss of t h e t y peof cou p l i n g u sed , wh et h er e l e ct r i cal o rm e c h a n i c a l , s y s t e m c o m p l e x i ty w i ll i n c r e a s e .

A t h i rd , an d p referab l e , so l u t i on t o t h ea e r o d y n a m i c r o l l - y a w c o u p l in g p r o b l e m ist o e l i m i n at e i t a t i t s so u r ce by mean s of as y m m e t r i c a l r u d d e r d e s i g n .

h. Yaw Due to Ro ll

Rol l i n g an a i r f rame wh i ch h as an an g l eof at tack , o r mushing velo ci ty , tends toro t a t e i t s mu sh i n g v el oci t y v ect o r i n t o t h ey aw p l an e, p ro p o r t i o n al t o t h e p rod u ct o f

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mu sh i n g v el oci t y an d ro l l ra t e . If ro l l d u et o y aw ex i s t s a l so , t h i s cou p l i n g resu l t s i na c l o sed l oop wh ich may b e u n st ab l e fo r s o mef l i g h t con d i t i on s , s i n ce ro l l mot i on p rod u cesfu r t h e r y awi n g. Red u ct i on of t h e mag n i t u d eof t h i s t e rm can b e ach i ev ed b y op era t i n gwi t h t h e l owest accep t ab l e r o l l r a t es , fo r ag i v en mi ss i l e d es i g n ; o r b y red u ci n g t h emu sh i n g v el o ci t y . P i v o t ed wi n g s fo r d ev e l o p ing li ft could be use d to avoid the nec ess i tyfor lar ge body an gles of at tac k, but with theat t en d an t d i sad v an t ag e t h at t h e b od y can n otbe used to produce l i f t .

O t h e r c r o s s - c o u p l i n g t e r m s w h i c h m a y b eo f i n t e r e s t a r i s e f r o m a i l e r o n de f le c t io n int h e p r e s e n c e o f a n a n g l e - o f - a t t a c k , w h e r e i na y awi n g momen t i s i n t rod u ced ; o r f rom s i d el o a d s o n a n u n s y m m e t r i c a l t a i l d u e t o r o l lr a t e . T h e s e t e r m s s ho u ld b e c o m p a r e dn u meri cal l y wi t h t h e o t h ers p resen t i n t h es y s t e m f o r t h e e x p e c t e d r a n g e o f m i s s i l emo t i on s , an d i n clu d ed in t h e t h r ee -d i men s i on al eq u at i on s if n e ce ss ar y . In t h ecas e of y aw d u e t o ro l l ra t e , a fin o r ru d d e rc o n f i g u r a t i o n w h i c h i s s y m m e t r i c a l a b o v eand below the m is si le lo ngitudinal ax is is ofc o u r s e d e s i r a b l e .

3 . STRUCTURAL CHARACTERISTICS

In h i g h - p e r f o r m a n c e m i s s i l e s , r e l a t i v e l yh i g h g a i n c o n t r o l l o o p s a r e n e c e s s a r y t oob t a i n t he sp eed of res p o n se and d eg r ee ofd a m p i ng o f m i s s i l e m o t io n w h i ch i s r e q u i r e d .T h e o v e r a l l d e s i g n o f a c o n t r o l s y s t e m f o rsu ch a mi ss i l e may b eg i n wi t h sep ara t es y s t e m d e s i g n s fo r t h e p i t c h - , r o l l - , a ndy a w - a x i s c o n t r o l s y s t e m s , u s i n g a n a lo gequipment and analyt ica l te chn ique s; and then

p r o c e e d t o a c o m p l e te t h r e e - d i m e n s i o n a la n a l o g c o m p u t e r s i m u l a t i o n o f t h e s y s t e m ,i n c l u d i n g a l l t h e i m p o r t a n t i n e r t i a l , a e r o d y n a m i c , a n d e l e c t r i c a l c r o s s c o u p l i n g s t h a ta r e p r e s e n t i n t h e s y s t e m . T h e s e s t u d i e sare b ased u p on t h e i n er t i a l an d aerod y n ami c

p a r a m e t e r s o f t h e m i s s i l e , a s o b t a in e d b ywei g ht cal cu l a t i o n s , wi nd t u n n el t e s t s , an dp e r h a p s p r e l i m i n a r y f l i g h t t e s t s o f s o m es o r t . A e r o d y n a m i c c o e f f i ci e nt s m a y o r m a yn o t r e c o g n i z e a e r o e l a s t i c e f f e c ts , d ep e n d in gu p o n t h e d e s i g n e r ' s a s s e s s m e n t o f t h e i ri m p o r t a n c e .

T h e o u t co me of t h e ab ov e s t u d i e s i s as p e c i fi c a ti o n o f c o n t r o l - l o o p t r a n s f e r f u n ct i on s b ased u pon t h e b eh av i o r of t h e mi ss i l e , u su al l y a ssu m ed t o b e a r i g i d b od y ,t h rou g h o u t t h e en t i r e v el oci t y and a l t i t u d eo p er at i n g en v el op e of t h e m i ss i l e . Hi ghp e r f o r m a n c e s y s t e m s m a y r e q u i r e b o t h bo dya n g u l a r r a t e s a nd th e i r d e r i v a t i v e s a s d a m p i ng s i g n a l s , i n o r d e r t o o b t a in t he d e s i r e dp e r f o r m a n c e .

T h e t r e a t m e n t o f th e a i r f r a m e a s a r i g i db ody in d efi n in g sy s t em r eq u i r em en t s i s al o g i c a l a n d n e c e s s a r y f i r s t s t e p . T h a t t h i sd e p a r t s c o n s i d e r a b l y f r o m t h e tr u t h w i ll b eap p a re n t , h o wev er , u pon ex ami n at i on of t h er e s o n a n c e c h a r a c t e r i s t i c s o f t he a v e r a g es t r u c t u r e . In s o m e s t r u c t u r e s e x c it a ti o nof co n t ro l i n s t r u m en t s b y b ody b en di n g o rl o c a l s t r u c t u r a l r e s o n a n c e s m a y i nt r o d u c ep a r a s i t i c l o o p s w h ic h i m p o s e m o r e s e v e r e

r e s t r i c t i o n s u po n s y s t e m g a i n s t ha n t h o s ei m p o s e d b y t h e c o n t r o l l o o p s t h e m s e l v e s .E x t er n al ap p en d ag es , su ch a s en g i n e p od sa nd t h e i r a t t a c h m e n t s , m a y a l s o f o r m r e s o n an t s y s t e m s w h ic h p r o d u c e t h e s a m e r e s u l t s .

In o r d e r t o v i su al i ze t h e n at u r e of th i sp r o b l em , a m i ss i l e i n f li gh t can b e co n s i d e r e d a s a s o - c a l l e d f r e e - f r e e e l a s t i c b a ro r b e a m . F o r c e s o r t o r q u e s a p pl ie d t o s u cha b e am e x c i t e t r a n s v e r s e v i b r a t i o n s w h i ch ,

in t h e s i mp l e c ase of a u n i fo rm b a r , a r egiven by the relat ionship (Ref. 1) ,

f = I T ( 3 . 0 I I 2 2 , 5 2 , 7 2 , 9 2 . X

(5)

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w h e r e

f = f req u en cy of v i b ra t i on

X - length of ba r (cm )

c = wav e v el oci t y in b ar m at e r i a l( c m / s e c )

k = rad i u s of g y ra t i on of b ar (cm)

T h e hi g h e r f r e q u e n c i e s a r e no t h a r m o n i c sof the fundam ental , even in the unifor m ba r ;an d mod e sh ap es of t h e f i rs t t wo f req u en ci esar e as shown in Fig . 2 . Only the o dd-n u m b e r e d f r e q u e nc i e s a r e s y m m e t r i c a lab ou t t h e cen t er o f t h e b a r . E x ci t a t i on oft h ese mo d es in t h e m i ss i l e bod y d u r i n gf li gh t m a y a r i s e f r o m s e v e r a l s o u r c e s : *

(1) A e r o d y n a m i c f o r c e s o r t o r q u e so n t he c o n t r o l s u r f a c e s .

(2) R e a c t io n t o r q u e s o n t h e s t r u c t u r ea s a r e s u l t o f c o n t r o l - s u r f a c ea n g u l a r a c c e l e r a t i o n s .

T h e c l o s e d p a r a s i t i c lo o p w h ic h c o n s i s t sof th e mi s s i l e b od y , t h e co n t ro l i n s t ru me n t ,t h e c o n t r o l s y s t e m e l e c t r o n i c s , a n d t h e c o n t ro l su rfa ce act u at o r , i s shown i n F i g . 4 ,

a l on g wi th t h e n o rm al co n t ro l l oo p . T h e i n flight gain m ar gin of this sys tem may bep re d i c t ed on t h e b as i s o f t es t s wh i ch a r ec a r r i e d o u t in t he l a b o r a t o r y . F i r s t , a t r a n s f e rc h a r a c t e r i s t i c b e t w e e n t h e c o n t r o l s u r f a c ean d co n t ro l i n s t r u m en t , u s i n g t h e i n er t i a lf o r c e p r o d u c e d by s u r f a c e a c c e l e r a t i o n a l o n e ,i s ob t a i n ed b y e l e ct r i ca l l y d r i v i n g t h eact u at o r . T h e ra t i o o f t o t a l i n - fl i g h t ex c i t at ion to that due to inert ial effects alone mayt h en b e u sed t o p r ed i c t i n - f li g h t b eh av i o r .

Excitat ion of the missi le body by ther e a c t i o n t o r q u e o f t h e s u r f a c e h a s no t p r o v e nt o b e t r o u b l e s o m e w i th in t h e a u t h o r ' s e x p e r i e n c e . H o w e v e r , if r e a c t i o n t o r q u e s w e r et ro u b l e so me , t h e in f lu en ce of t o rq u es an dl i n e a r f o r c e s c o u l d p r e s u m a b l y b e s e p a r a t e db y t e m p o r a r i l y p r o v i d i n g a r t i f i c i a l m a s sb al an ce ab o u t t h e co n t r o l -su r fac e h in g e l in eso that only a to rq ue wo uld be pro duced bys u r f a c e a c c e l e r a t i o n ; o r b y a p p l y i n g a l i n e a rfo rc e, o nly, to the f ixed sur fac e at the hingel i ne .

(3) L i n ear fo rc es ap p l i ed to t h es t r u c t u r e a s a r e s u l t o f c o n t r o l -s u r f a c e m a s s u n b a l a n c e a n da n g u l a r a c c e l e r a t i o n .

T h e g en er at i o n of t h ese fo rc es whi chp rod u ce b od y ex ci t a t i on may b e v i su al i zedby ref er r i n g t o F i g . 3 . Act u al mo d e sh ap e swi ll d if fe r som ewh at f ro m t h o se sh own i nFig . 2 bec aus e of the uneven st iffness and

m a s s d i s t r i b u t i o n o f t h e m i s s i l e b o d y .

* S ou rces of ex ci t a t i on , su ch as en g i n e n oi se ,which do not enter into a closed loop includi ng c o n t r o l e l e c t r o n i c s , a r e n e g l e c t e d .

T h e r a t i o o f t o t a l fo rc es wh ich ex i s t d u r i n gf li gh t t o th o se whi ch a r e p r esen t d u r i n g g rou n dt e s t s m a y b e r e a d i l y c a l c u l a t e d . T h i s r a t i o ,f o r s m a l l s in u s o i d a l a n g u l a r s u r f a c e sm o t i o n s , i s

Fm

+ F A l i f t ) 2 + Z g

m Jt O J 2

(6)

w h e r e

FA = a e r o d y n a m i c f o r c e d ue t o s u r f a c ed ef l ect i on

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Fm - ine r t ia l fo r c e due to s ur fa c ea n g u l a r a c c e l e r a t i o n

m - equivalent m as s at sur face e .g.

% - dis tan ce of sur face e .g. fro mhinge line

the l inear force coefficient perradian of surface deflection

<D - 2 v (frequency of su rf ace exci tatio n)

Recognizing that body bending may be ap r o b l e m , s e v e r a l m e a n s a r e a t o u r d i s p o s a lfo r a vo iding t r o uble :

(1) W e ma y de s ign the m is s i le s t r uc tu r e s uc h tha t i t s r e s o na nt f r e que nc ie s a r e a bo ve the pa s s ba ndo f t h e c o n t r o l - l o o p e l e m e n t s .

(2) W e ma y lo c a te the c o nt r o l ins t r uments , i f possible, a t nodalp o i n t s .

The in-fl ight b ehavio r of the sys tem can

the r e fo r e be p r e dic te d f r o m the r e s ul ts o f as imple te s t which is c a r r ie d o ut in the la bo r a to r y , p lus kno wn a e r o dyna m ic c o e ff ic ie nts .

(3) We may use spec ial metho ds for

pr o duc ing c o nt r o l s igna ls whic hr e p r e s e n t r i g id b o dy m o t i o n s ,but which a r e no t d i r e c t m e a s u r e m e n t s .

Since the pa ra sit ic loop gains ar e high onlyf o r h ig h -Q s t r u c t u r a l r e s o n a n c e s , t h e f r e que nc ie s o f c o nc e r n a r e o nly tho s e whichc o r r e s po nd to the na tur a l be nding f r e que n c ie sof the body.

An e xpe r ime nta l te c hnique fo r o bta in ingga in-m a r g in v a lue s whic h imm e dia te lya p p e a r s a t t r a c t i v e c o n s i s t s o f i n c r e a s i n g t h ega in o f the c o nt r o l lo o p e le c t r o nic s , unde rla bo r a to r y c o ndi t io ns , unt i l ins ta bi l i tyo cc ur s , and then mo difying the gain m ar ginfigure thus obtained by the rat io given byEq. (6) . This ma y o r ma y no t g ive the c o r r e c tr e s u l t s , s i n c e t h e i n e r t i a l f o r c e s i n c r e a s e(for a given am plitude) as the sq ua re of thes ur fa c e e xc i ta t io n f r e que nc y, whi le a e r o dyna mic fo r c e s r e ma in a lmo s t c o n s ta nt a t

low freq uenc ies . Co nsequently, the frequencyat which instabil i ty occurs due to the iner t ialfo rce alone is not n ece ssa r i ly that at whichinstabil i ty would occur during fl ight, s incethe s t r uc tur e wi l l ha ve s e ve r a l r e s o na ntm o d e s o f v i b r a t i o n .

With regard to (1) above, weight l imitat io ns m a y pr e c lude a s t r uc tur a l de s ign wi thr e s o na nt f r e que nc ie s o uts ide the pa s s ba nd o fthe e le m e nts in the s e r vo s ys t e m . Fi l te r ingc o u ld b e u s e d to d e l ib e r a t e l y d e c r e a s e p a s s -ba nds c o ns is te nt wi th the a l lo wa ble c o nt r o l -

loop phase shifts , but usually only with somed e g r a d a t i o n i n s y s t e m p e r f o r m a n c e . T h e r e f o r e a " q u i e t " l o c a ti o n f o r t he c o n t r o l i n s t r u m e n t s b e c o m e s a t t r a c t i v e .

R e fe r e nc e a ga in to Fig . 2 indic a te s a po s sible locatio n at point A wh ere body att i tud e,bo dy a ngula r r a te , a nd bo dy a ngula r a c c e le r a t io n me a s ur e me nts whic h a r e f r e e f r o m bo dyvibration at the fundamental body-bendingfrequency may be taken. Sim ilar ly, l inear

a c c e l e r a t i o n t r a n s d u c e r s c o u l d b e l o c a t e d a tpoint B. Unfo r tunately, no com mo n nodalpoint ex is ts for both the fi r s t and secon dbe nding f r e que nc ie s in e i the r c a s e . (Thismay be possible in the case of the actualm i s s i l e s t r u c t u r e , w h ic h i s m o r e co m p l e x. )

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On e p o ss i b l e ap p ro ach i s t h at o f co mb i n i n g(1) and (2) abo ve. The sele ct io n of o pt imumt r an sd u c er l oc at i on s may be su ffi c ien t a t f r equency f i ; and excita t io n at the seco nd bodyre so n an t f req u en cy can v er y l i k ely b e

at t en u at ed b y f i l t e r s , s i n ce i t wi ll p ro b ab l yl i e ab ov e t h e p a ssb an d s of t h e co n t ro le l e m en t s . At t en ti on h as n ot b een g i v en t ot h e h i g h er f req u en cy mod es , s i n ce t h e a t t en u a ti o n c h a r a c t e r i s t i c s of c o n t r o l i n s t r u m e n t s ,a c t u a t o r s , a nd o t h e r c o m p o n e n t s c a n b eexpe cted to pr eve nt diff iculty at the se f r eq u e n c i e s .

A secon d ap p r o ach wh i ch h as b een u seds u c c e s s f u ll y in o v e r c o m i n g p a r a s i t i c - l o o pi n s t ab i l i t y i s t h at o f ob t a i n i n g " c o m p u t ed "

bo dy a n g u l a r - r a t e a nd a n g u l a r - a c c e l e r a t i o ns i g n a l s fr o m c o n t r o l s u r f a c e d e f l ec t io n s o v e ra l i mi t ed f req u en cy ban d . F o r i n s t an ce, t h ea n g u l a r a c c e l e r a t i o n o f a m i s s i l e a b o u t i t sp i t ch ax i s i s g i v en b y t h e ex p ress i on :

9 -- M q q + M w w + Mg 8 E

(7 )w h e r e

q - p i t ch an g u l ar r a t e

S E = e l e v a t o r d e f l e c t i o n

w - mus hing velo ci ty

and M q , M g , a n d M w a r e th e a n g u la r a c c e l e r a t i on coef f i c i en t s d u e t o mi ss i l e p i t ch i n g ra t e ,e l e v a t o r d e f l e c t i o n , a n d m i s s i l e m u s h i n gv e l o c i t y , r e s p e c t i v e l y , f o r t he p a r t i c u l a ra i r f r a m e i n q u e s t i o n .

T h e p r e s e n c e o f th e t e r m M g • S E p r o v i d e sa m e a n s o f co m p u ti n g Q , to a c l o s e a p p r o x i ma t i on , if t h e o t h er t e r m s a r e su f fi c ien t lysm al l . (The co efficient M w w i l l b e r e c o g nized as depending upon the distance between

t h e a e r o d y n a m i c c e n t e r a n d t h e a i r f r a m ee.g. , and would be small for near neutral lys t ab l e m i ss i l es . ) In a s i m i l a r fash io n , i n t eg ra t i on of su rface d ef l ect i on wou l d p ermi t anap p ro x i mat i o n of b ody an g u l ar r a t e . T h e

m e t h o d d e s c r i b e d h e r e w o u l d n o t , o f c o u r s e ,b e su i t ab l e a t ze r o f req u en cy wh er e s t ead y -s t a t e s u r f a c e t r i m a n g l e s a r e r e q u i r e d .

S ev eral met h od s h av e b een ou t l i n ed ab ov efor o v erc o mi n g d i f fi cu lt i es asso ci a t ed wit hs t r u c t u r a l r e s o n a n c e s . T h e s e r e s o n a n c e s , iflow enough in f requenc y, ma y define the upperl i m i t o f c o n t r o l - l o o p g a in s d u e t o t he p r e s e n c eo f p a r a s i t i c l o o p s . M e t h o d s f o r o v e r c o m i n gthis difficulty usually lead to inc r eas ed c o n

t r o l s y s t e m c o m p l e x i t y ; t h e r e f o r e s t r u c t u r a lr i g i d i t y wh i ch i s comp at i b l e wi t h t h e p erfo rma n c e r e q u i r e m e n t s o f th e m i s s i l e s h o u ld b ed esi g n ed i n t o t h e a i r f rame i f p oss i b l e .

4 . CONCLUSIONS

In o r d e r t o av o id a d v e r s e c o n t r o l c h a r a c t e r i s t i c s w h i c h m a y l a t e r c o m p l i c a t e c o n t r o ls y s t e m d e s i g n , c a r e f u l c o n s i d e r a t i o n s h o u l db e g iv e n t o t h o s e m i s s i l e p a r a m e t e r s w h ic h

affect co n t ro l in th e p r e l i m i n ar y d es i g n p h aseo f a m i s s i l e p r o g r a m . T h i s i s p a r t i c u l a r l yd e s i r a b l e in th e c a s e o f h i g h - p e r f o r m a n c ei n t e r c e p t o r s , w h e r e a c c u r a c y an d s p e e d o fr e s p o n s e a r e o f p r i m a r y i m p o r t a n c e .

S t a t i c s t ab i l i t y req u i remen t s sh ou l d b erev iewe d o bject ively and should not be undulyi n f l u en ced b y con v en t i on al a i rp l an e p ract i ceu n l e s s i t i s d e s i r e d to o p e r a t e t he a i r f r a m ef o r p e r i o d s o f t i m e w i t h o u t a c o n t r o l l e r .At t en t i on sh ou l d a l so b e g i v en t o mi n i mi zi n gch an g es in s t a t i c s t ab i l i t y wh ich may r esu l tf r o m s h i ft s in t h e a i r f r a m e a e r o d y n a m i cce nte r with Mach num ber and shifts in e.g .wi th ch an g es i n fuel l oad i n g o r o t h er fo rm sof wei g h t d i sp osal .

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Also of m a j o r c o n c e r n are s uc h i te ms as

in te r a xis c o upl ings and n o n l i n e a r b e h a v i o r ofa e r o dyna m ic c o e ff ic ie nts . Th e s e may leadto s ys te m ins ta bi l i ty at s o me f l ight c o ndit io ns unle s s s ys te m de s ign and a n a l y s i s is

v e r y c o m p l e t e ; and at be s t wi l l c o mpl ic a tedesign or c o m p r o m i s e p e r f o r m a n c e .

S t r u c t u r a l c h a r a c t e r i s t i c s s h o u l d be c o m patible with the gain leve ls which are r e q u i r e dt o o b t a in t he d e s i r e d c o n t r o l s y s t e m p e r f o r m a n c e .

C o m p e n s a t i n g s c h e m e s can usually be

found for o v e r c o m i n g the effects of a d v e r s e

c o n t r o l c h a r a c t e r i s t i c s on s y s t e m d e s i g n .Unfo r tuna te ly , the s e le a d ge ne r a l ly in the

d i r e c t i o n of i n c r e a s e d s y s t e m c o m p l e x i t ywith its a t t e n d a n t d e c r e a s e in s y s t e mr e l i a b i l i t y .

5. ACKNOWLEDGEMENT

The e ffo r ts of m e m b e r s of the BoeingAir plane Com pany Applied Ph ysic s Staff,

A e r o d y n a m i c s Staff, and XIM-99 GuidanceP r o j e c t , in c o nt r ibut ing ide a s and r e vie wingt h e m a t e r i a l for t h i s p r e s e n t a t i o n , are g r a t e fully acknowledged.

R E F E R E N C E S

1. K i n s l e r , L. E., and F r e y , A. R., " F u n d a m e n t a l s of A c o u s t i c s , " W i l e y and So ns , 1950.

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x iuo 3Et o

A N G L E O F A T T A C K

F i g . 1. Un d esi r ab l e n on l i n ear re l a t i o n sh i p b etween p i t ch i n g mo men t an d an g l e of a t t ack .

LLI

UJ

o<_ lCLO

D I S T A N C E

F i g . 2 . T r a n s v e r s e b en d in g i n a u n if o r m e l a s t i c b a r .

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y .

T A t J = AERODYNAMIC PLUS INE RT IAL TORQUE

FA = SURFACE L IF T FORCE

=SURFACE ANGULAR ACCELERATION« SURFACE DE FLE CTIO N"FORC E DUE TO LINEAR

ACCELERATION OF EQUIVALENT MASS m.

= DISTANCE OF SURFACEe.g. FROM HINGE LINE

S

F

Fig . 3 . To rqu e s a nd fo r c e s due to c o n t ro l su rfa c e a ngula r a c c e le ra t io n a nd de fle c tio n .

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i

CONTROL SYSTEM

ELECTRONICS

PARASITIC LOOP

CO NTRO L

I N S T R U M E N T S

L. —, — — — -

ACTUATORS AND

CONTROL SURFACES " " 1

FORCE AN D/O R TORQUE [

RESONANT

STRUCTURE

AIRFRAM E

( R I G I D )

- T

11•

1|

NORMAL CONTRO L LOOP

F i g. 4 . P a ra s i t i c l oop due t o body bendi ng.

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GEOMETRICAL STABILIZATION BASED ON SERVODRIVEN GIMBALSAND INTEGRATING GYRO UNITS

C ha r le s S . Dr a pe r a nd R o ge r B . W o o dbur y*

SUMMARY

The pa pe r d e s c r i be s the ge n e r a l p r o ble m tha t mus t be s o lve d by ge o me t r ic a l s ta bi l iz a t ion equipment to be ca r r ie d by land, se a, and ai r veh icle s . A solutio n for the s tabil izatio np r o b l e m by m e a n s o f s i n g l e - d e g r e e - o f - f r e e d o m i n t e g r a t in g g y r o u n it s c a r r i e d by a t h r e e -de gr e e -o f-f r e e d o m s e r v o dr iv e n g imba l s ys te m is de s c r ibe d by ge o me t r ic a l and func tio na ldia gr a m s . A c o mp r e h e ns ive s y s te m o f c o nc e pts , te r mino lo gy , a nd s ymb o ls a da pte d fo r s e t t ingu p a c o m p l e t e m a t h e m a t i c a l f o r m u l a t io n o f th e p e r f o r m a n c e e q u a ti o n s f o r a t h r e e - g i m b a l -a xis s ta bi l iz a t io n s ys te m is appl ied to a n i l lus t r a t ive s ta b i l iz a t io n s ys te m . The de s ignf e a t u r e s a nd p e r f o r m a n c e c h a r a c t e r i s t i c s of s i n g l e - d e g r e e - o f - f r e e d o m i n t e g r a ti n g g y r o u n it ss ui ta ble fo r us e in s ys t e m s o f th is k ind a r e r e vie we d . Fr o m info r ma t io n s um ma r iz e d in thefunc t io na l d ia gr a m fo r a typica l g imba le d s ta bi l iz a t io n s ys t e m , the pe r fo r m a nc e e qua t io n fo r

motion about a s ingle gimbal axis is set up in a form that may be conveniently adapted for thec o m ple te s ys te m when dr ive s fo r a l l th r e e a xe s a r e in s imul ta ne o us o pe r a t io n . To i l lu s t r a teo ne a ppl ic a t io n o f the pe r fo r ma nc e e qua t io n , i t i s s ho wn tha t the us e o f ge a r - t r a in d r ive s fo rthe g imba ls i s no t a s fa vo r a ble fo r h igh-qua l i ty s ta bi l iz a t io n p e r fo r m a nc e a s the us e o f d i r e c t -d r i v e m o t o r s .

SOMMAIRE

C e t te no te de c r i t le p r o ble me ge ne r a l qui do i t e t r e r e s o lu pa r 1 ' e quipe me nt de s ta bi l i s a t io n g e o m e t r i q u e , d ev a nt e t r e t r a n s p o r t e p a r r o u t e , m e r e t a i r . U n e s o l u t io n a u p r o b l e m e d es ta bi l i s a t io n pa r le mo ye n d ' a pp a r e i ls g yr o s c o pique s in te gr a nt a un s e ul de gr e de l ibe r te p o r tepa r un d is po s i t i f a s s e r v i de m is e a d i r e c t io n f ixe a t r o is de gr e s de l ibe r te e s t de c r i te a 1'aide

de d ia g r a m m e s ge o me t r iqu e s e t fo nc t io nne ls . Un e ns e m ble c o m pr e he ns i f de c o nc e pts , dete r m ino lo g ie e t de s s ym bo le s a da pte s po ur une mis e e n e qua t io ns ma the ma t iqu e s c o mp le tede s e qua t io ns de pe r fo r m a nc e po ur un s ys te m e de s ta bi l i s a t io n pa r un d is po s i t i f de m is e ad i r e c t i o n f ix e p a r t r o i s a x e s , e s t a p p li q u e p o u r i l l u s t r e r u n s y s t e m e d e s t a b i l i s a ti o n . L e sc a r a c t e r i s t i q u e s e t u d i e e s e t l e s c a r a c t e r i s t i q u e s d e p e r f o r m a n c e d ' a p p a r e i l s g y r o s c o p i q u e si n t e g r a t e u r s a u n d e g r e d e l i b e r t e c o n v e n a bl e s p o u r 1'utilisation da ns de s s ys te me s de c ege n r e , so nt r e v ue s . A pa r t i r de 1 ' info r ma t io n r e s u m e e da ns le d ia g r a m me fo nc t io nne l d ' uns ys te me de s ta bi l i s a t io n typique p a r d is po s i t i f de mi s e en d i r e c t io n f ixe, 1'equation depe r fo r m a nc e po ur le de pla c e me nt s uiva nt un s e ul a xe de mi s e e n d i r e c t io n f ixe e s t e c r i teda ns une fo r m e qui pe ut e t r e a da pte d ' une ma n ie r e in te r e s s a n te po ur le s ys te m e c o mple t ,quand de s de pla c e me n ts s uiva nt le s t r o is a xe s s ' o pe r e nt s imu l ta ne m e nt . Po ur i l lu s t r e r uneapplication de 1'equation d e p e r f o r m a n c e , i l e s t m o n t r e q u e 1'utilisation d e t r a i n d ' e n g r e n a g e spo ur 1 ' e ntr a ine me nt de s mis e s en d i r e c t io n fixe n ' e s t pa s a us s i fa vo r a ble po ur une e xc e l le nte

p e r f o r m a n c e d e s t a b i l i s a t i o n q u e 1'utilisation d e m o t e u r s e n tr a i n a n t d i r e c t e m e n t .

• M a s s a c h u s e t t s I n s t i t u t e o f T e c h n o l o g y , C a m b r i d g e , M a s s a c h u s e t t s .

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1. INTRODUCTION

A c c u r a t e i n fo r m a t io n o n g e o m e t r i c a lo r i e n t a t i o n i s n e c e s s a r y f o r t h e g u i d a n c e a ndc o n t r o l o f m o d e r n v e h i c l e s . R a d i a ti o n li n k s

e xt e ndi ng t o o b j e c t s o u t s i de m o vi ng ve h i c l e sa r e us e f ul , but do no t , in ge ne ra l , s uppl yp r e c i s e a nd c o n t in u o u s g e o m e t r i c a l r e f e r e n c ei nf o rm a t i o n . D a t a o f t h i s k ind m a y be ba s e don an ine r t ial sp ac e, wi thout the need ofe x t e r n a l p h y s i c a l c o n n e c t i o n s , b y t h e a p p l i c a t io n of g y r o s c o p i c p r i n c i p l e s . T h i s i s p o s s i b l e b e c a u s e N e w t o n ' s l a w s o f d y n a m i c s a r et r u e w i th r e s p e c t t o i n e r t i a l s p a c e , s o t h a ta s p i nni ng ro t o r t e nds t o ho l d i t s p l a ne o fro t a t i o n i n t h i s s pa c e a nd m a y be us e d a s a no r i e n t a t i o n a l r e f e r e n c e f o r a ny n o n i n e r t i a l

c o o r d i n a t e s y s t e m t h a t e x i s t s i n a m o v i ngve h i c l e . Ex a m pl e s tha t i l l us t r a t e t h i s us e o fg y r o s c o p i c a c ti o n a r e c o m m o n in p r e s e n t - d a yf li ght i ns t r um e n t s f o r a i r c r a f t a nd na vi ga t io ne q u i p m e n t f o r m a r i n e v e s s e l s . A i r c r a f t t h a ta r e l i c e ns e d f o r a l l - w e a t he r f ly i ng m u s t ha veg y r o t u r n i n d i c a t o r s in a d d it io n t o c l i m b - a n d -ba nk i ndi c a t o rs , a nd a l l e xc e pt s m a l l s h i psc a r r y g y r o c o m p a s s e s . T h e p u r p o s e o f t h e s es e l f - c o n t a i n e d g y r o s c o p i c d e v i c e s i s t o e s t a b l i sh o r i e n t a ti o n a l r e f e r e n c e c o o r d i n a t es y s t e m s t h a t a r e c o n t i n u o u s l y a v a i l a b l e f o ri n d i c a t i n g o r a u t o m a t i c a l l y c o n t r o l l i n g o r i e n t a t i o n o f t he c ra f t i n w hi ch t he re f e r e n c e i sc a r r i e d .

U n t i l r e c e n t i m p r o v e m e n t s i n a i r - a n ds e a - v e h i c l e p e r f o r m a n c e i n t r o d u c e d m o r es t r i n g e n t r e q u i r e m e n t s o n o p e r a t i o n , g y r o s c o pi c de vi c e s de pe nde d o n t he p r i nc i p l e o fo v e r p o w e r i n g i n t e r f e r e n c e e f fe c t s in g i m b a ls u s p e n s i o n s b y t h e u s e o f h e a vy r o t o r ss pi nni ng a t h i gh s pe e d. R e c e nt de ve l o pm e nt sha ve s ho w n t ha t i t i s po s s i b l e to a c hi e veg e o m e t r i c a l s t a b i l i z a t i o n o f im p r o v e d q u a l it yf r o m s m a l l g y r o s c o p i c u n i ts t h at g e n e r a t ee l e c t r i c a l s i g n a l s f o r s e r v o d r i v e m o t o r s o fs u ff ic ie n t p o w e r to o v e r c o m e i n t e r f e r e n c ee f f e c ts in g i m b a l be a r i ng s . In p r a c t i c e , t h i s

g y r o s c o p e s e r v o d r i v e n g i m b a l c o m b i n a t i o nm u s t p r o v i d e t w o f u n c t i o n s :

(1) Stab i l izat io n, that is , ho lding ag e o m e t r i c a l r e f e r e n c e f r e e f r o m

r o t a t io n w i th r e s p e c t t o i n e r t i a ls p a c e in th e p r e s e n c e o f d i s t u r b i n g t o r q u e s a n d a r b i t r a r y m o v e m e n t s o f t h e s u p p o r t m e m b e rt h a t c a r r i e s t h e s t a b i l i z a t i o ne q u i p m e n t .

(2 ) R e s p o n s e t o o r i e n t a t i o n a l c o m m a n d s , t h a to r i e n t a t i o n

i s , changing theo f t h e r e f e r e n c e

m e m b e r w i t h r e s p e c t t o i n e r t i a ls p a c e in r e s p o n s e t o c o m m a n d

i n p u t s .

T h e a i r c r a f t t u r n i n d i c a t o r i s a n e x a m p l eo f a de vi c e t ha t p ro vi de s bo t h t he s e f unc t i o ns .Th i s i ns t ru m e nt s t a bi l i z e s t he i ndi c at i o n o fn o r t h o r a ny s e l e c t e d a z i m u t h d i r e c t i o na ga i ns t o s c i l l a t i o ns o f t he a i r p l a n e in w hic hi t i s c a r r i e d . It a l s o c h a n g e s t h e r e f e r e n c ed i r e c t i o n in r e s p o n s e t o c o m m a n d s f r o mp il o t d i r e c t o r e q u ip m e n t o r t o m a n u a l a d j u s t m e nt s o f a s e t t i ng kno b.

I t i s t he purpo s e o f t h i s pa pe r t o d i s c us st he m e c h a n i c a l f e a t u r e s a nd o p e r a t i n g c h a r a c t e r i s t i c s o f o r i e n ta t io n a l r e f e r e n c es y s t e m s b a s e d o n s i n g l e -d e g r e e - o f - f r e e d o mi n t e g r a t i n g g y r o u n i t s s u p p o r t e d b y s e r v o -dr i ve n g i m ba l s a nd de s i gne d t o be c a pa bl eof changing the or ientat ion of the indicatedg e o m e t r i c a l r e f e r e n c e in r e s p o n s e t oc o m m a n d i n p u t s . In o r d e r t o m a k e it p o s s i b l e f o r th e c o m p l e x g e o m e t r i c a l , i n s t r u m e nt a l , a nd dyna m i c a l s i t ua t i o ns i nvo l ve d t obe e f f e c t i ve l y unde rs t o o d a nd a na l yz e d, self-

defining notat ion and funct ional concepts areus e d in s e t t i ng up t he phys i c a l and t he o re t i c a las pe cts of the pr o blem (Ref . 1). As a mea nso f ke e pi ng t h i s pa pe r t o a re a s o na b l e l e ngth ,t h e p h a s e s o f o p e r a t i o n t h a t a r e a s s o c i a t e d

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w ith responses to command inputs w il l no tbe con sider ed, leaving the avai lab le spacef ree f o r developing the gen era l ized theo ryof one type of gimb aled s tabi l izat io n sy s tema n d a l i m i t e d d i s c u s s i o n o f c e r t a i n p h a s e so f i t s p e r f o r m a n c e .

2 . TH E P RO BLEM O F G EO METRI CA LSTABILIZATION

F i g . 1 s h o w s t h e e s s e n t i a l c o n c e p t s a s s o cia ted w ith ge o me tr i cal s tabi l izat io n w ithoutre gar d to the mec hanica l det a i ls by w hichw o rking equipm ent may be achieved inp r a c t i c e . T h e s u p p o r t m e m b e r i s t h e p a r to f a v e h i c l e t h a t c a r r i e s t h e s t a b i l i z a t i o nequipm ent . F o r conv enience in develo pingg e o m e t r i c a l r e l a t i o n s h i p s , a s e t o f r i g h t -

h a n d e d C a r t e s i a n c o o r d i n a t e s , X (sm) , Y( sm) ,Z(sm ) , i s a sso cia te d w ith the suppo r tm e m b e r .

In F ig . 1 , the ref e re nc e ax es , X ( r e f ) ,Xef) , Z ( r e f) , r e p r e s e n t t he r e f e r e n c e o r i e n ta t ion of a se t o f axes . X ( c r n ) , Y (cm) , Z ( c m ) ,f ixed to the co nt ro l led mem ber that w oulde x i s t i n a p e r f e c t l y o p e r a t i n g s t a b i l i z a t i o ns y s t e m . T h e r e f e r e n c e o r i e n t a t i o n t h a t i savailable in any actual system is given by

t h e i n d i c a t e d r e f e r e n c e a x e s , X (re f) (ind) >Y(ref)(ind) . Z ( r e f ) ( i n d ) , which ar e identic alw ith the axes f ixed to the co nt ro l led me m be r .In o p e r a t i o n , t h e d e s i r e d r e s u l t o f s t a b i l i z a t io n s y s t e m o p e r a t i o n i s t o k e e p t h e c o n t r o l l e dm e m b e r a x e s a c c u r a t e l y a l i g n e d w i t h t h er e f e r e n c e a x e s .

car r ies the gimbals and w hich, in turn, im o u n t ed o n a s u p p o r t m e m b e r . T h e g e n e r aar ra ng em en t is that desc r ib ed in U nitedS tates pate nts issued to the auth o rs o f thipa pe r (R efs. 2 and 3) .

T h e c o n t r o l l e d m e m b e r i s fi xe d t o t h

i n n e r g i m b a l . T h i s m e m b e r h a s r o t a t i o n af reedo m w ith res pe ct to the inner gimbaabout an axis o f sym m etr y, Z ( c r n ) , whichis the d i re ct io n cho sen f o r the z- a xis o f thi n n e r g i m b a l , Z (, g j . The inner gimbal ic a r r i e d b y t h e m i d d l e g i m b a l w ith r o t at ional f reedo m about an axis p erp end iculato Z( |g) . The z-a xi s of the mid dle gim bal

1 (mg) is taken a s id entical with the z- ax iof the inner gimb al . The y- a xis o f thmiddle gimbal , Y (mg ) , is taken along thaxis of ro ta t ion betw een the middle gimba

and the inne r gimb al . The x- a xis o f thmidd le gimb al , X (mgj , is chosen so that icompletes a r ight- handed set w ith Y ( m g)and Z( m g) and so that it is par all el to X (cm )w h e n Y ( c m ) i s p a r a l l e l t o Y ( m g ) .

T h e m i d d l e g i m b a l i s c a r r i e d b y a n o u t eg i m b a l w i th r o t a t i o n a l f r e e d o m a b o u t a d i r e ction at r ight angles to Y(mg) . T h i s d i r e c t i o n

is chosen as X (og) the x- axi s o f the o uteg i m b a l . T h e y - a x i s of t h e o u t e r g i m b a lY ( o g ) , is taken along Y ( m g ) ; and Z ( o g ) f o r m

t h e t h i r d m e m b e r of a r i g h t - h a n d e d - a x i sse t with X (og) and Y ( o g ) The axes of theb a s e w h ic h c a r r i e s th e g i m b a l s y s t e m , X ( b )

[ ( b ) • ( b l are ident ical w ith the commond i r e c t i o n s o f t h e g i m b a l a n d c o n t r o l l e dm e m b e r a x e s w he n a l l t h e g i m b a l s a r ero ta ted to po s i t ions in w hich a l l the x- ax esy - a x e s , a n d z - a x e s a r e a l i g n e d .

3, GIMBAL SYSTEM FOR CONTROLLEDMEMBER S U P P O RT

T h e d i a g r a m o f F i g . 2 s h o w s a t h r e e -g i m b a l s y s t e m f o r s u p p o r t i n g a c o n t r o l l e dm e m b e r h a v in g t h r e e d e g r e e s o f a n g u l a rf reedom w ith r esp ect to the base w hich

4. S I N G L E - D E G R E E - O F - F R E E D O MINTEGRATING GYRO UNITS

S t a b i l i z a t i o n o f a c o n t r o l l e d m e m b e r w i t hr e s p e c t t o i n e r t i a l s p a c e r e q u i r e s so m em e a n s f o r r a p i d l y a n d a c c u r a t e l y d e t e c t i n g

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of the c o n t r o l l e d m e m b e r w i t hto s o m e d e s i r e d i n e r t i a l s p a c e ref

c e o r i e n t a t i o n . B e c a u s e i n e r t i a l s p a c enot ha ve any n a t u r a l r e f e r e n c e s in

the f ixe d s ta r s , whic h are not

ul wi tho ut l ine -o f-s igh t info r m a t io n , any

for the c o n t r o l l e dof a s t a b i l i z a t i o n s y s t e m m u s t be

by e l e m e n t s w i t h i n the s y s t e mTT)is me a ns tha t gyr o uni ts for s t a b i l i

a ref

by o p e r a t i o nt he g y r o r o t o r s . S i n g l e - d e g r e e - o f - f r e e d o m

as r o t a a s ingle d i r e c t io n f ixe d to the

To g e n e r a t e the s i g n a l s r e q u i r e d to

f - f r e e d o m g y r o u n i ts are r e q u i r e do r a t h r e e - a x i s g i m b a l s y s t e m l ik e t h a t of

2. The g e n e r a l f e a t u r e s of g y r o u n i t sfor use in s y s t e m s of this kind are

S i n g l e- d e g r e e -o f - f r e e d o m i n t e g r a t i n g g y r oare a d e v e l o p m e n t of the I n s t r u m e n t a

at the M a s s a c h u s e t t s I n s t i of T e c h n o l o g y . T h e s e u n i t s h a v e b e e n

in p u b l i c a t i o n s ( R e f s . 4 and 5)

of s o u r c e s of t h e o and p r a c t i c a l i n f o r m a t i o n on g y r o

and i n s t r u m e n t s u s in g g y r oand are a l s o d e s c r i b e d in United6 and 7). A b r i e f

of the f e a t u r e s an d e s s e n t i a l p e r f o r m of the s i n g l e - d e g r e e -

m i n t e g r a t i n g g y r o u n it is includedto s ho w how u n i t s of t h i s t y p e o p e r a t e

of s t a b i l i z a t i o n s y s t e m s .

F i g . 3 is an i l l u s t r a t i v e p i c t o r i a l d i a g r a mthe f e a t u r e s of the M. I. T. i n t e

A g y r o r o t o r s p u n at

by an a l t e r n a t i n g - c u r r e n tis m o u n t e d in a g i m b a ln c l o s e d by a h e r m e t i c a l l y se a l e d c y l i n d r i c a lhell f i l led with helium to act as a n e u t r a l

a t m o s p h e r e and to s e r v e as a m e d i u m for

p r o d u c i n g an e v e n t e m p e r a t u r e d i s t r i b u ti o no v e r the p a r t s of the s h e l l . Th is f lo a te dg i m b a l has its we ight a djus te d so that it is

a l m o s t p e r f e c t l y s u p p o r t e d by b u o y a n t f o r c e sin a fluid that completely fi l ls the c l e a r a n c evo lume be twe e n the f lo a t a nd the he r me t ic a l lys e a l e d c a s e of the uni t . The float is pivo te dwithin the c a s e by w a t c h - j e w e l - t y p e b e a r i n g stha t , be c a us e of the lo w r e s i d u a l l o a d s , i n t r o d u c e s u b s t a n t i a l l y no f r i c t i o n to r e s i s t f l o a tr o t a t i o n .

T h e p i v o t s are c a r r i e d o n a x i al e x t en s i o n sf r o m the f l o a t . On one end, the e x t e n s i o nc a r r i e s the r o t o r of an a l t e r n a t i n g - c u r r e n ts i g n a l g e n e r a t o r of the m i c r o s y n (Ref. 8)

type tha t p r o duc e s s igna ls of the p h a s e

r e v e r s a l t y pe h av i ng m a g n i t u d e s p r o p o r t i o n a lto the a n g u l a r d e v i a t i o n of the f lo a t f r o m the

p o s i t i o n in which the o utput vo l ta ge has its

nul l le ve l . The end of the f lo a t a wa y f r o mt h e s i g n a l g e n e r a t o r c a r r i e s the r o t o r of a

m i c r o s y n t o r q u e g e n e r a t o r w h ic h r e c e i v e se l e c t r i c a l c u r r e n t i n p u t s and a p p l i e s a c o r r e s p o nding to r q ue to the f lo a t a bo ut the g imb a la x i s . B a l a n c e n u t s a d j u s t a b le f r o m o u t s i d et h e c a s e are u s e d to put the float into ac

c u r a t e r o t a t i o n a l b a l a n c e .

In any a c t u a l s i n g l e - a x i s g y r o u n i t , an u m b e r of f e a t u r e s m u s t be i n c o r p o r a t e dtha t , in the in te r e s ts of s i m p l i c i t y , are o m i t t e df r o m Fig. 3. For e x a m p l e , the flexible leadsus e d to m a k e e l e c t r i c a l c o n n e c t i o n s f r o m the

e x t e r n a l t e r m i n a l s to the g y r o d r i v e m o t o ra r e not s h o w n . It is i m p o r t a n t t h a t t h e t o r q u ea ppl ie d by t h e s e r o t o r - d r i v e p o w e r c o n n e c t i o n s to the g i m b a l be k e p t w i t h i n t o l e r a b l el e v e l s .

T h e r e q u i r e d p e r f o r m a n c e is a c hie ve d byus ing th in , f la t wi r e s of g o o d m e c h a n i c a lp r o p e r t i e s f o r m e d i n t o s e m i c i r c l e s w h i c h , in

t h e i r u n d i s t o r e d s h a p e , a c c u r a t e l y t o u c h the

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t e r m i n a l s o n t h e cas e an d on t h e g i mb al f l oa t .In p r ac t i ce , l ead - i n sy s t e m s wi th t h esef e a t u r e s i m p o s e s a t i s f a c t o r i l y lo w t o r q u e son t h e g i mb al . Neg l ect i ng t h e acc el er a t i o nt o r q u e ef fect s t h at ac t d u r i n g t ra n s i e n t

p er i od s , an an g u l ar v el oci t y of t h e u n i t caseabout the input axis, which is at r ight anglesto the gimbal axis and the spin axis when thesignal output is at i t s null level , ca us es theg y r o s c o p i c r o t o r o u t pu t t o r q u e t o a c t o n t heg i m b a l .

T h i s r o t o r o u tp u t to rq u e i s t h e re su l t o ft h e t en den cy of t h e r o t o r to t u rn to war d th einput axis thr o ugh the sm all es t angle in thed i rect i on t h at wou l d f i n al l y cau se t h e ro t o r

sp i n v elo ci t y t o h ave t h e sam e d i r ect i o n asthe input axis angular veloci ty i f the gimbalwere a l l owed t o t u rn i n t o comp l et e a l i g n men twith the input ax is . Stat ic equ il ibr ium isreach ed wh en t h e res i s t i n g t o rq u e ap p l i edt o t h e g i mb al by t h e i n t eg ra t i n g d am p er d u eto the fluid in the cle ar an ce between thef l oat an d t h e case j u s t b al an ces t h e g y ros c o p i c o u tp ut to r q u e . T h i s s i t ua t i o n o c c u r swhen the r at e of ro tat io n of the gim bal withresp ect t o t h e case i s su f f i c i en t t o g en erat ea r e s i s t i n g t o r q u e d u e t o v i s c o u s s h e a r i n g

action that is equal in magnitude to themag n i t u d e of t h e t o rq u e f rom t h e g y roscop i ce l e m e n t .

u n i t i s o p e r a t i n g p r o p e r l y * i s p r o p o r t i o n a l t ot h e an g u l ar d i sp l acemen t of t h e case ab ou tt h e i n p u t ax i s t h at occu rs d u r i n g t h e samet i m e i n t e r v a l .

In mak i n g p ract i cal ap p l i cat i on s , i t i sc o n v e n ie n t t o a s s o c i a t e c e r t a i n d e f i n i ti o n s ,c o n v e n t i o n s , a nd s y m b o l s w it h s i n g l e - d e g r e e -o f - f r e e d o m i n t e g r a t i n g g y r o u n i t s , w h i c h a r ea l s o c a l l e d s i n g l e - a x i s i n t e g r a t i n g g y r o u n i t s .T h e s e d e f i n it io n s a n d s y m b o l s a r e s u m mari zed i n F i g . 4 .

On e - fo rm of th e p erfo rm an ce eq u at ionfor the in tegr at ing gy ro unit given in Ref. 4ap p ear s as E q . (1 ) o f In fo rmat i on S u mm ary 1

in terms of the angular veloci ty input -v o l t ag e ou t p u t p er fo rm an ce fun cti on fo r t h eg y ro u n i t . In t h i s eq u at i on , t h e p erfo rm an cefunction has a role that is identical with thet r an sfe r fu ncti on (Ref. 9) o f L ap l ace t r an sfo rm t h eo ry . Of t h e sev er a l i n p ut s t o wh icht h e g y ro u n it re sp o n d s , E q . (1) i n cl u d es t h eeffects of only th r ee . The actuat ing inputsfo r the unit a r e the angular velo ci ty of thecase wi t h resp ect t o i n er t i a l sp ace ab ou t t h einput axis, which acts through the gyror o t o r , a nd t he i np ut c u r r e n t , w hic h s e r v e s

t o a p p l y o r i e n t a t i o n a l c o n t r o l c o m m a n d s t ot h e u n i t t h r o u g h t h e t o r q u e g e n e r a t o r .

The rate of change of the output vol tagef r o m t he s ig n a l g e n e r a t o r i s d i r e c t l y p r o p o r t i on al t o t h e an g u l ar v el oci t y of t h e g i mb alwi th re sp e ct t o th e cas e . T h e n et r esu l t o fthe act ion of al l the components except thet o r q u e g e n e r a t o r i s to p r o d u c e a r a t e o fchange of the gyro unit output signal that is

p r o p o r t i o n a l t o t h e r a t e o f c h a n g e o f t h e c a s eo r i e n t a t i o n w i t h r e s p e c t t o i n e r t i a l s p a c eabout the input ax is . Integr at io n of theinput ang ular velo ci ty and the o utput vo l tagerate shows that the change in the outputs i g n al fo r an y t i me i n t e rv al wi t h i n wh i ch t h e

An g u l ar a cce l er a t i o n of t h e ca se wit hr esp ec t t o i n e r t i a l sp ace ab ou t t h e ou t p u tax i s i s an i n t e rfe r i n g i n p u t t h at p rod u ces anu n d e si r ab l e ou t p u t s i g n al com p o n en t . T h i ss i g n al comp o n en t ex i s t s b ecau se t h e i n er t i are act i o n of t h e g i mb al c au s es i t t o l ag b ehi ndt he c a s e r o t a t io n a g a i n s t t h e a c c e l e r a t i n gt o rq u e of t h e v i scou s sh ear cou p l i n g b et ween

the case and the f loat .

*When the gimbal is in contact with stops setin the case, the damper is effect ively locked,an d t h e u n i t i s n o t op era t i n g p rop er l y .

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F o r t he p u r p o s e s o f s t a b i l i z a t io n , th ed es i r ed fu nct ion of a g y r o u n it i s t o p ro d u cean output signal that depends only on angularmo tion of the ca se about the input ax is . Theg y r o u n i t p e r f o r m a n c e e q u a t i o n i s r e s t r i c t e d

t o t h i s s i t u at i on by o mi t t i n g t h e i n p u t -cu r re n tt e rm f rom E q . (7) o f In fo rmat i on S u mmary 1 .T h e ab sen ce of o t h er t e r m s i n t h i s eq u at io nw h ic h w o u ld r e p r e s e n t v a r i o u s d r i f t -p ro d u ci n g t o r q u e co mp o n en t s t h at ex i s t i np r ac t i ce m ean s t h at such ef fects a r e assu m edt o b e n eg l i g i b le f rom t h e s t an d p oi n t o f s t ab i l i z a t i o n p e r f o r m a n c e .

E q s . (5) through (14) of Information Summ a r y 1 o u t l i n e t h e s t e p s r e q u i r e d t o c h a n g et h e g y r o p e r f o r m a n c e e q u a t i o n t o a f o r m

ad ap t ed fo r d escr i b i n g t h e b eh av i o r o f t h eg y r o u n i t u n d e r s t e a d y - s t a t e s i n u s o i d a l v a r i a t io ns of input and output qu ant i t ies . In afo r m t h at i s o f t en usefu l, t he p er fo rm an cefun cti on d ep en d s on t h e ch ar ac t er i s t i c t i m e.fo r c i n g f req u en cy p ro d u ct as t h e ess en t i a ld i m e n s i o n l e s s p a r a m e t e r . F o r c o n v e n ie n c e,t h e p er fo rm an ce fun cti on i s b ro k en down i n tot h e a n g l e - v o l t a g e r e f e r e n c e s e n s i t i v i t y , th ea n g l e - v o l t a g e d i m e n s i o n l e s s s e n s i t i v it yr a t i o , a nd th e a n g l e - v o l t a g e d y n a m i c r e s p o n s ea n g l e .

T h e d ef i n i t i on s an d sy mb ol s fo r t h eseq u an t i t i es a r e g i v en in co mp l et e fo rm fo rt h e i n t eg ra t i n g g y ro u n i t b ecau se t h ey i l l u st r a t e a p at t e r n t h at i s g en e ra l l y u sed fo rd e s c r i b i n g t h e p e r f o r m a n c e o f a l l c o m p o n e n t st ha t a r e r e q u i r e d to f o r m a g e o m e t r i c a ls t a b i l i z a t i o n s y s t e m .

T h e secon d t e rm wi t h i n t h e b rack et s ofE q . (9 ) of In fo r mat i on S u mm ary 1 sh ows t h eeffect of angular displacement about the

o u t p u t a x i s a s c o m p a r e d t o a n g u l a r d i s p l a c e men t ab ou t t h e i np u t ax i s . T h e o p e r a t o r ,p , indic ates that the effect is 90 de gr ee s outof p h ase wit h i np u t ax i s d i sp l ace m en t s . T h ed i m e n s i o n l e s s f o r c i n g f r e q u en c y - a n g u l a ra c c e l e r a t i o n - a n g u l a r v e l o c i ty r e f e r e n c e

s e n s i t i v i ty r a t i o p r o d u c t i s a m e a s u r e of th emag n i t u d e of t h e ou t p u t ax i s an g u l ar accel erat ion ef fect . In p r ac t i ce , t h e sen si t i v i t y r a t i oi s sm al l , so th at ou t p u t ax i s d i sp l ace me n tef fect s wi l l o rd i n ar i l y not b e se r i o u s ex cep t

at h i g h fo rc i n g f req u en ci es .

5 . ILLUSTRATIVE SINGLE-AXIS SERVO-DRIVEN CONTROLLED MEMBERWITH INTEGRATING GYRO

T o i l l u s t r a t e t h e b as i c act i o n of an i n t eg r a t i n g g y r o s c o p e s e r v o d r i v e c o m b in a ti o nfo r p ro v i d i n g s t ab i l i za t i o n . F i g . 5 i s d raw na s a l i n e s c h e m a t i c r e p r e s e n t i n g t he e s s e n t i a lf e a t u r e s o f a c o n t r o l l e d m e m b e r c a r r y i n g a

single integ ra t ing g yr o unit with i ts input ax isal on g th e ax i s of ro t a t i on of t h e ser v o d r i v esy s t em . W i th n o i np u t cu r r en t ap p li ed t o th egyro unit and the gimbal in the posi t ion atwhich the gyro output signal has i ts nulll ev el , an y ro t a t i on of t h e con t ro l l ed memb erabout the input axis wil l cause the gimbalt o t u rn wi t h re sp e ct t o t h e ca se abou t t h eo u t pu t a x i s . T h e r e s u l t i n g a n g u l a r d i s p l a c e me nt is picked up by the signal g en er ato ran d sen t t h ro u g h s l i p r i n g s to t h e p ow erc o n t r o l s y s t e m .

T h e ou t p u t of t h e p ower con t ro l sy s t ema c t s o n t h e d r i v e m o t o r t o p r o d u c e a to r q u eo n th e co n t r o l l e d m e m b e r t o c a u s e r o t a t i o nin t he p r o p e r d i r e c t i o n t o f o r c e t h e g y r og i mb al b ack t oward i t s n u l l -ou t p u t v o l t ag ep o si t i on . T h e con t i n u al rep et i t i on of t h i ss e r v o a c t i o n c a u s e s t h e c o n t r o l l e d m e m b e rt o re m ai n c l o se t o a p os i t i on t h at i s t h ei n e r t i a l s p a c e r e f e r e n c e o r i e n t a t i o n f o r t h eg y r o u n i t c a s e .

T h e r e f e r e n c e o r i e n t a t io n o f t h e c o n t r o l l e d m e m b e r m a y b e c h an g ed by i n t r o ducing a co mm and signa l in the for m of ani np ut cu r r en t t o t h e g y r o u n i t . T h i s cu r re n tc a u s e s t h e t o r q u e g e n e r a t o r t o ap pl y a t o r q u et h at s t a r t s t o t u r n t h e g i mb al away f rom i t s

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n u ll s ig n a l p o s i ti o n a n d p r o d u c e s a c o r r e s ponding gyr o unit output sign al . Th is signali s t h e i n p u t fo r t h e serv o sy s t em an d cau sest he d r i v e m o t o r to t u r n t he c o n t r o l l e d m e m b e ri n t h e p r o p e r d i r e c t i o n f o r t h e g y r o r o t o r t o

g en er at e an ou t p ut t o rq u e act i n g ag ai n s t t h eo u t p u t f r o m t h e t o r q u e g e n e r a t o r .

F o r a c o n s t a n t i n p u t c u r r e n t , t h e s t e a d y -s t a t e con d i t ion i s re ach e d wh en t h e se r v od r i v e s t h e c o n t r o l l e d m e m b e r a t a r a t e f o rwh i ch th e g y ro r o t o r t o r q u e ou t p u t j u s tb a l a n c e s t he t o r q u e g e n e r a t o r o u tp ut to r q u e .By mak i n g u se of t h i s ac t i on , t h e con t ro l l edm e m b e r r e f e r e n c e o r i e n ta t io n m a y b echanged at wil l . In any given si tuat io n, whent h e d es i red o r i en t a t i on i s reach ed , t h e i n p u t

c u r r e n t i s r e d u c e d t o z e r o ; t he g y r o u ni tserv o comb i n at i on wi l l t h en s t ab i l i ze t h ec o n t r o l l e d m e m b e r w i th r e s p e c t t o th e n ewi n e r t i a l r e f e r e n c e o r i e n t a t i o n .

6. GYROSCOPE AND SERVODRIVECOMBINATION FOR STABILIZATIONOF A GIM BAL -S UP P ORT E DCONT ROL L E D M E M BE R W IT H T HRE EDE GRE E S OF F RE E DOM

G e o m e t r i c a l s t a b i l i z a ti o n o f a c o n t r o l l e dm e m b e r w i th t h r e e d e g r e e s o f f r e e d o m m a yb e a c h ie v e d w it h t h r e e s i n g l e - d e g r e e - o f -f r e e do m g y r o u n i ts m o u n t ed o n a c o n t r o l l e dm em b er wi th t h ei r i np u t ax es a l on g t h r eem u t u a l ly p e r p e n d i c u l a r a x e s . T h e s e r v oact i on ab ou t each ax i s i s g en eral l y s i mi l a rt o t h at d escr i b ed i n t h e l as t sect i on ex cep tfo r t h e fact t h at, wi th t h r ee d eg re es of f r ee d o m , a n u m b e r o f g e o m e t r i c a l i n t e r a c t i o n sa m o n g t h e t h r e e c o m p o n e n t g y r o u n i t , s e r v o d r i v e co mb i n at i on s mu st b e t ak en i n to

a c c o u n t .

F i g . 6 i l l u s t r a t e s t h e r e s u l t of a s y s t e m a t i c p r o c e d u r e f o r o r i e n t i n g t h r e e g y r o u n i t sw it h r e s p e c t t o t h e g i m b a l - s u p p o r t e d c o n t r o l l e d m e m b e r s h o w n i n t h e d i a g r a m o f

Fig . 2 . The X- gyr o unit is f ixed with i tsinput axis pointed along the di rect ion ofX (cm ) , i ts output axis along Y(cm) , and itssp i n referen ce ax i s a l on g Z (cm ) . The f igures h o w s c o r r e s p o n d i n g o r i e n t a t i o n s f o r t h e

Y-g y ro u n i t an d t h e Z -g y ro u n i t .

In o r d e r t o u t i l i ze th e ou t p ut s i g n al sf rom an a r r an g em en t of g y r o u n i t s l ik e t h atof F i g . 6 , se rv od r i v es mu st b e mou n t ed sot h at s t ab i l i z i n g t o rq u es may b e ap p l iedb et ween t h e su ccess i v e g i mb al s of t h e sy s t em.In Fig. 6 , one drive is shown between themi d d l e g i m b al an d t h e i n n er g i m b al ; a seco n dd r i v e o p e r a t e s b e t w e e n t h e o u t e r g i m b a l a n dt h e mi d d l e g i mb al ; an d a t h i rd d r i v e a ct sb et ween t h e b a se an d t h e o u t er g i mb al .

7 . RESOLUTION OF GYRO UNIT OUTPUTSIGNALS FOR OPERATION OF THED R I V E S I N A T H R E E - D E G R E E - O F -FREEDOM GIMBAL STABILIZATIONSYSTEM

In sp ecti on of F i g . 6 i mm ed i at e l y sh o wsthat , with the exceptio n of the inner gim bald r i v e , t h e ou t p u t s i g n al s f rom t h e g y r o u n i t scan n ot b e d i r ect l y u sed as i n p u t s fo r t h e

s e r v o d r i v e s . T h e d r i v e a b o u t t h e z - a x i s o ft h e con t ro l l ed memb er i n comb i n at i on wi t ht he Z - g y r o u n it r e p r e s e n t s a s i m p l e o n e - a. xi ss i t u at i on l ik e t h at i l l u s t r a t ed i n F i g . 5 . T h eou t p u t f rom t h i s g y ro u n i t can b e d i rect l yc o n n e c t ed t o t h e Z - s e r v o d r i v e i np u t. O n t h eot h er h an d , t h e X- an d Y-g y ro u n i t ou t p u ts i g n al s mu st b e p rop er l y d i v i d ed b et weent h e mi d d l e g i mb al d r i v e an d t h e ou t er g i mb ald r i v e . T h e n at u r e of t h i s d i v i s i on of s i g n al sand the equations that must be fulfilled byt h e op era t i on of a s i g n al d i s t r i b u t i on sy s t ema r e d ev el op ed in Der i v at i on S u m mar y 1 .

T h e f i rs t s t ep of th e d ev el op m en t d es c r i b e d i n D e r i v a t i o n S u m m a r y 1 i s t o a s s u m ean an g u l ar v el oci t y of t h e con t ro l l ed memb erw ith r e s p e c t t o t he i n e r t i a l s p a c e r e f e r e n c e

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p osi t i on fo r t h i s m em b e r th at wou ld ex i s twi t h t h e ou t p u t s f rom al l t h ree g y ro u n i t ssim ultan eo usly at thei r nul l leve ls and thent o r e s o l v e t h i s a n g u l a r v e l o c i t y i n t o c o m p o n e n t s a l o n g t h e c o n t r o l l e d m e m b e r a x e s ,x

(cm) .Y

(cm) .a n d

Z ( c m j . F o r s t a b i l i z a t i on , t h e t h r ee s er v o d r i v e s wou ld h av e t on u l l i fy t h ese t h ree an g u l ar v el oci t y comp o n en t s on t he b a s i s o f t h e t h r ee s i g n al s f romt h e g y ro u n i t s t h at wou l d b e p r o p o r t i o n al tot h e an g u l ar v el oci t y comp o n en t ab ou t t h e t h re ec o n t r o l le d m e m b e r a x e s . T h e p r o c e d u r eou t l i n ed in Der i v at i on S u m ma ry 1 fo r f in d in gt h e r e q u i r e d s e r v o d r i v e a n g u l a r v e l o c i t yc o m p o n e n t s i s to p r o j e c t t he c o n t r o l l e dm e m b e r c o m p o n e n t s o n t o t he g i m b a l d r i v ea x e s . T r i g o n o m e t r i c e q u a t i o n s f o r t h e d r i v e

a x e s a n g u l a r v e l o c i ty c o m p o n e n t s a r e gi ve nas Eqs. (1) through (4).

T h e an g u l ar d ev i at i on co mp o n en t s of t h ec o n t r o l l e d m e m b e r w i th r e s p e c t t o t h e ref

e r e n c e o r i e n t a t i o n t h at d e v e lo p d u r i n g an yt i me i n t e r v al can b e fou nd b y in t eg r a t i o n sof th e an g u l ar v elo ci t y co mp o n en t s . W h ent h e s e i n t e g r a t i o n s a r e s t a r t e d f r o m a n i n s ta n twh en t h e con t r o l l ed m em b er i s a t r es t i nt h e referen ce o r i en t a t i on an d a t t en t i on i sr e s t r i c t e d t o th e s h o r t t i m e i n t e r v a l s r e

q u ir e d f o r a ny s e r v o d r i v e s t o o v e r c o m e ad ev i at i on i n on e d i rect i on , a l l t h e an g l esi nv ol v ed re m ai n sm al l . Bec au se of t h i s fact ,d e v ia t io n a n g l e s m a y be t r a n s f e r r e d f r o mc o n t r o l le d m e m b e r a x e s to s e r v o d r i v e a x e sby p r o j e c t i o n s s i m i l a r t o t h o s e us e d fo ra n g u l a r v e l o c i t y c o m p o n e n t s .

T h e e q u a t i o n s c o r r e s p o n d i n g t o t h e s ep r o j e c t i o n s a r e g i v e n i n D e r i v a t i o n S u m m a r y1 f o r t h e c o r r e c t i o n a n g l e c o m p o n e n t s , w h ic hrep resen t t h e an g l es t h rou g h wh i ch t h e con t r o l l e d m e m b e r w o u ld h a v e t o b e r o t a t e d ino r d e r t o b r i n g it i n to co i n ci d en ce wi t h i t sr e f e r e n c e o r i e n t a t i o n a s d e t e r m i n e d by n ul ll ev el s i g n a l s f ro m th e g y r o u n i t s . F i g . 6i n cl u de s r e p r e s e n t a t i o n s o f r e s o l v e r s .

mou n t ed so t h at t h ey recei v e an g u l ar d i sp l ac em en t s b et ween t he g i mb al s on e i t h e rside of the inner gimbal drive and the middleg im b a l d r i v e . T h e s e r e s o l v e r s a r e u se d t od i s t r i b u t e t h e X- an d Y-g y r o ou t p ut s i g n a l s

t o t h e mi d d l e an d o u t er g i mb al d r i v e s i na c c o r d a n c e w ith t h e r e l a t i o n s h i p s d e v e lo p e di n Der i v at i on S u mmary 1 .

8. FUNCTIONAL DIAGRAM FOR TH RE E-D E G R E E - O F - F R E E D O M G I M B A LSTABILIZATION SYSTEM

Fig , 7 gives a functional dia gr am for thet h r e e - d e g r e e - o f - f r e e d o m s t a bi li z a ti o n s y s t e m r e p r e s e n t e d i n F i g . 6 . T h e o u t e rgimb al is m ounted on the bas e with angu lar

f reed om ab ou t t h e X( 0 ) - a x i s . T h e m i d d l eg i m b a l i s s i m i l a r l y c a r r i e d b y t h e o u t e rgimb al with ang ular f reedo m about the Y ( o g ) -a x i s , and the inner gimbal , to which thec o n t r o l l e d m e m b e r i s r i g i d l y a t t a c h e d , i sca r r i e d b y t h e mi d d l e g imb al wit h an g u l ar

f reed om ab ou t t h e Z (mg) - a xi s .

T h e ou t er g i mb al d r i v e g ear t ra i n an d t h eo u t e r g i m b a l d r i v e m o t o r h a v e t h e i r h o u s i ng smounted on the base and apply torque to theo u t e r g i m b a l . T h e o u t e r g i m b a l s e c a n tmu l t i p l i e r (Refs . 1 0 an d 11) re cei v es t h eangular posi t ion of the middle gimbal withresp ect t o t h e ou t er g i mb al an d mu l t i p l i esan elect rical signal input by the secant of ther e l a t i v e g i mb al an g l e . T h e mi d d l e g i mb ald r i v e m o t o r a nd g e a r t r a i n c o m b i n at io nap p l i es t o rq u e t o t h e mi d d l e g i mb al , wi t h t h eo u t e r g i m b a l p r o v i d in g t h e n e c e s s a r yr e a c t i o n . T h e i n n e r g i m b a l d r i v e m o t o r a ndg e a r t r a i n c o m b i n a t i o n p e r f o r m s a s i m i l a rfunction for the inner gimbal with respect tot h e mi d d l e g i mb al . T h e mi d d l e g i mb alr es o l v e r i s o f t h e co n v en t ion al sy n ch ro t y pe

an d recei v es t h e an g l e of t h e i n n er g i mb alwi th re sp e ct t o t h e o u t er g i mb al an d mu l t i p l i eseac h of two el ec t r ic al input sign als by thes i n e an d cos i n e of t h e re l a t i v e g i mb al an g l er e s p e c t i v e l y .

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The x-a xis gyr o unit y-a xis gyr o uni t ,a nd z -a x is gy r o uni t a r e m o unte d o n thec o n t r o l l e d m e m b e r w i t h t h e o r i e n t a t i o n sshown in Fig 6. Th e ele t ica l output sign alfro m the z axis g yr o unit is used as the feed

ba c k input fo r the z -a xis d r ive po we r c o nt r o ls ys te m, whic h s uppl ie s vo l ta ge va r ia t io ns tothe inne r g imba l d r ive mo to r tha t de te r minethe to r que o utput o f th is mo to r . This d r ivepo w e r c o nt r o l s ys te m c o ns is ts o f f ive c o mpo ne nts c o nne c te d in s e r ie s . The pr e a mpl i f ie r r e c e ive s the t a xis gyr o o utputs igna 'vo ltage and chan ges i t to a for m su itable asthe input for the dem o dula to r . The output oft h e d e m o d u l a to r i s a d i r e c t - c u r r e n t v o l t a geinput fo r the mo dif ie r , whic h pr o vide s f i l te r ing a c t io n a nd in t r o duc e s the pha s e c ha nge s

ne c e s s a r y to g ive dyna mic s ta bi l iz a t io n fo rt h e i n n e r g i m b a l d r i v e s e r v o s y s t e m l o o p .

The po s s ib le a c t io ns o f e le c t r ic a l mo difying c i r c ui ts a r e d is c us s e d in ma ny bo o ksde a l ing wi th the the o r y o f e le c t r ic a l ne tw o r ksused in feedback s ys te m s (Ref. 12). Th et h e o r y o f s e r v o m e c h a n i s m s , w h i c h i s as pe c ia l pa r t o f fe e dba c k the o r y , i s t r e a te din many bo ok s (Refs 13 to 18) that m ay beus e d a s s o ur c e s o f info r ma t io n fo r de s igningthe e le c t r ic a l c o mpo ne nts a s s o c ia te d wi th

gimba l d r ive s . In Fig . 7 , the mo dif ie r r e pr e s e n t s an a r r a n g e m e n t o f e l e c t r i c a l c o m po ne nts to in t r o duc e the dyna m ic a l e ffe c tsn e c e s s a r y f o r a s a t i s f a c t o r i l y w o r k i n g s e r v o dr ive s ys te m. The o utput f r o m the mo d if ie ris the input for the remodulator , which, intur n , g ive s a n o utput tha t a c ts to ge ne r a tethe voltage input for the dr ive amplifier .This a mpl i f ie r r a i s e s the po we r le ve l a ndpr o vide s a n o tpu c ur r e nt tha t c a u s e s theinne r g imba l d r ive mo to r to a pply s ta bi l iz ingto r que to the inne r g imba l .

The feedback c hain s for the midd le gim baldr ive a nd o ute r g imba l d r iv e fo llo w thepattern of the closed chain for the innergimba l d r ive e xc e pt fo r the d i f fe r e nc e s

r e q u i r e d by t he t r i g o n o m e t r i c t r a n s f o r m a t io ns tha t a r e ne c e s s a r y to c o m bine the x-a x isa nd y-a xis gyr o o utput s igna ls in a c c o r da nc ewith the r e s ul ts o f De r iva t io n Summa r y 1 .The s e s igna l c o mbina t io n s in s ur e tha t the

middle g imba l s e r vo dr ive s y s te m a nd theo u t e r g i m b a l s e r v o d r i v e s y s t e m w i l l o p e r a t ewith loop gains that are not affected by relat ive po sit io ns of the gimb als as long as therelative angle of the middle gimbal withr e s pe c t to the o ute r g imba l r e m a ins s o me wha tl e s s t h a n 9 0 d e g r e e s .

9. STABILIZATION PERFORMA NCEEQUATION FOR INNER GIMBALDRIVE SYSTEM

T h e c o n t r o l l e d m e m b e r t e n d s t o m a i n t a i na f ixe d o r ie nta t io n wi th r e s pe c t to ine r t ia lspac e because of i ts own ine r t ia . I t can alsobe de mo ns t r a te d tha t no to r que s ne e d beapplied to the gim bals to main tain the co nt r o l le d me mb e r in a f ixed o r ie nta t io n in thepr e s e nc e o f a ngula r mo t io n o f the ba s e ,pro viding that the Z( c m ) and Z<b) - a x e s a r ee s s e n t ia l ly p a r a l le l a nd tha t I (0g)X - l(og)Y r

1/2 I(og)Z • (Th is imp lie s that the ou terg imba l i s e s s e nt ia l ly a r ing ) Unde r the s ec o n di t io ns , the d is tu r bing to r qu e s tha t mu s tbe canceled by the gimbal dr ive motor fal lin to the fo l lo wing c la s s e s :

(1) To r que c o m po ne nts due to no nvis c o us f r ic tio n in the be a r in gs .

(2) Vis c o us f r ict io n to r que c o mp o n e n t s d u e t o b e a r i n g s a nd a r m a t u r e r e a c t i o nt o r q u e s f r o m t h e m o t o r .

(3) To r que c o m po ne nts due to l ine a ra c c e le r a t io ns a c t ing o n unba la nc e d g im ba ls .

(4) To r qu e c o mpo ne nts r e q ui r e d toa c c e le r a te the d r ive ge a r t r a ins whe n theba s e i s s ubje c te d to a ngula r a c c e le r a t io ns .

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T h e p e r f o r m a n c e e q u a t i o n f o r t h e s t a b i l i zat i on of t h e con t ro l l ed memb er b y t h e i n n erg i mb al d r i v e sy s t em i s g i v en i n E q u at i onS u m ma ry 1 . If, i n ad d i t io n t o t h e p rev i o u sl ys t a t ed assu mp t i on s t h at t h e Z ( cm ) - and

Z ( o g ) -ax e s a r e p ar a l l e l an d t h at I ( o g ) x .Xg)Y . a r , d 1/2 I(og)z a r e eq ua l, i t is as su m edfor co n v en i en ce t h at th e x - , y - , an d z-a x e so f t h e v a r i o u s g i m b a l s a r e c l o s e t o b e i n gp a r a l l e l , t he n t he p e r f o r m a n c e e q u a t i o n s f o rt h e mi d d l e g i mb a l an d o u t er g i mb a l d r i v es y s t e m s m a y b e o b t a i n e d b y c y c l i c p e r m u t a t i o n o f t h e s u b s c r i p t s o n t h e v a r i o u s s y m b o l sto identify the ax es and the gim bal d r ive si n v ol v ed . T h e g y r o u n it ou t p u t s i g n al s fo rthe X(cm ) - and Y(cm) - a x e s m u s t b e c o m b i n e d b y t h e t r i g o n o m e t r i c r e l a t i o n s h i p s o f

Der i v at i on S u m ma ry 1 , an d t h e g i mb alm o m e n t s o f i n e r t i a m u s t b e i n tr o d u c e d a sr e q u i r e d b y d i ff e r e n t g i m b a l o r i e n t a t i o n s ,

E q . (1 ) o f E q u at i on S u mmary 1 con t a i n st e r m s fo r t h e i n er t i a l ef fects t h e d amp i n geffect s , an d th e e l ec t ro ma g n e t i c ef fects t h atc o m b i n e t o p r o d u c e th e r e s u l t a n t t o r q u e w h ic hd e t e r m i n e s t h e r o t a t i o n a l m o t i o n o f t h e c o n t r o l l e d m e m b e r . I n t e r f e r e n c e e f fe c t s a r et a ke n i n to a c c o u n t b y a s e p a r a t e t e r m r e p r e s e n t i n g t h e r e s u l t a n t u n d e s i r a b l e t o r q u e

c o m p o n e n t f r o m n o n v i s c o u s b e a r i n g f r i c t i o n ,s t i ck i n g of mech an i cal p ar t s , t h e act i on ofl i n e a r a c c e l e r a t i o n a nd g r a v i t y o n u n b al a nc e dm a s s e s in t h e c o n t r o l l e d m e m b e r a ndg i m b al s , an d t o rq u es t h at may act on t h es y s t e m f r o m a ny o t h e r s o u r c e s .

Eq. (2) is the form taken by Eq. (1) undercon d i t i on s i n wh i ch ro t a t i on al v el oci t i es an da c c e l e r a t i o n s a r e s o s m a l l th a t a l l t h e t e r m scon t ai n i n g p as a fact o r a re n eg l i g i b l e i nc o m p a r i s o n t o th e a n g u la r d i s p l a c e m e n t t e r m

on the left hand side of Eq. (1) and the interfe r en ce t o rq u e t e r m on th e r i g h t h and s i d e .In t h i s s i t u at i on , d y n ami c act i on s h av e n oappreciable effect , and the f requency funct io n s a s s o c i a t e d w i th t h e p e r f o r m a n c e f u nc t i on s b eco me eq u al t o u n i t y . T h i s m ean s

t h at t h e p er fo rm an c e fu n ct ion s r ed u c e t os e n s i t i v i t i e s . T h e r e s u l t a n t e x p r e s s i o n t h atap p ea rs a s E q . (2 ) sh ow s t h at t h e ra t i u o f t h ec o n t r o l l e d m e m b e r a n g le t o t h e i n t e r f e r e n c et o r q u e i s e q u a l t o t h e r e c i p r o c a l o f t h e p r o d u c t

o f t he s e n s i t i v i t i e s o f t he o p e r a t i n g c o m p o n en t s of t h e sy s t e m . T h i s ex p re ss i o n i s , i neffect , the elast ic coefficient of the system inres i s t i n g t o rq u e ap p l i ed i n t h e d i rect i on oft ak in g t h e co n t ro l l ed m em b er away f rom i t si n e r t i a l s p a c e r e f e r e n c e p o s i t i o n .

Eq. (3) is the form taken by Eq. (1) whent h e fo r c i n g mo t i on of t h e mi d d l e g i mb al h asa s t e ad y -s t a t e s i n u so i d al sh ap e of su ch h ig hf req u en cy t h at t h e t e r m s of E q . (1) d ep en d i ng on i n er t i a l ef fect s , i . e . , t h e t e r m s co n

taining p2

a s a f a c t o r , a r e m u c h l a r g e r t h a nal l t h e o t h er t e r m s. In t h i s ca se , E q . (1)r e d u c e s t o a r a t i o o f c o n t r o l l e d m e m b e r d i s p l acem en t t o mi d d l e g i mb al d i sp l acem en t t h atd ep en d s on l y on fact o rs t h at con t a i n mo me n t so f i n e r t i a . T h e n u m e r a t o r is t h e p r o d u c t o ft h e g e a r t r a i n s e n s i t i v i t y f o r c o n t r o l l e dm e m b e r a n g l e t o r o t o r a n g l e , a f a c t o r t h a tcontains this sensi t ivi ty minus unity , and them o m e n t o f i n e r t i a o f t h e r o t o r . T h e d e n o m i n a t o r i s t h e s u m o f t h e c o n t r o l l e d m e m b e rm o m e n t o f i n e r t i a a n d d ie r o t o r i n e r t i a

mu l t i p l i ed b y t h e sq u a re of t h e g ea r t r a i nsen si t iv i ty . Eq. (3) m ea ns that at high f r eq u e n c i e s t h e c o n t r o l l e d m e m b e r a m p l i t u d ei s a l way s a con st an t f ra ct i on of t h e am p l i t u d e of t h e mi d d l e g i mb al mot i on .

T h e r e s p o n s e s o f th e c o n t r o l le d m e m b e ro f an i l l u s t r a t i v e g e a r - d r i v e n s t a b i l i z a t io ns y s t e m t o s i n u s o i d a l i n t e r f e r i n g t o r q u e s a nds i n u s o i d a l b a s e d i s p l a c e m e n t s o v e r a r a n g eof fo r c i n g f req u en ci es a r e sh own b y t h e t won o n d i m e n s i o n a l l o g a r i t h m i c s c a l e c u r v e s o fF i g . 8 . T h e c u r v e r e p r e s e n t i n g th e r e s p o n s eo f t he c o n t r o l l e d m e m b e r t o i n te r f e r i n gt o r q u e s h a s a n o r d i n a t e t h at i s p r o p o r t i o n a lto the co nstan t low freque ncy st i ffness o f thes e r v o u p t o t h e r e s o n a n t f r e q u e n c y r e g i o n o ft h e sy s t em . In t h e reg i o n of t h e re so n an t

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f requen cy, the low frequency st i ffness due tot h e s e r v o d r i v e a n d t h e c o n t r o l l e d m e m b e ra nd g e a r t r a i n i n e r t i a i n t e r a c t t o p r o d u c e apeak in the cu r ve . Above this f reque ncyreg i on , t h e cu rv e d rop s of f wi t h a s l op e

having a magnitude of two.

T h e c u r v e r e p r e s e n t i n g t h e r e s p o n s e o ft h e c o n t r o l l e d m e m b e r t o s i n u s o i d a l m i d d l eg i m b a l m o t io n r i s e s a s th e f r e q ue n c yi n c r e a s e s u p t o t h e r e s o n a n t f r e q ue n c y r e g i o n .A bo v e t hi s r e g i o n , t h e c u r v e c o r r e s p o n d s t oa c o n s t a n t r e s p o n s e a m p l i tu d e r a t i o . T h em a g n i tu d e o f t h e r e s p o n s e a m p l i tu d e i s d e t e r mi n ed b y t h e ra t i o o f t h e ef fect i v e g ear t ra i na nd r o t o r i n e r t i a t o t h e s u m o f t he c o n t r o l l e dm em b er i n er t i a and t h e ef fect iv e r o t o r an d

g e a r t r a i n i n e r t i a .

I n p r a c t i c e , t h e o v e r a l l r e s p o n s e o f t h ec o n t r o l l e d m e m b e r i n a s t a b i li z a t i o n s y s t e mi s d et er m i n ed b y t h e re su l t an t o f t h e ef fect sr e p r e s e n t e d b y t he tw o c u r v e s o f F i g . 8 . F o ra d i r e c t - d r i v e s e r v o , t h e r e i s s u b s t a n t i a l l y

n o c o n t r o l l e d m e m b e r r e s p o n s e d u e t o r o t a t ion of the base; the only motion of the cont ro l l ed m em b er i s t h at p ro d u ced b y t h ei n t e r f e r i n g t o r q u e s . E x a m i n a t io n o f t h i sc u r v e c l e a r l y i n d i c a t e s t h a t, w ith a d i r e c t -d r i v e s e r v o s y s t e m , t he s e r v o n ee d o p e r a t eeffect ively only up to a f requency sufficientlyh ig h t o i n su re ad eq u at e s t ab i l i zat i o n b y t h ec o n t r o l l e d m e m b e r i n e r t i a . T h i s r e d u c e s t hes e r v o p r o b l e m f r o m o n e o f m a i n t a i n in g hig hst i f fn ess o v er t h e en t i r e f req u en cy ran g e t oone of maintaining a high st i ffness in only

t h e low f req u en cy ra n g e .

REFERENCES

1. Dr ap e r , C. S. , M cKay , W al t e r , an d L e es , S i d ney , " In s t r u m en t E n g i n eer i n g , " M cG raw -Hil l Book Co mpany , Inc. , New Y o rk, Vo l . I , 1952; Vo l . 11, 1953; Vo l . I l l , P ar t 1 , 1955;Vol . I l l , P ar t 2 , b ei n g wri t t en .

2. D r a p e r , C . S „ a nd W o o d b u r y, R . B . , " G y r o s c o p i c A p p a r a t u s , " U . S . P a t e n t 2 , 7 5 2 , 7 9 2 ,Ap p l i cat i on d at e M arch 2 2 , 1 9 5 1 .

3 . D r a p e r , C . S . , H u t z e n la u b , J . F . , a nd W o o d b u r y, R . B . , " G y r o s c o p i c A p p a r a t u s , " U . S .P at en t 2 , 75 2 , 79 3 , Ap p l i cat i on d at e M arch 2 2 , 1 9 5 1 .

4. Dr ap e r , C. S . , W ri g l ey , W al t e r , an d Gro h e, L . R. , " T h e F l oat i n g In t eg ra t i n g Gy r o an dI ts A p p li c at io n t o G e o m e t r i c a l S t a b i li z a ti o n P r o b l e m s o n M o v i ng B a s e s , " S h e r m a n M .F a i r c h i l d P u b l i c a t i o n F u n d P a p e r N o . F F - 1 3 , In s t i t u t e of t h e Aeron au t i cal S ci en ces ,New York , Jan u ary 1 9 5 5 .

5 . Dr ap e r , C. S ., W ri g l ey , W a l t e r , an d Gr o h e, L . R. , " T h e F l o at i n g In t eg ra t i n g Gy r o an dI ts A p p l ic a ti o n t o G e o m e t r i c a l S t a b il iz a t io n P r o b l e m s o n M o v i ng B a s e s , " A e r o n a u t ic a lEngineering Review, Volume 15, No. 6 , June 1956.

6 . D r a p e r , C . S „ " G y r o s c o p i c A p p a r a t u s , " U . S . P a t e n t 2 , 7 5 2 , 7 9 0 , A p p li ca ti o n d a t eAugust 2, 1951.

7 . J a r o s h , J . J . , H a s k e l l , C . A „ a nd D u n n el l, W . W . , J r . , " G y r o s c o p i c A p p a r a t u s , " U . S .P at en t 2 , 75 2 , 79 1 , Ap p l i cat i on d at e F eb ru ary 9 , 1 9 5 1 .

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8. Ref. 1 , Vo l . I l l , P ar t 1, Fi gu r e 35 -5 .

9 . G a r d n e r , M . F . , a nd B a r n e s , J . L . , " T r a n s i e n t s in L i n e a r S y s t e m s , " J o h n W i le y & S o n s ,Inc. . New York, 1942.

10. G r e e n w o o d , I . A. , J r . , H o l d a m , J . V . , J r . , a nd M a c r a e , D u n ca n , J r . , ( e d i t o r s ) , " E l e c t r o n i c I n s t r u m e n t s , " R a d ia t io n L a b o r a t o r y S e r i e s , 2 1 , M c G r a w - H i l l B o o k C o m p a n y , I n c.New York, 1948.

11. A h r e n d t , W , R . , " S e r v o m e c h a n i s m P r a c t i c e , " M c G r a w - H i l l B o o k C o m p a n y, I n c .,New York, 1954.

12 . Bo d e, H, W . , " N et wo rk An al y si s and F eed b ack Amp l i f ie r D esi g n , " D. Van No st ran dCompany, Inc. , New York, 1945.

13. M a c C o l l , L . A , , " F u n d a m e n t a l T h e o r y o f S e r v o m e c h a n i s m s , " D . V a n N o s t r a n d C o m p a n y ,Inc. , New York , 1 9 4 5 .

14. B r o w n G . S ., a nd C a m p b e l l , D . P . / ' P r i n c i p l e s o f S e r v o m e c h a n i s m s , " J o h n W il ey & S o n s ,Inc. , New York, 1948.

1 5. J a m e s , H . M . , N i c h o l s , N . B . , a nd P h i l l i p s , R . S ,, ( e d i t o r s ) , " T h e o r y o f S e r v o m e c h a n i s m s , " R a d i a ti o n L a b o r a t o r y S e r i e s , 2 5 , M c G r a w - H i l l B o o k C o m p a ny , I n c ., N ew Y o r k ,1947.

16. Ah ren d t , W . R. , and T ap l i n , J . F . , " Au t o m at i c F eed b ack Co n t r o l , " M cG raw-H i l l BookComp an y , In c . , New York , 1 9 5 1 .

17. C h e s t n u t, H a r o l d , a n d M a y e r , R . W . , " S e r v o m e c h a n i s m s a nd R e g u l at in g S y s te m D e s i g n , "

Vo l . I-I l , Jo hn Wiley & So ns, Inc. , New Yo rk , 1 951, 1955.

18. T r u x a l , J . G ., " A u t o m a t i c F e e d b a ck C o n t r o l S y s t e m S y n t h e s i s , " M c G r a w - H i ll B o o kCompany, Inc. , New York, 1955.

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B a t e d o n th e d e v el o p m e n t g i v e n i n D e r i v a t i o n S u m m a r i e s o f R e f e r e n c e " w i th m e c h a n i c a l a nd e l e c

t r i c a l i n a c c u r a c y p r o d u c i n g e f f e ct ? c o n s i d e r e d a s n e g l i g i b l e , t h e p e r f o r m a n c e e q u a t i o n o f t h e i n t e g r a t i n g

g y r o u n i t m a y b e w r i t t e n i n t h e f o r m

"(gu)

w h e r e

l P F l ( g u ) | W ; e | [ W f I - ( c a ) ] ( I A ) " ( S R ) ( g u ) | i ; W | ( r e f ) l ( l n ) f c u ) " ( S R ) ( g u ) | A ; W | ( r e f ) A [I - (ca)](OA )J

(1 )

(gu) g y r o u n i t o u t p u t v o l t a g e

•* *Fj Vizu)IW-el = a n g u l a r v e l o c i t y i n p u t - v o l t a g e o u t p u t p e r f o r m a n c e f u n c t i o n * o f t h e

g y r o u n i t

W[ I - ( c a ) ] ( I A ) = a n g u l a r v e l o c i t y o f g y r o u n i t c a s e a b o u t t h e i n p u t a x i s w i th r e s p e c t t o

i n e r t i a l s p a c e

- g y r o u n i t i n p u t c u r r e n t ( i n pu t c u r r e n t t o t h e t o r q u e g e n e r a t o r )' ( i n ) ( g u )A II - (ca)UOA ) ' a n

8u l a r a c c e l e r a t i o n o f g y r o u n i t c a s e a b o u t t h e o u t p u t a x i s w i th r e s p e c t

t o i n e r t i a l s p a c e ( N o t e : ( ) = d / d t )

(SR) ( g u ) | i ; W | ( r e f )

W [ I - ( c a ) K l A )

' ( i n ) ( g u )

• c u r r e n t - a n g u l a r v e lo c i ty r e f e r e n c e s e n s i -

( e • 0 ) t i v i t y r a t i o m a g n i t u d e o f t h e g y r o u n i t ( 2 )

• r a t i o o f a n g u l a r v e l o c i t y i n p ut m a g n i t u d e

t o i n p u t c u r r e n t m a g n i t u d e r e q u i r e d t o h o l d

o u t p u t v o l t a g e c o n s t a n t

(SR) ( g u ) | A ; W |

I P F ' (gu) |W; (

VOIWiclfref)

| W l l - ( r . a ) ] ( l A ) |

| A [ I - (ca)) (O A ) |

= o u t p ut a x i s a n g u l a r a c c e l e r a t i o n - a n g u l a r

( e = 0 ) v e l o c i t y r e f e r e n c e s e n s i t i v i t y r a t i o o f t h e

g y r o u n i t

• r a t i o o f a n g u l a r v e l o c i t y i n p u t m a g n i t u d e

t o a n g u l a r a c c e l e r a t i o n a b o u t o u t p u t a x i sr e q u i r e d l o h o l d o u t p u t v o l ta g e c o n s t a n t

' ( g u l l W j e K r e f )

p| l - P(CT) (gu)J

'"(BU)

W[I - (ca)J(IA )

(3 )

( u n d e r s t a t i c c o n d i t i o n sg i v i n g r e f e r e n c e v a l u eo f r a t i o )

(4 )

a n g u l a r v e l o c i t y - v o l t a g e

r a t e r e f e r e n c e s e n s i t i v i ty * *

m a g n i t u d e o f t h e g y r o u n i t

*A p e r f o r m a n c e fu m t io n m a y be a s s o c i a t e d w i th a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r m o f p e r f o r m a n c ee q u a t i o n s o l h al it r e p r e s e n t s t h e e f fe c t o f t h e d i f f e r e n t i a l e q u a t i o n in e s t a b l i s h i n g r e l a t i o n s h i p s b e t w e e nt h e i n d e p e n d e n t v a r i a b l e a n d t h e d e p e n d e n t v a r i a b l e . T h e p e r f o r m a n c e f u n c ti o n i s p a r t i c u l a r l y u se f u l b e c a u s e it r e d u c e s d i f fe r e n t i a l e q u a t i o n m a n i p u l a t i o n s to t he p r o c e s s e s o f a l g e b r a . F o r e x a m p l e , a t y p i c a ld i f f e r e n t i a l e q u a t i o n T v + v a u l i a s a p e r f o r m a n c e f u n c t i o n [ P F J. w h e r e v • [ l ' F ] u ; s e e D r a p e r , M c K a y

a n d I . e e s . I n s t r u m e n t E n g i n e e r i n g [ ^4 ) . \ o l . I I . C h a p t e r 1 7 .* * S e e f o o tn o te o n n e x t p a g e .

I n fo r m a t io n S u m m a r y 1. P e r f o r m a n c e e q u a ti o n f o r a s i n g l e - a x i s i n t e g r a t i n g g y r o u n i t .I l lus t r a t ive s um m a r y o f pe r fo r m a nc e funct io n de f in i t io ns , c o nve nt io ns , a nd no ta t io n .

(Pa ge 1 of 3.)

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N o t e t h a i

S = S ,( g u ) | W ; e | ( i c f ) • a ( g u ) | A ; e | ( r e f )Ku = a n g l e - v o l t a g e r e f e r -

( u n d e r s t a t i c c o n d i t i o n s , . „ . , . . . _g i v i n g r e f e r e n c e v a l u e e n c e ^ n s . t i v . t y m a g m -o f r a t i o ) t u d e o f t h e g y r o u n i t

(CT) (gu)

' ( g i m l ( O A )

(gim )(O A )

3 ( v s ) | A ; M |

I - (ca)](IA)

c h a r a c t e r i s t i c t i m e o f g y r o u n i t

m o m e n t o f i n e r t i a o f t h e g i m b a l f l o a t a b o u t t h e o u t p u t a x i s

SM

( v J ) | A ; M |(vd) v i s c o u s d a m p e r t o r q u e a c t i n g o n g i m b a l

a n g u l a r v e l o c i t y o f g i m b a l w i t h r e s p e c t t o c a s e a b o u to u t p u t a x i s[ ( c a ) - ( g i m ) ] ( O A )

i n g u l a r v e l o c i t y in p u t - t o r q u e o u t p u t s e n s i t i v i t y of v i s c o u s d a m p e r

jw f tW h e n a t t en t i o n is r e s t r i c t e d t o s t e a d y - s t a t e s i n u s o i d a l c h a n g e s ,

jw f te (gu) - ' e ( g u ) a €

d_

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1

W , • w .[ I - ( c a ) ] a A ) = " ( I - ^ a J j a A J a 4

j w , » ) 2 i n , ; w . = 2 n n , ; n , = f o r c i n g f r e q u e n c y

T f = — • f o r c i n g p e r i o d ; n , • ( F F )n f

W h e n

V [ I - ( c a ) ] ( I A ) i b w r U t e n a s p A [ I - ( c a ) ) ( I A ) " 1 2 " n f A [ I - ( c a ) ] ( l A )

l [ I - ( c a ) K O A ) i s b i t t e n a s p 2A [ , _ ( c a ) ] ( O A ) - - ( 2 » n , ) 2 A r , _ ( c a ) ] ( O A )

(5 )

(6 )

* * T h e g e n e r a l i z e d d e f i n i t io n o f t h e c o n c e p t o f s e n s i t i v i t y f o r a n o p e r a t i n g c o m p o n e n t i s a s s o c i a t e d w i tht h e c h a n g e o f t h e o u t p u t c a u s e d b y a c h a n g e i n a n i n p u t t o a n o p e r a t i n g c o m p o n e n t .

I n p u t Q u a n t i t y

' ( i n )

I d e n t i f i e d O p e r a t i n g

C o m p o n e n t( i o c )

O u t p u t Q u a n t i t y

• • ( o u t )

5 < i o c ) [ q ( i n ) ; q ( o u t ) r d q

' O J ' = i n p u t q u a n t i t y - o u t p u t q u a n t i t y s e n s i t i v i t y o f t h e i d e n t i f ie d

( i n ) o p e r a t i n g c o m p o n e n t

W h e n t h e in p u t q u a n t i t y - o u t p u t q u a n t i t y r e l a t i o n s h i p i s l i n e a r o v e r t h e o p e r a t i n g r a n g e c o n s i d e r e d ,

c _ q < o u t )

" ^ ' X n r V X " q (in)

T h e s y m b o l f o r t h e c o r r e s p o n d i n g s e n s i t i v i t y m a g n i t u d e i s

S ( i o c ) | q ( i n J : q ( o u t ) | | S ( ^ c ) [ q ( i n ) ; q ( o u t ) ]

4 ( o u t )

q ( l " )

i n p u t q u a n t i t y - o u t p u t q u a n t i t ys e n s i t i v i t y m a g n i t u d e o f t h e 'i d e n t i f i e d o p e r a t i n g c o m p o n e n t

R e f e r t o V o l u m e 1 o f I n s t r u m e n t E n g i n e e r i n g b y D r a p e r , M c K ay an d L e e s [ ^ 4] f o r t h e g e n e r a l i z e d c o n v e n t i o n s u s e d in d e f i n in g c o n c e p t s an d f o r m i n g s y m b o l s .

I n f o r m a t i o n S u m m a r y 1 . P e r f o r m a n c e e q u a t i o n fo r a s i n g l e - a x i s i n t e g r a t i n g g y r o u n i t.I l l u s t r a t i v e s u m m a r y o f p e r f o r m a n c e f u n c t i o n d e f i n i t i o n s , c o n v e n t i o n s , a n d n o t a t i o n .

(Page 2 of 3 .)

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a n d f o r t h e p u r p o s e s of s t a b i l i z a t i o n p e r f o r m a n c e

' ( l n ) ( g u ) = °

t h e p e r f o r m a n c e e q u a t i o n m a y b e w r i t t e n in t h e f o r m

S ( g u ) | A ; o | ( r e f )

< g U ) X . ( C T , ( g u )

D e f i n e

1 n lS R i •• * I - ( c a ) ( O A )i - P ( S R ) ( K u ) | A ; W | ( r e f ) - J —

A

[ I - (ca)) ( IA ) J

A i I - ( c a ) | ( I A ) ( 7 J

| ( F F ) ( S R ) P ) ( ( , u ) | ^ . W | ( l . e f ) - n f ( S R ' ( g u ) | A ; W | ( r e f ) " f o r c i n B fr e q u e n c y - a n g u l a r a c c e l e r a t i o n -

a n g u l a r v e l o c i t y s e n s i t i v i t y r a t i o p r o d u c t

N o t i n g t h a t

A l l - ( r a l K I A i = a n g l e of c a s e w i t h r e s p e c t t o i n e r t i a l s p a c e r e f e r e n c e a b o u t i np u t a x i s

A l l - f c a l l f O A l ' a n f i ' e ° ' c a s e w i t h r e s p e c t to i n e r t i a l s p a c e r e f e r e n c e a b o u t o u t p u t a x i s

L e t

p — j 2 » n f • J 2 i — and < C T ). u ) n , « | ( C T ) ( F F ) P ] = c h a r a c t e r i s t i c t i m e

f f o r c i n g f r e q u e n c y

p r o d u c t(8 )

e ,

S ( g u ) | A ; e | ( r e f )( g u ) ' 1 + j 2 » ( ( C T ) ( F F ) P ]

B y d e f i n i t i o n

l P F J X i . l A ! e l = S ( 1 ? u ll A !e W r e f l ." r ^ T T T ^ T T ^ ^ T T = S

I - } » . [ ( F F ) ( S R ) P ] ( g u ) l X ; W | ( r e f ) P - ( « ) i e o A )A ( l - ( c a ) ] ( I A )

A f I - ( c a ) ] ( I A )

(8 )

] ( D R A ) ( g u ) ( A ; e )

( g u ) | A ; e | = ° ( g u ) | A ; e K r e f ) 1 + j 2 , ( ( C T ) ( F F ) p ] " <gu)|A;e K r e f ) ^ ^ u { { C T ) ( T F ) p ) ] i

(10)

- s e1 ( D R A ) ( g u ) ( A ; e )

" S ( g u ) | A ; e l e

S ( g u ) | A ; e | " a n * l e " v o l t a«

e s ^ l t l v l t y o f g y r o un it - S ( g l l ) | A . e | ( r c ( ) ( S ( H S ) R l ( g l l ) | A . e | ( I t )

c

[ S ( R S ) R ] ( g u ) | A . e i • ( g ") lA ; -- 'J ' * s e n s i t i v i t y - r e f e r e n c eS ( g u ) | A ; e | ( r e f ) V l * [ 2» ( ( C T ) ( F F ) F ) ) 2 s e n s i t i v i t y p r o d u c t o f g y r o u n i t

( 1 2 )

N o t e| S ( R S ) R ] ( ) | A c i = ' ( S R ' ( g u ) I A ; e | = a n g l e - v o l t a g e d i m e n s i o n l e s s r e f e r e n c e s e n s i t i v i t y

o l t h e g y r o u n i t(13)

(DRA)( w A . t , j • a n g l e - v o l t a g e d y n a m i c r e s p o n s e a n g l e o f g y r o u n i t ( 1 4)

I n fo r m a t io n S u m m a r y 1. P e r f o r m a n c e e q u a ti o n f o r a s i n g l e - a x i s i n t e g r a t i n g g y r o u n it .I l lus t r a t ive s umm a r y o f pe r fo r m a nc e func tio n de f in i t io ns , c o nve nt io ns , a nd no ta t io n .

(P ag e 3 of 3.)

167

7/22/2019 AGARD-AG-21


A t l - ( . - ) W ( c » , , i n * ( Y ( . f ) . V ( e - ) l * V t . - . l X , . . . , " * * ( V C . t ) - y , „ , l " * ( ' - ( . - ) l ( X Y ) ( „ , l Y ( | . t )

km -

• X u » ) l x ( c „ ) ' i " A [ r ( m s ) . Y ( c

A l i - (c») lz (c „ ,Y ( « , ) V ( c m ) '

^ . . ( . - ) l Y ( c . , " ' * | Y ( . t ) . Y u . ) l

k | --< c m> lZ(mg) =l ' -

(cm,lZM

(1)

• A , cos A i - A,,T»-<«»PW

="X ^cm) l

Y(mg)-Y(cml-"H-(cm)lXcmsnAY(mK)_ ^

(2 )

A X < < - ) r < ( o g ) = S e C A l Z ( o g ) - Z ( m g ) lA

n - <c

« > Pr( c m )

8 t o An

rc » ^ -

i r( € « ) i

c o s A , v _v 1+ A H - ^ ) ) X ( c m ) C O l Y ( m g ) - Y ( c m ) l .

A | I - ( c m ) t Z ( u g ) "C t n A I Z ( o g ) - Z ( m g ) l [ A X ^ r v ( c m )S i n A l Y ('(cn:

+ A

(3 )

(4 )

X ( c m ) ] X ( c m )c °s A | Y ( m g ) - Y ( c m ) l ]

a) Angular velocity compo nents about the gimbal dr ive axes of the co ntro lled mem ber w ith r espe ct

to inertial space in terms ot angular velocity components about the gyro unit input axes

D e r i v a t i o n S u m m a r y 1 . C o n t r o l l e d m e m b e r a n g u l a r c o r r e c t i o n s a b o u t t h e g i m b a l d r i v e a x e s i nt e r m s o f c o r r e c t i o n c o m p o n e n t s w i t h r e s p e c t t o i n e r t i a l s p a c e r e f e r e n c e o r i e n t a t i o n a b o u t

c o n t r o l l e d m e m b e r a x i s . ( P a g e 1 o f 4 .)

16 8

7/22/2019 AGARD-AG-21


Integ rat ing E qs . (1), (2), (3) and (4)

t = t 2dt (5)^ A [ I - ( c m ) ] Z ( m g )

d t = / A [ I - ( c m ) ] Z ( c m ) '

t = t 2 . } ' * i

S A [ I - ( c m ) l Y ( m g ) d t = / = t A X ( c m ) ] Y ( c m ) - s A [ Y ( m g ) - Y ( c m ) ] ^

t = t 2

- ( A P - ( « - ) l X ( c i I l ) ' t a A l T ( l I l g ) - T ( e m ) l «l _ l l

(6 )

t = t 2 t = t 2

/ = t j ^ - ( ^ I X ^ = / ^ A ^ ^ ^ ^ s l n A ^ ^ . ^ ^ j s e c A ^ ^ ^ ^ j d t

+

/ . t ,A

tI

-( C m )

lX

( c m )C O S A

lY

( m g ) ^ ( c m ) lS e C A

[Z

( o g ) -Z

( m g ) ld t

(7)

t = »2 t = t 2

/ = t j V < « - ) P ( o g ) * " t { t j ^ - ( c m J l Y ^ ^ ^ V ^ - Y ^ j l ^ X ^ - Z ^ ] ^

t = t 2

• / A H - ( c m ) l X ( c m ) - A | Y ( m g ) - Y ( c m ) l ^ A [ Z ( o g ) - Z ( m g ) ] d t

1 (8)

For convenience the t , may be taken a t an instant when the contro l led member is co incident with i ts

i n e r t i a l s p a c e r e f e r e n c e o r i e n t a t io n a nd i s n o t r o t a t i n g w i th r e s p e c t t o t h i s p o s i t i o n . F o r t h e p u r p o s e s of

c o n t r o l s y s t e m o p e r a t i o n a t t e n t i o n i s d i r e c t e d t o w a r d t h e s ma l l a n g u l a r d e v i a t i o n s i n c o n t r o l l e d me mb e r

or ienta tion that occur in such shor t t ime inte rvals that the gimbal o r ienta tion angles Ar v v il Y ( m g ) - Y ( c m ) l

and A,™ r , imay be t reate d as co nstan ts in car ry ing out the integr a tio ns of Eq s. (5) , (6) , (7)lZ(og) " Z ( m g ) l

and (8) . Under these condit ions the integr a tio ns give

A U - ( c « ) l Z ( n l g ) " A | I - ( c m ) ] Z ( c m ) <9>

A H - ' • "" r V ( m g ) X - ' « • • "> ! * ( . • « ) * * lY (mg) " Y(cm)l " *!• " <"»> l*(c«J ^ lY (mg) " Y ( c m ) l

(10)

D e r i v a t io n S u m m a r y 1 . C o n t r o l l e d m e m b e r a n g u l a r c o r r e c t i o n s a b o u t th e g i m b a l d r i v e a x e s i nt e r m s o f c o r r e c t i o n c o m p o n e n t s w i t h r e s p e c t t o i n e r l i a l s p a c e r e f e r e n c e o r i e n t a t i o n a b o u t


169

7/22/2019 AGARD-AG-21


A ( I - (< " ) r V g ) ' A ! ' " « " ^ ( c m ) ^ A | Y ( m g ) " Y ( c m )- " " A | Z ( o g ) " Z ( m g , l

* A H - ( " " ) l X m ) C ° S A t Y ( m g ) - Y ( c m ) | S e C A l Z ( o g ) " Z ( m g ) I

( H )

A H - < ^ ) | Z ( o g ) = A l ' - < ^ l Y ( c m )S i n A I Y ( m g ) - Y ( c m ) ] < - ' n A | Z ( o g ) - Z ( m g ) |

+ A H " < c m ) l * ( c m ) C ° S A l Y ( m g ) " Y ( c m ) l ^ A f Z ( o g ) " Z ( n , g ) l

(12)

b ) A n g u l a r d i s p l a c e m e n t c o m p o n e n t s a b o u t g i m b a l d r i v e a x e s o f c o n t r o l l e d m e m b e r a n g l e s fr o m t h e

i n e r t i a l s p a c e r e f e r e n c e o r i e n t a t i o n

F o r th e p u r p o s e s of s t a b i l i z a t i o n t h e c o n t r o l l e d m e m b e r d i s p l a c e m e n t c o m p o n e n t s w i th r e s p e c t

t o t h e I n e r t i a l r e f e r e n c e o r i e n t a t i o n a r e d e v i a t i o n a n g l e s to b e c o r r e c t e d b y o p e r a t i o n o f t h e s t a b i l i z a t i o ns y s t e m .

I n g e n e r a l , c o r r e c t i o n s a r e e q u al in m a g n i t u d e an d o p p o s i t e i n s i g n t o t h e c o r r e s p o n d i n g d e v i a t i o n s .

F o r e x a m p l e ,

L [ i - W ] z ( .' 6 ' r e s p e c t t o t h e i n e r t i a l s p a c e r e f e r e n c e o r i e n t a t i o n

- ( D ) A r i / _ m \v » = (C ) A r , , „ _ i i r . • c o n t r o l l e d m e m b e r c o r r e c t i o n a n g l e a b o u t t h eH - t c m i | ^ / m „ ) [I - t c m ) ] ^ i m s \

i m g ' , m w m i d d l e g i m b a l Z - a x i s(13)

w h e r e ( D) ( ) i s t h e s y m b o l f o r a d e v i a t i o n a s s o c i a t e d w i t h t h e q u a n t i t y r e p r e s e n t e d b y a n y s y m b o l

p l a c e d i n t h e o p e n p a r e n t h e s i s a n d

( C) ( ) i s th e s y m b o l f o r a c o r r e c t i o n a s s o c i a t e d w i t h t h e q u a n t i ty r e p r e s e n t e d b y a n y s y m b o l

p l a c e d in t h e o p e n p a r e n t h e s i s .

F r o m E q s . ( 9) a n d (1 3 )

( C ) y V < - ) ] Z ( m g , " - ( D ) A P - < - ) ] Z ( m g ) • ( C ) A [I-(cm,]Z ( c m ) <">

S i m i l a r l y , f r o m E q s . ( 1 0 ) , ( 1 1 ) , (1 2 ) a n d (1 3 )

( C ) A [ l - ( c m ) ] Y ( m g ) " ( C ) A P - ( c m ) l Y ( c m ) « " A [ Y ( m g ) - Y ( c m ) ]

- < C ) A [ I - ( c m ) ] X ( c m ) ^ A [ Y ( m g ) - Y ( c m ) ] 0 5 )

D e r i v a t io n S u m m a r y 1. C o n t r o l l e d m e m b e r a n g u l a r c o r r e c t i o n s a b o u t t h e g i m b a l d r i v e a x e s i nt e r m s o f c o r r e c t i o n c o m p o n e n ts wi th r e s p e c t t o in e r t i a l s p a c e r e f e r e n c e o r i e n t a ti o n a b o u t


17 0

7/22/2019 AGARD-AG-21


( C , Al ' - ' 1 1 S « | " ^ - <• ""I X, , . ,

S m V , m g ) - Y(c„.,J " * AlZ,og) - W

* (O A , . _ ( r . m ) ] X ( c m ) c o . A [ Y Y ( c m ) ] s e c A { Z M _ ^ .

(16)

( C ) A I « - < ' m » Z ( , . g ) ' < C , AI - - « - ™ > l » « c « , M R A l Y ( m g ) " Y ( c m ) l C t " A | Z ( o g ) " Z ( m g ) l

+ ^ - < c m > l * ( c m ) C ° S A ' Y ( m g ) " Y ( c m ) i * A ( Z ( o g ) " Z( m g , l

(IT)

c ) C o r r e c t i o n s f o r c o n t r o l l e d m e m b e r d e v ia t i o n s f r o m th e r e f e r e n c e o r i e n t a t i o n

D e r i v a t io n S u m m a r y 1. C o n t r o l l e d m e m b e r a n g u l a r c o r r e c t i o n s a b o u t th e g i m b a l d r i v e a x e s int e r m s o f c o r r e c t i o n c o m p o n e n t s w i t h r e s p e c t t o i n e r l i a l s p a c e r e f e r e n c e o r i e n t a t i o n a b o u t

c o nt r o l le d m e m be r a x is . (Pa ge 4 o f 4 . )

17 1

7/22/2019 AGARD-AG-21


T h e p e r f o r m a n c e e q u a t i o n o f t h e i n n e r g i m b a l s t a b i l i z a t i o n d r i v e m a y b e w r i t t e n a s

( h ^ X m ) + ^ ( ^ ( ^ ^ ( ^ K i g J ^ r o t X d m X i g ) ] ^

* | ( S ( d m ) l A ; M | ( i g ) + C ( r o t ) ( d m ) ( i g ) ) S ( d g t ) | A ( c m ) ; A ( r o t ) ] ( i g ) + C ( c m ) Z J p

+ S ( d m ) h ; M | ( i g ) f P F l ( d p c & ) [ e ; i ] ( i g ) [ P F l ( g u ) ( A ; e l Z S ( d g t ) | A ( c m ) ; A ( r o t ) l ( i g ) J A [ I - ( c m ) ) Z ( c m )

= M ( i n t f r ) ( c m ) ( i g ) + | P [ ( S ( d m ) l A ; M | ( i g ) + C ( r o t ) ( d m ) ( i g ) ) S ( c i g t ) l A ( c m ) ; A ( r o t ) J ( i g ) + C ( c m ) Z J

+ p 2 l ( " - ° t ) ( d m ) ( i g ) S ( d g t ) [ A ( c m ; : A ( r o t ) l ( i g ) [ S ( J g t ) [ A ( c m ) ; A ( r o t ) ] ( i g ) " * ] } A [ I - ( m g ) ] Z ( c m )

(1 )w h e r e

I I - ( c m ) I Z = a n5

u l a r d i s p l a c e m e n t o f t h e c o n t r o l l e d m e m b e r a b o u t t h e Z ( » - a x i s w i t h r e s p e c t

* t o t h e i n e r t i a l s p a c e o r i e n t a t i o n f o r w h i c h t h e Z - a x i s g y r o o u t p u t s i g n a l h a s

A | i - v c m , j t ( ( . m )

i t s nul l level

' ( c m ) Z = m o m e n t o f i n e r t i a o f c o n t r o l l e d m e m b e r a b o u t t h e Z / » - a x i s

^ I X m l X r o t ) ] ^ )i n n e r g i m b a l d r i v e g e a r t r a i n ( i d e n t i c a l w i t h t h e lo w s p e e d to h i g hs p e e d g e a r r a t i o o f t h e g e a r t r a i n )

' ( r o t U d m K i e ) =f"

0"1611

' ° ' i n e r t i a o f t h e i n n e r g i m b a l d r i v e m o t o r a b o u t i t s a x i s o f r o t a t i o n .

T h i s i n c l u d e s a c o m p o n e n t t h a t d e p e n d s o n t h e e f f e c t i v e i n e r t i a o f t h e g e a r t r a i n

r e d u c e d to r o t o r s p e e d

S ( d m ) I A - M l ( i e ) = a n8

u l a r v e l o c i t y i n p u t - t o r q u e o u t p u t s e n s i t i v i t y m a g n i t u d e o f d r i v e m o t o r .

T h i s s e n s i t i v i t y i s d ue to r e d u c t i o n i n m o t o r t o r q u e by i n t e r n a l e l e c t r i c a l

e f f e ct s a s s p e e d i n c r e a s e s

^ t r o t ) ( d m K i n ) " c o e f ) ' i c ' e n t o f v i s c o u s d a m p i n g a c t i n g o n i n n e r g i m b a l d r i v e m o t o r r o t o r -

r e f e r r e d t o r o t o r s p e e d

C / ) z = c o e f f i c i e n t o f v i s c o u s d a m p i n g a c t i n g o n c o n t r o l l e d m e m b e r a b o u t Z / c _ v - a x i s -

r e f e r r e d t o c o n t r o l l e d m e m b e r s p e e d

^ ( ' m )l i -M l( ie)=

v o l t a g e i n p u t - t o r q u e o u t p u t s e n s i t i v i t y m a g n i t u d e o f i n n e r g i m b a l d r i v em o t o r u n d e r z e r o s p e e d c o n d i t i o n

E q u a ti o n S u m m a r y 1 . P e r f o r m a n c e e q u a ti o n f o r s t a b i l i z a t io n of t h e c o n t r o l l e d m e m b e r by t h ei n n e r g i m b a l d r i v e s y s t e m . ( P a g e 1 o f 2 .)

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l P F l ( d p c s ) ( e , i ] ( i g ) " S ( d p c s ) [ c ; i ] ( r c i ) ( i g ) ( B F ) ( d . , c s ) l e ; i ] ( i g ) * v o l t a g e i n p ut - c u r r e n t o u t p u t

p e r f o r m a n c e f u n c t i o n o f t h e i n n e r

g i m b a l d r i v e p o w e r c o n t r o l s y s t e m

^ (dpcs) le- i l ( ig) = v o l t a g e in p u t - c u r r e n t o u t p u t r e f e r e n c e o e n s i t i v i t y o f t h e i n n e r g i m b a l d r i v e

p o w e r c o n t r o l s y s t e m

C ' X i i c s U e - i l f l e i : = v o l t a g e i n p u t - c u r r e n t o u t p u t f r e q u e n c y f u n c t i o n o f t h e i n n e r g i m b a l d r i v e' ( J P c s ) [ e ; i ] ( i g )p o w e r c o n t r o l s y s t e m . W h e n f r e q ue n c y — — 0 , ( F F ) ( d u c s ) r e - i l ( i i ; , — * " 1

l P F l ( g u ) ( A ; e ] Z ' S ( g u ) [ A ; e ) ( r e f ) Z ( F F ) ( g u ) [ A ; e ] Z " a n g l e i n p u t " v o l t a g e o u t p u t P e r f o r m a n c efunct io n o f the Z -axis gyr o uni t

^ ( e u l l A - e K r e f l Z = a n g l e i n p ut - v o l t a g e o u t p u t r e f e r e n c e s e n s i t i v i t y o f t h e Z - a x i s g y r o u n i t

(F F ) / w . i~ • angle input - vo l tage o utput f r equ ency funct io n o f the Z -a xis gyr o uni t .

W hen f r e q u e n c y - ^ " , ( F F ) ( g u ) j A ; e j z — 1

M ( i n U r U c m ) Z = i n t e r f e r e n c e t o r q u e a c t i n g o n t h e c o n t r o l l e d m e m b e r a b o u t t h e Z - , , - a x i s

A ( I - ( m e ) l Z = a n8

u l a rd i s p l a c e m e n t o f t h e m i d d l e g i m b a l a b o u t th e Z / c n > - a x i s w i t h r e s p e c t

t o a r e f e r e n c e o r i e n t a t i o n i n i n e r t i a l s p a c e i n w h i ch th e Y - a x i s o f t h e g i m b a l

i s a l i g n e d w i t h t h e Y - a x i s o f t h e c o n t r o l l e d m e m b e r w h e n t h e c o n t r o l l e d

m e m b e r i s i n t h e r e f e r e n c e o r i e n t a t i o n d e t e r m i n e d b y t h e n u l l l e v e l s i g n a l

o f t h e Z - a x i s g y r o u n i t

a ) P e r f o r m a n c e e q u a t io n fo r th e i n n e r g i m b a l s t a b i l i z a t i o n l o o p w h en t h e o n l y m o t i o n s w i th r e j e c t

to iner t ia l spa ce of the m iddle gim ba l and the co nt r o l led m em be r a r e a lxiut the Z . . , -axi : -

W hen the syst em is s ta t io na r y , i . e . , a t zer o f r eq uency and w i th A i , _ « , . . , = 0[ 1 t n ' r - ' K ^ c i n )

A | I - ( c m ) ] 7 . ( ( , m ) ^ ,

(2 )

(3 )

M ( i n t f r ) ( c m ) ( i g ) S ( d m ) | l ; M | ( r e f ) ( i g )S ( a p r s ) | e ; l ) ( r e f ) ( i g )S ( g u ) | A ; e ) ( r c f ) Z S ( d g t ) | A ( ( . n l ) ; A ( r o ( ) l ( i . )

= a n g u l a r s t i f f n e s s o f s t a b i l i z a t i o n c o n t r o l i n r a d i a n s p e r un i t t o r q u e

b) T he effect o f d is tu r bi ng to r q ue a t lo w f r equ enc ies

2A t v e r y h ig h f r e q u e n c i e s - w h e r e p t e r m s a r e s o g r e a t t ha t a l l o t h e r s a r e n e g l i g i b l e

A [ l - ( c m ) | 7 ( ( r m ) ^ S ( d g t ) | A ( c m ) ; A ( r o t ) l ( i g ) [ S ( d g t ) | A ( c m ) ; A ( r u t ) ] ( i g ) ~ ' | ' ( r o Q M m H i . ; )

H - «««>J

8

(c«) '(••m)Z,, r n| *

S

(^ ' ' ) |A ( ( m );A ( l . o t ) l(u) »(«*)( ..„,)(...)

r ) T h e e ff ec t o f b a s e m o t io n i n t e r f e r e n c e a t h i gh f r e q u e n c i e s

Equa t io n Summ a r y 1 . Pe r fo r m a nc e e qua t io n fo r s ta bi l iz a t io n o f the c o nt r o l le d me m be r by theinne r g imba l d r iv e s ys te m . (Pa ge 2 o f 2 . )

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VECTOR POINTS IN D IR EC T IO N OF

AD VAN C E OF RIGHT-HAND SCREW

T U R N ED IN SAME SENSE AS ROTA

T IO N AL Q U AN T IT Y AC T S.

RIOHT-HAND SCREW

SENSE OF

R O T AT IO N AL Q U AN T IT Y RIGHT HAND

SENSE OF R O T AT IO N AL Q U AN T IT Y

VECTOR POINTS IN

D IR EC T IO N OF

R IG H T - H AN D T H U MB

WHEN FINGERS L INE

U P AS IN D IC AT ED

WITH SENSE IN WHICHT H E R O T AT IO N AL

Q U AN T IT Y AC T S.

THE RIGHT

HAND RULE

IS

IN A RIGHT

H AN D ED SET

OF AXES

r.

INDICATED

REFERENCE X-AXIS

X ( ' . 0 ( i » < l )CONTROLLED MEMBER

X-AXIS

X ,

REFERENCE X-AXIS-

CONTROLLED MEMBER

X-AXIS ANGLE

AL L AX IS

SYSTEMS

USED IN

THIS PAPER

ARE RIGHT

H AN D ED .

REFERENCE

X-AXISSUPPORT MEMBER

X-AXIS

X ,

COINCIDENT WITH X,hV

THE BASE X-AXIS

SUPPORT

MEMBER(.rr,)

REFERENCE Z-AXIS-

CONTROLLED MEMBER

Z-AXIS ANGLE

A(Z„

REF ERE NCE Y-AXIS-CONTROLLED MEMBER

Y-AXIS ANGLE

[ Y ( r . 0 - Y ( e m ) '

INDICATED

REFERENCE Y-AXIS

( , . f ) ( l n d )

CONTROLLED MEMBER

Y-AXISY ( c m )

SUPPORT MEMBER

Y-AXISY

( . m )

COINCIDENT WITH Y, M,THE BASE Y-AXIS1

'

SUPPORT MEMBER

Z-AXIS

Z,

INDICATEDiEFERENCE Z-AXIS

Z ( r . O ( l n d )

CONTROLLED MEMBER

Z-AXIS

Z ( c n , )

COINCIDENT WITH Z,THE BASE Z-AXIS l b ) '

N O T E S i THE C O N T R O L L E D M E M B E R IS C A R R I E D BY A S Y S T E M M O U N T E D ON A B ASE NOT SHOWN INT H I S D I A G R A M . THE B A S E IS

C A R R I E D BY THE S U P P O R T M E M B E R .

S U B S C R I P T L E T T E R ON A N Y M A I N S Y M B O L I D E N T I F Y TH E E N T I T Y W I T H W H IC H T H E Q U A N T I T Y R E P R E S E N T E D BY THE MAIN

S Y M B O L IS A S S O C I A T E D - T H U S X, . > IS THE X - A X I S A S S O C I A T E D W I T H T H E C O N T R O L L E D M E M B E R ( S Y M B OL ( c m ) ) .\ c m l

T H E S Y M B O L * ( ( r , ( ) . ( c m . , d ) ] R E P R E S E N T S THE A N G L E A, M E A S U R E D F R O M T H E R E F E R E N C E D I R E C T I O N ( ( »» IN T H E

S U B S C R I P T ) TO T H E C O M P A R E D D I R E C T I O N ( ( c m pr d ) IN THE S U B S C R I P T )

R E F E R E N C E

D IR EC T IO N " « 5

( r e f ) / - •

C O MPAR ED

D IR EC T IO N

( c mp r d )

A ( ( r . f ) - ( c mp r d ) l

F i g . 1. G e o m e t r i c a l q u a n t i t i e s a s s o c i a t e d w i th the b a s i c p r o b l e m of g e o m e t r i c a l s t a b i l i z a t i o n .

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INNER GIMBALX-AXIS

X

OUTER GIMBAL[x

(oorx(m0)l

X-AXIS

INNER GIMBAL

Y-AXIS

Y( i g )

Y ( e n . )

^ ( r n g J - Y f c m ) 1

MIDDLE GIMBAL

Y-AXIS

Y ( m g )

OUTER GIMBAL

Y-AXIS

Y ( o » )

MIDDLE

GIMBAL

(mg)

OUTER GIMBAL

Z-AXIS

Z,SUPPORT MEMBER

(»m)

N O T E S : INPRA CT ICE ALL GIMBALS WOULD

USUAL L Y BE COMPLETE AND SYM

METRICAL AS INDICATED BY THE

SKE T CH. OPE N GIMBAL S ARE USE DH E R E TO SIMPL IFY T HE DIAGRAM.

THE CONTROLLED MEMBETI IN T HISFIGURE HAS T HRE E DE GRE E S OF

FRE E DOM WIT H RE SPE CT TO T H E

BASE AND MAY T H E RE FOR E BE

COMPL E T E L Y ISOL AT E D FROM ANY

ANGULAR ROTATION OF THE BASEOR T HE SUPPORT ME MBE R.

MIDDLE GIMBALZ-AXIS

Z(".»)

INNER GIMBALZ-AXIS

Z

THIS DIAGRAM IS BASE D ONF I G . * OF U. S. PAT E NT 2,752,792 AND FIG. 5 OF U. S. P A T E N T

2,752,793, DAT E D JUL Y 3, 1956.

F i g . 2. L i n e s c h e m a t i c d i a g r a m of gimba l s ys te m for s t a b i l i z a t i o n .

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GYRO ROTOR

DAMPER CLEARANCE

TORQUE GENERATOR ROTOR , v GIMBAL FLOAT SHELL

TORQUE GENERATOR STATOR

SIGNAL GENERATOR ROTOR

- J

OUTPUT AXIS(OA)

GIMBAL BEARING

SIGNAL GENERATOR STATOR

GYRO MOTOR STATOR

N s X ^ S P I N REFERENCE AXIS* } (SRA)

/ ^ SPIN AXIS

(SA)GIMBAL ANGLE

BALANCE NUTSINPUT AXIS

(IA)

THE VOLUME BETWEEN THE OUTSIDE OF THE FL O ATAND THE INSIDE OF THE CASE IS CO MPL ETEL Y FIL L ED

WITH A FLUID HAVING THE PROPERTY OF NEWTONIANVISCOSITY. THAT IS, THE CO EFFIC IENT OF VISCOSITY

IS INDEPENDENT OF SHEAR GRADIENT.

GIMBAL

BEARING

BALANCE NUT

ADJUSTMENT FORKS

THIS DIAGRAM IS B A S E D ON U. S. P A T E N T 2 , 7 5 2 , 7 9 1 , FIG. 1 OF U. S. P A T E N T 2 , 7 5 2 , 7 9 0 AND FIG. 12 OF T HE SHERMAN M. F A I R C H I L DP U B L I C A T I O N F U N D P A P E R N O . F F - 1 3 , I N S T I T U T E OF THE AERON AUTICAL SCIEN CES, NEW YORK, JANUARY 1955. (USED W ITH PERM ISSION)

F i g. 3. P i c t o r i a l d i a g r a m for the s i n g l e - a x i s i n t e g r a t i n g g y r o u n i t .

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GYROSCOPICELEMENT

( § • )

SIGNALGENERATOR

( • 9 )

GIMBALBEARING

(gb)

GIMBALTORQUE

GENERATOR(t f )

GIMBAL BEARING( g b )

CASE( c o )

OUTPUT AXIS(OA)

' F I X E D T O C A S E \P A R A L L E L T O I

. G IMBAL BEA R IN G /

SPIN REFERENCE AXIS(SRA)

F I X E D T O C A S E A T R I G H T \A N G L E S T O O U T P U T A X I S

AND C OIN a DEN T WITH T HESPIN AXIS OF T HE

I Y R O S C O P IC E L E M E N T W H EN IT H E O U T P U T S I G N AL F R O M /T H E S I G N A L G E N E R A T O R I S /

1

INPUT AXIS( IA)

I F I X E D T O C A S E \A T R I G H T A N G L E S \

T O T H E O U T PU T IAXIS AND THE I

S P I N R E F E R E N C E /k AXIS /

(F I X E D T O \G I M B A L P A R A L L E L \

T O A X I S A B O U T 1W H I C H G Y R O R O T O R /

S P I N S /

"SL

AT IT S N U L L L EV EL /

GIMBAL ANGLEA ( g i i » )

A [ (co )- (gim )r/ A N G L E F R O M ( S R A ) \^ TO (SA) )

R E F E R E N C E D I R E C T I O N -C O M P A R E D D I R E C T I O N

A N G L E

1. POSIT IVE SENSE S SHOWN B Y TH E AR R OWS AR EC HOSEN SO TH AT (IA) , (SR A), AND (OA) FOR M AR I G H T - H A N D E D S Y S T E M . ,

2. T H E . G Y R O U N I T T E M P E R A T U R E C O N T R O L P O W E RI S S U P P L I E D T O A M O U N T IN G B L O C K A D A P T E DT O R E C E I V E T H E G Y R O U N I T C A S E . T H E F L O WO F P O W E R B C O N T R O L L E D B Y T H E D A M P E RT E M P E R A T U R E S E T T I N G .

, THE SYMBOL A r ( r - ) . ( c l l p r d ) )

R E P R E S E N T S T H E A N G LE AM E A S U RE D F R O M T H E R E F E R E N C E D I R E C T I O N ( ( r » I )I N T H E S U B S C R I P T ) T O T H EC O M P A R E D D I R E C T I O Nl ( c m p d ) IN T H E S U B S C R I P T ) . *

R E F E R E N C ED I R E C T I O N

(rW)

> ^^MWeinxfll

x~O M P A R E D f r ,

D I R E C T I O N 4 )

( c m p d )

C A S E - ( c a ) - T H E S T R U C T U RE T H A T G I V E S S U P

P O R T F O R T H E I N T E R N A L W O R K I N GP A R T S O F T H E G Y R O U N I T , E N C L O S E ST H E P A R T S , A N D C A R R I E S P H O V I -S I O N S F O R E X T E R N A L C O N N E C T I O N SO F A L L K I N D S .

TORQUEGENERATOR

- ( t g )C O M P O N E N T F O R R E C E I V I N G I N P U TS I G N A L S A N D P R O D U C I N G C O R R E S P O N D I N G O U T P U T T O R O U E A P P L I E DT O T H E G IM B A L A B O U T T H E O U T P U TAXIS.

D A M P E R - ( d m p ) - S U B SY S T E M R E C E I V IN G A N G U LA RV E L O C I T Y C F T H E G I M B A L W I T H R E S P E C T T O T H E C A S E A S I T S I N P U TA N D P R O D U C I N G A S O U T P U T A R E T A R D I N G T O R O U E A C T I N G O N T H EG I M B A L A B O U T T H E O U T P U T A X I SW I TH A M A G N I T U D E P R O P O R T I O N A LT O T H E M A G N I T U D E O F T H E A N G U L A RV E L O C I T Y O F T H E G I M B A L W I T H R E S P E C T T O T H E C A S E .

GENERATOR

G Y R O U N I T - ( g u ) - T H I S E N T I T Y M A D E U P O F T H E CO M

P O N E N T S R E P R E S E N T E D I N T H I SDIAGR AM AND ALL THE ADDITIONALF A R T S N E C E S S A R Y F O R A S IN G L EP A C K A G E T O C A R R Y O U T T H E F U N C T I O N S O F A G Y R O U N I T .

S I G N A L _ . . C O M PO N E NT F O R R E C E IV I N G T H E1 • ' ~ A N G L E O F T H E S P I N A X I S W I TH R E

S P E C T T O T H E C A S E A S I N P U T A N DP R O D U C I N G A C O R R E S P O N D I N G S I G N A L T H A T S E R V E S A S T H E O U T P U TS I G N A L F R O M T H E G Y R O U N I T .

G I M B A L - ( g i m ) - S T R U C T U R E C A R R Y IN G T H E B E A R IN G SF O R T H E S P I N N I N G R O T O R O F T H EG Y R O S C O P I C E L E M E N T , R O T O R S F O RT H E T O R Q U E G E N E R A T O R A N D S I G N A L G E N E R A T O R , P A R T O F T H ED A M P E R , F L O A T S E A L S A N D S T R U C T U R E , B A L A N C E A D J U S T M E N T S , S T O P S ,P I V O T S , E T C .

• A D I SC U S SI O N O F G E N E R A L I Z E D C O N V E N T I O N S F O R S E L F - D E F I N I N G S Y M B OL S O F W HI CH * f ( _ n . / a _ r l ) l I s A N E X A M P L E I S G I V E N B Y D R A P E R ,M cK AY A ND L E E S I N I N S TR U M E N T E N G I N E E R IN G [ 2 4 ] , V O L . I . u r t l M O i p o i l

THIS DIAGRAM IS BASED ON FIG . 13 OF THE SHERMAN M. FAIRC HILD PUB LICATIO N FUN D PAP ER NO. F F - I 3 , INST IT UT E OF T HE

AERONAUTICAL SCIENCES, NEW YORK, JANUARY 1955. (USED WITH PERMISSION)

Fig . 4 . Line s c he ma t ic fo r the s ing le -a x is in te gr a t ing gyr o uni t .

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CONTROLLED MEMBERDRIVE POWER

CONTROL SYSTEM( c m d p c i )

OO

GYRO UNITCOMMAND SIGNAL

(S« ' ( e m d ) ( g u )

CONTROLLED MEMBER(em )

GYRO UNITCASE

(co)(gu)

SIGNALGENERATOR

(>g)

GYROSCOPIC ELEMENT( g . )

SPIN REFERENCEAXIS(SRA)

CONTROLLED MEMBERDRIVE MOTOR

(cm df f l )

BAC.F SUPPORTING* , MEMBER(ba)

INERTIAL SPACEROTATION MEAS

URING SYSTEMINPUT ANGLE

A( i n ) ( i » r m i )

A[l-(b«)J(IA)/ A N G U L A R D I S P L A C E - X/ M E N T A B O U T T H E 1| I N P U T A XI S O F T H E I

B A S E W I TH R E S P E C T IT O I N E R T I A L S P A C E /

ANGULAR VELOCITYINTEGRATING SYSTEM

INPUT AXIS

INDICATING SYSTEMGEAR TRAIN

( i t g t )

BASE ROTATION SCALE READING(BR)

vTORQUESPIN AXIS GENERATOR

(SA) (fg)

GIMBAL ANGLE

* ( • • • )

N O T E l B A S E IS R I G I D L Y C O N N E C T E D T O S U P P O R T I N GMEMBER FOR W HICH ROTAT ION IS TO BE MEA SURED.

THIS DIAGRAM IS BASED ON FIG. 14 OF THE SHERMAN M. FAIRCH ILD PUB LICATIO N FUND PAPER NO . FF -I3 , INSTIT UTE O F THE AERONA UTICALSCIENC ES, NEW YORK, JANUARY l»S5. (USED W ITH PERMISSION)

F i g . 5 . L i n e s c h e m a t i c d i a g r a m o f s i n g l e - a x i s s e r v o - d r i v e n c o n t r o l l e d m e m b e r w i th i n t e g r a t i n g g y r o .

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Vcm)m.

t 1 '(cm)

y(SRA) ifrfl

(A>(9U,Y(gu)Y

X^ ) ^ ( I A ) ( 9 U ) X

MIDDLEGIMBALDRIVEMOTOR(mgdm)

(drp)(mg)

MIDDLEGIMBALANGLE

RESOLVER(mgor)

(or)(mg)

OU TER GIMBALDRIVE MOTOR

(ogdm)

(dm)(og)

T HIS DIAGRAM IS BASE D ON FIG. 4 OF U. S . PAT E NT 2,752,792 AND FIG. 5 OF U. S . PAT E NT 2,752,793, DAT E D

JUL Y 3, 1956.

Fig. 6 . Line s c he m a t ic d ia gr a m o f g imb a l s ys te m fo r s ta bi l iz a t io n wi th s ing le -a x isi n t e g r a t i n g g y r o u n i t s , s e r v o d r i v e s , a n d r e s o l v e r s .

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CWTIKXLIDMfN4t>lT4*K.I I4TiaNl rST|M [a.

CO

o

rwn*£ft4uiii4UMS carte * wi i P IT IM T - - U I I I M I S i w i i fii-rnT i m ti IIA» i

F i g. 7. F u n c t io n a l d i a g r a m for t h r e e - g i m b a l g e o m e t r i c a l s t a b i l i z a t i o n s y s t e m b a s e d o n s e r v o d r i v e s a c t u a t e db y t h r e e s i n g l e - d e g r e e - o f - f r e e d o m i n t e g r a t in g g y r o u n i t s .

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0 0

TJ•18cnr.i&—

• <

cin

irC

M n

—it-1

c

rtantr- l

n-(^cc)aE L

~*<oli .

—P

o

53n-t

c9n l

d' -s—-U i

3r r•—

• t• J - 5

o c

r-*

-;—•—cLf t

Si

—•

c<D3l- l

E Dr-l

0ai0—1

•J.

B

— •

N

w—

IPaIB

•-<•r-t

c5—*o-1n0a-fHI'J

—':.3r.~.

W•ioc-t

~3

• oo- I

3

ato- i

<OCUJ

Cm

3

Q .

5

-6 .0

-3 .0 -2 .0 -1 .0 • 0 1.0 2.0 3.0 4.0

REFERENCE PERIOD - FORCING PERIOD RATIO [(R P) (F P) R] (LOG SCALE)

[ A ( , . ( e m )] , ] - AMPLITUDE OF CONTROLLED MEMBER OSCILLATION ABOUT Z ( c m ) AXIS

^. -( m gj lZ f J " AMPLITUDE OF MIDDLE GIMBAL OSCILLATION ABOUT Z ( e m ) AXIS

[ M / , n t f r ) z 1 - AMPLITUDE OF INTERFERENCE TORQUE APPLIED TO CONTROLLED( c m ) • MEMBER ABOUT Z ( c m ) AXIS

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SAMPLED-DATA SYSTEMSJ o hn R , R a ga z z ini*

SUMMARY

The ba s ic the o r y , de s ign, a nd a ppl ic a t io n o f s a m ple d- da ta guida nc e a nd c o nt r o l s ys te m sa r e d e s c r i b e d . T h i s d e s c r i p t i o n in c l u d e s t h e m a t h e m a t i c a l c h a r a c t e r i z a t i o n of t h e s a m p l i n gp r o c e s s , u s e o f t h e z - t r a n s f o r m , s t a b i l i t y o f s a m p l e d s y s t e m s , p u l s e d t r a n s f e r f u n c t i o n s , a n da d is c us s io n o f the a dva nta ge s a nd d is a dva nta ge s o f th is type o f c o nt r o l .

SOMMAIRE

L a t h e o r i e d e b a s e , 1'etude e t l a p p l i c a t i o n d e s s y s t e m e s de c o n t r o l e e t d e g o u v e r n ep a r i n f o r m a t i o n s p u l s e e s so n t d e c r i t e s . C e t t e d e s c r i p t i o n c o m p r e n d 1'etude: d e s c a r a c te r is t iqu e s m a the m a t ique s du pr e c e de ' de puls a t io n , de 1'utilisation d e l a t r a n s fo r m a t i o n " z " ,de la s ta bi l i te de s s ys t e m e s pu ls e s , de s fo nc t io ns de t r a ns fe r t de s s ys te m e s puls e s e tf ina lle me nt une d is c us s io n s ur le s a va nta ge s e t le s de s a va nta ge s de c e type de c o nt r o le .

1. INTRODUCTION

Sa mple d-da ta s ys te ms pla y a n impo r ta ntr o l e i n g u i d a n c e a n d c o n t r o l s y s t e m s f o rm i s s i l e s , p r o c e s s e s , a n d a i r c r a f t . T h e c h a r a c te r iz ing fe a tur e o f s uc h s ys te m s is thefact that the s ignal data appear at one orm o r e po ints o f the s ys te m a s a puls e s e que nc eo r a s a s e que nc e o f nu mb e r s . Any in te r mi t te nt e le me nt s uc h a s a t im e -s ha r e d da ta

l ink, a d ig i ta l c o mpute r , a s c a nning r a da r ,o r o t h e r d a t a g a t h e r i n g t r a n s d u c e r w i l l c a u s ethe c o nt r o l s ys te m inc o r po r a t ing s uc h a ne l e m e n t t o b e a sa m p l e d - d a t a s y s t e m .

S o m e t i m e s s a m p l e d - d a t a s y s t e m s e x i s tbecause one of the elements is unavoidablys a m p l e d . F o r i n s t a n c e , a s c a nn i n g r a d a rused for control will yield a fix on the targeto r c o n t r o l l ed a i r c r a f t o n c e ea c h s c a n ni n gpe r io d . B e twe e n s uc h s a m ple s , da ta mu s tbe in te r po la te d by s o me me a ns o r o the r . On

t h e o t h e r ha nd , s o m e s y s t e m s a r e d e l i b e r a t e l yma de s a mple d whe n the y inc o r po r a te a d ig i ta lc o m p u t e r a s a c o n t r o l l e r . T h e c o m p u t e r c an

a c c e p t n u m b e r s a n d co m p u t e s t e e r i n g c o m ma nds pe r io dic a l ly but no t c o nt inuo us ly . Theinc r e a s e d s o phis t ic a t io n o f c o nt r o l whic h c a nr e s ul t by us e o f a d ig i ta l c o mp ute r whic h ma ybe r e qui r e d fo r da ta r e duc t io n in a ny c a s ema ke s the de s ign o f a n o the r wis e c o nt inuo uss y s t e m a s a s a m p l e d - d a t a s y s t e m e n t i r e l yd e s i r a b l e .

Fr o m the v ie wpo int o f a na lys is a nd de s ign,

s a m p l e d - d a t a s y s t e m s p r e s e n t u ni qu e p r o b l e m s . Without going into detail a t this point ,i t i s evide nt tha t the s a mpl ing pr o c e s s r e duc e sthe info rm ation co ntent of the function beings a mp le d. Fo r ins ta n c e , if a s c a nning r a d a ri s u s e d t o d e t e r m i n e t h e t r a c k o f a n a i r b o r n eve hic le , the po s i t io n i s de te r min e d o nc e e a c ht i m e t h e r a d a r b e a m s w e e p s p a s t t h e t a r g e t .B e twe e n s c a ns , no po s i t ive info r ma t io n i sobtained so that if the target undertakes atu r n o r ma n e uve r , th is i s no t de te c te d unti lthe next and subsequen t sc an s. Had the

ta r g e t be e n c o nt inuo us ly i r r a d ia te d by me a nso f a t r a c k ing r a da r , the in i t ia t io n o f ama ne uve r wo uld be imme dia te ly de te c te d .

• C o lumbia Unive r s i ty , Ne w Yo r k , Ne w Yo r k .

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A n o t h e r f a c t o r o f i m p o r t a n c e t o a p p l i c a t io n o f s a m p l e d s y s t e m s t o f e ed b a ck c o n t r o li s t h e ef fect t h e p r o ce ss of sam p l i n g h as ont h e s t a b i l i t y o f t h e s y s t e m . L o w - p a s s c o n t r o lsy s t e m s t en d t o b e u n st ab l e d u e t o t i m e l ag s

i n t h e l oo p . T h e i n t ro d u ct i o n of sam p l i n gswi t c h es m ea n s t h at ch an g es i n t h e s i g n al a r en o t r e c o g n i z e d u n t i l o n e o r m o r e s a m p l i n gi n t e r v a l s l a t e r . T h u s , c o n t i n uo u s s y s t e m sw h ic h a r e s t a b l e w i th a c o m f o r t a b l e m a r g i nc a n b e c o m e u n s t a b l e i f a s a m p l i n g o p e r a t i o ni s i n t r o d u c e d in s o m e p a r t o f t h e l o o p . T h e o r yw h ic h d e s c r i b e s t he p e r f o r m a n c e of s a m p l e d -d at a sy s t ems b ot h f rom t h e v i ewp oi n t o fs t ab i l i t y an d resp on se h as b een d ev el op ed i nv ar i o u s co u n t r i es as wi l l b e seen in t h e l i s to f r e f e r e n c e s . S u b se q u en t s e c t i o n s w i ll d e a l

wi th t h e s t a t u s of t h eo ry and ap p l i cat i o n s t ot h e a n a l y s i s a n d s y n t h e s i s o f s a m p l e d - d a t af e e d b a c k c o n t r o l s y s t e m s .

2 . E L E M E N T S O F A S A M P L E D - D A T ASYSTEM

W h i l e t h ere i s n o u n i q u e fo rm of samp l ed -d a t a s y s t e m , a l l s u c h s y s t e m s c o n t a i nc o m m o n e l e m e n t s a r r a n g e d in v a r i o u s c o n f i g u r a t i o n s . T h e s e e l e m e n t s c a n b e c a t e

g o r i z e d a s s a m p l e r s , d a t a h o l d s , d i g it a lc o n t r o l l e r s , a n d c o n t i n u o u s l i n e a r ( o r n o n l i n e a r ) p l a n t s . T h e s e e l e m e n t s w i ll b ed e s c r i b e d i n d i v i d u a l l y a n d t h e i r c h a r a c t e r i s t i c s d e s c r i b e d b o th q u a l it a t iv e l y a ndm a t h e m a t i c a l l y . T o i n d ic a t e w h e r e e a c h o ft h es e e l em en t s may b e foun d, a t y p i cal con f i g u rat i on i s sh own i n F i g . 1 fo r i l l u s t r a t i v ep u r p o s e s . A s w i l l b e s ho w n l a t e r , t h e r e a r em a n y o t h e r p o s s i b l e c o n f i g u r a t i o n s .

T h e f i r s t e l em en t wh i ch wi l l b e co n si d e re di s t h e sa m p l e r wh i ch i s sh own h e r e as ame ch an i ca l swi t ch on l y t o sy m b o l i ze t h eo p e r a t i o n . R e f e r r i n g t o F i g . 2 , t h e c o n t i n uo u sfunction which i s being s am pled is e, ( t ) .T h i s fu n cti on i s sam p l ed fo r a v er y sh o r tt im e r and the output of the switch is a

s e q u e n c e o f s h o r t p u l s e s w h o s e a r e a i s p r o p or t i on al t o t h e v al u e of t h e t i me fu n ct i onat i n t eg ra l m u l t i p l es of t h e sam p l i n g p er i o dT . T h e p u l s e s e q u e n c e w h ic h r e s u l t s i sidentif ied by the function e*(t) . The resul tant

p u l se seq u en ce may b e ob t a i n ed mat h emat ical ly by mult iplying the co ntinuo us functione( t ) b y a car r i e r s i g n al con si s t i n g of ap e r i o d i c s e q u e n c e o f p u l s e s r e f e r r e d t o h e r eas p ( t ) . T h u s ,

e f t . ) • e ( t ) p ( t ) .

(1)

Since p(t) is a periodic sequence, i t mayb e e x p r e s s e d a s a F o u r i e r s e r i e s s o t ha t

+ 00e " ( t l • e ( t ) Z C ke

k : - o o K

; 2 i r kJ T

(2 )

w h e r e C k ' s r e p r e s e n t t h e F o u r i e r c o e f f i c i en t s of t h e v ar i o u s t e r m s. It i s n o t ed h e r ethat i f the sampling function consists of verys h o r t p u l s e s w h o s e d u r a t i o n l i m e T i s n e g l i g i b l e , t h e v ar i ou s F ou r i e r coef f i c i en t s Ckt en d to b e eq u a l . T h i s con d i t i on i s reco g n i zeda s t h e " i m p u l s e s a m p l i n g " a p p r o x i m a t i o n

u s e d b y m o s t i n v e s t i g a t o r s t o c h a r a c t e r i z et h e sam p l i n g o p er a t i o n . It may b e p oi n t edou t h ere t h at t h i s assu mp t i on i s i d eal b u ta l so t h at i t f i t s t h e p ract i cal s i t u at i on v erywel l an d i s v ery co mm o n l y u sed in an al y t i calt r e a t m e n t s o f s a m p l e d - d a t a s y s t e m s .

T r a n s f o r m m e t h o d s p la y a n i m p o r t a n t p a r ti n t h e an a l y s i s o f l i n e ar sy s t e m s so t h at i t i sp r o f i ta b l e t o e x a m i n e th e F o u r i e r a nd L a p l a c et r a n s f o r m s o f s a m p l e d t i m e f u nc t io n s . T h eF o u r i e r t r a n s f o r m o f t h e s a m p l e d f u nc ti o ngiven in Eq. (2) is

2 7rkF*(jw) • T C k F( jw+ .

k = - c o

(3 )

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whe r e F(jw) i s the Fo ur ie r t r a n s fo r m o f thecon tinuous t ime function befor e sam pling andF*(jw) i s the Fo ur ie r t r a n s fo r m o f thesam pled function and k has only int egr alva lu e s . This r e s ul t i s i .s eful fo r i l lus t r a t ing

the band l imiting effects of the samplingo pe r a t io n . The r e s ul t i s s ho wn gr a p hic a l lyin Fig . 3 whe r e the s pe c t r u m o f the s a m ple dfunction e*(t) is plotted.

It is seen that the spec tru m of the o r ig ina ls i g n a l p r i o r to s a m p l i n g i s r e p e a t e d p e r i o d ically after s am plin g. If the pulse whichs a m ple s the s igna l i s ve r y s ho r t , the m a g ni tude s o f the r e pe a te d s pe c t r a a r e a lmo s tthe s a m e ma g ni tude . Thu s , the s a mp le ds igna l ha s a Fo ur ie r s pe c t r u m F*(w) whichc o nta ins c o mpo ne nts wi th f r e que nc ie s no tc o nta ine d in the o r ig in a l s igna l . In o r d e r toe xt r a c t the o r ig ina l s igna l f r o m the puls et r a i n r e s u l t i n g f r o m t h e s a m p l in g o p e r a t i o n ,a low -pas s fi l ter m ust be emp lo yed. If thes pe c t r a a r e f in i te , tha t i s , if they te r m ina tec o mple te ly a t s o me f in i te ma ximum f r e que nc yF , i t fo l lows that a perfect f i l ter can conc e i v a b l y s e p a r a t e a n d r e c o v e r t h e o r i g i n a ls ignal pro vided the samp ling frequency istwice the maximum frequency F containedin the s igna l s pe c t r um . This i s the we l l -known Shannon sam pling th eo re m and hassignificance in this applicatio n that if thes a mpl ing r a te i s to o lo w r e la t iv e to the f r e quen cies co ntained in the s ignal beings a m ple d, lo s s in info r ma t io n r e s ul ts due tothe impo s s ib i l i ty o f c o mple te ly s e pa r a t ingo ut the s igna l s pe c t r um by f i l t r a t io n . Th us ,a go o d r u le i s tha t in p r a c t ic a l s i tua t io ns , i ts ho uld no t be e xpe c te d to de s ign c o nt r o ls ys te ms ha ving a n e ffe c t ive f r e que nc y r e s p o n s e m uc h g r e a t e r t h an o n e - q u a r t e r t o o n e -tenth the frequency of the sam pling o per atio n.

e qua l to uni ty . Th is ma the m a t ic a l a pp r o xima t io n i s e xt r e me ly us e ful be c a us e i t i sma the ma t ic a l ly c o nve nie nt a nd e xpe die nt .E x p r e s s e d m a t h e m a t i c a l l y , p ( t ) b e c o m e s

+cop (t ) « Z 8 ( t - n T ) .n= -co

(4 )

T h u s , a signal e(t) which is being sampledby a sam plin g function of th is type is given by

coe*(tl = I e (nT ) S ( 1 - n T ) .

n = o

(5)

If the original signal function e(t) has aLa p la c e t r a ns fo r m E(s ) , the n the s a mple dfunction e*(t) has a Laplace t ransform E*(s)which is given by

+ c 0 -nTsE*(s) -- E e(nT)e

n =o

(6)

I t i s fo r tuna te tha t fo r mo s t s ys te ms theinf in i te s e r ie s c a n be e xp r e s s e d in c lo s e d

f o r m t h u s m a k i ng m a t h e m a t i c a l o p e r a t i o n sr e l a t i v e l y s t r a i g h t f o r w a r d . F o r i n s t a n c e ,if the function e(t) is a step function, thene(nT) is a sequence of unit values for a l lpo s i t ive va lue s o f n . E*(s ) be c o me s fo rt h i s c a s e

E'(s)

which is equal to

oo• 1 e '

n = onTs

(7)

It i s c o mmo n in s a mple d-da ta s y s te m s touti l ize a sampling function which is veryshort so that i t may be expressed by as e que nc e o f de l ta funct io ns who s e a r e a s a r e

E"(s )I - e - T S

(8)

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S i m i l a r l y , i t i s p o s s i b l e t o e x p r e s s a llseq u en ce s of t h i s t y p e fo r s i g n a l s wh o seL a p l ac e tr a n s f o r m s a r e r a t i o s o f r a t i o n a lp ol y n omi al s i n s i n c l osed fo rm as a ra t i oo f r a t i o n a l p o l y n o m i a l s i n e T s . F o r t h i s

r e a s o n , t a b l e s o f t r a n s f o r m s o f i m p u l s esam p l ed t r an sfe r fu n ct io n s can b e d ra f t ed .B e c a u s e t h e t r a n s f o r m s o f t h e s a m p l e dsi g n a l s a l wa y s co n t a i n t h e co mp l ex f req u en cys in the form e T s , t h e l a t t e r i s rep l aced b ya n a u x i l ia r y v a r i a b l e " z " w hi ch h a s b e enused sin ce the ea r ly wo rk in the f ield byHu rw icz (Ref. 1). Th us Eq . (8) can ber e w r i t t e n a s

E { z ) « I

- z

(9 )w h e r e z h a s b e e n r e p l a c e d e Ts

T r a n s f o r m s o f s a m p l e d t i m e f un c ti o n sa r e th us r e f e r r e d t o a s " z - t r a n s f o r m s " a nda r e w r i t te n i n t e r m s o f z r a t h e r t ha n e T s .T ab l e 1 co n t a i n s a l i s t o f co mm o n l y u sedz - t r a n s f o r m s . R e f s . 2 a nd 11 g iv e m o r ec o m p l e t e t a b l e s . T h e u s e o f c l o s e d t r a n s f o r m s t o d e s c r i b e t h e s a m p l e d f u n c ti o n s a tt h e ou t p u t of t h e sa m p l e r i s a p owerfu l t oo lu s e d t o a n a l y z e a n d s y n t h e s i z e s a m p l e d - d a t a

s y s t e m s .

Refer r i n g on ce ag ai n t o t h e sy s t em sh owni n F i g . 1 , it i s se en t h at t h e sam p l ed e r r o rfu n ct i on i s ap p l i ed t o a d i g i t a l con t ro l l e rw h ic h p r o c e s s e s t h e p u l se s e q u e n c e a ndd e l i v e r s a m od i f ied p u l se seq u e n ce t o t h edata ho ld. Much m o r e will be said abo ut thed i g i t a l con t ro l l e r l a t e r an d a t t h i s p o i n t i twi l l su f f i ce t o s t a t e t h at a d i g i t a l p rog rami s i mp l emen t ed b y t h i s con t ro l l e r su ch t h att h e ou t p u t seq u en ce cau ses t h e sy s t em t o

p e r f o r m in s o m e p r e s c r i b e d m a n n e r o v e r a l l .T h e o u t p ut o f t he n u m e r i c a l o p e r a t i o n s

c a r r i e d o u t by th e d i g it a l c o n t r o l l e r i s as e q u e n c e o f s h o r t p u l s e s , o r s ta t e d dif

f e r e n t l y , a s e q u e n c e o f n u m b e r s . W e r e t h e s e

sh o r t p u l se s to b e ap p l ied d i rec t l y t o a p l an twhich mus t be co ntr o l led, i t is evident thatt h e p l an t wou l d b e su b j ect ed to se v e r e sh o ck s .T h e " d u t y cy cl e" of t h e comman d s i g n alw o u l d b e t o o l o w f o r p r a c t i c a l p u r p o s e s . F o r

i n s t an ce, i f t h e con t ro l l ed p l an t were ane l e c t r i c m o t o r , t he a r m a t u r e w o uld b e s u b j ect ed t o a seq u en ce of sh or t p u l ses wh i chwou l d p rod u ce on l y a smal l av erag e p owero r t o r q u e . In o r d e r t o o p e r a t e t he p la n t m o r eeff i c ien t l y , i t i s n e ce ssa r y t o smo o t h ou t th epuls e se que nce , o r f i l l in the vo ids betweensamp l es wi t h an ex t rap ol a t ed t i me fu n ct i ont h at a p p r o x i m a t e s a c o n t in u o u s f un ct io n p r i o rt o s a m p l i n g .

T h e s y s t e m t h a t a c c o m p l i s h e s t h i s d a t ar e c o n s t r u c t i o n o r e x t r a p o l a t i o n i s k no w n a s

a " d a t a h o l d , " " d a t a e x t r a p o l a t o r , " o r" d e sam p l i n g f i l t e r . " T h e fun cti on of t h i sd ev i ce may b es t b e d es cr i b ed by s t a t i n g t h ati t a t t e m p t s t o r e c o n s t r u c t t h e o r i g i n a l c o n t inuous t ime function f rom which the pulseseq u en c e wa s d er i v ed . In g en er al , i t can n otbe expected that the function be accuratelyr e c o n s t r u c t e d s o t h a t o n ly a n a p p r o x i m a t i o nw i l l r e s u l t .

T h e s i m p l es t t y p e of d at a h o l d sy s t em i st he " c l a m p " c i r c u i t w h ic h a s s u m e s th a t t h e

value of the function during the samplinginterval is equal to the value at the samplingi n s t an t a t t h e s t a r t o f t h e i n t e rv al . T h u s , t h erecon st ru ct ed fu n ct i on wi l l ap p ear as a" s t a i r c a s e " f u nc ti o n a s s ho w n in F i g . 4 .T h i s i s t h e t y p e of recon st ru ct i on ob t a i n edb y s t o r i n g t h e l a t e s t p u l se in a d i g i t a l re g i s t e ru n t i l t h e on set o f a n ew samp l e wh i ch ref resh es t h e n u mb er h el d b y t h e reg i s t e r t ot h e n ew v al u e. T h e re co n st r u ct e d fu n ct ioncon t ai n s sp u r i ou s s i g n al comp on en t s wh i chcan b e reco g n i zed as p er i o d i c wit h a f req u en cyo f t h e s a m p l e r a nd i t s h a r m o n i c s . T h e s e

s p u r i o u s f r e q u e n c i e s a r e c o l l e c ti v e l yr e f e r r e d to a s " r i p p l e , " O n e o f t he i m p o r t a n tp r o b l e m s in s a m p l e d - d a t a c o n t r o l s y s t e m si s t o mai n t a i n t h e r i p p l e wi t h i n sp eci f i cat i o n s .

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A m a t h e m a t i c a l r e p r e s e n t a t i o n o f t he d a t ahold is needed to understand its effect inc o n t r o l s y s t e m s . A s a n i l l u s t r a t i o n , t h et r a ns fe r func t io n o f the s imple c la mp c i r c ui twill be o btained. As s tated pr evio usly , the

clam p cir cuit r e ta in s the value of the functiona s tha t of the p r e vio us s a m ple . It i s a s s um e dh e r e t ha t th e i m p u l s e m o d u l a ti o n r e p r e s e n t a t io n o f the s a mpl ing pr o c e s s i s s uff ic ie nt lya c c u r a te fo r a na lys i s . In th is c a s e , the c la m pc i r c ui t ha s a n impu ls ive r e s po ns e whic h isas shown in Fig . 5. An imp ulse of unit ar eais applied to the clamp circuit whose outputr is es to a unit value and then dr o ps off tozero at the end of the sampling interval tobe reset to i ts new value upon application ofthe ne xt s a mple .

It is seen that the broad pulse result ingfrom the applicatio n of an imp ulse to theclam p cir cui t can be com po sed by addingtwo unit s tep function s, a po sit ive one init iateda t z e r o t ime a nd the o the r ne ga t ive s te p a pplied at a delay equal to the sam pling in ter va l .Thus the impuls ive r e s po ns e o f the c la mp isgiven by

aX ) ; u ( t ) - u ( t - T )h

(10)

wh e r e u(t ) r e pr e s e nt s the unit s te p func t io n .

Obta ining the La pla c e t r a ns fo r m o f theimpuls ive r e s po ns e y ie lds the t r a ns fe r func t io n o f the c la mp c i r c ui t . This i s

G h(s) • - eTS

(11)

Eq. 11 is useful in dete rm inin g the effecto f the c la mp c i r c ui t in a c o mple te c o nt r o lsy ste m . It is inte re stin g to note the frequency

r e s p o n s e o f a c l a m p c i r c u i t . T h i s i s o b tained by r epla cing the co mplex frequency sby jw. Doing so and applying s im ple t r ig o n o m e t r i c t r a n s f o r m a t i o n s , t h e r e r e s u l t s t hee x p r e s s i o n

• h « w • T [w-]-jwT/2

(12)

The a mpl i tude a nd pha s e r e s po ns e o f theclam p cir cu it a r e shown in Fig . 6. Effectivelythe r e s po ns e i s that o f a lo w-p a s s f i l te r a sh a d b e e n p r e d i c t e d f r o m p r i o r d i s c u s s i o n s .

On the other hand, the clamp circuit is not ape r fe c t lo w-pa s s f i l te r be c a us e i t a t te nua te sin the pa s s ba nd a nd a dm its s pur io us h ighe rs a mpl ing f r e q ue nc ie s . The la t te r c o m bineto fo r m the r ipple in the r e c o ns t r uc te ds igna l . The pha s e la g whic h c o nt r ibu te s toinstabil i ty is s een to r ea ch a value of irr a dia n s a t f r e que nc ie s e qua l to the sa mpl ingfrequ ency. Stated differently, t his phase lagr e p r e s e nt s a t ime la g e qua l to o ne -ha l f thes a m p l i n g i n t e r v a l .

It should be pointed out here that mores o phis t ic a te d me tho ds o f r e c o ns t r uc t ing thes igna l f r o m a s e que nc e o f s a mple s e xis t(Refs . 13 and 27). In ge ne ra l , such datah o l d s r e q u i r e a c o n s i d e r a b l e n u m b e r o f p a s ts a m p l e s to a c c o m p l i s h t h e r e c o n s t r u c t i o nwith a r e s ul ta nt te nde nc y to c a us e o s c i l la t io n in lo w-pa s s fe e dba c k c o nt r o l s ys te m se mplo ying the m . It i s e mp ha s iz e d he r e tha tthe m ain pu rp o se of the data hold in a feedba ck c o nt r o l s y s te m is to imp r o ve the duty

c yc le o f the c o nt r o l le d s ys te m r a th e r tha nthe a c c ur a te r e pr o duc t io n o f a s igna l . Thel a t t e r p r o b l e m i s m o s t i m p o r t a n t in o p e n -c y c l e c o m m u n i c a t i o n s s y s t e m s w h e r e r e c o ve r y o f a s igna l r e g a r dle s s o f the t imed e l a y i s t h e m a j o r p r o b l e m .

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I n f e e d b a c k s y s t e m s w h e r e t h e s a m p l e ran d d at a h ol d a re i n cl u d ed i n s i d e t h e con t ro ll o o p , t i m e l a g s r e s u l t i n g f r o m m o r e s o p h i s t i c a t e d r e c o n s t r u c t i o n o f t h e s i g n a l f r o ms a m p l e d d a t a g e n e r a l l y r e s u l t in m o r e dif

f icu lt y t h an i s w ar ra n t e d . As a r esu l t , i t i sa n a l m o s t u n i v e r s a l c u s t o m t o c o n s i d e r th ecl am p ci r cu i t a su f f ic i en t , t hou g h n ot p er fect ,d at a h o l d sy s t e m . In sy s t em s emp l o y i n g ad i g it a l c o m p u t e r a s a c o n t r o l l e r , i t w o u ld bediff icul t to at tempt to include a system whichd o es m o r e t h an i n cl u d e t h e con t en t s of ar e g i s t e r a s t h e e x t r a p o l a t e d s a m p l e d t im efunction.

T h e r e m a i n i n g e l e m e n t s o f t h e t y p i c a l

s a m p l e d - d a t a c o n t r o l s y s t e m s ho w n in F i g . 1ar e t h e con t ro l l ed p l an t wh ich i s an an al ogd ev i ce by i t s v er y n at u r e . W h i le n o t a l way sl i n e a r , i t i s g e n e r a l l y c o n s i d e r e d s o in o r d e rt o e x p e d i t e a n a l y s i s a nd s y n t h e s i s . G e n e r a l l y ,f o r t h o s e c a s e s w h e r e t h e p l a n t i s n o n l i n e a r ,i t c a n b e s tu d i e d by m e a n s o f l i n e a r p e r t u r b a t ion m o d el s . In t h e b ody of t h eo ry a v ai l ab l eat t h e p r ese n t t i m e, t h e p lan t G i s co n si d e re da l i n e ar sy s t e m d es cr i b ed fu ll y b y i t s t r an sfe rf un c ti o n e x p r e s s e d a s a L a p l a c e t r a n s f o r m .

T h e o t h er e l em en t sh own in F i g . 1 i s t h ee r r o r e l e m e n t . W h il e s ho w n h e r e a s as i mp l e d ev i ce wh i ch t ak es t h e d i f fe ren ceb et ween a referen ce i n p u t an d a con t ro l l edou t p u t , i t may be a re l a t i v e l y co mp l ex co m p u t er , d i g i t a l o r an al og , wh i ch comp u t es ac o n t r o l e r r o r . F o r p e r t u r b at io n m o d e l s,t h e r e e x i s t s a s i m p l e l i n e a r r e l a t i o n b e t w e e nthe input and output which is symbolized int h e f i g u r e . In o r d e r to a n a l y z e t he p e r f o r m a n c e o f s y s t e m s c o n t a i n in g a s a m p l i n g o p e r a t i on , i t i s n e ce ss ar y t o h av e av ai l ab l e th es a m e t e c h n i q u e s a s t h o s e w h i c h a r e a v a i l a b lef o r l i n e a r c o n t i n u o u s f e e d b a c k s y s t e m s . S u b seq u en t s ect i o n s d eal wi t h t h e con v o l u ti onsu mmat i on an d t h e t ran sfer fu n ct i on of t h el i n ear sy s t e m s i n clu d ed i n t h e co n t ro l l oo p ,

t he c o m b i n a t o r i a l t h e o r e m s w h ic h m a ke p o s s i b l e th e s t a t e m e n t o f o v e r a l l r e s p o n s e ,co n d i ti on of s t ab i l it y , and t h e r eq u i r em en t si n a d i g i t a l c o n t r o l l e r , D .

3. THE PULSE TRANSFER FUNCTION

On e of th e mo st p owerfu l r e l a t i o n sh i p swh i ch can b e d er i v ed fo r a l i n ear sy s t em i sthe o ne r ela t ing the output and input t imefu n ct io n s . T h e con cep t i s fa i r l y com mo n b yn ow i n t h e case of con t i n u ou s sy s t ems an d t h et r an sfe r fun cti on w h i ch i s n o rm al l y u sed t oc h a r a c t e r i z e s u c h s y s t e m s i s t he L a p l a c et r a n s f o r m o f t h e i r im p u l s iv e r e s p o n s e . L e s sc o m m o n i s t h e c o r r e s p o n d i n g r e l a t i o n i n

sam p l ed sy s t e m s wh er e an ou t p ut p u l seseq u en ce i s re l a t e d t o t h e i np u t p u l se seq u en cethrough a function known as the "pulse (orp u l sed ) t r an sfe r fu n ct i on " (Refs . 2 an d 5 ) .Hav i n g t h i s re l a t i on sh i p i n a comp act fo rmm a k e s t h e p r o b l e m o f a n a l y s i s o f s a m p l e d -d a t a s y s t e m s n o m o r e d if fi cu lt o r c o m p l e xt h a n f o r c o n t i n u o u s s y s t e m s .

T o d e r i v e t h e p u l se t r an s fer fun cti onr e f e r e n c e i s m a d e t o F i g . 7 . H e r e a l i n e a r

sy s t em w h o se con t i n u ou s t ra n sf er fun cti oni s G (s) i s su b j ect ed t o a seq u en ce of p u l s esr*(t) de riv ed by sam pling the input functionr ( t ) . The output c(t) is then sampled synch ro n o u sl y t o p ro d u ce an ou t p u t p u l ses e q ue n c e c * (t ). T h e t r a n s f o r m s c o r r e s p o n d i ng t o t h ese q u an t i t i es a r e R(s) , R* (s) , C(s)an d C* (s) resp ect i v el y .

T o o b t ai n th e d e s i r e d r e s u l t , a c o n v o l u t i on p rocess i s u sed wh i ch i s c l a r i f i ed b yre fer en c e t o F i g . 8 . E ach p u l se of t h e i np u tseq u en ce i s assu m ed t o b e sh o r t en ou g h sot h at an accep t ab l e ap p rox i mat i on t o t h eresp on se of G(s) t o t h i s p u l se i s t ak en as t h er e s p o n s e t o a n i m p u l s e w h o s e a r e a i s e q u a lt o t h e mag n i t u d e of t h e co r resp on d i n g f i n i t e

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pul se. Th us, a t any samp ling instant m T, thevalue of the output time function c(t) is

k c a n be c ha nged f r o m -n to z e r o . He nc eEq. (16) c a n be r e a r r a nge d to

c ( « T l - ? g [ ( m - n ) T ] r ( n T ) c « ( s ) J { g ( k T ) e - k T s " | [ ? r ( n T ) e - n T s l

(13)

w h e r e g ( m - n ) T i s t h e i m p u l s i v e r e s p o n s eafter an elapsed t im e (m - n)T and r(nT )is the n ' th input p ulse . The low er l imit of nis taken a s z er o s inc e i t Is assu m ed thatr ( t ) i s a ppl ie d a t z e r o t ime .

The output pulse t rain c*(t) can then beexpressed as in Eq. (5) by

oo

c " (n • Z c ( m T ) S { t - m T ) .m = o

(14)

Ta king the La pla c e t r a ns fo r m o f th isi m p u l s e s e q u e n c e t h e r e r e s u l t s

C" (s ) • Z c ( m T )e " m T S

m = o

(15)

Substi tuting for c(mT) the summation In

Eq. (13),

OO CO

C *(s ) • Z Z g [ ( m - n ) T ] r ( nT )erTUO n = o L J

• mTs

(16)

making a change of var iabl e by Intro ducingk = (m - n), Eq. (15) can be rewrit ten

C*(s ) * Z Z g lkT ) r ( nT )k=-n n=o "

kTs -nTse e

(17)

It is noted that s ince g(kT) is zero forne ga t ive a r gume nts due to the r e qui r e me ntfo r phys ic a l r e a l iz a b i l i ty , the lo we r l imi t o f

(18)

The im po r ta nt po in t Is tha t the s e c o nds umma tio n i s r e c o gniz e d to be l ike tha t o fEq. (6) and is the pulse sequenc e r* (t) . Th ef i r s t s umma tio n i s the La pla c e (o r z -)t r a ns fo r m o f the impuls ive r e s p o ns e o f thel i n e a r s y s t e m . T h e e x p r e s s i o n in E q . (1 8)c a n no w be r e wr i t te n a s

o r

C*ls) • G * ( s ) R * { s )

C(z) - 6(2) R (z)

(19)

(20)

wh e r e G*(s ) o r G(z) Is the puls e t r a n s fe rfunction.

E x a m p l e : C o n s i d e r t h e l i n e a r s y s t e m

who s e c o nt inuo us t r a ns fe r func t io n G(s ) i s

si») - i U

a n d w h o s e i m p u l s i v e r e s p o n s e i s

g ( t ) • e - a t .

The pulse t ransfer function Is then

G(z) - Z e - n a T z ' n .n = o

(21)

(22)

(23)

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T h e i n f in i t e su m ma t i o n so o b t a i n ed canb e e x p r e s s e d in c l o s e d f o r m

G{z) -l - e - a T z " 1

(24)

T h e resu l t cou l d h av e b een ob t a i n ed d i r e c t l y f r o m T a b l e 1 j u s t a s f o r o r d i n a r yL a p l a c e t r a n s f o r m s .

4. INVERSION OF Z-TR/JMSFORMS

It i s d e s i r ab l e t o b e ab l e t o i n v er t

z - t r a n s f o r m s r e a d i ly to o b ta in t he c o r r e s p o n d i ng p u l s e s e q u e n c e . A g e n e r a l e x p r e s s i on fo r t h e n ' t h p u l se of t h e seq u e n ce r* ( t )has be en der ive d (R efs. 6 , 11 , and 12) andis given by

r ( n T ) =2 T T r / r "

(z )zn - i

dz

(25)

wh er e T i s a p at h of i n t eg r a t i o n i n t h e z-p l an e

t h at en cl oses a l l t h e s i n g u l ar p o i n t s o f R(z) .T h i s p a th i s t h e u n it c i r c l e fo r s t a b l e s y s t e m s .

F o r p r a c t i c a l p u r p o s e s , h o w e v e r , t h eex p an si on of R(z) i n t o a p ower se r i es i nz" 1 by a s i m p l e p r o c e s s o f l o n g d i v is i o n(Refs. 5 and 15) can be used fo r in ve r sio n.T h i s p r o c e d u r e i s p a r t i c u l a r l y u s e f u l f o rcomp u t i n g t ran s i en t s s i n ce i t y i e l d s t h eini t ial values of the pulse sequence by thea p p l i c a t i o n o f a s i m p l e n u m e r i c a l p r o c e d u r e .T h e p r o c e s s i s b e s t i l l u s t r a t e d by a n e x a m p l e .

E x a m p l e : T h e s y s t e m u n d e r c o n s i d e r a t i o ni s sh own i n F i g . 9 . T h e l i n ea r sy s t em co n s i s t s o f a c l a m p c i r c u i t an d a n i n t e g r a t o r

w h o s e c o m b i n e d c o n t i n u o u s t r a n s f e r f u n c t i o ni s

•« • [ * £ ] [+1-(26)

U s e o f T a b l e 1 g i v e s t h e p u l s e t r a n s f e rfu n ct i on co r resp on d i n g t o E q . (2 6 )

GUI - (I-.-1 [ j & f ] -

(27)

If the input r( t) is a unit st ep , the

z- t r an sfo r m of t h e in p ut i s f ro m E q , (9)o r T a b l e 1 ,

RM - r X

T h e o u t p u t z - t r a n s f o r m i s t h u s

C(z) = G(z)R (z)

. T 2 - '

(28)

2z + z"

(29)

T h e i n v ers i on of C(z) y i e l d s t h e ou t p u tseq u e n ce, ap p l y i n g a p ro ce ss of l on g d i v i s i onto Eq. (29),

- 2 z - '. 2 ' + 2 z 2 + 2 z 3 + . . .

l 2 - , - 2 z - X 2 - s

2 Z " 2 - 5 - 4

2 2 - 3 - 2 z " 4

2 2 - 3 - 4 z ' 4 + 2 z " 9

2 z " 4 .. etc-

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T h e r e r e s u l t s a p o w e r s e r i e s in z '

C(2) • T [z ' ' + 2z" 2 + 2 z ' 3 + . . . ] .

(30)

Each coefficient of the sequ ence in Eq.(30) is recognized as the magnitude of theo utput s a m ple a t a s a m ple t ime c o r r e s po ndingt o t h e o r d e r o f z - ' . It is seen that the outputis the s e que nc e o f s a mple s r e s ul t ing f r o msampling a ramp t ime function which is as i tshould be. Th e input and output pulses e que nc e s a r e p lo t te d in Fig . 10 .

The e xa m ple i l lu s t r a te s ho w the f i r s tgro up of values of a pulse seque nce can beo bta ined r e a di ly by a s imp le p r o c e s s o f lo ng

divis io n . This is a num e r ic a l p r o c e du r e whic hc a n be c a r r i e d o ut by huma n c o mp ute r s usingha nd c a lc ula t ing ma c hine s o r by me a ns o fla r g e s c a le d ig i ta l c o mp ute r s whic h ha ve be e np r o g r a m m e d t o im p l e m e n t t h e p r o c e s s o flong divis io n. On the o ther hand, wh er e age ne r a l e xpr e s s io n fo r the n ' th puls e i sde s i r e d , the us e o f the c o nto ur in te gr a l o fEq . (25) can be eva lua ted by the me tho d ofr es id ue s. It should be pointed out als o thatta ble s s uc h a s Ta ble 1 c a n be us e d fo r inv e r s ion if the z- t r an sf o r m is l is ted . If the

t r a n s fo r m is no t l i s te d , i t i s pe r m is s ib let o d e c o m p o s e t h e c o m p l e x t r a n s f o r m i n t opa r t ia l f r a c t io ns a nd to e va lua te e a c h f r a c t io ns e pa r a te ly a nd a dd the r e s ul ts .

5. COMBINATORIAL THEOREMS

The s ys te m s whic h a r e o f ge ne r a l in te r e s tin the de s ign o f s a mple d-da ta c o nt r o l s ys te msg e n e r a l l y h a ve m o r e c o m p l ex c o n f i g u r a t io n sthan the s imp le c a s c a ding o f e le m e nts . Them o r e c o m p l e x r e l a t i o n s h i p s w h i c h r e s u l t f o rfe e dba c k s ys te ms o f va r io us type s c a n bede r ive d by us ing a few s imp le ba s ic r e la t io ns h i p s . These will now be outl ined:

a. The pulse t ra ns fe r function of twoc a s c a de d e le m e nts whic h a r e s e pa r a te d by asynchronous switch is the product of the

puls e t r a ns f e r funct io ns o f e a c h o f the e l e m e nt s . R e fe r r ing to Fig . 11 , i t i s s e e n tha tthe pu lse seq uenc e C, (z) is re late d to R(z)b y G , ( z ) :

C.(z) -- 6, (z) R (z )(31)

A l s o , the pulse sequence C 2 (z ) i s r e la te dto the pulse sequence C, (z) at the input to thes e c o nd e le me nt by

C,{z) -- G 2 ( 2 ) 6 , ( 2 )

(32)

T h u s , the o ve r a l l r e la t io ns hip be twe e n theoutput and input sequences is

C»(2l • [ G , ( Z ) G 2 ( Z ) ] R ( Z )(33)

b. The pulse t r an sfe r function of twoc a s c a de d e le me nts no t s e pa r a te d by a s wi tc hi s th e z - t r a n s f o r m c o r r e s p o n d i n g t o t h epr o duc t o f the c o nt inuo us t r a n s fo r m s . It i snoted that the pulse t ransfer function of sucha co mb ination is not the pro duct of the twoindividua l puls e t r a ns fe r func t io ns . Thu s ,r e fe r r in g to Fig . 12 , a nd r e c a l l ing tha t the

puls e t r a ns fe r func t io n i s the t r a ns fo r m o fthe impuls ive r e s po ns e o f the l ine a r s ys te m,so m e reflectio n will show that thez - t r a ns fo r m o f the c a s c a de d pa i r i s g ive n by

C 2 (z) = [G, 6 2 ( 2 ) ] R ( z )(34)

w h e r e G , G 2 (z ) indic a te s the puls e t r a ns fo r mc o r r e s p o n d i n g t o t h e s y s t e m w h o s e c o n ti n uo u str an sfe r function is G, (s)G 2 (s). It is em pha sized o nce again that G, (z)G 2(z) is not equal

t o G | G 2 (z) as can readily be ver ified byt r y ing a fe w e xa mple s .

By applying the basic theorems givena bo ve , i t i s po s s ib le to o bta in o ve r a l l r e l a tio ns between the input and output of feedback

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c o n t r o l s y s t e m s . F o r i n s t a n c e , t a k in g t h ee r r o r - s a m p l e d s y s t e m g i ve n in F i g . 1 3 , t h erelat ion between input R(z) and output C(z)can readily be shown to be

C( z>6 ( 2 )

+ GH(2)R(2)

(35)

w h e r e C ( z ) an d R ( z) a r e t he z - t r a n s f o r m sco r r esp o n d i n g t o t h e ou t p u t an d i np u tsequences, and G(z) and GH(z) are the pulset r a n s f o r m s c o r r e s p o n d i n g t o t h e f e e d f o r w a r dt r an sfe r fu ncti on G(s) an d l oop t r an sfe r fu n ct ion G(s)H(s).

In a s i m i l a r m a n n e r , th e o v e r a ll _ r e l a t i o nbetween input and output of the system shownin F i g . 1 co n t a i n i n g a d i g i t a l s t a b i l i ze r i s

6. STABILITY OF FEEDBA CK CONTROLSYSTEMS

Of p r i m a r y i n t e r e s t in f ee d ba c k c o n t r o ls y s t e m s i s a s i m p l e m e t h o d f o r a s c e r t a i n i n g

t h e s t ab i l i t y of t h e sy s t e m . In l i n ear co n t inuous sy st em s, i t can be state d that if thet ra n sfe r fu nct ion of t h e sy s t em h as p o l es wi t hn e g a t i v e r e a l p a r t s o r s t a t e d d i f f e r e n t l y , a r elo cated on the left half of the co mplex f r eq u en cy p l an e, th e sy s t em i s s t ab l e . T h i ss t a t e m e n t i s a l s o a p p l i c a b l e t o s a m p l e d - d a t asy st e m s ex ce p t t h at t h e t r an sfe r fu nct ion ofa s a m p l e d - d a t a s y s t e m i s t h e r a t i o o f p o l y n omi al s i n e T s ra t h er t h an s an d t h e n u mb erof p o l es t o b e su rv ey ed i s i n f in i t e . T h u s ,s u c h t e s t s a s t h e R o u t h -H u r w i c z c r i t e r i o n

can n ot b e ap p l i ed d i r ect l y wi th ou t so m e co m p l i cat i on . T h e Ny q u is t c r i t e r i o n can b eapplied with sl ight modificat ion and is quiteu sefu l i n th e f i el d of sam p l ed -d at a sy s t em s.

C{2) -D(z)G(z)

I + 0 ( 2 ) 6 ( 2 )R(2]

(36)

wh ere D(z) i s t h e p u l se t r an sfe r fu ncti on of

t h e d i g it a l s t ab i l i z er wh ose d et a i l ed s i g n i f i can ce wi l l b e sh own l a t e r .

T h e i mag i n ary ax i s of t h e s -p l an e map sinto the unit ci rcle in the e T s - o r z - p l a n e .T h e r e f o r e , fo r r o o t s o n t h e s - p l a n e h a v in gn eg at i v e r ea l p a r t s , t h e mag n i tu d e i n t h ee T s - o r z-p l a n e i s l es s t h an u n i t y . S i m i l a r l y ,f o r t h o s e r o o t s h a v in g p o s i t iv e r e a l p a r t s ,

the magnitude in the e

T s

- o r z-p l an e i sg r e a t e r t h a n u n i t y .

T h e r e a r e m a n y o t h e r p o s s i b l e c o n f i g u r a t io n s o f s a m p l e d - d a t a s y s t e m s p o s s i b l eb e s i d e s th e e r r o r sa m p l e d s y s t e m s d e s c r i b e da b o v e . T h e o v e r a l l p u l s e t r a n s f e r f u nc ti o n sf o r a n u m b e r o f s u c h s y s t e m s a r e t a b u la t e di n T ab l e 2 . Gen era l l y sp eak i n g , t h e d es i g nof feedb ack co n t ro l sy s t e m s h av i n g su ch co n f i g u rat i on s i n v ol v es t h e ch oi ce of e l emen t sw h ic h r e s u l t i n s t a b l e s y s t e m s w h o s ed y n am i cal p er fo rm an c e fu lf il ls sp e ci f i cat i o n ss e t b y th e r e q u i r e m e n t s o f t h e s y s t e m . T h econ d i t i on s of s t ab i l i t y an d t h e t ech n i q u es fo ra c h i e v i n g a c c e p t a b l e d y n a m i c a l p e r f o r m a n c ewi l l b e d i sc u ss ed i n su b seq u en t s ec t i o n s .

In g e n e r a l , t h e r e f o r e , t h e c o n d i ti o n f o rs t a b i l i t y o f a l i n e a r s a m p l e d - d a t a s y s t e mi s t h at t h e p o l es of t h e p u l se t ra n sf er fu n ct ion relat ing input and output must l ie insidethe unit ci r cl e of the z- pla ne . Stated dif-fe ren t l y , t h e rooT s of t h e d en o mi n a t o r p o l y n o mi al o f t h e p u l se t ran sfe r fu n ct ion mu sth av e mag n i t u d es wh i ch a re l es s t h an u n i t y .As i n t h e case of t h e Ny q u i s t c r i t e r i on fo rl i n e a r c o n t i n uo u s s y s t e m s , t h e a p p l ic a t io nof on e of t h e Cau ch y t h eo r em s can b e u sedt o asc er t a i n t h e lack of p o l es ou t s i d e t h e u n itc i r c l e t o p r o v e t h e l i n e a r s a m p l e d s y s t e ms t a b l e .

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The the o r e m is a ppl ie d by ma pping acontour shown in Fig. 14 in the G(z)-planewhere G(z) is the pulse t ransfer function ofthe s ys te m unde r s c r u t in y. Any po le s o fG(z) which are outs ide the unit c i rcle will be

lo c a te d in the s ha de d a r e a a nd a r e e nc lo s e dby the c o nto ur . The C a uchy the o r e m whic his us e d s ta te s tha t the ma p o f th is c o nto uro n the G (z )-pla ne wi l l e nc lo s e the o r ig in o fthe G(z)-plane a number of t imes equal tothe difference between the ze r o s and po les ofG(z ) s o e nc lo s e d.

The type s o f c o n f igur a t io ns o f in te r e s t a r ef ee d ba ck s y s t e m s w h o s e o v e r a l l t r a n s f e rfunct io ns a r e g ive n in Ta ble 2 . Th e s e e xp r e s s io ns ha ve a n o ve r a l l t r a ns f e r func tio n ve r y

s im ila r to tho s e fo r c o nt inuo us fe e dbac ks ys te m s in tha t the s ing ula r i t ie s o f in te r e s ts t e m f r o m c h a r a c t e r i s t i c e q u a ti o n s o f t h efo rm 1 + GH(z) = 0. As in the cas e of co nt i n u o u s s y s t e m s , t h e p r o c e d u r e t o d e t e r m i n es ta bi l i ty i s to a s c e r ta in the numbe r o f t im e sthe ma p o f GH(z ) e nc lo s e s the c r i t ic a l po in t-1,0 in the G(z) plane.

The ma p o f the c o nto ur s ho wn in Fig . 14o n the GH(z ) p la ne i s r e fe r r e d to a s the pu ls et r a ns fe r lo c us a nd ha s the s a m e s ta tu s in

s a mple d-da ta s ys te ms a s do e s the Nyquis tplot for co ntinuo us sy ste m s in that thee nc lo s ur e o f the c r i t ic a l po in ts -1 ,0 i s e qua lto the d i ffe r e nc e be twe e n the numbe r o f z e r o sa nd po le s in the r e gio n o u ts ide the unit c i r c l e .If the num ber of po les of GH(z) o uts ide theu n it c i r c l e i s e i t h e r z e r o o r s o m e k no w nfinite num be r , the numb e r of z e r o s o f1 + GH(z) which ar e so located can bea s c e r ta ine d . Ge n e r a l ly s pe a king, in de a l ingwith e le me nts whic h a r e the ms e lve s s ta ble ,the s tabil i ty of the feedback control system

i s m e a s u r e d b y t h e n o n - e n c l o s u r e o f t h ec r i t ic a l po in t -1 ,0 .

T o i l l u s t r a t e t h e p o i n t , a t r a n s f e r l o c u sfo r a s imple s a m ple d -da ta fe edba ck c o nt r o lsys tem is shown in Fi g. 15. The block

dia gr a m o f the s ys te m is s ho wn in th is f igur eand it is seen that the puls e t r an sfe r lo cusdo e s no t e nc lo s e the c r i t ic a l po in t -1 ,0 .Since GH(z) i tself co ntains no po les o uts idethe unit c i r cl e i t is concluded that the o ver all

sy ste m is s tab le. I t is fur ther no ted that ifthe fe e dfo r wa r d ga in we r e r a is e d by a fa c to ro f s l ight ly o ve r two , the s ys te m wo uld be c o meuns ta ble s inc e the t r a ns fe r lo c us wo uld bee nla r ge d to e nc lo s e the c r i t ic a l po in t .

V a r i o u s p a p e r s h a v e c o n s i d e r e d t h e p r o b le m o f s ha ping the puls e t r a ns fe r lo c us bya dding c o nt inuo us ne two r ks N(s) to the c o ntinuous elem ent G(s) (Refs 4, 11, and 16).In a ll c a s e s , t h e p r o c e d u r e b e c o m e s o n e o ft r i a l a nd e r r o r d u e t o t h e u n f o r t u n at e c i r c u m stance that the pulse t ransfer function of thecom bined elem ent s N(s)G(s) is not equal tothe p r o duc t o f the two puls e t r a ns fe r func tio ns N(z) and G(z). Th is m ean s that theins e r t io n o f a c o mpe ns a t ing ne two r k c a nno tbe a s s e s s e d unt il the e n t i r e new puls e t r a ns f e rf u n c t i o n i s r e c o m p u t e d a n d r e p l o t t e d . N e v e r the le s s , de s pi te th is d i f f ic ul ty , the s ta bi l iz a t io n o f s a mple d-da ta s ys te ms by a dding c o nt inuo us c o mpe ns a t ing ne two r ks in the fe e dfo r wa r d e le me nt ge ne r a l ly fo l lo ws a lo ng thel ine s a nd c o nc e pts de ve lo pe d fo r c o nt inuo us

s y s t e m s .

B y fa r the mo r e impo r ta nt me tho d o fs ta bi l iz a t io n whic h is a ppl ic a ble to s a mple d-da ta s ys te ms is tha t e mplo ying a d ig i ta lc o m p u t e r w h i c h i m p l e m e n t s a r e c u r s i o nfo r mula c ho s e n fo r i t s c a pa c i ty no t o nly tos ta bi l iz e the s y s te m but a ls o to s ha pe i t sr e s po ns e in the time do m a in . It i s po in tedo ut he r e tha t e ve n tho ugh s ys te ms mighto the r wis e be c o nt inuo us , it ma y pa y to a c hie ve

p r e c i s e t r a n s i e n t a nd s t e a d y - s t a t e p e r f o r m a nc e to c o nve r t the m in to s a mple d-da tad i g i t a l l y - s t a b i li z e d s y s t e m s . T h e n e xtsectio n de als with the design of the sy ste me mplo ying a d ig i ta l c o mpute r to s ta bi l iz ea nd s ha pe a s a mple d-da ta s ys te m.

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7. DESIGN OF DIGITALLY -STABILIZEDFEEDBACK CONTROL SYSTEMS

A s m e n t i o n e d p r e v i o u s l y , o n e o f t h e m o s ti m p o r t a n t a p p l i c a t i o n s o f s a m p l e d - d a t a c o n

cep t s i s i n t h e an a l y s i s and sy n t h es i s o fs y s t e m s w h ic h i n c lu d e a d i g i t a l o r a n i n t e r m i t t e n t a n a l o g c o m p u t e r f o r c o m p e n s a t i o n(Ref s. 2 , 17, 18, 21 , and 22). Such a co m p u t er i s ca p ab l e of ac cep t i n g a seq u en ce ofn u m b e r s a nd o p e r a t i n g o n t h e m in s o m el i n e a r o r n o n l i n e a r m a n n e r . B y i m p l e m e n t i n ga r e c u r s i o n r e l a t i o n s h i p b e tw e e n th e o u t pu tan d i n pu t seq u en c es , i t i s p o ss i b l e to s t ab i l i zea n d c o m p e n s a t e l i n e a r s y s t e m s .

W h i l e t h e n u mb er of p oss i b l e con f i g u ra

t i on s i s l a r g e , t h e on e whi ch wi l l b e co n si d e re dh as b een sh own i n F i g . 1 an d i s r ep r o d u ce di n somewh at mod i f i ed fo rm i n F i g . 1 6 fo rp u r p o s e s o f d i s c u s s i o n . T h e s t r u c t u r e i s f o ra n e r r o r - s a m p l e d a n d e r r o r - c o m p e n s a t e df e ed b ac k c o n t r o l s y s t e m . F o r p u r p o s e s o fa n a l y s i s t h e p l a n t o r c o n t r o l l e d e l e m e n t i scomb i n ed wi t h t h e d at a h o l d p reced i n g i t an di s c h a r a c t e r i z e d b y a n o v e r a l l c o n t i n u o u st r a n s f e r f un ct io n G ( s ) . T h e d i g i ta l c o n t r o l l e rh as a p u l se t r an s fer fu nct ion D(z) wh i chr e l a t e d t h e n u m b e r s e q u e n c e E * at i ts outputt o t h e n u m b er se q u en ce E * a t i t s i n p u t .T h i s r e l a t i o n s h i p n e e d s s o m e e x p l a n a t i o n a st o i t s s i g n i f i ca n ce.

I t i s assu med t h at t h ere ex i s t s a l i n earrecu rs i on re l a t i on sh i p b et ween t h e i n p u t an doutput sequences given by

e 2[nT] + b,e 2 [ ( n - l )T ]+b 2 e 2 [ ( n -2 )T ]+ . . .

= a 0e,[nT]+ a,e, [ (n- l ) ]T + a 2 e , [ ( n - 2 ) T ] + . . .

(37)

w h e r e e 2 (nT) is the output sample at then 'th in stant et c. , and e,( nT ) is the outputsample at the n ' th instant etc. , and the coef

f icients a 0 , a , , et c. , and b 0 , b , e t c . , a r ec o n s t a n t s . T h e r e c u r s i o n r e l a t i o n g iv en

In E q . (3 7) mak es p oss i b l e t h e so l u t i on ofthe n ' th output sample e 2 (n T ) b y a s i mp l el i n e a r c o m b i n a ti o n o f w e ig h te d p r e v i o u ssamples at the input and output .

T o ob t a i n t h e p u l se t ran sfer fu n ct i onwh i ch i mp l em en t s t h i s re l a t i o n sh i p an d u s i n gt h e i m p u l s e a p p r o x i m a t i o n f o r t h e s a m p l i n gp r o c e s s , t h e s u m m a t i o n o f t he L a p l a c e t r a n s -f o r m s o f a s e q u e n c e o f i m p u l s e s w h o s e a r e ai s eq u al t o t h e v ar i o u s e , ' s and e 2 ' s o v e ra l l i n s t an t s n i s t ak en :

oo oo __, oo -n-zZ e 2 ( n T ) z + b | Z e 2 ( n T ) z + b, L e,(nT) z

n=o n-o z r\-o

oo oo= a 0 Z r te,( nT )z " + a, Z e, (nT) z

- n - i

n^o

+ a 2 Z e . ( n T ) 2n = 0

( 3 8 )

w h e r e t h e i n c r e a s e d n e g a t iv e o r d e r s o f zare u sed fo r t h e p u rp ose of sh i f t i n g t h e

s e q u e n c e s t h e n e c e s s a r y i n t e g r a l n u m b e rof samp l e t i mes t o t ak e i n t o accou n t t h ed el ay ed p u l ses i n d i cat ed i n t h e recu rs i onf o r m u l a .

It i s p o ss i b l e t o fact o r ou t t h e su m ma t i o n sas shown in the fol lowing:

00 n r . . 2 "1

Z e 2 ( n T ) z ' | i + b ,2 + b 22 + . . .J-

co -n r . 2 1= n Z e , (nT )2 | a 0 + a,2 + a z z ' + . . .V

( 3 9 )

T h e su mmat i on s wh i ch h av e b een fact o redou t i n E q . (39 ) a r e rec o g n i zed t o b e t h ez- t r an sfo r m s of t h e ou t p u t an d i np u t p u l ses e q u e n c e s , E 2 ( z ) a n d E , ( z ) r e s p e c t i v e l y .

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Replacing the infinite summations by thesee q u i v a l e n t s a n d r e a r r a n g i n g ,

E2(2){l + b.z"' + b2z-*+ • ••}

» E,(*) fa+„,*-'• atz-«+ •••}(40)

which is e xp r e s s e d a s a puls e t r a ns f e rfunction defined as fo l lows:

- s a 0 + 0,2"' + a 2 z - l +

I + b,2-' + b t 2 - * +

(41)

The puls e t r a ns f e r func tio n o f a l ine a rd i g i ta l p r o c e s s s o e x p r e s s e d i s c o n v e n ie n tl yha ndle d in a na lys is a nd s ynthe s is . The l ine a rr e c u r s i o n f o r m u l a w h i ch i t r e p r e s e n t s c a nr e a di ly be imple m e nte d by a n in te r m it te ntc o mpute r whe the r i t be o f the d ig i ta l o ra n a lo g f o r m o r s o m e c o n v e n ie n t c o m b i n a ti o n .The pr o ble m o f s ynthe s is r e s o lve s i t s e l fto s pe c i fy ing the va r io us c o ns ta nts , a 0 , a , ,a 2 , . . . and b , , b 2 , w h ic h a r e r e q u i r e d t o a c h ie v e s o m e o v e r a l l p e r f o r m a n c es pe c if ic a t io n .

T h e o v e r a l l r e s p o n s e o f th e s y s t e m i se x p r e s s e d b y t h e o v e r a l l p u l s e t r a n s f e rfunction C(z)/R(z) which is wr it ten as K(z).D e s i r a b l e f o r m s o f K ( z ) a r e k n o w n a s p r o t o types and can be specified on the basis ofr e q u i r e m e n t s o f t h e s y s t e m p e r f o r m a n c e ,such as the abil i ty to sett le in a specifiedt i m e , the abil i ty to fo l low perfectly cer tainspecified inputs in the s teady s tat e and t imed o m a i n, a nd o t h e r s p e c i f i c at i o n s . T h e i m p o r

ta nt po int i s tha t o nc e a n o ve r a l l puls e t r a ns fe rfunction K(z) is specified, it is possible toa c hie ve I t by me a ns o f a d ig i ta l s ta bi l iz e r ,s ubje c t to c e r ta in l imi ta t io ns whic h wil l bed i s c u s s e d l a t e r .

T o d e m o n s t r a t e ho w t h is i s d o n e , c o n s i d e rthe o ve r a l l r e s p o ns e o f the s ys te m s ho wnin Fig. 16,

K(Z) -- ^ - D(z ) G<

z )

' R(2) ' 1 + 0 (2 )6 (2 )(42)

Since D(z) is adjustable and under thec o nt r o l o f the d e s ign e r Eq. (43) i s s o lve dfor D(z) in terms of K(z):

D(2) --I K(2)

6(2) I - K(2)

(43)

T h e r e l a t i v e l y s i m p l e p r o c e s s o u t l i n e da bo ve is subje c t to c e r ta in l im i ta t io ns whic hwill be given lat er . Befo re going into the sel imi ta t io ns , i t i s wo r thwhi le to d is c us s no wt h e f o r m s o f t h e d e s i r e d o v e r a l l p u l se t r a n s f e rfunctions K(z) which ar e to be achieved .

T h e o v e r a l l r e s p o n s e K (z) d e t e r m i n e s t h emanner in which the output of a feedbacksys tem is affected by the input. Fo r instan ce,in a d u p l i c a t o r c o n t r o l s y s t e m , t h e d e s i r e dr e s po ns e i s tha t the o utput r e p r o duc e theinput a t a l l t im e s . Tr a n s ie n ts whic h o c c ur

s ho uld d is a ppe a r a s quic kly a s po s s ib le o rshould fulfi ll set s pecific atio ns of r id e t im ea nd o ve r s ho o t . On the o the r ha nd, in thec a s e o f a r e g u l a t o r w h e r e t h e i n p u ts a r eunde s i r a ble d is tu r ba nc e s , the o utput s ho uldbe c o m ple te ly ins e ns i t ive to the d is tu r ba nc einputs . Any t r a n s ie n ts s ho uld l ike wis e beminimu m o r fu lf il l s e t s pe c i f ic a t io ns . Th us ,K(z ) c o nta ins the o ve r a l l r e qu i r e m e nts o f thec o nt r o l s y s te m a nd it c a n be r e a l iz e d byi n tr o d u c i n g a d i g it a l c o n t r o l l e r .

An i m p o r t a n t c l a s s o f o v e r a l l r e s p o n s efunctions is the one which can be refer redto as having finite sett l ing t im e. In a sy ste mof this type, th e t r an sie nt decays in a finitet i m e . To cla r ify the point , i t should be notedtha t the t r a ns ie n t r e fe r r e d to i s the s a mple d

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output so that se t t l ing tak es pl ace at thes a m p l e t i m e s o n l y . B e t w e e n s a m p l i n ginstants, the output wil l contain r ipple, whichi n m o s t s y s t e m s w i l l a l s o d i s a p p e a r g r a d u a l l y . F i n i t e s e t t l in g t i m e s y s t e m s a r e

c h a r a c t e r i z e d b y r e s p o n s e f u nc t io n s K( z)w h ic h a r e p o l y n o m i a l s . F o r i n s t a n c e , t h ev ery s i mp l es t fu n ct i on i s

K(z)(44)

F o r a fun cti on of t h i s t y p e, t h e res p o n ses e t t l e s t o i t s s t e a d y - s t a t e v a l u e a f t e r o n es a m p l e i n t e r v a l . A c t u a ll y , a s a g e n e r a lr u l e , t h e con t i n u ou s ou t p u t c( t ) osci l l a t esa bo u t t h e s t e a d y - s t a t e v a lu e b ut c r o s s e st h e s t e a d y - s t a t e v a l u e a t s a m p l i n g i n s t a n t s .

T h e t e r m " f i n i t e s e t t l in g t i m e " u s e d int h e s e d i s c u s s i o n s r e f e r s t o th e o u t pu t a sr e a c h e d a t s a m p l e i n s t a n t s o n l y r a t h e r t ha nt h e con t i n u ou s ou t p u t itself.

T h e r e q u i r e m e n t t h a t r e s p o n s e f u n c t i o n sof d u p l i cat o r sy s t ems fo l l ow su ch i n p u t sa s s t e p f u n c t io n s , r a m p o r c o n s t a n t v el o c i t yi n p u t fu n ct i on s , an d p arab ol i c o r con st an tacc el er a t i o n fu n cti on s can b e me t b y co n s i d e r i n g t h e c o n t r o l e r r o r E , in F i g . 1 6 .It can readily be shown the E, (z) is given by

£ ( * ) • R(2 ) [ l -K (2 ) ] ,(45)

If the system fol lows the input perfect lyi n t h e s t e a d y s t a t e , t h e c o n t r o l e r r o r g o e st o z er o . T h u s ap p l yi n g t h e f in al v al u e t h eo re mfor z- t r an sf o r m s (Ref. 1 1 ), t h e f in al v al u eo f t h e c o n t r o l e r r o r E , i s g i ve n b y

• [ ( •

ss

) E . ( 2 ) l z - i • (46)

Now for the c as e w her e the input is as t e p , r a m p , o r p a r a b o l a ,

R(2) 'M ( 2 )

( I - z- ' ) n

wh ere M (z) i s a p o l y n omi al i n z H and n isthe d eg r ee of the input (n = 1 fo r s tep , 2 fora r a m p e t c . ) . T h u s , t he s t e a d y - s t a t e e r r o r i s

's s

[l ~ K(2)]

( I - Z " 1 ) " " 1M ( 2 ) .

(48)

(47)

No w in o r d e r f o r t h e s t e a d y - s t a t e e r r o ri n fo l lowi n g t o b e ze r o , i t i s n e ce ssa r y t h at[ 1 - K(z)] co ntain the fac to r (1 - z' 1 ) tot h e on e h i g h er d eg ree as t h at o f t h e d en ominato r of the input function o r equal in de gr eet o t h e d e n o m i n a t o r o f t he z - t r a n s f o r m o f t h einput , R(z).

F o r c o n v e n i e n c e , a ta b u l at io n o f m i n i m a lp r o t o t y p e s w h i c h s a t i s f y t h e r e q u i r e m e n t sabo ve a r e given in Ta ble 3 (Ref. 17). Th es er e s p o n s e f u n c t i o n s a r e c a l l e d m i n i m a lb ecau se t h ey rep r es en t t h e s i m p l es t fu nct ion st h at can b e u sed wh i ch wi l l sa t i s fy t h e r eq u i r eme n t . It i s u n d ers t o o d t h at mu ch m o reco mple x functio ns can be dev ised which wil ls a t is f y t h e r e q u i r e m e n t s i n c e t he l a t t e r s t a t e sonly that the term (1 - z*1 ) mu st b e a fact o rof [ l - K(z)] i n t h e co r r ec t d eg re e b ut t h ata n y o t h e r f a c t o r s m a y be c o n t a i n e d if d e s i r e d .I t i s t h u s p oss i b l e t o sp eci fy ov era l l t ran sferfunctions having f ini te s et t l in g t im e and thec a p a c i ty f o r f o l lo w i n g s t e p s , r a m p s , o r o t h e rp ol y n omi al i n p u t s i n t h e s t ead y s t a t e .

Gen eral l y sp eak i n g , sy s t ems h av i n g f i n i t eset t l i n g t i me an d con d i t i on s wh i ch p ermi t t h eper fect stead y sta te fol lowing of a givenh i g h e r o r d e r p o l y n o m i a l s u c h a s a c o n s t a n taccel era t i on i n p u t wi l l ex h i b i t p oor t ran s i en tp e r f o r m a n c e w h en s u b j e c te d t o a p o l y n o m i a lo f l o w e r o r d e r s u c h a s a s t e p f un c ti o n . F o rs y s t e m s w h ic h a r e s u b je c t e d to a v a r i e t y o fi n p u t s , t h e u s e o f a " s t a l e n e s s f a c t o r " i n t h eo v e r a l l t r a n s f e r f u n c t i o n i s r e c o m m e n d e d .

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T h e s t a l e n e s s f a c to r i n t r o d u c e s a s m o o t h ing e ffe ct in the o ve r a l l r e s po ns e ve r y s im i la rto tha t p r o duc e d in a c o nt inuo us s ys te m by al o w - p a s s r e s i s t a n c e - c a p a c i t a n c e n e t w o r k .T h e t e r m w a s o r i g i n a l l y i n tr o d u c e d b y B a r k e r

(Ref. 2) and extended by the work of Bertram(Re f. 19) . The pr o to type o ve r a l l t r a n s fe rfunc tio n mus t s a t i s fy a l l the r e qu i r e m e nt so ut l ine d pr e vio us ly fo r the fo l lo wing o f apo lyno mia l but a ls o c o nta in a de no mina to ras shown below

K(z) -( I - 2 - ' ) " F (z )

( I - d 2 - ' ) m

(49)

w h e r e a i s k no w n a s th e " s t a l e n e s s f a c t o r . "A l i t t le thought will show that a response ofthis type does not have a finite sett l ing t imeeven at sam pling i nsta nts but will app ro ach as te a dy- s ta te c o ndi t io n a t inf ini te t ime a s i sthe c a s e in a l l c o nt inuo us l ine a r s y s te m s .

B e r t r a m s ho ws tha t the c o ns ta nt a s ho uldhave value s ran ging fro m 0.4 to 0.6 and thatthe o r de r o f the s ta le ne s s fa c to r te r m in theden o m inato r of K(z) need be not high er thant h e f i r s t a l t h o u g h B a r k e r r e c o m m e n d s h i g h e r

o r d e r s . In a ny c a s e , t h e c o m p r o m i s e b et w e end e s i r a b l e s t e p , r a m p , a n d c o n s t a n t a c c e l e r a t ion response functions can be achieved bythe in t r o duc t io n o f s ta le ne s s fa c to r s a s indi c a te d . I t i s e mp ha s iz e d tha t the o ve r a l lre spo ns e functions can be achieved by theint r o duc t io n o f a d ig i ta l s ta b i l iz e r e le me nta nd the s ubs e que nt m a te r i a l indic a te s ho wthe t r a ns fe r func t io ns fo r s uc h de vic e s a r eo b t a i n e d .

The basic specifica tio n of the dig ital

c o nt r o l t r a ns f e r func tio n i s c o nta ine d in Eq.(44). Onc e the o ve r a l l t r a ns fe r func t io nK(z) is specified and the plant t ransfer function G(z) is known, the re qu ire d digitals ta bi l iz e r t r a ns fe r func t io n i s a ls o s pe c i f ie d .T h e r e a r e h o w e v e r , c e r t a i n l i m i t a t i o n s i n

the fre e cho ice of K(z). After satisfying ther e qu i r e m e n ts fo r fo l lo wing the po lyno mia linputs in the s teady s ta te , the addit ionalr e qu i r e m e nt s which K(z ) mus t s a t i sfy a r egiven as follo ws (Re fs. 17 and 19).

a . The o ve r a l l puls e t r a ns fe r func tio nK(z ) mus t c o nta in a s z e r o s a l l tho s e z e r o so f G(z ) which a r e o n o r o uts ide the uni tc i r c l e i n t h e z - p l a n e .

b . Th e function [l - K(z)l mu st co ntain as ze r o s al l tho se po les of G(z) whichl ie o n o r o uts ide the unit c i r c le in the z -pla ne .

Onc e the s e l imi ta t io ns ha ve be e n me t ,the d ig i ta l p r o gr a m fo r s ta bi l iz a t io n a nd

shaping, D (z), can be specified . The designo f s uch a c o nt r o l le r c a n be s t be i l lus t r a te dby me a ns o f a n e xa mple .

Exa m ple : The s ys te m to be de s igne d isshown in Fig. 17 wh ere i t is seen that thepla nt to be c o nt r o l le d Is a s ingle in te gr a to rwith two equal t ime de lay s. The plant ispr e c e de d by a c la m p ho ld c i r c ui t and thee r r o r i s s a m p l e d a nd p r o c e s s e d in a d i g i ta ls ta b i l iz e r ma r k e d D(z ) . It c a n be r e a d i lya s c e r ta ine d tha t e ve n if the s ys te m we r e

c o nt inuo us w i th the s a m ple a nd ho ld o pe r a t io n r e mo ve d a nd r e pla c e d wi th a d i r e c tc o nne c t io n , the s ys te m wo uld be uns ta b le .In t r o duc t io n o f s a mpl ing ma ke s the s ys te me ve n mo r e uns ta ble tha n the c o nt inuo uss y s t e m .

I t i s de s i r e d to de s ign the d ig i ta l p r o gr a mrepresented by D(z) which will make thesy ste m sta ble , have a finite sett l in g t ime andbe capable of fo l lowing a constant velocity(ra mp ) input per fectly in the s teady s ta te .The pulsed t r an sfe r function G(z) of the

plant and da ta ho ld co mbin ed is given by

6(2) = /I - e'Ts

s « ( s - l )

(50)

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T h i s t r a n s f o r m can be found in the c o m plete tables given by B a r k e r (Ref. 2) or by

e x p a n s i o n of G(s) i n t o p a r t i a l f r a c t i o n s and

ob t ai n i n g the p u l s e t r a n s f e r f u n ct io n s f r o mt h e m o r e r e s t r i c t e d T a b l e 1. T h i s p u l s e

t ran sfer fu n ct i on is

6(2)( I - 2 . 3 4 Z - ' ) ( 1 -» Q . I 6 Z ' ' ) Z "

(I - 2 _ l ) ( l - 0 . 3 6 8 2 " ' )

(51)

E x a m i n i n g the p u l se t r an sfe r fu n ct ion ,it is s e e n t h a t t h e r e is a z e r o in G(z) whichl i e s at 2.34 or o u t s i d e the u n it c i r c l e . H e n c et o m e e t the l i m i t a t i o n set f o r t h p r e v i o u s l y ,it is n e c e s s a r y t h a t the o v e r a l l t r a n s f e rfunction K(z) c o n t a i n t h i s z e r o as one of its

z e r o s . T h u s , K(z) m u s t be

K(2) -- (l - 2.342-')(a,2-'+ a22"2)

(52)

w h e r e a, and a2 are u n sp eci f i ed coeff i c i en t s .

By making K(z) a p o l y n o m i a l , a finiteset t l i n g t i me is a s s u r e d . In o r d e r to m e e tt h e sp eci f i cat i on t h at the sy st em fo l l ow a

ramp fu n ct i on p erfect l y in the s t e a d y s t a t e ,[ 1 - K(z)] m u s t c o n t a i n (1 - z"

1) to the

s e co n d o r d e r . T h i s c o n d it io n r e s u l t s int h e r e q u i r e m e n t t h a t

[ l - K(z)] » (I - z" ' ) ( | + bzH)

w h e r e b, is an unspecified coefficient .

(53)

T h e o v e r a l l p u l s e t r a n s f e r f u n c t i o n K(z)

mu st sa t i s fy s i mu l t an eou sl y the c o n d i t i o n si m p o s e d by Eqs. (52) and (53). SubstitutingK(z) f rom Eq. (52) in Eq. (53), t h e r e r e s u l t s

the equali ty

I - a(z- ' -( 2.34 0. + aj z "2

- 2.34 a , z

i + (b ( - 2)z-' + (i -2b , )z -2

+ b,

(54)

wh i ch can be sat i s f i ed if the coeff i c i en t s of

t e r m s of l i k e o r d e r are eq u al , t h u s resu l t i n gin the f o l l o w i n g s i m u l t a n e o u s c o n d i t i o n s :

b, - 2 = - a,

2 b , - l = 2.34a, + a2

b, = - 2.34a2

(55)

T h e r e b e i n g t h r e e e q u a t i o n s and t h r e eu n k n own s, a solut ion is obtained giving valuesfo r the coeff i c i en t s of K(z) as f o l l o w s :

a, = 0.81

a2 = - o.5i

(56)

It is noted that the m i n i m u m n u m b e r of

a r b i t r a r y c o e f f i ci e n ts w e r e c h o s e n to sat i s fyt h e con d i t i on s of the p r o b l e m . If m o r et e r m s in h i g h e r o r d e r s of z"' w e r e u s e d in

K(z) resu l t i n g in a l o n g e r s e t t l i n g t i m e , t h e r ewou l d h av e b een more con st an t s t h an con d i t i o n s and all coeff i c i en t s ex cep t two cou l db e a r b i t r a r i l y s p e c i f i e d . T h i s m e a n s t h a tt r a n s i e n t r e s p o n s e c o u l d be c o n t r o l l e d at

t h e ex p en se of m o r e t e r m s in K(z). As

s o l v e d h e r e , the s y s t e m is a m i n i m a l ones i n c e the set t l i n g l i me is at a m i n i m u m .

With the coeff i c i en t s of K(z) used asgiven in Eq. (56), the o v e r a l l r e s p o n s e oft h e s y s t e m is

K ( 2 ) = 0 . 8 I 2 - 1+ \ 3 B Z ' e

- I . I 9 Z ' S.

(57)

If the input function R(z) is a unit constantv e l o c i t y ( r a m p ) the output of the s y s t e m C(z)

is given by

6(2) •-( 0 . 8 I 2 " 1

+ I . 3 8 2 " 2- I . I 9 2 _ S ) 2 - '

( I - 2-')1 , 2

(58)

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T h e p u l s e s e q u e n c e c o r r e s p o n d i n g t o t h i sr esu lt is plo t ted in F ig . 18 wh er e i t is seentha t the s ys te m pr o duc e s a z e r o fo l lo winge r r o r a t s a m p l e i n s t a n t s a f t e r t h r e e s a m p l einte r v a ls . The c o m ma nd to the p la nt E 2 (z )

is also plotted where i t is seen that an offsetvalue is pr o duced at s teady s tate . I t is thisoffset which makes i t possible for the systemto fol low a ramp input perfectly even thoughthe plant has only a s ingle integration whichwo uld no r m a l ly c a us e the fo l lo wing e r r o r tobe s o me f in i te c o ns ta nt .

A b e t t e r c o m p r o m i s e i s r e a c h e d b y u s i n gs ta le ne s s fa c to r s a s s ugge s te d by B a r k e r a ndBe r tr am (Refs . 2 and 19). Ano ther point ofin te r e s t i s tha t the s ys t e m fo l lo ws the r a m ppe r fe c t ly o nly a t s a mple ins ta nts . B e twe e n

the s e ins ta nts , the s ys te m o s c i l la t e s in a ndo ut of the c o r r e c t po s i t io n a t s a mpl ing f r e quency. Th is effect is known as r ip ple andme a ns a r e a va i la ble to s tudy i t s ma gni tude(Refs . 21 , 22, 23, and 24). Ripple effectsa r e g e n e r a l l y r e d u c e d w h en s t a l e n e s s f a c t o r sa r e u s e d .

The d igi ta l s ta bi l iz e r whic h is r e qui r e dis o btained by sub stitu tio n of K(z) and G(z)in Eq. (43). Doing thi s , the pulse t r an sfe rfunction of the digital s tabil izer D(z) becomes

0(2) "-Q8I - I.I06Z'

1 + 0 . 4 8 5 2 ' 2 - 0 . 6 9 I 2 - 5

I + 0 .3522" 1 - I . I 5 9 2 " 2 - 0 . I 9 3 2 " 3

While the emphasis has been placed on agive n pr o to type e mplo ying the d ig i ta l s ta bi l i z e r i n t h e e r r o r l i n e , o t h e r f o r m s c a n b es imila r ly a na lyz e d a nd s ynthe s iz e d whe n the

digital s t ab il iz er is in the feedback l ine o rwi th c o m bina t io n s ys te m s wh e r e the d ig i ta ls t a b i l i z e r b y p a s s e s a c o n t i n u o u s e r r o r l i n e(Ref. 20).

(59)

The s ignif ic a nc e o f th is puls e t r a ns fe rfunc tio n in te r m s o f a r e c ur s io n fo r mula i sgiven in Eq. (37) and a system diagrams ho wing the s e que nc e o f num e r ic a l o pe r a t io nsis sho wn in Fig . 19 . T his d ia gr a m i l lus t r a te s

the p r o c e s s o f ho ld ing e a c h s a mple da tum,mul t ip ly ing i t by i t s a ppr o pr ia te we ight,delay ing it, and adding all the weigh tednum be r s to o bta in the p r e s e n t o utput s a m plee 2 ( n T ) .

The e xa mple g ive n a bo ve is a n i l lus t r a t io n o f ho w a d ig i ta l c o m pute r c a n be p r o g r a m m e d t o s t a b i l i z e a nd p r o d u c e d e s i r a b l ed y n am i c r e s p o n s e c h a r a c t e r i s t i c s i n a l i n e a rs ys te m . The pr o to type K(z ) which wa s c ho s e n

is a minimal finite settling type in that itfo l lows the input perfectly after a t ransiento f f in i te dur a t io n . Ge ne r a l ly s pe a king, s y s t e m s o f t h i s t y pe h a v e e x c e s s i v e o v e r s h o o t swhen subjected to inputs for which they ar enot specifically d esign ed.

B e ha vio r o f s a m ple d -da ta s ys te m s in thepr e s e nc e o f r a ndo m inputs ha s be e n s tudie dby B a r k e r , Fr a nkl in , a nd Le e s (R e fs . 25 ,26, and 27). The theo ry will not be pr esen tedhe r e exc ept to s ta te that many of the functionsc o mmo nly us e d in the a na lys is o f c o nt inuo us

s ys te m s a r e fo und in s a mple d s ys te m s a swe l l . Fo r ins ta nc e , the r e a r e de fined thec o r r e la t i o n func tio n a nd the po w e r s pe c t r u mfo r s a mple d r a ndo m func t io ns . Opt im iz a t io n p r o c e dur e s by minimiz a t io n o f the me a ns q u a r e s y s t e m e r r o r a r e a v a il a b l e . I t m a ybe s ta te d ge ne r a l ly tha t s a mple d da ta c o nt r o ls ys te m s c a n be o pt imiz e d in muc h the s a mema nne r a s the c o nt inuo us s ys te m s , po s s ib lywith mo r e f le xibil i ty s inc e the r e i s ge ne r a l lya d ig i ta l s ta b i l iz e r whic h c a n imple me nto v e r a l l d e s i r e d t r a n s f e r f u n c t i o n s a s r e q u i r e d

i n t h e o p t i m i z a t i o n p r o c e d u r e .

O t h e r s a m p l e d - d a t a p r o b l e m s w h i c h h a v ebeen s tudied ar e tho se in which a mu ltiplici tyo f s a mp l ing o pe r a t io ns a t d i f fe r e nt s a mpl ingfr e que nc ie s a r e p r e s e nt (R e fs . 28 a nd 29) .

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T h i s s i t u a t i o n a r i s e s i n s y s t e m s h a v i n gs i m u l t a n e o u s l y a s a m p l e d - d a t a i n fo r m a t i o ng at h er i n g d ev i ce su ch as a ra d a r ; a d i g i t a lcomp u t er wh ose ou t p u t d at a ra t e may b el o w e r , e q u a l t o , o r h i g h e r t ha n t h e r a d a r d a ta

r a t e ; a nd o n e o r m o r e i n t e r m i t t e n t d i g it a ld at a l i n k s . W h i l e t h e an a l y s i s o f su ch sy st em s i s n o t s i m p l e , it i s su f f ic i en tl y re g u l art o m a k e t h e d e s i g n o f s u c h s y s t e m s p r a c t i c a l .

8 . AP P L ICAT ION OF S AM P L E D-DAT ASYSTEMS TO THE GUIDANCEOF MISSILES

It i s t h e p u r p o se of t h i s sect i o n t o p o i n to u t t h e a p p l ic a t i o n o f s a m p l e d - d a t a s y s t e m si n t h e g u i d a n c e a n d c o n t r o l o f m i s s i l e s .Gu i d an ce sy s t e m s may b e ro u g h l y d i v id ed i n tot wo c l a ss es , t h o se in wh i ch d at a g at h e r i n gan d cou rse comp u t at i on i s i mp l emen t ed b yd e v i c e s c a r r i e d b y t h e m i s s i l e ; an d t h o s e inwh i ch su ch d ev i ce s a r e s i t u at ed on t h e g ro u n d ,mi x ed sy s t ems h av e comb i n at i on s i n wh i chb o th m i s s i l e - b o r n e a n d g r o u n d e q u ip m e n t a r eu s e d . T y p ic a l m i s s i l e - b o r n e s y s t e m s a r et h ose emp l oy i n g su ch g u i d an ce sy s t ems asp r e p r o g r a m m i n g , h o m i ng , b e a m - r i d i n g , a ndi n e r t i a l an d ce l es t i a l n av i g at i on . Du e t oweight and space l imitat ions, i t is not l ikelyt h at m i s s i l e - b o r n e d ig i t al d a t a a nd p r o c e s s i n gd e v i c e s b e u s ed t o c o m p u t e c o n t r o l c o m m an d s. On t h e o t h er h an d , sy s t e m s u s i n gg r o u n d - b a s e d i n s t r u m e n t a t i o n o f te n h a v es a m p l e d e l e m e n t s s u c h a s d a t a - g a t h e r i n gr a d a r s , d a t a - p r o c e s s i n g c o m p u t e r s , a n dg r o u n d - t o - a i r d a t a l i n k s. F o r t h is r e a s o n ,t h e ex p ect ed in ci d en ce of sam p l ed -d at a sy st em s i s h i g h est i n rad i o co mm an d g u i d an ces y s t e m s .

On e of t h e ad v an t ag es of g rou n d -b asedsy st e m s i s t h at t h ey p ro v i d e a k nown ands t a b l e g r o u n d r e f e r e n c e o n w h ic h c o m m a n d scan be based . Also , the po ssib i l i ty of usingl a r g e r c o m p u t e r s m a k e s p r a c t i c a l t he i m p l e m e n t a t i o n o f m o r e s o p h i s t i c a te d d a t a -p r o ces s i n g , g u i d an ce, and d eci s i o n mak i n g

p r o c e d u r e s . N e v e r t h e l e s s , e v e n th o ug h s p a c ean d wei g h t a re n o t maj o r l i mi t a t i on s , t h ec o m p u t e r s h o u l d b e a s s i m p l e a s p o s s i b l e i nt h e i n t e r e s t s o f e c o n o m y a nd r e l i a b i l i t y .T h e r e f o r e , u n n e c e s s a r i l y h i g h- s p ee d c o m

p u t ers wh i ch wou l d resu l t i n a h i g h ou t p u td at a ra t e a re t o b e av oi d ed an d t h e l owestd a t a r a t e s c o n s i s t e n t w i th s a t i s f a c t o r y o v e r a l lp e r f o r m a n c e o f th e m i s s i l e s y s t e m s ho u l db e u sed . It i s p r ec i se l y u n d er su ch co n d i t i on sof low d at a o r sam p l i n g ra t e s t h at sam p l edd a t a t h e o r y w i l l p r o v i d e a s o u n d a s s e s s m e n to f p e r f o r m a n c e , s t a b i l it y , a nd a c c u r a c y o fg u i d a n c e s y s t e m s .

As an i l l u s t ra t i o n of t h e ap p l i cat io n ofs a m p l e d - d a t a t e c h n i q u e s , c o n s i d e r a h y p o t h et i cal sy s t em emp l oy i n g a d i g i t a l comp u t er

fo r d at a red u ct i on an d fo r t h e comp u l at i on ofs t e e r i n g c o m m a n d s . T h e s y s t e m i s a s s u m e dt o b e a g r o u n d - t o - a i r d e f e n s i v e m i s s i l es y s t e m e m p l o y i n g a r a d i o c o m m a n d g u i d a n c es y s t e m . It i s a s s u m e d t ha t t h e g r o u n d - b a s e dr a d a r d a t a g a t h e r i n g s y s t e m s a r e p e n c i lb e a m t r a c k i n g r a d a r s d i s p o s e d o n t h e g r o u n dt o f o r m a t r i a n g u l a t i o n s y s t e m f o r l o c a t i n gb ot h d efen si v e an d of fen siv e m i s s i l e s . T h ep r i m a r y d a t a a r e r a n g e f r o m e a c h r a d a r s i t et o b o t h mi ss i l es so t h at t h e d i g i t a l comp u t ermu st re d u ce t h ese d at a t o ob t a i n f i x es re l a t i v e

t o a r e f e r e n c e s e t o f g r o u n d c o o r d i n a t e s .A m i ss i l e t r ack i s es t ab l i sh ed an d t h e fu t u rep o si t i on of t h e of fen siv e mi s s i l e p red i c t edby t h e c o m p u t e r . A m i d c o u r s e t r a c k fo r t hed efen si v e m i ss i l e i s co mp u t ed and s t o re d i nt h e c o m p u t e r .

T h e ob j ect o f t h e mi d cou rse g u i d an ce i st o s t ee r t h e d efen si v e m i ss i l e t o fo ll ow t h et r a c k a s a c c u r a t e l y a s p o s s i b l e . D e v i a ti o n so f t h e m i s s i l e f r o m t h i s d e s i r e d t r a c k a r ec o n v e r t e d in to a c o n t r o l e r r o r p e r i o d i c a l lya n d a s t e e r i n g c o m m a n d i s t r a n s m i t t e d t o

t h e m i s s i l e e v e r y c y c l e t i m e o f t h e c o m p u t e r .T h e s t e e r i n g c o m m a n d s a r e c o m p u te d by ad i g i t a l p rog ram D(z) ob t a i n ed b y t ech n i q u esd e s c r i b e d i n p r e v i o u s s e c t i o n s l e a d i n g t ob ot h a s t ab l e an d re l a t i v el y d oci l e sy s t em.

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It is noted that a s teer ing command canbe de l ive r e d by the c o mp ute r o nly a t s a mpl ingi n t e r v a l s d e t e r m i n e d b y t h e c y c l e t i m e r e qui r e d to r e d uc e da ta to o bta in a p r e s e ntpo s i t io n fix, imple m e nt the r e c ur s io n fo r mulawhic h y ie lds the s te e r ing c o mm a nd, a ndt r a n s m i t t h i s c o m m a n d t o t h e m i s s i l e . T h et ime r e qui r e d to c o mple te the c yc le i s thes a mpl ing t ime w hic h, in the in te r e s t o fe c o no my, s ho uld be a s la r ge a s po s s ib lec o n s i s t e n t w i t h r e s p o n s e r e q u i r e m e n t s o ft h e m i s s i l e . T h i s s y s t e m i s a s a m p l e d -data sys tem and must be designed as such.

It sho uld be po inted out that not alls a mple d-da ta s ys te m s a r e o f the ma gni tudeo r com plexity of the one des cr ibe d in thehypo the tic al e xa m ple g iven a bo v e . Fo rins ta nc e , a d ig i ta l t im e -s ha r e d da ta l ink c a nt r a n s m i t a c o m m a n d t o a m i s s i l e o n l ype r io dic a l ly , thus in te r po s ing a s a mpl ingo p e r a t io n in the c o nt r o l lo o p . It s ho uld a ls obe po inte d o ut tha t d ig i ta l c o nt r o l le r s ne e dn o t e m p l o y c o m p l e x l a r g e s c a l e d i g i t a l c o m pu te r s . Simple a nd c o m pa c t s pe c ia l pur po s es a mple d c o nt r o l le r s c a n be de ve lo pe d whichlake adv antage of the re lativ ely s low datar a te tha t i s r e qui r e d in mo s t s y s te m s a ndwhic h us e o pt imum c o mbina t io ns o f d ig i ta l

a nd a nalo g te c hnique s . Ho we ve r , whe r e thec o m p u t e r i s r e q u i r e d fo r t h e p u r p o s e o fr e duc ing da ta a s de s c r ibe d a bo ve , i t i s jus ta s we l l to p r o gr a m the c o nt r o l func t io n a spa r t o f the c yc le .

As s e e n f r o m the b r ie f d is c us s io n a bo ve ,s a m p l e d - d a t a c o n t r o l s y s t e m s a r e a n a t u r a le xte ns io n o f the c o nc e pt o f g r o und c o mma ndguida nc e e mplo ying d igi ta l c o m put e r s . Onthe o the r ha nd, wi th the a c c e le r a te d de ve lo pment in the design and packaging of digital

de v ic e s , i t i s no t be yo nd the r e a lm o f ima gina t io n t o c o n c e iv e o f m i s s i l e - b o r n e d i g i t a lde vic e s fo r the p r o c e s s ing o f da ta o bta ine dd i r e c t l y f r o m i n s t r u m e n t a t i o n c a r r i e d b y t h em i s s i l e a nd t o g e n e r a l e p e r io d i c s t e e r i ng c o m m a n d s . F o r i n s ta n c e , a n i n e r t i a l

s y s t e m r e q u i r i n g t h e d o u b le i n t e g r a t io n o ft h e o u tp u t o f a c c e l e r o m e t e r s to o b t a in p o s i t io n c a n be c a r r i e d o ut by num e r ic a l me tho dse m p l o y i n g o p t i m u m q u a d r a t u r e f o r m u l a s .Th is o p e r a t io n c a n be imple me n te d by me a nso f a d ig i ta l c o mpute r a s c a n o the r da tap r o c e s s i n g o p e r a t i o n s a n d , i n a d d i t i o n , c o m pute s te e r ing c o mma nds whic h wi l l s ta bi l iz ea nd c o nt r o l the m is s i le o n i t s c o ur s e . Ift h i s b e t h e c a s e , e ve n a i r b o r n e s y s t e m swil l r e qu i r e a na ly s is and s ynthe s is by me a nso f s a m p l e d - d a t a t h e o r y .

The e mplo yme nt o f d ig i ta l de vic e s fo rguida nc e a nd c o nt r o l pe r m its the us e o fi n c r e a s e d s o p h i s t ic a t io n i n c o n t r o l . T h epo s s ib i l i ty o f de s igning a da pt ive s ys te m swhic h a djus t the p r o gr a m fo r c o mput ing thes t e e r i n g c o m m a n d in a c c o r d a n c e w i the x t e r n a l c o n d i t io n s i s by n o m e a n s r e m o t e .It i s no w po s s ib le to de s ign l ine a r t im ev a r i a n t s y s t e m s w h ic h a r e p r o g r a m m e d t oc h a n g e t h e i r c h a r a c t e r i s t i c s w i th ti m e ( R e f s .30 and 31). Such sy ste m s can be ma dee c o no mic a l in e xpe ndi tur e o f c o nt r o l e ffo r ta nd te r m ina te the mis s i le f light a c c u r a te ly .In a d d it io n , c o m p u t e r s c a n b e p r o g r a m m e dt o i m p l e m e n t s i m p l e lo g i c a l d e c i s i o n s r e

quir ed befo re and durin g the fl ight of them is si le . It is beca use of the flexibili ty ofthe d ig i ta l c o mpute r tha t s uc h a dva nc e d s ys te m s c a n be im ple me nte d e i the r by us e o f ag r o u n d - b a s e d c o m p u t e r o r , l e s s r e a d i l y ,b y m e a n s o f a m i s s i l e - b o r n e c o m p u t e r .

9. CONCLUSIONS

T h e s a m p l e d - d a t a c o n t r o l s y s t e m h a s aplac e in the g ene ra l ar ea of guidance and

c o n t r o l o f m i s s i l e s . T h e t h e o r y w h ic h h a sbe e n pr e s e nte d i s o nly a s ma l l pa r t o f thebody of theory which is now available andwhich is in the p r o c e s s of de ve lo pme nt .B e c a us e o f the fa c t tha t s a mple d-da ta s ys te msoften go hand in hand with digital computers .

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i t i s l i k ely t h at a h i g h er d eg re e of sop h i s t i c at ion can b e ach i ev ed wi th su ch sy s t em s. Ont h e o t h er h an d , t h e d es i g n er i s o f t en facedwi t h th e n ece ss i t y of d es i g n i n g a sy s t em asa s a m p l e d - d a t a t h eo r y b e c a u s e o n e o r m o r e

o f t h e e l e m e n t s , s u c h a s d a t a g a t h e r i n gr a d a r s a nd d a t a l i n k s , i s i n t e r m i t t e n t i nn a t u r e .

T h e s a m p l e d - d a t a s y s t e m h a s a c l o s er e l a t i o n s h i p t o s a m p l e d s y s t e m s w hi ch a r es t u di e d i n t h e t h e o r y c o m m u n i c a t i o n s . A si n t h e l a t t e r , s o m e i n f o r m a t i o n i s l o s t o rd e t e r i o r a t e d b y t h e p r o c e s s o f s a m p l i n g a s i sindicated by a simple applicat ion of thes a m p l i n g t h e o r e m . Y e t , in m o s t c a s e s , aco mp l et e l y co n t i n u ou s sy s t em h as t h e

c a p a b i l i t y o f t r a n s m i t t i n g a f a r g r e a t e r

a m o u n t o f i n fo r m a t i o n t ha n i s r e a l l y r e q u i r e d .T h e c h o i c e o f s a m p l i n g r a t e s c a n b e a r r i v e dat ra t i o n al l y u n l e ss d i c t a t ed b y o t h er co n s i d e r a t i o n s . H a v in g s e t t l e d o n t h i s f i g u r e ,r a t i o n a l d e s i g n p r o c e d u r e s a r e a v a i l a b l e f o r

t h e d es i g n of l i n ear t i me v a r i an t an d t i m e-i n va r i a n t c o n t r o l a nd t r a n s m i s s i o n s y s t e m s .Not yet ful ly understood is the behavior andd esi g n of n o n l i n ear an d ad ap t i v e sam p l e d -d at a sy s t ems a l t h ou g h some ef fo r t s an dr es u l t s h av e b een ob t a i n ed i n t h i s d i re ct i o n(Ref. 3 2 ). In co n cl u s i on , th e sam p l ed -d at ac o n t r o l s y s t e m h a s c o n s i d e r a b l e p r e s e n tan d p o t en t i a l i m p o r t a n ce i n th e f ie l d ofg u i d an ce an d co n t ro l o f m i ss i l e s an d anu n d er s t an d i n g of th e op e ra t i o n of su ch sy st em s sh o u l d b e of co n si d e ra b l e v al u e t o t h e

d e s i g n e r .

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2 4. J u r y , E . I . , " S y n t h e s i s a nd C r i t i c a l S tu dy o f S a m p l e d - D a t a C o n t r o l S y s t e m s , " T r a n s .A. I . E . E . , P ar t I I , P ap er No. 5 6 -2 0 8 , Ju l y 1 9 5 6 .

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t r o n i c s R e s e a r c h L a b o r a t o r i e s , D e p a r t m e n t of E l e c t r i c a l E n g i n e e r in g , C o l u m bi a U n i v e r s i t y , New York 2 7, N. Y. , Decemb er 1 9 5 4 .

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29 . K r a n c , G , M . , " M u l t i - R a t e S a m p le d S y s t e m s , " T e c h n i c a l R e p o r t T - 1 4 / B , E l e c t r o n i c sR e s e a r c h L a b o r a t o r i e s , D e p a r t m e n t o f E l e c t r i c a l E n g i n ee r i n g , C o l u m b ia U n i ve r s i ty ,New York 27, N. Y. , May 7, 1956.

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B - 3 . S eal , H. L . , " T h e H i s t o r i ca l Dev el op m en t of t h e U se of Ge n er at i n g F u n ct i o n s i nP r o b a b i l i t y T h e o r y , " M i t t ei lu n g e n d e r V e r e in i g un g S c h w e i z e r i n c h e r V e r s i c h e r u n g sM at h emat i k er , Bern , S wi t zer l an d , Vol . 4 9 , p p . 2 0 9 -2 2 8 , 1 9 4 9 .

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Hill Book Co. , New York, N. Y., 1955, Chapter 9.

B - 7 . S a m u e ls o n , P . A . , " F o u n d a t i o n s o f E c o n o m i c A n a l y s i s , " (b o o k ) . H a r v a r d U n i v e r s it yP r e s s , C a m b r i d g e , M a s s a c h u s e t t s , 1 9 4 7 .

B - 8 . M iller , K . S. and Schw arz, R. J . f " A n a l y s i s of S a m p i e d -D a t a S e r v o m e c h a n i s m s , "Journal of Applied Physics , Vol. 21, No. 4, April 1950, pp. 290-294.

B - 9 . B r o w n, R , G. a nd M ur phy, G. L. , " An Appr o xima te T r a ns f e r Func t io n fo r the Ana lys isa nd De s ign o f Puls e d S e r vo s ," T r a n s . A. 1. E . E. , Vo l . 7 1 , Pa r t I I, 1952, pp . 435-440,(1953 section).

B-1 0. Joh nso n, G. W, and Lindorff, D , P . , " T r a n s i e n t A n a l y s is o f S a m p l e d -D a t a C o n t r o lSys te ms ," Tr a ns . A. I . E. E. , Pa r t I I , Vo l . 7 4 , J u ly 1954, pp . 147 -153.

B - l l . J u r y , E . I ., "T he Effec t o f R o o t Lo c a t io ns o n the Tr a ns ie nt R e s po ns e o f Sa mp le d-Da ta Sys te ms ," Tr a ns . A. I . E. E. , Pa r t I I , Vo l . 7 5 , M a r c h 1955,

B - 1 2 . L a g o , G . V , , " A d d i t io n s t o Z - t r a n s f o r m a t i o n T h e o r y f o r S am p l e d - D a ta S y s t e m s , "T r a n s , A. I. E. E. , Par t II , Vol. 75, January 1955, pp. 403-408.

B - 1 3 . T e i c h m a n n , T . . " C l o s e d - L o o p C o n t r o l S y s te m C o n t a in in g a D i g it a l C o m p u t e r , " T r a n s .I . R . E . G r o u p o n E l e c t r o n i c C o m p u t e r s , V o l . E C - 4 , N o . 3 , S e p t em b e r 1 9 55 , p p . 1 0 6 - 1 1 7 .

B -1 4. Sa lz e r , J . M , , "F r e que nc y Ana lys is of Digi tal C o mp ute r s Ope r a t ing in R e a l T im e , "P r o c . I . R. E„ Vol. 42, No. 2, February 1954, pp. 457-466.

B - 1 5 . C h o w , C . K . , " C o n t a c t o r S e r v o m e c h a n i s m s E m p l o y i n g S a m p l e d - D a t a , " T r a n s .A. I . E. E. , Par t II , Vol. 74, March 1954, pp. 51-62.

B -16. R us s e l l , F . A. , " De s ig n C r i te r io n fo r Sta bi l ity o f Sa mple d-Da ta On-Off Se r vo -m e c h a n i s m s , " D o c t o r a l T h e s i s , D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g , C o l u m b i aUniversi ty, New York 27, N. Y., June 1953.

B - 1 7 . J u r y , E . I ., " D i s c r e t e C o m p e n sa t io n o f S a m p le d - D a ta a nd C o n t in u o u s S y s t e m s , "T r a n s . A. I. E. E. , Paper 56-644, 1956.

B - 1 8 . F r e e m a n , H e r b e r t , " M u l t i p o l e S a m p l e d - D a ta C o n t r o l S y s t e m s , " T e c h n i c a l R e p o r tT - 1 2 / B , E l e c t r o n i c s R e s e a r c h L a b o r a t o r y , D e p a r tm e n t o f E l e c t r i c a l E n g i n e er i n g ,Co lumbia U niver si ty, New Yo rk 27, N. Y., Septem ber 30, 1955.

205

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(3 )

(4 )

T a b l e 1 . A b b r e v i a t e d T a b l e o f L a p l a c e a nd z - T r a n s f o r m s

L a p l a c e T i m e

T r a n s f o r m F u n c t i o n z - T r a n s f o r m sF(s) f(l) F * ( z )

(I) 1 S(t) 2

(2 ) e ' n T s S ( t - n T )

I

(7 ) . u ; Sin at

2 0 6

-o

1 - 2 - '

T z"

8 * ( 1 - 2 - ' ) *

(5) —!— e -Q t Iw ' s + a e

( l - e - a T 2 - ' ,

(6 ) — 9 — (| -e-ot) 2 - ( i - e - ° T )s(s + a) ( l - z - ' ) ( l - e - a T z - ' )

s in oTz -1

s * + a 2 l - ( 2 c o s a T ) 2 - ' + z -2

(8) F( s + a) e" a t f ( t ) F*(e"aT

z)

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Ta ble 1 . Abbr e via te d Ta ble o f La pla c e and z -T r a ns fo r m s (C o nt inue d)

L a p l a c e T i m eT r a n s f o r m F u n ct io n z - T r a n s f o r m

F(s) f(t) F*(z )

(9 ) e - » tF ( s ) f ( t - T ) 2 " ' F * ( 2 )

(10) j af/T

s - Y { n a

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2 - a

- cos aTz"

s 2 + a 2 I - ( 2 c o s a T ) 2 - ' + 2"«

(12) - ^ l { a t - , + e - a t )s 2 (s + a) a d - 2 - ' ) 2 a ( l - 2 - ' ) ( l - e - Q t 2 - ' )

(13) b ^ ja t

_b t

e ' o T - e - b T

(s + a ) ( s + b ) ( I - e - a T 2 - ' ) ( l - e - b T 2 - ' )

{ l 4 ) s + o , e -a t cos b t I - 2 - 'e - "T c o s b T

(s + o) 2 + b 2 I - 2 2 - ' e - a T c o s b T + e - 2 a T 2 ' 2

207

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T a b l e 2 . O u tp ut T r a n s f o r m s fo r B a s i c S a m p l e d - D a t a S y s t e m s

S y st emL a p l a c e T r a n s f o r m

of Output C(s)

z - T r a n s f o r mof Output

C*(z)

( I ) l - y . R * ( « ) R * ( z )

G R * U ) G R * ( z )

( 3 ) 6 { » ) R * { « ) G * ( 2 ) R * ( z )

/ +( 4 )

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( 5 )G * ( t ) R * ( « )

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f +( 8 )

* / - \ o * #0 K ( i ) 6 , , , r ( i ) R v ( s ) |

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20 8

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O v e r a l lT r a n s m i s s i o n

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Fo r Ste p Input Te s t Func t io n , R (s ) = l / s

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7 - S

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( l - z - ' ) ( l + z- ')

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'

z

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Fo r Ac c e le r a t io n Input Func t io n , R (s ) = l / s 3

* - I , - Z - . 3 3 Z ' ' - 3 2 ' 2 + Z " 5

32 ' - 32 2 + 2 5 etc . ( I - z"2)s

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data sampler

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digitalcontroller

data continuoushold plant

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p ( t )

- 2 T - T 0

t ime(a)

w *

w *

w *

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e?(t)

Fig . 2. (a) Pe r i o dic sam pling function, p(t) .

(b) Sampled-signal function, e*(t) .

211

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tor—IO

4 fT 2 JT M l

frequency (w)

4 JT

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rot—co

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e ( t )

reconstructedfunct ion

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T 2T 3T 4T

time

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1.0

g h ( t )

o —

( a )

t ime

g h ( t )

u(t)

( b )

t ime

u ( t - T )

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(b) Deco mp o si t io n of i mp u l s i v e res p o n se i n t o t wo s t ep fu n ct i o n s .

21 4

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t o1—en

Gh(jw)

Ang.Gn(jw)

f requency

F i g . 6 . F r e q u e n c y r e s p o n s e of a c l a m p ( o r z e r o - o r d e r d a t a h o ld ) s y s t e m .

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t or—

C(t)-O

C*(s)

F i g. 7 . T ypi cal l i near samp l ed- d at a sys t em showi ng i nput s and ou t put s .

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t o

2 T

t ime

S . C om bi nat ion of i mpul s i ve r espo nse f unct ions used t o obt a i n the respo nse of sys t em t o an i nput r ( t ) .

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r(t)O —

00 T

r*(t)

R(z)

CLAMPCIRCUIT

F i g . 9 . S y s t e m u s e d in e x a m p l e t o d e m o n s t r a t e u s e o f z - t r a n s f o r m m e t h o d .

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tor—'

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4 .0

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/

/

/

//

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r

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(

: i i

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t imeFig. 10. Output of sys tem shown in Fig. 9 for input step function.

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1—6,(2) * • G.(z)

V\

t o \

\

\

R(s) R(z)

v

C(z)

\

C,(z)

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G, G2(z)

toto

R(s) R(z)

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H(s)

11

• 1

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Z-PLANE

F i g . 14. Locus in z-plane used to map in C(z)-plane for determination of stability.

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G ( z ) - plane

transfer-locus

F i g . 1 5 . T r a n s f e r l o c u s f o r t y p i c a l s a m p l e d - d a t a f ee d b ac k c o n t r o l s y s t e m .

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R'- O

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totocn

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dig ital data holdco ntro ller and plant

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\

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F i g. 17. T ypi ca l

D(z)

\\

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s

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in ex.

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8

7

6

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DIGITAL TECHNIQUES IN MISSILE GUID/iNCE SYSTEMSSidney Darlington*

SUMM/iRY

This pa pe r de s c r ibe s wa ys in which d igi ta l te c hnique s a r e us eful in guida nc e a nd c o nt r o ls ys te m s , fo r mis s i l e s a nd a i r c r a f t . It i s s ho wn tha t c o ns ide r a b le info r m a t io n o r in te l l ige nc em u s t u s u a ll y b e i n c o r p o r a t e d i n a g u i da n c e s y s t e m , c o n c e r n i n g s u it a b l e t r a j e c t o r i e s , t a c t i c s ,e t c . In a ddi tio n , num e r ic a l info r m a t io n fur nis he d by da ta c o l le c t ing ins t r u me nts mus t ge ne r a l lybe pr o ces sed in fair ly co mp licated w ays . Th ese two functions can be com bined in a s ingled i g it a l c o m p u t e r . E x t e r n a l a nd i n t e r n a l c h a r a c t e r i s t i c s o f d i g i ta l c o m p u t e r s , a p p r o p r i a t e f o rguida nce a ppl ic a t io ns , a r e d is c us s e d in ge ne r a l te r m s . The n the p r o gr a m min g o f typic a lguida nce ma the m a t ic s i s e xa mine d, a nd i t s r e la t io n to the ful l th r e e -d ime ns io na l guidanc ep r o b l e m .

SOMMAIRE

C e t te no te de c r i t le s m a ni e r e s d a ns le s q ue l le s le s te c hnique s d ig i ta le s s o nt in te r e s s a nte sda ns le s s ys te me s de go uve r ne e t de c o nt r o le po ur le s mis s i le s e t 1'aviation. II est demontrtSqu ' ha bi tue l le me nt une info r ma t io n c o ns ide r a b le , o u de 1 'in tel l ige nc e , do i t e t r e inc o r p o r e e da nsun s ys te m e de go uve r ne , e n vue d ' o bte ni r de s t r a je c to i r e s , de s ta c t ique s , e tc c o n-ve na b le s . De p lus , 1 'info r ma t io n num e r ique fo ur nie pa r le s ins t r um e nts r a s s e m bla nt le sd o n n e e s , d o i t e t r e g e n e r a l e m e n t t r a i t e e o u i n t e r p r e t e d d e m a n i e r e a s s e z c o m p l e x e .C e s de ux fo nc t io ns pe uvent e t r e c o mbin e e s da ns une s e ule c a lc u la t r ic e d ig i ta le . Le s c a r a c t e ' r i s t i q u e s i n t e r i e u r e s e t e x t e r i e u r e s d e c a l c u l a t r i c e s d i g i t a l e s , a p p r o p r i ^ e s a u x a p p l i c a t i o n sde go u ve r ne , s o nt t r a i t e e s e n te r m e s ge"ne 'r aux. La m is e e n p r o gr a m m e d ' e qua t io n s ma th e -ma t iqu e s de go u ve r ne typique e t s a r e la t io n a u p r o ble m e c o m ple t de go uve r ne e n t r o isd i m e n s i o n s s o n t e n s u i t e e x a m i n e e s .

1. INTRODUCTION

This pa pe r i s c o nc e r ne d wi th the us e o fdigital technique s in auto ma tic guidance andc o n t r o l s y s t e m s f o r m i s s i l e s o r a i r c r a f t .The r e a r e ma ny di f fe r e nt d ig i ta l me c ha niz a t io ns no w in e xis te nc e , o f c o ur s e , wi th c ha r a c te r is t i c s which d i f fe r in ma ny de ta i ls .Th e r e a r e a ls o ma ny di f fe r e nt guida nc e a ndc o n t r o l p r o b l e m s , w i t h r e q u i r e m e n t s w h i c hd i ff er in m an y d e t a i l s . F o r p r e s e n t p u r p o s e s ,ho wever , i t wil l be sufficient to take a very

ge n e r a l po in t o f v ie w. In wha t wa ys a r edigi ta l te c hnique s ge ne r a l ly a ppr o pr ia te inguida nc e a nd c o nt r o l s ys te ms o f us ua l s o r ts ?

S e ct io n 2 d e s c r i b e s s o m e g e n e r a l p r o p e r t ie s o f guida nc e a nd c o nt r o l s y s te m s . Its h o w s h ow g u id a n c e s y s t e m s c o m m o n l y r e q u i r e i n f o r m a t io n s t o r a g e ( in f o r m a t io n

r e g a r d i n g t r a j e c t o r i e s a n d t a c t i c s ) a n dn u m b e r p r o c e s s i n g ( p r o c e s s i n g of o b s e r v e dm is s i le be h a vio r ) . The two fe a tur e s c a n be

* B e l l T e l e p h o n e L a b o r a t o r i e s , I n c . , M u r r a y H i l l , N e w J e r s e y .

229

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comb i n ed i n a s i n g l e d i g i t a l comp u t er . Ot h erap p l i cat i on s of d i g i t a l t ech n i q u es a re p oss i b l e , b u t t h e y a r e r e l a t i v e l y m i n o r a n d a r ea l s o r e l a t i v e l y s p e c i a l i z e d .

S e ct io n 3 d e s c r i b e s e x t e r n a l c h a r a c t e r i s t i c s o f di g it a l c o m p u t e r s , o f s o r t s a p p r o p r i a t e f o r g u i da n c e a nd c o n t r o l a p p l i c a t i o n s .S e ct io n 4 d e s c r i b e s i n t e r n a l c h a r a c t e r i s t i c s ,in v e r y g e n e r a l t e r m s . S e ct io n 5 d e s c r i b e sh o w d i g i t a l c o m p u t e r s c a n b e p r o g r a m m e dt o p e r f o r m t h e m a t h e m a t i c s u s u a l ly r e q u i r e df o r m e c h a n i z i n g t h e t h r e e - d i m e n s i o n a l g u i d a n c e p r o b l e m .

2 . S OM E GE NE RAL P ROP E RT IE S OF

GUIDANCE AND CONT ROL SYSTEMS

T h i s s e c t i o n f i r s t d e s c r i b e s w h a t m a y b ecal l ed g u i d an ce t act i cs ; it th en d r aw s v ar i o u sc o n c l u s i o n s r e g a r d i n g c h a r a c t e r i s t i c s n ee d edin g u i d a n c e a nd c o n t r o l s y s t e m s .

a . Gu i d an ce T a ct i c s

C o n s i d e r f i r s t t h e t r a j e c t o r y o f a g r o u n d -t o - g r o u n d m i s s i l e , d i r e c t e d a g a i n s t a f i x e dt a r g e t . W i th o l d - f a s h io n e d a r t i l l e r y t h e r ei s n o g u i d an ce (b ey on d t h e g u n b ar re l ) , y eto n e a c h i e v e s a c c u r a c i e s o f , s a y , 1/1000 oft h e r a n g e . T h i s i s p o s s i b l e b e c a u s e t h eu n g u i d ed t ra j ect o r i es can b e p red i c t ed t oh ig h a c c u r a c y . W he n a g r o u n d - t o - g r o u n dmi ss i l e i s g u i d ed , t h e g u i d an ce g en eral l yi s u s e d t o a t t a i n a c c u r a c i e s w h i c h a r e n o tat tain able w ith unguided f l ight. (The m is si le 'sunguided f l ight may be less predictable thant h at of a n a r t i l l e r y p r o j e c t i l e ; o r t h e r e q u i r e d a c c u r a c y m a y be e v e n g r e a t e r t ha n t h ata t t a i n ab l e wi t h a r t i l l e r y . ) An ad d i t i on al u se

of t h e g u i d an ce may b e t o ach i ev e t ra j ect o rys h a p e s r a d i c a l l y d i f f e r e n t f r o m t h e t r a j e c t o r i es of f ree f l i g h t as a mean s of i mp rov i n gp r o p u l s i o n e f f i c i e n c y , r e d u c i n g a e r o d y n a m i cd r a g s , e t c .

T h e o re t i cal l y , g u i d an ce may be ap p l iedo n e i th e r an " o p e n - lo o p " o r a " c l o s e d - l o o p "b a s i s . G e n e r a l l y , h o w e v e r , a c c u r a c y c o n s i d e r a t i o n s d i c t a t e c l o s e d - l o o p g u i d a n c e . T h efu n d amen tal i d ea of c l o sed - l o o p g u i d an ce i s ,

of co u r se , as fo l l ow s: T h e f li gh t o f t h e m i ss i l ei s o b s e r v e d by s u i t a b l e d a t a c o l l e c t i n g i n s t r u -m en t s . If i t d e p a r t s s i g n i fi can t ly f ro m as u i t a b l e t r a j e c t o r y l e ad i ng t o t h e t a r g e t ,g u id a n c e o r d e r s d i r e c t i l o n t o a t r a j e c t o r ywh i ch d o es l ead to th e t a rg et . S i n ce man yt r a j e c t o r i e s l e a d t o t he t a r g e t , t h i s m a y o rma y not be a single t ra jec to r y, f ixed ina d v a n c e .

T h e d e g r e e o f a r b i t r a r i n e s s i s i l l u s t r a t e din F i g . 1 . T h e m i s s i l e i s s t a r t e d o n t r a j e c t o r y 1 f rom t h e l au n ch er a t M , t o t h e t a r g et

a t T , Dev i at i o n s in t h e p ro p u l s i on o r co n t ro lsy s t em c au s e i t t o wan d er off t ra j ec t o ry 1and i t is observed by the guidance system atpoint M 2 .* If al lowed to co ntinue withoutg u i d an ce, i t wi l l m i s s t h e t a r g et b y a wi d em a r g i n . T h e g u i da n c e s y s t e m , h o w e v e r ,s t e e r s i t o n t o t r a j e c t o r y 2 , w h ic h p a s s e sthro ugh the tar ge t . It would be po ssib le tod esi g n t h e g u i d an ce sy s t em t o s t e er t h em i s s i l e b a c k o n to t r a j e c t o r y 1 . T h i s i s n o tn e ce ssa ry , h o wev er , an d i t may b e i n ef f ic i en t.

In t h e case of a l on g ran g e b al l i s t i c mi ss i l e , the guidance ends at the end of propuls i on . By t h e en d of t h e p ro p u l s i on p h a se ,t h e g u i d an ce sy s t em mu st h av e s t eered t h em i ss i l e on t o a f ree f l ig h t t ra j ect o ry t o th et a r g e t . W ith a s h o r t e r r a n g e , a e r o d y n a m i cal l y s t eered mi ss i l e , g u i d an ce can b eap p l i ed ev en wh en t h e mi ss i l e i s c l ose t o t h et a r g e t . T h i s d o es n ot me an , h o wev er , t h atf l ight path dev iat io ns can be igno red ate a r l i e r t i m e s . T h e m i s s i l e m u s t b e k e pt

*If the "guidance loop" is t ight enough, andt h e o b s e r v a t i o n s g o o d e no u g h , t he m i s s i l ewi l l h av e n o o p p o r t u n i t i es t o wan d er of ft ra j ect o ry 1; b ut t h ese co n d i t io n s a r e n o tg e n e r a l l y r e a l i z e d i n p r a c t i c e .

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r e a s o n a b l y c l o s e t o s o m e s o r t o f s u i t a b let r a j e c t o r y a t a l l t i m e s s o t h at e r r o r s w i llnot accumulate beyond the capacity of theg u i d a n c e s y s t e m t o m a k e c o r r e c t i o n s .

W he n the ta r ge t i s a n a i r c r a f t (o r a no the rm i s s i l e ) , t h e s i t u a t i o n b e c o m e s e v e n m o r eco mp licated. The actua l point of im pact willnow depend, of co ur se , o n the mo tion of theta r g e t f r o m pr e s e nt t ime unt i l impa c t a ndthe guidance system must make a suitablepr e dic t io n o f th is mo t io n . If the ta r ge tma ne uv e r s in wa ys no t p r e vio us ly p r e d ic te d ,the p r e dic t io n fo r the r e m a ining t ime o ff light mus t be c o r r e c t e d . Thus a ma ne u ve r ing ta r ge t c a l ls fo r a c o nt inuo us a djus tme nto f the t r a je c to r y a lo ng whic h the mis s i le

is s te er ed , even though i ts own fre e flightis pe r fe c t ly p r e dic ta ble .

The s i tuation is i l lustrated in Fig. 2. Themissile is a t point M 2 when the target is a tT 2 . T h e t h r e e m i s s i l e t r a j e c t o r i e s a r e th ena p p r o p r i a t e , r e s p e c t i v e l y , f o r th e t h r e e dif

ferent target f l ight paths.

It is a function of the guidan ce sys tem tofigure out the tact ical s i tuatio n and to guidethe m is s i le a c c o r dingly . W ithin l im i ts , thepr e dic t io n o f ta r ge t mo t io n c a n be fa i r ly

c r u d e s i n c e c o n t in u o u s c o r r e c t i o n s a r e p o s s ib le a lmo s t up to impa c t . The l im i ts ,ho we ve r , de pe nd o n the ma ne uve r a bi l i ty o fthe mis s i le r e l a t ive to tha t o f the ta r ge t .I f the ma r gin i s s ma l l , fa i r ly s o phis t ic a te dp r e d i c t i o n s m a y b e i n o r d e r . *

b. Info r ma t io n Sto r a ge a nd Numbe rP r o c e s s i n g

It is c lea r that the guidance tac tic s will ,in fact, be followed only if suit abl e inf o r

m a t i o n , o r " i n t e l l i g e n c e , " i s s t o r e d w i t h i n

the guida nc e s ys te m . If t r a je c to r ie s a r e tobe muc h di f fe r e nt f r o m f r e e t r a je c to r i e s ,the guida nc e s ys te m mus t s to r e e no ughi n t e ll i g en c e f o r t he g e n e r a t i o n o f t r a j e c t o r i e so f the de s i r e d s o r t . Eve n if f r e e t r a je c

t o r i e s a r e s u i t a b l e , e x c e p t f o r p e r t u r b a t i o n s ,the guidance system must know, at eachi n s t a n t , w h e t h e r o b s e r v e d c o n d i t o n s c o r r e s po nd to a n a c c e pta ble f r e e f l ight t r a je c to r y . In a n a nt ia i r c r a f t s ys te m , a ddi t io na lin te l l ige nc e mus t be s to r e d to e na ble them i s s i l e t o c o u n t e r t a r g e t e v a s i o n .

In or de r to apply the buil t-in intell igenceto the o bs e r v e d s i tua t io n , the r e mus t beme a ns fo r p r o c e s s ing the qua nt i ta t ive da tac o l le c te d by o bs e r va t io n , o r da ta c o l le c t ing

i n s t r u m e n t s .

In ma ny guida nc e s ys te ms , the guida nc eta c t ic s de pe nd o n bo th po s i t io ns a nd ve lo c i t i e s . Us ua l ly , the da ta c o l le c t ing ins t r um e ntsgive o nly po s i t io ns o r v e lo c i t ie s . The n theguida nc e s ys te m mus t e i the r d if fe r e nt ia te thepo s i t io ns o r in te gr a te the ve lo c i t ie s . W he npo s i t io ns a r e d i f fe r e nt ia te d , f i l te r ing o r da tas mo o thing is us ua lly r e qui r e d to r e d uc ee ffe cts o f f luc tua t io ns in o bs e r va t io na le r r o r s . * T h u s , i n t e g r a ti o n , d i f fe r e n t ia t io n ,

f i l te r ing , a nd da ta s mo o thing a r e l ike ly tob e i m p o r t a n t d a t a p r o c e s s i n g o p e r a t i o n s i ng u i d a n c e s y s t e m s .

T r a n s f o r m a t i o n s o f c o o r d i n a t es a r ea n o t h e r s o r t o f d a t a p r o c e s s i n g w h ic h a r ea l s o l i ke ly to b e i m p o r t a n t . P o s i t io n s o rv e l o c i t i e s a r e u s u a ll y o b s e r v e d in t e r m s o fa c o o r dina te s ys te m which is c o nve nie nt fo rthe da ta c o l le c t ing ins t r u m e nt s . On theo the r ha nd, the s te e r ing ins t r uc t io ns de r ive dby the guida nc e s ys te m mus t r e fe r to ac o o r d i n a t e s y s t e m a p p r o p r i a t e fo r t he s t e e r ing me a ns .

*One s tar t ing point fo r a s tudy of impactpo int p r e dic t io n i s the p r e dic t io n the o r y o fW iener and Ko lmogo ro ff. (See Ref. 1.)

*Data smo o thing is a fundamental pa r t oft h e t h e o r y o f p r e d i c t i o n r e f e r r e d t o a b o v e .

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V e c t o r p o s i ti o n s o b s e r v e d by a r a d a r ,f o r i n s t a n c e , a r e u s u al ly m e a s u r e d a s c o m b i n at i on s of s l an t ran g e, e l ev at i on an g l e ,a nd a z i m u t h a n g l e . A f r a m e o f r e f e r e n c eof t h i s so r t i s i l l u s t r a t e d i n F i g . 3 a . On

t h e o t h e r h a n d , s t e e r i n g i s a c c o m p l i s h e dby s e t t i n g u p a c c e l e r a t i o n s in d i r e c t i o n sd et e rm i n ed b y t h e yaw and p i t ch s t e er i n gm e a n s . T h u s , s t e e r i n g i n s t r u c t io n s g e n e r a l l y a r e r e f e r r e d t o a c o o r d i n a te s y s t e mlike that shown in Fig. 3b.

A d d i t i o n a l c o o r d i n a t e s y s t e m s m a y a l s ob e n eed ed t o faci l i t a t e i n t eg ra t i o n s , d at as m o o t h in g , a nd t h e l i k e . F o r e x a m p l e ,t h e n a v i g a t i o n o f a n a i r c r a f t o r a n a i r b o r n em i s s i l e m a y c a l l f o r t h e c o m p u t a ti o n o f

l a t i t u d es an d l on g i t u d es b y su i t ab l e i n t e g r at i o n s o f m i s s i l e v e l o c i t i e s .

S u b st an t i a l ad d i t i on al d at a p rocess i n g i sa l s o l i k ely t o b e n eed ed , b ut o f so r t s n o t soeas i l y cod i f i ed. F o r on e t h in g , t h e t act i cali n t e l l i g e n c e a c t u a l l y i s s t o r e d a s m a t h e m a t i cal fo rm u l as t o be ap p li ed t o t h e o b ser v edd a t a . O t h e r d a t a p r o c e s s i n g m a y h a v e t od o wi t h t h e g en erat i on of d i sp l ay s u sed i nm o n i t o r i n g t h e g u i d a n c e .

I n f o r m a t i o n s t o r a g e a n d n u m b e r p r o c e s s i n g can b e comb i n ed i n a d i g i t a l comp u t er .B e f o r e c h o o s i n g a d i g it a l c o m p u t e r fo r t h i sp u r p o s e , h o w e v e r , o n e m u s t m a k e s u r e th a ti t i s fas t en o u g h . Act u al l y , t h e sp eed re q u i r em e n t s a r e r a t h e r m o d e s t b e c a u s e o f t h e w a yi n w h i c h v a r i o u s f e e d b a c k p r o c e s s e s i n t e r a c twi t h in t h e o v er a l l g u i d an ce an d co n t ro ls y s t e m .

g u i d an ce o r d e r s . Act u al l y , t h e t y p i cal g u i d an ce an d con t ro l sy s t em h as ad d i t i on al feed b ack p at h s wi t h i n t h e ov era l l " g u i d an cel o o p . " T h i s i s i l l u s t r a t e d in F i g . 4 .

T h e in n e r m o s t lo o p r e p r e s e n t s o n e o rm o r e s e r v o d e v i c e s w h i c h s e t t h e a c t u a ls t e e r i n g c o n t r o l s ( a e r o d y n a m i c s u r f a c e s o rj e t d ef l ecti on m ean s) . S o t h at ac t u at o rs n eedn ot b e acc u r at e l y ca l i b r a t ed , t h e act u ald e f le c ti o n s a r e m e a s u r e d an d a r e c o m p a r e dw it h t he d e f le c t i o n s o r d e r e d .

T h e d e f l e c t i o n s o f t h e s t e e r i n g c o n t r o l sa r e o r d e r e d i n a c c o r d a n c e w it h a s e c o n df ee d ba c k p a t h . H e r e m i s s i l e o r i e n t a t i o n so r a t t i t u d e an g l es a r e fed b ack . (Or i en t a

t i o n s m a y b e m e a s u r e d e i t h e r w i t h r e s p e c tt o t he a e r o d y n a m i c s l i p s t r e a m o r w i thr e s p e c t t o s o m e fix ed r e f e r e n c e d i r e c t i o n .T h e y m a y be m e a s u r e d e i t h e r in d e g r e e so r in t e r m s o f a c c e l e r a t i o n s p r o d u c e d b yt h e d ef l ect i o n s . ) T h e o r i e n t a t i o n s fed b acka r e c o m p a r e d w i t h o r i e n t a t i o n s a s k e d f o rby the guidance system itself. T h e dif

f e r e n c e s p r o d u c e d e f le c t io n s o f t h e s t e e r i n gd ev i ces , su ch as to k eep t h e d i f fe r en cess m a l l .

T h e t w o i n n e r l o o p s , t o g e t h e r , a r ec o m m o n l y c a l le d t he a u t o p i l o t . T h e o r i e n t a t ions set up by the autopilot , together witht h e sp ee d s se t u p b y t h e p ro p u l s i o n , d e t e r m i n eh ow t h e m i ss i l e act u al l y f l i es . In o t h erw o r d s , t h e y d e t e r m i n e w h a t p o s i t io n s a ndv el o ci t i es a r e act u al l y ach i ev ed b y t h e m i s s i l e . T h e s e a r e o b s e r v e d b y t h e g u i d a n c es y s t e m , a nd a r e u s e d in o r d e r i n g f u r t h e rc h a n g e s i n t h e o r i e n t a t i o n .

c . F e e d b a c k P a t h s

T h e c l o s e d - l o o p g u i d an c e s y s t e m e x h i b i t s ,o f c o u r s e , t he u s u a l c h a r a c t e r i s t i c s o f" f e e d b a c k . " T h e g u i da n c e o r d e r s d e pe n don t h e f li g ht o f t h e m i ss i l e as o b ser v ed , an dthe f l ight , as observed, is modified by the

Ver y r o u g h l y , t h e v ar i o u s feedb ack l oop sa r e i n t e r re l a t e d in t h e fo l lowi n g way : T h ei n n e r l o o p i s s i m p l y a n e l e c t r o m e c h a n i c a lse r v o fo r se t t i n g a sh aft p o s i t i on . It h asa r e l a t i v e l y s h o r t t i m e - c o n s t a n t . T h e o r i e n t a t i on of t h e mi ss i l e , on t h e o t h er h an d ,c o r r e s p o n d s t o s o m e s o r t o f i n te g r a t io n

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o f the de f le c t io ns o f the s te e r ing c o nt r o ls(s inc e tu r n ing mo me n ts p r o duc e o nly a ngula ra c c e le r a t io ns ) . (The e xa c t fo r m o f ther e la t io ns hip de pe nds upo n whe the r the s te e r ing i s a c c o mpl is he d by a e r o dyna m ic s ur fa c e s

o r by je t de f le c t io ns . ) Ve lo c i ty c o m po ne ntsa n d d i r e c t i o n s o f v e c t o r v e l o c i t i e s c o r r e s po nd to s o me s o r t o f in te gr a t io n o f them i s s i l e o r i e n t a t i o n s ( s i n c e t h e o r i e n t a t i o n sd e t e r m i n e o n ly a c c e l e r a t i o n s ) . F i n a l ly ,p o s i t i o n s c o r r e s p o n d t o i n t e g r a t i o n s ofv e l o c i t i e s .

One result of this s i tuation is as fo l lows:Auto pi lo t e r r o r s do no t le a d to s ignif ic a ntt r a j e c t o r y e r r o r s u n l e s s t he y a r e le ft u n c o r r e c t e d f o r a n a p p r e c i a b l e t i m e . A s a

c o r o l la r y , guida nc e ne e d no t be c o nt inuo usno r ne e d i t us e po s i t io n a nd ve lo c i ty da tawhich ar e s t r ic t ly up to da te. I t is onlyn e c e s s a r y t h a t a u to p i l o t e r r o r s w i ll n o ti n t e g r a t e i n to s i gn i fi c an t g u id a n ce e r r o r sduring the effective lag t imes.

As an example of the effect of temporarya c c e l e r a t i o n e r r o r s , c o n s i d e r t h e f o ll o w in g :An a c c e l e r a t i o n e r r o r o f a s m u c h a s I g ,la s t ing fo r 1 /2 s e c o nd , p r o duc e s a ve lo c i tye r r o r o f 16 fe et pe r s e c o n d. Suppo s e the

ve lo c i ty e r r o r i s o bs e r ve d a t the e nd o fthe 1/2 s eco nd, and is wiped out by ap ply ing a ne t a c c e le r a t io n o f - l /2g dur ing then ex t 1 /2 s e c o n d . T h e p o s i t io n e r r o r a c c u mu lated dur ing the full 1 second of theve lo c i ty pe r tu r ba t io n i s o nly 8 fe e t .

Fr o m a n a c c u r a c y sta ndpo int , the n, aguida nc e c o mpute r ne e d fur nis h s te e r ingins t r u c t io ns o nly in te r m i t te nt ly at a r a t eo f pe r ha p s o nc e e ve r y 0 .1 to 0 .5 s e c o nd.We sha ll se e (in Sec tion 5) that this is a

r e a s o na ble r a te o f c o mputa t io n fo r d ig i ta lc o mpu te r s in mo s t guida nc e a ppl ic a t io ns .One mu st also c o nsid er the s tabil i ty of theoverall guidance loop, but i t turns out thatdelays of the order of 0.1 to 0.5 secondc a n us ua l ly be a c c o mmo da te d.

3 . EXTER NAL PR OPER TIES OF DIGITALGUIDANCE COMPUTERS

W e ha ve s e e n tha t info r ma t io n s to r a gea nd da ta p r o c e s s ing a r e impo r ta nt func t io ns

pe r fo r me d wi th in mo s t mis s i le guida nc esy st em s, and we have noted that thes e functions can be com bined in a digital co m pu ter .T h i s s e c t i o n d e s c r i b e s s o m e o f t h e e x t e r n a lc h a r a c t e r i s t i c s o f d i g i t a l g u i d a n c e c o m p u t e r s . S e ct io n 4 d e s c r i b e s s o m e a p p r o p r i a t e i n t e r n a l a r r a n g e m e n t s by m e a n s o fwhic h the e xte r na l c h a r a c te r is t i c s c a n bea c hie ve d.

In a c tua l a ppl ic a t io ns , s o me o f the a uto pilo t functions may also be included in the

digi ta l o pe r a t io ns pe r fo r m e d by the d ig i ta lc o mp ute r . As a m a t t e r o f fa c t , in a ninte gr a te d s ys te m the l ine be twe e n a uto pi lo ta nd o ute r guida nc e lo o p ma y be so m e wha tfuzz y. The r e m a r ks of th is s e c t io n s t i l lapply in a general way, however , eventho ugh the y a r e he r e d i r e c te d e xpl ic i t ly a tthe guidance loop itself.

a . R e a l Tim e Ope r a t io n

The guidanc e c o m pute r i s o f c o ur s e a" r e a l t i m e " c o m p u t e r . It m u s t k e e p u pwith events as they o cc ur . We have alr ead ye x a m i n e d c o m p u t i n g s p e e d r e q u i r e m e n t s .W e wi l l e xa mine c o mput ing s pe e d c a pa bi l i ties in Section 5, and will find that they arelikely to be adequate if a reasonably highs p e e d e l e c t r o n i c c o m p u t e r i s u s e d .

b . Sa mpling In te r va l

The d igi ta l c o mpu te r o p e r a te s o n a c yc l ic a lb a s i s . It takes in new data cyclically, compute s new s te e r ing ins t r uc t io ns o n the ba s isof the new data, and de liv er s the new in str uc t io ns to the a uto pi lo t . In the s imp le s t a r r a n ge me nt , e v e r ything is r e p e a te d e xa c t ly o nc e

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d u ri n g each co mp u t i n g cy c l e . T h en the

m a c h i n e t a k e s in new d at a ex act l y on ce eachcy cl e and p u t s out new i n s t r u c t i o n s e x a c t l yo n c e e a c h c y c l e . M e a s u r e m e n t s m a d e at

i n t e r m e d i a t e t i m e s are i g n o r e d .

The effect of the f i n i t e comp u t i n g i n t e rv alm ay be c o m p a r e d w i t h " s a m p l i n g " in an al ogd e v i c e s . The effect i v e samp l i n g i n t e rv al is,

o f c o u r s e , the c o m p u t i n g i n t e r v a l itself. It

is a m e a s u r e of the effect ive speed of the

mach i n e wh i ch we h a v e a l r e a d y d i s c u s s e dan d wi l l d i scu ss fu r t h er in S ect i on 5.

I f cer t a i n i n p u t s or ou t p u t s v ary s i g n i f i cantly within the o v e r a l l c o m p u t i n g i n t e r v a la m o r e c o m p l i c a t e d p a t t e r n of c y c l e s may

be used to ob t ai n a higher effect ive speedw h e r e s p e e d is n eed ed . For e x a m p l e , s e v e r a lc y c l e s of a s i m p l e e x t r a p o l a t i o n c o m p u t a t i on may be u sed to " u p d a t e " a r a p i d l yv ary i n g ou t p u t q u an t i t y sev era l t i mes wi t h i ne a c h o v e r a l l c o m p u t i n g i n t e r v a l .

c. Dynam ic Inputs and Outputs

The function of the g u i d an ce l oop is to

t r a n s l a t e o b s e r v e d p h y s i c a l v a r i a b l e s i n t o

s u i t a b le s t e e r i n g i n s t r u c t i o n s . The p h y s i c a lv a r i a b l e s are b asi cal l y an al og in n a t u r e( p o s i t i o n s and v e l o c i t i e s ) . The s t e e r i n gi n s t r u c t i o n s are ob ey ed in b asi cal l y an al ogt e r m s ( d i s p l a c e m e n t s of s t e e r i n g d e v i c e s ) .T h u s , s o m e s o r t of a n a l o g - t o - d i g i t a l c o n v e r s i o n s and d i g i t a l - t o - a n a l o g c o n v e r s i o n sare fu n d amen t al to the use of a d i g i t a lc o m p u t e r .

M o s t of the s e n s i n g or d at a co l l ect i n gi n s t r u m e n t s w h i c h are now av ai l ab l e fu rn i sh

a n a l o g r e p r e s e n t a t i o n s of p h y si c al v a r i a b l e s .T h at is, t h e i r o u t p u t s are usually shaftr o t a t i o n s , a n a l o g v o l t a g e s , or s y n c h r o s i g n a l sr e p r e s e n t i n g s ha ft r o t a t io n s . T h e s e can,

in fact, be t r a n s l a t e d i nto d ig i ta l r e p r e s e n t a t i o n s , but the t r a n s l a t i o n is l ikely to be

r a t h e r e x p e n s i v e in t e r m s of s y s t e m c o m p l e x it y . S e n s in g i n s t r u m e n t s w h ic h m e a s u r ep h y s i c a l v a r i a b l e s d i r e c t l y in d i g i t a l t e r m swould be a g r e a t h e l p , but are not now

g e n e r a l l y a v a i l a b l e . The e x c e p t i o n s are

g e n e r a l l y q u i t e s p e c i a l . (An i l l u s t r a t i o n isth e use of an o s c i l l a t o r and c y c l e c o u n t e r to

m e a s u r e t i m e i n t e r v a l s . )

G e n e r a l l y s p e a k i n g , it is p o s s i b l e to

c o n v e r t the an al og ou t p u t s of the i n s t r u m e n t si n t o d i g i t a l fo rm wi t h ou t any s i g n i f i can t l osso f a c c u r a c y . T h a t is, the d i g i t a l r e p r e s e n t a t ion is not s i g n i f i c a n t l y p o o r e r t h a n the

analog quanti ty itself. C l e a r l y , h o w e v e r , the

a c c u r a c y of the a n a l o g i n s t r u m e n t ' s o u t p u twil l not be e x c e e d e d w i t h o u t r e c o u r s e to a

d i g it a l i n s t r u m e n t . S i m i l a r r e m a r k s a p pl yt o o u t p u t c o n v e r s i o n s f r o m d i g i t a l - t o - a n a l o gs t e e r i n g i n s t r u c t i o n s . R o u g h ly , the a c c u r a c yis l imited by the acc u r acy t o wh ich the an al ogquanti ty can be handled in a n a l o g o p e r a t i o n s .

Within the d i g i t a l c o m p u t e r , of c o u r s e ,c o m p u t a t i o n s can be r e l a t i v e l y v e r y p r e c i s e .A c c u r a c i e s of n u m b e r p r o c e s s i n g are l i mi t edon l y by c o s t and s p e e d c o n s i d e r a t i o n s .

d. P r e s e t C o n s t a n t s

Within the c o m p u t e r , the t r a j e c t o r y and

t a c t i c a l i n f o r m a t i o n is s t o r e d in the f o r m of

v a r i o u s f o r m u l a s , to be applied to the dy

n ami c i n p u t s . In m a n y a p p l i c a ti o n s c o n s t a n t sa p p e a r i n g in t h e s e f o r m u l a s m u s t be adjustedb efore each f i r i n g to coincide with the c u r r e n tt a c t i c a l s i t u a t i o n . In a g r o u n d - t o - g r o u n dg u i d a n c e s y s t e m , for e x a m p l e , the t r a j e c t o r y i n fo r mat i o n m u st tak e accou n t of the

r a n g e and d i r e c t i o n f r o m the l au n ch er to the

p a r t i c u l a r t a r g e t w h i c h is to be a t t a c k e d .(In a g r o u n d - t o - a i r m i s s i l e , t a r g e t l o c a t i o nm u s t c o m e in as a dynamic input .) In g e n e r a l ,then, it m u s t be p o s s i b l e to supply the

c o m p u t e r w i t h a n u m b e r of con st an t s wh i cha r e to be d i f fe ren t for d i f fe ren t sh ot s .

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The s im ple s t s ys te m ma y be s uc h tha teach individual co nstant is set by hand.If the r e a r e mo r e tha n a fe w c o ns ta nts , ho we v e r , r e l i a b i l i t y c o n s i d e r a t i o n s a r e l i k e l yt o d ic t a t e a m o r e a u t o m a t i c s y s t e m . A

punche d c a r d s y s te m is o ne po s s ib i l i ty ,o f c o u r s e , w i th o n e c a r d o r s e t o f c a r d sr e c o r d i n g a l l c o n s t a n t s e t t i n g s f o r o n e t a r g e t .C ar ds can be cut and checked for a l l init ialc o ndi t io ns l ike ly to be o f in te r e s t . The no p e r a t o r e r r o r s c an e n t e r o nly in t he s e l e c t io n o f c a r ds r a the r tha n in the s e t t ing o fe a c h c o ns ta nt .

e . Displays

In ge ne r a l , c e r ta in va r ia ble s mus t bed i sp l ay e d b ef o r e o n e o r m o r e o p e r a t o r ss o tha t the pe r fo r ma nc e o f the who le mis s i les ys te m c a n be mo ni to r e d dur ing the fl ighto f the m is s i le . Va r io u s te c hnique s a r ea va i la ble fo r de r iv ing s ui ta ble d is pla ys f r o mdigi ta l qua nt i t ie s wi th in the c o mp ute r . I tcan per hap s be said, ho wev er , that none oft h e p r e s e n t t e c h n i qu e s a r e e n t i r e l y s a t i s fa c to r y . The t r o uble i s tha t the d ig i ta lnum be r s a r e us ua l ly in b ina r y fo r m a ndm u s t b e c o n v e r t e d e i t h e r t o d e c i m a l n u m b e r s

o r to a na lo g va r ia bl e s be fo r e the y c a n ma kes e n s e t o a h u m a n o p e r a t o r .

f. F le xibi l ity o f Pr o gr a m mi ng

The digi ta l c o mp ute r do e s i t s jo b bysubjecting the input data to a specifics e q u e n c e o f e l e m e n t a r y o p e r a t i o n s . T h e s einc lude the t r a ns fe r o f num be r s f r o m o nepa r t o f the ma c hine to a no the r a nd the a p p l ic a ti o n o f o r d i n a r y a r i t h m e t i c o p e r a t i o n s

such as addit ion, mu ltiplicatio n, e t c . I ti s po s s ib le to fix the s e que nc e of o pe r a t io nso r p r o g r a m b y p e r m a n e n t i n t e r c o n n e c t i o n s .Then the program can be changed only bytaking the machine apart and making internalmo dif ic a t io ns . I t i s ge ne r a l ly p r e fe r a ble ,

ho we ve r , to ha ve a f lexible p r o gr a m . It i sp r e f e r a b l e t o h a v e s o m e m e a n s o f s t o r i n ga n y p r o g r a m w i t h i n s o m e r e a s o n a b l e l i m i t s ,in a way a na lo go us to the s to r in g o f p r e s e tc o ns ta n ts . One o f the Impo r ta nt a dva nta ge s

o f d i g i t al c o m p u t e r s r e l a t i v e to an a lo g c o m pu te r s i s tha t they lend the m s e lve s m o r en a t u r a l l y to f le x ib l e p r o g r a m m i n g .

W hen the p r o gr a m is s to r e d in a n e a s i lychanged way, the guidance program can ber e p l a c e d b y " d i a g n o s t i c r o u t i n e s " d u r i n gequipm ent che ck s. Chang es in the guidancee qua t io ns r e f le c t ing f ie ld e xp e r ie n c e o rc ha nge s in mis s i le de s ign c a n be in t r o duc e dwitho ut r e bui ld ing the c o mp ute r . W ithinl i m i t s , t h e o n e c o m p u t e r b e c o m e s a " g e n e r a l

p u r p o s e g u id a n ce c o m p u t e r " a p p l ic a b le t odi f fe r e nt guida nc e s ys te ms wi tho ut mo dif ic a t io n . The l im i ts a r e s e t by a numbe r o f" d i m e n s i o n s " s u c h a s c a p a c i t i e s f o r s t o r a g eo f p r o g r a m s t e p s , p r e s e t c o n s t a n t s , a nddyna mic va r ia ble s ; numbe r o f inputs a ndnum be r o f o utpu ts ; numbe r of d ig i ts (de te r mining c o mputa t io n a c c ur a c ie s ) , c o mput ings pe e d, e tc .

g. Reliabil i ty

A guida nc e c o mp ute r m us t be r e l ia b lein a wa y no t us ua l ly r e qui r e d o f a ge ne r a lp u r p o s e l a b o r a t o r y c o m p u t e r . If a l a b o r a t o r y c o m p u t e r m a k e s a m i s t a k e t h e w o r kcan usually be done o ver a gain. I t is onlyne c e s s a r y to kno w whe n m is ta ke s ha ve , infact , been ma de, and to be able to affordthe ma c hine t ime ne c e s s a r y fo r r e pe a t ingthe c o mpu ta t io ns . If a guida nc e c o mp ute rma ke s a mis ta ke , ho we ve r , i t ma y mis guidethe missile in a way which spoils i ts chancesof hit t ing the tar ge t . Then the m iss ile i tself

is wasted as well as the co mputing t im e ofthe guida nc e c o mpute r .

Ac c o r dingly , the de s ign o f d ig i ta l c o mpute r s fo r mis s i le guida nc e i s ve r y s t r o nglyinfluenc e d by r e l ia b i l i ty c o ns ide r a t io ns .

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h . C o m p u t e r E n v i r o n m e n t s

C o m p u t e r e n v i r o n m e n t s are quite dif

f e r e n t in d i f fe ren t g u i d an ce ap p l i cat i on . In

a " c o m m a n d " s y s t e m in which a m i s s i l e

is guided from the g r o u n d , the c o m p u t e r m a ybe in a t r a i l e r or even in an a i r - c o n d i t i o n e dbuilding. In an a i r b o r n e s y s t e m , it may be

i n s i d e the cab i n of an a i r c r a f t . In o t h e rs y s t e m s it may ev en be i n s i d e the m i s s i l eitself, su b j ect to r e s u l t i n g e x t r e m e s of ac

c e l e r a t i o n , v i b r a t i o n , and t e m p e r a t u r e f l u c t u a t i o n s . In g e n e r a l , d i g i t a l m e c h a n i z a t i o n scan be d esi g n ed for all s o r t s of d i f fe ren tcon d i t i on s j u s t as a n a l o g m e c h a n i z a t i o n sc a n . If anything, digi tal techniques shouldb e l e s s s e n s i t i v e to e n v i r o n m e n t a l c o n d i t i o n s

s i n c e p r e c i s i o n c a l i b r a t i o n s are n o t r e q u i r e d .

4 . I N T E R N / L L CHARACTERISTICS

F r o m the s t an d p oi n t of i n t e r n a l o p e r a t i o n s , t h e r e are two q u i t e d i f fe ren t fo rm so f d i g i ta l c o m p u t e r s . B o t h of t h e s e are

r e c o g n i z e d as ap p l i cab l e to g u i d a n c e p r o b l e m s . One of t h e s e may be d e s c r i b e d as

a r i t h m e t i c a l , the o t h e r as i n c r e m e n t a l .

A n a r i t h m e t i c a l m a c h i n e can f o r m s u m s ,p r o d u c t s , etc. of c o m p l e t e n u m b e r s in a

s i n g l e c o m p u t i n g c y c l e . An i n c r e m e n t a lma ch i n e, k nown al so as a d i g i t a l d i f fe ren t i a l a n a l y z e r , can on l y mak e smal l ch an g esi n p r e v i o u s r e s u l t s . ( E ve n m u l t i p l i c a t io n sa r e p e r f o r m e d by s e q u e n c e s of s m a l l c h a n g e sc o m p a r a b l e to the m e c h a n i c a l i n t e g r a t i o n su sed for m u l t i p l i c a t i o n in the e a r l i e s t dif

f e r e n t i a l a n a l y z e r s . ) I n c r e m e n t a l m a c h i n e scan be u sed in g u i d a n c e s y s t e m s b e c a u s et h e s i g n i f i can t n u mb ers do not, in fact ,ch an g e v ery mu ch in any one c o m p u t i n gc y c l e .

In g e n e r a l , i n c r e m e n t a l c o m p u t e r s are

s i m p l e r t h a n a r i t h m e t i c a l m a c h i n e s , but

they do h a v e s o m e r a t h e r s e r i o u s d i s a d v an t ag es . Al so , t h ey are h a r d e r to u n d e r s t an d . Accord i n g l y , we w i l l r e s t r i c t our

at t en t i on to a r i t h m e t i c a l c o m p u t e r s . It

sh ou l d be b o r n e in mi n d , h owev er , t h at the

i n c r e m e n t a l m a c h i n e s do a l s o e x i s t .

a. F u n ct i on al S ch emat i c of a GuidanceC o m p u t e r

F i g . 5 i l l u s t r a t e s the m a j o r p a r t s w h ic hare l i k el y to a p p e a r in a d i g i t a l c o m p u t e ras u sed in a g u i d a n c e s y s t e m .

T h e a r i t h m e t i c u n i t d o e s the a c t u a lp r o c e s s i n g of n u m b e r s . It a d d s , s u b t r a c t s ,a n d m u l t i p l i e s . It may a l s o d i v i d e and

e x t r a c t s q u a r e r o o t s . G e n e r a l l y , it a l s op e r f o r m s v a r i o u s o t h e r o p e r a t i o n s s u c h as

l imit ing, shif t ing of d e c i m a l (or b i n a r y )

p o i n t s , e x t r a c t i n g s p e c i a l d i g i t s (or b i n a r ybi t s ) , and t e s t i n g n u m b e r s for s i g n s . All

n u m b e r s p r o c e s s e d may be e i t h e r p o s i t i v eo r n e g a t i v e , and a n s w e r s are g i v en comp l et ewi t h s i g n s .

I n g e n e r a l , al l n u m b e r s are r e p r e s e n t e di n b i n a r y f o r m , e x c e p t p o s s i b l y n u m b e r su sed in ou t p u t s to d i s p l a y s . G e n e r a l l y ,b i n a r y n u m b e r s are r e p r e s e n t e d by c o m b i n a t i o n s of e l e c t r i c a l p u l s e s , e i t h e r s e r i a l l yo n s i n g l e w i r e s , or p a r a l l e l w i s e on c o m

b i n at i on s of w i r e s .

T h e p a r t i c u l a r o p e r a t i o n w h i c h the a r i t h m e t i c u n i t p e r f o r m s at a g i v en t i me is

d e t e r m i n e d by a s i g n al f rom the u n i t wh eret h e p r o g r a m is s t o r e d . The s a m e u n i t c o n t r o l s the s o u r c e s of the n u m b e r s t h a t are

o p e r a t e d on, and the d i s p o s i t i o n of the new

n u m b e r s g e n e r a t e d by the o p e r a t i o n s . The

n u m b e r s are mo v ed f ro m on e p oi n t t o an o t h erby switching or r o u t i n g m e a n s ( c o m p u t e rc o n t r o l c i r c u i t s ) c o n t r o l l e d by the p r o g r a m

u n i t . In the c o n s t a n t s t o r a g e u n i t and in thed y n a m i c s t o r a g e u n it t h e r e are many different" s l o t s " or a d d r e s s e s . A n u m b e r is takeno ut of a p a r t i c u l a r s l o t , or is s t o r e d a w a yin a p a r t i c u l a r s l o t , all in a c c o r d a n c e w i t hi n s t r u c t i o n s f r o m the p r o g r a m u n i t .

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Num be r s in the c o n s ta nt s to r a ge a r e no tchanged during the guidance of a missile .Numbers can be taken out, but cannot be putin , e xc e pt in the p r e pa r a t io n o f the c o mpute rbe fo r e la unc h. Num be r s in the dyna mics to r a ge , ho we ve r a r e a l l s ubje c t to c ha nge .T h e d y n a m i c s t o r a g e i s s i m p l y a " s c r a t c hp a d m e m o r y " w h e r e i n t e r m e d i a t e r e s u l t sa r e w r i t t e n d o w n .

The s e pa r a t io n o f the c o ns ta nt s to r a gea nd the dyna mic s to r a ge i s fa r f r o m a c a d e m ic .Th e r e i s a s e r io us que s t io n o f r e l ia b i l i tyas well as a difference in acc essi bil i ty towr i te - i n . W hen a numbe r in the dyna m ics t o r a g e i s r e c o m p u t e d e v e r y c o m p u t in g c y c l e ,an acciden tal change in the num ber while

in the s to r a ge uni t wi l l p r o ba bly be c o r r e c te ddur ing the next cyc le. If a co nstan t isc ha nge d, ho we ve r , i t wi l l r e m a in in e r r o rth r o ugho ut the r e s t o f tha t m is s i le f l ight .As a r e s ul t , qui te d i f fe r e nt me c h a niz a t io n sma y be in o r de r fo r the c o ns ta nt a nd dyn a m i c s t o r a g e u n i t s .

As is indicated in the figure, the computerc o n t r o l c i r c u i t s a l s o r e c e i v e i n s t r u c t i o n sf r o m the a r i thm e t ic uni t. F i r s t , the s e ma ybe t iming s igna ls te l l ing the p r o gr a m uni t

whe n the a r i thme t ic uni t i s r e a dy fo r fu r the rins t r uc t io ns . In a ddi t io n , the s e ma y inc lude" c o n d i t io n a l t r a n s f e r " i n s t r u c t i o n s . A t apa r t ic ula r s te p in the p r o gr a m, the ne xt s te pma y be e i the r o f two a l te r n a t iv e s . T he c ho ic ede pe nds o n the r e s ul t o f the p r e v io u s a r i th m e t i c a l o p e r a t i o n a s r e p o r t e d by t hea r i t h m e t i c u n i t .

b . C o m p o n e n t s

T h e a r i t h m e t i c o p e r a t i o n s m a y b e m e c h a niz e d a s s u i ta ble c o m bina t io ns o f e le m e n

t a r y " l o g i c c i r c u i t s " of v a r i o u s s o r t s ( " a n dc i r c u i t s , " " o r c i r c u i t s , " " u n it d e l a y s , "e t c . ) . The s a me is t r ue o f the s wi tc hing o rr o u t i ng c i r c u i t s . ( T r a n s m i s s io n r o u t e s a r ede te r mine d by lo gic c i r c ui ts r a th e r tha n bys witc he s wi th me c ha nic a l mo ving pa r ts . )

E l e m e n t a r y l o g i c c i r c u i t s c a n b e m e c h a niz e d in va r io us d i f fe r e nt wa y s , us ing e i th e rva c uum tube s o r s e mic o nduc to r s (d io de s a ndt r a n s i s t o r s ) . F o r g u id a n ce a p p l i c a ti o n s ,h o w e v e r , t h e r e a r e s t r o n g a r g u m e n t s i n f a v o ro f se m i c o n d u c t o r c i r c u i t r y o f o n e s o r t o ra no the r . It i s s up e r io r to va c uum tubec i r c ui t r y in at le a s t th r e e wa y s . It i s s m a l l e ra n d m o r e c o m p a c t , i t r e q u i r e s v e r y m u c hless power, and when suitably designed, i ti s m o r e r e l i a b l e .

The s i tua t io n i s muc h le s s c le a r in r e ga r dt o t h e o t h e r m a j o r p a r t s o f t h e c o m p u t e r .M a g n e ti c c o r e m a t r i c e s a r e a t t r a c t i v e f o rt h e d y n a m i c s t o r a g e . T h e r e a d o u t o f c o r em a t r i c e s i s d e s t r u c t i v e , h o w e v e r , a nd e a c h

r e a d o u t m u s t b e a c c o m p a n i e d b y r e g e n e r a t io n if the numbe r r e a d o ut i s to be r e ta ine din s t o r a g e . A s a r e s u l t , c o r e m a t r i xr e l ia bi l i ty i s c o ns ide r e d do ubtful fo r thes to r a g e o f the c o n s ta nts a nd o f the p r o gr a ms te p s (whic h mus t be r e ta ine d wi th no e r r o r sa t a l l th r o ugh ma ny c o mput ing c yc le s ) .

P r o g r a m s a nd c o n s t a n t s c a n b e s t o r e d a si n t e r c h a n g e a b l e w i r e m a t r i c e s o r p l u gbo ar ds , but thes e b eco me unwieldy in anyb u t q u i t e s i m p l e a p p l i c a t i o n s . B o t h p r o g r a m s

a nd c o ns ta nts c a n a ls o be s to r e d a s numbe r so n m a g n e t i c d r u m s o r t a p e s . A c t u al ly ,d r u m s a r e i n p r e t t y go o d r e p u t e . T h e y d opr e s e n t a t iming pr o ble m , ho we ve r , and th isma y r e s t r ic t the e ffic ie nc y o f the p r o gr am m ing , in a way which is explained in thene xt s e c t io n .

Input and o utput dev ices for t r an slat ingbe twe en digi ta l a nd a na lo g va r ia ble s a r emany and var ied . We will not a t tem pt tod e s c r i b e t h e m in a ny m o r e d e t a il h e r e .

c . W o und vs . Unwo und Pr o gr a m s

C e r t a i n s e q u e n c e s o f o p e r a t i o n s a r e l ik e l yto be us e d s e v e r a l t im e s in a s ingle p r o gr a m . As a n e xa m ple , s uppo s e tha t the s ine s

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o f s e v e r a l d i f f e r e n t a n g l e s a r e t o b e c o m p u t ed . An ap p ro x i m at e fo rm u l a wi l l b e u sed ,s u c h a s a p o w e r s e r i e s , w h e r e b y th e s i n eof a g i v en n u mb er i s con st ru ct ed ou t o f as e r i e s o f a r i t h m e t i c a l o p e r a t i o n s a p pl ie d t o

t h e g i v en n u mb er an d a se t o f s t o red con s t a n t s . E a c h t i m e a s i n e i s c o m p u t e d,e x a c t l y t h e s a m e p r o c e s s i s u s e d e x c e p tthat i t is ap plied to a differen t num berr e p r e s e n t i n g a d i ff e r e n t a n g l e .

A f req u en t l y re p ea t ed seq u en ce su ch a&t h a t d e s c r i b e d a b o v e i s c a l l e d a s u b r o u t i n e .I n s o m e k i n d s o f p r o g r a m s t o r a g e u n i t s as u b r o u t i n e s e q u e n c e n e e d n o t b e s t o r e d m o r ethan once even though i t is used many t imesin t h e c o m p l e t e p r o g r a m . In o t h e r t y p e s i t

mu st b e s t o r ed ag ai n an d ag ai n , as man yl i m es as i t i s t o b e u sed in t h e co mp l et ep r o g r a m .

I n o p e r a t i o n a p r o g r a m c o n t r o l u n i t g o e sf r o m s t e p t o s t e p, o r " s t a t e " t o " s t a t e , " a st h e p r o g r a m p r o g r e s s e s . At e a c h s t e p o rs t a t e i t i ss u e s o n e se t o f i n s t r u ct i o n s to t h ear i t h met i c u n i t an d t h e swi t ch i n g c i rcu i t s . I tco n t i n u es i n t h i s man n er u n t il th e p r o g r am i scomp l et ed , an d t h en s t a r t s ov er ag ai n i n an ew c o m p u t in g c y c l e . If t h e p r o g r a m m e r c a nad v an ce on l y o n e s t a t e a t a t i m e an d can n ev e rr e t u r n t o p r e v i o u s s t a t e s ( e xc e p t w he n i tr e t u r n s t o s t a t e o n e a t th e b eg in n i n g of a n ewc o m p u t in g c y c l e ) , a s u b r o u t i n e m u s t b e s t o r e don ce fo r each t i me i t i s u sed i n t h e comp l et ep r o g r a m . If, h o w e v e r , t he p r o g r a m m e r ca nr e t u r n r e p e a t e d l y t o e a r l i e r s t e p s , a s u b r o u t i n e s e q u e n c e n e ed b e r e c o r d e d i n t h ep r o g r a m m e r o n ly f o r i t s i n it ia l u s e . F o ra d di ti o n a l u s e s , t h e p r o g r a m m e r c an r e t u r nt o t h e sa m e seq u e n ce of s t a t es . In t h e u su alc o m p u t e r t e r m i n o l o g y , th e f i r s t p r o g r a m m e rc a n s t o r e o n l y " u n w o u n d " p r o g r a m s , w h il et he s e co n d c a n a c c e p t " w o u n d " p r o g r a m s .

A t r i g o n o m e t r i c s u b r o u t i n e , w h ic h w i ll b ede sc r ibe d in Sect ion 5, of ten can be used tot a k e c a r e o f a l a r g e p a n o f a g u i d a n c e c o m p u t er p r o g r a m . U n de r t h e se c i r c u m s t a n c e s .

i t t a k e s m u c h l e s s s t o r a g e c a p a c i t y t o s t o r et h e p r o g r am i n wou nd fo r m t h an i n u nwou n df o r m . If t h e p r o g r a m i s s t o r e d o n a m a g n e t i cd r u m , s y n c h r o n i z a t i o n p r o b l e m s m a y d i c t a t es t o r a g e in t h e u nw o u nd f o r m . T h e s t o r a g e

c a p a c i t y o f m a g n e t i c d r u m s i s s o g r e a t ,h o w e v e r , t h a t i t m a y b e c h e a p e r t o s t o r e t h eu nwou n d p r o g r am on a d r u m t h an t h e wou ndp r o g r a m in a m o r e v e r s a t i l e m e c h a n i z a ti o n .

5 . PROGR/iMMING REPRESENTATIVEGUIDANCE MATHEMATICS

We no ted in Sect ion 2 that a guidance co m p u t e r ' s c a p a b i l i t i e s m u s t g e n e r a l l y i n c lu d et h e fo l l owi n g : s t o ra g e of t ra j ect o ry an d

t a c t ic a l i n f o r m a t i o n , t r a n s f o r m a t i o n s of c o o r d i n a t e s , a n d c o m p u t a t i o n s s u c h a s i n t e g r a t i on , d i f fe ren t i a t i on , f i l t e r i n g and d at asmo o t h i n g . L et u s re ex a mi n e t h ese in t h el ight of what has just been said about digi talc o m p u t e r s .

a . S t o r a g e o f T r a j e c t o r y an d T a c t i c a lI n f o r m a t i o n

T h e i n f o r m a t i o n , o r i n t e l l i g e n c e , s t o r e dwi t h i n t h e comp u t er can u su al l y b e rep re sen t ed b y fu n cti on s of on e se t o f v ar i ab l eso r a n o t h e r . In p r a c t i c e , t h e f u nc ti o n s a r ea p p r o x i m a t e d b y c o m b i n a t i o n s o f a r i t h m e t i co p e r a t i o n s ( p o w e r s e r i e s , i n te r p o l a t io nro u t i n es , an d t h e l i k e) . Var i o u s t ech n i q u esa r e a v a i la b l e f o r p r o g r a m m i n g m a t h e m a t i c sof t h i s so r t , as a resu l t o f t h e wi d esp readu s e o f g e n e r a l p u r p o s e co m p u ti n g m a c h i n e s .

I n g u i d a n c e a p p l i c a t i o n s , r a t h e r c r u d ea p p r o x i m a t i o n s a r e g e n e r a l ly p e r m i s s i b l e

wh en t h e mi ss i l e ' s f l i g h t can b e ref i n ed asi t a p p r o a c h e s th e t a r g e t . A s a r e s u l t , t h ec o r r e s p o n d i n g d e m a n d s u po n t h e c o m p u t e ra r e l i k e l y t o b e q u i t e r e a s o n a b l e i n r e g a r dt o c o m p u t i n g t i m e , p r o g r a m s t e p s , a ndn u m b e r s o f c o n s t a n t s .

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b. T r a n s f o r m a t i o n s o f C o o r d i n a t e s

The two s ubr o ut ine s de s c r ibe d be lo w c a nbe us e d fo r mo s t o r a l l o f the c o o r dina tet r a n s fo r m a t io ns ne e ded in a guida nc e c o m

put e r . W he the r they a r e a c tua l ly s o us e d inpr a c t i c e ma y de pe nd upo n whe the r wo und o runwo und pr o gr a m s a r e to be s to r e d . In a nye ve nt , the s ubr o ut ine s g ive a r o ugh me a s ur eo f the d if ficulty o f the c o o r dina te t r a ns fo r m a t io ns in te r ms o f c o mput ing t ime a nd numbe ro f p r o g r a m s t e p s .

In a s imple e xpe r ime nta l c o mpute r thea utho r us e d pr o gr a m s o f the fo l lo wingle ngths , fo r the two s ubr o ut ine s : fo r thef i r s t s u b r o u t i n e , 5 3 s i n g l e a d d r e s s s t e p s ;for the second subroutine, a to tal of 72

(including the use of the fi rs t subroutine ina feedback lo op).* (This included co mp utat i o n o f t h e n e c e s s a r y t r i g o n o m e t r i c f u n c t io ns . ) If the a r i thm e t ic uni t had bee n s o m e w h a t m o r e v e r s a t i l e i n r e g a r d t o c o n d i t i o n a lt r a n s fe r s a nd l imi t ing, the num be r s c o uldha ve be e n a l i t t le s ma l le r .

The f i r s t a nd m o r e ba s ic s ubr o ut inec a r r i e s o u t t h e t w o - d i m e n s i o n a l t r a n s f o r m a -tion i l lu str ated in Fig , 6. It is equivalent toa " r e s o l v e r " in a n a n a lo g m a c h i n e . In t h ef igur e , x a nd y a r e c o mpo ne nts o f a two -dime ns io na l ve c to r R in a g ive n f r a m e ofr e fe r e nc e . The n x ' a nd y ' a r e c o m po n e nts o fthe s a m e R in a new f r a m e of r e fe r e nc eo bta ine d by r o ta t ing the x , y f r a m e th r o ugh agive n a ngle 9 . The s ubr o ut ine de te r m ine sx' and y' when given x, y, and 9.

Note that 9 , R are the components of R ina r e c t a n g u l a r f r a m e o f r e f e r e n c e s u c h t h atthe y a xis i s a lo ng the ve c to r . T hus the s a m e

s ubr o ut ine c a n be us e d to r e s o lve a two -dime ns io na l ve c to r in to c o m po ne nts x a nd y,whe n i t s ma gni tude a nd d i r e c t io n a ngle a r egive n.

T h e s e c o n d s u b r o u t i n e p e r f o r m s a n i n ve r s e s o r t o f o pe r a t io n . It s ta r t s with thec o m po ne nts x a nd y a nd de te r m ine s the c o r r e s p o n d i n g t w o - d i m e n s i o n a l v e c t o r m a g n i tude and d ir ec tio n. One way in which it canbe me c ha niz e d us e s the f i r s t s ubr o ut ine in

a feedback o r se rv o loop within the co mputingp r o g r a m . An a n g le o f r o t a t io n , m ak es x' = 0 (and y'po sit ive) . Then y ' and <P ar e the de sir edma gni tude a nd d i r e c t io n .

Fo r the ful l th r e e -d ime ns io na l guida nc ep r o b l e m , s e q u e n c e s o f t w o - d i m e n s i o n a lt r a n s fo r m a t io ns c a n be us e d. The fo l lo winge xa mple wi l l i l lus t r a te the me tho d: Suppo s ea t a r g e t p o s i t i o n r e l a t i v e t o a n a i r c r a f t o rm is s i le i s give n a s N o r th a nd Ea s t c o mpo nents and differe nce in al t i tude, and suppo sethat the deflection angle, e levation angle,a nd s la nt r a nge a r e de s i r e d r e la t ive to a xe sfixed in the ai r f r am e. It is assu m ed that thea i r c r a f t he a ding, p i tc h , a nd r o l l a ngle s a r ege n e r a l , but a r e kno wn. The c o m puta tio n c a nbe a c c o mpl is he d by th r e e o pe r a t io n s o f thef i r s t s ubr o ut ine , a nd two o f the s e c o n d.

The a bo ve me c ha niz a t io n a s s ume s tha theading, pitch, and ro l l angles are defineda s the va r io us g imba l a ngle s o f a g imba lmo unted s ta ble p la t fo r m . Ai r f r a m e a ngle sa r e l ike ly to be me a s ur e d in te r m s o fgimba l a ng le s , a nd g imba l a ngle s a r e mo r ea p p r o p r i a t e f o r t h e u s e o f o u r s u b r o u t i n e st h a n , s a y , d i r e c t i o n c o s i n e s .

Ano the r i l lus t r a t io n w i ll s ho w the ve r s a t i l i ty of the two sub ro utin es. Suppose ana i r c r a f t o r m is s i le i s kno wn to be a t lo n gi

tude ty. and lati tud e 9, and that it is to

*In ge n e r a l , i t ta ke s two s ingle a d dr e s s s te p sto add, subtract , multiply, o r divide twon u m b e r s .

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b e flown al on g t h e g r ea t c i r c l e co u r se t o atarget at longitude ty z and lat i tude 9 2 . Whati s th e b e a r i n g o f th e g r e a t c i r c l e c o u r s eat ty , , 8 , , and what is the gr ea t ci r cl e d is t an ce t o t y2 , 6 , 19,, ? T h i s p ro b l em al s o can b e

sol v ed b y t h ree ap p l i cat i on s of t h e f i rs tsu b r o u t i n e an d t wo of t h e seco n d . T h es e q u e n c e o f o p e r a t i o n s i s d e r i v e d b y r e p resen t i n g t h e d es t i n at i on as comp on en t s i n ar e c t a n g u l a r ( x , y , z ) s y s t e m o f c o o r d i n a t e swi t h i t s o r i g i n a t t h e ce n t er o f t h e ea r t h .T h e p r o b l e m i s s o l v e d by s u i t a b l e r o t a t i o n so f t h i s c o o r d i n a t e s y s t e m .

c . In t eg ra t i o n , Dat a S mo ot h i n g, E t c .

I n o r d i n a r y g u i d a n c e c o m p u t a t i o n s , i n t e g r a t i o n s a r e t o be c a r r i e d o u t w i th r e s p e c tt o t i me . T h ey can t h en b e ap p r o x i m at ed wi t hs u m s o f s u i ta b l e i n c r e m e n t s a d d ed o n c e e a c hc o m p u t i n g c y c l e .

T h e i n t e g r a t i o n p r o c e s s i s e x t r e m e l ysi m p l e ex cep t fo r t h e p o ss i b i l i t y of se r i o u sa c c u r a c y t r o u b l e s . T h e a c c u r a c y p r o b l e ma r i s e s b e c a u se s m a l l r o u nd -o f f e r r o r s ini n di v id u a l i n c r e m e n t s m a y a c c u m u l a t e a s

l a r g e e r r o r s in t h e s u m s . T h e a c c u r a c yp ro b l em i s an o l d on e , h o wev e r , an d can b esol v ed i n v ar i ou s wel l -k n own way s.

Di f feren t i a t io n i s g en e ra l l y com b i n ed wi t hd at a smo o t h i n g . F i l t e r i n g an d d at a smo o t h i n ga r e u s u a l l y l i n e a r o p e r a t i o n s r e p r e s e n t a b l eb y l i n ear d i f fe ren t i a l eq u at i on s .

T h e d i f fe ren ce eq u at i o n s can b e so l v edby m e a n s o f a s e r i e s of i n t e g r a t i o n s c o r r e s p o n d i ng t o t h e i r v a r i o u s t e r m s . A l t e r n at i v el y , t h e so l u t i on s can b e ex p ressed assu m s of so l u t i on s of sy s t e m s of f i rs t and

s e c o n d o r d e r e q u a t io n s o f t he f o r m i l l u s t r a t e db e l o w .

(1) Typ ical differ ential equation:

1 ^ ^t x - o d t

y •<r = o o t

( 2 ) C o r r e s p o n d i n g s y s t e m o f dif

f e r e n c e e q u a t i o n s :

y = y0 + y, + y2 + • • • + y, + • • •

w h e r e

v0n • A 0 x n

an d e i t h er

% . • A S * n + B S y ,n S*n T"S'S(n-l)

o r

'SnA S x n + B

S V s ( n - l )

[CS

X(n-D

+ D S y S (n -2 )]

If the solut ions of the differential equat i o n s c o r r e s p o n d t o s a m p l e d f o r c i n g f u n c ti o n sa nd a r e t h e m s e l v e s s a m p l e d a t r e g u l a r i n t e r v a l s c o r r e s p o n d i n g t o t h e c o m p u t i n g c y c l e sof t h e g u i d an ce comp u t er , t h e samp l es a remat ch ed ex act l y b y so l u t i on s of d i f fe ren ceeq u at i on s d er i v ab l e f rom t h e d i f fe ren t i a le q u a t i o n s .

T h e i n d iv i d ual f i r s t and secon d o r d e req u at i o n s can b e so l v ed i n a ro u t i n e way .T h e so l u t i on of an n ' t h o rd er d i f fe ren t i a leq u at i on wi t h con st an t coef f i c i en t s req u i reda tota l of 2n m ult ipl ic at io ns and from n to 2na d d i ti o n s p e r c o m p u t i n g c y c l e . It r e q u i r e sn s l o t s i n t h e d y n ami c memory , an d u p t o2 n d co n s ta n ts .

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Tab le 1. Co mputing Speed of the TRADIC Phas e One Co mp uter

Type o f Ope r a t io n

Add o r s u b t r a c t t w o n u m b e r s( inc luding r e a do ut f r o m a c c umula to r )

M ul t ip ly o r d iv ide two numbe r s( inc luding r e a do ut f r o m a c c umula to r )

R o t a t i o n ( r e s o l v e r ) s u b r o u t i n e(x1 , y1 fro m x, y, 9 )

Magnitude and direction from x, y(o ne inc r e me nt e a c h c o mput ing c yc le )

No . O p e r a t i o n sin 1/2 Second

8000

1600

100

80

d . P e r m i s s i b l e P r o g r a m L e n g th s

The c o mple te p r o gr a m ma y inc lude asub stantia l amo unt of addit ion al co mpu tingbe s id e s the c o mputa t io ns de s c r ib e d in s ub s e c t io ns b a nd c a bo ve . This i s to o va r ie d inc h a r a c t e r , h o w e v e r , t o b e c o n s i d e r e d f u r t h e rh e r e .

In co nclusio n. Ta ble 1 is used to show

that, in fact , a digital co mp uter can do a lo to f c o mput ing, wi th in a c o mput ing in te r va l

o f to le r a ble le ngth . Ta ble 1 i l lu s t r a te s then u m b e r o f o p e r a t i o n s o f v a r i o u s s o r t s w h i c hma y be pe r fo r m e d in a 1/2-second i n t e r v a l .Ope r a t io ns o f the s ubr o ut ine s de s c r ibe d insubs ectio n a a r e included, a s well as theo r d i n a r y a r i t h m e t i c o p e r a t i o n s . In p r a c t i c e ,o f c o ur s e , the num be r s mu s t va r y wi th thes pe c i f ic c a pa b i l i t ie s o f the s pe c i f ic c o mp ute r se mplo ye d. The num be r s in the f igur e ,h o w e v e r , d o c o r r e s p o n d t o a n a c t u al c o m

pute r , the s o -c a l le d Pha s e One TR ADICComputer (Ref. 2).

REFERENCES

Bo de, H. W., and Shannon, C . E . , "A Simplified Der ivatio n of Li ne ar Le ast Squ are sSmo o thing The o r y ," Pr o c e e dings o f the Ins t i tu te o f R a dio Engine e r s , Apr i l 1950.

B e l l T e l e p h o n e L a b o r a t o r i e s , " T R A D I C P h a s e O n e S u m m a r y E n g i n e e r i n g R e p o r t V o l u m eI," Ai r R e s e a r c h a nd De ve lo pm e nt C o mm a nd, U. S . Ai r Fo r c e R e po r t N o -21536-5,July 1954.

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Fig. 5 . Block diagram of a guidance computer.

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F i g . 6 . R o t a ti o n o f r e c t a n g u l a r c o o r d i n a t e s i n t wo d i m e n s i o n s .

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THE USE OF DIGITAL COMPUTER TECHNIQUES INMISSILE DESIGN AND CONTROL*

D, H. Gr id le y**

SUMMARY

The c o mbine d impa c t o f h igh-s pe e d e le c t r o nic d ig i ta l c o mp ute r s and the us efultechniques der ived fro m the ir develo pment is now being felt in the ar ea s of m iss ile designa nd c o nt r o l . Two e xa m ple s o f d ig i ta l p r o c e s s in g a r e d is c us s e d a s they a pply to the p r e flight and po stflight phase s of m iss ile syste m design and an aly sis . A thir d example is aimeda t the c o nt r o l a r e a wi th a p r e s e nta t io n o f a ppl ic a ble me tho ds fo r the c o nve r s io n be twe ena na lo g a nd d igi ta l qua nt i t ie s a s r e qui r e d in the d ig i ta l c o nt r o l s ys te m.

SOMMAIRE

L' ut i l i s a t io n c o mbine e de s c a lc ul a t r ic e s e le c t r o nique s d ig i ta le s a g r a n de v i te s s e et de s

in te r e s s a n te s te c hnique s de r ive e s de le u r de ve lo ppe me nt e s t ma inte na nt to mbe 'e da ns ledo ma ine de 1'etude de s mis s i le s e t de le ur c o nt r o le . De ux e xe m ple s d 'a ppl ic a t io n de p r e c e d e sdigitaux comme ils sont applique 's a 1'etude et a 1'analyse d e s p h a s e s d ' a v a n t e t d e p o s t - v o ld ' un s ys te me de m is s i le , so nt d is c ute s . Avec la p r e s e nta t io n de s me tho de s a ppl ic a ble s po urune c o nve r s io n e nt r e le s qua nt i te s d ig i ta le s e t a na lo gue s c o mm e c e la e s t ne c e s s a i r e e nm a t ie r e de s ys te m e de c o nt r o le d ig i ta l , un t r o is ie m e e xe mple e s t do nne da ns le do ma ine duc o n t r o l e .

1. INTRODUCTION

Over the past decade and a half the elect r o nic d ig i ta l c o mpute r ha s be e n br o ught o uto f th e " g i a n t - b r a i n " c a t e g o r y a nd p l a ce d i n toa n e xt r e me ly us e ful a nd pr a c t ic a l a r e a fo rus e by bo th bus in e s s a nd s c ie n c e . Th e s emachines have in turn been reduced in s ize,but increased in both speed and rel iabil i ty.As a resu lt of the grow th per io d we have a lsole a r ne d c o ns ide r a ble a bo ut the me tho ds bywhich they can be used mo st efficiently.Ano ther outgro wth of this er a has been the

pr ac tic al applicatio n of the digital technique

to fields o ther than efficient computation.Th ese fac to rs as they apply to the design

a nd c o n t r o l o f m is s i le s wi l l be the to pic sco ver ed in this pap er . An attem pt will bem a d e t o p o i n t o u t t h r e e s e p a r a t e a r e a swhe r e the d ig i ta l c o mpu te r a nd i t s te c hnique sa r e a ppl ic a ble - de s ign c o mputa t io n , da tah a n dl in g , a nd t r a n s d u c e r s ; y e t y o u r im a g i nation will add o th er s to this l is t which willbe to p ic s fo r fu tur e d is c us s io ns . The f i r s ttwo a r e a s a r e a ime d a t the de s ign fa c to r s

*The o pinio ns e xpr e s s e d in th is pa pe r a r e no t ne c e s s a r i ly tho s e of the De pa r tm e nt o f De fe ns e,the U, S . Na vy, o r the Na va l R e s e a r c h La bo r a to r y .

"Na va l R e s e a r c h La bo r a to r y , W a s hingto n , D. C .

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of mi ss i l e d es i g n , on e on a p ref l i g h t b as i sand t h e o t h er on a p os t f li g h t b a s i s , wh e re ast he l a t t e r f a c t o r i s a im e d a t th e c o n t r o l a r e a .

T h e d es i g n -co mp u t at i o n u se of d i g i t a l

c o m p u t e r s i s p r o b a b l y t h e f i r s t a r e a in w h ic hs u c h m a c h i n e s w e r e u s e d f o r m i s s i l e w o r k .And, al though we are st i l l learning how toap p ly t h ei r cap ab i l i t i e s , i t ap p e ar s t h at ap l an fo r t h e o rg an i zed u se of t h ei r faci l i t i esis co min g into being. At f i r st the task wasthat of producing machines that would do thej o b . H e r e p r o g r a m m i n g o f p r o b l e m s w a s o fs e c o n d a r y i n t e r e s t ; b ut t o d a y , w i th m a c h i n e sav aila ble , the task of pro viding efficientp r o b l e m p r o g r a m m i n g h a s b e c o m e f o r m i d able and is faced equally by al l pe rs o nn el

u s i n g t h e m ach i n e s . In p a r t i cu l ar f i e l d s ,su ch as mi ss i l e d es i g n , wh i ch may a l soi n cl u d e an y v eh i cl e d es i g n , t h ere a re cer t a i nb a s i c s e t s o f e q u a t i o n s t h at d e s c r i b e av e h i c l e ' s g e n e r a l p e r f o r m a n c e i n r e l a t i o n t oi t s co n f i g u rat i o n , b al an ce, p o wer , l if t, a t m o sp h e r i c d r a g , an d o t h e r e x t e r n a l f o r c e s .

The task of set t ing up and checking outa p rog ram al l owi n g fo r t h i s mu l t i t u d e ofv a r i a b l e s i s a l en g th y j o b f o r s e v e r a l e x p e r i e n c e d m a t h e m a t i c i a n s a n d p r o g r a m m e r s .

However, i t can probably be said that thissam e p ro b l em h as b een wo rk ed ou t i nn u m er o u s g r o u p s faced wi th t h e n eed fo r as i m i l a r s o l u t i o n . It i s h e r e t h at c o o r d i n a t e deffo r t b y t h es e g ro u p s can , an d h as , p ro v ent o b e v al u ab l e .

In d i v e r s e g r o u p s , b ut w h e r e s i m i l a rt y p es o f c o m p u t e r s a r e a v a i l a b l e , a n e x c h a n g eo f b a s i c p r o g r a m s p e r m i t s a n e ff ic i en t u s e o fh i g h g r a d e p e r s o n n e l a n d r e l i e v e s t h e m o ft e d i o u s d u p l i c a t i o n o f p r o g r a m m i n g t a s k s ,

k f r e e s t h e m f o r m o r e d e t a i le d s t u d i e s ,p e r h a p s e v e n o n a r e a s d e s c r i b e d g e n e r a l l yby t he e x ch a n g ed p r o g r a m s . T h i s t y pe o fex ch an g e n eed n ot b e l i mi t ed t o g ro u p sp o s s e s s i n g s i m i l a r e q u i p m e n t , s i n c e i n t e r p r e t i v e r o u t i n e s h a v e b e e n w o r k e d f o r

ch an g i n g b et ween p rog rams of s i n g l e an dt r i p l e ad d ress cod i n g , an d i n man y i n s t an cesb et w ee n v e r y d i s s i m i l a r m a c h i n e s . N e v e r t h e l e s s , c o n t i n u i t y b e t w e e n c o m p u t e r o r d e r s ,wo rd l en g t h s , an d ad d re ss i n g met h o d s sh ou l d

b e con si d ered b y mach i n e d es i g n ers wi t ht h i s p rob l em i n mi n d .

I t i s fe l t t h at p rov en g en eral p u rp ose p rog r a m s d e s ig n e d f o r f l e xi b il it y o f p a r a m e t e r s ,d es cr i b ed e i t h er by s i n g l e v alu ed i n p ut s o rb y v ar i a b l e v al u es dev el op ed in b ran ch su b r o u t i n e s , m a y p r o v e t o b e a s i m p o r t a n t t ot h e mi ss i l e d es i g n ers as t h e wi n d t u n n elh as p ro v en in t h e p as t . But o rg an i zed an dc o o p e r a t i v e e ff o r t i s r e q u i r e d if p r o g r e s s i st o b e m a d e .

An at t emp t a t su ch a p rog ram i s t h atd e s c r i b i n g a s i n g le s t a g e r o c k e t t r a j e c t o r y .F i g . 1 s h o w s t he b a s i c f a c t o r s c o n c e r n e di n t h e t r a j e ct o ry . F i g . 2 i n t u rn sh ows t h ei n t e r d e p e n d e n c e o f t h e g e n e r a l p r o g r a m o nt h e c o n t r i b u t in g p a r a m e t e r s . M o s t o f t h e s ec a n b e c o n s i d e r e d i n t h e p r o g r a m a s v a r i a b l e swi th each b ei n g d esc r i b ed , p o ss i b l y byse p a r at e su b r o u t i n e s . It can b e seen t hat t h eb a s i c p r o g r a m s e t - u p , a s s u m i n g e x t e r n a l l ya p p li e d p a r a m e t e r s , c a n b e v a r i e d by a l t e r i n g

t h e s u b r o u t i n e s a nd t h e i r f o r m u l a t io n .Although nu m er o us diff icul t ies may exis t int h e i n s t ru men t at i on of su ch an ap p roach , t h ef in al re su l t s a p p ea r t o o f fer man y ad v an t ag esand impr o ved efficienc ies in the use ofc o m p u t e r f a c i l i t i e s .

T h e fo reg oi n g d i scu ss i on i s a su g g est edpa tte r n of at ta ck aimed at eff icient use oft h e com p u t at i on al c ap ab i l i t i es of t h e d i g i t a lco mp u t er wh i ch , if g i v en ear l y i mp et u s , canb e ex t r em el y u sefu l as t i me g o es on . T h e

f o r e g o i n g , h o w e v e r , i s c o n c e r n e d p r i m a r i l ywith effo rts in the prefl ight design phase ofm i s s i l e v e h i c l e s o r s y s t e m s . T h e n ex t i t e mt o b e d i scu sse d i s a l so re l a t ed t o d es i g n ,but occurs in the postf l ight phase whenev er y o n e s i t s a ro u n d an d a r g u e s ab ou t why

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the missile performed as i t did or why i t didno t o pe r a te a s e xpe c te d. Po s t f l ight a na lys ise n c o m p a s s e s t h e p r o c e s s i n g o f n u m e r o u stype s o f da ta ; ho we ve r , th is d is c us s io n i sr e s t r i c t e d t o t h e a r e a o f t e l e m e t e r e d d a t a .

2 . DATA HANDLING

Until r e c e nt ly , te le m e te r e d da ta ha ve be e nha ndle d pr im a r i ly o n a ba s is whe r e by a na n al o g s t r i p r e c o r d p r e s e n t a t i o n h a s b e enma de av ailable fo r ana lysi s by tes t ande n g i n ee r i n g p e r s o n n e l . A r e a s o f i n t e r e s tcan be quickly dete rm ined fro m such ap r e s e n t a t i o n ; n e v e r t h e l e s s , d e t a il e d a n a l y s i s

o f the s e a r e a s r e q ui r e d tha t they be r e a da t fr e que nt in te r v a ls , ' num e r ic a l va lue so bta ine d, l ine a r iz ing fa c to r s a ppl ie d , a nd theda ta r e plo t te d be fo r e a c c ur a te qua nt i ta t ivea na lys is c a n be pe r fo r m e d. Eve n wi th s o m emac hine aid this is a lengthy and tedio uspr o c e s s c o nt inual ly e nc umbe r e d wi th thehuman ele m ent. If the t ime schedule of ate s t p r o gr a m is c o mpa c t ( fe w a r e no t ) , l i t t leo r no t ime for this type of ana lysi s can bea l lo w e d, a nd o nly the o bvio us fa i lu r e s a r epr e s e nte d, but o f te n wi tho ut a de qua te r e a s o ns

f o r t h e i r o c c u r r e n c e . C o n s e q u en t ly , a r a p i dbut c o mple te ly a uto ma t ic s ys te m is de s i r e dthat w i ll a l le via te the p r o ble m fo r te le m e te r e dda ta r e duc t io n .

In the solution of this problem we have-again made use of digital techn ique s. Thede s i r e to us e the ge ne r a l pur po s e d ig i ta lc o mp ute r a nd i t s ve r s a t i l i ty i s , a t f i r s t g la nc e ,a n o bvio us s o lu t io n but a f te r a fu r the r b r e a k down of the tasks to be accomplished, the

p r o b l e m a p p e a r s t o o s i m p l e f o r s u ch amac hine and do es no t mak e efficient use ofi t s c a pa bi l i t ie s . The e quipme nt pha s e i sfur the r c o mpl ic a te d by a de s i r e tha t thereduction system be designed for f ield usea nd be t r a i le r mo unte d.

A c o mple te s ys te m de s ign fo r the ta s k o fr e duc ing a ny ( fm/fm, pd m/fm , a nd ppm /a m )t e l e m e t e r e d d a t a h a s r e s u l t e d in a n a r r a yof digital data reduction equipment that willbest me et the field o per atio n needs and s t i l l

p r e s e r ve the po s s ib i l i ty o f us ing the da tao bta ine d in a ge ne r a l pu r po s e c o mpu te r fo rthe s m a l l a mo unt o f a na ly s is no t a de qua te lypro vided for in the field unit . Data red uctio nt im e s a r e r e duc e d fr o m mo nths to tha t o fh o u r s a nd c o n s i s te n c y o f d a t a i s a s s u r e d .

The f i r s t p r o c e s s fo r th is e quipme nt top e r f o r m i s t h a t o f c o n v e r t i n g t h e t e l e m e t e r e da na lo g qua nt i t ie s to d ig i ta l da ta fo r s ubs e que nt r e c o r ding in the d ig i ta l fo r m o nmul t ic ha nne l ma gn e t ic ta pe . This o pe r a t io n

is i l lu str ate d in Fig. 3. The input data ar ea c c e pte d in r e a l - t ime , o r p la ye d ba c k ind e l a y e d - t i m e f r o m v i de o t a pe r e c o r d i n g swhic h we r e ma de in r e a l - t im e . In e i th e rcase the input data are sampled sequentiallya nd di r e c te d to o ne o f the two fo r ms o fqua n t iz e r s - vo l ta ge , o r puls e width . Theppm /a m da ta i s a c tua l ly the d is ta nc e be twe e na z e r o r e fe r e nc e puls e a nd a n info r ma t io npulse which is converted on the ground topulse width data for quantization by the pulsewidth qua n t iz e r . The vo l tage qua n t iz e r c a n

o pe r a te o n input da ta s a mple d a t r a te s upto 20 kc.- Th e pulse width data quan tizero pe r a te s a t the input da ta r a te o f 900 o r5,000 s a m p l e s p e r s e c o n d .

We have cho sen to quantize the data to1 p a r t in 256 o r 8 b ina r y p la c e s fo r u l t im a teda ta a c c ur a c ie s o f ± 1 /2 pe r c e n t , a l tho ugh a l lr e c o r d s a r e no t e q ua l t o t h i s p r e c i s i o n . T h eeight bits of quantized data, sample by sample,a r e p la c e d o n the ma gn e t ic ta pe in pa r a l le lcha nn els . Additional ma gnetic tape chann els

a r e r e s e r v e d f o r c o d e d - t i m e v a l u e s a ndf r a m e s y n c h r o n i z a t io n p u l s e s . R e c o r d i n gta pe s pe e ds a r e in f ive s e le c ta ble s te ps , f r o m60 inc he s pe r s e c o nd to 4-1 /2 inc he s pe rs e c o nd, o ne o f whic h is s e le c te d fo r c o mpatibil i ty with the incoming data rate .

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Once the data have been quantiz ed andst o re d i n d i g i t a l fo r m on t h e ma g n et i c tap et he y a r e i n a c o m m o n f o r m a t f o r u s e i nt h e d a t a r e d u c t i o n p r o c e s s o r f o r u l t i m a t ei n s e r t i o n i n t o a d i g i t a l c o m p u t e r . A n i l l u s

t r a t i v e s a m p l e o f t h e s e t a p e d a t a i s p r e s e n t e di n F i g . 3 wh er e 1 6 ch an n e l s of p p m / am d at aa r e r e c o r d e d a t t he 5 k c r a t e .

B e f o r e d e s c r i b i n g t h e r e d u c t i o n e q u i p m ent , a loo k at the fo rm of the usu al o utput(F i g . 4) wi l l i l l u s t r a t e b et t e r wh at mu st b eacc o mp l i sh ed . S hown i s a sect i o n of v i su alf i x ed -s t y l i t y p e of ou t p u t record h av i n gsp a ce fo r a t o t a l o f 6 0 0 s t y l i ev en l y sp ac edfor 11 inches of the 12-inch paper width.2 5 6 s t y l i (o r 5 12 s t y l i , if ev e ry o t h er s t y l u s

i s us e d ) a r e r e s e r v e d f o r t h e b a s i c d a t ac u r v e w i t h a l l , o r s e l e c t e d d a t a p o i n t s ,p r i n t ed l i n ear l y d own t h e t ap e. L i g h t v er t i call i n es a r e wr i t t en b y ev e ry 1 0 th o r 2 0 ths t y l u s of t h e 2 5 6 scal e , t o p rov i d e rap i dv i s u a l r e f e r e n c e . A s e l e c t i o n i n t h e s p e e dof t h e ou t p u t re co r d (con t i n u ou sl y v ar i ab l ef ro m 3 / 4 i n ch to 10 i n ch es p er seco n d ) canc o m p r e s s o r e x p a n d t h e d a t a i n t o s p e c i f i e dl e n g t h s c a l e s f o r e a s e o f r e a d i n g .

T o t h e l ef t o f t h e record , cod ed t i me

m a r k e r s a r e u s e d t o n o t e e a c h s e c o n d o ft i m e f r o m 0 to 9 99 9 s e c o n d s o r , if d e s i r e d ,i n zon e l i m e, wi t h h o u r s , mi n u t es , ands e c o n d s r e c o r d e d t o t h e n e a r e s t s e c o n d .L i g h t h or i zon t a l l i n es n ot e t h e 1 / 1 0 secon dr e a l - t i m e i n t e r v a l s . T o t he r i g h t a r e 4 -c h a r a c t e r A r a b i c n u m e r i c a l s a nd a f id u c ia lm a r k . T h e s e n u m e r i c a l c h a r a c t e r s a r efo r med i n a 5 x 7 d ot p at t e r n t h at can b ewri t t en ou t as of t en as on ce ev ery 1 / 1 0s e c o n d o f r e a l - t i m e . T h e f i du c ia l m a r k e rn o t e s t h e e x a c t c u r v e r e f e r e n c e p o i n t a t

w h i c h t h e n u m e r i c a l r e a d i n g i s m a d e . T h e4 - c h a r a c t e r n u m e r i c a l d a t a a r e c a r r i e d b yth e s y st e m in o r d e r to p e r m i t " e n d - o r g a n "r e f e r e n c e r e a d i n g s t o b e p o r t r a y e d . B o tht h e c u r v e d a t a an d t h e e n d - o r g a n v a l u e s a r ep r e s e n t e d i n t h e l i n e a r i z e d f o r m .

F i g . 5 p o r t r a y s t h e r e l a t i o n b e tw e e n i np uta nd o u tp u t b i n a r y o r e n d - o r g a n d a t a , a c o n v e r s i o n t h a t i s p e r f o r m e d w i th in t h e r e d u c t i o np o r t i o n o f t h e s y s t e m .

Ref er en ce t o t h e red u ct i on eq u ip men t cannow b e mad e (see F i g . 6 ) . F i r s t i s sh ownthe magnetic tape playback equipment whichh a s s i m i l a r p h y s i c a l a n d t a p e t r a n s p o r tc h a r a c t e r i s t i c s a s t he r e c o r d e r u se d p r e v i o u s l y . T h e d i g i ta l r e c o r d t a p e s a r e r e p l ay ed a t se l ec t ed sp eed s b ut a r e u n d erc o n t r o l o f t h e P r o g r a m m e r U n i t , w h i c ho p e r a t e s f r o m c o d e d ti m i n g t r a c k d a t a . O nt h e P r o g r a m m e r U n it c o n t r o l p a n e l, s e l e c ti o ncan be made of the channel number to bet r a n s c r i b e d , t h e t i m e to s t a r t , t h e t i m e t o

s t o p , t r a n s c r i p t i o n , a nd t h e r e c o r d i n g r a t e(ev er y d at a p o i n t, ev e ry o t h er p o i n t , e t c . )o f d a t a a s t h e m a g n e t i c t a p e r e c o r d s .

W o rk i n g in con j u nct ion wi th t h e P r o g r a m m e r U n it i s t h e S u b - C o m m u t a t e dCh an n el S el ec t o r , w h ich i s b r ou g h t i n to u seo nly when data of this type a r e use d. I tsho uld be no ted that the po ssibi l i ty ex ist sthat the selected output data may be quitelo w i n r e f e r e n c e t o a r e a l - t i m e r e c o r d ;thus i t is p o ssib le to spe ed up the playback

tape unit and reduce a single channel atb e t t e r t h a n r e a l - t i m e .

S e l ec t e d d a ta p a s s e d by t h e P r o g r a m m e rU n i t a n d / o r S u b - C o m m u t a t e d C h a n n e lS e l e c t o r a r e p r e s e n t e d t o t he l i n e a r i z e r f o ra l t e r a t i o n i n to t h e v a r i o u s f o r m s to b e u s e da s o u t p u t. T h e l i n e a r i z e r i s a m a g n e t i ccore mat r i x of 2 4 p l an es , each wi t h 2 5 6c o r e s o n a p l a n e . T h e l i n e a r i z e r i s s e tu p wi t h d at a ob t a i n ed f rom a ch ar t s i mi l a rt o F i g . 5 , p r i o r t o p l ay i n g b ack a sp eci f i cc h a n n e l . T h i s s e t - u p i s p e r f o r m e d b yau t o ma t i cal l y rea d i n g a p u n ch ed p ap e r t ap e,p r e p a r e d p r e v i o u s l y f r o m t h e c h a r t d a t a .T h i s re ad - i n t i m e i s ab ou t 1 mi n u t e . Att h e o u t p u t o f t h e l i n e a r i z e r a r e t h e c o n v e r s i o neq u i p men t s fo r p rod u ci n g t h e f i x ed -s t y l i

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r e c o r d e r d r i v i n g s i g n a l s a n d f o r p r o d u c i n ga punched paper tape (with format) in afor m that can be used dir ectl y by a high-speede le c t r o nic d ig i ta l c o mpute r (e .g . , NR L' sNAR EC ) i f a ny fur the r p r o c e s s ing is de s i r e d .

T h e A r a b i c n u m e r a l g e n e r a t o r i s a l s o ama gne t ic c o r e de vic e , wi th e a c h c ha r a c te rhaving a 5 x 7 co r e plane asso ciat ed withi t . One o f te n po s s ib le c ha r a c te r fo r mingw i r e s a r e e n e r g i z e d fr o m t h e l i n e a r i z e r s i g n a l s , the r e by s e t t ing up s e le c te d c o r e s . Thec h a r a c t e r i s d e ve l o p e d i n p r o p e r fo r m b ys y n c h r o n i z i n g t h e m a t r i x r e a d o u t p r o c e s swith the cha rt sp eed. When all seve n-r ead o ut s te ps ha ve be e n c o mple te d , the ma t r ixha s be e n c le a r e d a nd a ne ut r a l a r r a y i s

available for receiving the next input s ignal.

3 . CONVERSION PROCESSES

Digital techniques have not been usede xte ns ive ly wi th in dyna mic c o nt r o l lo o psp r i m a r i l y b e c a u s e a d e qu a te t r a n s d u c e r s h a venot been availab le; yet today, with the advento f r e l ia ble p r e c is io n e nd ins t r ume nts a ndminia tu r iz e d c o mpo ne nt r y , the ma ny a dva nta ge s a nd ve r s a t i l i ty o f the va r ia ble

p r o g r a m m e d d i g i t a l s y s t e m h a s a l t e r e d t h ethinking in ma ny m is s i le c o nt r o l g r o ups .T h e p r o g r e s s i n t h e a r e a o f p r o d u c i n go pe r a t io na l e quipme nt ha s be e n s lo w be c a us eo f the o ve r whe lming s impl ic i ty o f a na lo ginstr um entatio n. Yet, with a view to thefutur e, the digital technique is now develo pedto a po in t in the o r y , p r a c t ic e , a nd phys ic a lcompatibil i ty (s ize, weight, and power) thatit s ho uld be c o ns ide r e d mo r e s e r io us ly a s apa r t o f the s to r e ho us e o f te c hnique s to bedr a wn upo n whe n s ys te m de s igns a r e be ing

de ve lo pe d.

M a ny c o nt r o l p r o ble ms c a l l fo r the s o lu t io no f ba s ic a l ly s imp le p r o ble m s ; ho we ve r , weall know that within any of the confines ofmo de r n de fe ns ive v e hic le s s uc h a s a i r c r a f t ,

s h i p s , o r m i s s i l e s , t h e c o m p l e x it y o f m a t h e m a t i c a l t a s k s to b e p e r f o r m e d m a k e s t h e" c o m p u t e r - f o r - e a c h - t a s k " a lt it ud e o b s o l e t e.This o bs e r v a t io n c o me s no t o nly f r o m thema ny a ddi t io na l ta s ks tha t mus t be pe r fo r me d,

but a ls o f r o m the r e a l iz a t io n tha t the s e ta s ksa r e c o mpo s e d of da ta s e le c t io n and e va lua t io n ,bo th fo l lo we d by c o mputa t io na l o pe r a t io nsr e q ui r in g s to r a g e c a pa c i ty a nd pr e c is io n fa rbeyond the sco pe of availa ble o r knowna na lo g te c hnique s .

No s ingle data input is now assumedde s i r a bl e , but by s e le c t io n o f da ta f r o m thema ny po s s ib le s o ur c e s o f in te l l ige nc e , a nevaluatio n is made and the data ar e used inthe be s t c o mbina t io n fo r the s i tua t io n p r e

sented at the mo me nt. Ba sic data, as in thepa s t , a r e the p r ima r y input to a ny c o mput ingsy ste m , but l ike the human sy ste m , the manyc o nt r ibut ing but no r ma l ly no ne s s e nt ia l da taf o r m s o f i n fo r m a t io n a r e n e c e s s a r y t o m a k ea n a uto ma t ic s ys te m func tio n wi th the p r o pe r ," e d u c a t e d g u e s s " i n o r d e r t o o b t a in th epr o pe r de c is io n . The huma n e le me nt o fc o m ma nd ta ke s in to a cc o unt the ba s ic fa c to r sof any s i tuatio n; yet by knowledge of num ero uso the r c o nt r ibut ing fa c to r s the individua lma ke s h is de c is io n , whic h is no r ma l ly c a l le d

" o u t g u e s s i n g " th e a d v e r s a r y . It i s t h i sa dde d info r ma t io n a nd h is a bi l i ty to pe r fo r ma lo gic a l me nta l r e duc t io n o f a l l th is info r mation that makes the individual f i t o r notf it fo r c o mm a nd. Like w is e , a n a uto ma t ics ys te m mus t ma ke o pt imum us e o f a l ls o ur c e s o f in te l l ige nc e a va i la ble to i t .

Ad ve r s a r ie s in the pa s t ma de u s e o f thefact that only the human could make suchde c is io ns o f c o nt r o l a nd, a s a r e s ul t , ha vea c tua l ly us e d hum a ns e xpe nda bly a s c o n t r o l

e le m e n ts . In the mo de r n wo r ld the s pe e da nd a c c ur a c y o f de c is io n r e qui r e d in c o nt r o lha s in ma ny ins ta nc e s o utmo de d the huma n.C o ns e que nt ly , the ne c e s s a r y e le me n t i s as ys te m, vo id o f huma n de f ic ie nc ie s but c a pa ble o f us ing o nly the ne c e s s a r y da ta , p r o pe r ly

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p rocess i n g i t , an d t h en mak i n g a l og i cal an dp r o p e r d e c i s i o n , t h u s c a u s i n g t h e a c t i o n t ob e c a r r i e d o u t .

Di g i t a l t ech n i q u es as p resen t l y d ev el op ed

d o n o t o f fe r t h e a n s w e r t o a l l o f t h e s e r e q u i r e m e n t s , y e t p e r h a p s t h e y d o o f f e r m o r ean s we r s t h an we h av e ev er k nown i n t h e p as t .

T h e u n u su al sp ee d s of h an d l in g i n fo r mat i o nb y su ch d ev i c es an d t h e i r h i gh p r eci s i o n mak et h e m c a p a b l e o f r a p i d p r o c e s s i n g , a n d d e p en d i n g on t h e comp l i cat i on of t h e p rocessan d th e r eq u i r ed r a t e of d at a ou t p u t n e ce ssa r yfo r t h e end u s e , i t i s o f t en p o ss i b l e t o o p er a t et h e s e d e v i c e s in a p r o b l e m s h a r i n g m o d e .T h i s fact ex p l ai n s wh y d i g i t a l t ech n i q u es a reso ap p e al i n g ; on e d ev i ce can b e u sed fo r an u m b er of t ask s , t h e reb y e l i m i n at i n g t h e n eedfor d u p l i c at e , w ei g h ty , an d p ow er an d sp a cec o n s u m i n g s e p a r a t e b u t d i s t i n c t e l e m e n t s .

A l t h o u g h s p e c i a l p u r p o s e t y p e c o m p u t e r su s i n g d i g i t a l t ech n i q u es a re cal l ed fo r i ns u c h a p p l i c a t i o n s , t h e r e s t r i c t i v e n a m eapplied he r e should not be co nfusing. Digitalcomp u t ers of sp eci a l d es i g n n eed n ot b e t ood i ff e r e n t i n t e r n a l l y f r o m t h e s o - c a l l e d g e n

e r a l p u r p o s e ty p e . H o w e v e r , o n l y s u c hequipm ent a s that needed for the type ofp r o g r a m m i n g t o b e h a n dl e d i s d e s i g n e d i n tot h e s p e c i a l p u r p o s e e q u i p m e n t . T h e c o n t r o lp r o v i d e s b a s i c a r i t h m e t i c o p e r a t i o n s a n d t h ea s s o c i a t e d m e m o r y c a n p r o v i d e t he g e n e r a la s p e c t o f t h e s y s t e m c o n t r o l , th u s p e r m i t t i n ga l t e r a t i o n s o f p r o b l e m f o r m u l a t i o n w h e nr e q u i r e m e n t s a r e c ha n g e d . T h i s ca n b e d o n ew i t h o u t m a k i n g t h e s y s t e m o b s o l e t e .

S e v e r a l a n a l o g - t o - d i g i t a l c o n v e r s i o n d e

v i ces h av e b een d i scu sse d a s t h ey ap p ly t ot h e d ata r ed u ct i o n t ask . T h e v o l t ag e q u an t i zerh as a t p resen t a p reci s i on of ab ou t 0 . 0 5p ercen t b u t t h e p u l se wi d t h q u an t i zer , o rv a r i a t i o n s o f i t , h a s p r o v i s i o n l i m i t e d p r i m a r i l y b y t h e t r a n s d u c e r u s e d t o g e n e r a t e

t h e p u l s e d u r a t i o n . O p t i c a l l y - r e a d c o d ewh eel s of fe r h i g h p reci s i on , y et l eav e mu cht o b e d e s i r ed i n ru g g ed n ess fo r ac t u alo p e r a t io n a l u s e . A l so r e q u i r e d o f m o s t o p e r a t i on al con v ers i on eq u i p men t s i s an ou t p u t

f o r m c a p a b l e of b e in g c a r r i e d t h r o u g h m u l t i p l ex i ng o r t i m e s h a r i n g c i r c u i t r y w i th o u t l o s so f q u a l i ty . F o r l o w e r p r e c i s i o n d a t a , v o l t a g eswitching can be handled by diode gates witha s i n g l e q u an t i zer u sed fo r a l l i n p u t s . F i g . 7sh ows h ow t h i s sy s t em wou l d ap p ear .

F or h i g h l y p reci se d at a t h i s met h od fa l l ssh o r t . On e of th e mo st rea d i l y h an d led anda c c u r a t e m e t h o d s o f i n s t r u m e n t i n g t h i s a c t io ni s acco mp l i sh e d b y u s i n g s i m p l e p h ase

s h i f t e r s s u c h a s c o n v e n ti o n a l e l e c t r i c a l r e s o l v e r s . T h i s p r o c e s s i s i l lu s t r a t e d inF i g . 8 . T h e d r i v i n g f req u en cy fo r t h e p h ases h i f t e r i s g e n e r a t e d f r o m d i g it a l c i r c u i t r ywi t h i n t h e comp u t er o r i f t h e b as i c comp u t eri s g r o u n d b a s e d , a m i n i m u m o f a i r b o r n e c i r cu i t ry . E ach b ase cy cl e of t h i s f req u en cy i sp ro d u ced b y cou n t i n g a t som e b i n ary mu l t i p l efrequen cy (x 256, 512, 1024, etc .) .

At t h e ou tp u t of each p h ase-sh i f t i n g t r a n s

d u c er a t t ach ed t o a co n t ro l e l em en t wi llapp ea r a new signal of identical f requency butp h ase sh i ft ed i n an amo u n t p ro p o r t i o n al t othe input shaft dis pla ce m en t. Qua ntizat ion ofthis phase difference by the digi tal systemp r o v i d e s i n p u t f o r f u r t h e r d i g i t a l p r o c e s s i n g .By sel ec t i n g h i gh d r i v i n g f r eq u en c i es (40 0 o r1000 c ps) the pha se lag due to high inputv e l o c i t i e s a n d a c c e l e r a t i o n i s m i n i m i z e d .F o r h i g h e r p r e c i s i o n o p e r a t i o n t w o - o rt h r e e - s p e e d s y s t e m s m a y b e r e q u i r e d . Int h e s e i n s t a n c e s t h e m u l t i p l e t r a n s d u c e r u n i t sa r e g e a r e d t o g e t h e r in r a t i o s r e l a t e d t o t h en u m b e r s y s t e m u s e d w i th in t he c o m p u t e r .Amb i g u i t y b et ween d i g i t read i n g s , d er i v edf r o m t h e s e v e r a l t r a n s d u c e r s in a m u l t i -sp eed sy s t em i s reso l v ed l o g i cal ly wi t hi nt h e d i g i ta l s y s t e m .

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Th e co m puter output (Fig . 9) can be ei th era b ina r y-c o de d gr o up o f s igna ls tha t a r ec o nv e r te d in to a puls e po s i tio n r e la t iv e to theba s e f r e que nc y z e r o po int o r i t c a n ge ne r a tethe puls e po s i t io n in te r na l ly . In e i the r

ins ta nc e the e xte r na l e nd ins t r um e nt d r ive sto ma tc h the puls e po s i t io ns f r o m the c o mpute r a nd tha t o f z e r o c r o s s o ve r po in t o f thet r a ns d uc e r o utput s igna l . The a dva nta ge o fthis method of conversion is that in both thea n a l o g - t o - d i g i t al a nd d i g i t a l - to - a n a l o g c o n v e r s i o n p r o c e s s , t h e i n f o r m a t i o n i s c a r r i e din a pu lse fo rm which can be handled to ahigh de g r e e o f p r e c is io n by s imple lo gic a ld io d e g a ti n g a r r a y s .

Only ge ne r a l me tho ds ha ve be e n de s c r ibe dhe r e and no at tem pt h as been ma de todis c us s the ma the ma t ic a l the o r y o f s a mple dda ta s e r vo lo o p s . Ye t , in ge n e r a l , it i spointed out that dynamic equalization can bepr o vide d e i the r wi th in the c o mpu te r o r bya n a l o g c o r r e c t i v e n e t w o r k s w i t h i n t h e e x t e r nal feedback lo o p. In al l pro babil i ty bothc o uld be us e d i f s uc h c o mpl ic a t io ns a r er e q ui r e d; but wi th the c o mp ute r p r o vidingda ta c o mpo s e d o f g r o s s va lue s a nd no nl ine a rfunction, analog netwo rks would handle the

m o r e c o n v e n ti o n a l l i n e a r m o d i f ie r s r e l a t i n gto the usual physical dynamics of a vehicle.Consequently, we have at our disposal a

too l for mixing the com mo n with the un c o m mo n da ta . It s ho uld no t be o v e r lo o ke dthat a digital system of this type also affordsme a ns fo r inc o r po r a t io n o f a r e a s o na bly h ighp r e c i s i o n d a t a t r a n s m i s s i o n s y s t e m be tw e e n

mis s i le a nd gr o und o r f ixe d ba s e e le me nts .By use of these techniques a coded s ignalc a n b e r e c e i v e d o r p r o c e s s e d f o r t r a n s m is s io n . Se c ur i ty o f the s e s igna ls i s a ls oimpr o ve d by me a ns o f c o de r e dunda nc yfa vo r a bly us e d fo r c he c king and e r r o rc o r r e c t i n g .

Many control functions, navigation, o rp r o g r a m m e d s lo w m a n e u v e r s n ee d n o t i m p o s ethe h igh-f r e que nc y c a pa bi l i t ie s upo n the c o m pute r sys tem and can re adily be handled ata muc h s lo we r p r o ble m s o lu t io n r a t e by thec o mp ute r but in te r la c e d with o the r , po s s ib lyu n r e l a t e d s o l u t i o n s .

In sum m ar y, i t is pointed out that thephy sical instr um entatio n and the abil i ty toa pply the d ig i ta l te c hnique s a r e p r o gr e s s ingin a pa r a l le l fa s hio n . The fu tur e ho lds muc hp r o m i s e f o r a m o r e c o m p l e te a p p li c at io n o fthe i r c a pa bi l i t ie s . Ye t , a wo r d o f c a ut io nmu st be added. Digital sy ste m s ar e not thea n s w e r t o o r a p a n a c e a f o r a ll p r o b l e m s ;c o ns e que nt ly , a tho r o ugh s ys te m de s ign, a nd

e va lua tio n o f p r o po s e d a ppl ic a t io ns s ho uld bereviewed with an open mind and selections ho uld be ba s e d o n o ve r a l l m e r i t .

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THE APPLICATION OF NOISE AND FILTERTHEORIES TO GUIDANCE PROBLEMS

R . J , Pa r ks a nd R o be r t M . Ste wa r t*

SUMMyVRY

This pa pe r p r e s e nts the a ppl ic a tio n o f r e c e nt no is e a nd f i l te r the o r ie s to the de s ignof guidance sy st em s. It includes the application of W iener s ingle and mu ltiple input fi l terthe o r y to guida nc e s ys te m de s ign, p a r t ic ul a r ly whe r e th e r e i s mo r e than o ne s o ur c e o fguida nce info r m a t io n wi th d if fe r e nt c h a r a c te r is t ic s a nd a ls o to the c a s e whe r e o ne o r mo r eo f the s o ur c e s o f info r ma t io n i s r e c e ive d in the fo r m o f s a mple d da ta . Typic a l e xa m ple s o fthe a ppl ic a t io n o f the s e te c hnique s to p r a c t ic a l p r o ble ms a r e g ive n.

SOMMAIRE

C e t te no te p r e s e n te 1 'a pplic at io n de s th e o r ie s r e c e n te s r e la t ive s a ux br ui ts e t a u

f i l t r a ge a 1'etude de s s ys te m e s de go uve r n e . El le c o mpr e nd 1 'a pplic at io n de la thdo r iedu fi l t r e d 'entr £e unique ou multip le de WIENER a 1'eXde d e s s y s t e m e s d e g o u v e r n e , e npa r t ic ul ie r da ns le c a s d ' une s o ur c e d ' info r ma t io n de go uve r ne a ve c d i f fe r e nte s c a r a c t e r is t iq ue se t a us s i da ns le c a s o u une o uplus ie r s de s s o ur c e s d ' info r ma t io n s o nt r e c ue s s ur la fo r me dedo nn^ e s pu ls a t o i r e s . De s e xe m ple s typique s de 1 ' appl ica t io nde c e s te c hnique s a de s p r o ble m e sp r a t i q u e s s o n t d o n n e s .

1. INTRODUCTION

This pa pe r d is c u s s e s the a ppl ic a t io n o fs o me o f the r e c e nt no is e a nd f i l te r the o r ie sto the design of cer tain types of guidance

s y s te m s . The c o m ma nd type guida nc e s ys te mfo r c e r ta in a ppl ic a t io ns wo uld, in ge n e r a l ,be the one in which these concepts have themo st di re ct applicatio n, but they might wellfind application in direct o r modified form too the r type s o f guida nc e s y s te m s , o r in fa ctto ma ny o the r r e la te d f ie lds .

2. THE DISTORTIONLESS FILTERPR OB LEM

Inherent in any guidance system is them e a s u r e m e n t o f o n e o r m o r e p h y si c a l p a r a m e te r s s uc h a s po s i t io n o r ve lo c i ty . It

i s i n g e n e r a l r e q u i r e d t h a t t h e s e m e a s u r e me nts be qui te a c c u r a te a nd a ls o f r e e o f a nys ignif ic a nt f i l te r la gs o r t ime de la y s . The s etwo r e qui r e me nts a r e o f te n c o nf l ic t ing ina s

muc h a s the h igh a c c ur a c y r e qui r e me nt o f te ndic ta te s c o ns ide r a ble f i l te r ing o f the da tawhic h, o f c o ur s e , wi l l in ge ne r a l c a us e s o mel ag , de la y, o r d is to r t io n o f the da ta . Th ispr o ble m ha s le d to the c o nc e pt o f qua s i -d is to r t i o nle s s f i l te r de s ign in whic h, fo rc e r t a i n f o r m s o f s ig n a l s ( f o r i n s t a n c e , s i g na ls which co uld be expanded in a po wers e r ie s in t ime wi th a l imi te d numbe r o fte r m s ) a f i l te r c a n be de s igne d tha t wi l lc a us e no d is to r t io n . If the a c tua l s igna lto be fi l tere d dev iates at a l l f rom the for ma s s um e d, the f i l te r wi l l d is to r t the s igna lto a de gr ee which obvio usly depends upon themagnitude of the deviation from the assumed

*J e t Pr o puls io n La bo r a to r y , C a l i fo r n ia Ins t i tu te o f Te c hno lo gy, Pa s a de na , C a l i fo r n ia .

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fo rm . It depe nds upon the applicat io n as toh ow wel l t h e act u al s i g n al s can b e ap p ro x ima ted by the fo r m s which wo uld al low ther e q u i r e d a m o u n t o f q u a s i - d i s t o r t i o n l e s sf i l te r i n g , b ut c a s e s d o e x i s t w h e r e t h e r e

a p p e a r s t o b e v e r y l it t l e t h e o r e t i c a l j u s t i f i cat i on fo r t h i s ap p roach .

If t he p h y s i c a l p a r a m e t e r o f i n t e r e s t c a nb e measu red b y t wo t ech n i q u es wh i ch h av es i g ni f ic a n t d if f e r e n c e s in t h e c h a r a c t e r i s t i c so f t h e s e e r r o r t e r m s , p a r t i c u l a r l y in r e l a t io nt o t he s p e c t r a l d i s t r i b u t i o n s of t h e e r r o rt e r m s, a f i l t e r i n g t ech n i q u e i s p o ss i b l e wh i chwi l l p er m i t co mp l et e l ack of d i s t o r t i o n of t h es i g n al n o m at t e r wh at i t s fo r m mi g h t b e .

F o r i n s t a n c e , v e l o c i t y c a n be m e a s u r e d b yd i f fe ren t i a t i n g a p os i t i on m eas u r em en t an d b yi n t e g r a t i n g a n a c c e l e r a t i o n m e a s u r e m e n t .T h e e r r o r c h a r a c t e r i s t i c s o f t h e se tw o m e a s u r e m e n t s w o u l d b e c o n s i d e r a b l y d i f f e r e n t .T h e f i rs t s i g n al , b ecau se of t h e d i f fe ren t i at io n, wo uld tend to have much re lat iv elyh i g h -f r e q u e n c y n o i s e a n d l i t tl e e r r o r c l o s e t oze r o f req u en cy wh i l e t h e seco n d , b eca u se oft h e i n t eg ra t i on , wou ld t en d t o h av e an e r r o ro f j u s t t he r e v e r s e c h a r a c t e r i s t i c s w i th m o s tof i t s e r r o r b ei n g of t h e n at u r e of a low

freq u en cy d r i f t .

It is to be noted that, if F r + p F a = 1, thes y s t e m w i l l b e d i s t o r t i o n l e s s ; i . e . , t h e fi r s tt e r m , w h i ch r e p r e s e n t s an e r r o r t ha t i s afu n ct i on of t h e t i me h i s t o ry of t h e t ru ev e l o c i t y , b e c o m e s z e r o . U n d e r t h i s c o n d i ti o n ,

the choice of FQ and F r a nd th e e r r o r inest im ati ng V do not depend upon V i tself butonly upon NQ and N r ; a l s o , t he e r r o r w i llh av e zero mean i f N a and N r h a ve z e r o m e a n ,a n d t h e e r r o r w i ll t en d t o z e r o if N a and N r

a p p r o a c h z e r o .

I f V h ad zero mean an d resu l t ed f rom as t a t i o n a r y r a n d o m p r o c e s s a n d it w a s d e s i r e dt o m i n i m i z e t h e t i m e a v e r a g e o f t h e s q u a r e de r r o r , t he d i s t o r t i o n l e s s r e s t r i c t i o n w o u ld ,i n g e n e r a l , m a k e i t i m p o s s i b l e t o o b t a i n a ss m a l l a n e r r o r a s wo u ld o t h e r w i s e b ep o ss i b l e . E v en in t h i s i n s t an c e, h o wev er ,t h e d i f fe ren ce b et ween t h e ab sol u t e op t i mu man d t h e d i s t o r t i on l ess op t i mu m i s n eg l i g i b l ei f t h e e r r o r i n t h e a c c e l e r o m e t e r i s s m a l lc o m p a r e d w i t h t h e a c t u a l a c c e l e r a t i o n .F u r t h e r , i n m an y c a s e s , t he f o ll o w in g f a c t o r sap p l y :

(1) V d o es n ot in g en er al h av e ze r ome an an d i s n o t re p r ese n t a t i v e of

a s t a t i o n a r y r a n d o m p r o c e s s .

Co n si d e r , t h en , t h e d i ag ra m shown i nFi g. 1. F 0 and F r a r e f i lt e r t r a n s f e rf u nc ti o n s f o r th e a c c e l e r o m e t e r a n d p o s i t i o nd a t a , r e s p e c t i v e l y ; N a i s t h e e r r o r o r n o i s eo f t h e a c c e l e r o m e t e r ; N r i s th e e r r o r o rno ise in the differ ential po si t io n data andV* i s t h e es t i m at e of V mad e by t h i s c i r cu i t .

I t can be e as i ly shown by an aly sis of theci r cu i t in F i g . 1 t h at t h e e r r o r in t h e

e s t i m a t i o n i s g i v en b y t h e t r a n s f o r m o ro p e r a t i o n a l e x p r e s s i o n

(2) In many in sta nc es, the value ofi n t e r e s t m a y b e t h e e n s e m b l ea v e r a g e o f t h e e r r o r ( a s o p p o s e dt o t h e t i me av erag e) sh or t l y af t e ra n ear d i scon t i n u i t y i n V.

(3 ) U s in g t he d i s t o r t i o n l e s s r e s t r i c t i on , a d e cr ea se i n t h e n o i se wou ldr e s u l t in a l a r g e r p e r c e n t a g e d r o pi n t h e e r r o r , w it h a l o w e r l i m i tof z e r o e r r o r .

«„«V -V [F r + p F a - l ] v + F rN r + F aN aF o r t h ese r ea so n s , i t i s b el iev ed t h at t h ed i s t o r t i o n l e s s c r i t e r i o n i s a u se fu l a nd v a l -

(1) uable o ne.

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It will now be assumed that N r and N a ha vez e r o m e a n a n d a r e s t a t i o n a r y a n d r a n d o m .W ith the po s s ib le e xc e pt io n o f the s ta t io na r yr e s t r i c t i o n , t h e s e a s s u m p t i o n s a r e g e n e r a l l yr e a l i s t i c f o r N r , the e r r o r in the d i f fe r e nt iate d

po s i t io n da ta . Eve n the s ta t io na r y a s s umpt io nis no t unr e a l is t ic s inc e i t i s a c tua l ly r e qu i r e donly that N r b e a p p r o x i m a t e l y s t a ti o n a r y o v e rthe weighting per iod of F r . As a ge n e r a lr u l e , N a wil l no t ha ve z e r o me a n; ho we ve r ,the DC s hif t o r m e a n-v a lue e r r o r ca n bea na lyz e d s e pa r a te ly a nd by c o nt r o l l ing thea c tua l le ve l of the z e r o s h if t no t pe r m it te dto ha ve a s ignif ic a nt c o nt r ibut io n .

T h e s e a s s u m p t i o n s a n d t h e d i s t o r t i o n l e s sc r i te r ia r e s ul t in the s pe c t r a l de ns i ty <> o ft h e v e lo c i t y e r r o r fv -by the e xpr e s s io n

V, being given

* . I F |Z <J> +

^v In r

P X *°

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where rj>r and <t>0 a r e t h e s p e c t r a l d e n s i t i e sof N r and N a , re sp ec tiv ely , and a> - 1-rri,

whe r e f i s the f r e que nc y.

S in ce t he m e a n - s q u a r e e r r o r a nd t he e r r o rs p e c t r a l d e n s i t y a r e r e l a t e d b y

co2 2

a a €v v • 1 % df

-co(3 )

i t is possible to f ind the form of F r that willminimiz e o -v if <J>0 and <J>r a r e kno wn.

The pr o c e s s o f c a lc ula ting the be s t fo r mof F r is given by Norbert Wiener (Ref. 1).T h e f o r m o f t he e q u at io n fo r t h e e r r o r s p e c t r a ldensity is the sam e as the equation (in thefrequency do main) so lved by W iener for asingle input channel.

A s ignif ic a nt pa r a me te r o f in te r e s t wo uldbe the a c c u r a c y impr o ve me n t a ffo r de d by thec o m b i n a t i o n m e a s u r e m e n t o v e r a n o p t i m i z e ds ys te m us ing o nly the po s i t io n da ta . Th isc o mpa r is o n i s d i f f ic ul t , ho we ve r , s inc e a

po s i t io n m e a s u r e m e nt a lo ne wi th a ny f i l te r inga t a l l c a nno t be ma de d is to r t io nle s s , a nd thei m p r o v e m e n t a ff o r d ed b y t h e d i s t o r t i o n l e s sfe a tur e de pe nds he a vily o n the V pr o gr a m o ft h e p a r t i c u l a r s y s t e m u n d e r c o n s i d e r a t i o n .Ho we ve r , in p r inc iple , i t wo uld be po s s ib let o p r o g r a m t h e s t a n d a r d v a l u es ( en s e m b l ea ve r a ge ) o f a c c e le r a t io n a nd to us e the s es igna ls in l ie u o f the a c c e le r a t io n m e a s u r e m e n t .

If th is p r o gr a m min g w e r e do ne , the

e ffe c t ive a c c e le r a t io n e r r o r wo uld be thediffe r e nc e be twe en the a c tua l a c c e le r a t io n a ndthe s ta nda r d a c c e le r a t io n . B y de f in i tio n , th ise r r o r w o u l d h a v e z e r o m e a n in t h e e n s e m b l ese ns e, and if, in o r de r to gain a quali tat iv efe e ling fo r the impr o ve m e nt , i t i s fu r the ra s s u m e d t h a t t h e e r r o r i s r a n do m a nds ta t io na r y a nd ha s a f la t s pe c t r a l de ns ity $ T

o ve r the f r e que nc ie s o f in te r e s t , a nd tha t thes p e c t r a l d e n s i t i e s o f th e p o s i t i o n e r r o r an da c c e l e r a t i o n e r r o r a r e a ls o fl at , it c a n b es ho wn that the r a t io o f the ve lo c i ty e r r o r fo r

the c o mbina t io n m e a s u r e m e nt to tha t o f thea l l - r a d i o m e a s u r e m e n t i s g iv e n b y t h ee x p r e s s i o n

W comb

n adio" ( f ) '

(4 )

In ge ne r a l , the n, i f the a c tua l a c c e le r a t io ndeviated by about 10 pe r c e nt f r o m i ts s ta nda r dv a l u e , t h e u s e o f a n a c c e l e r o m e t e r a c c u r a t e

wi thin 1 pe r c e nt wo uld r e s u l t in a fa c to rof about 5.6 improvement in the velocitya c c u r a c y , a n d t h e u s e o f a n a c c e l e r o m e t e ra c c ur a te wi th in 0 .1 pe r c e nt wo uld r e s ul t ina fa c to r o f a bo ut 30 impr o ve me nt in ve lo c i tya c c u r a c y .

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In p ract i ce , i t wou l d of t en b e i mp ract i cala c t u a ll y t o p r o g r a m t h e s t a n d a r d v a lu e o fa c c e l e r a t i o n , u n l e s s i t c o u ld b e a p p r o x i m a t e dby a co n st an t o v er t h e reg i o n of i n t e re s t . Ift he s t a n d a r d a c c e l e r a t i o n i s n o t p r o g r a m m e d ,

o r if it i s on l y ap p ro x i ma t ed , a n d / o r if t h ea c c e l e r a t i o n d e v ia t io n f r o m s t a n d a r d i s n o ts t a t i o n ar y (as i t wi l l n o t b e i n ce r t a i na p p l i c a t i o n s w h e r e t h e m e a s u r e m e n t i s t o b et ak en sh o r t l y af t e r a n ea r d i sco n t i n u i t y i na c c e l e r a t i o n ) , t h e i m p r o v e m e n t a f f o r d e d b yt h e c o m b i n a t i o n m e a s u r e m e n t w i l l a c t u a l l yb e u n d eres t i mat ed b y E q . (4 ) .

T h e i m p r o v e m e n t a f fo r d e d by t h e c o m b i n at ion f i l tering is very defini tely a functionof t h e d eg ree of d i f fe ren ce b et ween t h e

s p e c t r a l d e n s i ty o f t h e t w o m e a s u r e m e n tsy st ems, an d t h e ab ov e resu l t s ap p l y on l y i ft h e assu m p t i o n s mad e ap p l y . Ob v i ou sl y , ift h e t wo s y s t e m s h ad i d e n ti c a l e r r o r s p e c t r a ld e n s i t i e s , t he c o m b i n at io n e s t i m a t e e r r o rcould be, at m o st , a facto r of the >/2 le sst h an t h e es t i mat e t h at cou l d b e ob t a i n ed f rome i t h e r m e a s u r e m e n t a lo n e . E v e n t h i si mp r o v em en t wou l d b e a t t a i n ed on l y if th etw o m e a s u r e m e n t e r r o r s w e r e i nd e pe n de n t.

Although velo ci ty has been use d as an

e x a m p l e h e r e , t h e s a m e p r o c e s s c o u l d b eap p l i ed t o th e me asu re m en t of an y p h y si calp a r a m e t e r t ha t co u ld be m e a s u r e d in m o r et h an on e way wi th su ff i c ien t ly d i f fe ren t e r r o rb e h a v i o r . F u r t h e r t h e r e i s n o l i m i t a t io n o nt h e n u m b e r o f m e a s u r e m e n t s w h i c h c o u l d b ec o m b i ne d . If t h r e e o r m o r e m e a s u r e m e n t sw e r e m a d e o f t he s a m e p a r a m e t e r , th e y c o u ldal l b e com b i n ed s i mu l t an e o u sl y t o re su l t i na n e s t i m a t i o n w h o s e a c c u r a c y w o u ld b e b e t t e rt h an co u l d b e o b t a i n ed if an y on e o r se v e r a lw er e n ot u sed . T h e d eg re e t o wh i ch an y

o n e m e a s u r e m e n t w o uld i m p r o v e t he e s t i m a t ewould a gain depend heavily on how differenti t s e r r o r s p e c t r a l d e n s i ty w e r e f r o m t h a t o fa n y o f t h e o t h e r m e a s u r e m e n t s a n d, o fco u r se , on t h e re l a t i v e m ag n i t u d e of t h ee r r o r o f t h a t s o u r c e .

3 . T HE AP P L ICAT ION OF W IE NE R'SRESULTS TO CLOSED LOOP SERVOSOR GUIDANCE SYSTEMS

T h e b a s i c p r o b l e m c o v e r e d by N . W i e n e r

i n Ref. 1 i s , i n s i mp l i f i ed fo rm , es sen t i a l l ya s f o l lo w s . C o n s i d er a s e r i e s o f m e a s u r e men t s of a s i g n al a l l con t ami n at ed b y n oi se( in g en er al , d i f fe ren t n o i se i n each ch an n el)in which each signal wil l be independentlyf i l t e red an d t h en a l l ad d ed t og et h er asshown in Fig. 2 . What is the fo rm of eachof the f i l te r s Yj that wil l re su l t in am i n i m u m r . m . s . d i ff e r e n c e b e tw e e n t he o u t put and the t r ue signa l? The fol lowinga s s u m p t i o n s a r e m a d e i n W i e n e r ' s s o l u t io n tot h e p r o b l e m :

(1 ) T h e f i l l e r s a r e a l l l in e a r .(2 ) T h e s i g n al an d t h e v ar i o u s n o i ses

a r e a l l s t a t i o n a r y r a n d o m s e r i e swi th ze r o mean and h av e k nownc o r r e l a t i o n f u n c t i o n s o r s p e c t r a ld e n s i t i e s .

W i e n e r ' s b o o k p r e s e n t s a p r o c e d u r e f o rso l v i n g fo r t h e v ar i o u s Y: . Co n si d er n owt h e feedb ack sy s t em re p r ese n t e d b y F i g . 3 .

It can be shown easily that, if Y2 is knownan d i t i s d e s i r ed t o d et e rm i n e Y, so as t o m i n i mi ze t h e r . m. s . d i f fe ren ce b et ween e 0 andS , th e e r r o r c a n b e r e p r e s e n t e d in t h e f o r m

V * ' I + Y, Y,( Y . N . - S ) +

I + X Y,N,

(5 )

which, if we let Y = (Y, Y2 )/ (l + Y, Y 2 ) willbe of the sa m e fo rm as the sing le input

W i e n e r p r o b l e m w h e r e Y i s t h e u nk no w n ,Y 2 N 2 -S is the equivalent signal , N, is theequivalent noise and, as such, can be solvedby the technique o ut l ined by W iener . Once Yi s d e t e r m i n e d , Y, c a n b e e a s i l y d e t e r m i n e d ,since Y, is known.

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C o ns ide r the p r o ble m o f guiding a d r o neair plan e down a s t r aig ht l ine by m ean s Of ag r o u n d b a s e d m e a s u r e m e n t o f t h e d r o n e e r r o rin posit ion and through the use of a commandl ink to the d r o ne to c a us e i t t o ma n e uve r to

r e m o v e a n y e r r o r .

The system shown in Fig. 3, then, can bei n t e r p r e t e d to c o v e r t h i s p r o b l e m . T h es igna l S i s ide nt ic a l ly z e r o s inc e it i s de s i r e dt o m i n i m i z e t h e s i d e d i s p l a c e m e n t( r e p r e s e n t e d b y e 0 ) under the effects of botht he n o is e o r e r r o r in t he g r o u nd m e a s u r e m e nts N, a nd the d is tu r b a nc e s a c t ing o n thed r o n e r e p r e s e n t e d by N 2 . T h e d i s t u r b a n c e sin ge ne r a l wo uld be c a us e d pr im a r i ly by s idewind e ffe c ts a nd ina c c ur a c ie s o f the d r o ne

a uto pi lo t .

The t r a ns fe r func t io n Y2 w o u l d r e p r e s e n tthe effects of th ese dis t ur ba nce s on thea c tua l s ide d is pla c e m e nt . Sinc e the s e e ffe c tswo uld be de te r m ine d by the a i r f r a m e a ndpropulsion unit of the drone, they could bekno wn fo r a ny pa r t ic u la r d r o ne . The unkno wnY, would r ep r es en t the fi l ter in g that shouldb e d o n e o n t h e m e a s u r e d e r r o r s i g n a l b e f o r eit is t ransmitted to the drone and mixed intothe dro ne autopilo t input. If m o re than one

dis tur ba nc e s o ur c e e xis ts , the y c a n a l l ber e pr e s e nte d by a n e quiva le nt wind d is tu r ba nc e , fo r ins ta nc e .

Fo r e xa mple , i f i t i s a s s ume d tha t

Y* = p ( l + TP)

which would be representative of a winddisturbance in which a sudden wind would

f i r s t c a us e a s ide a c c e le r a t io n whic h wo uldgra dua lly decay unti l the dr o ne too k up theside velocity of the wind, and that N, andN 2 both had flat sp ec tr al dens ity ofmagnitude t 2 a nd 1 r e s p e c t ive ly (a s s uminga unit spectral density for € z c a u s e s n o l o s s

o f ge ne r a l i ty s inc e i t me r e ly de f ine s theunits being used in the problem), the solutionfor Y, is

+ ( / « * + 2 « T - « ) p

This s o lu t io n indic a te s tha t no f i l te r ingof the gro und s ignal should be done eventhough i t is noisy; in fact , the gro und s ign alshould have a der ivative term added whichwil l ma ke i t e ve n m o r e no is y . The ga in o fthe g r o und o pe r a t io n i s s pe c if ie d a s l /e a ndis the r e fo r e a funct io n o f the g r o und no is e .Th e r e s ul t indic a te s tha t the f i l te r ing i s

r e a l l y d o n e b y t h e d r o n e a i r f r a m e itself. Eve ntho ugh the c o mm a nd s igna l t r a ns m it te d to thedr o ne i s e xt r e m e ly no is y , the a c tua l s idepo sit ion of the dr o ne will be ver y l i t t le affectedby the no ise d ue to th e fact that the sid eposit ion of the drone cannot be changedr a pidly a nd i t the r e f o r e do e s no t r e s po nd to o rf i l te r o ut the no is e in the c o mma nd s igna l .

This s ys te m wo uld , in fa c t , be impr a c t ic a la s i t s ta nds , s inc e the r .m.s . va lue o f the

n o i s e t r a n s m i t t e d o v e r t h e c o m m a n d c h a n n e lwo uld, in the exam ple, be infinite . Th issituatio n is due to the fact that the no ises p e c t r a l d e n s i t i e s w e r e a s s u m e d t o b e f l a tout to infinite fre qu enc ies . In pr ac tic e, theycould be fi l ter ed with fi l t er s with t im ec o n s t a n t s s m a l l c o m p a r e d t o t h e r e s t o f t h es ys t e m s o a s to ha ve a ba ndwidth la r gec o mpa r e d to the ba ndwidth o f the o ve r a l ls e r v o lo o p but s t i l l no t be inf in i te . Fu r th e r m o r e , t h e r e e x i s t s a s t r a ig h t fo r w a r de xte ns io n o f the te c hnique us e d whic h pe r m its

the o pt imiz a t io n o f the e r r o r s igna l in thes a me s e ns e a s be fo r e but s ubje c t to ther e s t r i c t i o n t h a t t h e r . m . s . v a l u e o f t h e n o i s ein the c o mm a nd c ha nne l o r a ny o th e r po in tin the circ uit be equal to o r less than so m es pe c if ie d va lue . Obvio us ly , the m o r e the

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r . m . s . c o m m a n d n o i s e l ev e l i s r e s t r i c t e d , t hem o r e t he r e s t r i c t e d o p t im u m w i ll be w o r s et ha n t h e u n r e s t r i c t e d o p t i m u m . In g e n e r a l ,h o w e v e r , a c o m p r o m i s e c a n b e fo u nd th a tresu l t s i n an accep t ab l e comman d n oi se l ev el

w i th o u t s e r i o u s l y r e d u c i n g t h e o v e r a l lg u i d a n c e a c c u r a c y .

Y2 s o a s t o m i n i m i z e t h e r . m . s . d if f e r e n c ebetween S* and S, if

Y, "-Y Y' l l '22

l

+

{Y„ + Y |2 )Y 2 i

S u p p ose fu r t h er t h at t h e d ron e h ad ab oarda g y r o c o m b i n e d w it h a t y pe o f d r i f t m e a s u r e men t o r h ad ab o ard a s i m p l e fo r m of ani n er t i a l g u i d an ce sy s t em t h at wou l d p ermi tso m e fo rm of cr ud e but not sufficientlyac cu r at e g u i d an ce ev en in t h e ab se n ce oft h e g r o u n d c o m m a n d s y s t e m . T h e q u e s ti o nt h en n at u ra l l y a r i ses of h ow mu ch of eachguidance signal should be used and of howt h e s i g n al s sh ou l d be mi x ed t o p er m i t t h em i n im u m gu id a nc e e r r o r . T h i s p r o b l e m c anbe handled by an extension of the abovet ech n i q u e wh i ch i s p ara l l e l t o W i en er d u ali n p u t p r o b l e m .

T h i s p r o b l e m c a n b e r e p r e s e n t e d d i a -g r a m m a t i c a l l y by F i g . 4, w h e r e Y 22 i skno w n a nd r e p r e s e n t s t h e a i r f r a m e r e s p o n s e ,N 3 r e p r e s e n t s t h e d i s t u r b a n c e s a f fe c ti ngthe drone, and N, and N 2 r e p r e s e n t t he n o i s e

o r e r r o r o f th e tw o g u id a nc e m e a s u r e m e n t sr e s p e c t i v e l y . T h e t r a n s f e r fu n c ti o n s YM andY l2 are the unknown, and i t is desi red toc h o o s e t h e m s o a s to m i n i m i z e t he r . m . s .value of the difference between e„ and S.

In t h i s p ar t i cu l ar ca se , S i s ag ai n i d en t i cal l y ze r o , s i n c e, i n t h i s ex a mp l e , i t i sd e s i r e d t o g u id e t he d r o n e d o w n a s t r a i g h tl i n e (if t h e d es i r ed p ath w er e so me t h i n g o t h erthan a st r aig ht l ine, the signal S wouldr ep r es en t th e d i f fe ren ce b et ween t h e act u al

d es i red p at h an d a s t ra i g h t l i n e) .

I t can be easi ly shown that this problemcan b e red u ced t o t h e p rob l em sh own i nF i g . 5 , wh e re i t i s d e s i r ed t o ch o o se Y, an d

and

VY Y'12 '22

l + ( Y M + Y | 2 ) Y 2 2

(6)

and Y 22 N 2 is the equivalent signal .

T h e p r o b l em shown in F i g . 5 i s , o f co u r se ,the sp eci al du al input ca se of the ge ne r alpr o blem which was shown in Fig . 2 andsolved by Wiener (see Ref. 1) .

F ol l o wi n g t h e p r o ce d u re s gi v en in Ref. 1 ,i t is po ssib le to so lve for Y, and Y2 an dt h e r e f o r e f o r Yn and Y l2 f ro m Eq. (6) . I t

i s f u r t h e r p o s s i b l e , i f t h e p a r t i c u l a r p r o b l e mw ar r an t s i t , t o o b t a i n a con d i t ion ed o p t i mu msol u t i on t o t h i s p rob l em su b j ect t o t h el i m i t a ti o n t h a t t h e r . m . s . e r r o r a t s o m e p o i nti n t h e c i rcu i t , say a t t h e comman d ch an n elp oi n t , b e n o g r ea t e r t h an so me sp eci f ieda m o u n t .

T h i s p a r t i c u l a r p r o b l e m c a n b e f u r t h e rextended to include any num ber of inputs ; i ti s n o t l i mi t ed t o on e o r t wo . Un fo r t u n at e ly ,e x c e p t i n s o m e s p e c i a l c a s e s , t h e m e c h a n i c sof so l v i n g t h e p rob l em b ecome more d i f f i cu l tw h e n e v e r tw o o r m o r e i np u t s a r e r e q u i r e d , b utt h e p ro ce d u r es a r e o u t l in ed an d can , in an ys p e c i fi c c a s e , b e c a r r i e d t hr o u g h w i th s o m el a b o r .

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4. AN APPLICATION OF WIENER'SM U L T I P L E - I N P U T F I L T E R I N GTHEORY TO DIVERSITY SYSTEMS

A te c hnique o f c o ns ide r a ble in te r e s t inc o n t e m p o r a r y c o m m u n i c a ti o n , c o n t r o l , a ndi n s t r u m e n t a t io n s y s t e m s i s i l l u s t r a t e d inFig . 6 .

W ie ne r ' s ge n e r a l a na lys is o f s uc hpr o ble m s show s that , if S, N, , N 2 , . . .N n a r emutua l ly inde pe nde nt s ta t io na r y r a ndo mt i m e - s e r i e s , t h e i m p u l s i v e - r e s p o n s e s y . ( T )o f the l ine a r f i l te r s whic h minimiz e the me a n-s q u a r e e r r o r in e s t im a t i n g th e " s i g n a l " S b yS* satisfy the fol low ing s im ultan eo us set ofinteg ra l equatio ns: ( j • 1, 2, . . . n)

* S( T )

" - / { U s( r

-< r )

J t Y i ( a )

o U l

+ [ * N X - o - , ]Y j(o - , }do -

(7)

fo r T- > 0, where the d> t e r m s a r e a u t o c o r r e lat ion functions of the s ignal and v ar io usn o i s e s .

Th e s e e qua t io ns a r e e quiva le nt to the s e t

rO O

* S ( T ) - / { [ * , ( T - « T ) ] £ Y, W' • J j , < -L - 8

- C O i =

+ [ t \ ( T - o X y c r ) } do - -- h ( r )

(8)

with the understanding that

Y,(o-) • 0

(9)

fo r o- < 0 , ( the " r e a l iz a b i l i ty " c o ndi t io n) a nd

h ( r ) -- 0

(10)

for r > 0.

F o u r i e r - t r a n s f o r m i n g b o th s i d e s o f E q .(8) then gives:

• , {» ) [ l -Y Y, (to)] - * (•) Y (to) -. Hj(«0

(11)

wh er e the <t> 's a r e sp ect ra l d en sit i es ofs igna l a nd the va r io us no is e s , the Y ' s a r ef r e que nc y r e s po ns e func t io ns o f the o pt imumfilt er s having (from Eq . (9)) no po les in thelower half-plane, and the H's having (fromEq . (10)) no po les in the upper half- plan e.

A frequently encountered s i tuation is thatin which the no ises in the var io us channe lsa r e s t a t i s t i c a l l y s i m i l a r , e x c e p t p o s s i b l y f o r

a mpl i tude . The n,

Nj J * N 0

(12)

Substituting Eq. (12) into Eq. (11) theng i v e s :

* . [ ' - I ^ l l -K

i * NYx

Hi

(13)

Subtracting each s ide of this equation fora r b i t r a r y j f r o m tha t fo r j • 1 , e .g . , g ive s

V [ K ! V K ' Y ' l ' H ' - v(14)

U s i n g W i e n e r ' s s p e c t r a l - f a c t o r i z a t i o nt h e o r e m .

<JJ) (to) = \J>(t0) ty (u ) = ty tyN ° (15)

w h e r e ty * a n d t y' h a v e p o l e s an d z e r o e s o n l y

in the uppe r a nd ' lo we r ha l f -p la ne s r e s pe c t ive ly . Thus ,

f [ K ' Y - K*Y,] . H . ~ H i

(16)

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Th e left sid e of Eq . (16) can then havepoles only in the upper half-plane, while ther i g h t s i d e h as p o l es o n ly i n t h e l ow er half-

p l a n e ; h e n c e .

w h e r e

\J>*[K? Y - Kf Y,] = constant

(17)

It may be ea si ly shown that as to— oo,

^ [ K j Y j " K«Y,] - 0

(18)

an d t h u s , i n g en eral

K] Yj - K* Y, = 0

(19)

o r

Yj (to) -

K fY. (to).

(20)

E q . (2 0 ) i n d i cat es t h at t h e o p t i mu m sy st e mmay b e mech an i zed wi t h j u s t on e f req u en cy -sen si t i v e e l e me n t as sh ow n in F i g . 7 . T h i s

i m p l i e s t h e s o m e w h a t s u r p r i s i n g r e s u l t t h a tt h e op t i mu m b an d wi d t h fo r each ch an n el i si d en t i cal , ev en t h ou g h t h e s i g n al - t o-n oi sera t i os may b e q u i t e d i f fe ren t .

T h e b es t fo rm of Y, may b e d et e rm i n edas fo l low s: Subst i tut io n of Eq. (20) intoEq. (13) gives

o r ,

* s " [ * . + * J ] Y,'= H,

•.. n i

(23)

and

Kty](24)

Eq . (22) may be so lved fo r Y, by themet h od ap p l i cab l e t o a s i n g l e- i n p u t f i l t e r ,and then Y, may be found fro m Eq . (24).

5. SOME NEW RESULTS CONCERNINGTHE STATISTICAL DESIGN ANDANALYSIS OF SAMPLED-DATASYSTEMS

a . R e s t o r a t i o n o f S a m p le d D a t a

Ref. 2 d e sc r i b e s a s t a t i s t i c al t ech n i q u ea p p l i c a bl e t o t h e p r o b l e m o f r e s t o r i n gsam p l ed d at a . In t h i s sect i o n we wi l lm e r e l y s u m m a r i z e th e p r i n c ip a l r e s u l t s

o b t a i n e d .T h e p r o b l e m t r e a t e d c a n b e r e p r e s e n t e d

b y t h e b l ock d i ag ram i n F i g . 8 . Accord i n gt o t h e wel l -k n own samp l i n g t h eorem, i fm(t ) h as n o f req u en cy co mp o n en t s h i g h erthan f0, a l l o f m( t ) can be p erfe ct l y re co n s t r u c t e d f r o m s a m p l e s t a k e n p e r i o d i c a l l yat an y f req u en cy g r ea t e r t h an 2 f0. Sincethis is an ideal izat ion of the usual t rues i t u at i on , i t i s o f i n t e r es t t o ex am i n e t h ere l a t i on sh i p b et ween samp l i n g f req u en cy an d

(2 1) mi n i m u m at t a i n ab l e e r r o r s in t h e ou t p u t fo r

m e s s a g e s h av in g a r b i t r a r y s p e c t r a .

U s i n g m e a n - s q u a r e e r r o r a s a m e a s u r eof f i d eli t y , th e fo l lowi n g maj o r r es u l t s a r eo b t a i n e d f o r t h e c a s e o f n o i s e - f r e e s a m p l e

(2 2 ) p u l ses :

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(1) As t r a i g h t f o r w a r d p r o c e d u r e is

de ve lo pe d fo r de te r m ining the o pt imum phys i c a l ly r e a l i z a b l e t i m e - i n v a r i a n t l i n e a rs mo o thing f i l te r . It i s s ho wn tha t the e r r o rspectral density is given by

* € ( f ) -- | Y - l |Z*m(f ) + | Y | 2 * s ( f )

w h e r e

Y • f, Y

and should equal zero for x = D + n A,w h e r e n is a ny in te ge r o the r tha n z e r o , a ndA is the s a mpl ing in te r va l . Fo r phys ic a lr e a l i z a b i l i t y , y ( r ) = 0 fo r r < 0 (see Fig. 10).

(4 ) If the s p e c t r a l d e n s i t y of theo r i g i n a l d a t a or m e s s a g e is of the fo r m

(25) K / (f 0 + f )an d if t h e s e o p t i m u m r e c o v e r yf i l t e r s are used, the sampling frequencym u s t be of t h e o r d e r of10

4fo ino r d e r t o

o bta in 1p e r c e n t e r r o r int h e r e p r o d u c t i o n .

(26)

and the sideband sp ec tr um <1>S is

* s ^ s Z [ * m ( f - n f , ) + * m ( f + n f , ) ] .n=i

(27)

(5) If the s pe c t r a l de ns i ty of the

o r i g i n a l m e s s a g e at f r e que nc ie s ne a r ha l fthe s a mpl ing f r e que nc y a nd a bo ve is a ppr o ximately of the form K/f n t h e r . m . s . e r r o r o - e

va r ie s wi th s a mpl ing f r e que nc y fs a p p r o x

ima te ly a s fs'(n_l/2).

This s pe c t r um is i l lus t r a te d in Fig , 9 . It

may be no ted that Eq . (25) is identical in fo rmw ith th e w e l l- k no w n e x p r e s s i o n f o r e r r o rs pe c t r a l de ns i ty fo r c o nt inuo us s mo o thing o fme s s a ge p lus no is e if the no is e we r e inde pendent of the message and had a s p e c t r a ldensity equal tothe s ide ba nd s pe c t r um Eq.(27). '

He nc e W ie ne r ' s me tho d ma y be a ppl ie ddir ectl y to give the optimum (in the sense ofm i n i m u m r . m . s . e r r o r ) l i n e a r f i l t e r f u n c t i o nY' (and Y = Y ' / f s ) f ° r a n Y g i v e n s a mpl ingr a t e f s . Eq. (25) can be eas ily mo dified toa l lo w fo r de la y in r e c o ve r ing the me s s a ge i fs u ch d el ay i s d e s i r a b l e f o r g r e a t e r a c c u r a c y .

(2) The r e is n o t i m e - v a r y i n g l i n e a rf i l t e r ( w h o s e c h a r a c t e r i s t i c s v a r y p e r i o d ically in s ync h r o nis m with the s a mpl ingfrequency) which is better than the optimum

t i m e - i n v a r i a n t l i n e a r f i l t e r .

b . Sa mpled Da ta in C lo s e d-Lo o p Sys te m s

C o n s i d e r t h e c l o s e d - l o o p c o n t r o l s y s t e mindicated by Fig. 11,

B y me tho ds s im i la r to tho s e o f R ef . 2 , iti s e a s i l y s h o w n t ha t t he F o u r i e r t r a n s f o r m so f e r r o r * a nd d is tu r b a nc e D o ve r a lo ng butfinite t ime -T — + Ta r e r e l a t e d a p p r o x i

mately* by

e T ( X Y m ( f ) [ D T ( f ) - f s Y c ( f ) V c T ( f - n t s ) ] .- 0 0

(28)

T h i s r e l a t i o n s h i p m a y be inve r te d to

o b t a in a n e x p r e s s i o n f o r t h e e r r o r in t e r m sof the d i s t u r b a n c e in the following way:Substi tute for f.

f =f - mf s

(29)

(3) Th e w eighting function y( r ) ofsuch ano pt imum f i l te r s ho uld a lwa ys be

equal to unity for r =D, wh er e D is thede s i r e d de la y inr e c o v e r y of t h e m e s s a g e .

*The effe c t o f the a ppr o xima t io n d is a p pe a r sa s T —oo, a s w e s h a l l l a t e r d o .

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Then, for every integral m,

C T ( f - m f s ^ YM ( f - n V { D T < f - m t s )

-f

s Y X - m f s ) f < T [ f - ( n + m ) f s ] }(30)

But

»C0

fn=-oo

I • T [ f - ( n + m ) f , ] » g - T ( f - n f s ) .n^oo

(31)

Th us, if we w r i te an infini te set ofeq u at i on s , u s i n g ev ery i n t eg ra l v al u e of mi n t h e ex p ress i on ab ov e on ce, an d t h en su m,

f > « T ^ - n f , ) ' f Y M ( f - r V , ) D T ( f - r V , )- 0 0 - 0 0

" f s [ f Y M ( f - n f s ) Y c ( f - n f s ) l f ^ T ( f - n f s ) .L -0 0 J -c o

(32)

H e n c e ,

•oo

l M f - " f S > =

- 0 0

+ 00

I- 0 0

l Y M ( f - n f s ) D T ( f - n f s )

• CO

I- 0 0

l + f s Z V f ^ n f s ) Y c ( f - n f s l

(33)

and then f rom Eq. (28)

oo* T ( f l = [ l - X ( f ) ] Y M ( f )D T ( f ) + [ x ( f ) ] X

x [ YM ( f - n f s ) D T ( f - n f s ) + Y M ( f + n f s ) D T ( f + n fs ) ]

(34)

w h e r e

X( f ) =

fsY c(t)YM(f )

i 03

I- 0 0

• + f s Z Y M ( f -n f s )Y c ( f -n f s )

(35)

Using the methods out l ined in Ref. 2 , i tfol lows f rom Eq. (34) that , i f D i s a s t a t i o n a r yr a n d o m t i m e s e r i e s h a vi ng a s p e c t r a l d e ns i tyfunction D (f) , the spectral densi ty of thee r r o r i s g iv e n b y :

* € ( f ) • | I X

^ r { i v

f

> r * D < « }n^H'fiv^xirv^fs)

+ | y f + nfsM v

f + fw

(36)

T h e i n t eg ra l o f t h i s fu n ct i on ov er a l lf r e q u e nc i e s g i ve s t he m e a n - s q u a r e e r r o r

and may be so used to evaluate an exist ingo r c o n t e m p l a t e d s y s t e m . It i s a l s o o b v i o u s l yo f t h e s a m e f o r m a s t h e e x p r e s s i o n f o re r r o r s p e c t r a l d e n s i ty i n t h e c a s e o f f i lt e r i n gindependent signal and no ise and, henc e,W i en er ' s m et hod fo r t h i s cas e may b e u seddir ec t ly to find the best X, given Y m (f),and 4>D(f)- i n this case, Eq. (35) must beinverted to f ind Y c( f ) , t h e req u i red op t i mu mfeedback frequency function. Th is can bedone by the same technique used above too btain Eq. (34) f ro m Eq. (28). The r esu l t is :

YC(H*sYJf )

X( f )• co

I - Z X ( f - n f s )-00

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R E F E R E N C E S

1. W ie ne r , N. , "Ext r a po la t io n , In te r po la t io n , a nd Smo o thing o f Sta t io na r y Time Se r ie s , wi thEngineering Applications," New York, John Wiley and Sons, 1949.

2. Ste wa r t , R . M . , " Sta t is t ic a l De s ign a nd Eva lua tio n o f F i l te r s fo r the R e s to r a t io n o fSampled Da ta, " P ro cee din gs of the I .R.E ., 44 (No. 2), pp. 253-257 , 1956.

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Di f ferent ia tedPosit ion Data V + N r

Accelera t ionData

V + N a Fa t

F i g . 1 . V e l o c it y f r o m a c c e l e r o m e t e r an d p o s i t i o n d a t a .

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S + N,

S + Ns

&

S + N nYn

Fig. 2 . Inde pende nt f i l te r ing o f a s e r ie s o f m e a s ur e m e n ts o f a s igna l .

S + N,

<HN 2

<±H""vJ

Fig. 3 . Typic a l fe e dba c k s ys te m .

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JX_^,-©—© i

4 > — © -

N 3

© ©"

"1 2

> Z 2

e,

F i g . 4 . D i a g r a m o f e x a m p l e .

S+ N,

&

S+ N,

F i g . 5 . D i a g r a m o f e x a m p l e .

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q 1 M ,° T I N vy i^N

• T, \ nS + Nr » Y i to/^.

S + Nn . ~~L• Tn *V !/

s *

F i g . 6 . D i v e r s i t y s y s t e m .

S + N,

S + N 2

S + N n

2

K,2

K2

„

*V

<

* { -* \ . 5

Y.loi)S *

F i g . 7 . F i l t e r i n g s o l u t io n o f d i v e r s i t y p r o b l e m .

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^ l l

^ — 1

m^^m*

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1

3

AMPE

ITH

OUENCY

PONS

co H UJ cno j *" or UJ

o u. or*—

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i •

—

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RANSM

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55Z

Ul UJ

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oorUlN

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X

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or<

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0cn

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tn3

ro

to3

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c oC O

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O

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S(DbOt8COC OC D

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.5?

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Disturbance YM

-Yc

ERRORI

SAMPLE

1

F i g . 1 1 . S a m p le d e r r o r c o n t r o l s y s t e m .

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REC ENT D EVELOPM ENTS IN FIXED AND ADAPTIVE FILTERINGA. G. C a r l to n a nd J . W . Fo l l in , J r .*

SUMMARY

C la s s ic a l o p t ima l f il te r ing m e tho ds ha ve be e n e xtende d to a la r ge c la s s of p r o ble m s inwhic h the input ha s inc o mple te ly s pe c i f ie d c ha r a c te r is t ic s . B y minima x pr inc iple s the o pt ima lf i l t e r a nd th e b e st i np ut a r e d e t e r m i n e d . T w o p r o b l e m s o f t i m e - v a r y i n g f i l t e r s a r e c o n s i d e r e d ,fi r s t the optim al se tt l in g of f i l te rs to s teady s tate and second the design of adaptive fi l t er swhich adjust to var ying or unknown envir o nm ent.

SOMMAIRE

Le s me tho de s c la s s iq ue s de f i l t r a ge o pt imum o nt e te e te ndue s a une la r ge c la s s e dep r o b l e m e s , d a n s l a q u e ll e l a g r a n d e u r d ' e n t r e e a d e s c a r a c t e r i s t i q u e s i n c o m p l e t em e n ts pe 'c i f ie e s . En s e ba s a nt s ur de s p r inc ip e s "m in im a x ", le f i l t r e o pt imum e t la g r a nde ur

d ' e n t r e e la p lus in te r e s s a nte s o nt de te r m ine s . De ux pr o ble m e s de f l i t r e s e n fo nct io n du te m ps ,s o nt c o ns id e r e s , e n p r e m ie r 1 'a juste me nt o pt imum de f i l t r e a u r e gim e pe r m a ne nt e td e u x i e m e m e n t 1'etude de fi l t r es ada ptes qui ajustent au milieu inconnu ou va r i ab le.

1. INTRODUCTION

Line a r f i l te r ing the o r y i s la r ge ly bas e d o nthe funda me nta l wo r k o f W ie ne r , "Ext r a po lation, Interpo latio n, and Smo othing ofSta t io na r y Tim e S e r ie s , " 1950, which inma ny r e s pe c ts pa r a l le le d the inde pe nde nt

w o r k o f K o l m o g o r o v .

It may be well to review the problem cons ide r e d by W ie ne r . He a s s um e d tha t the r ewa s a va i la ble the e nt i r e pa s t h is to r y o f at i m e s e r i e s c o n s i s t i n g o f s i g n a l - p l u s - n o i s epo s s ib ly c o r r e la te d wi th the s igna l , tha t a l lp r o c e s s e s i nv o lv ed w e r e s t a t io n a r y a nd i n de e d e r go d ic to the se c o nd o r d e r , wi th kno wna u t o c o r r e l a t i o n a n d c r o s s - c o r r e l a t i o n f u n c t i o n s . He wis he d to de te r mine the r e a l iz a blelinea r f i l ter to apply to the s ignal in o r de r to

minim iz e the m e a n -s qu a r e d i f fe r e nc ebetween the o utput and the m ess ag e t r ans late dby an assigned posit ive or negative t ime

inte r va l . W ie ne r s o lve d th is p r o ble m byusing var iat ional analysis on the weightingfunction to obtain an integral equation, thenby us ing s ubt le Fo ur ie r a na ly s is to s o lvethe in te gr a l e qua t io n . M ul t iple t ime s e r ie swer e handled by an extensio n of this techn ique.

It will be noted that the problem solvedby W ie ne r c o nta ins two r e s t r ic t io ns be yo ndthe a s s u mp t io ns : f i r s t , o p t imiz a t io n i sr e s t r ic te d to l ine a r func t io ns ; s e c o nd, theloss function whose expectation is minimizedi s t he s q u ar e d e r r o r .

In at tempting to extend fi l ter ing theory, i ti s a p p r o p r i a t e t o mo d ify o r e l i m i n a t e v a r i o u so f t h e s e a s s u m p t i o n s o r r e s t r i c t i o n s . V a r i o u sinv estig ato rs have widened the field of p er mis s ib le f i l te r s a nd de a l t wi th a l te r na t iv e

l o s s f u n c ti o n s . N o n s ta t io n a r y p r o c e s s e s h a veb ee n c o n s i d e r e d ; s o m e t r i v i a l o b v i o u s r e s u l t shave been o btained, and som e adaptive fi l ter s

*Appl ie d Phys ic s La bo r a to r y , The J o hns Ho pkins Unive r s i ty .

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app ea r sui tab le, but l i t t le has been done thati s b o t h s ig n i f ican t an d r i g o r o u s . In t h i s p ap e rw e s h a l l r e m o v e t h e a s s u m p t i o n o f c o m p l e t ek n owl ed ge of t h e co r re l a t i o n fu n cti on s an da l s o i n d ic a t e s o m e m i n o r e x t e n s i o n s o f t h e

b asi c t h eory an d t ech n i q u es , con si d er t h eo p t i mu m f i l t e r wi th on l y p o r t i o n s of t h es i g n al h i s t o ry av ai l ab l e , an d a t t emp t t oc l ass i fy t h e t y p es of ad ap t i v e f i l t e rs .

2 . THE FREQUENCY APPROACH

T h e r e a d e r o f W i e n e r ' s w o r k w i l l n o t et h at a l t h ou g h h i s b as i c p rob l em i s fo rmu l at edin t e r m s o f t i m e s e r i e s , c o r r e l a t i o n f un c ti o n s ,an d wei g h t i n g fu n ct i on s , h i s so l u t i on s a re

e x p r e s s e d i n t e r m s o f s p e c t r a l d e n s i t i e s a n dt r a n s f e r f u n ct io n s . It a p p e a r s r e a s o n a b l e ,co n seq u en t l y , t o se t up t h e p ro b l em in th ef r e q u e n c y d o m a i n .

F r o m t he e l e m e n t a r y p r o p e r t i e s o fs p e c t r a l d e n s i t i e s , w e h a v e

. 2 . F r M Fr N d o i

(1 )

w h e r e c r2 i s th e m e a n - s q u a r e e r r o r , F (w ) t h ef i l t e r t r an sfe r fu nct ion , a t h e t i me t r a n s l a t io n , an d M an d N t h e s i g n al -n o i se s p e ct r a ld e n s i t i e s , s o n o r m a l i z e d t h a t th e s i g n alp o w e r i s TM(CU) d w . All i n t eg ra l s a r e t ak eno v e r t he e n t i r e r e a l fr e q u e nc y a x i s u n l e s so t h er wi se i n d i cat ed , an d t h e d ep en d en ce ofthe va r ia bl es on cu will usually no t be ind i cat ed . It i s ass u m ed h er e an d h en cefo r t hthat the signal and no ise a r c independe nt;

t h i s en t a i l s n o l o ss in g en e ra l i t y i n t h ec l a s s i c a l d e v e l o p m e n t s , w h e r e M c a n b ere g ar d ed as t h e su m of th e s i g n al an ds i g n a l - o n - n o i s e s p e c t r a l d e n s i t i e s , N a s t h es u m o f t h e n o i s e a nd n o i s e - o n - s i g n a l s p e c t r a ld e n s i t i e s .

It i s u sefu l t o co n si d e r t h e sp e ct r a ld e n s i t i e s a s r e s o l u t i o n s o f t h e s i g n a l p o w e ri n t o a con t i n u u m of f req u en cy comp on en t s .F rom t h i s s t an d p oi n t i t i s c l ear t h at e rg od i cp r o p e r t i e s a r e n o t r e l e v a n t t o th e p r o b l e m o f

l i n ear f i l t e r i n g , a l t hou g h th e o p t i mal f i l t e rm a y b e no n l i n e a r if t h e s e c o n d - o r d e r c h a r a c t e r i s t i c s a r e n o t e r g o d i c .

B e f o r e a p p ly in g v a r i a t i o n s t o m i n i m i z e o -2

b y ch oi ce of F , l e t u s i n d i cat e so me ex t en s i o n s o f t h i s r e l a t i o n t o p r o b l e m s b ey o n dthe o rig inal on e. In the f i r st p lace it wil lbe noted that Eq. (1) is valid even thought h e p ower of s i g n al , n o i se , o r b o t h i s u n b ou n d ed ; i t i s n o t n e ces sar y t h at t h e c o r r e

lation functio n e xi st . It will be see n thatt h i s m a y b e o f i m p o r t a n c e .

A t r i v i a l g en eral i zat i on of E q . (1 ) i s t or e p l a c e e l 0 j a b y t h e F ou r i e r t ran sfo rm Y(cu )o f a n y d e s i r e d l i n e a r o p e r a t o r o n t h e s i g n a l ,e . g . by io) fo r t h e d er i v at i v e . T h u s ,

X i Y - F 2 M + F• }

(2 )

An o t h er easy ex t en si on i s t o t h e ca se inw h ic h it i s d e s i r e d t o w e ig h t e r r o r s u n eq u al l y fo r v a r i o u s freq u en cy co mp o n en t s . Asy mmet r i c n on n eg at i v e fu n ct i on W(OJ), s ono r m ali ze d that fW(cu) dto = l , could bei n s e r t e d t o o b t a i n

- . ' •/ fr- M + F -} W d c

(3 )

wh i ch i s o f cou rse fo rmal l y eq u i v al en t t o

Eq. (2) with M and N r ep lac ed by MW and NW.

T h e n ext ex t en si o n i s t o mi n i m i ze o-2 s u b j ect t o a re s t r i c t i o n on t h e mean p ow er of t h eou t p u t o r so me o t h er l i n ear fun cti on of t h e

28 6

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o utput. As an exa m ple, if o- 2 mus t beminimiz e d s ubje ct to the r e s t r ic t io n tha t theo utput a c c e le r a t io n po we r mus t be le s s tha n/3 , i.e ., that

/ ( M + N) ]F |

2

<"

4

d w < / 3 .(4 )

we s ho uld minimiz e

o-2 + X/3 = / { | Y - F|2 M + |F | 2N + MM + N)|F|ZCJ4| dcu,

(5)

and selec t X to satisfy Eq. (4). In ce r t ainc a s e s , e s pe c ia l ly wi th no n-Ga us s ia np r o c e s s e s , r e s t r a i n t s s u c h a s E q . ( 4 ) m a y

preferably be applied only to the noise. Inthis case the integ ra l to be min imize d can bereduced to Eq. (2) by suitable definit ion ofN . This type o f s ide c o ndi t io n c a n be in t r o duc e d fo r ma l ly a s he r e , o r c a n be us e d inthe definit ion of the class over which F iso pt imiz e d. Se ve r a l s imul ta ne o us s ide c o ndit io ns can be intro duced in the sam e way.

The final e xtensio n to be co nsid er ed is tor e p l a c e t h e c l a s s o f r e a l i z a b l e t r a n s f e rfunc t io ns by o the r c la s s e s , s a y y i , a s a ppr o

p r i a t e .

3. MINIMIZATION OF o-2 BY Fc J<

We now consider the selection of F withint h e c l a s s J i t o m i n i m i z e o - 2 . An inc r e me n tJ o r F wi ll p r o duc e an inc r e m e nta l u 2 o f

o-f + F - <r* -- / | j |2 (M + N) d«

+ f J [ ( M + N) F - MY ] doj.

(6 )

a re latio n o btained by noting that eve rytr an sfe r function has an even r ea l par t anda n o dd ima gina r y pa r t .

The t ransfer function F will be optimalin 7 ^ t n e r ight hand integ ra l of Eq. (6)i s n o n n eg at iv e f o r e ve r y J s uc h th at J + F c ?a n d J | J | 2 (M + N) dcu is finite.

The a bs o lute ly o pt ima l F i s , f r o m E q.(6), evidently

. MYo ' M + N

(7)

The o pt ima l r e a l iz a b le F i s g iven by

I

(M + N ;

MY

(M + N) '(8)

whe r e the new s ymbo ls de no te fa c to r iz a t io na nd de c o m po s i t io n o f a me r o mo r p hic funct io na s

' H' -- H„+ H. ,H

(9)

H * being analytic and without poles orze r o es in the low er half plan e, H" the co njuga te of H*; H + and H. have no poles in thelow er half plane and upper half plane, r e s

pe c t ive ly . Po lyno m ia l te r m s o f H a ppe a r inH ,. If M + N is not facto rab le, F R = F 0 .

To check the validity of Eq. (8), substi tuteit in Eq. (6), obtaining

o f + F - a-2 = / " I j l 2 (M + N) dcuJ R r D J I l

+ J J (M + N] MY{M + NT

dcu ;

(10)

the la t te r in te gr a l i s z e r o , by c o nto ur in te gra tio n o ver the upper half plane,fo r J s uff ic ie nt ly ' c o nve r ge nt , i . e . , fo rf \ j \ 2 (M+ N) dcu finite.

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As an ex amp l e l e t u s con si d er t h e sp ect ra

M -- $ / „ • i N • ^ i wi th Y = I ,

(11)

t h e ab sol u t e op t i mu m ( real i zab l e wi t h i n f i n i t edelay) is then

Fo(p> - T V J V 9 -(12)

an d t h e op t i mal real i zab l e F i s

F„(p) *V I P (*/fl)'/4

\ + Vz p(4>/e)'u + P 2 ^ / 0 ' / 2

(13)

T h i s t r an s fer fu ncti on i s t h e ze r o v el oci t ylag loo p with .7 cr i t ic al dam ping which wil lb e d i s c u s s e d b e lo w . T h e c o r r e s p o n d i n ge r r o r s a r e

<r02 - . 3535 <p3/< 0'A

o-2 . 1.414 * * « flV

(14)

A very useful relat ion can be obtained by

man i p u l at i on of th e seco n d i n t eg r a l o f E q . (6 )a s f o l l o w s :

/ j [ ( M + N ) F - M Y ] dcu

= / J / F [ ( M + N ) | F | 2 - M Y F ] dcu

= / j / F [ ( M + N) | F | 2 - ReMYF" m MYF]d<u.

(15)

T h i s i n t eg r a l i s ze ro fo r su f f ic i en tl y co n v erg en t rea l i zab l e J i f

FD

2

(M + N) = Re MYF + Re: lm M YF ,(16 )

w h e r e R e : l m H r e p r e s e n t s t h e r e a l c o m p l e ment to the given imaginary function such ast o r e n d e r H r e a l i z a b l e .

T h i s c a n be m o r e f o r m a l l y e x p r e s s e d byt h e B o d e r e l a t i o n

h J ' ^ N ) ' £ / .w ( M Y F ) w d w

(17)

w h e r e J i n d i c a t e s d i s r e g a r d o f p o l e s a tw = ± tu , pro vided MY doe s not div er ge fo rlarge o>.

S ol u t i on s of E q . (1 6) fo r s i m p l e fo r m s ofMY a r e r e a d i l y o b t a i n ed ; f o r e x a m p l e .

MYi if MY is constant

(M + N ) = • MYF (ia), i f MY --a z + iu z

MY + k, if MY = c 0 + c2<u 2 + c4<u 4

(18)

[ K de ter m ine d by J log |F I dcu - o l .

By sy m m et r y u n d er t h e i n t e rch an g e ofM — N and F—(Y - F) we obtain

l Y " F J 2 (M + N) -- N Y,

i f NY is constant .(19 )

F o r o rd i n ar y f i l t e r i n g , wi th Y - 1 , t h es ed ef in e t h e op t i mal sp e ct r a l d en si t y of t h ee r r o r s i g n a l in a s e r v o t yp e f i l te r , a nd c a nb e u sed t o con st ru ct an ad ap t i v e f i l t e r wh i chwi l l b eco me op t i m al fo r an y s i g n al sp e ct r u m .

E x p r e s s i o n s f o r | F R | , base d on Eq . (16)a r e q uite useful in o pt im izing f i l te r s with sidec o n d i t i o n s s u c h a s l i m i t e d m e a n - s q u a r e o u t p u t , s i n c e t h e e x p r e s s i o n s f o r | F R | a r ef req u en t l y more con v en i en t t o u se t h an t h osef o r F R i n d e t e r m i n i n g t h e L a g r a n g e m u l t i p l i e r s .

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4. MINIMAX FILTERING

Le t us c o ns ide r no w s o m e typica l ly s t a t i s t ic a l p r o ble m s , in whic h o ne ha s inc o mple teknowledge of the spe ctr um of no ise, of

s igna l , o r bo th . W e dis c us s belo w a da pt ivequ a s i - l ine a r f i l te r s w hich a ppe a r s u i ta ble fo rc a s e s in whic h o ne s pe c t r um is c o mple te lyunknown, and which can cope with ca se s in volving a spectral density of known form butunknown magnitude.

T h e p r o b l e m s c o n s i d e r e d b y W i e n e r w e r ee s s e nt ia l ly p r o ba b i l i s t ic , i . e . , the s ys te m isc o m p l e te l y d e s c r i b e d in t e r m s o f a p p r o p r i a t ep r o b a b i li t y m e a s u r e s ; t h e p r o b l e m s w e a r eno w c o ns ide r ing a r e s ta t i s t ic a l , in tha t wea r e dealing with a syste m defined by pr o ba bi l i t ie s , s o me o f whic h a r e unkno wn. Ourpr o ble m in th is c a s e i s o ne o f s ta t i s t ic a lestimation of a function of the s ignal. Oure s t im a t ing func tio n s ho uld be o pt ima l in s o m es e n s e . One o f the mo s t lo gic a l c r i te r i a fo ran estimate, developed by Abraham Wald, istha t i t min imiz e s the ma ximum e xpe c te dl o s s ; tha t i s , e a c h f i l te r i s a s s e s s e d o n theba sis of the expected lo ss with the po ssib les ys te m whic h is le a s t fa vo r a ble fo r thegiven fi l ler , and the o ptim al fi l ter is thatf i l te r fo r whic h the ma xim um e xpe c te d lo s sis minimu m. Th is fo r mula t io n o f s ta t i s t ic a ld e c i s i o n t h e o r y i s v e r y s i m i l a r t o t w o - p e r s o ngame theory, independently developed by vonNe uma n. W e a do pt th is c r i te r io n a nd c o ns ide ra s o pt ima l the minima x f i l te r , wi th the lo s sf un cti o n p r o p o r t i o n a l to t h e s q u a r e d e r r o r .In a c o mple te ly p r e s c r ibe d s ys te m, theminima x l ine a r f i l te r i s the W ie ne r o pt ima ll i n e a r f i l l e r .

M inimax the o r y o ffe r s a s t r o ng jus t i fication for the use of l ine ar f i l te r s . Ifthe d is t r ibut io n func tio ns of the p r o c e s s e sare not known but the class of possibled i s t r i b u t i o n s i n cl u d es G a u s s i an d i s t r i b u t i o n s ,the minim ax fi l ter is l ine ar , s ince with a

l i n e a r f i l te r th e m e a n - s q u a r e e r r o r i s i n dependent of the form of the dis t r ibutionfunction, and with a no nlinear f i l ter the m ea n-s q u a r e e r r o r e x c e e d s t h at w it h a l i n e a rf i l t e r w h e n t h e p r o c e s s e s a r e G a u s s i a n .

T y p i c a l p r o b l e m s e n c o u n t e r e d i n p r a c t i c einvo lve s i tua t io ns in whic h the no is e p r o c e s sis known to be l imited in po wer , in me ans q u a r e v e l o c i t y , e t c . :

J N dcu • C 0 , J Ncu2 dtu = C,, e tc .

(20)

Such r e s t r ic t i o ns c a n be put in the ge n e r a lf o r m

/ i M f l d<u = | ,

(21)w h e r e 9 i s a p r e s c r i b e d s y m m e t r i c n o n -negative function. I t can be shown str aig ht fo rw ardly that the var iat io n in cr2 due tovar iat io n n in N is a no nnegative function ofn, ze r o if and only if n = 0, p lus

! • M,NI xe do t

(22)

w h e r e F M N is the o ptima l t r an sfe r function

with M and N, and X is a Lag ran ge mu ltiplie r to be selec ted to satisfy Eq. (21). Fr o mEq . (22) and the fact that n + N m ust be no n-negative , i t fo l lows that the maximum N isN 0 given by

M ,N CXfl

\ 9

N o >

w . M

(23)

This r e s ul t i s e a s i ly ge ne r a l iz e d to thec a s e o f s e v e r a l i n e q u a l i t y r e s t r i c t i o n s

K. j N0 d<u -. l ( j * 1,2..., k)

(24)

with the Kj not specified but S 1.

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T h e m a x i m u m N i s N 0 sat isfying We have thus shown that

< Tx.K. f l .w=N

0-- 0 .

(25 )

with the X:, K, sat isfyin g Eq. (24) and al so

X . ( K . - I ) -- 0 . (j = 1,2..., k).

(26 )

S i m i l a r r e s u l t s a r e o b t a i ne d f o r c a s e s in

wh i ch th e s i g n al s p e ct r a l d e n si t y i s su b j ectt o o n e o r m o r e e q u a l i t i e s o r i n e q u a l i t ie s ,an d wh e re b oth sp e ct r a a r e l i m i t ed on l y b ys u c h r e s t r i c t i o n s .

T h e resu l t s j u s t g i v en h av e d er i v ed t h em a x i m i n s p e c t r u m o r s p e c t r a , b ut o u r o b j e c ti s t o d e t e r m i n e th e m i n i m a x f i l t e r . F o r t h i sp u r p o s e w e n o w p r o v e t h a t

Min Max <r2c - Max Min cr2

F N N , F N F N '

and that the minimax F is F M . N ,

T o p r o v e t h i s , w e o b s e r v e t h a t

Min Max a •' < Max o-*

F N N , F N , F " . " 0

Max

N

/{N

|FM.N (

FM N J

d w

= 7 X, + / " M I Y - F , , I'dai "- Max Min cr2

* * i J ~ M,N0I N p N,F

Min

F

Max

NN.F

Max

N

Min

FN,F

(27 )

Bu t by th e fu n d amen tal t h eo re m of g am et h e o r y .

Min

F

Max

N'N ,F

Max

N

Min

FN.F

(28 )

f rom wh i ch i t fo l l ows a t on ce t h at t h e " g a m e "i s d et e rmi n ed , wi t h t h e mi n i max f i l t e r b ei n gF M . N Q

a n c i t n e ma xim in N being the N 0 p r e viously defined.

T h e b as i c resu l t o f t h e mi n i max ap p roacht o o p t i m u m f i lt e r i n g i s t h at t h e e r r o r s d ep e ndi n t h e secon d o r d e r o n th e sp ec t r a an d t h efo r m of t h e f i l t e r . As a co n seq u en ce, if as u i t a b l e a p p r o x i m a t i o n t o t h e o p t i m a l f o r mi s u s e d a n d t h e p a r a m e t e r s a r e a d j u s t e dp r o p e r l y t h e r e s u l t i n g s y s t e m w il l b e s a t i s f a c t o r y .

5. T I M E - V / L R Y I N G F I L T E R S

L et u s n ow co n si d e r t h e p ro b l em offi l ter ing when only a f ini te and pe rh ap sf r a g m e n t a r y h i s t o r y o f th e s i g n a l s i s a v a i l a b l e . In t h i s c a s e th e f i l te r p a r a m e t e r s a r ev a r i a b l e , a nd w e m u s t a s s u m e a p a r t i c u l a rfo r m of t r an sfe r fu n cti on . In g en er al , t h es t e a d y - s t a t e o p t im a l f i l te r w it h v a r i a b l eb an d -p a ss an d d amp i n g i s b es t . W e mayat t ack t h i s p rob l em i n t h e t i me d omai n b yco n si d er i n g t h e r a t e of ch ang e of t h e f i l t e r i n ge r r o r s a nd ad ju s ti ng p a r a m e t e r s to m a x i m i z et h e r a t e o f d e c r e a s e o f t h e e r r o r .

F o r a l i n e a r s y s t e m t h e t r a c k i n g a c c u r a c ym a y b e d e s c r i b e d in t e r m s o f t h e v a r i a n c e so f t he tr a c k i n g e r r o r a nd e r r o r r a t e . A san ex amp l e l e t us co n si d e r t h e s i m p l e zer o

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velocity lag feedback system in Fig . 1. Theinput co nsis t ing of s ign al and no ise x(t) +x n ( l )is at the left, the output, x c , a t the r ight.The e qua t io ns o f the s ys te m a r e

dx c •W - x c + b ( x - x c ) + bx n

dx-

w -- °(*-\. +" „

(29)

Note, in explanation of the notation, thatx c ^ x c •

Ac tua l ly the p r in c ipa l in te r e s t c e nte r supo n t he e r r o r s «c = x - x c , fj. . x - x -c . Inthe second diag ra m of Fig. 1 is shown thee r r o r lo o p e quiva le nt to the o r ig ina l signa ll o o p . The s ignal x now appears as ana c c e le r a t io n input to the f i r s t in te g r a to r . Thisis a significant advantage when, as is oftenthe c a s e , the a c c e le r a t io n s pe c t r um o f thesignal is known.

So fa r a , b a r e unr e s t r ic te d . If x a nd x n

both have flat spe ctr al de nsit ies 9, 4> r e s pectively then this f i l ter is o ptimal with theva lue s of a a nd b p r e v io us ly de r iv e d. Thepr e s e n i pur po s e i s to exte nd the o pt imiz a t io nto the t ra nsie nt per io d. The gains a , b inthis case a r e t ime functions and the re sult ings ys te m, whi le no t ne c e s s a r i ly o pt imum a mo nga l l po s s ib le s y s te m s , i s the be s t o bta ina blewith a g ive n s t r uc tur e .

The key to the so lution l ies in sett ing

up the differential equations relating thev a r i a n c e s a nd c o v a r i a n c e s o f t h e i n t e g r a t o ro utputs e c , « j . W r i te Eq s . (29) in te r m so f the e r r o r s a nd e x pr e s s the s o lu t io n in thene ighbo r ho o d o f t a s a po we r s e r ie s in At .

T e r m s beyo nd the f i r s t de gr e e in At a r e no t

r e qui r e d . The r e s ul t i s mo s t e a s i ly o bta ine dby direct use of the second figure

• t + A tx n d t + 0 (At 2 )

"co ' ' 60

o o

(30 )

l I T U I / > t + A t

• / x d t + a / x n d t + 0 ( A t 2 ) .

(31)

Square Eq, (30) and average over theensemble of inputs x, x n . Then denotingr e s u l t i n g v a r i a n c e s a nd c o v a r i a n c e s by^ , T t , T c b . r e s p e c t i v e l y

X - % • 2 " * : . A t - Z b " * ; A t + b2< rSAt + 0 (At 2 )

(32)

wh er e is der ive d as fo l lo ws: <£(a)isa u t o c o r r e h

/-t + A t r i

J, ' Xt + At

* nd t

. t + At / . t + At

X t i t [X u X n M ] <k. dv

/•At /»At- I j <> (LI - v) du dv

Jo Jo

/•At M t- 4> / / S (u -v )d u dv

Jo Jo

= 4> A t .

Dividing Eq . (32) by A t and letting At — 0

" d f - 2 ^ c c - 2 b ? c + b N > .

(33)

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S i m i l a r l y , a s s u m i n g t h e s p e c t r a l d e n s i t y9 of x is flat

* S = - 2 o T , + o ^ . f l

(34)

d Tc<d t

a T - b T . + a b XC C C '

(35)

T h e s e v a r i a n c e a n d c o v a r i a n c e e q u a t io n smay be used to adjust the gains to geto p t im u m t r a c k i n g . T h i s w i l l s u r e l y r e s u l tif functio ns a, b can be found m akin g the ri ghthand members of Eqs. (33), (34), and (35)s i mu l t an e o u sl y m i n i mu m fo r t h i s wi l l mak eT c a nd "fj d e c r e a s e a t m a x i m u m r a t e .

T h i s s i m u l t a n e o u s m i n i m u m d o e s o c c u rand at

<r * -T(36)

as can b e seen by se t t i n g th e p ar t i a l d e r i v a t i v e s o f a ll t h r e e r i g h t m e m b e r s w i thr e s p e c t t o a a n d b e q u a l t o z e r o . T h er e s u l t i n g s y s t e m h a s o p t i m u m t r a c k i n g a ndrat e of se t t l i n g an d t h e v ar i an ces faci l i t a t ee v a l u a t i n g p e r f o r m a n c e o f t h e s y s t e m .

T h i s o p t i m i z a ti o n o f t h e t r a n s i e n t b e h a v i o rh as a b y p ro d u ct , t h e k nown s t e ad y -s t a t er es u l t . F o r in t h i s ca se t h e l ef t h and m em b e r sof Eqs. (33), (34), and (35) vanish whenusing Eq, (36)

• { i t'/ 4

>' (f)(s t ead y -s t a t e) (3 7)

F o r s i m p l i c i t y t h e i np u t a c c e l e r a t i o n s p e c t r u m h a s b e e n a s s u m e d f la t . T h e r e a r ese v e r al way s t o d eal wi th n on fl at s p e ct r a .F or ex amp l e i t can b e sh own t h at fo r ag en er al 9 (cu) wi th au t o co r r e l a t i o n fu ncti on

<jS(t), 9 in Eq . (34) wo uld be r ep lac ed by2 / 0 W (t. r ) d. ( t - r ) d r . He re W ( t . - r )is the im pulsiv e r es po nse of *•- to x . Or af la t sp r ec t r u m cou l d b e f il t e red b y9 (<u)* [ f lM " = 0 M * 0 H ' ] t o g i v e a s i g n alo f s p e c t r u m Q (<u) into the int eg r ato r .

T h i s l a s t i s p a r t i c u l a r l y s i m p l e in t h eM ark off ca se 9 = 9 0 / { C 2 + <"2 ) w h e r e9* -- 1 /(C + jcu) , T h e o r i g i n al e r r o rt r ac kin g loo p could be mo dified as shown int h e t h i rd f i g u re t h u s in t ro d u ci n g on e ad d it i on ali n t e g r a t o r . P r o c e e d i n g a s b e fo r e , s i xv a r i a n c e - c o v a r i a n c e e q u a ti o n s r e s u l t . Ing e n er al , wi th a sy s t e m i nv ol v i n g n i n t e g r a t o r s( n ' t h o r d e r d i f f e r e n t i a l o p e r a t i o n ) , t h e r e w i l lb e on e -h al f n (n + 1) v ar i an c e eq u at i on sa l t h o u g h g e n e r a l l y s o m e a r e t r i v i a l .

T h e t r a n s i e n t f i l te r i n g p r o b l e m h a s b e e nd i sc u ss ed on t h e b as i s t h at t h e n o i se an dsi g n al sp e ct r a a r e k n own . T h i s l ead s t o aso l u t i on of t h e op t i mu m set t l i n g t i me of af i l t e r an d t o t h e b es t co mb i n at i on of p a r am et er s ev en if th e t ra n sf er fun cti on i s n o to p t i m al . If t h e n o i se an d s i g n al sp ec t r a a r enot known then the above technique of comp u ti n g v a r i a n c e s f a i ls a nd o t h e r m e t h o d s m u s tbe use d. If sufficient tim e is av aila ble it isp o s s i b l e t o m e a s u r e th e s p e c t r a l d e n s i t i e s an du s e t h e v a r i a n c e m e t h o d s , b u t l e s s c u m b e r s o m e m e t h o d s a r e d e s i r a b l e .

6. ADAPTIVE FILT ER S

T h e d i scu ss i on wh i ch fo l l ows wi l l con cern

ad ap t i v e sy s t ems, i . e . , t h ose wh i ch ch an g ep a r am et er s o r ad ap t as a fun cti on of t h ee n v i r o n m e n t . In g e n e r a l , t h e r a t e o f c h a n g eo f p a r a m e t e r s i s s lo w c o m p a r e d t o t he d a t arate of the input so that they may be t reated

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a s t i m e - v a r y i n g r a t h e r th a n n o n l i n e a rs y s te m s . If the r e s po ns e o f the a da pt inglo o p is fa s t , pa pe r a na lys is i s impo s s ib leand s imulatio n techniq ues ar e needed. Inde s igning a n a da pt ive s ys te m i t i s ne c e s s a r y

to c o ns ide r the r e s po ns e a nd s ta bi l i ty o f thelo o p a s we l l a s the s o ur c e o f in te l l ige nc eto be employed in adjusting the sys tem .

Ada pt ive s ys te ms ma y be c la s s i f ie da c c o r d i n g t o s e v e r a l d i s t i n c t c r i t e r i a a sfo l lo ws :

a. Object of adap tive loop

(1) Sett ing of gain o r t r an sfe rfunction.

In addit ion, a l l adaptive systems may bec la s s i f ie d a c c o r ding to s ta nda r d s e r v op r a c t i c e a s e l e c t r i c a l o r m e c h a n i c a l , d i g i t a lo r a na lo g , e tc . , but s uc h d is t inc t io ns a r e no td e s i r a b l e fo r t h e p r e s e n t p u r p o s e . T h e

c l a s s o f a d a p ti v e s e r v o s r a n g e s f r o m s t a n d a r dAGC a nd AFC lo o ps to s e r vo d r ive n a uto -t r a n s f o r m e r s f o r v o l t a g e r e g u l a t i o n t o m o r es o phis t ic a te d o pt ima l f i l te r ing lo o ps .

Let us now look at examples of the threediffe r e nt a da pt ive s ys te ms l i s te d unde r thef i r s t c r i t e r i o n ( a ) . T h e s e a r e t h e z e r ove lo c i ty la g t r a c king lo o p me nt io ne d e a r l ie r ,a s imi la r f i l te r wi th l imi te d o utput a c c e le r a t io n , a nd a s ys te m fo r ma inta in ing a s e r voloop as t ight as possible without instabil i ty

when the loo p gain i s s lo wly var ying o r notkno wn a c c ur a te ly .

(2) Adjusting o utput level o r o t herp a r a m e t e r .

(3) Adjusting s ta bil i ty m ar gi ns ofma in lo o p ,

b. So ur c e o f info r ma t io n

(1) M e a s ur e me nts o f no r ma l input

o r o u t p u t .

(2) Injection of t r ac er s ignal o uts ideba nd-pa s s o f no r ma l input .

(3) T i m e s h a r i n g t r a c e r s i g n a l .

(4) Amplitude o r phase of self-

e x c i t e d o s c i l l a t i o n s .

c . Type of sys tem

(1) Open loo p, i .e . , sys tem ad justeda c c o r d i n g to m e a s u r e m e n t s o nthe input .

(2) Clo sed loo p, i .e . , sys temm e a s u r e s s i g n al o r t r a c e r o u t p ut .

Fig . 2 sho w s a s imp le s e r v o whe r e o nlythe gain is var ied and the loo p gain is to bema xim iz e d (po s s ib ly to minim iz e the e ffe c tso f a va r ia ble ba c k to r que f r o m the lo a d) .The unknown gain is assumed to be in thes e r v o . The ba nd-pa s s f i l te r pa s s e s thefrequ ency at which instab ility is expec ted andthi s sign al is det ecte d and used to adjust thega in to da mp the o s c i l la t io ns .

Let X0 be the gain at which the loop isneu trally s ta ble . Then the r ate of buildupo r de c a y o f o s c i l la t io ns i s p r o po r t io ne d to( X / X 0 - 1). Th e am plitu de, z, of theo s c i l la t io ns s a t is f ie s the e qua t io n

z - k , z ( X - X 0 ) / X 0 = k , z 0 ( X - X 0 ) / X 0 .

(38)

If the signal is picked off at point A, the

c o nt r o l e qua t io n i s

X = - g (s) (z - z 0)

(39)

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while if i t i s p icked off at po int B, we have

i « g (s) X (z - z0)

- - x 0 g (s) (z - z 0 ) .

(40)Co mbining the se equatio ns we f ind that in

c a s e A

s2 + X - 1 9 W

wh i l e fo r case B

X - - k , z 1 X 0 g ( s ) ( J - ) ,

(41)

X

T IS

X 0

' X

c a na s

: , fl

b et h e

(s) X0 .

(42)

m a d e s t a b l e ,s te a d y - s t a t e

s 2 + k.z. g(s)

While both systeand both have X =s o l u t i o n , t h e t r a n s i e n t r e s p o n s e o f t h e a d a p t i v e l oo p d ep en d s on t h e re q u i r ed g ain i nca se A, but in cas e B the t r an sie nt is fixedan d t h e sy s t em h as zero v el oci t y l ag wi t hr e s p e c t t o v a r i a t i o n s o f X 0 . If X0 i n c r e a s e sl ine ar ly with t im e then X= X0 but z>z, withz = 0 . T h u s a c l o sed - l o o p ad ap ti v e sy s t e m i sb e t t e r h e r e . T h e s e s t a t e m e n t s a r e s u b j e c t t o

mod i f i cat i on i f t h ere ex i s t s d ead sp ace o rf r i c t i on in t h e mai n o r ad ap t i v e l oo p ;a c t u a l ly s i m u l a t i o n i s t he n r e q u i r e d t o d e t e r m i n e t h e b e h a v i o r .

If the main loop shaping network f(s) isp r o p e r l y c h o s e n t he n a v e r y g o o d s e r v oresp on se i s ob t a i n ab l e b u t i n p u t s i g n al s a tt h e l oop reson an ce f req u en cy mu st b e av oi d ed .T h i s d es i g n i s esp eci a l l y u sefu l fo r a reg u lato r, i .e. , when X| = 0 and the se r vo isd es i g n ed t o cou n t er t h e b ack t o rq u e.

F i g . 3 i s a d i a g r am of an ad ap t i v e f i l t e ri n wh i ch t h e r . m. s . ou t p u t accel era t i on d u et o n o i se i s l i mi t ed . T h e ad ap t i v e l oop i svery simple and, with the gains as shown,h as a r esp o n se wh i ch i s i n d ep en d en t of i n p ut n o i se sp ec t r a l d en si t y . T h i s can b e

s e e n f r o m t h e f ac t t ha t th e e r r o r in r . m . s .a c c e l e r a t i o n i s p r o p o r t i o n a l t o A X /X a ndh en ce t h e con t ro l eq u at i on i s

A X ,(43)

if the filter g(s) is unity.

If i t is de si r ed to have the adap tive loo pt i m e c o n s t a nt p r o p o r t i o n a l t o t h e m a i n l o o pt i me con st an t t h en t h e X mu st b e rep l acedby X2 an d t h e con t ro l eq u at i on i s

X ~ X A X .(44)

If the noise is f lat or of known spectralsh ap e i t i s p oss i b l e t o me as u r e i t s amp l i t u d e

at the input to the f i l ter and co mpute ther . m . s . a c c e l e r a t i o n fr o m t h e k no w n t r a n s f e rfunction of the f i l t er . I t is then po ssib le touse an o pen loop metho d of adjust ing X but,wh i l e t h i s e l i mi n at es ad ap t i v e l oop s t ab i l i t yp r o b l em s i t d o es not h av e t h e acc u r acy inadjust ing X . If the no ise do es not have theexpec ted sp ec tr um the f i l ter f(s) wil l not beo p t i m a l i n s h a p e ; h o w e v e r t h e p e r f o r m a n c eof the system wil l deviate in the secondo r d e r if X i s c o r r e c t , bu t e r r o r s in X m a yg iv e fi r s t o r d e r e f fe c ts o n s y s t e m p e r

f o r m a n c e d u e t o v i o l a t i o n o f t h e c o n s t r a i n t s .While this loop is very simple we have

d i s c u s s e d i t b e c a u s e s o m e s i m u l a t o r s t u d i e sof t h e ef fect s o f n o n l i n ea r i t i es may b e ofi n t e r e s t . T h e n o n l i n e a r i t y c o n c e r n e d is t h a td u e t o a f i x ed l i mi t e r i n ser t ed i n t h e l oopas shown at the bo t tom of Fig . 3 . Thea n a l y s i s w a s c a r r i e d o u t u s i n g v a r i o u s f i x e dv al u es of X and n o i se r a t h er t h an c lo s i n gt h e ad ap t i v e l oop as ab ov e.

F i g . 4 sh ow s t h e r es u l t s o b t a i n ed . T h esol i d u p p er cu rv e sh ows o-2 vs. X for anu n l i mi t ed sy s t e m . If t h e l i m i t e r i s i n se r t e dt hen t h e d ash ed cu r v e i s ap p ro p r i a t e ; t h emi n i mu m i s on l y a few p er cen t ab ov e t h eo p t i mu m at t h e mi n i m u m. T h e l ow er h al fof the f igur e sho ws the effect of the l im ite r

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o n the o utput a c c e le r a t io n be fo r e a nd a f te rthe l imi t . The r e s ul t o f the s im ula to r s tudywas that the min im um in a-2 o c c u r s a l m o s texactly at a , = L, hence the s imple adaptives y s t e m j u s t d e s c r i b e d f o r m s a v e r y s o p h i s

t ic a te d t r a c king lo o p .Fig . 5 i s a b lo c k d ia gr a m o f the z e r o

velo city lag t ra cki ng lo op in which we do notknow the s ignal o r noise spectra al though wea s s u m e the no is e to be ne a r ly f la t . In o r de rto a djust ba nd -pa s s we ma y us e the r e s ul t ,Eq. (18) tha t the e r r o r s igna l s pe c t r u m ispr o po r t io na l to the no is e s pe c t r um when thelo o p i s o pt ima l . W hi le the t r a ns fe r o f theloop is not co r r ec t if the no ise is not f lator if the s ignal is not that assumed, i tis s t i l l t rue that adjusting the loo p ban d-p ass

s o tha t the e r r o r s igna l s pe c t r u m is f la t i sn e a r l y o p t i m u m .

The me tho d o f me a s ur e me nt i s to us e twofi l te r s f, (s ) , a lo w-p a s s f i l te r c o ve r ing theban d-p ass of the m ain loo p, and f2(s ) c o ve r ingan equal band-pass just above the main loopa nd ta ke the r a t io o f the o utputs . The o pt imumshape of such fi l ter s has not been de ter m ine db ut s i m u l a t o r r u n s sh o w s u i ta b l e p e r f o r m a n c efo r s imple f i l te r s . f2(s) should have a finiteban d-p ass beca use the actual high frequency

no ise is unim po rtan t; only the no ise in thevicinity of the m ain loo p ba nd -pa ss isi m p o r t a n t .

In the loop as shown the obvious scalefa c to r s ha ve be e n ins e r te d to ma ke ther e s po ns e f r e que ncy ( r e la t ive to the ma in lo o p)independent of the value of the sp ec tr a . Thef i l te r f3(s ) de te r m ine s the ba nd -pa s s o f thea da pt ive lo o p a nd, in o r de r to ha ve minimumr . m . s . e r r o r s i n X , f3(s) must be adjustedso that the la gs in following cha nge s in the

spectra are balanced by the fluctuations inthe no is e o ut o f the de t e c t o r s . Fr o m th isc r i t e r i o n w e c a n d e t e r m i n e t h e a d a p t i v e l o o pband-pass as <"A ~ X x / T , whe r e 1 /T~ $/4>is the effective t ime constant relat ing to thechange in input sp ec tr a . Ho wever , if s tep

chan ges in the input s ignals ar e contem platedthen the ada ptive loo p should be as tight asstabil i ty dictates and the gain sett ings in thef i g u r e a r e c o r r e c t .

The r a t io o f the f i lte r o utputs minus o ne i spr o po r t io na l to 8 X/X s o that

' ~ X A X (45)

and the ba nd -pa ss in the adaptive loop ispr o po r t io na l to tha t in the ma in lo o p .

In a l l o f the a da pt ive s ys te ms c o ns ide r e dit is easy to specify the gain changes to keepthe lo o p dyna mic a l ly s imi la r fo r d i f fe r e ntinputs , but it is h ar de r to specify the exactba n d-pa s s o r the s ha pe o f the f i l te r s in the

adaptive loo p. It is a lways po ssible to makea l ine ar s tabil i ty an aly sis , if the inputs ar efixed, and the no ise out of the squa rin g c ir cu itscan be computed for Markovian noise but noge n e r a l the o r y o f o pt ima l de s ign e x is ts .

Th e r e a r e ma ny wa ys o f ins t r um e nt inga da pt ive s e r vo s whic h g ive a de qua te pe r fo rm ance and the effects of no nline ar i t ies andc o m ple xi ty m us t be c o ns ide r e d c a r e ful ly if as a t is fa c to r y de s ign is to be o bta ine d. Fo re xa mple , the u s e o f s mo o the d a bs o lute va lueins te a d o f r .m.s . le a ds to o nly a fe w pe r c e nt

m o r e no is e in the a da pt ive lo o p . As a no the rexam ple the divis io n in the last exam ple maybe re plac ed by a subtr actio n if the dynam icra nge is not too gr ea t . At low input s ig nal sthe loo p is then s luggish but the t rack inge r r o r i s s ma l l due to the s ma l l input .

Unde r ly ing the de s ign o f a da pt ive s e r vo sis the a s s umpt io n o f r e la t ive ly s lo w, o r o nlyo c c a s io na l , c ha nge s in e nvi r o nme n t . If r a pidc ha nge s o c c ur a d if fe r e nt ma in lo o p , po s s ib lyno nl ine a r , i s r e qui r e d . Ada ptive lo o ps ma y

be c a l le d qu a s i - l i ne a r but be c a us e they a r eno nl ine a r no ge ne r a l me tho d of a na lys is ha se m e r g e d t o d e t e r m i n e o p t i m a l p e r f o r m a n c ea s a s ta nda r d o f c o mp a r is o n wi th s pe c i f icl o o p s , o r t o c he c k i n s t r u m e n t a t io n a p p r o x i m a t i o n s .

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PRACTICAL PROBLEMS ENCOUNTERED INMISSILE GUIDANCE AND CON TRO L DESIGN

R. E . Whiff en*

SUMMARY

T h i s p a p e r d i s c u s s e s v a r i o u s p r o b l e m s t o b e o v e r c o m e d u r i n g th e p r o c e s s o f d e si g n in g ,ma king, g r o und te s t ing, and s uc c e s s ful ly f ly ing mi s s i le s . The impo r ta nc e o f r e l ia b i l i ty i ss t r e s s e d . Em pha s is i s o n de s igning a nd te s t ing the m is s i l e itself, s pe c i f ic a l ly the e le c t r o nicpa r ts of guida nc e a nd c o nt r o l s ys t e m s . Pr o ble m s dis c u s s e d a r e the o r ga n iz a t io n o f the guide dm i s s i l e d e p a r t m e n t a nd t h e r o l e o f a s p e c i a l " r e l i a b i l i t y g r o u p ; " m e e t i n g t h e r e q u i r e m e n t se nc o unte r e d by a m is s i le th r o ugho ut i t s li fe ; the de s ign l imi ta t io ns impo s e d by a va i la blec o mpo ne nts ; te s t e quipme nt r e q ui r e m e n ts ; a nd gr o und a nd pr e f l ight te s t ing te c hnique s .

SOMMAIRE

C e t te no te d is c ute de s p r o ble m e s va r i e s de va nt e t r e s o lu t io nne ' s dur a nt 1'etude, lar e a l is a t io n , le s e s s a i s a u s o l e t le vo l r e u s s i de s m is s i le s . El le a c c e ntue 1 ' impo r ta nc e dela s e c ur i te de fo nc t io nne me nt , e t po r te s ur 1'etude et 1'essai du m i s s i l e l u i - m e m e , s p e X a l e m e n tl e s p a r l i e s e l e c t r o n i q u e s d e s s y s t e m e s d e c o n t r o l e e t de g o u v e r n e . L e s p r o b l e m e s t r a i t e ss o nt : 1 ' o r ga nis at io n du de pa r te m e nt du m is s i le a go uve r ne e t le r o le d ' un gr o upe s pe c ia l" s e c u r i t e d e f o n c t i o n n e m e n t " r e u n i s s a n t l e s e x i g e n c e s r e n c o n t r e e s a t r a v e r s l a v i e d ' unm is s i le ; le s l imi ta t io ns d ' d tude impo s e 'e s pa r le s o r g a ne s d is po nible s ; le s e xige nc e s de1 'e quipe me nt d ' e s s a is ; le s te c hnique s d ' e s s a is a u s o l e t e n p r e -v o l .

1. INTRODUCTION

T h e m o r e c o m p l e x a p r o b l e m , t h e m o r e

highly o r ga niz e d mus t be a t te mpts to wa r ds as o l u t io n . T h i s p a p e r d i s c u s s e s t he p r o b l e mof successfully design ing, man ufactur ing, andte s t ing a guida nc e a nd c o nt r o l s ys te m fo r ar e l ia ble guide d mi s s i le . It i s no t wr i t te n topr o vide a s s is ta nc e to the de s ign e ngine e r inde vis ing o r c ho o s ing a mis s i le guida nc e a ndc o nt r o l s c he m e . Tha t to pic i s t r e a te d byo the r pa pe r s inc lude d in th is vo lume .

B a s ic a l ly , th is pa pe r i s d i r e c te d to me nw ho a r e r e s p o n s i b l e f o r a n e n t i r e m i s s i l ep r o j e c t w h i c h s p r e a d s a c r o s s t h e b o u n d a r i e s

o f e ngine e r ing, ma nufa c tur ing, a nd te s t ing.The in te nt i s to c o ve r ma ny impo r ta nt a s p e c tso f the p r o ble m but to t r e a t ea c h in no mo r e

de ta i l tha n is ne e ded to e s ta b l is h the i r p r o pe rpe r s pe c t ive In the who le . M o s t o f the to pic sd i s c u s s e d a r e " j u s t c o m m o n s e n s e " an d

a l m o s t " s e l f - e v i d e n t . " T h e a c h i e v e m e n t o fa go o d ba s ic guida nc e a nd c o nt r o l de s igndo e s no t gua r a n te e u l t ima te s u c c e s s fo r thep r o j e c t . A c a r e f u ll y c o n s i d e r e d b a l a n ce m u s tbe achieved between such factors as thede s ign a nd i t s e xe c ut io n; the fo r m o f o r ga nization and the availab le qualified per so nn el;and the desire for new invention and i ts highc o s t in t ime a nd mo ne y.

Experience shows that a sufficiently highle ve l of r e l ia b i l i ty o f m is s i le e le c t r o nic s i s

exceedingly difficult to at tain and is rarelya c hie ve d. St r o ng e m pha s is i s p la c e d th e r e fo r e o n the p r o ble m o f a t ta in ing r e l ia b i l i ty .

"Bendix Aviation Corp. , Mishawaka, Indiana.

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2 . ORGANIZATION OF THE GUIDEDMISSILE GROUP

a . I m p o r t a n c e o f C e n t r a l C o n t r o l a n dS u p e r v i s i o n

F i r s t l e t u s c o n s i d e r t he o r g a n i z a t i o nr e q u i r e d t o d e v e l o p a n d p r o d u c e a s u c c e s s f u lm i s s i l e . Be cau se of t h e mag n i t u d e of t h etask involved, the job is of ten spl i t upa m o n g s e v e r a l g r o u p s . W h en t h i s i s t h ec a s e , s p e c i a l e m p h a s i s m u s t b e p u t o nr e t a i n i n g in o n e c e n t r a l g r o u p t h e c o n t r o la n d s u p e r v i s i o n o f t h e v a r i o u s c o n t r i b u t i n go r g a n i z a t i o n s . F r e q u e n t ly t h e r e a r e a l t e r n a t e a p p r o a c h e s t o t h e s o l u t i o n of a p r o b l e mn ei t h er o f wh i ch h as an y ob v i o u s ad v an t ag es

o v e r t h e o t h e r . U n l e s s t h e r e i s a s t r o n gcen t ra l g ro u p t h at can m ak e a d eci s i o n ande n f o r c e a p a r t i c u l a r a p p r o a c h , v a lu a b l e t i m ei s l o s t . S i n ce t i m e i s mo n ey , mo n ey wi l la l s o b e w a s t e d .

b . Resp on si b i l i t y - T r i ad of Desi g n -Manufacturing - Quali ty

In t h e b u si n ess of d es i g n i n g an d man u fact u r i n g g u i ded m i s s i l e s , i t i s t h e fu ncti on

o f t h e e n g i n e e r i n g d e p a r t m e n t to c r e a t e ad esi g n . T h e fu nct ion of t h e man u fact u r i n gd ep ar t m en t i s t o ex ecu t e t h i s d es i g n . I feeli t wi se i n t h e g u i d ed m i ss i l e b u si n e ss t o se tu p a l so a t h i rd d ep ar t m en t an d to g i v e i t as t a t u r e c o m p a r a b l e t o t h e e n g i n e e r i n g an dm a n u f a c t u r i n g d e p a r t m e n t s . T h e t h i r d p a r tof th e t r i ad i s a q u al i t y d ep ar t m en t . F i g . 1s h o w s t h e t h r e e d e p a r t m e n t s .

(1 ) E n g i n eer i n g

( a) I n t e r p r e t s p e r f o r m a n c e r e q u i r e m e n t s o f t he c u s t o m e ra n d t r a n s l a t e s t h e s e i n t o a

sp eci f i ed work i n g d es i g n .

(b) Dev el op s t h e ex p er i m en t a lm o d e l s a nd t r a n s l a t e s t h e i rd e s i g n i n t o w o r k i n g d r a w i n g s an d sp eci f i cat i on s .

(2) Manufacturing

( a ) D e v e l o p s t o o l s a n d p r o c e s s e s a nd a c q u i r e s n e c e s

s a r y m a t e r i a l s .

(b ) F a b r i c a t e s p a r t s a nd a s s e m b l e s m i s s i l e .

(3) Quality

(a) Develo ps test and inspec tio np r o c e d u r e s a n d p r o v i d e s a l ln e c e s s a r y t e s t e q u i p m e n t .

(b) Co nducts al l inspe ctio ns and

t e s t s .

(c) -Mo nito rs pro duct quali ty andr e l i a b i l i t y .

T h e ab o v e o u t l in e of re sp o n si b i l i t y i ssh own i n b l ock d i ag ram fo rm i n F i g . 1 .

c . Re sp o n si b i l i t i es of In d iv i d ual

D e p a r t m e n t s

W i t h t h e t r i ad of en g i n eer i n g , man u factu rin g, and quali ty defined, let us exa m inet h e i n t e r r e l a t i o n s h i p s . R e d u ce d t o s i m p l e s tt e r m s , w e h a v e :

d . P er fect i o n Un d er Di f f i cul t ies ;Need fo r Rel i ab i l i t y

G u i d e d m i s s i l e s a r e p e r h a p s t h e m o s tcomp l ex f l y i n g mech an i sms y et a t t emp t edb y m an . If t h ese u n i t s a r e t o wo rk u n man n edu n d er ex t remel y d i f f i cu l t en v i ron men t s , ane x t r e m e d e g r e e o f p e r f e c t i o n i n t h e d e s i g n

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a nd i t s e xe c utio n i s r e q ui r e d . A high de gr e eo f r e l ia bi l i ty i s a ls o r e qui r e d . It i s no tenough to have developed a guidance systemc a pa ble o f the de s i r e d pe r fo r ma nc e a ndsm all enough to fi t into the me age r volum e

begrudgingly assign ed it by the a ir fr am ede sig ne r . It is not enough that il be sufficiently l ight to come within the weightl imi ta t io ns im po s e d in o r d e r that s uffic ie ntfuel can be ca r r ie d. No r is i t enough thati t be easy to manufacture and test in spiteo f a te r r i b le c o mple xi ty . In a ddit io n to a l lthe above vir tues, the guidance unit has towo rk faithfully eve ry t im e, not just par t ofthe t im e. It mu st have a very high re l iab il i ty .The achievement of a high rel iabil i ty does notcome by fiat and, to my knowledge, no organiz a t io n ha s ye t d is c o ve r e d a ny r o ya l r o a d tos uc c e s s in the a t ta inme nt o f mis s i le r e l iabil i ty. The guidance sys tem with al l ofi ts e le ctr o nic s is one of the mo st difficultp a r t s o f a m i s s i l e t o m a k e r e l i a b l e .

e . Two Br o ad A re as in AchievingHigh Reliabil i ty

F o r o r g a n i z ed a t t a ck , t h e p r o b l e m o fattaining high rel iabil i ty can be broken downi n t o t w o b r o a d a r e a s :

(1) Co nceiving a desig n that is inh e r e n t l y r e l i a b l e .

(2) Building a device bas ed on thisdesign in a manner to enhancei ts inhe r e nt r e l ia bi l i ty .

The e ngine e r ing de pa r tme nt i s ba s ic a l lyr e s po ns ible fo r the f i r s t a nd the ma nufa c tur ing de pa r tme nt fo r the s e c o n d. Thequa l ity de pa r tme nt mus t a s s is t . It a s s is tsthe engineering department by feeding backanalyses of fai lure data to be used in designi m p r o v e m e n t ; i t a s s i s t s th e m a n u f a c t u r in gdepartment by continual inspections and byp e r f o r m i n g a l l r e q u i r e d t e s t s .

f. Re liab ility Is a Vi rtu e

C o n s c i o u s n e s s o f t h e i m p o r t a n c e o f r e l ia bi l i ty might be c o ns ide r e d a s a v i r tuetha t ne e ds s pe c ia l e mp ha s i s in o r d e r fo r a

guide d mis s i le g r o up to be s uc c e s s ful . Fr o mthe very outset of the project , the thinkingo f the t r ia d o f e ngine e r ing -ma n ufa c tur ing-q u a li ty m u s t al w a y s b e " r e l i a b i l i t y o r i e n t e d . "I t mus t be ve r y c le a r ly e s ta bl is he d a ndr e a l iz e d tha t the r e s po ns ibi l i ty fo r a c hie vingr e l ia b i l i ty i s s h a r e d by e ve r yo ne in thee nt i r e guide d m is s i le g r o up. A s ignif ic ant lyr e l ia ble mis s i le c a n be a t ta ine d o nly byr e qui r ing e a c h ma n to th ink a nd a c t in te r mso f ma ximiz ing r e l ia bi l i ty .

g. Special Reliab il i ty Gr o up

The r e a r e ma ny t ime s whe n the e a s y wa yis to igno r e the de ma nds im po s e d by s t r ic ta d h e r e n c e t o r e l i a b i l i t y r e q u i r e m e n t s . As pe c ia l me a ns o f ke e ping the " r e l ia bi l i tyo r i e n t a t i o n " a l i v e i s n e e de d . T o d o t h i s , as pe c ia l r e l ia b i l i ty g r o up c a n be s e t up .Fu r t he r r e a s o ning jus t ify ing the us e o f ar e l ia bi l i ty g r o up is tha t , w hi le the r e s po ns ibi l ity fo r the a t ta inme nt o f r e l ia bi l i tymus t be s ha r e d by the who le o r ga niz a t io n ,a c tua l ly a t ta in ing r e l ia bi l i ty i s s o imp o r ta n tthat a spec ial gro up is useful to em ph asi zea nd a s s is t in r e a c hing a s a t is fa c to r i ly h ighgo a l o f r e l ia bi l i ty .

The dut ie s o f s uc h a g r o up a r e de s c r ibe dbe lo w.

(1) It has the res po nsib il i ty of co nstantly emphasizing the need fora nd the impo r ta nc e o f h igh r e l ia bil i ty.

(2) It mu st mo nito r the re l iab il i ty ofthe product and function as ac e n t r a l s o ur c e o f da ta o n te s ts o ft h e m i s s i l e a nd i t s v a r i o u s p a r t s .

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(3) It m ust c o ntinually ana lyze thetes t da ta and r ela y the f indings tot h e o p e r a t i n g d e p a r t m e n t t o f a c i l i t a t e e l i m i n at i o n of t h e sp eci f i cf a i l u r e s .

It i s e x t r e m e l y i m p o r t a n t t h a t t h e r e l i ab i l i ty g ro u p b e s t a r t ed a t t h e b eg i nn i n g oft he p r o j e c t . T h e r e a r e s e v e r a l r e a s o n s f o rt h i s . T h e i m p o r t a n t o n e s a r e : d a t a o neq u i p men t fa i l u r e can b e o b t a i n ed i n a co n s i s t e n t m a n n e r ; s u c h d a t a a r e a v a i l a b l e t ot h e d e s i g n e r s in r e t a i n i n g g o o d p a r t s o f t h ed esi g n and e l i mi n at i n g t h e b ad ; an d an ear l ys t a r t c a n b e m a d e t o i m p r e s s t h e i m p o r t a n c eo f r e l i a b i l i t y o n t h e e n t i r e o r g a n i z a t i o n . S i nc ea m i s s i l e p r o j e c t o f a n y s i z e c a n v e r y e a s i l y

fin d i t se l f co n f ron t ed wi t h l a rg e am o u n t s ofd at a o n r e l i ab i l i t y , an ea r l y ad op t i on ofm a c h i n e m e t h o d s o f d a t a s o r t i n g a n d c o m p i l a t i on i s u sefu l b o t h t o i n su re accu racy an dt o m i n i m i z e e x p e n d i t u r e o f v a l u a b le m a n p o w e r .

3 . E N V I R O N M E N T S R E Q U I R E M E N T S

a . E n v ir o n m e n ta l P a r a m e t e r s M u stBe Def i n ed E ar l y

B e f o r e d e s i g n o f a m i s s i l e g u i d a n ce s y s t e mc a n m a k e t h e t r a n s i t i o n f r o m b r e a d b o a r d top r o t o t y p e , t he e n v i r o n m e n t a l p a r a m e t e r smu st b e l i s t ed an d ra n g e s of v al u es e s t i m at e d .Th e fol lowing type s of diff icul t ies a r e l ikelyt o b e e n c o u n t e r e d .

(1 ) T e m p e r a t u r e . U n d er o p e r a t i n gc o n d i t i o n s , c h a n g e s m a y c a u s ed r i f t i n c i r c u i t p e r f o r m a n c e d uet o t e m p e r a t u r e c o e f f i c i e n t s o fc a p a c i t o r s , r e s i s t o r s , i n d u c t o r s ,e t c .

(2 ) Al t i t u d e. P o ss i b i l i t y of a r c i n gat h ig h a l t i t u d es m u st b e p r ev e n te d by p r o p e r d e s i g n .

(3 ) A c c e l e r a t i o n a nd d e c e l e r a t i o n( t r a n s i e n t o r c o n s t a n t ) c a u s e sd a m a g e s u c h a s w i r e b r e a k a g ea n d s h o r t i n g o f t e r m i n a l s .

(4 ) V i b r a t i o n . H ig h a c c e l e r a t i o n sd a m a g e v a c u u m t u b e s , r e l a y s ,g y r o s , e t c .

(5) Humidity.

Al l o f t h e ab o v e fact o rs m u st b e co n sid er ed in the l ight of the effects of co m bina t i o n s o f t h e a b o v e o c c u r r i n g s i m u l t a n e o u s l yan d i n t h e fu r t h er en v i ro n m en t of t h e v ar i o u sl i k el y serv i ce l i fe p h ases of :

(1) Shipment.

( 2 ) S t o r a g e .

(3 ) T a c t i c a l s t o w a g e .

(4) Han dling and launc hing.

(5) Fl ight .

T h e f l i g h t en v i ron men t i s t h e ob v i ou s on et o wh i ch t h e a t t en t i on of t h e d es i g n er i sn a t u r a l l y a t t r a c t e d . N e v e r t h e l e s s , t h e o t h e re n v i r o n m e n t s c a n b e e q u a l ly d a m a g i n g . F o rex a m p l e , t h e u su al g u i d an ce sy s t em i se x t r e m e l y c o m p a c t . N o r m a l l y , t h e f li gh tp h a s e i s s o s h o r t t h a t h e a t i n g g e n e r a t e dw i th in t he e l e c t r o n i c s i s n o t a p r o b l e m .H o w e v e r , s u s t a i n e d o p e r a t i o n f o r m a n y h o u r su n d e r t e s t m a y r e q u i r e l i m i t in g t h e a m o u n tof t es t i n g a t on e t i me, a p rod u ct i on i n con v e n i e n c e a nd e x p e n s e . S i m i l a r l y , v i b r a t i o n sen co u n t er ed i n f li g ht a r e man y oct av e s ab ov e

t h e l ow f req u en cy v i b r a t i o n s i n t rod u ced b yt r a n s p o r t a t i o n by r a i l r o a d o r t r u c k ; W iths o m e t y p e s o f s h o c k m o u n t s f o r e l e c t r o n i ceq u i p men t , t h i s l ow f req u en cy v i b ra t i on canc a u s e s e v e r e d i f f i c u l t i e s .

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b . Envi r o nme nta l a nd Life Te s tEva lua t io n

(1) P r o t o t y p e T e s t s

As soon as a prototype of the guidancesys tem is availab le i t should be put thro ugha c o m p l e te s e r i e s o f e n v i r o n m e n t a l t e s t s .The s o o ne r the s e te s ts c a n be r un the be t te rs inc e a ny t r o u ble s unc o ve r e d wi l l r e qu i r eimm e dia te c ha nge s in de s ign; i t i s e s s e nt ia lthat this infor matio n be obtained long befo reexpen sive inve stm ents have been made inc o m p o n e n t s o r t o o l i n g .

Dur ing the s e ve r y e a r ly te s ts a s muc hpe r fo r m a nc e info r ma t io n s ho uld be o bta ine da s i s po s s ib le . It i s no t e no ugh to de te r m ine

whe the r the de s ign me e ts s pe c i f ic a t io ns .Specifications should be met with sufficientma r gin to ta ke c a r e o f ma te r ia l a nd ma nufa c tur ing to le r a nc e s . Info r ma t io n f r o m the s ee a r ly te s ts wi ll pe r m it r e m e dia l s te ps to betaken while this can s t i l l be done rea so nab lye c o no mic a l ly , i .e . , be fo r e p r o duc t io n s c he dules must be met.

h i g h t e m p e r a t u r e s o f r u b b e r s , l u b r i c a n t s ,p l a s t i c s , e t c . , h e l p s o n l y m o d e r a t e l y i n p r e dic t ing u l t ima te s ur viva l o f s uc h m a te r ia l s .Ho wever , the technique may be used on ar e l a t i v e b a s i s a n d c o m p a r a t i v e r e l i a b i l i t i e s

ma y be de te r mine d be twe e n va r io us po s s ib lec ho ic e s o f m a te r ia l by s ubje c t ing a l l to theide nt ic a l te s t .

(3) Life T es ts

Include d in a ny e nvi r o nme n ta l te s t p r o gr am should be judic io us l ife test ing ofm i s s i l e p a r t s a nd c o m p o n e n t s . T h i s a p p l ie sp a r t i c u l a r l y t o a ny e l e c t r o m e c h a n i c a l d e v i c e s

no r m a l ly s ubje c te d to we a r a nd c o ns e que ntde gr ad atio n. Although the flight life of amis s i le ma y be o nly s e c o nds o r po s s ib lym i n u t e s , i t i s n e c e s s a r y f o r m i s s i l e d e v i c e st o o p e r a t e s a t i s f a c t o r i l y f o r m a n y h u n d r e d sof ho ur s dur ing facto ry test and final checkp r i o r t o l a u n c h i n g .

(2) Ac c e le r a te d Tim e Te s ts

One type of know ledge the m iss ile de sign er

would l ike to have ea r l y is how well hisde s ign wi ll s ta nd up unde r r e pe a te d te s ts o rp r o lo nge d s to r a ge . It i s de s i r a ble to c o nduc t" a c c e l e r a t e d t i m e " t e s t s in a n a t t em p t topr edic t the pr o bable effect of lengthy pe r i o dso f s to r a ge o r o pe r a t io na l use in the f ie ld .In the s e a c c e le r a te d te s ts , the us ua l a ppr o a c his to s ubs t i tu te e x t r e m e s of e nvi r o nme nt fo rt i m e . T h u s , f o r e x a m p l e , s u c c e s s i v e t e s t srun alter nat ely f i rs t under high heat and thenunder high humidity and which would take aweek to run, might be the equivalent of

s e v e r a l m o n t h s ' e x p o s u r e t o t r o p i c a l c o n d i t i o n s .

The va l id i ty o f a c c e le r a te d e nvi r o nme nta lte s t s ha s no t be e n we l l e s ta bl is he d . Fo rins ta nc e , the p r o lo nge d s o a king a t e xt r e me ly

R o ta t ing e quipme nt , s uc h a s mo to r s , dyna -m o t o r s , an d i n v e r t e r s , m u s t b e d e s ig n e d t ode l ive r full po w e r dur ing ma ny r e pe a te dcyc les of te st . Switching dev ices , such asr e la y s , s ho uld be li fe te s te d unde r lo a dsc lo s e ly s imu la t ing the s wi tc hing lo a ds . Itis not necessary to conduct l ife tests unti lc o mple te fa i lu r e e xc e pt in unus ua l c i r c ums ta n c e s . The l ife o f a m is s i le ma y bespecified as 500 cyc les of on-off o per atio nof five-m inute dur atio n. Hence, the l ifetest of 1500 cycles of f ive-minute durationgives a safety facto r of thr ee without thene c e s s i ty o f te s t ing to fa i lu r e . M o vingc o nta c ts in po te nt io me te r s mus t be te s te do ve r the fu l l r a nge o f c o nta c t po s i t io ning

unde r the a nt ic ipa te d e le c t r ic a l lo a d fo rs im i la r duty c yc le s . The s e a r e a fewe xa mple s o f the k inds o f de vic e s tha t r e qui r el i fe te s t ing a bo ve the no r ma l r a nge o f e nv i r o n m e n t a l t e s t s .

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(4 ) C o m b i ne d E n v i r o n m e n t a l T e s t s

T h e u s u a l a p p r o a c h t o e n v i r o n m e n t a lt e s t in g i s t o c o n s i d e r o n e p a r a m e t e r a s t hev a r i a b l e an d t o e s t a b l i s h s t a n d a r d c o n d i t io n s

f o r a ll o t h e r s . T h u s , f o r e x a m p l e , a m i s s i l ew o u l d b e t e s t e d u n d e r v i b r a t i o n o v e r t h e f r e q u en cy ran g e of 2 0 -5 0 0 cy cl es p er secon d ,wh i l e t h e t em p e r at u r e i s 2 2 °C , t h e h u mi d i t yn o m i n a l , a n d t h e a t m o s p h e r i c p r e s s u r e 7 6 0m m of Hg. I t is much m o r e difficul t to do ,but i t has been found that much valuablei n f o r m a t i o n c a n b e d e r i v e d b y o p e r a t i n g t h em i s s i l e u n d e r c o m b i ne d e n v i r o n m e n t a l c o n d i t i o n s . Hen ce, wh i l e t h e m i ss i l e i s i n ana l t i t u d e - t e m p e r a t u r e - h u m i d i t y c h a m b e r i t sa l t i t u d e i s ch an g ed f rom sea l ev el t o , say ,35,000 feet (760-180 mm of Hg.) , i ts ambientt e m p e r a t u r e f r o m 2 2 ° C t o - 5 5 ° C , a nd w i thess en t i a l l y n o h u mi d i t y , a t t h e sa m e t i me t h em i s s i l e i s s u b j e c te d to v i b r a t i o n t e s t i n g .Un d er t h e comb i n ed con d i t i on s , t h e c l i mb i n gof t h e mi ss i l e f rom sea l ev el t o h i g h a l t i t u d ei s m o r e fa it h fu l ly s i m u l at e d .

4 . LIMITATIONS PLACED ON THEDESIGNER BY AVAILABLECOM P ONE NT S

a. M ee t i n g a S h o r t Desi g n S ch ed u l eF o r c e s U s e o f A v a i l a b l e C o m p o n e n t s

S u c c e s s o f a g u id e d m i s s i l e d e s i g n e r d e pends on combining the ski l ls and knowledgeo f a t r e m e n d o u s n u m b e r o f v e r y s p e c i a l i z e dd e s i g n e r s . T h i s i s p a r t i c u l a r l y t r u e i n t h ec a s e o f t h e e l e c t r o n i c s d e s i g n e r w h o m u s tmak e u se of man y comp on en t s i n h i s d es i g n .In a m i ss i l e t h e r e m ay b e as man y as 3 0 0 ,0 0 0c o m p o n e n t s . T h e e l e c t r o n i c s p a r t o f t h eg u i d an ce an d con t ro l sy s t em may t y p i cal l yh a v e :

100 to 200 vacuum tubes in 10 to 15d i f fe ren t t y p es ,

9 0 0 to 1 20 0 r es i s t o r s i n 20 0 t o 2 2 5different types,

3 0 0 t o 6 0 0 cap ac i t o r s i n 1 0 0 t o 12 5d i f fe ren t t y p es ,

75 t o 1 00 p o t en t i o me t er s i n 2 5 t o 35d i f fe ren t t y p es ,

6 0 t o 8 0 c r y s t a l d i o d es i n 1 0 t o 1 5d i f f e r e n t t y p e s ,

7 0 to 9 0 r a d i o o r i n t e r m e d i a t e f r e q u e n c yco ils and cho kes in 30 to 50 different ty pes,

50 to 60 plugs and co nne cto rs in 15 to 20different types (750 to 1000 individualc o n n e c t i o n s ) ,

4 0 t o 50 t r a n s f o r m e r s i n 1 5 t o 20 d i f fe ren tt y p e s ,

20 to 25 r el ay s in 5 to 10 different types,

2 to 4 gyr o s in as m any different types,

5 5 0 0 t o 6 0 0 0 so l d er j o i n t s .

F r o m a c o n s i d e r a t i o n of t h e t o t a l n u m b e ri n v ol v ed , it b eco m es ap p are n t t h at n o l a r g en u mb er of comp on en t s may b e d ev el op eds p e c i a l l y . B a s i c e l e c t r o n i c c o m p o n e n t s a r eq u i t e s t a n d a r d i z e d , c o m m e r c i a l l y a v a i l a b l e ,u su al l y r eas o n ab l e in co s t , an d av ai l ab l e t os p e c i f i c a t i o n s . A s a m a t t e r o f e c o n o m yb ot h i n t i me an d mon ey , t h e d es i g n er wi l lfin d t h e m o st re aso n ab l e ap p ro ach to b e on ewh erei n h i s i n i t i a l se l ect i on i s mad e f roma l r e a d y a v a i l a b l e c o m p o n e n t s .

b . Redundancy

W i t h on l y a few ex cep t i on s , t h e l i mi t a t i on si mp o sed on t h e m i ss i l e d es i g n er by wei g ht an ds i z e r e q u i r e m e n t s p r o h i b it t he w i d e s p r e a du s e o f r e d u n d a n cy i n e l e c t r o n i c c i r c u i t s a s a

m e a n s o f p r o d u c in g a m o r e r e l i a b l e s y s t e m .Al most a l way s t h e comp on en t s i n t h e v ar i ou sc i r c u i t s a r e c o n n ec te d in s e r i e s . D u r i ngfl ight , the fai lure of a single component int h i s s e r i e s c h a i n i s u s u a l l y c a t a s t r o p h i c .

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One e xce pt io n , ho we ve r , whe r e r e dunda nc ya s a te c hnique o f imp r o ving r e l ia b i l i ty i savailable to the designer is in the applicationo f r e l a y s . W h e r e a r e l a y c o n t a c t f o r m s th elink in the s ignal circuit chain, the designer

should sel ect a m ultiple po le re lay andpa r a l le l the c o nta c ts . If the p r o ba bi l i ty o ffailur e of a s in gle co ntact is 0.0 1, two suchc o nta c ts in pa r a l le l , unde r the mo s t fa vo r a b l e c i r c u m s t a n c e s , h a v e a f a i l u r e p r o b abil i ty of 0.0001.

c. Ov era ll Reliab il i ty Related toComponent Reliabil i ty

W ith s uc h a la r ge numbe r o f c o m po ne ntsinvolved in a typical missile guidance andco ntr o l design and the fact that fai lur e of anyo ne o f ma ny c o uld be c a ta s t r o phic to afl ight, i t is instr uct ive to co nsid er how theo v e r a l l r e l i a b i l it y r e l a t e s to t h e v a r i o u sc o m po ne nts . Le t us c o ns ide r a typic a lmis s i le e le c t r o nic s de s ign wi th a numbe r o fc o m po ne nts a s indic ate d a bo ve . A c r u dee s t im a te o f the o ve r a l l r e l ia b i l i ty o f a ne le c t r o nic s ys te m c a n be c a lc ula te d a sfo l lo ws :

Fig . 2 s ho ws the p r o ba bi l i ty o f s ys te mfa i lu r e a s a func t io n o f fa i lu r e r a te s fo rgr o ups o f c o mpo ne nts , ba s e d o n the a bo vef o r m u l a .

d . C o m p o n e n t E v a l ua t io n P r o g r a m

Eve ntua l ly , the mi s s i l e de s ign e r wi l la r r i v e a t a s o - c a l l e d a p p r o v e d l i s t o f c o m ponents around which his design is based.Ho we ve r , to de la y de s ign a wa i t ing the c o mpletion of this approved l is t would be towa s te va lua ble t im e . Th e r e i s a lwa ys aba s e f r o m which to s ta r t and a us ua llys a t is fa c to r y a ppr o a c h is to build up th isl is t as you go alo ng. A few po inte rs may bei n o r d e r o n c o m p o n e n t e v a l u a t i o n .

(1) R e que s t e a r ly info r m a t io n o nfa i lu r e s f r o m the r e l ia bi l i tyg r o u p .

(2) Us e the e va lua t io n p r o c e s s toc o m p a r e e q u a l l y a v a i l a b l e c o m po ne nts to de c ide whic h c o m ponent is the best .

*.- W X X - W"N

w h e r e R 0 = o v e r a l l r e l i a b i l i t y

q, - (i - P;)"1

p. • pro babil i ty of fai lur e of i- thc o m p o n e n t

q. - pro babil i ty that the i-th

component will not failnj = number of i- th type components

i n s e r i e s

N • num ber of different types ofc o m p o n e n t s i n s e r i e s

(3) Eva lua te c o mpo ne nts o n a p r i o r i ty ba s is by wo r king o n thep o o r e s t o r p o t e n ti al t r o u b l e m a k e r s f i r s t . E x p e r i e n c e w o u lda s s ign to th is c a te go r y va c uumt u b e s a n d r e l a y s .

(4) Ev aluate co mp o nents not only inr e g a r d t o t h e i r e l e c t r i c a l s p e c i f ic a tio ns but a ls o in r e ga r d tothe e nvi r o nme nta l s pe c i f ic a t io ns

unde r whic h the y mus t o pe r a te .

(5) Es ta b l is h a p r o gr a m o f r a t ing tor e v e a l p r e f e r r e d s u p p l i e r s ( d e pendable and co nsis te nt quali ty ).

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e . E n l i s t i n g t h e C o o p e r a t i o n a n dS u p p or t o f Comp on en t S u p p l i e rs

A m o n g o t h e r t h i n g s , t h e m i s s i l e e l e c

t r o n i c s d e s i g n e r i s d e pe n d en t fo r h i s s u c c e s son h i s sk i l l i n mak i n g g ood u se of t h e av ai l a b l e c o m p o n e n t s . A v e r y p o s i t iv e s t e p c a n b et a k e n t o w a r d s f u r t h e r i n g h i s s u c c e s s b yen l i s t i n g t h e coo p era t i on an d su p p o r t o f t h ev a r i o u s c o m p o n e n t s u p p l i e r s . A r e s e r v o i r ofsp eci a l i zed k n owl ed g e an d ex p er i en ce i nc o p i n g w i t h t h e p r o b l e m s o f a p a r t i c u l a rco mp o n en t i s u su al l y t o b e foun d on t h et ech n i c al st af fs o f t h e b e t t e r su p p l i e r s . Byp r e s e n t i n g t h e i r e n g i n e e r s w i th a s t a t e m e n to f r e q u i r e m e n t s a nd a n e x p la n a t io n o f t h e i r

s t r i n g en cy , on e can of t en ch al l en g e t h e en g i n eers ' p r i d e wi t h t h e resu l t t h at t h ey wi l lo u t do t h e m s e l v e s t o de v e lo p a m o r e s a t i s f a c t o r y c o m p o n e n t .

5. T E S T E QUIP M E NT RE QUIRE M E NT S

a. T e st E q u i p men t

T h e faci l it y wi th wh ich a m i ss i l e e l e c

t r o n i c s d e s i g n e r c a n e v a l u a t e h i s d e s i g n s i sp rob ab l y af fect ed more b y t h e av ai l ab i l i t y ofsat i s fa ct o ry t es t eq u i p m en t t h an an y o t h ers i ng l e f a c t o r . T h e e l e c t r o n i c s d e s i g n e r i sd e a l in g w i th c o m p l e x v a r i a b l e s s u c h a s e r r o rvol tage output as a function of phase of as u b c a r r i e r a m p l i tu d e m o d u la ti o n o n a r a d i ob e a m o f m i c r o w a v e fr e q u e n c y . H e n c e , i na d d it io n t o t h e m o r e o r l e s s s t a n d a r d p i e c e so f e l e c t r o n i c t e s t e q u i p m e n t s u c h a s o s c i l l o s c o p e s a n d s i g n a l g e n e r a t o r s , t h e d e s i g n e ri s d e p e n d en t o n s u c h s p e c i a l i t e m s a s r a d a r

b e a m s i m u l a t o r s an d a l ti tu d e s i m u l a t o r s .L a t e r i n p r o d u c t io n t e s t , p e r h a p s b o t h o ft h e s e w i ll b e t ie d t o g e t h e r w i th a p r o g r a m m e rt o c o m m a n d t h e s e s p e c i a l p i e c e s o f e q u i p men t t o p rod u ce cer t a i n ou t p u t s a t sp eci f i edt i m e s .

b . T est E q u i p men t Ob j ect i v es

In test equipment design, i t has long beent h e p r a c t i c e t o a t t e m p t a p r e c i s i o n t e nt i m e s g r e a t e r t h a n t h e d e v i c e b e i n g c h e c k e d .

P r a c t i c a l l i m i t a t i o n s o f th e s t a t e o f t h e a r tof t en p rev e n t a t t a i n m en t of t h i s o b j ect i v e .I f , f o r e x a m p l e , t h e m i s s i l e e l e c t r o n i c sdesig n is deal ing with a high de gr ee ofs t a b i li t y i n a p a r t i c u l a r o s c i l l a t o r a nd i t i sre ach i n g t h e l i m i t s o f t h e s t a t e of t h e a r t t oa t t a i n t h i s , i t i s n o t l i k el y t h at a comp ar i sont e s t o s c i l l a t o r c a n b e t en t i m e s b e t t e r .

c . T e st E q u i p men t Desi g n S t ar t ed i nP a r a l l e l w i t h M i s s i l e D e s i g n

E v e n b e f o r e t h e m i s s i l e e l e c t r o n i c s b l o c kd i a g r a m s a nd in s t r u m e n t a t i o n d i a g r a m s s ho wso me ev i d en ce of se t t l i n g down , th e t es t e q u i p men t d es i g n g ro u p mu s t g o t o wo rk . Id eal l y ,f rom t h i s t i me fo rward , t h e t es t eq u i p men tm u s t p r o g r e s s i n p a r a l l e l w i t h t h e m i s s i l edes ign. Ac tual ly , at be st , the tes t equipmentcan on l y k eep u p wi th t h e m i ss i l e d es i g n .Certainly i t is not pract ical for i t to lead them i s s i l e d e s i g n .

Desi g n of t es t eq u i p men t i s n o t as a t t ract i v e as d es i g n wo rk on t h e m i ss i l e . Un l ess

m anag em ent is al er t , it wil l f ind that tes tequipm ent is not given adeq uate at tent io n andt he e n t i r e p r o g r a m i s d e l a y e d . A s p o i nt edo u t e a r l i e r i n t h i s p a p e r , m a n y o r g a n i z a t i o n sp r efe r t o m ak e t h e d es i g n of t es t eq u ip men ta r esp o n si b i l i t y of t h e q u al it y d ep ar t m en t .T h i s i s b e c a u s e o n e of t h e b a s i c r e s p o n s i b i l i t i es of t h e q u al i t y d ep ar t men t i s t op e r f o r m t e s t i n g of t h e m i s s i l e . S pe ed yd esi g n an d co n st ru ct i o n of t es t eq u i p men t i ses se nt ial if the quali ty dep ar tm ent is to fulf i lli t s r e s p o n s i b i l i t i e s .

d . Spec ial ized Te st Sets OftenW o r t h w h i l e

It i s u su al l y p o ss i b l e to s t a r t wi th s t an d ar dp i e c e s o f t e s t e q u ip m e n t an d t o a r r a n g e a n d / o r

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modify them so that just about any specialelec tro nic tes ts can be conducted in thel a b o r a t o r y . T h e f a c to r y t e s t r e q u i r e m e n t sa r e a g r e a t de a l d i f fe r e nt . In fa c to r y te s ts ,i t i s ne c e s s a r y to p r o vide s pe c ia l iz e d p ie c e s

o f te s t e quipme nt whe r e m o r e o f the r e qui r e dintell igence is buil t into the equipment andle s s i s r e qui r e d o f the o pe r a to r . Ofte n i t i sw o r t h w h i l e t o s t a r t c o n s t r u c t i o n o f s p e c ia l iz e d te s t e quipme nt a s s o o n a s po s s ib le .Th is i s pa r t ic u la r l y t r u e whe n te s t ing s pe c ia lfunc t io na l c i r c ui t r y .

6. GROU ND AND PREFLIGHT TESTINGTECHNIQUES

a . Te s t ing Is An Impo r ta nt P a r t o fDesign

The cost of a missile fl ight test is high.T h e r e a r e m an y r e a s o n s f o r t h i s . F o r o n ething, no satisfactory way has yet beend e v i se d to f l i g h t - te s t s u p e r s o n i c m i s s i l e sm o r e tha n o nc e . The gr e a t e r the te nde ncyto fail when in flight or the lower ther e l ia bi l i ty , the mo r e e xpe ns ive i t be c o me s toget flight data. Th e flight seq uen ce of amis s i le might be a s fo l lo ws :

(1) Launch and boo st to supe rs o nics pe e d,

(2) Stabil ize in ro l l ,

(3) Lead into the guidance beam ,

(4) Acq uire the tar ge t , and

(5) Ho m e o n the ta r ge t .

If fai lures are sustained in phases (1)through (4) in nine out of ten flights, then ittak es ten m iss ile fl ights to get one set ofdata on phase (5), the hom ing ru n. The co sto f the ho ming da ta be c o me s im pr e s s ive .

One method of redu cing this high co stis exhau stive gro und testin g. Star t ing witha l l t h e m i s s i l e e l e c t r o n i c s b i t s a nd p i e c e sas the ba se of a py ra mi d, a syst em ofs uc c e s s ive 100 pe r c e nt te s ts mu s t be de vis e d

that has as i t s apex the prefl ight test pr i o rto la unc h. Suc c e s s ive te s ts mus t be ba s e do n a s a t is f a c t o r y a nd c o n c l u s i v e p r i o r t e s t ;the who le p r o c e s s mus t be c a r e ful ly wo r ke dout so that nothing im plic it in the planneds e que nc e o f e ve nts c a n do a nything to inv a l id a t e t h e e a r l i e r t e s t s . In o t h e r w o r d s ,t h e t e s t i n g p r o c e s s m u s t r e m a i n u n d e r c o n t r o l . W hen ine vi ta ble t r o uble s a r i s e a ndr e p l a c e m e n t s a r e m a d e t o e l i m i n a t e t h et r o u b l e s , o n l y sa t i s f a c t o r i l y t e s te d p a r t s o fs u b a s s e m b l i e s o r a s s e m b l i e s m u s t be u s e d

in ma king the r e p la c e m e nt . J us t a s thes u r g e o n s c r u p u l o u s l y s c r u b s b e f o r e t h e o p e r a ti o n i n o r d e r n o t t o i n t r o d u c e g e r m s , s oc o n s t a n t c a r e m u s t b e e x e r c i s e d in o r d e r no tto in t r o duc e unte s te d e le me nts to the c o mple teunit. When dealin g with as many as 2,000p a r t s , the above r ule is not a lways easy tofollow. The lack of t imely availab il i ty oft e s te d p a r t s i s p e r h a p s t h e m o s t c o m m o nda nge r fa c e d.

b. S p e c i f i c T e s t P r o c e d u r e s M u s tB e W r i t te n

As e a r ly a s po s s ib le , s pe c i f ic s ta nda r diz e dte s t p r o c e dur e s s ho uld be wr i t te n fo r a l lp a r t s , s u b a s s e m b l i e s , a s s e m b l i e s , a n d f i n a lg u id a n ce a nd c o n t r o l a s s e m b l i e s . T h e o b je c t ive i s a lwa ys to a t ta in c o ns is te nc y o ve rthe test ing methods which must be dependedu po n t o d e m o n s t r a t e a d eq u a te p e r f o r m a n c eo n th e g r o u n d . T h e sp e c i fi c p r o c e d u r e s m u s t

be adh er ed to . I t is not uncom mo n to finda techn ician who feels he knows a be tterwa y to r un a pa r t ic u la r pha s e o f a te s t andha bi tua lly t r ie s d i f fe r e nt wa ys r a the r thant h at p r e s c r i b e d in t h e s t a n d a r d t e s t p r o c e d ur e . A "b e t te r wa y" i s f ine, but i t mus t

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b e so re co g n i z ed . It may v er y wel l h ap p ent ha t w h e n s o m e o n e e l s e c o m e s t o r u n th et e s t , v e r y i n c o n s i s te n t r e s u l t s a r e o b t a in e d .

F o r m a n y e l e c t r o n i c s d e s i g n e r s , t h e r e i s

n ot mu ch fu n i n wri t i n g t es t p roced u res , an dt h e r e i s a t en d en cy t o d e l ay t h e d ay of w ri t i n gt h e m . O n e w a y o f o v e r c o m i n g t h i s t r o u b l eis by divisio n of labo r. It should be ther e s p o n s i b i l i t y o f t h e q u a l it y d e p a r t m e n t t oc r e a t e t h e te s t p r o c e d u r e s un d e r t he s u p e r v i s i on of t h e d es i g n en g i n ee r s . T h e q u al i t yd e p a r t m e n t w i ll u s e t h e s e p r o c e d u r e s inp e r f o r m i n g t e s t s a n d s o t he y h a v e a n i n c e n t i v e t o g e t th e s e p r o c e d u r e s w e l l e s t a b l i s h e d a s e a r l y a s p o s s i b l e . I t i s u n r e a s o n a b l et o t es t i n a way t h at d o es n ot co r r esp o n d t o

t h e i n t en t i on s of t h e d es i g n er . T o av oi dm i s u n d e r s t a n d i n g , t he t e s t p r o c e d u r e w r i t e rs h o u l d s u b m i t a l l p r o c e d u r e s f o r a p p r o v a l o ft h e r e s p o n s i b l e d e s i g n e r . A n a d d i ti o n a l p o i n ti s t h e i m p o r t a n c e o f k e e p in g t h e p r o c e d u r e sup to da te . I t is difficul t to get them w ri t tenan y way , an d i t i s p ar t i cu l ar l y h ard t o g ett h e m k e p t up t o d a t e . O n ly s u p e r v i s i o n ' si n s i s t e n c e w i l l s o l v e t h i s p r o b l e m .

c . F i n al T e st Imp o r t an t an d S hou ldBe S en si t i v e

The f inal test should be given carefulc o n s i d e r a t i o n . It i s p a r t i c u l a r l y i m p o r t a n tto ma ke a car eful study of the f inal tes tp r o c e d u r e t o d e t e r m i n e w h e t h e r o r n o t a l lcon d i t i on s t h at cou l d cau se op era t i on al f l i g h tfa i l u r e w i l l b e d et ect ed b y t h i s f in al se r i e sof t e s t s . If, fo r e x am p l e , p u l se j i t t e r i nt h e g u i d a n c e r a d a r c o u l d c a u s e t h e g u i d a n c eu n it t o fa il i n p er fo rm an c e, each g u i d an cep ack ag e h ad b et t e r b e t es t ed on t h e g ro u n dt o as ce r t a i n t h at i t i s su f f ic i en tl y i n sen si t i v et o p u l s e j i t t e r . A s e c o n d e x a m p l e : e v e n

t h ou g h t h e t h eo ry of d ep en d en ce on p r ev i o u st es t s i n t h e t es t p y rami d h as a l read y b een

a g r e e d t o , t h e r e m a y b e c e r t a i n c r i t i c a lc o m p o n e n t s t h a t s h o u l d b e r e t e s t e d e v e r yt i m e a f in a l t e s t i s r u n . A t h i r d e x a m p l e :the t es t should be so devise d that a re layu s e d t o ef fe c t c h a n g e o v e r f r o m b e a m - r i d i n gt o h om i n g i s e i t h er i mp l i c i t l y o r ex p l i c i t l yc h e c k e d e a c h t i m e .

T h e d e s i r a b i l i t y o f a n o v e r a l l t e s t s i m u l at i n g on t h e g rou n d t h e en t i re seq u en ce ofevents eventual ly taking place in the f l ighti s o b v io u s . T h e r e a r e c e r t a i n c o m p r o m i s e st h at mu st b e m ad e, h o wev er , p ar t i cu l a r l y i nt h e l a t e r p ro d u ct i on t e s t p h a se s . In l au n ch in gan d i n f l i g h t , ev ery mi ss i l e su s t a i n s a cer t a i namo u n t of sh ock an d v i b r a t i o n . T h e q u est io ni s , sh ou l d eac h m i ss i l e b e sh ock an d v i b ra t i o n

t es t e d ? A su r v ey wou ld f in d m i ss i l e t es t i n gacco mp l i sh ed wi th an d wi t h ou t . P ro b ab l y nos a t i s f a c t o r y g e n e r a l r u l e s a p p ly u n l e s s i t i st he r u l e , " t a k e a c o n s e r v a t i v e a p p r o a c h . "I f sh ock an d v i b ra t i on t es t s a re con si s t en t l ysh ow i n g up t ro u b l e , i t sh ou l d b e mad e cer t a i nt h at t h e sh ock an d v i b ra t i on l ev el s a re n o te x c e s s i v e . E x c e s s i v e s h o c k a nd v i b r a t i o nl e v e l s d u r i n g t e s t i n g m a y e a s i l y o c c u r b e c a u s e t h e m e t h o d s i nv o l ve d a r e a r b i t r a r y a tb e s t . H a v in g p r o v e d p r o p e r e n e r g y le v e l sa n d t r o u b l e s a r e s t i l l b e in g s h o w n , c o n s i d e r

t h e sh ock an d v i b ra t i o n t es t s as u sefu lad j u n ct s t o t h e o t h e r s t a t i c t e s t s in f i l te r i n go u t i n c i p i e n t t r o u b l e s .

7. CONCLUSION

T h i s p a p e r d i s c u s s e d f a c t o r s t h a t c a naffect t h e su c ce ss of a p ro j ect t o d es i g n ,mak e, t es t , an d sa t i s fact o r i l y f l y a g u i d edm i ss i l e . It i s h op ed t h at co n si d er a t i o n of t h es i g n i f i can ce of each i n t h e p a r t i cu l ar s i t u at i ont h at con f ro n t s t h e re ad er may b e h elp fu l i n

i n cr eas i n g an d sp eed i n g su c ce ss of t h ep r o j e c t .

REFERENCES

W i l so n , B , J . , " A n a l y z i n g M i s s i l e E l e c t r i c S y s te m R e l i a b i l i t y , " T r a n s . A l E E , 7 5 ,II , 1956, pp. 206-213.

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GLOSSARY OF TERMS

B r e a d b o a r d T h e a r r a n g e m e n t u s ed t o t e m p o r a r i l y c o n n e ct e l e c t r o n i cc i r c u i ts in o r d e r to de te r m ine ho w the y wo r k . Li t t le o r no

phys ic a l r e s e mb la nc e to the f ina l p r o duc t e xis ts in ab r e a d b o a r d .

C o mpo ne nt The s m a l le s t funct io na l e nt ity in to whic h e le c t r o nic se quipme nt i s d iv ide d. Exa m ple s o f c o m po ne nts a r ec a p a c i t o r s , fix ed r e s i s t o r s , p o t e n t io m e t e r s .

Pr o to ty pe The f i r s t mo de l tha t i s bo th e le c t r ic a l ly a nd me c h a nic a l lylike the expected final product.

Redundancy A technique of adding in pa ra llel to a needed elem ent,addit io nal e lem ent s that con tinue to function should theo r ig ina l e le me nt fa i l .

Reliabil i ty The pro babil i ty that a device will function success fullyunde r a l l e nvi r o nme nta l c o ndi t io ns in s e r v ic e .

System Often in guided m is si le s a functional enti ty used to ser ves o m e p a r t i c u l a r p u r p o s e ne ed ed by t he o v e r a l l m i s s i l e .Ex am ples would be a guidance system that gives them i s s i l e d i r e c t i o n s t o t he t a r g e t , a c o n t r o l s y s t e m th a ttakes action by making the missile fl ight obey the

guida nce s igna ls , a p r o puls io n s ys te m tha t p r o vide s thethr us t fo r f l ight , e tc .

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APPLICATION OF METHODS OF SCIENCETO THE PROBLEM OF RELIABILITY

C, Raymond Knight*

SUMMARY

M o de r n s ta t i s t ic a l te c hnique s pe r m it qua nt i ta t ive s tudy o f r e l ia b i l i ty p r o ble m s by thecl as si c pr o ced ur es of the scientific metho d: (1) Definit ion of the pr o blem , (2) s tudy ofavailab le data, (3) fo rm ulation of hyp o theses, (4) test in g of the hypo theses by ex pe ri m en tat ion , and (5) r eeva luatio n of the pro blem in the l ight of r es ul ts of the exp er im ent. Applicationof the scientific method is i l lustrated in an experiment conducted by Aeronautical Radio, Inc. ,to te s t the hypo the s is tha t ma inte na nc e pr o c e du r e s ba s e d pr im a r i ly o n tube te s t ing a dv e r s e lyaffect tube and sys tem re liabil i ty, and fur th er , that bette r re l iab il i ty could be ob tainedthr o ugh m a r gin a l - te s t ing pr o c e du r e s a ppl ie d to the e quipme nt a s a who le . R e s ul ts o f thee xpe r ime n t c o n f i r me d the the o r y tha t tube r e l ia bi l i ty i s a dve r s e ly a ffe c te d by ma inte na nc eba s e d p r im a r i ly o n tube te s t ing; ho w e ve r , the te nta t ive the o r y that s ys te m r e l ia bi l i ty i ss im i la r ly a ffe cted wa s no t c o nf i r m e d in the pa r t ic u la r e xpe r im e nt . The r e s ul t s a ls o indic a tethat ma r gin a l - te s t ing te c hnique s , to be e ffec t ive in r e l ia b i l i ty im pr o ve m e nt , mu s t be c a r e fully c ho s e n in r e la t io n to the c h a r a c te r is t ic s o f the s pe c i fic e quipm e nts . T he o r ga niz e dme tho d o f s tudy e xe mpl i f ie d in th is e xpe r ime n t i s e s s e n t ia l to r a pid p r o gr e s s in the s o lu t io no f r e l ia b i l i ty p r o ble m s in the f ie ld o f mis s i le guida nc e a nd c o nt r o l , a s we l l a s o the r a r e a so f e l e c t r o n i c s .

SOMMAIRE

Le s te c hnique s mo de r ne s de la s ta t i s t ique pe r me t te nt d ' e tudie r qua nt i ta t ive me nt le spr o ble m e s de la s e c ur i te de fo nc t io nne me nt e n s uiva nt le s de m a r c h e s c la s s iqu e s de la me tho descientifiqu e: (1) Definition du pr o blem e, (2) Exam en des donne'es dispo nible s , (3) Fo rm ulatio ndes hypo theses, (4) Ver ification des hypo theses pa r 1 'experience et , (5) No uvelle appr eciatio ndu pr o ble mc a la lum ie r e de s r e s u l ta t s de 1 ' e xpe r ie nce . L ' a ppl ic a t io n de la me tho des c ie nti f ique e s t i l lus t r e e pa r une e xpe r ie n c e fa ite pa r 1 ' Ae r o na utic al R a dio Inc , po ur ve r i f ie r1'hypothese s e lo n la que l le le s p r e c e d e s d ' e nt r e t ie n ba s e s e s s e nt ie l le m e nt s ur la ve r i f ic a t io nde s la mpe s a ffe c te nt de fa vo r a ble me nt c e l le s -c i e t la s e c ur i te de fo nc t io nne me nt du s ys te m e ;e t de p lus po ur ve r i f ie r qu ' une mc i l le ur e s e c ur i te de fo nc t io nne me nt po uva i t -e t r e o bte nue pa rde s p r o e ddds de ve r i f ic at io n ma r g ina le a ppl ique s a 1 'e quipeme nt c o ns ide r e c o m me un to ut .Le s r e s ul ta ts de 1 ' e xpe r ienc e c o nf i r m e r e nt la the o r ie s e lo n la que l le la s e c u r i t e de fo nt io nn-e me nt de s la mp e s e s t de fa vo r a ble me n t a ffec te e pa r 1'entretien ba s e e s s e nt ie l le me nt s ur lave r i f ic a t io n de s la m pe s ; c e pe nda nt la the o r ie p r o vis o i r e s e lo n la que l le la s e c u r i te defonctionnement est affectee de la me m e m an ier e ne fut pas co nfirm ee dans cette m em ee xpe r ie nc e . Le s r e s u l ta ts m o nt r e nt a us s i que le s te c hnique s de ve r i f ic a t io n ma r g ina le ,po ur e t r e e ff ic ac e s e t a ppo r te r une a m e l io r a t io n a la s ^c ur ite" de fo nc t io nne me nt , do ive nte t r e c ho is ie s s o igne us e m e nt e n r e la t io n a ve c le s c a r a c te r i s t iq ue s de s e quipe me n ts e tudie ' s.C e t te e xpe r ie nc e do nne un e xe mple de la me tho de r a t io nne l le d ' e tude qui e s t e s s e n t ie l le po ur

pr o gr e s s e r da ns la r e s o lu t io n de s p r o ble m e s de s e cur i te " de fo nc t io nne me nt a us s i b ie n e nm a t i e r e d e g o u v e r n e e t d e c o n t r o l e d e s m i s s i l e s , q u e d a n s d ' a u t r e s d o m a i n e s d e 1'elec-

t r o n i q u e .

• Ae r o na ut ic a l R a dio , Inc . , Di r e c to r , R e l ia bi l i ty R e s e a r c h D e pa r tm e nt , W a s hingto n , D. C .

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1. INTRODUCTION

T h e t e r m " r e l i a b l e " h a s b ee n a p a r t o fo u r v o c a b u l a r y f o r m a n y c e n t u r i e s ; a n d , a sco mm o n l y u sed fo r q u al i t a t iv e o r i n t u it i v e

d e s c r i p t i o n , i t h a s b e c o m e w i d e l y u n d e r s t o o d . W e b s t e r ' s U n a b r id g e d D i c t i o n a r yd e f in e s " r e l i a b l e " a s " S u i t a b l e o r f it t o b er e l i e d o n . W o r t hy o f d e p e n d e nc e o r r e l i a n c e . T r u s t w o r t h y . "

T h i s g r o s s , i n t u i ti v e m e a n i n g m a y p r o v i d ean ad eq u at e m ea su r e wh en we ar e t a lk i n ga b o u t r e l i a b i l i t y o r t r u s t w o r t h i n e s s o f p e o p l e ,o r ev en of i n an i m at e o b j ect s , if we a r ewil l ing to acc ept the stat us quo . If, on theo t h er h an d , we feel i mp el l ed t o i mp ro v e th e

r e l i a b i l i t y o f t h e s e i n a n im a t e o b j e c t s o r t h e s ep e r s o n s , a n d t o do s o w i th a l l p o s s i b l e s p e e d ,t h e n a f a r m o r e p r e c i s e , q u a n t i t a t i v e m e a s u r eof re l i ab i l i t y i s es se n t i a l . " W h en y ou canm ea su r e wh at y ou a r e sp eak i n g ab ou t , an de x p r e s s i t i n n u m b e r s , " s a i d L o r d K e l v i n ," y o u k no w so m e t h i ng a b o u t i t . " R e a l p r o g r e s s in s c i e n c e an d e n g i n e e r i n g h a s a lw a y sd e p e n d e d u p o n m e a s u r e m e n t p e r f o r m e d i ns u c h a w a y t h a t r e l a t i o n s h i p s b e tw e e n v a r i o u sf a c t o r s m a y b e e x p r e s s e d m a t h e m a t i c a l l y a n dt h at o t h e r e x p e r i m e n t e r s m a y d u p l ic a t e r e

s u l t s t h r o u g h s i m i l a r e x p e r i m e n t s .

We need not bela bo r the imp ell ing need forr a p i d p r o g r e s s i n r e l i a b i l i t y i m p r o v e m e n t i nal l f i e l d s of e l e ct r o n i c s , an d esp eci a l l y int h e f ie ld of m i ss i l e g u i d an ce an d co n t r o l . I ti s t h e p resen t u rg en cy of t h i s n eed t h atr e q u i r e s u s t o f in d w a y s o f m e a s u r i n g inq u an t i t a t i v e t e r m s b o t h t h e re l i ab i l i t y t hatwe h av e and t h e re l i a b i l i t y t h at we mu sta c h i e v e . M e a s u r e m e n t a l o n e , h o w e v e r , i so b v i ou sl y n ot en ou g h . An o rg a n i zed meth od

of study of the quanti tat ive effects is evenm o r e i m p o r t a n t .

The method of study that I s h a l l r e c o m m e n dis not new; i t is spel le d out in the his to ryo f m o d e r n t e c h n o l o g i c a l p r o g r e s s . I h a ve

summed i t up in the t i t le of this paper:" Ap p l i cat i on of M et h od s of S ci en ce t o t h eP r o b l e m o f R e l i a b i l i t y . "

It may seem s t r an g e th at I sh ou l d d ev o t et h i s p a p e r t o a c o n c e p t a s e l e m e n t a r y a s t h escie ntif ic me tho d. Actual ly , my subject isthe applicat ion of this old method with thea id o f a r e m a r k a b l e n ew t o o l - m o d e r ns t a t i s t i c s . F i g . 1 i l l u s t r a t e s t h e b a s i cel emen t s of t h e sc i en t i f i c met h od .

Start ing at the top of the cycle shown inFig. 1 , we have;

(1) Co mp l et e , co n ci se d ef i n it io n oft h e p r o b l e m .

(2) Invest igat ion (accumulat ion andstudy of al l facts and data avai la b l e ; ,

(3 ) Gen eral i zat i on , i n t h e fo rm oft e n t a t i v e c o n c l u s i o n s o r h y p o t h eses d rawn f rom t h e sp eci f i cfacts and data.

( 4) V e r i f i c a ti o n , th r o u g h e x p e r i m e n t a t i on d es i g n ed t o t es t , an d t h u ss u p p o r t o r r e f u te t h e g e n e r a l i z a t i o n s .

(5 ) E i t h e r r e p e t i t i o n o f t h e e x p e r i m e n t u n d e r v a r i e d c o n d i t io n s o rr e e v a l u a t i o n o f t h e p r o b l e m o rt h e g e n e r a l i z a t i o n s , d e p en d in g o nw h e t h e r t h e e x p e r i m e n t a t i o n h a ss u p p o r t e d o r r e f u te d t he g e n e r a l i z a t i o n s .

In es se n c e, t h i s met h od i s a self-

c o r r e c t i n g p r o c e s s o f i n q u ir y a nd c o n t r o l , a ndon e can re ad i l y see th e an al og y t o t h e com mo ns e r v o l o o p .

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My main int er est , howev er , is in the factthat enti re new ar ea s of inquiry ar e nowo pe n to th is we l l -e s ta bl is he d a nd t r a di t io na lpr o ced ur e. It is to point out that , with thene w to o ls p r o vide d by mo de r n s ta t i s t ic a l

techniqu es, the scien tific method can beapplied with encouraging effectiveness to thes o lut io n o f the p r o ble m o f r e l ia bi l i ty .

Dur ing the e a r ly ye a r s o f the g r o wth o fs c i e n c e , p r o g r e s s c a m e m o s t r a p i d l y in t h efield of physical sci en ce . Th er e w er e twoi m p o r t a n t r e a s o n s f o r t h i s s i t u a ti o n . T h efirs t is that in the physical sciences, precisem e a s u r e m e n t , t ha t i s , m e a s u r e m e n t w it hs m a l l v a r i a b i l i t y f r o m o n e m e a s u r e m e n t t othe next, could be achieved. The secondr e a s o n is tha t , in the phys ic a l s c ie nc e la bo r a t o r y , e x p e r i m e n t s c o u l d b e c a r e f u l l y c o n t r o l l e d t o k e e p e x t r a n e o u s v a r i a b l e s f ix edand thus per m it the effect of var i at io n in asingle factor to be rep ro ducib ly s tudied. Ino the r f ie lds o f s c ie nc e , pa r t ic ula r ly b io lo gic a ls c i e n c e , e a r l y p r o g r e s s w a s h a m p e r e d by t h eg r e a t e r c o m p l e x i ty o f t he e x p e r i m e n t a lm a te r ia l a nd, in ma ny c a s e s , by the c o n s e que nt ina bi l i ty o f the e xpe r ime nte r toc o n t r o l e x t r a n e o u s e x p e r i m e n t a l f a c t o r s . Itwa s la r ge ly in th is a r e a o f b io lo gic a l s c ie nc etha t the mo de r n s ta t i s t ic a l me tho ds we r e

developed - methods which now permit quanti tat ive , meaningful ex per im entatio n with ane nt i r e ly new c la s s o f c o m ple x e xpe r im e nta lm a t e r i a l .

In s ho r t , dur ing a 30-ye a r s pa n, thede ve lo pme nt o f the s e s ta t i s t ic a l te c hnique shas opened to scientific s tudy enti re newc l a s s e s o f p r o b l e m s — p r o b l e m s w h ic h ,be c a us e o f the i r c o mple xi ty a nd pr a c t ic a li m p o s s i b i li t y o f p r e c i s e c o n t r o l , w e r e e a r l i e rconsidered outside the scope of meaningful

e x p e r im e n t . " E x p e r i m e n t , " a s t he t e r m i sus e d he r e , i s the ve r i f ic a t io n type o f e xpe r i ment refer red to in the fourth s tep of Fig. 1.

The s ignificance of the new experimentaltechniques can be seen in their effect on the

study of re l iab il i ty . In a rel ia bil i ty inv est iga t io n , the fundame nta l " l a b o r a t o r y" i s thefarflung ar ea wh ere the pro duct is being putto i ts intended end us e. The task of co nt r o l l i n g a n e x p e r i m e n t in s u ch a " l a b o r a t o r y "

wo uld be v i r tua l ly ins ur mo unta ble we r e i t no tfo r the mo de r n s ta t i s t ic a l me tho ds me nt io ne d.P a r t i c u l a r l y v a lu a b le a r e t h o s e te c h n iq u e sa s s o c ia te d wi th e xpe r ime nta l de s ign a ndana lysis of va r i an ce . In this pap er , it wouldb e i m p o s s i b l e t o d e s c r i b e t h e s e s t a t i s t i c a lto o ls in de ta i l . The ma jo r pur po s e he r e i sto ge ne r a te s uff ic ient in te r e s t in the po s s i b i l i t ie s o f the i r a ppl ic a t io n to e nc o ur a ge awid er u se of the scientific metho d in s tudyinga nd f inding s o lu t io ns to the p r o ble m s o f r e l ia bi l i ty .

2. DEFINITIONS LEADING TOQUANTITATIVE CONCEPTS OFRELIABILITY AND THE IMPORTANCEOF MEASUREMENT

The im po r ta nc e o f qua nt i ta t ive me tho dsin a scientific s tudy has been s t ressed and i thas been po inted out that the basic s ta r t ingpoint for such a s tudy is concise definit iono f the p r o ble m a nd o f the te r m s to be us e d

in the ana lysi s of the pr o blem . With the sefundam entals in mind, definit ion s of se ve ra lo f the mo s t impo r ta nt te r ms e nc o unte r e d inr e l ia b i l i ty wo r k wi ll be p r e s e n te d. Th e s ear e definit ions fo rm ulated with the intent thatthey sha ll be highly specif ic and usa ble forq u a n ti t at iv e m e a s u r e m e n t . T h ey a r e p r e s e nte d, o f c o ur s e , fo r the pur p o s e o fi l lus t r a t io n r a th e r tha n a s a n e xha us t ivel i s t .

The f i r s t de fin it io n i s o f the te r m " r e

l i a b i l i t y " itself. W i t h m i n o r v a r i a t i o n s ,the following definition is becoming widelya c c e pte d: "R e l ia b i l i ty i s the p r o ba bi l i ty tha ta p r o d u c t w i ll g i ve s a t i s f a c to r y p e r f o r m a n c efor a given per iod of t ime under s tatedc o n d i t i o n s . "

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T h e m a t h e m a t i c a l fo r m u l a t io n of t h e defi n i t i on i s as fo l l ows:

Un rel i ab i l i t y i s merel y t h e an t i t h es i s o fre l i ab i l i t y ; i t can b e s i mp l y ex p r ess ed as

R ( X c E ; t ) = P ( a < X > b ) = / f ( X , t ) d >

(1 )

w h e r e X * ( x i . x z - - xk ) = a v e c t o r

o f p r o d u c t p e r f o r m a n c ec h a r a c t e r i s t i c s

c ? " i s c o n t a i n e d i n "

E = the regio n of po ints defined ass a t i s f a c t o r y p e r f o r m a n c e

a - (a , , a 2 .

b i ( b , , b t .

• a k )

X )

v e c t o r s s p e c i f y ing the upper andl o w e r p e r f o r m a n c e s t a n d a r d s

f (X; t) = t h e jo i n t p ro b ab i l i t y d i s t r i bution function of the perf o r m a n c e c h a r a c t e r i s t i c s

t = the length of the inte rv al ofo b s e r v a t i o n .

T h ou g h p rob ab i l i t y i t se l f i s n o t d i rect l y

m e a s u r a b l e , i t c a n b e e s t i m a t e d f r o m m e a s u r ab l e q u a n t i t i es . Co n seq u en t l y , t h e ab ov ed e fi n it io n m e e t s b a s i c r e q u i r e m e n t s . It i si n t e r e s t i n g t o n o t e t h a t t i m e , o r m o r esp eci f i cal l y , t h e t i m e p er i o d d u r i n g wh i chp e r f o r m a n c e i s s a t i s f a c t o r y i s g e n e r a l l y t h eq u a nt it y m e a s u r e d . W h e r e t h e d u r a t i o n o fa m i s s i o n i s r e l a t i v e l y f ix e d, t h e p r o p o r t i o no f t o t a l m i s s i o n s w h i ch a r e s a t i s f a c t o r y( th at i s , t h e re l i ab i l i t y ) i s o f t en e s t i m at e dd i r ect l y . T h i s d ef i ni t ion of re l i ab i l i t y , t hou g hmu ch more sp eci f i c t h an t h e d i c t i on ary

d e f i n i t i o n , i s n e v e r t h e l e s s c o n s i s t e n t w i t hi n t u i ti v e co n cep t s of t h e t e r m . S i n ce p r o b ab i l i t y i s e x p r e s s e d o n a s c a l e o f z e r o t o o n e ,and re l iab i l i ty by this defini t io n is ap ro b ab i l i t y , t h en re l i ab i l i t y w i l l a l so b ee x p r e s s e d o n a s c a l e o f z e r o t o o n e .

U( t ) « I - R( t )

(2 )

w her e U(t) * unr el iab i l i ty (a function of t im e)

R(t) t r el ia bi l i ty (a function of t im e) .

T h i s m a t h e m a t i c a l f o r m u l a ti o n i s s ho w n infu n ct i on al n o t a t i on as a remi n d er t h at b o t hr e l i a b i l i t y a nd u n r e l i a b i l i t y a r e i n t r i n s i c a l l yfu n cti on s of t i m e. P r o p o sed d ef i n it i on s oft w o a s s o c i a t e d t e r m s t h a t a r e n o t s o w i d el yu sed b ut a r e n o t ewor t h y h e r e , n amel y

" m a i n t a i n a b i l i t y " a n d " o p e r a t i o n a l r e a d i n e s s , " r e a d a s f o l l o w s :

" M ai n t a i n ab i l i t y i s t h e p rob ab i l i t y t h atwh en m ai n t en an ce act i on i s t ak en , a sy s t emw i l l b e r e s t o r e d t o s a t i s f a c t o r y o p e r a t i n gcon d i t i on wi t h i n a g i v en p er i od of t i me; " an d" o p e r a t i o n a l r e a d i n e s s i s t h e p r o b a b i l it y t h a tt h e s y s t e m w i ll p e r f o r m s a t i s f a c t o r i l y a t a nyp oi n t i n cal en d ar t i me, '*

All of these defini t ions lean heavily upont h e c o n c e p t o f s a t i s f a c t o r y p e r f o r m a n c e .

I n t h e p r e s e n t s t a t e o f m i s s i l e d e v e l o p m e n t , i t i s p r a c t i c a b l e t o m o n i t o r t he m i s s i l et r a j e c t o r y w ith r e a s o n a b l e p r e c i s i o n ; c o n s e q u e n t l y , i n t h e e s t i m a t i o n o f m i s s i l e r e l i ab i l i ty , a d ef in i ti on of sa t i s fa ct o ryp e r f o r m a n c e b a s e d up o n t h e c l o s e s t a p p r o a c ht o t h e t a r g et h as g ai n ed fa i r l y wid e acc ep t a n c e . T h u s , a f t e r t h e m i s s i l e i s c o n s i d e r e dre ady and the f inal dec isio n ha s been m adet o f i re i t , a c l ear d ef i n i t i on of sa t i s fact o ryp e r f o r m a n c e c a n u s u a l l y b e f o r m u l a t e d .

O n t h e o t h e r h a nd , s a t i s f a c t o r y p e r f o r m a n c e a s a p p l i e d t o m i s s i l e o p e r a t i o n a l r e a d i n e s s i s n o t e a s i l y m e a s u r e d . A h ig h d e g r e eo f v a r i a b i l i t y in t h e m e a s u r e m e n t of r e a d i n e s si s ap p a re n t wh en we rea l i ze t h at th e d eci s i o nt o c l a s s if y a m i s s i l e a s " r e a d y to f i r e "

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mus t be ba s e d o n a r e la t ive ly g r o s s s ys te m-check at the pre launch s tage.* Th is finalco nclusive dec is io n in effect is a co m po siteof many individual dec is io ns by a num ber ofpe o ple , ba s ed o n r e a din gs ta ke n f r o m va r io us

type s o f te s t e quipme nt . The la r ge de gr e eo f va r ia bi l i ty which r e s u l ts do e s no t p r e c ludeme a ningful m e a s u r e m e nt . It do e s , ho we ve r ,me a n tha t many me a s ur e m e nt s m us t be ma deto pe r mit the va r ia bi l i ty to be a s s e s s e d.E x p e r i m e n t s c o n d u c te d to ev a l u at e i m p r o v e me nt mus t ta ke th is va r ia bi l i ty in to p r o pe rc o n s i d e r a t i o n .

Ope r a t io na l r e a d ine s s , whi le o f o nly s e c o nda r y in te r e s t dur ing a de ve lo pme nt pha s e ,b e c o m e s a p r i m a r y f ac et o f l a t e r m i s s i l eut i l iz a t io n . As s uc h, it de ma nds gr e a te ra t te nt io n tha n i t ha s he r e to fo r e r e c e ive d.M o s t of the c u r r e n t f igur e s o n m is s i ler e l ia bi l i ty a r e c o ndi t io na l upo n the mis s i lebe ing in o pe r a t io na l r e a di ne s s ; in o the rwo r ds , g ive n a mis s i le tha t i s o pe r a t io na l lyr e a dy to f i r e , i t ha s a c e r ta in p r o ba bi l i ty o fs uc c e s s ful fl ight to ta r ge t . The s e r e l ia bi l i tyf i g u r e s , o r " c o n d i t i o n a l r e l i a b i l i t y " f i g u r e s ,thus are applicable only after the decis ionhas been made that the mis sile is rea dy tof i r e . They take no account of whether threem i n u t e s o r t h r e e d a y s w e r e r e q u i r e d t o r e a d y

the mis s i le o n the la unc hing r a c ks .

In the l ight of the definit ions pr ese nte de a r l ie r , we no w s e e tha t the p r o duc t o f the" c o n d i t i o n a l r e l i a b i l i t y " a nd o p e r a t i o n a lr e a din e s s o f a g ive n m is s i l e be c o me s theprobabil i ty that i t can be fi red at any part icular t ime and that i t wil l then come withina minimum pr e s pe c i f ie d d is ta nc e o f thet a r g e t .

• C o ns ide r a ble s tudy a nd d is c us s io n a r e be ingd e vo t e d t o t h e d e t e r m i n a t i o n o f a n " o p t i m u m "amount of complex test equipment for them e a s u r e m e n t o f o p e r a t i o n a l r e a d i n e s s .(Stanley and Tampico, Associated MissileP r o d u c t s C o r p o r a t i o n , P o m o n a , C a l i f o r n i a . )

T h e r e a r e m a ny m o r e t e r m s r e q u ir i n gdefinit ion in the s tudy of re l iab il i ty . Thete r m s c ho s e n fo r de f in it io n h e r e ha ve be ens e l e c t e d t o i l l u s t r a t e t h e t y p e s o f m e a s u r e me nt tha t a r e ne c e s s a r y if we a r e to ma ke

va l id a ppr a is a ls o f impr o ve me nt , o r va l idc o m p a r i s o n s o f d i ff e r en t s y s t e m s . In d i s cussi ng the var iabil i ty that mus t be recko nedwith in the s e me a s ur e me nts , the in te nt ha sbeen to i l l us tr ate the need for and thei m p o r t a n c e o f s t a t is t i c a l t o o l s . T h e s e t o o l sa nd the e le c t r o nic c o mpute r ha ve ma depossible scientific s tudy in fields where, afe w ge ne r a t io ns a go , the phys ic a l s c ie nt is twould have given up in co mp lete desp air .

It is not implied that the methods of

m e a s u r e m e n t a nd t h e e x p e r i m e n t a l t e c h nique s no w a va i la ble r e pr e s e nt the u l t ima tein efficiency. Imp ro vem ent is cer tain lypo s s ib le , but we c a n ne v e r th e le s s p r o c e e du po n t h e p r e s e n t b a s i s . T h e p r i m a r y g a i nto be e xpe c te d f r o m fur the r r e f ine me nt i s ar e d u c t io n in t h e n u m b e r of o b s e r v a t i o n s r e q u i r e d t o r e a c h v a l i d c o n c l u s i o n s .

3. RELIABILITY INVESTIGATION

Pr e l im ina r y inve s t iga t io n a nd s tudy o fb a ck g r o u n d m a t e r i a l r e l a t i n g to th e p r o b l e mof int er est is a fundamental s tep in thes c ie nt i f ic me tho d.

M o s t o f the r e l ia bi l i ty info r ma t io n a va i l a ble to da y is be s t de s c r ibe d a s a la r ge ma s so f o bs e r va t io ns , inc luding: the r e la t ivef r e qu e nc y of c o m p o n e n t p a r t s " f a i l u r e s , "t i m e - t o - p a r t s - " f a i l u r e s , " t i m e - b e t w e e n -e q u i p m e n t - " f a i l u r e s , " a nd m e a s u r e d e n vi r o nme nta l c o ndi t io ns whic h ha ve be e nc o n s i d e r e d s e v e r e o r e x c e s s i v e - t o m e n t i o n

a few. The im po rta nce of such investigatio nalinfo r ma t io n , whe n c a ut io us ly us e d, mus t no tbe o ve r lo o ke d, fo r i t i s the o nly ba s is fo rgo o d hypo the s e s . Publ ic a t io n o f th is ma te r i a l ,wi th in the r e s t r ic t io ns o f s e c ur i ty , mus t bee n c o u r a g e d .

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F o r a n i l l u s t r a t i o n o f i n v e s t i g a t i o n a l f i n d i n g s of t h i s t y p e, re fe r en c e i s mad e t oG e n e r a l R e p o r t N o . 1 o f A e r o n a u t i c a l R a d i o ,Inc. (ARINC), dated J an ua r y 4, 1954. In thei n v est i g at i o n re p o r t e d on , i t was foun d th at ,

o n t h e a v e r a g e , o n e - t h i r d o f t h e t u b e s r e m o v e d f r o m e l e c t r o n i c s y s t e m s i n m i l i t a r ys e r v i c e a n d r e t u r n e d t o A RIN C f o r a n a l y s i sc o n t a i n e d n o d i s c e r n i b l e d e f e c t . T h e i n v e s t i g a t o rs a t f i rs t ass u m ed t h at , s i n ce t u bed efect s v ar y ac co rd i n g t o t y p e an d ap p l i cat i ono f t h e t u b e s , t he c h a r a c t e r i s t i c d e fe c t p a t t e r n sa m o n g t u b e s r e m o v e d f r o m s i m i l a r e q u i p m e n t s s h o u ld b e r e c o g n i z a b l e r e g a r d l e s s o ft h e l ocat i o n of t h e eq u i p me n t s . T h ey o b s e r v e d , h o w e v e r , t h a t t h e o p e r a t i n g p r o c e d u r e s a n d m a i n t e n a n c e p r a c t i c e s a t a ny

g i v e n m i l i t a r y b a s e m a s k t h e s e s i m i l a r i t i e s ;an d t h at t u b e remov al s f rom a s i n g l e t y p eof eq u i p men t t en d t o b e more an al og ou s t ot h e a g g r e g a t e r e m o v a l s f r o m a l l e q u ip m e n tt y p es a t t h e b ase t h an t o remov al s f romc o m p a r a b l e e q u i p m e n t s a t o t h e r b a s e s .

A p o r t i o n of t h e d at a f ro m wh i ch t h i st en t a t i v e co n cl u s i o n w as d raw n i s sh ownin F i g s . 2 a nd 3 . T h e s e c h a r t s p r e s e n tc o m p a r a t i v e d e f e ct d i s t r i b u t i o n s o f t ub e r e t u r n s f o r s i m i l a r t y p e s o f e q u i pm e n t u s e dat d if fe ren t b a ses an d fo r a l l eq u i p m en t su n d e r s u r v e i l l a n c e a t e a c h b a s e . F o r t h i san al y s i s , th e t u b es a r e g rou p e d i n to fou rb r o a d c l a s s e s : t h o s e w ith m e c h a n i c a l , e l e c t r i c a l , a nd m i s c e l l a n e o u s d e f e c t s , an d t h o s ewi th no d i sc er n i b l e d efect .

F i g . 2 c o m p a r e s t u be r e t u r n s f ro m f i r ec o n t r o l e q u ip m e n t s a t t h e F o r t B l i s s A r m yb as e an d i n Navy i n s t a l l a t i o n s a t No rfo l k .T h e c h a r t a l s o c o m p a r e s t h e s e r e t u r n s w ith

t h e p a t t e r n o f r e t u r n s f r o m a l l ty p e s o feq u i p m en t s a t t h e re sp e ct i v e b a se s . InF i g . 3 , s i m i l a r c o m p a r i s o n s a r e m a d e fo rt u b e r e t u r n s f r o m r a d a r b o m b i n g s y s t e m sat Car sw el l Ai r F o rc e B ase an d a t M acDi l lA i r F o r c e B a s e .

T h e g r e a t e r s i m i l a r i t y in r e t u r n s f r o m a l leq u i p m en t s a t o n e b ase as co mp ar ed tor e t u r n s f r o m e q u i p m e n t s o f t h e s a m e t y p e a tdifferent ba se s is quite eviden t . He nce, thet en t a t i v e con cl u s i on t h at op era t i n g an d mai n

t e n a n c e p r o c e d u r e s o v e r s h a d o w e q u i p m e n ts i m i l a r i t y . It i s a l so ev i d en t , h ow ev er , t h atmore t h an on e h y p ot h esi s cou l d b e d ev el op edf r o m t h e d a t a in t h e s e c h a r t s . M o r e o v e r ,an y on e o r a comb i n at i on of t h ese h y p ot h esesm i g h t b e t r u e i n v a r y i n g d e g r e e s .

In t h e r e m a i n d e r o f t h i s p a p e r , t h e s t e p -b y -s t ep p roced u res wh i ch were fo l l owed i nt h e fu r t h e r s t u dy of t h i s p ro b l em wi l l b ed e s c r i b e d .

4. GENERALIZATION AND PREDICTION

As a sc i en t i f i c g en eral i t y d ev el op s , i t i sp r o g r e s s i v e l y l a b el e d a h y p o t h e s i s , a t h e o r y ,and a law, depending upon how su cce ssful lyand how bro adly it has been ver if ied. W hate v e r i t s s t a t e o f d e v e l o p m e n t , a n y g e n e r a l i t ywi l l en comp ass u n ob serv ed ev en t s - wh i chinclud es al l futur e eve nts - as well aso b se rv ed , o r p a s t , ev e n t s . In ef fect , th en ,wh en we g e n er al i z e , we p r ed i c t . T h u s t h e

t e r m " p r e d i c t i o n " a s a p p l i e d t o r e l i a b i l i t yi s m o r e t ha n a m e r e c a t c h w o r d ; t h e r e i sjust if icat io n in applying the te r m to anyg e n e r a l i t y .

In the early phase of a scientif ic study,t h e i n v e s t i g a t o r f i r s t a s s e m b l e s t h e a v a i l ab l e k n o wl ed ge p er t i n en t to t h e p r o b l em ,t h en i n v est i g at es t h e i mp l i cat i o n s of t h ef r e q u e n t l y a p p a r e n t l y - u n r e l a t e d o b s e r v a t i o n s .T h e n ex t n at u r a l s t ep i s t o l ook fo r g en e ra lc a u s e - a n d - e f f e c t r e l a t i o n s h i p s w h ic h " e x

p l a i n " th e o b s e r v a t i o n s . T h i s i s t h e s t a g eo f h y p o t h e s i s f o r m u l a t i o n .

A good hypothesis is one which can bep u t to ex p er i me n t al t es t on a q u an t i t a t i v ean d u n amb i g u o u s b a s i s . S u ch a h y p o t h esi s

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c a n va r y in na tur e f r o m the s imple s t type ,w h i c h w i l l p r o v i d e o n l y a y e s - o r - n o a n s w e r ,to the h ighly s o phis t ic a te d fo r m whe r e in am a t h e m a t i c a l m o d e l r e l a t i n g v a r i o u s f a c t o r sin the e xpe r im e nt i s te s te d . To i l lus t r a te

the s imple r type o f hypo the s is , the e xa mplepr e vio us ly in t r o duc e d wi l l be e xpa nde d.

As was shown in that example, ear lyARINC findings indicated that maintenancepr o c e du r e s a nd the e xpe r ie nc e le ve l o f ma inte na nc e pe r s o nne l a ppa r e nt ly ha ve a nimp o rtant effect on elec tro n tube and equ ipment r el iab il i ty. On the bas is of the sef indings , the fo l lo wing hypo the s e s w e r e fo r mula te d:

Hypo thesis 1. Th at, with com plex equ ipm ent, it is difficult and often im po ssib lefo r ma inte na nc e me n o f a ve r a ge te c hnic a lbackground to identify the t r ue so ur ce oft r o u b l e . M o r e h ig h ly t r a i n e d p e r s o n n e lwould be expected to delve more deeplyand accurately into the actual causes, andfewer tubes and other par ts would ber e m o v e d u n n e c e s s a r i l y .

(The tendency to change tubes rather thandelve into the t rue cause also has aps yc ho lo gic a l ba s is in tha t tube s a r e e a s yto c ha nge a nd a r e c o ns ide r e d r e la t ive lyc h e a p . F o r t h e s e r e a s o n s , t h e r e i s as t r o ng inc lina tio n to c ha nge tube s f i r s ta nd pe r fo r m c i r c ui t inve s t iga t io n o nly a sa last resor t , in the event the tube changef a i l s t o " c u r e " t h e t r o u b l e . )

This hypothesis could be tested for at r u e o r fa ls e a ns w e r by c o mpa r ing r e l ia bi l i tya c hie ve d o n ide nt ic a l g r o ups o f e quipme nts

us e d in e s s e nt ia l ly the s a m e type o f o pe r a t io nbut maintain ed, in one ca se , by no rm almil i ta r y ma inte na nc e pe r s o nne l , a nd, in theother case, by civil ian technicians withe ngine e r ing e duc a t io n a nd ma inte na nc ee xpe r ie nc e o n the e quipme nt .

Hypo the s is 2 . Tha t ma inte na nc e pr o c e d ur e s ba s e d pr im a r i ly o n the us e o ff ie ld tube te s te r s ha ve a de t r im e nta leffect on tube re l iabi l i ty and on sys temr e l ia bi l i ty ; fu r the r , tha t p r e ve nt ive ma in

te na nc e , ba s e d upo n ma r gina l te s t ing pr o ce du r es applied to the equipment a s awho le , wo uld g ive be t te r tube -a nd s ys te m-r e l ia bi l i ty tha n wo uld ma s s tube - te s t ing.

This hypo the s is c o uld a ls o be e xpe r i m e n t a l ly te s t e d f o r a t r u e - o r - f a l s e a n s w e rby a s s igning d i f fer e nt m a inte na nc e pr o c e d ur e s to g r o ups o f ide nt ic a l e quipme ntsa t a s ingle lo c a t io n .

B o th o f the s e s ugge s te d e xpe r ime nts ha ve

be en c o nduc te d. The a na lys is of the r e s u l tsof the fi r s t i s not yet co mp lete; I sha llc o mme nt fu r the r o n the s e c o nd e xpe r ime ntin th is pa pe r .

An e xa mple o f a mo r e s o phis t ic a te d hypo the s is i s c ur r e nt ly be ing e va lua te d byARINC in a new exp erim ent re cen tly in it iated .A pur p o s e of the e xpe r ime n t i s to te s t thee ffe c t ive ne s s o f a minia tu r e tube -s hie ld inser t in prolonging tube l ife by reducing bulbt e m p e r a t u r e s .

I n c r e a s e d t e m p e r a t u r e g e n e r a l l y a c c e l e r a t e s t ub e d e t e r i o r a t i o n th r o u g h c h e m i c a lr e a c t i o n s , d i f f u s i o n p r o c e s s e s , d e s o r p t i o n o fg a s e s , a n d o t h e r p h y s i c a l - c h e m i c a l p r o c e s s e s , t h e r e a c t i o n r a t e s fo r w hic h a r e a llr e a s o na bly we l l de s c r ib e d by 4 \ r r he nius ' La w.A ma the m a t ic a l mo de l ta king the s e fa c to r sin to a c c o unt ha s be en a s s u me d. This mo de lhypo the s iz e s tha t the r e m o va l r a te o f tube swill be equal to some constant independento f t e m p e r a t u r e p l u s a n o t h e r c o n s t a n t t i m e s

the ba s e o f the na tur a l lo ga r i th ms to thepo w e r - b /k t ; that i s ,

U - c, + c2

b / « t

(3 )

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wh ere b i s t h e act i v at i on en erg y fo r t h ed e s o r p t i o n o f t h e g a s e s f r o m t h e e n v e lo p e( b e l i e v e d t o b e a p p r o x i m a t e l y 0 . 1 6 e l e c t r o nv o l t s ) , T i s t h e t e m p e r a t u r e in d e g r e e sKel v i n , an d k i s Bol t zman n 's con st an t .

E n ou g h ex p er i en ce i s av ai l ab l e on t h ee l e c t r o n i c s y s t e m c o n c e r n e d t o p e r m i ta s s i g n m e n t o f a p p r o x i m a t e v a l u e s t o t h ec o n s t a n t s . T h u s , th e c h a n ge i n t ub e r e m o v a lr a t es d u e t o t h e u se of t h e ' sh i e l d s i s b ei n gp r e d i c t e d i n a d v a n c e f o r e x p e r i m e n t a l v e r i f i cat i on .

E x p e r i m e n t s o f c o n s i d e r a b ly g r e a t e r c o m -

p l ex i ty a r e b ei n g p l an n ed . On e of p a r t i c u l a ri n t e res t wi l l b e d es i g n ed t o t es t a comp l exm e t h o d o f p r e d i c t i n g r e l i a b i l i t y o f e n t i r es y s t e m s , i n c l u di n g c o n s i d e r a t i o n o f s u c hf a c t o r s a s p r i m a r y s u p p ly v o l t a g e a n d r e g u la ti o n , c o m p o n e n t - p a r t s t e m p e r a t u r e s , p r o v i s i on fo r ea se of m ai n t en a n ce, an d man yo t h e r f a c t o r s .

T h e d an g er of u n cr i t i ca l l y accep t i n g g en e r a l i z a t i o n s b a s e d o n i n v e s ti g a t io n w it ho u tr e p e a t e d e x p e r i m e n t a l v e r i f i c a t i o n s e e m sa l m o s t a t r u i s m . E x a m p l e s o f t h i s d a n g e rar e l eg ion i n t h e an n a l s of sc i en ce. In d eed ,

i n t h i s h ome cou n t ry of Gal i l eo , i t may seeman aff ron t t o co n si d e r t h i s p o i n t an y t h in go t h e r t ha n s e l f - e v i d e n t . T h e m e r e c o m p l ex i t y of t h e re l i ab i l i t y p rob l em, h owev er ,m a k e s i t p e r i l o u s l y e a s y t o i m a g i n e c a u s e -an d -ef fect re l a t i o n sh i p s an d t hen t o accep tt h em as sc i en t i f i c fact .

T h e fo l l owi n g ex amp l e of an ex p er i men tconducted by ARINC, with the cooperat ion oft h e Un i t ed S t a t es M i l i t a ry S erv i ces , h as b een

c h o s e n f o r t h i s p r e s e n t a t i o n f o r t w o p r i m a r yr e a s o n s , f i r s t , t o i l lu s t r a t e ho w a r e l i a b i li t ye x p e r i m e n t c a n b e c a r r i e d o u t; a nd s e c o n d ,t o d e m o n s t r a t e t ha t n e g at iv e r e s u l t s a r es o m e t i m e s j u s t a s i m p o r t a n t a s p o s i t i v er es u l t s . It i s i mp o r t an t t o recog n ize t h atw e l e a r n t h r o u g h m a k i n g m i s t a k e s , e v e n t h o s em i s t a k e s w h i c h , i n r e t r o s p e c t , s e e m o b v i o u sa f t e r t h e e x p e r i m e n t i s c o m p l e t e .

E X P E R I M E N T

E x p e r i m e n t a l t e s t o f h y p o t h e s e s i s g e n era l l y more d i f f i cu l t i n t h e f i e l d of re l i ab i l i t yt h an i n man y areas of t h e p h y si cal sc i en ces;i n con seq u en ce, n o t en ou g h ef fo r t h as b eend e v o t e d t o e x p e r i m e n t . W e w h o a r e w o r k i n gi n t h e f ie l d of re l i ab i l i t y a r e u n for t u n at e l yfaced with the problem of developing thesci e n ce a t t h e sam e t i me t h at we a r ea t t e m p t i n g t o e n g i n e e r r e l i a b l e p r o d u c t s .Since the function of eng inee rin g is to apply

t h e k n owl ed ge of s c i e n c e, we h av e a d i l e m m a.Rel i ab i l i t y en g i n ee rs m u st reco g n i ze t h e factt h at t h ey mu st d ev el op t h e sc i en ce as wel l asapply i t , and they m ust have the supp o rt ofm a n a g e m e n t i n t h e i r e f f o r t s t o w a r d t h e s ee n d s .

T h i s ex p er i me n t was d es i g n ed t o t es t t h e

s e c o n d o f t h e tw o h y p o t h e s e s s ta t e d e a r l i e r ,n a m e l y , " t h a t m a i n t e n a n c e p r o c e d u r e s b a s e dp r i m a r i l y o n t h e u s e o f f ie ld tu be t e s t e r sh av e a d e t r i m en t a l ef fect on t u b e re l i ab i l i t yan d on sy s t em re l i ab i l i t y ; fu r t h er , t h atp r e v e n t i v e m a i n t e n a n c e , b a s e d u p o n m a r g i n a lt e s t i n g p r o c e d u r e s a p p li e d t o t h e e q u ip m e n tas a wh ol e , wou l d g i v e b et t e r t u b e- an ds y s t e m - r e l i a b i l i t y t h an w o u ld m a s s t u b e -t e s t i n g . "

A g r o u p of 6 0 g e n e r a l - p u r p o s e m i l i t a r yr a d i o r e c e i v e r s w e r e c h o s e n a s a s u i t a b l ev e h i c l e f o r t he t e s t . P a i r s o f t h e s e r e c e i v e r s a r e u s e d in s p a c e d i v e r s i t y r e c e i v i n gsy st e m s a t a f ix ed g rou n d rec ei v i n g s i t e ofa n A r m y r a d i o s t a t i o n . T h e 6 0 e q u i p m e n t s

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we r e d ivided in to th r e e g r o ups o f 20 e quipme nts e a c h. A dis t inc t ma inte na nc e pr o c e dur e wa s a s s igne d to e a c h gr o up o f 20e quipme n ts . A br ie f de s c r ip t io n o f e a c hma inte na nc e pr o c e du r e i s s ho wn in Ta ble 1 .

With the exception of t ime required forpr e ve nt ive a nd e me r ge nc y ma inte na nc e , a l lr e c e i v e r s i n e a c h g r o u p a r e o p e r a t e d o n a24 ho ur pe r day ba s is . This type o f o pe r a t io nr e s u l te d in a l l r e c e iv e r s ha ving ve r y n e a r lythe s a m e o pe r a t ing t ime fo r the dur a t io n o fthe te s t , c o m pute d to be a ppr o xima te ly 437 0o p e r a t i n g h o u r s . R e c o r d s w e r e k ep t o n a l lr e p o r te d ma lfunc t io ns . The o bs e r va t io n unitused in calculatin g the rel iabil i ty functionof an equipment, that is , the mathematical

exp re ssio n of i ts re l iab il i ty as a function ofa specified t im e int er va l , is the t ime betweenma lfunc tio ns o the r tha n tho s e d is c o ve r e d a ts c he dule d mo nthly o r qua r te r ly ma inte na nc ep e r i o d s . P e r i o d s o f t i m e f r o m a n e q u ip m e n tmalfunction to a scheduled maintenance andthe t ime from the beginning of the test tof i r s t ma lfunc t io n o r s c he dule d ma inte na nc e

we r e inc lude d in the c o mputa t io ns a s inc o mp l e t e o b s e r v a t i o n s . In a d d it io n t o t he r e c o r d so n the e quipme nt , r e c o r d s w e r e ke pt o n thet ime s a t whic h e le c t r o n tube s we r e r e mo ve d,a nd o n the numbe r o f ma inte na nc e m a n-h o ur sr e q u i r e d f o r e a c h g r o u p o f e q u i p m e n t s .

In e s ta bl is hing the ma inte na nc e pr o c e d u r e s t o b e u s e d , s e v e r a l a s s u m p t i o n s w e r ema d e : (1) I t wa s a s s um e d tha t a c ha r a c te r i s t i c o f p r i m a r y i m p o r t a n c e t o t hep e r f o r m a n c e o f t h e s e r e c e i v e r s w a s t hes e ns i t iv i ty ; (2) i t wa s a s s um e d tha t de te r i o r a t io n o f the r e c e i ve r s e ns i t iv i ty in t ime wa sp r i m a r i l y d u e t o e l e c t r i c a l d e t e r i o r a t i o n o fthe tube s used in the equipm ent; and (3) it

wa s a s s ume d tha t r e duc t io n in e le c t r o n- tubehe a te r vo l ta ge wo uld s imula te a c o ndi t io n o fde te r io r a t io n to be e xpe c te d la te r in l i feand, co nseque ntly, would tend to give ap e r f o r m a n c e p r e d i c t i o n . T h e r e s u l t s s ho wtha t a s s um pt io ns 1 a nd 3 do no t a ppe a r to bejus t i f ie d . Ne ve r the le s s , va lua ble c o nc lus io nscan s t i l l be drawn from the test .

Ta ble 1 . Th r e e Type s o f M a inte na nc e Fo r M il i ta r y R a dio R e c e ive r s

Equipme ntG r o u p

N u m b e r *

I

II

III

M o nthly Se ns i t iv i ty M e a s ur e me nts :C r i t e r i a f o r P r e v e n t i v e M a i n t e n a n c e

A t N o r m a l H e a t e rVo l ta ge

When input signalneeded for

specified output is4 m i c r o v o l t s o rg r e a t e r

At He a te r Vo l ta geReduced by 10%

None

When inputsignal neededfo r s pe c i f ie d

output is twicethat needed atn o r m a l h e a t e r

vo l ta geNone

Sc he dule d Qua r te r lyM a inte na nc e

All tubes tested onHickok 539A tester

None

None

C r i t e r i a F o rTube R e pla c e me nt

B e l o w - m i n i m u mr e a ding o n Hic ko k-

539A te s te r

Adverse effect on

r e c e i v e r p e r f o r m a n c e , as shown bytube substi tution

*20 e quipme nts in e a c h gr o up.

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T h e act u al re l i ab i l i t y fu n ct i on s es t i mat edfor each eq u i p m en t g r o u p a r e sh own i n F i g .4 . In each o f the Equ ipm ent Gr o ups II andIII, the r el ia bi l i ty function f i ts an expo nentialc u r v e v e r y c l o s e l y . T h e m e a n t i m e t omal fu n ct io n i s 9 97 h o u r s fo r Gro u p I I, an d9 69 h o u r s fo r G ro u p I II . W i th t h e n u mb ero f o b s e r v a t i o n s a v a i la b l e , a n a p p r o p r i a t es t a t i s t i ca l t es t d o es n ot show t h e d i ffe ren c ebetween Groups II and III to be significant .T h e re l i ab i l i t y fu n ct i on of t h e eq u i p men t si n Gro u p 1 d o es n ot f it t h e ex p o n en t i a l d i st r i b u t i o n , t h u s i n d i cat i n g a p o ss i b l e d i f fe ren cei n re l i a b i l i t y b et ween eq u i p m en t s in t h i sg r o u p a nd t h e o t h e r tw o g r o u p s . M e a n t i m eb et ween mal fu n ct i o n s , as es t i m at ed b y n o n -p a r a m e t r i c m e t h o d s , i s 1 01 0 h o u r s . S in c et h i s i s n o t a s i n g l e-p aramet er fu n ct i on , t h em e a n a l o n e d o e s n o t a d e q u a te l y d e s c r i b e i t ;t h e r e f o r e , a c o m p a r i s o n o f m e a n s b e tw e e nGr o up I and Gr o ups II and III is not ana d e q ua t e c o m p a r i s o n . An a p p r o p r i a t e s t a t i s t i ca l t e s t a t 70 0 h o u r s i n d i ca t es a s t at i s t i ca l l y s i g n i fi can t d i f fe ren ce i n t h e r e l i ab i l it y b et ween G ro u p s I an d t h e co mb i n edGr o ups II and III , Gro up I having the higherr e l i a b i l i t y i n t h e a p p r o x i m a t e r a t i o o f 4 t o 3 .

I t i s o f i n t e res t t o ex ami n e t h e g ross t u b er e l i a b i l i t y a n d m a i n t e n a n c e m a n - h o u r sp i c t u r e f o r t h e s e s a m e e q u ip m e n t g r o u p s .

P r e l i m i n a r y d a t a , ba s e d u po n t h e f i r s t s i xm o n t h s ' s u r v e i l l a n c e , a r e p r e s e n t e d i n T a b l e2.

Here we f ind that over an equivalent t imep e r i o d , t u b e r e m o v a l s f o r G r o u p I e q u i p m e n t sa r e a p p r o x i m a t e l y t w i c e t h e r e m o v a l s i ne i t h e r o f t h e o t h e r g r o u p s . S i m i l a r l y , m a i n t e n a n c e m a n - h o u r s f o r t h e G r o u p 1 e q u i p m e n t s a r e a l m o s t d o u b le t h e m a i n t e n a n c em a n - h o u r s f o r e i t h e r o f th e o t h e r tw o g r o u p s .

It may be c o nclude d that, up to 700h o u r s , b et t e r re l i ab i l i t y i s ob t a i n ed i n eq u i p m en ts m aintaine d as in Gr o up I , but i t isachieved at a substantial cost in tubes andm a i n t e n a n c e m a n - h o u r s . W e ig hin g t h e a d v a n t ag es t o b e g ai n ed b y t h e i n cr eas ed re l i ab i l i t yagainst the cost of the tubes and maintenancetime is a task beyond the scope of thisp r ese n t at i o n ; h o wev er , i t i s a p h ase of t h ep r o b l e m t h a t p r e s e n t s a n i n t e r e s t i n g a n dp o s s i b l y i m p o r t a n t f i e l d f o r t h e o p e r a t i o n sa n a l y s t .

T h e r e s u l t s o f t h e t e s t a l s o e m p h a s i z e t h ee c o n o m i c j u s t i f i c a t i o n f o r f u r t h e r w o r kt oward t h e d ev el op men t of more ef f i c i en tm e a n s o f m a r g i n a l t e s t i n g . It w a s d e t e r m i n e dd u r i n g t h e cou rse of t h e ex p er i men t t h atr e c e i v e r s e n s i t i v it y ha d r e l a t i v e l y l i t tl e c o r r e l a t i o n w ith t he c h a r a c t e r i s t i c s r e s p o n s i b le

T ab l e 2 . T u b e Rem o v al s and M ai n t en an ce T i m e fo r M i l i t a ryR a d i o R e c e i v e r s , U n d e r T h r e e T y p e s o f M a i n t e n a n c e *

E q u i p m e n tG r o u p

N u m b e r

I

II

II I

N u m b e ro f

E q u i p m e n t s

2 0

2 0

2 0

A p p r o x . H o u r so f O p e r a t i o n

P e r E q u i p m e n t

4 3 7 0

4 3 7 0

4 3 7 0

T u b e R e m o v a l s

N o . o f T u b e s

4 4 1

2 3 3

2 2 6

% o f T o t a l

4 9 . 0

2 5 . 9

2 5 . 1

M a i n t e n a n c e T i m e

N o . o f H o u r s

6 1 7 . 5

3 3 3 . 0

3 4 0 . 7

% o f T o t a l

4 7 . 8

2 5 . 8

2 6 . 4

* P r e l i m i n a r y d a t a b a se d o n f i r s t 6 m o n t h s o f t e s t .

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fo r m o s t o f the r e p o r te d ma lfunc t io ns . R e c e ive r no is e a nd, c o ns e que nt ly , s igna l - to -n o i s e r a t i o , s e e m s t o b e a c h a r a c t e r i s t i c o fc o n s i d e r a b l y g r e a t e r i m p o r t a n c e . It i sbelieved that much of this noise may be

c a us e d by le a ka ge pa ths fo r ming o n mic ainsulation in the tube spacers , but adequateme tho ds o f me a s ur ing th is le a ka ge , o r thenoise which i t causes, are not now available.Repeti t ion of an experiment of this typewith r e vis e d ma r gina l te s t ing te c hnique sa p p e a r s d e s i r a b l e .

The r e s ul ts o f the e xpe r ime nt de f in i te lyc o nf i r me d o ne po r t io n o f the hypo the s is .This was the par t which holds that main

tenance based mainly upon tube test ing adve r s e ly a ffe c ts tube r e l ia bi l i ty . Othe r e xpe r iments conducted by ARINC also support thistheo ry . On the o the r hand, the po r t io n of thehypo the s is de a l ing wi th e quipme nt r e l ia bi l i tywas not ver ified, and, as a matter of fact ,the re sul t was nega tive. Our belief thatr e vis e d ma r gina l - te s t ing te c hnique s c a n besuccessful is nothing more than a newhypothesis which must also be subjected toe x p e r i m e n t . I r e f e r h e r e to m a r g i n a l - t e s t i n gte c hnique s whic h wi l l g ive g r e a te r c o ns ide r

a t i o n t o s i g n a l - n o i s e c h a r a c t e r i s t i c s o f t h er e c e i v e r s .

wo r ds , i t i s po s s ib le to ha ve a n impr o ve me nt in c o mp o ne nt-pa r t r e l ia b i l i ty , wi th an et d e c r e a s e in s y s t e m r e l i a b i li t y . T h i ss e e ming pa r a do x s te ms f r o m the fa c t tha tde f in it io ns o f s a t i s fa c to r y pe r fo r m a nc e fo r

s y s t e m s a n d c o m p o n e n t p a r t s a r e n o tn e c e s s a r i l y c o n s i s t e n t .

The f i r s t o f the s e two pr im a r y impl ic a t io nsmay seem self-ev iden t. Yet, the findingstha t ha ve be e n pr e s e nte d s e e m to indic a tethat this obvio us conclusio n has not alwaysbe e n c o mple te ly o bvio us . In the m a r gin a l -testing system which was evaluated in thise x p e r i m e n t , t h e r e c e i v e r c h a r a c t e r i s t i c stested w er e the sa m e a s tho se which had beenc he c ke d fo r p r e ve n t ive ma inte na nc e pur po s e s

a t the s a m e lo c a t io n fo r ma ny ye a r s . B utthe f indings s ugge s t tha t the s e c ha r a c te r is t ic shad not been as carefully selected (formax imu m e ffectiveness) as they might havebe e n.

The second implication (that there is nota o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n c o m po ne nt r e l ia bi l i ty a nd s ys te m r e l ia bi l i ty) ma ya ls o s e e m s e l f -e vide nt . B ut i t no w s e e m sc le a r tha t a n undue pr o po r t io n o f the b la mefo r s ys te m unr e l ia bi l i ty ha s be e n p la c e d o n

the electron tube s imply because of a lack ofa ppr e c ia t io n o f the t r u th o f th is s ta te me nt .

It i s impo r ta nt to r e c o gniz e the p r im a r yimpl ic a t io ns o f th is e xpe r ime nt :

m * + * * *

F i r s t , it i s to o b r o a d a g e n e r a l i z a t i o nm e r e l y to s t a t e t h at m a r g i n a l - t e s t i n gp r o c e d u r e s i m p r o v e s y s t e m r e l i a b i l i t y ;t h e p a r t i c u l a r m a r g i n a l - t e s t i n g p r o c e dures for any specific equipment must be

c a r e ful ly c ho s e n.

S ec o n d, t h e r e i s n o t a o n e - t o - o n e c o r r e s po nde nc e be twe e n a ppa r e nt c o mpo ne ntr e l ia bi l i ty a nd s ys te m r e l ia b i l i ty ; in o the r

T h e e x a m p l e s o f r e l i a b i l i t y a n a l y s i s -te c hnique s p r e s e nte d in th is pa pe r we r echo sen with the ho pe that they would em pha s iz e the impo r ta nc e o f the o r ga niz e d

metho d of s tudy. No where is the need forth is s c ie nt if ic a ppr o a c h m o r e c o mpe l l ingthan in the field of m is si le g uidan ce andc o nt r o l . W he the r i t be the m is s i le f ie ld o r

.a ny o the r a r e a o f e le c t r o nic s a ppl ic a t io n ,e n g i n e e r i n g i n t u i t i o n o r h i t - o r - m i s s t a c t i c s

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can no longer be solely depended upon if wea r e t o m a k e i n t e l l i g e n t l y p l a n n e d p r o g r e s s .At a l l s t ag es of t h e ap p r o ach t o t h e p r o b l emof u n r el i ab i l i t y , t h e d i sc i p l i n es of t h es c i e n t i fi c m e t ho d a r e e s s e n t i a l . It i s h o p e d

t h at t h e i l l u s t ra t i o n s g i v en i n t h i s p a p erwi l l b e su g g est i v e of ex p er i men t s t h at wi l ln o t on l y a i d i n t h e so l u t i on of p ar t i cu l arre l i ab i l i t y p r o b l em s b ut a l so ad d t o t h eg e n e r a l s t o r e o f s c i e n t i f i c k n o w l e d g e .

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Fig. 1 . Ele m e nts o f the s c ie nt i f ic me tho d.

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FORT BLISS NORFOLK

PERCENT

100 i -

8 0

6 0

4 0

2 0

NOtDEFECT

MISC.

EL EC.

MECH.

FIRE

CONTROL

FIRE

CONTROL

NOtDEFECT

MISC.

ELEC.

MECH

F i g . 2. Defect d i s t r i b u t i on of r e m o v e d t u b e s :F i r e c o n t r o l s y s t e m vs. all e q u i p m e n t s at b a s e .

32 8

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CARSWELL M A C D I L L

PERCENT100 r-

80

6 0

4 0

20

A L LE Q P T S ,

ELEC.

MECH.

BOMBS Y S T E M

BOMBS Y S T E M

A L LEQPTS.

NODEFECTS

MISC.

ELEC

MECH

Fig. 3 . De fe ct d is t r ibu t io n o f r e m o ve d tube s :R a da r bo mbing s ys te m vs , a l l e quipme nts a t ba s e .

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UJ

co

V)

«r• = >

oX

z

a•8cuVH

o

=a2

C O

H

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c o

I4- 1

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RELIABILITY OF GUIDED MISSILESEdwin A. Speakman*

SUMMARY

In th is d is c us s io n o f guide d m is s i le r e l ia b i l i ty , s pe c ia l e m pha s is i s p la c ed o n theimpo r ta nc e o f a t ta in ing r e l ia b le s ys te m s o f guidanc e a nd c o nt r o l . W ith the a dvent o f s upe r s o n ic guide d mis s i le s who s e guida nc e a nd c o nt r o l s ys t e m s r e pla c e the huma n pi lo t a nd h isa t te nda nt judgme nt , go ve r nme nt a nd indus t r y a r e fa c e d wi th s c ie nt i f ic p r o ble ms ne ve r be fo r ee nc o unte r e d . The o r ga niz a t io na l pa t te r n o f the De pa r tme nt o f De fe nse a nd the wo r k i t ha spr o mo te d in indus t r y to s o lve the s e impo r ta nt p r o ble m s is de s c r ib e d. In a ddi tio n , c e r ta ins pe c i fic a r e a s a r e o ut l ine d whe r e c o nc e nt r a te d e ffo r t s ho uld y ie ld p r o gr e s s ive im pr o ve m e nt .The s e r e c o mm e nda t io ns a r e c i te d: (1) R e duc e c o m ple xi ty , (2) de vis e me a n s by whic h thea dv e r s e c o ndi t io ns e nc o unte r e d by guide d m is s i le s fr o m de ve lo pm e nt , th r o ugh pr o duc t io n ,t ra nsp o rta t io n, launching and actu al f light can be anticipated and dete rm ined , and (3) develo pplanne d p r o gr a m s o f te s t ing c o mpo ne nts to fa i lu r e with e m pha s is o n c o ndi t io ns o f ma ximum

s t r e s s .

SOMMA1RE

Da ns c e t te d is c us s io n s ur la s e c u r i t e de fo nc tio nne me nt de s m is s i le s a go uve r ne ,1'auteur de no te 1 ' impo r ta nc e d ' o bte nt io n de s ys te m e s de go uve r ne e t de c o nt r o le do nt lefo nc t io nne me nt s o i t s u r . Avec la ve nue de mis s i le s a go uve r ne s upe r s o nique s do nt le ss ys t e m e s de go uve r ne e t de c o nt r o le r e m pla c e n t le p i lo te huma in e t s o n p r o pr e juge me nt , lego uve r ne me nt e t 1'industrie s o nt mis e n fac e de p r o ble m e s s c ie nt i f ique s ja m a is r e nc o nt r e sa upa r a va nt . Le mo de le d ' o r ga nis a t io ndu M inis te r e de la De fe nse e t le t r a va i l qu ' i l a e nc o ur a g eda ns 1'industrie p o u r r e s o u d r e c e s i m p o r t a n t e s q u es t io n s s o n t d e c r i t s . D e p l u s , c e r t a i n sd o m a i n e s p a r t i c u l i e r s o u un e ff o r t c o n c e n t r e d e v r a i t a m e n e r u n e a m e l i o r a t i o n p r o g r e s s i v e

s o nt de c r i t s . C e s r e c o mm e nda t io ns s e de no mbr e nt : (1) R e dui r e la c o mple xi te , (2) p la nnif ierle s mo ye ns pa r le s que ls le s c o ndi t io ns de fa vo r a ble s r e nc o nt r e e s pa r le s mis s i le s a go uve r nedu de ve lo ppe me nt , a t r a v e r s la p r o duc t io n , le t r a n s po r t , le la nc e me nt e t le s c o ndi t io nsp r e s e n t e s d e v o l p e u ve nt e t r e a n t i c i p e e s e t d e t e r m i n e e s , (3 ) d e v e l o p p e r d e s p r o g r a m m e s d epla nnific at io n c o n c e r na nt le s e s s a is de s o r g a ne s a ux de fe c t io n s e n s e r e fe r a nt a ux c o ndi t io nsd ' a c c e ntua t io n ma ximum.

1. INTRODUCTION and guidance and co ntr o l a r e re qu ir ed whichhave n ever been achieved in any m ili ta ry

W ith m o de r n s upe r s o nic a i r p la n e s and ve hic le . Of s pe c ia l in te r e s t i s the p r o ble mm is s i le s we a r e fa c ed with s c ie nt if ic p r o b- o f guida nc e a nd c o nt r o l . In guide d m is s i le s ,

le m s no t he r e to fo r e e nc o unte r e d . New a nd c o m ple x e le c t r o nic de vic e s mu s t r e pla c e ther a dic a l de s igns in p r o puls io n , a e r o dyn a mic s huma n pi lo t a nd h is po we r o f r e a s o n a nd

*Vic e Pr e s id e nt a nd Ge ne r a l M a na g e r , Fa i r c hi ld Guided M is s i le s Divis io n , Fa i r c hi ld En gine e r ing a nd Ai r p lane C o r po r a t io n .

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co ntr o l . If the guidance of a m is si le in f l ighti s l os t , i t i s v ery ra re l y reg ai n ed an d t h ewh ol e p u rp o se of o p er a t i o n i n clu d i n g t h em i s s i l e i s l o s t . It i s , t h e r e f o r e , m o s t a p p r o p r i a t e t h a t t o d a y w e g i v e a t t e n ti o n t o p r o b l e m s

of r e l i ab i l i t y as re l a t ed t o g u i d an ce an dc o n t r o l o f m i s s i l e s .

2 . GE NE RAL S COP E

C o n s i d e r a b le p r o g r e s s h a s b e en m a d ed u r i n g t h e p as t few y ear s i mp r o v i n g t her e l i a b i l i t y g e n e r a l l y o f m i l i t a r y e l e c t r o n i ceq u i p me n t . T h i s h as b een b ro u g h t ab ou t b yst u d y an d con t i nu ed i m p ro v em en t i n e l ec t r o n i c c o m p o n e n t s s u c h a s e l e c t r o n t u b e s ,e l e c t r o n i c p a r t s , a nd a d v a n c e s i n e q u ip m e n t

d e s i g n , t e s t i n g , a n d p r o d u c t i o n m e t h o d s .Ho wev er , wh en t h e g u id ed m i ss i l e i sex am i n ed , re l i ab i l i t y t ak e s on a n ew s i g n i f i can ce b eca u se t h e wh ol e t echn ol o g y of m i s s i l es i s n ov el an d t h e fu n ct i on of mi ss i l esi n v o l v e s c o m p l e x m e c h a n i c a l a n d e l e c t r o n i cd e v i c e s . T h e s e m u s t o p e r a t e u n d e r s e v e r ee n v i r o n m e n t a l c o n d i t io n s . S o , w h i le w er e c o g n i z e t h e e n o r m o u s e f f o r t i n w h i c hg o v ern men t an d in d u st r y a r e en g ag ed , t h ep r o b l e m o f s o l v i n g t h e s e q u e s t i o n s w i t h p a r t i c u l a r r e f e r e n c e t o gu id ed m i s s i l e s r e m a i n swith us.

In t h e fo l l owi n g sect i on s t h e re l i ab i l i t yp r o b l e m i n g e n e r a l w i ll b e d i s c u s s e d w i ths o m e r e f e r e n c e t o w o r k d o n e by t he D e p a r t men t of Defen se. E x a mp l es an d d at a wi l l b ec i t e d t o i l l u s t r a t e t h o s e a r e a s w h e r e c o n c e n t ra t ed effo r t w i l l y i e ld i mp ro v em en t . S i n cet h e gu i d an ce an d co n t ro l o f g u id ed m i ss i l e si s p r i m a r i l y e l e c t r o n i c i n n a t u r e , m o s t o f t h ep ap er wi l l ap p l y t o t h e e l ect ron i c sy s t ems ofg u i d a n c e a n d c o n t r o l .

3 . RELIABILITY GROUPS

In 1 95 0 t h e r e w as es t a b l i sh e d an ad h ocg r o u p o n r e l i a b i l i t y o f e l e c t r o n i c e q u ip m e n tu n d e r t h e R e s e a r c h a nd D e v e l o p m e n t B o a r d

of t h e Dep ar t m en t of Defen se. As v i c e-c h a i r m a n o f t h e R e s e a r c h a n d D e v e l o p m e n tBo ard d u r i n g t h e p er i o d f r o m 1 9 5 0 t o 1 9 52 ,t he a u t h o r o r g a n i z e d t h i s p r o g r a m a nd e s t a b l i sh e d th e n e c e s s a r y p o l i c i e s f o r c a r r y i n g o n

a n a c t iv e r e l i a b i l i ty p r o g r a m . A t t h at t i m e ,t h ou g h man y ag en ci es were aware of t h er e l i a b i l i t y p r o b l e m , i t w a s fo u nd t h a t c o o r d i n a t i o n o f t h e s e a c t i v i t i e s w a s n e c e s s a r y .(See T ab le 1.)

T ab l e 1. Ad Hoc Gro u p on Rel i ab i l i t yo f E l e c t r o n i c E q u i p m e n tDep ar t men t of Defen se

T a s k

1. Qu an t i t a t i v em e a s u r e m e n t s

2 . D e v e l o p m e n tt e s t s

Object ive

E s t a b l i s h m i n i m u mr e l i a b i l i t y f i g u r e s

D e t e r m i n e d e s i g nc r i t e r i a

3 . P r o d u c t i o n t e s t s V e r if y r e l i a b i l i t y inp r o d u c t i o n

4 . D e v e l o p m e n tp r o c e d u r e s

5. S p eci f i cat i on s

Veri fy re l i ab i l i t y i nd e s i g n

E s t a b l i s h m e t h o d sfo r sp eci fy i n gr e l i a b i l i t y

6 . P r o c u r e m e n t a nd D e t e r m i n ec o n t r a c t u a l c o m p a t ib i l it y w i thr e g u l a t i o n s r e l i a b i l i t y

o b j e c t i v e s

7. Pac kagin g andt r a n s p o r t a t i o n

8 . S t o r a g e

Avoid damage insh i p men t

I m p r o v e s t o r a g em e t h o d s

9 . S e r v i c e e x p o s u r e I n c r e a s e r e l i a b i l i t yb y b e t t e r m a i n t e n a n c e m e t h o d s

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After intensive s tudy the ad hoc gro up onr e l i a b i l i t y m a d e s e v e r a l r e c o m m e n d a t i o n ss um ma r iz e d a s fo l lo w s : (1) Fa i lu r e da ta a ndpe r fo r m a nc e da ta o n a l l type s o f m i l i ta r ye quipme nt s ho uld be c o l le c te d , s um m a r iz e d ,

a nd e va lua te d , (2) v igo r o us impr o ve me nt p r o g r a m s f o r e l e c t r o n t u b e s a nd a s s o c i a t e de quipme nts s ho uld be c o nt inue d, (3) p r o c u r e me nt s pe c i f ic a t io ns o f the mi l i ta r y de pa r t me nts s ho uld inc lude a r e l ia b i l i ty s e c t io n ,a nd (4) r e qui r e me nts a s to the de g r e e o frel iabil i ty should be adopted in mili tarys pe c i f ic a t io ns . W hi le the s e a r e b r o a d a ndg e n e r a l r e c o m m e n d a t i o n s t h ey s e r v e a s t h ef r a m e w o r k f o r p r o g r a m s i ni ti a te d by n u m e r o u s m i l i t a r y a n d i n d u s t r i a l g r o u p s an d p r o v i d ea ba s e fo r the c o mme nts tha t fo l lo w.

M o r e r e c e n t l y t h i s a d h o c g r o u p o n r e l i a bi l i ty wa s r e o r g a niz e d a nd r e e s ta b l is he d a sthe Advis o r y Gr o up o n R e l ia bi l i ty o f Ele c tronic Equipment within the Department ofDe fe ns e . Al tho ugh i t i s s o m e wha t p r e m a tu r eto l i s t the a c c o mp l is hme nts o f th is g r o up, a ndits working panels , i t can be s tated definitelytha t i t ha s s t imu la te d ma ny indus t r ia l a s s o c ia t io ns a nd individua l c o mpa nie s to unde r ta ke inc r e a s ing e ffo r t to s o lve the r e l ia bi l i typ r o g r a m s w i th p a r t i c u l a r e m p h a s i s o n g u id edm i s s i l e s .

So much fo r the o r ga niz a t io na l ba c kgr o undwithin the Department of Defense and thewo r k i t ha s p r o mo te d in indus t r y . W e c o nsid er the tas ks which have been und ertak enby th is g r o up.

4. RELIABILITY GROUP TASKS

T h e f i r s t c o m p l e t e p r o g r a m f o r t h e g r o u phas been fully developed and approved by theAss is tant Se cr etar y of Defense. Wo rk is nowc ur r e nt ly under wa y in a numbe r o f ta s k g r o u ps

e nume r a te d be lo w:

(1) Develop minim um quanti tat ivef igur e s fo r r e l ia bi l i ty o f the va r io us type so f mi l i ta r y e le c t r o nic e quipme nt . The ba s isupo n which the f igur e s a r e de te r min e d s h a l l

i n cl u de f a c t o r s of o p e r a t i o n a l r e q u i r e m e n t s ,ma inte na nc e , c o mple xi ty , a nd o the r s ignif i c a n t f a c t o r s .

(2) De ve lo p ba s ic r e qui r e m e nts fo rte s ts to be a c c o mpl is he d o n de ve lo pme ntmodels which will prove that the design isc a pa ble o f me e t ing minimum a c c e pta bi l i tyr e l i a b i l i t y .

(3) De velo p ba s ic r e qui r e me nts fo rte s t s to be a c c o mpl is he d o n p r o duc t io n m o de lswhich will prove that the equipment will meetthe minimum f igur e fo r r e l ia bi l i ty .

(4) Inve s t iga te a nd r e c o mme nd me tho ds o f s pe c i fy ing de ve lo pme nt p r o c e dur e s toins ur e that equipm ent de sign s will have ther e l i a b i l i t y r e q u i r e d . S o m e f a c t o r s a r e

(a) r e l ia bi l i ty p r e d ic t io n , (b) c o mpo ne nt s e le c t io n r e la te d to s pe c i f ic c i r c u i t a nd e nvi r o nme nt r e qu i r e m e n ts , (c ) a de qua te s igna l le ve l ,(d) effects of mec han ical shock, vibr atio n andt e m p e r a t u r e .

(5) E s ta bl is h c r i te r ia a nd me tho dsfor specifying the re l iab il i ty of com ponentp a r t s a nd t u b e s in t e r m s o f f a i l u r e r a t e .

(6) Study pr e s e nt p r o c ur e me nt a ndc o nt r a c t ing p r a c t ic e s a nd r e gu la t io ns tod e t e r m i n e t h e i r c o m p a t i b l i t y w i t h r e l i a b i l i t y .

(7) Inve s t iga te p r e s e nt p r a c t i c e s o fpa c ka ging fo r s h ipme nt a nd r e c o mm e ndimpr o ve me nts whic h wi l l e nha nc e r e l ia bi l i ty .

(8) Investigate the effects of s to r ag ea nd r e c o m m e n d i m p r o v e m e n t s .

(9) R e vie w pr e s e nt me tho ds a nd pr o c e du r e s to a s s u r e tha t the r e l ia bi l i ty o f e quipment in service is kept up to the inherentde s ign le ve l . Fa c to r s included a r e : (a)m a i n t e n a n c e b as e d o n p e r f o r m a n c e m e a s u r e

me nts , (b) ma r gina l te s t ing , a nd (c ) pe r s o nne lt r a i n i n g .

The in te r e s t e vide nce d in th is p r o gr a mthroug hou t industr y has been ver y gr atifyingand helpful. The so lution of the pr o ble m s

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involved wil l be d i ff icu lt an d wi l l r eq u i r e co n t in u ed c l o s e c o o p e r a t i o n b e tw e e n t h e m i l i t a r yand the e l e c t r o n i c a g e n c i e s .

Having given a p i c t u r e of the D e p a r t m e n t

o f D e f e n s e o r g a n i z a t i o n a l b a c k g r o u n d forp r o g r a m s in r e l i a b i l i t y g e n e r a l l y , a t t e n t i o ni s now d i r e c t e d to m o r e p r a c t i c a l a s p e c t s of

t h e r e l i a b i l i ty p r o g r a m .

3 . GUIDED MISSILES VS, AIRCRAF T

It has b een s t a t ed t hat g u id ed m i ss i l e sa r e m e r e l y a i r c r a f t w i t h o u t a h u m a n p i l o t .T h i s l e a d s to the e r r o n e o u s a s s u m p t i o n t h a tsince human l i fe is not involved, the m i s s i l eneed not be as r e l i a b l e as a i r c r a f t . J u s t the

o p p o s i t e is t r u e b e c a u s e the p i l o t in ana i r p l a n e can adjust and s u b s t i t u t e for m a l functioning of c o m p o n e n t s , w h e r e a s in a

g u i d e d m i s s i l e all c o m p o n e n t s of the guidancean d con t ro l sy s t em mu st fu n ct i on wi t h ou tbenefi t of h u m a n a s s i s t a n c e . T h e y m u s tt h e r e f o r e be c o m p l e t e l y r e l i a b l e .

T h e r e are o t h e r f a c t o r s w h i c h add

e m p h a s i s to t h i s c o n c l u s io n . M o s t of the

c o m p o n e n t s of g u i d an ce and c o n t r o l are in

s e r i e s . F a i l u r e of any one single unit can

c a u s e the m i s s i l e t o m i s s t h e t a r g e t an d u n d e rt h e s e c o n d i t i o n s it b e c o m e s i m p o s s i b l e to

r e c o v e r , e x a m i n e , or r e u s e the m i s s i l e . It

i s o b v i o u s t h e r e f o r e t h a t c o m p o n e n t s in m i s s i l e s m u s t be m u c h m o r e r e l i a b l e t h a n t h o s ei n a i r c r a f t .

7 . V-2 ROCKE T S

With th is defini t ion in mind , let us e x a m i n et h e r e c o r d of a w e l l - k n o w n G e r m a n m i s s i l e ,th e V-2 r o c k e t . (See T a b l e 2.) The d at a

p r e s e n t e d are b a s e d on t e s t s m a d e in theUn i t ed S t a t es . In t h e s e t e s t s , a t o t a l of 68,

V - 2 r o c k e t s w e r e l a un c h ed a n d of t h i s n u m b e r3 p e r c e n t w e r e p a y l o a d f a i l u r e s , 21 p e r c e n tp r o p u l s i o n f a i l u r e s , and 29 p ercen t g u i d an cef a i l u r e s . It is i n t e r e s t i n g to n ot e the l a r g en u m b e r of f a i l u r e s a t t r i b u t e d to the g u i d an ces y s t e m . Of the t o t a l n u mb er l au n ch ed , 47

p e r c e n t w o r k ed c o r r e c t l y and 67 p e r c e n tw e r e u s a b l e .

In view of the h i s t o r y of the V-2 and the

fact that some 3000 or m o r e m i s s i l e s hadbeen used by the G e r m a n s in o p e r a t i o n sag ai n s t E n g l an d and B r u s s e l s , one mi g h tcon cl u d e t h at the t e s t s d e s c r i b e d h e r e are

n o t i m p r e s s i v e . On the o t h e r h a n d , we m u s ta s s u m e t h a t the V-2 mi ss i l es mi g h t h av e b eenaffect ed ad v ersel y by shipping, handling, and

g e n e r a l age at t h e t i m e of l a u n c h i n g . I n f o r m a t i on av ai l ab l e , h owev er , i n d i cat es t h at t h i sr e c o r d of r e l i a b i l i t y is not a p p r e c i a b l y dif

fe ren t f rom t h at in G e r m a n y .

8. COMPONENTS AND COMPLEXITYT h e o v e r a l l r e l i a b i l i t y of a g u id e d m i s s i l e

o r e l e c t r o n i c s y s t e m w i t h p a r t i c u l a r ref

e r e n c e to g u i d an ce is dependent upon the

6. DEFINITION OF RE L IABIL IT Y

But what is r e l i a b i l i t y ? The E l e c t r o n i cR e l i a b i l i t y C o m m i t t e e of the R a d i o E l e c t r o n i c T e l e v i s i o n M a n u f a c t u r i n g A s s o c i a t i o ndefined it r e c e n t l y as f o l l o w s :

" R e l i a b i l i t y is the p r o b a b i l i t y of a d e v i c ep e r f o r m i n g its p u r p o s e a d e q u a t e l y for the

p e r i o d of t i me i n t en d ed u n d er the o p e r a t i n gc o n d i t i o n s e n c o u n t e r e d . "

T a b l e 2. U. S. T e s t s of V-2 R o c k e t s

T ot al n u mb er l au n ch ed 68

P a y l o a d f a i l u r e s 3%

P r o p u l s i o n f a i l u r e s 21%

Gu i d an ce fa i l u res 29%

W o r k e d c o r r e c t l y 47%

Usab l e 67%

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va r io us c o m po ne nts a nd uni ts whic h ma ke upt h e c o m p l e t e s y s t e m . In p a r t i c u l a r , t h ec o m ple xity o f a mis s i le we ighs ve r y he a vi lyin de te r m ining r e l ia bi l i ty . C o mple xity ge ne r a l ly ma y be me a s ur e d by the numbe r o f

m iss ile co mpo nents which by individualfa i lu r e c o uld c a us e the fa i lu r e o f the c o mple teguida nc e s ys te m.

The s e c o mpo ne nts ma y be c la s s i f ie d ge ne r a l l y a s " s e r i e s " c o m p o n e nt s a s d i s t i n guished from those components which if theyfailed would not nece ssa r i ly c ause the failur eo f the c o m ple te guida nc e s ys te m . The s ela t te r c o mpo ne nts a r e c la s s i f ie d a s" p a r a l l e l " c o m p o n e n t s . T h e f a i lu r e o f as ingle pa r a l le l c o mpo ne nt us ua l ly ha s l i t t le

influence on the o ver all re l ia bil i t y. It is forth is r e a s o n tha t we mus t c o nc e nt r a te o ura t te n t io n p r i m a r i l y o n t he s e r i e s ty p e c o m po ne nts s inc e the y a r e mo s t c r i t ic a l in the i rinfluence on the ove ra ll r e l iab il i ty of theguida nc e s ys te m.

9. EXAMPLES

U n l i k e m o r e c o n v e n t i o n a l s y s t e m s , r e l i a bi l i ty o f the mis s i le guida nc e s ys te m de pe ndsno t upo n the a ve r a ge but r a the r the p r o duc t

of the rel iabil i ty of each of the components .I t i s the r e fo r e o bvio us tha t the o ve r a l l r e l ia bi l i ty o f the mis s i le de c r e a s e s r a pidly wi tha n inc r e a s e in the numb e r o f c o mpo ne nts .

Fo r e xa mple , if a m is s i l e c o nta ins 100components each having a rel iabil i ty of 99pe r c e nt , the r e s u l t ing r e l ia bi l i ty o f the s ys te mwould be only 36 per cen t. If a m is si le co ntains 1000 co mp o nents each having a r el ia bi li ty of 99 pe r c e n t , the r e s u l t ing o v e r a l lrel iabil i ty of the system would be only .02

p e r c e n t .

The fo r m ula upo n which the s e o bs e r va t io nsa r e ba s e d is a s fo l lo ws :

Ov era ll Reliabil i ty » R, X R 2 XR, R ,

R e fe r e nc e to th is fo r mula s ho ws tha t ifwe s ho uld ha ve 400 c o mpo ne nts in a mis s i le ,a nd th is i s no t unus ua l , a nd a r e qui r e d r e l ia bil i ty of 80 percent for the system, we couldto le r a te o n the a ve r a ge no t mo r e tha n o ne

fa i lu r e in 1800 c o mpo ne nts .

While the se ar e s implified e xam ples , theyd o i l l u s t r a t e t h e p a r a m e t e r s a n d e n o r m o u spr o ble m s m is s i le e ngine e r s ha ve in de s igningr e l ia b le m is s i le guida nc e a nd c o nt r o ls y s t e m s .

In a ddi t io n to the s e c o ns ide r a t io ns muc hdata have been accumulated based on the testa nd o pe r a t io n o f mi l i ta r y e quipme nt in thef ie ld . Th is info r m a t io n i s ve r y pe r t ine nt tothe s ubje c t o f r e l ia bi l i ty . Fo r e xa mp le , c o m

prehensive s tudies have been made and dataa na lyz e d to de te r mine the impa c t a nd c o s t o fe quipme nts tha t do no t o pe r a te . The Ai r F o r c er e p o r t s t h at t h e c o s t of m a i n t a in i n g e l e c t r o n i cequipmen t each ye ar is about twice theo r ig ina l c o s t o f the e quipme nt . F o r the e nt i r el ife of the equipment, i t is es t imated that thema inte na nc e c o s t i s te n t ime s the o r ig ina lpu rc ha se co st . (See Fig . 1.)

In t h e A i r F o r c e o n e - t h i r d of t h e o p e r a t i n ge xpe ns e i s fo r m a inte na nc e and o ne - th i r d of

the i r pe r s o nne l i s e nga ge d in th is a c t iv i ty ,even though a large amount of maintenancei s d o n e b y i n d u s t r i a l c o n t r a c t o r s .

W hile the s e o bs e r va t io ns ha ve r e fe r e nc eto e le c t r o nic e quipme nts , ge ne r a l ly , the y a r ee qua l ly pe r t ine nt to mis s i le guida nc e a ndc o nt r o l , which de pe nds s o muc h upo n e l e c t r o n i c c o m p o n e n t s a n d s y s t e m s . A p r o m i n e n ts ta ff ge ne r a l in o ne o f the mi l i ta r y de pa r t me nts has s tated that the availabil i ty of m is s i l e s f o r a m i l i t a r y o p e r a t i o n m a y b e d e t e r

mine d mo r e by the a va i la bi l i ty of ma inte na nc efa c i l i t ie s a nd s ki l l s tha n by the me r e numbe ro f m is s i le s . He r e , the n, i s a n a r e a whe r elarge dividends in the form of improvedwe a po ns c a n be r e a l iz e d by inc o r po r a t ingr e l i a b l e c o m p o n e n t s a n d d e s i g n s .

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10. MANUFACTURING RELIABILITY 12. COST OF ATTAINING RELIABILITY

With this backg ro und of data let us lookfu r t h er i n t o t h e p rob l em of re l i ab i l i t y wi t hs p e c i a l r e f e r e n c e t o m a n u f a c t u r i n g t e c h n i q u es . Co mp o n en t s mu st b e man u fact u redp r o p e r l y a nd h a v e p r o v i s i o n f o r 1 00 , 2 0 0 ,o r ev en 3 00 p er cen t i n sp ect i o n . In v iew ofthe quanti tat ive data ci ted above as appliedt o s e r i e s c o m p o n e n t s , i t i s o b v i o u s t h a t as m a l l p e r c e n t d e f e c t i v e c o m p o n e n t s c a n n o tb e t o l e r a t e d . W e can n ot af fo rd t o h av e am i ss i l e co s t i n g $ 1 5 0 ,0 0 0 fai l b eca u se of t h efai l u re of a 5 o r 1 0 cen t co mp o n en t . T h ep r i m a r y c o n c e r n i s t h e r e f o r e in m a n u f a c t u r i n g p r o c e s s e s ; c o m p o n e n ts c a nn o t b e m a d et o o r e l i a b l e . A s p e c i a l s t a t i s t i c a l q u a li tyc o n t r o l s y s t e m m u s t b e i n s t it u t e d f o r t he

m a n u f a c t u r e o f c o m p o n e n t s t o b e e m p l o y e di n g u i d e d m i s s i l e s .

11. DESIGN RELIABILITY

O f e q u a l i m p o r t a n c e i s t h e m a t t e r o fd es i g n . A com p on en t may b e fab r i cat ede x a c t l y a c c o r d i n g t o t h e d e s i g n e r ' s s p e c i f i c a t ions but may fai l when subjected to the condit io ns of f l ight . In such a ca se i t is o bvio ust h at a com p on en t h as n ot b een p r o p er l yd e s i g n e d . D e s i g n m u s t t a k e i n to c o n s i d e r a

t i o n a l l o f t h e e n v i r o n m e n t a l p a r a m e t e r s .T h e r e i s , h o w e v e r , a p r o b l e m h e r e w h i c h w em u s t r e c o g n i z e . D e s ig n w e a k n e s s e s a r e f r e q u en t l y d et ermi n ed b y f l i g h t t es t s an d t h esed at a a re i n f req u en t l y av ai l ab l e b ecau se t h em i s s i l e i s n o t r e c o v e r a b l e . A s a r e s u l t , a nu n sat i s fac t o ry com p on en t may f in d i t s wayi n t o m a s s p r o d u c t i o n b e f o r e i t h a s r e a c h e dt h e req u i red l ev el o f re l i ab i l i t y .

Desi g n r e l i ab i l i t y i s t h er efo re a u n iq u e an dc r i t i c a l p r o b l e m in t h e c a s e o f m i s s i l e s . A s a

r e s u l t o f th i s s i tu a t i o n , i t i s n e c e s s a r y t o r e e x a m i n e t h e e n v i r o n m e n t a l c o n d i t i o n s a n dl i mi t i n g t es t v al u es . T h ese may b e care fu l l yd et ermi n ed b y sc i en t i f i c t es t s wi t h u n u su al l yh i gh safet y fact o rs ev ol v ed by n u m er o u s t es t ing of units up to and including fai lur e.

B a s e d o n m y r e m a r k s s o f a r it s h o u ld b eob v i ou s t h at mi ss i l e re l i ab i l i t y can n ot b ee v o l v ed m e r e l y b y t r i a l a nd e r r o r o v e r a

p er i o d of t i m e b u t mu st be ach i ev ed b y aw e l l - p la n n e d p r o g r a m . S uc h a p r o g r a m m a yc o s t a l o t o f m o n e y .

T h i s q u est i on was ex ami n ed by Dr . E r i chP i e r u s c h k a o f t h e R e d s t o n e A r s e n a l a n d t h ef o ll o w in g r e s u l t e d . T w o g u id e d m i s s i l e p r o g r a m s w e r e c o m p a r e d . In e a c h o f t h e s ep r o g r a m s , a t o t a l o f o n e h u n d r e d g o o d r e l i a b l em i s s i l e s w o u ld b e o b t a i ne d . P r o g r a m " A "was b ased on a p l an n ed ex p en d i t u re of on e

m i l l io n d o l l a r s , f o r r e l i a b i l i t y o u t o f a to t a lo f t w o h u n dr e d s i x ty - e i g h t m i ll i o n d o l l a r s .P r o g r a m " B " c o n t e m p l a te d a n e x p e n d it u r eof s i x mi l l i on d ol l a rs fo r re l i ab i l i t y ou t o f at o t a l o f on e h u n d red fo r t y -ei g h t mi l l i on . Asi l l u s tr a t e d in F ig . 2, P r o g r a m " B " w a s byf a r t h e m o r e p r e f e r a b l e a p p r o a c h t o t h ep r o b l e m .

R e c e n t l y , t h r e e A i r F o r c e j e t t r a i n e r sl an ded a t an Ai r F o rc e b ase in T ex as wi t hi na f ew m i n u t e s o f e a c h o t h e r . G e n e r a l H o l t n e r ,

p i l o t o f t h e f i rs t a i r c r af t l an ded wi th ou t diff i cu l ty . Ge n er al S ch r i e v er , p i l o t o f t h es e c o n d a i r p l a n e , ha d a n e l e c t r i c a l p o w e rf a i l u r e a nd ha d t o u s e h i s e m e r g e n c y s y s t e mt o g et h i s l an di n g g e ar d ow n . Ge n er al Dav i s ,p i l o t o f t h e t h i rd a i rp l an e, fou n d t h at h i sl an d in g g e ar i n d i cat o r g av e a faul ty r ead i n ga nd h e h a d t o r e c h e c k in o r d e r t o a s s u r esafe landing.

In c o m m e n t i n g u po n t h i s o p e r a t i o n ,G e n e r a l D a v i s s a i d , " H e r e w e r e t h r e e o f o u rm o s t m o d e r n a nd r e l i a b l e ty p e p l a n e s an dtw o o f t he t h r e e h ad e l e c t r i c a l s y s t e m t r o u b l e .If t h ey h ad b een m i s s i l es , t wo of t h em c e r tainly would have been in t ro uble and pro bablyf a i l e d c o m p l e t e l y . "

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Ge ne r a l Da vis c o mme nte d fur the r wi thr e g a r d t o t h e i m p o r t a n c e of m i s s i l e r e l i a bil i ty and the extensive and complicatedt e s t s n e c e s s a r y to i n s u r e r e l i a b i l i t y . In o n ec a s e whic h he de s c r ib e d, a c o nt r a c to r ha dr e p o r t e d o n t h e b a s i s o f l a b o r a t o r y c o m p o n e n tt e s t s , that the mi ss ile would be 60 per cen tr e l ia ble , whe r e a s a c tua l f l ight te s ts s ho we da r e l ia bi l i ty o f o nly 10 pe r c e nt . Ge n e r a lDavis explained that f ield and environmentalco ndit ions can affect the o per atio n to such ade gr e e tha t s y s te m s r e l ia bi l i ty c a n bed e c r e a s e d t h r e e o r f o u r t i m e s . H e s t r e s s e dthe high cost of missile development butindicated that good rel iabil i ty justified suche x p e n d i t u r e s .

13. CONCLUSIONS ANDRECOMMENDATIONS

So me o f the impo r ta nt a s pe c ts of r e l ia bi l i tyas applied to the guidance and co ntr o l sy ste m so f guide d m is s i le s have be e n o ut l ined he r e .The o r ga niz a t io na l pa t te r n o f the De pa r t ment of Defense has been descr ibe d br ieflya s we l l a s c e r ta in indus t r ia l g r o ups whic h

a r e c o nt r ibut ing to the s o lu t io n o f th is impo r t a n t p r o b l e m . In a d d it io n , c e r t a i n a r e a swh e r e c o nc e nt r a te d e ffo r t s ho uld y ie ld p r o gr e s s ive impr o ve me nts ha ve be e n ide nt i f ie d .

In c o n c l u s io n , t h r e e r e c o m m e n d a t i o n s a r es t r e s s e d : F i r s t , o v e r a l l r e l i ab i l it y d e c r e a s e sve r y r a pidly wi th the inc r e a s e in numbe r ofc o m p o n e n t s . W e m u s t t h e r e f o r e r e d u c ec o mple xi ty . Se co nd, to o bta in r e l ia bi l i ty wemu st find so me way to anticip ate and det er mine the nume r o us a dve r s e c o ndi t io ns whic hguide d mis s i le s e nc o unte r f r o m de ve lo pme ntt h r o u g h p r o d u c t i o n , t r a n s p o r t a t i o n , l a u n c h i n ga nd a c tua l f l ight . Th i r d , r e l ia bi l i ty o f c o mpo ne nts c a n be impr o ve d by a p la nne d pr o gr a m o f te s t ing to fa i lu r e wi th e m pha s is o n

c o n d i t i o n s o f m a x i m u m s t r e s s .While i t is believed that missiles must

be c o me mo r e r e l ia ble tha n we no w kno w ho wto make them, i t is fel t that concentrationo n s impl ic i ty o f de s ign wi l l p r o vide ma jo rimpr o ve m e nt . The De pa r tme nt o f De fe nsea nd indus t r y a r e wo r king to ge the r a s a te a mto s o lve the r e l ia bi l i ty p r o ble m a nd thesolutio n will re qu ir e the best effor t that eachc a n p r o v i d e .

5.

6.

REFERENCESDa vis , B r ig . Ge n, Le ighto n I . , 1955 Ame r ic a n Ins t i tu te o f Ele c t r ic a l Engine e r s Te c hnic a lC o nfe r e nc e o n Ai r c r a f t Ele c t r ic a l Appl ic a t io ns .

B r idge s , J , M . , Di r e c to r o f El e c t r o nic s , Off ic e As s is ta nt Se c r e ta r y o f De fe ns e , W a s hingto n ,D . C , "E le c t r o nic s E quipm e nt ," p . 6 , Sutto n Publ is hing C o . , New Yo r k , J a nua r y 1956.

C le me nt , Le w is M ., "R e l ia bi l i ty o f M il i ta r y Ele c t r o nic Sys te m s a nd Equip m e nts ," So c ie tyo f Auto mo t ive Engine e r s me e t ing, 21 Apr i l 1955.

Ha vi la nd, R . P . , "R e l ia b i l i ty in Guided M is s i le s ," J e t Pr o puls io n , Vo l . 25 , No . 7 ,p p. 321-325, 330, July 1955.

M a r t in , W il l ia m H. , De puty As s is ta nt Se c r e ta r y o f De fe ns e , Ame r ic a n Or dna nc e As s o c ia

t ion meeting. May 1955.Lu s s e r , R o be r t , "R e l ia b i l i ty o f Guided M is s i le s ," Offic e R e ds to ne Ar s e na l , Hunts vi l le ,Alabama, September 1954.

Pie r us c hk a , D r . Er ic h , "Opt im um Allo c a tio n o f Funds fo r R e l ia bi l i ty of Guide d M is s i le s ,"R e ds to ne Ar s e na l , Hunts vi l le , Ala ba ma , J a nua r y 1955.

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I N I T I A LCOST

AVERAGEMAINTENANCE COST

1 2 3 4 5

Y E A R S

F i g . 1 . I n it i al c o s t v s . a v e r a g e m a i n t e n a n c e c o s t .

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PROGRAM "A"268 MILL ION4 0 % RELIABIL ITY

150BAD MISSILES

100GOOD MISSILES

PROGRAM " B "148 MILLION8 5 % R E L I A B I L I T Y

ill

18BAD MISSILES

100GOOD MISSILES

I MILL ION" R E L I A B I L I T Y "

D O L L A R S

6 M ILL ION ,R E L I A B I L I T Y

D O L L A R S

F i g . 2 . C o m p a r i s o n b et w ee n p r o g r a m s " A " a nd " B " .

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LABORATORY VS, FLIGHT EVALUATION OF AIRBORNE GUIDANCE COMPONENTSDr . W . H. C lo he s s y*

SUMMARY

Th is pap er defines evaluation to include sys tem li m its , re l iab il i ty, and effects of no ns ta nda r d e nvi r o nme nt . I t de s c r ibe s the o r ga niz a t io n a nd fa c i l i t ie s o f the Te s t Divisio n a ndE l e c t r o - M e c h a n i c a l L a b o r a t o r i e s a t t he W h it e S a nd s P r o v i n g G r o u n d . It d i s c u s s e s f a c t o r sfavo ring labo ra to ry and fl ight evaluatio n and fact o rs in the m ar gina l ar ea . It co ncludes witha d is c us s io n o f c ur r e nt t r e nd s in th is f ield .

SOMMA1RE

C e t te no te de fini t r e v a lua t io n ne c e s s a i r e po ur inc lui r e le s l imi te s du s ys te m e , las e c ur i te de fo nc t io nne me nt e t le s e ffe ts d ' un mi l ie u no n-no r m a l . El le de c r i t 1 ' o r ga nis a tio ne t l e s p o s s i b i l i t e s d e s L a b o r a t o i r e s d 'E l e c t r o - m e c a n i q u e d u d e p a r t e m e n t d ' e s s a i s du t e r r a i n

d ' e pr e uve de W hite Sa nds . El le t r a i t e de s fa c te ur s fa vo r is a nt r e v a lua t io n e n La b o r a to i r ee t e n vo l e t de s fa c te ur s m a r gina ux. El le s e te r mi ne pa r une d is c us s io n s ur le s e vo lut io nsc o u r a n t e s d a n s c e d o m a i n e .

T h e p r o b l e m d i s c u s s e d h e r e i s p e r t i n en tno t o nly to e va lua t io n o f a i r bo r ne guida nc ec o mpo ne nts but a ls o to the mo r e ge ne r a lque s t io n o f guide d mis s i le we a po ns s ys te msa s a who le . Th e na t ur e o f the p r o ble m ispart ly the apparent conflict between technic a l a nd e c o no mic c o ns ide r a t io ns in s uc h

e va lua tio n a nd pa r t ly ho w to ma ximiz e theinfor matio n ava ilab le at a f ixed expen ditureo f me n, mo ne y, m a t e r ia ls , and t ime . In ave r y b r o a d s e ns e the p r o ble m impl ie d in thet i t le i s no t "whic h o f the two is be t te r ? " butr a the r "wha t c o m bina t io n o f the two mo de s o fe va luat io n m a xim iz e s info r ma t io n fo r a f ixede c o n o m i c s i t u a t i o n ? "

The fi rs t s tep in evaluation is to definethe pur p o s e s o r o bje c t ive s o f the wo r k . Thes e c o nd is to de te r m ine the na tur e o f the

syste m to be test ed . Th is then allow s theselection of the specific things to be learnedo r que s t io ns to be a ns w e r e d , a nd the va r ia ble s

o r e n v i r o n m e n t s to b e i m p o s e d . F r o m t h e s ec a n be de te r m ine d te s t s to be pe r fo r m e d,m e a s u r e m e nt s to be ta ke n, a nd c a lc ula t io nsto be mad e. Executio n of these tes ts andme a s ur e me nts le a ds to da ta whic h mus t beanalyzed to o btain the an sw er s to theque stio ns. The above is idealized and

a s s u m e s n o v a r i a t i o n s in d e s ig n o r c h a r a c te r o f the s ys te m unde r te s t dur ing thete s t pe r io d . In mo s t c a s e s th is a s s umpt io nis not valid and the whole picture is cont inuo us ly unde r r e v is io n . The ba s ic s te psho we ve r r e ma in e s s e nt ia l ly unc ha nge d. The yme r e ly ha ve to be r e e va lua te d f r e que nt ly .

If, a t the point in the pr o ced ur e wh eret h e " t e s t s t o be p e r f o r m e d " a nd t he" m e a s u r e m e n t s t o be ta k e n " h a v e b ee nde te r mine d, we a s k que s t io ns a bo ut a c c ur a c y,c o mple te ne s s o f c o ve r a ge , a nd c o nf ide nc eand if we invite the s tat is t icians to prepares ui ta ble numbe r s o f t r ia ls a nd r e pe t i t io ns

*U. S. Army Ordnance Proving Ground, White Sands, New Mexico.

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f o r o u r e x p e r i m e n t s , n o o n e w i ll b e s u r p r i s e dat t h e eco n o mi c i n feas i b il i ty and t h e com p l et er i d i c u l o u s n e s s o f t h e s t a t i s t i c i a n ' s c o n t r i b u t i o n s . No co u n t ry 's n at i on al b u d get cou l dstand the impact of such a blow.

W e t h e r e f o r e i m p o s e ad d it io n a l r e s t r i c t io n s , o r r a t h e r , a l t e r n a t e r e s t r i c t i o n s . W ed r o p t h e a c c u r a c y an d c o n f id e n ce r e q u i r e m e n t s a nd u s e i n s t e ad n u m b e r s o f m i s s i l e so r c o m p o n e n t s a c t u a l l y a v a il a b l e a nd q u a n t i t i e s o f m e n , m o n e y , m a t e r i a l s , a n d t i m e a scan b e sp a re d . It i s h er e t h at t h e q u est i onof su b st i t u t i on of l ab ora t o ry fo r f l i g h t occu rsan d i t i s h e re t h at we re al l y d i g in t o t r yt o p l an an ev al u ati on p r o g r am wh i ch wi l lgive information in the quanti ty and of theq u a l i t y m e e t i n g m i n i m u m r e q u i r e m e n t s .

E c o n o m i c s , in t h e b r o a d s e n s e , d i c t a t e s o u rap p roach t o t h e p rob l em at h an d .

A m i s s i l e f l ig h t t es t i s a re l a t i v el y u n co n t r o l l e d e x p e r i m e n t . T h e i n -f li g h t e n v i r o n m e n t s a r e p a r t i c u l a r l y d i f f i c u l t t o d e t e r m i n ean d i t i s h ard t o a ss e ss t h ei r ef fect s . T h en u m b e r o f i n t e r n a l m e a s u r e m e n t s w h i c h c a nbe mad e i s l im ited by the type and num bero f t e l e m e t r y u n i t s w h i c h c a n b e c a r r i e d .Al so , each co mp o n en t of t h e t e l e m et r y sy s t emi s e x p o s e d t o t h e s a m e e n v i r o n m e n t a s t h em i s s i l e c o m p o n e n t s a n d t h e i r b e h a v i o r i s

h en ce mo d i fi ed b y t h at en v i ro n me n t i n mu cht h e s am e way a s th at o f t h e m i ss i l e co m p o n en t s . In ad d i t io n , t h er e i s a l way s t h ep oss i b i l i t y t h at t h e i n s t a l l a t i on an d op era t i onof t h e t e l em et r y wi l l af fect t h e p er fo rm an ceo f t h e m i s s i l e a d v e r s e l y . E x t e r n a l m e a s u r e m en t s (e l ect r o n i c an d o p t i cal t rac k i n g , e t c . )a r e d i ff icu lt t o m ak e re l i ab l y , an d v er yex p en si v e an d t h ey l ead t o l a rg e q u an t i t i esof d at a to b e p r o ces sed an d red u ced . T h eyd o , h o w e v e r , g i v e v e r y p r e c i s e v a l u e s o ft r a j e c t o r i e s , a t t i tu d e s , a nd m i s s d i s t a n c e s .

M e a s u r e m e n t s s u c h a s t h e s e m a y b e o fg r ea t u se in t h e ev al u at io n of sp eci f i c co m p on en t s su ch as t h e a i rb orn e g u i d an ce u n i t s .T h i s i s t r u e b e c a u s e t h e p e r f o r m a n c e o f a

g u i d an ce u n i t (o t h er t h i n g s b ei n g n ormal )d e t e r m i n e s t r a j e c t o r y , a t t it u d e , a n d m i s sd i s t a n ce . T h e n u mb er of v al u es of en v i r o n m e n t a l a n d p e r f o r m a n c e v a r i a b l e s fo r w h i chf l i g h t t es t s can b e mad e i s g reat l y l i mi t ed

b y t h e n u m b er of m i ss i l e s t o b e f lown . E v enw er e i t po ssib le to f ly sufficiently manym i s s i l e s t o c o v e r a ll n e c e s s a r y v a l u e s o f s u c hv a r i a b l e s t h e t ask of so r t i n g ou t t h e ef fectso f i nd iv id u a l o n e s o n s y s t e m p e r f o r m a n c ewou l d p r ese n t t h e an al y s t wi th i mm en se q u an t i t i e s of w o rk . Al l o f t h i s co s t s l a rg e q u an t i t i e s of mo n ey , men , an d t i m e. Bu t t h eg r ea t es t co s t o f a f li gh t i s t h e m i ss i l e i t se l fwh i ch , if r eco v er ed a t a l l , se r v e s on l y as av e r y p o o r s o u r c e o f i n fo r m a t i o n a bo u t t heflight itself.

T h e b i g g er t h e mi ss i l es t h e fewer wi l l b ea v a i l a b l e fo r t e s t . T h i s i s a s i m p l e e c o n o m i cfact . It is cle ar then that the big, lo ng- ra ngem i s s i l e s w ith t h e i r i m m e n s e r e q u i r e m e n t si n t e r m s o f r a n g e i n s t r u m e n t a t i o n , t h e i r v e r yp r e c i s e c o m p o n e n t s w h i c h s h o u l d b e m o n i t o re d i n f l ig h t , an d t h e i r g re at co s t mak e f li g htt es t i n g i n l a rg e n u mb ers p roh i b i t i v e an du n sa t i s fact o ry . E ach f li g ht mu st y i e ld g r ea tqua nti t ies of info rm atio n. But with a l imitedn u m b er of f l i g h ts t h e re a r e bou n d t o b e v as ta r ea s of v ar i a t i o n of a l l t h e co n d i t io n s an d

p aramet ers wh i ch sh ou l d b e i n v est i g at ed , fo ra comp l et e ev al u at i on , wh i ch can n ot b e ev entouc hed upon. In addit ion to this , is theq u est i on of i n - f l i g h t re l i ab i l i t y wh i ch can n otbe answ er ed with any gr ea t co nfidence by asm al l n u m b er of f l i g h t s . If t h e l ab o ra t o rym o d e o f t e s t i n g m i s s i l e s c a n n o t b e s u c c e s s ful ly explo i ted then we a r e do o med to" s h o o t i n g i n th e d a r k " o r t o e c o n o m i c r u i n .

L et u s ex ami n e t h e p oss i b i l i t i es fo r l ab or a t o r y t e s t i n g . I nc lu de d u n d e r " l a b o r a t o r yt es t i n g " i n t h i s p ap er a r e com p on en t b en ch

t e s t s , comp on en t f i e l d t es t s , s i mu l at i on ,c o m p l e t e m i s s i l e l a b o r a t o r y t e s t s a n d c o m p l e t e s y s t e m t e s t s u n d e r l a b o r a t o r y ( c o n t r o l l e d ) c o n d i t io n s . In s h o r t , w e m e a n e v e r y thing excep ting only act ual f l ights. If, dur ing

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a few fl ights , as much information aboutin-f l ight e nvi r o nm e nts a s i s po s s ib le i s o btained, then a component (say the guidancesection) of the missile can be subjected atle a s t a ppr o xima te ly , to the s e e nvi r o nme nts

ma ny t ime s in la bo r a to r y te s ts . Ac tualstati s t ic ally valid data m ay be o btained andthe compo nent in questio n, factor y re buil tupon failure, to serve again and again in tests .The functional inputs can be s im ulated bya na lo g c o mpute r s a nd the o utputs r e c o r de do r c o m p a r e d d i r e c t l y a g a i n s t r e c o r d e doutputs of units in actual f l ight.

The e ffec ts o f num e r o us c o ndi t io ns no texpe rien ced in fl ights but expected underbatt le use can be s imulated and added to the

no r m a l input s ign a ls . The r e pr o duc t io n o fsuch condit ions in fl ight tests might posea lmo s t ins ur mo unta ble o pe r a t io na l d i f f i c ul t ie s whi le the i r s im ula t io n in the la bo r a t o r y might be fa i r ly d i r e c t ly and s implyachieved . Exa mp les of such would be ba tt le f i e l d i n t e r f e r e n c e , c o u n t e r m e a s u r e s , o rs p e c i a l t e r r a i n o r w e a t h e r f e a t u r e s n o tpr e s e nt a t the p r o ving gr o unds . R e pr o duc ibleinputs can be used to detect var iat io ns inoutput due to internal var iabil i ty of theguida nc e s e c t io n o r to va r ying e nvi r o nme nta lc o n di t io ns . Als o , the num be r o f b r e a kdo w nso r ma lfunct io ns pe r unit o pe r a t ing t ime ca nbe de te r mine d in s ta t i s t ic a l ly s ignif ic a ntqua n t i t ie s . Te s ts to fa i lu r e c an be r un toa s s e s s f a il u r e p o i n t s a nd " s a f e t y m a r g i n s "o f e i t h e r m e c h a n i c a l o r e l e c t r i c a l p a r t s .E n v i r o n m e n t a l an d p e r f o r m a n c e v a r i a b l e sc a n be us e d o ve r the i r e nt i r e r a n ge s wi th a nydes ire d density of test po ints and at relat ivel ys m a l l c o s t .

The to tal cost of equipment and personnelfo r a l l suc h te s t s fo r a pe r io d c o ve r ing thee nt i r e e va lua t io n o f a we a po ns s ys te m ma ybe equal only to from 2 to 100 t imes the costo f o ne o f the mis s i le s invo lve d o r pe r ha pso ne o r two s e ts o f g r o und e quipme nt . Thes m a l l e r f i g u r e s a p p l y t o t h e l a r g e r m i s s i l e s .

o f c o ur s e . Th is , adde d to the fa ct tha t info r mation may be obtained which s imply is nota va i la ble in f l ights , ma ke s la bo r a to r y te s t inga ne c e s s a r y s upple me nt to f light te s t ing .

W hile th is p ic tur e o f la bo r a to r y e va lua t io nm ake s i t appar ent that f l ight te st s can bema de in l imi te d numbe r s a nd the r e qui r e dinfo r m a t io n s t i l l o b ta ine d, g r e a t c a r e s ho uldbe e xe r c is e d in the e xt r a po la t io n o f la bo r a t o r y r e s u l t s to f ie ld c o n d i t io n s . E n g i n e e r sa nd t e c h n ic i a n s in a l a b o r a t o r y c h a r a c t e r i s t ically t reat the components under test witha healthy respect while t roops in the fieldma ke r e pa i r s a nd s e t t ings a c c o r ding tos t a n d a r d r a p i d f i r e p r o c e d u r e s . R a nd o ms a m ple s wi l l no t ge ne r a l ly be us e d in la bo

r a to r y te s ts a nd individual i te m s ma y ber e pa i r e d a nd r e us e d f r e que nt ly . He nc e , thesta tis t ica l v alidity of data from such test sma y be que s t io na ble .

N o a m o u n t o f l a b o r a t o r y t e s t in g o r s i m u l a t io n c a n y ie ld r e a l ly r e l ia ble m is s d is ta nc ed a t a f o r a n t i a i r c r a f t m i s s i l e s . A l so , t h epr o ble m o f r e pr o duc ing a c c ur a te ly , in thela bo r a to r y , in-f l ight e nvi r o nme nts , s uc h a sshock and vibratio n, is a ver y difficult one.T h i s , added to the difficulties of in-flighte n v i r o n m e n t a l m e a s u r e m e n t s , p a r t i c u l a r l y o f

v i b r a t i o n s w i th a p p r o x i m a t e n o i s e s p e c t r a ,ma ke s ne c e s s a r y c o nf i r m a to r y f lights tocheck out predicted component and missiler e l i a b i l i t i e s .

Fina l ly , th e r e a r e tho s e who s e p r inc iple sdo no t pe r m it the m to be l ie ve in the r e s ul tsof anything but full scale field or flight testso f we a po ns . Fo r the m, f l ights a ga ins t r e a lta r ge ts a r e the o nly s ui ta ble me a ns o fevaluatio n. So we see that f l ights will neverbe done away with and the m o re fl ights thebe t te r i s a fa i r ly go o d r u le o f thumb.

I should l ike to de scr ib e b r iefly how theUnited Sta te s Ar my Or dna nc e C o r ps o p e r a te sat White Sands Pr o ving Gro und in the eva luation of guided missile weapons systems and to

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c o v e r s p e c i f ic a l l y th e s o r t o f l a b o r a t o r y a n df li gh t t e s t s m a d e t h e r e o n a i r b o r n e g u i d a n c ec o m p o n e n t s .

The ce nte r of act ivi ty in what is cal le d the

en g i n eer i n g ev al u at i on of g u i d ed mi ss i l eweap on s sy s t ems a t W h i t e S an d s P rov i n gGro u n d i s a t e s t p l an n in g an d co n t ro l staff.

T h i s c o n s i s t s o f p r o j e c t e n g i n e e r s , t e c h n i c a ls p e c i a l i s t s , a n d s y s t e m s a n a l y s i s p e o p l e .T h ey c o n t ro l t h e p l an n i n g an d ex ecu t i on of a l lt e s t s an d i n su re an i n t eg r a t ed f l i g h t -l a b o r a t o r y p r o g r a m . T h e o r g a n i z a ti o nex ecu t i n g t h e fl ig h t t es t s , t h e S y st em s T estDi v i s i on (S T D), co n si s t s p r i n ci p al l y of m i l i t a r y s i n c e i t i s o u r a i m t o t e s t m i s s i l e s u n d e ra s n e a r l y f i e ld tr o o p c o n d i t i o n s a s p o s s i b l e .T h i s d i v is i o n i s o r g a n i z e d by p r o j e c t s , byt e s t t e a m s o f f ir i n g c r e w s , a s s e m b l y c r e w sand a handful of civi l ian technicians forcon t i n u i t y .

T h e b as i c ass i g n men t of S T D i s t h es t o c k p i l e - t o - t a r g e t s e q u en c e o f o p e r a t i o n s .O p e r a t io n a l p r o c e d u r e s f o r c he c ko u t ,a s s e m b l y , f i r i n g , a n d m a i n t e n a n c e a r es t r i c t l y fo l l owed . W h en t h e m i ss i l e I sl au n ch ed i t b eco me s t h e p r o b l em of t h eW h i te Sa n ds P r o v i n g G r o u n d r a n g e p e r s o n n el t o co l l ect an d r ed u c e a l l t ra ck i n g d at a an d

t o t ran smi t su ch d at a t o t h e en g i n eer i n g t es ta g e n c y .

T h e W h i te S an d s P r o v i n g Gro u n d m i ss i l el a b o r a t o r i e s , k n o w n a s t h e E l e c t r o -M e c h a n ic a l L a b o r a t o r i e s , a r e f o u r in nu m b e r ,co n si s t i n g of a Gu id an ce an d Co n t ro l L a b o r a t o r y , p r i m a r i l y a n e l e c tr o n i c s l a b o r a t o r y ;a F l i g h t S i m u l a t i o n L a b o r a t o r y , c o m p o s e dl a r g e l y o f p h y s i c i s t s a n d m a t h e m a t i c i a n su si n g an al og an d d i g i t a l comp u t ers ; a Rock etV e h i c l e a n d W a r h e a d L a b o r a t o r y ; a n d a n

E n v i r o n m e n t a l a n d I n s t r u m e n t a t i o n L a b o r a t o r y . P l a n s f r o m t h e s ta ff a r e e x e c u te db y o n e o r b o t h o f t h e o p e r a t i n g d iv i s i o n sa nd fe e d er r e p o r t s a r e p r e p a r e d t h e r e . T h e s ear e co n so l i d at ed an d an al y zed b y t h e cen t ra l

s taff so as to i n su re max i mu m effect i v en essan d i n t eg r a t i o n . In t h i s way a coo rd i n at edt es t i s ach i ev ed , we l l -b al an ce d b et ween t h el a b o r a t o r y a n d f l i g h t a s p e c t s .

Gi v en a sp eci f i c weap o n s sy s t e m t oev al u at e , W h i t e S an d s P rov i n g Grou n d t ech n i cal p erson n el b eg i n b y ex ami n i n g t h e mi l i t a r y c h a r a c t e r i s t i c s d e s i r e d by o u r f ie ldf o r c e s . In t h e c a s e o f t h e a i r b o r n e g u i da n c es y s t e m , d e s i r e d a c c u r a c y , r e l i a b i l i t y , a ndv u l ne r a b i li t y t o c o u n t e r m e a s u r e s o r i n t e r fe r en ce a r e t h e q u al i t i es wh ich can b e d i r ect l yd e t e r m i n e d f r o m t h e m i l i t a r y s p e c i f i c a t io n s .H e n c e , o u r t e s t p r o g r a m c o n c e n t r a t e s he a v il yi n t h e s e a r e a s .

A t h e o r e t i c a l a n a l y s i s o f t h e s y s t e m i sm a d e a n d a l l o f t h e s u b a s s e m b l i e s a n d p a r t sa r e s u b j e c te d t o s c r u t i n y a s t o t h e i r r a n g e so f o p e r a t i o n , l i f e ti m e s e x p e c t e d , a nd e n v i r o n m e n t s a n t i c ip a t e d . In s o m e c a s e s t h i sa n a l y s i s p o i n t s d i r e c t l y t o w e a k s p o t s o rp r e d i c t e d o p e r a t i o n a l l i m i t a t i o n s . O n e o rm o r e g u i da n c e p a c k a g e s a r e p u t i n to o u re l e c t r o n i c s l a b o r a t o r i e s f o r f a m i l i a r i z a t i o na n d o p e r a t i o n a l t e s t s u n d e r a m b i e n t c o n d i t i o n s . At t h e sa m e t i m e, f l ig h t s a r e b eg u nw i t h a s m u c h i n s t r u m e n t a t i o n a s i s p o s s i b l e .In s t ru m en t at i o n fo r sh ock an d v i b ra t i o n an d

t e m p e r a t u r e e n v i r o n m e n t s a s w e l l a s f u n c t io n a l p a r a m e t e r s ( s e r v o l o o p g a in , e t c .)r e s u l t s in r e c o r d s o n a i r b o r n e t ap e r e c o r d e r so r in t e l e m e t e r e d d a t a . T h e v i b r a t io n d a t aa r e d i g i t i zed an d p ut on IBM ca r d s fo rp r o c e s s i n g i n d i gi ta l c o m p u t e r s t o d e t e r m i n et h e m a t h e m a t i c a l n a t u r e o f t h e v i b r a t i o n s ,1. e . f o r a gi v en r e c o r d t h e c o r r e l a t i o n f u n ct i o n s a n d a m p l i t u d e d i s t r i b u t i o n c u r v e s , e t c .

At t h i s s t ag e we h av e d et ermi n ed , i n sofar as p oss i b l e , t h e i n - f l i g h t en v i ron men t s

a nd o p e r a t i n g c h a r a c t e r i s t i c s o f th e g u i da n c ep ack a g e. Now, i n a s t a t i s t i ca l l y d es i g n eds e t o f l a b o r a t o r y t e s t s w e t r y t o m e a s u r eo p e r a t i o n a l l i m i t a t i o n s a n d r e l i a b l l i t e s u n d e ra l l e n v i r o n m e n t a l c o n d i t io n s c o n s i d e r e d t o b e

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p r o b a b l e e i t h e r i n s t o r a g e , in t r a n s p o r t a t i o n ,in checko ut and as sem bl y, o r in flight of theguidance packa ge. The data from such te stsa r e a na lyz e d by the s ta t i s t ic ia ns a nd e ngine e r s a nd a p r o ba ble f igur e i s a s s igne d to

the ove rall re l iabil i ty of the guidance sectio n.Similar ly, the l imits within which the sectionpe r fo r m s i t s r e q ui r e d funct io ns a r e de fined.If the r e a ppe a r to be o pe r a t io na l l im i ta t io n sinside of the region of normal functioning,fl ights under those specific condit ions willb e p e r f o r m e d t o c h ec k o u t s y s t e m p e r f o r m a n c e .

Te s ts to fa i lu r e wi l l a ls o be r un in de te r mining r e l ia bi l i ty fo r the fo l lo wing r e a s o n:If a given component operates too near the

a ve r a ge le ve l o f fa i lu r e (e i the r in vo l ta ge ,c u r r e n t , t i m e , o r o t h e r p a r a m e t e r ) th e n t her at e of random failu re s w ill be high andenough will o ccu r d urin g the above ou tl inedte s ts to e s t ima te the r e l ia b i l i ty unde r va r io use nvi r o nme nta l l imi ta t io ns . If, o n the o the rhand, the level of operation is far below theave ra ge level of fai lu re , the frequency offailur e in no rm al o per atio n will be low anda lo w e r l imi t e s t im a te o f r e l ia bi l i ty c a n bem a d e f r o m t h e t e s t - t o - f a i l u r e r e s u l t s d i r e c t ly . Exa m ple s of the s e two po s s ib i l i t ie s

wo uld be r e s pe c t ive ly a ma gne t r o n o pe r a t ingne a r ma ximum po we r o utput a nd the s a mema gne t r o n o pe r a te d a t o ne - th i r d tha t po we r .

La bo r a to r y te s ts o f th is k ind o f guida nc epa c ka ge ta ke r a th e r a lo ng t ime , po s s ib lya y e a r to r un a do z e n uni ts th r o ugh s e t s o ft e s t s i n v a r i o u s o r d e r s . At t h e s a m e t im e .

however , f l ights will be continuing and othertype s o f te s t s r un in the la bo r a to r y . Thee ffe c t, fo r e xa mp le , o f va r io us t r a c king d is tu r ba nc e s o n a n a nt i -a i r c r a f t ta r g e t t r a c kingr a da r c a n be de te r mine d in f ie ld te s ts a nd

the subsequent system effects s tudied bys imula t io n . The o utputs o f r e a l o r simula te dco m pu ter s can now be fed into the guidancepackage and i ts behavior analyzed.

Ano ther type of pro blem which can bestudied is the effect of var io us s im ulatio ns igna ls o f a ppr o xima te ly the s a me f r e que nc yo n the guida nc e pa c ka ge r e c e i ve r . In th iswa y the e ffe c ts o f c o unte r me a s ur e s o n them is s i le i t s el f o r f r ie ndly in te r fe r e nc e c a n bea s s e s s e d . Suc h te s ts ma de dur ing f l ightswo uld y ie ld o nly f r a gme nta r y info r ma t io n a ndmight amo unt to ve r y s u bs ta nt ia l f ie ld p r o g r a m s . In t h e l a b o r a t o r y , s i gn a l g e n e r a t o r sof al l types are available and can be used inv a r i o u s c o m b i n a t i o n s a n d s e q u e n c e s .

Na tur a l ly o nly a po r t io n o f wha t i s po s s ib le in la bo r a to r y te s ts ha s be e n d is c us s e dh e r e . It is believed, ho wever , that i t is c l ea rtha t s uch te s ts s upple me nt a nd in s o me c a s e sa c tua l ly r e pla c e fl ight te s t s . The numbe r o f

f l ights po s s ib le g r o ws s ma l le r a s the s iz ea nd c o mple xi ty o f the mis s i le s ys te ms inc r e a s e . B ut t h e r e q u i r e m e n t s f o r p r e c i s ea nd e xte nde d te s t da ta g r o w in p r o po r t io nto the s iz e o f the s ys te m s . Our o nly c o ur s eo f a ct io n in s uc h c i r c um s ta n c e s i s to ma kerapid and intell igent in cr ea se s in the num bera nd type s o f la bo r a to r y te s ts c o nduc te d.

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TRENDS IN FIELD TESTING OF GUIDED MISSILESD r . E r n s t A . Steinhoff*

SUMMARY

The pa pe r p r e s e nte d d e s c r i be s the va r io us e ffo r ts to w a r d impr o ve me nt o f te s t r a ng eins t r um e nta t io n a nd i t s a da pta tio n to a uto m a t ic da ta handl ing and a na lys is . The inc r e a s ingavailabil i ty of lar ge analog and digital com puting m achin es lends i tself to m as s handling ofr a w da ta a s e nc o unte r e d in the da ta r e duc t io n a nd a na ly s is fie ld . To ma ke us e o f th is c a pa bil i ty the output of the test r ang e ins tru me ntatio n m ust be adapted to dir ect input into thec o m put e r s . Gr e a t e ffo r ts a r e p r e s e nt ly be ing ma de to a c c o m pl is h th is a nd s e v e r a l p r o je c tsc o nc e r ne d wi th r e a l t ime da ta r e duc t io n a nd da ta a na lys is a r e in the de ve lo pm e nt s ta ge .Co nside rab le saving s in manpo wer in data handling and in o ver all co st of m iss ile test inga r e e x p e c t e d . T h e m a n p o w e r s a v ed w i l l b e c o n v e r t e d t o c o m p u t e r p r o g r a m m i n g a nd a s s o c ia te d a s s ignm e nts le ading to be t te r a va i la bi li ty o f impr o ve d e ngine e r ing info r m a t io n .

SOMMAIRE

La no te p r e s e n te e de c r i t le s d i f fe r e nte s te nta t ive s ve r s une a me l io r a t io n de1'appareil lage per m etta nt la se r ie d ' es sa is et son adaptation a la manipulatio n et a 1'analyse

a uto ma t ique de d o nne e s . La po s s ib i l i te c r o is s a n te de s g r a nde s c a lc u la t r ic e s a na lo gue s e td ig i ta le s c o nt r ibue e l le -m e m e a la ma nipula t io n ma s s ique de do nne e s b r ute s c o mm e c e las e r e nc o nt r e da ns le do ma ine de 1'analyse e t de la r e duc t io n de s do nne e s . En vue d ' u t i l i s e rc e t le po s s ib i l i te , la s o r t i e de 1 ' a ppa r e i l la ge pe r me t ta nt la s e r ie d ' e s s a is do i t e t r e a da pte e aI ' e nt r e e d i r e c te da ns le s c a l c ul a t r ic e s . De gr a n ds e ffo r ts s o nt a c tue l le me nt fa i t s da ns c e bute t p lus ie ur s p r o je ts e n r a ppo r t a ve c la r e duc t io n de s do nne e s da ns le te mps r e e l e t 1'analyse

d e s d o n n e e s so n t e n c o u r s d e d e v e lo p p e m e n t . O n e s p e r e r 6 d u i r e c o n s i d e r a b l e m e n t le c o u td ' e s s a i s d e s p r o j e c t i l e s e t 1'energie huma ine ne c e s s a i r e a la ma nipula tio n de s do nne e s . La

m a i n d ' o e u v r e a i n s i e c o n o m i s e e s e r a u t i l i s e e p o u r la m i s e e n p r o g r a m m e d e c a l c u l a t r i c ee t po ur de s t r a va ux a s s o c ie s me na nt a une po s s ib i l i te p lus g r a nde d ' a m e l io r e r le s c o n-n a i s s a n c e s d a n s c e d o m a i n e .

1. INTRODUCTION

In past y ea r s of f light test ing of so ph ist ic a te d guided m is s i le s a nd p i lo t le s s a i r c r a f t ,the ne c e s s i ty fo r mo r e tho r o ugh pr e f l ighttesting pr ior to actual launching has beende m o ns t r a te d a gain a nd a ga in . The inc r e a s

ing c o m ple xity o f m is s i le s fr e que nt ly r e d uc e sthe s uc c e s s c ha nc e s o f e xpe r ime n ta l f i r ingsin the ear ly development phases to an extent

that a po int of dim inishing ret ur n is quicklyr e a c he d. R e c o gniz ing th is , te s t ing te c hnique sa r e be ing deve lo pe d to r e v e r s e th is t r e nda nd to r e duc e fa i lu r e c ha nc e s o f c o mple tes ys t e m s to a n e co no mic le ve l . In p i lo te da i r c r a f t a p ilo t c a n , in c a s e o f a n e me r ge nc y,change hi s fl ight p lan and in many ca se s

r e duc e o r bypa s s a fa i l ing v i ta l c o mpo ne nt .A pi lo t le s s a i r c r a f t o r guide d mis s i le , ho we ve r , do e s no t ha ve th is a l te r na t ive .

• Te c hnic a l Di r e c to r o f R e s e a r c h a nd De ve lo pme nt , Ho l lo ma n Ai r De ve lo pme nt C e nte r .

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T h e f o l l o w i n g p a r a g r a p h s o u t l i n e s t e p sa n d a p p r o a c h e s t a k e n t o i n c r e a s e t h e s u c c e s sof fl ig h t t es t m i ss i o n s an d t o re d u ce th en u mb e r of mi s f i r i n g s of v alu ab l e t es tm i s s i l e s .

2. PREFLIGHT ANALYSIS OF NEWMISSILE SYSTEMS

In o r d e r t o a r r i v e a t r e a l i s t i c t e s t p l a n s ,a c o m p l e t e d y n a m i c p r e f li g ht s y s t e m sa n a l y s i s i s a d v i s a b l e t o d e t e r m i n e c r i t i c a ls t a b il it y a r e a s , s h o r t c o m i n g s in a e r o d y n a m i cp r o p e r t i e s a nd c o n t r o l s u r f a c e e ff ic a c y , a ndo t h e r p r o b l e m s . T h i s is d o n e g e n e r a l l y w itha n a lo g c o m p u t e r s . A f te r c o m p l e ti n g t h is" d y n a m i c s u r v e y , " a c t ua l p h y si c a l c o m p o n e n t s o f c o n t r o l s a n d g u i d a n c e s y s t e m ss u ch a s a c t u a t o r s , g e a r t r a i n s , a i r b o r n ec o m p u t e r s , c o n t r o l s u r f a c e l o a d s i m u l a t o r s ,e t c . , should be included in the open and clo sedl oop a n al y s i s t o s t u dy ef fects o f n o n l i n earc o m p o n e n t s a nd p o s s i b l e s a t u r a t i o n o fc i r c u i t s o n t h e o v e r a l l d y n a m i c s o f t h esy st e m . Af t er es t ab l i sh i n g t h e n o n l i n earp a t t e r n s of co mp o n en t s in c l o sed and op enl o o p a n a l y s i s , t h e s e p a t t e r n s c a n b e s i m u l a t e dag ai n i n an al og co m p u t e rs t o faci l i t a t e t h ei n v est i g at i on of t ren d s cau sed b y t h e m o d i f i cat i on of p erfo rman ce an d cap ab i l i t i es of

i n d i v i d u al comp on en t s .

Co mp l ex i n t e rac t i o n of co mp o n en t s andc o m p o n e nt p e r f o r m a n c e m o d i fi c at io n o n t h een t i r e d y n ami c sy s t e m can b e s t u d i ed andc r i t i c a l s t a b i li t y c a s e s m a p p e d f o r f u r t h e rex p l o ra t i o n in f li gh t t es t s . S i nce co mp o n en t sf r e q u e n t l y c h a n g e t h e i r o p e r a t i n g c h a r a c t e r i s t i c s w i th c h a ng i ng e n v i r o n m e n t a l c o n d i t i o n s , a t h o r o u g h e n v i r o n m e n t a l t e s t p r o gr am must be s et up and i ts effects upono v e r a l l d y n a m i c s a n a l y z e d .

3. COMPONENT RELIABILITY ANDENVIRONMENT>\L TESTING

As ou t l i n ed i n t h e p reced i n g p arag rap h ,en v i ron men t al con d i t i on s may ch an g e b as i cc o m p o n e n t p r o p e r t i e s a n d a l s o m a y h a v e

effects on th e o v er a l l com p on en t r e l i ab i l i t y .T h e r e f o r e c o m p o n e n t s , a s s e m b l i e s , an d e v ent h e co mp l et e sy s t em mu st be su b j ect ed tor i g i d and rea l i s t i c en v i ro n me n t al t e s t s . M an yof t h ese t e s t s can b e p erfo rm ed by s i mu l at i n g

e n v i r o n m e n t a l c o n d i t io n s in t e s t l a b o r a t o r i e s .

F r e q u e n t l y , f a i l u r e s o f c o m p o n e n t s o c c u ro n l y w h e n s e v e r a l c r i t i c a l e n v i r o n m e n t a lco n d i t i on s o ccu r s i mu l t an e o u sl y an d n ot wh encomp on en t s a re t es t ed i n d ep en d en t l y u n d eri n d iv i d ual en v i ro n me n t al t e s t s . T h i s i n d i c a t e s t h a t r e a l i s t i c si m u l a t io n o f e n v i r o n me n t al con d i t io n s mu st t ak e p l ace . T est i n gto fai lure of the individual components shouldest ab l i sh t h e act u al op era t i on al l i mi t s o f t h e

com p on en t in q u est i on b es i d e s es t ab l i sh i n gthe fact that the component meets the originals p e c i f i c a t i o n s .

Since the individual co mp o nents (seeFig. 1) must have a general rel iabi l i tyd eg re e of 1 in 1 0 5 o r b e t t e r in a c o m p l e xm i ss i l e in o r d e r t o resu l t in a h ig h d eg re eof sy s t e m s r e l i ab i l i t y , t es t s t o fa i l u re mu stbe extended to a high enough number ofsam p l e s t o es t ab l i sh t h i s . T h i s p h i loso p h yi n d i cat es t h e n ece ss i t y of s t an d ard i za t i onof su ccessfu l co mp o n en t s to red u ce t h en u m b er of n o n st an d ard i zed com p o n en t s on am i s s i l e t o a t o l e r a b l e m i n i m u m . D e v ia ti o nfrom this r at e is advis able only if this ism a n d a t o r y t o r e a c h s p e c if ie d o v e r a l l s y s t e m sp e r f o r m a n c e .

Resu l t s o f re l i ab i l i t y i n v est i g at i on s sh owi n g t h e b reak d own of v ar i ou s fa i l u re sou rcesar e i n d i cat ed in F i g . 2 . T h e d i ag r am i st y p i cal fo r a g r ea t n u mb er of in d i vi d ual

m i s s i l e s y s t e m s a nd s h o w s t ha t b e s i d e sc o m p o n e n t f a i l u r e s , w o r k m a n s h i p i n a s s e m bly and human shortcomings in handling ofm i s s i l e s d u r i n g t he l au n ch in g p r e p a r a t i o n sals o have an influence on the su cc es s of theo v e r a l l m i s s i o n .

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4. CAPTIVE FLIGHT TEST RANGES ASTOOLS TO MORE REALISTIC FLIGHTSIMULATION

In the past decades, models as well as

c o mple te m is s i l e s ha ve be e n te s te d in c a pt ivef l ights , c a r r i e d by a i r c r a f t . Adva nc e s in tothe s upe r s o nic s pe e d r a nge s ma ke c a pt ivef light s imula t io n m o r e a nd m o r e unr e a l is t ics inc e e nvi r o nme nta l c o ndi t io ns c a nno t e a s i lybe s imula te d c lo s e ly e no ugh, pa r t ic ula r lyi f h i g h p e r f o r m a n c e m i s s i l e s a r e c o n c e r n e d .Frequently the s ize of the missile involveddo e s no t pe r m it a i r bo r n e c a pt ive f lights i m u l a t i o n .

In the last few y ea r s , a new too l , thehigh speed t rack or as we call i t , the captivef l ight te s t r a nge , ha s ga ine d mo r e a nd mo r epo pular i ty. On this t r ac k, with a future lengthof 10.65 km, s le ds can be pro pelled far intothe s upe r s o nic s pe e d r a nge a nd a c c e le r a t io nhis t o r ie s pa t te r ne d a f te r the a c tua l inf lighta c c e l e r a t i o n a r e s i m u l a t e d . A c c e l e r a t i o n sup to 57 g ' s a nd de c e le r a t io ns up to 100 g ' sha ve be en r e a c he d . Ful l s iz e m is s i le s c a nb e t e s te d h e r e u n d e r m o r e r e a l i s t i c c o n dit io ns than po ssib le in captive free -fl ight

t e s t i n g .

One pa r t ic ula r ly a t t r a c t ive fe a tur e o f thet r a c k te s t ing i s the r e c o ve r y o f the te s te dmis s i le e ve n in the c a s e o f a s e ve r e c o mponent fai lu re . The s led design and them o d e l o r t e s t i t e m a r r a n g e m e n t a r e i m p o r tant if the interference between the groundsur face and the test i tem is to be avo ided.In th is c a s e e ve n the a e r o dyna mic mo de lt e s t s c a n b e p e r f o r m e d w i th g r e a t s u c c e s s .

The c a pt ive h igh-s pe e d f lu t te r a na lys iso f new a i r p la ne s a nd m is s i le s i s a n im po r tant application of such a captive test range.S l e d s a r c a l s o u s e d f o r l i n e a r a c c e l e r a t i o nte s ts o f in te r t ia l guida nc e c o mpo ne nts a nd

a s h ig h - s p e e d l au n c hi n g p l a t f o r m s f o r a i r -t o - a i r m i s s i l e s . T h e h i g h -s p e ed t r a c k a sa n a e r o dyna m ic to o l i s no t a new ide a .P r o f e s s o r W a l c h ne r , f o r m e r l y a t G o e t t in g e n ,investig ated and planned a high -spee d t rac k

dur ing W o r ld W a r I I .

5. TEST RANGE REQUIREMENTS ONOPTIC AL AND ELEC TR ONICINSTRUMENTATION

The inc r e a s e o f s pe e d and s la nt r a nge o fm i s s i l e s a n d p i l o t l e s s a i r c r a f t h a s d o u b t l e s s l y i n c r e a s e d p e r f o r m a n c e r e q u i r e m e n t so n o p t i c a l a n d e l e c t r o n i c t e s t r a n g e i n s t r u me n ta t io n . B e s id e s the impr o ve d t r a c king

fe a tur e s a nd d igi ta l iz e d s ha ft o utput p r o vis io ns , a s e mia uto ma t ic t r a c king , in whichhuma n judgme nt i s ma inta ine d, ha s p r o ve nt o be m o s t p r o m i s i n g . T o m a t c h t he c a p a bil i t ies of the human readout of as l i t t lea s 10 s e c o nds , the a uto ma t io n , in the o pt ic a lt ra cki ng field s t i l l m ust go a long way.

In the field of electronic t racking thes i tua t io n i s no t ve r y d i f fe r e nt . C o ns id e r a ble p r o gr e s s ha s be e n ma de by the a ppl ic a

t io n o f c ha in r a d a r s which a r e c o nt r o l le df r o m o ne c e n te r a nd pe r mi t te d to fo llo w them i s s i l e t h r o u g h o u t t h e v a r i o u s p o r t i o n s o f i t sfl ight (see Fig. 4).

T h e c h a n g e o v e r f r o m o n e r a d a r s i t e toa no the r i s a uto ma t ic a nd d ic ta te d by thequa l ity o f the r e c e iv e d da ta . M o r e po we r fulr a d a r s i n c r e a s e b o t h t h e r a n g e o f t r a c k i n ge ve n fo r s ma l l o bje c ts , a nd the t r a c kingacc ur ac y. Shaft o utputs have been digita l izedbut the y a r e no t s a t i s fa c to r y in a l l c a s e s .

E l e c t r o n i c t r i a n g u l a t i o n t yp e t r a c k i n gsy ste m s l ike MIRAN (F ig. 5) have beend e v el o p e d s o f a r th a t t r a j e c t o r y c o o r d i n a t e sc a n be t r a ns m it te d in a d ig i ta l iz e d fo r m a nd

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t he n d i r e c t l y t r a n s f e r r e d i nt o t he e l e c t r o n i cd i gi ta l c o m p u t e r s . F r o m t h e e x p e r i e n c e o ft h e M IRAN d i g i t a l i zat i o n , t h e d i g i t a l i zat i o nof Do p p l er d at a sh o u l d n ot cau se co n si d e ra b l ed i f f i cu l t i es . Ho wev er , t h e d i g i t a l i zat i o n

s h o u ld b e p e r f o r m e d a f t e r t h e p o s i t i o n c o o r d i n a t e s o f t he m i s s i l e o r a i r c r a f t a r e d e t e r m i n e d .

6 . T E NDE NCY T O F L IGHT T E S TANALYSIS IN REAL TIME

T h e p o ss i b i l i t y of ob t a i n i n g o p t i cal an de l e c t r o n i c t r a c k i n g d a t a i n d i g i ta l f o r m m a k e si t p o ss i b l e t o h an d l e t h ese d at a b y l a r g ed i g i t a l c o m p u t e rs t o o b t a in t h e f li gh t p ath

co o rd i n at es as fu ncti on of t i me ( l ocat i on ofob j ect ) , i . e . , v el oci t y , an d accel era t i on i nt e r m s o f s p a c e - f i x e d c o o r d i n a t e s , o n a o n e -t o - o n e t i m e b a s i s , o r r e a l t i m e . S i nc et e l emet er d at a can a l so b e d i g i t a l i zed , at r a n s i t i o n f r o m s p a c e - f i x e d c o o r d i n a t e s t oa m i s s i l e - f i x e d c o o r d i n a t e s y s t e m w i l l p e r m i tt h e c o r r e l a t i o n o f t r a c k i n g a nd i n t e r n a l d a t awi t h each o t h er .

T h i s p o s s i b i l i t y o p e n s t h e r o a d f o r a r e a l

t i m e f li gh t p e r f o r m a n c e a n a l y s i s . B y i n t r o d u c in g r e d u c e d t r a c k i n g d a t a , s u c h a s v e l o c i t i e s a nd a c c e l e r a t i o n r e f e r r e d t o t he m i s s i l ea x i s , i n t o t h e m i s s i l e e q u a t i o n s o f m o t i o n ,f li gh t p er fo rm an ce d at a as w el l as d y n ami cresp on se d at a , e . g . , t ran sfer fu n ct i on s an dF o u r i e r i n t e g r a l a n a l y s i s d a t a , c a n b e o b t a i n ed . T h i s fact p r o v i d es a co mp l e t e l y n ewb asi s fo r t h e econ omy of f l i g h t t es t i n g , s i n cet h e com p u t er i t se l f n ow can d et e rm i n e b y t h eu s e o f p r e s e t c r i t e r i a , w h e t h e r s t a b i l i t yp r o p e r t i e s a r e a s d e s i r e d o r m u s t b ei m p r o v e d . S i g n al s f r o m t he c o m p u t e r t o t h emi ss i l e can t h en ch an g e g ai n se t t i n g s t o i mp r o v e o r c h a n g e s t a b i l i t y p a r a m e t e r s t o w a r dm o r e f a v o r a b l e o r m o r e u n fa v o r a bl e c o n d i t i o n s , t he l a t t e r i n o r d e r t o d e t e r m i n ec r i t i c a l s t a b i l i t y c o n d i t i o n s .

T h e econ omi c asp ect o f su ch eq u i p men ti s o b v i ou s , s i n ce t h e r ea l t i me an a l y s i s y i e l d ss t ab i l i t y d at a i n a mu ch sh o r t e r t i m e. Ad d i t i on al mod i f i cat i on s b ased on ob t a i n ed d at acan b e i n t ro d u ced d u r i n g t h e sa m e f li g h t .

Reduc tion up to 30 pe rc ent of the f light te stsn o r m a l l y n e c e s s a r y i s e x p e c te d by t he r e a lt i m e p e r f o r m a n c e a n a l y s i s . S e v e r a l l a r g eU . S . p r o v i n g g r o u n d s a r e c o n v e r t i n g t h e i rd a t a r e d u c t io n f a c i l i ti e s f o r t h i s p u r p o s e .

7. PROCESSING OF DATA

T h e p r o c e s s i n g o f t r a c k i n g a n d a i r b o r n ed at a , semi au t omat i c t h u s fa r , can b e mad efu l l y au t omat i c i f t h e t rack i n g an d i n t e rn al

d at a can b e u sed j o i n t l y fo r red u c t i onp u r p o s e s . W h il e e x t e r n a l d a t a a r e h ig h lya c c u r a t e f o r p o s i t i o n i n f o r m a t i o n a n d i n c a s eo f D o p p l er d a t a a l s o f o r v e l o c i ty i n t h e e a r t h -f i x e d c o o r d i n a t e s y s t e m , t h e a c c e l e r a t i o ni n fo r m a t i o n i s m o s t l y v e r y u n s a t i s f a c t o r y .If th e a c c e l e r a t i o n s in m i s s i l e a x i s a r e d e t e r m i n ed by a i r b o r n e a c c e l e r o m e t e r s ( a c c u r a t eto one part in 10,000 or bet ter) and thenc o r r e l a t e d w i t h t h e e x t e r n a l t r a c k i n g d a t a ,h i g hl y d et a i l ed an d ac cu ra t e i n fo rm at i on canb e ob t a i n ed an d t h e v el oci t y d at a i mp rov ed

c o n s i d e r a b l y .

T h e b o t t len eck in d at a red u ct i o n wo rk isp r e s e n t l y t he r e a d i n g an d p r o c e s s i n g of r a wd a ta f r o m e i t h e r s o u r c e . P a r t i c u l a r l y ,t r a c k i n g d a t a r e q u i r e a c o n s i d e r a b l e a m o u n to f s m o o t h i n g b y h i g h e r o r d e r p o l y n o m i a lt e c h n iq u e s b e fo r e f u r t h e r p r o c e s s i n g , w h i lei n t e r n a l t e l e m e t e r e d d a t a p r e s e n t p r o b l e m sof no ise re m o val affect ing the quali ty of thed at a . T o imp ro v e t h e q u al i t y , p u l se wi d thmod u l at i on an d t h e p u l se cod e t y p e t e l emet rym e t ho d a r e u s e d , p r e m i t t in g h i g h er a c c u r a c y ,l ower n o i se an d , i n case of t h e l a t t e r , on -m i ss i l e d i g i t a l i zat io n of t h e s i g n al s . T oc o m p e n s a t e f o r p o o r g e o m e t r y o f t r a c k i n gst a t i on s an d p oor p erfo rman ce of i n d i v i d u al

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o p e r a t o r s , t h e d a ta a r e o v e r - d e t e r m i n e d byus ing mo r e t r a c king s ta t io ns tha n a bs o lute lyn e c e s s a r y , t h u s p e r m i t t i n g t h e d a t a p r o c e s s ing c o mp ute r s to c ho o s e a nd us e o nly thebest com bination of s tat io ns in each cas e.This p r o c e dur e e l im ina te s the te dio us ha nd-r e a ding o f la r ge qua nt i t ie s o f t r a c king da ta .

The c o mbina t io n o f a na lo g c o mpute r s wi thdigi ta l c o m pute r s fo r da ta p r o c e s s in g a ndd a t a a n a l y s i s r e d u c e s t h e o v e r a l l r e q u i r e ment on com putation equipm ent. The analogequipment fur nishe s d er iva tiv es of f l ight datains ta nta ne o us ly a nd a ls o s e r ve s to fu r n isht h e f i r s t t e r m s f o r i t e r a t i o n s p e r f o r m e d bythe digital po r t io n of the facil i ty. The digital

c o mpu te r in tu r n mo ni to r s the a c c ur a c y o f theanalog co mp uter to pr event dr ift ing of ther e s u l t s in c a s e s o f a m p l i f i er d r i f t s . T e l e m et er data, e . g . , wil l be available in digitalfo r m fo r the d ig i ta l c o mp ute r , a nd a t the s a m et ime in a na lo g fo r m fo r r e a do ut o r p r in to utpur po s e s a nd v is ua l d is pla y .

TREND IN TEST RANGEINSTRUMENTATION

B e s ide s c o nt inuo us v is ua l p r e s e nta t io n o ft e s t r e s u l t s , m a g n e t i c t a p e r e c o r d s a r emaintaine d to s to r e the init ial raw data andthe r e duc e d da ta fo r a n a ddi t io na l a na lys isif n ec ess ar y. Since the handling of 60 o rm o r e c ha nne ls s im ul ta ne o u s ly in o ne da tar e duc t io n s ys te m do e s p r e s e n t , e ve n fo r thela r g e s t c o mputa t io n fa c i l i ty , qui te a s t r a in ,s i m u l t a n eo u s t a p e r e c o r d i n g p e r m i t s l a t e rinvestigation of those phases which have notbe e n inc lude d in the p r ima r y a na lys is .

The f i rs t and imm ediately handled dataa r e o f the quic k- lo o k type to indic a te whe the rthe f l ight mis s io n wa s ba s ic a l ly a s uc c e s s o r

f a i l u r e . A r e a l t i m e d a t a p r o c e s s i n g s y s t e mno rm ally handle s between 30 and 50 per cen tof the incoming raw data only, s ince thatnumbe r i s ne e de d to indic a te the o ve r a l ls uc c e s s o r fa i lu r e o f a mis s io n a nd to ide nt ifythe o v e r a l l a r e a in whic h the fa i lu r e o c c u r r e d .The n a de ta i le d a na lys is i s pe r fo r me d tode te r mine the de ta i ls whic h c a us e d a pa r t ic ula r malfunction. In this way the availa bleequipment can be flexibly employed.

The ge ne r a l de ve lo pme nt t r e nd in te s tr a nge ins t r um e nta t io n i s to wa r d self-

c o nta ine d a uto ma t ic a l ly d ig i ta l iz e d da ta p r o duc tio n e l imina t ing the ne ed fo r ha nd p r o c e s s ing o r s e mia uto ma t ic p r o c e s s in g. Due to thes imul ta ne o us us e o f a n e xc e s s numbe r o fs tatio ns to t r ack the flying o bject , how ever ,a huma n o pe r a to r ma y mo ni to r the da ta a ndalso selec t the o ptimum combination of dataat each given t im e. Human mo nito r ing oftho s e te le me te r c ha nne ls whic h c o nta in theleast noise and the most essential data isano ther var iat io n in which human intell igence

can still effect full automatic handling.

P a r t i c u l a r l y d e s ig n e d i n s t r u m e n t a t i o nr a d a r s ha ving the r e q ui r e d fe a tur e s a nda c c ur a c y qu a l i t ie s a r e c o ming in to be ing.

The ba s ic t r a je c to r y da ta a c quis i t io n i ssupplemented by long focal length opticals ys t e m s to g ive ne c e s s a r y v is ua l de ta i lsa nd h igh f r a m e r a te c a m e r a s to pe r mit s lo wm o t i o n i n t e r p r e t a t i o n o f s t r u c t u r a l f a i l u r e s ,o f the s e pa r a t io n o f s ta ge s o f mul t is ta gem is s i le s , and o f ma ny o the r de ta i ls . Aninte r e s t ing a r e a i s the c o ns ta nt Impr o ve me ntof o ptical and electr o nic equipment tom e a s u r e m i s s d i s t a n c e f o r i n t e r c e p t e r t y p em i s s i o n s .

B a l l i s t ic type c a m e r a s a r e no t a da pta bleto the r e a l t ime digi ta l r e a do ut . Ho we ve r ,c o mpa r a to r s ha ve be e n de ve lo pe d fo r thedigital r eado ut and they ar e being fur therimpr o ve d. Th is info r ma t io n the n c a n be

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i n c lu d e d i n th e s e c o n d g o - o v e r i n d a t a r e d u c t i o n t o s u p p l e m e n t t h e r e a l t i m e i n f o r m a t i o n b y m o r e e x a c t a n d a c c u r a t e i n f o r m a t i o n .I n o r d e r t o o b t a i n o p t i c a l a t t i t u d e i n f o r m a t i o n t o s e r v e a s c a l i b r a t i o n f o r i n t e r v a la t t i t u d e r e f e r e n c e , m o d i f i c a t i o n s o n A s k a n i aC i n e t h e o d o l i te s h a v e b e e n p r o p o s e d t o p e r m i tt he o p e r a t o r to r o t a t e th e c r o s s h a i r s p a r a l l e lt o on e of t h e m ai n ax e s of t h e m i ss i l e an dd i g i t a l i ze t h e ou t p u t of t h i s ro t a t i o n . T ak i n gt h r e e i n s t r u m e n t s w i th t h i s f e a t u r e , a c o m p u t er cou l d comp u t e t h e a t t i t u d e of t h e mi ss i l e a n d c o m p a r e t h e r e s u l t s w i th t h e i n t e r n a li n fo r mat i o n . It i s ob v i o u s t h at m o re d et a i l ed

i n f o r m a t io n i s o b t ai n ed b y t he g y r o r e f e r e n c e ,and that the effects of the gyr o drif t c anb e c o r r e c t e d b y s u c h o p t ic a l a t t it u d ei n f o r m a t i o n .

I n o r d e r t o i m p r o v e t h e o p t i c a l a c q u i s i t i on of h i g h f l y i n g mi ss i l es , ch ai n rad areq u i p men t h as been u sed t o co mp u t e c u r re n ta c q u i s i ti o n c o o r d i n a t e s . S e v e r a l ty p e o p t i c a lt r a c k i n g c a m e r a s , a m o n g t he m A s k a n i a s ,h av e b een ad ap t ed t o t h i s met h od ; o t h ers wi l lf o l lo w . B a l l i s t i c t y pe c a m e r a s a r e t r i g g e r e da u t o m a t i c a l ly a s s o o n a s t h e o b j e c t r e a c h e stheir f ield of view.

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- 1.4 % MISSION POSTPONED

- 1 . 5 % LOSS OF TEST ITEM

- 5 . 1 % MISSION CURTAILED

2 . 1 % SLIGHT MATERIAL DAMAGE

4 4 % MISSION COMPLICATED

- 4 S % MISSION SLIGHTLY DELAYED

- 4 . 0 % MISSION CONSIDERABLY DELAYED

- 4 0 % MISSION FAILED 25 %

5 . 2 % MISSION FAILED 10 %

UNKNOWN 0.6%

NEAR MBS OF TARGET 0 3 %

DROPPING OF DEFECTIVE PART 0.9 %

CHANGE OF MISSION 0 3 %

" 1 2 . 8 % MISSION CANCELLED

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- 1 3 2 % MISSION FAILED 100%

- 2 2 . 8 % NO DIRECT EFFECT UPON MISSION

1 0 0 %

F i g . 2. Effect of f a i l u r e s (1200 f a i l u r e s ) .

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ELECTRICAL

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3.8% GENERAL

0 . 9 % OPTICAL FAILURES

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- 7 . 1 % UNAVOIDABLE

1.5% GENERAL

1.9% ASSEMBLIES

3 2 % SUBASSEMBLIES

- 3 . 4 % SYSTEMS

0 % SUBASSEMBLIES

- 9 5 % ASSEMBLIES

- 1 4 % COMPONENTS

- 1 9 . 3 % SYSTEMS

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Fig. 3 . Typ e s o f fa i lu r e s (1200 fa i lu r e s ) .

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LOW SIGNAL LEVEL MISSILE INSTRUMENTATIONL. G. deBey*

SUMMARY

•During the past decade i t has been practical to employ relatively unsophisticated radiot r a n s m is s io n a nd r e c e iving s ys t e m s be twe e n m is s i le s in f light a nd gr o und r e c o r din g e quipme ntin o r d e r t o a c h ie v e s at i s f a c to r y r e s u l t s f r o m t h e t r a j e c t o r y i n s t r u m e n t a ti o n s y s t e m s u t i li z in gthe s e t r a n s m is s io n l inks . R e c e nt t r e nds in the guide d m is s i le p r o gr a m ma ke i t c le a r tha tfu tur e ins t r um e nta t io n s ys t e m s wi ll be r e q ui r e d to o pe r a te a t ve r y lo w le ve ls o f r a dio s igna lener gy with a con sequent need for re vis io n of the techniqu es which have been emplo yed inthe p a s t . This pa pe r d i s c u s s e s th is ge ne r a l p r o ble m a nd s ugge s ts te c hnique s whic h s ho wp r o m i s e o f s u b s t a n t ia l i m p r o v e m e n t i n i n s t r u m e n t a t i o n s y s t e m p e r f o r m a n c e un d e r t h e s ea nt ic ipa te d a dve r s e c o ndi t io ns .

SOMMAIRE

D u r an t c e s d e r n i e r s d ix a n s , p o u r o b t e n i r d e s r e s u l t a t s s a t i s f a i s a n t s a p a r t i r u e ss y s t e m e s d ' in s t r u m e n t at i o n d e t r a j e c t o i r e , i l a e t e i n t e r e s s a n t d ' u t i l i s e r l e s l ie n s r e l a ti v e m e n tp eu co m p l iq u e s d e s s y s t e m e s d e r e c e p t i o n e t d e t r a n s m i s s i o n p a r r a d i o , e n t r e l e s m i s s i l e s e nvo l e t 1 'e quipe me nt d ' e n r e gis t r e m e nt du s o l . De r e c e n te s e vo lut io ns da ns le p r o gr a m me dem is s i l e go uve r ne indique nt c la l r e m e nt que le s s ys t e m e s d ' ins t r um e nta t io n fu tur s de vr o nto p e r e r a 1'aidede s ignaux pa r r a d io do nt le n ivea u e ne r ge 't ique s e r a t r e s ha s , c e qui im pl ique r apa r c o nse 'quent la ne c e s s i te d ' une r e v is io n de s te c hnique s e mplo ye e s da ns je pa s s e . C e t te no tet r a i t e d e c e p r o b l e m e g e n e r a l e t s u g g e r e d e s te c h n iq u e s q ui s e m b l e n t v o u l o i r p r o m e t t r ec e r t a i n e s a m e l i o r a t i o n s d e s p e r f o r m e n c e s d e s s y s t e m e s d ' i n s tr u m e n t a t i o n s o u s 1'effet p r e v ude c e s c o ndi t io ns de fa vo r a ble s .

The r is e in technolo gy following WorldW a r II pe r ta in ing to p r o puls io n a nd guida nc eo f mi s s i le s b r o ught a bo ut a g r e a t e xpa ns io nof effor t in the instr um enta tio n field. M ea sur em ent techniques applicable to the flighto f a r t i l l e r y p r o j e c t i l e s r e q u i r e d a g r e a t d e a lo f r e vis io n a nd e xte ns io n to pe r mit a de qua tec o l le c t io n of mi s s i le b a l l i s t ic da ta . The pa s tdecade has seen a s izeable number of newtype s o f ins t r ume nta t io n s ys te ms e ne r ge f r o m

t h e r e s e a r c h l a b o r a t o r i e s a n d t h e s e s y s t e m sa r e c u r r e nt ly p la ying a n impo r ta nt pa r t in theguide d mis s i le r e s e a r c h a nd de ve lo pme nt

pr o gr a m s . In s pi te o f the la r ge e ffo r t e xpended in developing and improving thesesy ste m s many of them a r e s t i l l a long wayfr o m be ing r e a l ly f i r s t c la s s te c hnic a l s o lut i o n s t o t h e i m m e d i a t e i n s t r u m e n t a t i o n p r o b le ms a nd fa l l fa r s ho r t o f me e t ing immine nta n d m o r e c o m p l e x r e q u i r e m e n t s .

Almo s t wi tho ut e xc e ptio n in s t r um e nta t io ns ys te ms invo lve the e s ta bl is hme nt o f a c o m

munication l ink between the missile in fl ighta n d o n e o r m o r e a p p r o p r i a t e g o u n d - b a s e do b s e r v i n g s t a t i o n s . T h e i n s t r u m e n t a ti o n

* B a l l is t ic R e s e a r c h L a b o r a t o r i e s , A b e r d ee n P r o v i n g G r o u n d , M a r y l a n d .

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s y s t e m s i n u s e t o d a y e m p l o y m o r e o r l e s sb r u t e f o r c e m e t h o d s t o i n s u r e t h a t t h e c o m mu n i cat i on sy s t ems wi l l y i e l d an ad eq u at es i g n al f ro m wh i ch t h e d e s i r ed i n t e l l i g en cem a y b e e x t r a c t e d . T h e s e s t a t e m e n t s a r e

p a r t i c u l a r l y t r u e w h en a p p li e d t o e l e c t r o n i ci n s t r u m e n t a t i o n s y s t e m s . In t he u s e o f r a d i oco mm u n i cat i on l i n k s b et ween m i ss i l e s an dg r o u n d s t a t i o n s t h e r e h a s b e e n a t r e n d t o w a r dt h e u se of g r ea t e r and g i v -at e r ra d i a t ed en erg yt o c o m p e n s a t e f o r t h e i n c r e a s e d r a n g e c a p a b i l it i e s o f im p r o v e d m i s s i l e s y s t e m s .A n t e n n a s y s t e m s o f g r e a t e r d i r e c t i v i t y a n dp o wer g ai n a r e b ei n g u sed t o fu r t h er au g men tt h e r e c e i v e d p o w e r . S uc h a n t e n n a s r e q u i r eei t h e r d et a i l ed a p r i o r i k n owl ed ge of t h e fl ig h tp a t h c h a r a c t e r i s t i c s o r c o m p l e x a u x i l i a r y

a n t e n n a - d i r e c t i n g e q u i p m e n t .

T h er e a r e o b v i ou sl y l i m i t s t o th e ex t en t t owhich such methods can be employed and theeffect s o f t h ese l i mi t a t i on s a re c l ear l y ev i d en ti n i n s t r u m e n t a t i o n p r o b l e m s w h i c h a r ea r i s i n g a s a r e s u l t o f m i s s i l e s y s t e m t e c h n ol o g i cal ad v an cem en t . On e h as on l y t oscan casu al l y t h e con t en t s of a l a rg e n u mb ero f t e c h n ic a l j o u r n a l s , p o p u l a r m a g a z i n e s , an dt h e n ewsp ap e r t o fin d a v ar i e t y of s t a r t l i n gc l a i m s p e r t a i n i n g t o t h e c a p a b i l i t i e s o f

m o d e r n m i s s i l e s y s t e m s . It b e c o m e s e v id e n tt h at t h e g eog rap h i cal b ou n d ar i es of t h e ear t hw i ll s o o n b e c o m e l i m i t a t i o n s to o u r t e s t r a n g ef a c i l i t i e s .

W e a r e , i n fact , en t er i n g t h e e r a of sp acea nd i n t e r p l a n e t a r y t r a v e l . T h e m i l e s t o n emarking the beginning of this new age wil l bet h e su cces sfu l l au n ch in g of t h e sm al l a r t i f i c i a l s a t e l l i t e s c u r r e n t l y b e i n g g r o o m e d f o rf l ig h t . He re we h av e m i ss i l e s of su ch ran g e

a nd p e r f o r m a n c e a s t o t a x c o m p l e t e l y t h ei n g en ui ty of t h e i n s t ru me n t at i o n s p ec i a l i s t .T h e f i rs t o f t h ese v eh i cl es wi l l b e v er y sma l la n d y e a r s w i l l p r o b a b l y e l a p s e b e f o r e signif

i can t l y l a rg e p ay l oa d s can b e p u t i n to o rb i t .T h u s , f o r p e r h a p s a d e c a d e , c o m m u n i c a ti o n

s y s t e m s w i l l be r e q u i r e d t o o p e r a t e o v e re x t r e m e r a n g e s a nd e x t en d e d p e r i o d s w i thv e r y s m a l l p e r m i s s i b l e r a d i a t e d p o w e r s .

T h e so l u t i on s t o me et t h ese an d fu t u rer e q u i r e m e n t s e x i s t o n l y i n t h e f o r m o f t h e o r e t i c a l p r o p o s a l s . S o m e o f t h e t e c h n iq u e sr e q u i r e d a r e a s t e p o r tw o b ey o n d t h e c u r r e n t s t a t e o f t h e a r t . T h e B a l l i s t i c R e s e a r c hL a b o r a t o r i e s o f t h e U . S . A r m y O r d n a n c eC o r p s h a v e p r o p o s e d a p a r t i a l s o l u t io n t o t h ep a r t i c u l a r p r o b l e m o f e s t a b l i s h i n g suf

f i c ien t l y ad eq u at e c o mm u n i cat i on wi t h a v er ysm al l a r t i f i c i a l ea r t h sa t e l l i t e t o sa t i s fy th ee x p e r i m e n t a l r e q u i r e m e n t s o f a n i o n o s p h e r er e s e a r c h p r o b l e m . T h i s p a p e r w i ll d e a l w i thso m e of t h e t ech n i cal asp ec t s of t h i s p ro p o sed

sy st e m . It sh ou l d b eco me ap p ar en t t h at t h ep h i l osop h y an d t ech n i q u es t o b e p rop osedwould be equally applicable to the problemof i n s t ru m en t i n g v e ry l on g ra n g e b al l i s t i cm i s s i l e s .

T h e B a l li s t ic R e s e a r c h L a b o r a t o r i e s h a vefor man y y e ar s b een p ro p o n en t s of t h e u seo f p h a s e m e a s u r i n g t e c h n i q u e s f o r e l e c t r o n i cb a l l i s t i c i n s t r u m e n t a t i o n w h e r e t h e u tm o s t Inp r e c i s i o n , i s r e q u i r e d . T h e D OV AP s y s t e m(DOppler Velo ci ty And Po si t io n) Instal led atW h i t e S an d s P rov i n g Grou n d i n ear l y 1 9 4 6 ,h a s c o n s i s t e n t l y m r n o d i n s u p e r i o r p e r f o r m an ce f rom b ot h t h e re l i ab i l i t y an d acc u r acyv i ew p o i n t s. F u r t h e r m o r e , i t s u s e a s ar o u t i n e b a l l i s t i c i n s t r u m e n t a t i o n s y s t e m h a sp ai d a d i v i d en d b y d emon st ra t i n g i t i s a l sou se fu l a s a r e s e a r c h t o o l f o r u p p e r a t m o s p h e r e r e s e a r c h . S p e c if ic a l ly , t h e D o p p le rd at a f ro m a l a r g e n u mb er of h i gh a l t i tu d ef i r i n g s h av e b een used t o d et e rm i n e i on i zat ion d en si t i e s of t h e a t m o sp h er e a s a fu nct ionof a l t i t u d e.

T h e i n s t r u m e n t a t i o n c o n c e p t s t o b ep r e s e n t e d i n t h i s p a p e r w e r e c o n c e i v e d t om e e t t h e r e q u i r e m e n t s i m p o s e d b y t h e u s eof this same Doppler technique in the UnitedS t a t e s e a r t h s a t e l l i t e p r o g r a m , P r o j e c tV a n g u a r d .

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The B e r ning (R e f . 1) io no s o phe r ic e xpe r i ment is being re adie d as one of the po ssiblea n d d e s i r a b l e u p p e r a t m o s p h e r e e x p e r i m e n t suti l izing the sa tell i te as u. vehicle fo r placinga s o u r c e o f e l e c t r o m a g n e t i c r a d i a t i o n o f

o pt imum wa ve le ngth a bo ve the mo s t de ns ep o r t i o n o f t he e a r t h ' s i o n o s p h e r e f o r a r e a s o n a bly lo ng pe r io d o f t im e . Pa ylo a d fo r thesate ll i te s tag e ha s b een l imited to two po unds,r e q u i r i n g t h a t t h e c r y s t a l c o n t r o l l e d t r a n s -mil te r a nd po we r s upply , ne e de d a s a s o ur c eof radiation, be held within this weight l imit .F u r t h e r m o r e , i t i s d e s i r a b l e t h at t he t r a n s mit te r be c a pa ble o f o pe r a t io n o ve r a suf

ficiently long per io d of t im e to yield a lar gen u m b e r o f o b s e r v a t i o n s .

In the belief that the satell i te will remainin o r b i t fo r a r e a s o na bly lo ng t ime , i f p la c e din o r b i t a t a l l , the io no s phe r e t r a n s m it te ri s be ing de s igne d to o pe r a te c o nt inuo us lyfor s ix mo nths at a frequency of 74 m eg ac y c l e s p e r s e c o n d . F o r a n u m b e r o f r e a s o n sthis f r e que ncy a pp e a r s to be ne a r o p t imum .Weight considerations and the efficiency oft r a n s i s t o r s d i c t a t e t h at t h e t r a n s m i t t e r c a nn o tde l ive r mo r e tha n a bo ut o ne mi l l iwa t t o fr a d io f r e que nc y e ne r gy to the s a te l l i te a ntenna

s y s t e m .

T h e D o p p le r i o n o s p h e r e e x p e r i m e n tde pe nds fo r i t s s uc c e s s upo n the me a s ur e ment of the apparent radial velocity of thes a te l l i te wi th r e s pe c t to a n o bs e r ving s ta t io no n the e a r th ' s s ur fa c e . It i s o bvio us tha ts u c h m e a s u r e m e n t s c a n b e m a d e m o s t p r e cisely when the apparent velocity is highest ,i . e . , when the sate ll i te is mo ving mo st near lyto wa r d the o b s e r vin g s ta t io n . It i s unfo r tuna tethat this condit ion fi rs t exis ts at the t imewhen the s a te l l i te r i s e s a bo ve the r a dioho r iz o n a nd la te r , dur ing the s a me o r b i ta lp a s s , when i t disappears below the oppositeho r iz o n. Fo r a n e l l ip t ic a l o r b i t o f twohundred miles per igee and eight hundred

mi le s a po ge e the s la nt r a ng e to the m is s i leo n t h e h o r i z o n f r o m a n o b s e r v i n g s t a ti o n o nthe ea r t h va r ie s between ro ughly fifteenhundr e d a nd th r e e tho us a nd mile s .

The difficulty of the communication problem now beco me s app ar ent. Indeed i t isv i r tua l ly impo s s ib le to de te c t the mis s i leradiation at the hor izons because of the highattenuation of the s ignal at grazing incidencewith the e a r th ' s s ur fa c e . Fig . 1 i s a p lo t o fthe p r o pa ga t io n a t te nua t io n vs . e le va t io nan gle . It is to be no ted that the attenua tio nat low angles of elevation is s ignificantlygr e a te r than fo r the c a s e o f f r e e s pa c epr o pa ga t io n . B a s e d o n the s e p r o pa ga t io ndata and the fact that the ionosphere experiment will not yield significant data at the

high satell i te al t i tudes where the ionizationdensity is very low, i t has been concludedthat no at tem pt should be m ade to detectthe m iss ile s ignal below elevatio n angles ofabout s ixtee n d eg r ees no r at a l t i tude ine xc e s s o f 300 m ile s . The ma ximum s la ntr a nge unde r the s e c o ndi t io ns wi l l va r y f r o mabout 500 miles for a 200-mile al t i tude toa bo ut 800 mi le s fo r a 300-m ile a l t i tude .

The m inimum r e c e ive d s igna l po w e r a tthe te r mina ls o f the a s s ume d 9-db a nte nna

will be 3.16 x 10'

1 6

wa tts o r -155 dbw. Thema ximum s igna l po we r , whe n the mis s i leis dire ctly o ver head , at the lowest al t i tude,will be 5 x 10 "" wa tts o r -143 dbw. Sincei t i s ne c e s s a r y to r e c e iv e s igna l c o nt inuo us lydur ing the e nt i r e p a s s a ge a bo ve e le va t io na ngle s o f 16 de g r e e s an e s s e nt ia l ly o mn idi r e c t io na l a ntenna m us t be us e d. A dipo le ,o ne -qua r te r wa ve le ngth a bo ve gr o und, i ss ui ta b le . The r e duc t io n in r e c e iving a nte nnagain at an elevatio n ang le of 16 de gr ee s isa ppr o xima te ly 6 db. Fu r t he r , the ma ximumgain of a dipole is 2 db less than for the 9-dbantenna assu m ed in ob taining the r es ul ts inFig . 1 . As s uming a t r a n s m it te r a nte nnapa t te r n fa c to r lo s s o f 15 db, the minimumrec eive d pow er will be -155 dbw - 8 db -15 d b = -1 7 8 dbw = 1.58 x 10'18 w a t t s .

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F i g . 2 , p r ep ar ed f ro m d at a p u b l ish ed b yt h e U. S. Nat i on al Bu r eau of S t an d ar d s , sh owst h e av er ag e l e v el s of n o i se to b e en co u n t ere das a function of f req uency . I t is to be notedt h at a t t h e p ro p o sed o p era t i n g f req u en cy of

7 4 m c p s c o s m i c n o i s e p r e d o m i n a t e s a nd i sabout 0 .1 Mv/m p er ki lo cycle of bandw idth.T h i s , fo r a r ecei v i n g sy s t em h av in g a 5 0 -oh minput , i s equivalen t to a no ise po wer of2 x 10"16 wat t s p e r k i l ocy cl e of b an d wi dt h .

A r i g o ro u s an a l y s i s o f t h e ef fect of an t en n ap a t t e r n o n a n te n n a t e r m i n a l n o i s e p o w e r h a sn o t b e e n c a r r i e d o u t , b ut b a s e d o n i n f o r m a t ion furn ished by V. H. Go er ke (Ref. 2) oft h e Nat i on al Bu r eau of S t an d ar d s , i t i ses t i m at ed t h at t h e ef fecti v e n o i se cap t u r e

area fo r a h or i zon t a l d i p ol e wi l l n o t d i f fe rg r eat l y f ro m t h at o f a v er t i ca l wh i p su ch asw a s u s e d t o m e a s u r e t h e c o s m i c n o i s e l e v e l s .Thus, i t may be expected that the effect iven oi se p ower a t t h e an t en n a t e rmi n al s wi l l b eabout 10 db above KTB or 4 x 10 ' l 7 w a t t sp e r k il ocy c l e of b an d wi d th . Re fer r i n g nowt o t h e ex p ect ed mi n i mu m s i g n al an t en n at er m i n a l p o wer 1 .5 8 x 1 0 '" wa t t s , i t i sseen t h at fo r a recei v i n g sy s t em of on e k i l ocycle bandwidth the SP /NP r a t i o w i l l b eap p rox i mat el y 0 . 0 4 wh i l e fo r t h e max i mu msi g n al case t h e S

P/N

Pra t i o wi l l b e ab ou t

125. The Sv/Nv r at io s w il l be about 0 .2an d 1 1 . 2 resp ect i v el y .

On e mo r e p i ec e of i n fo r mat i o n i s n eed edb e f o r e t h e r e c e i v e r d e s i g n m a y b e u n d e r t a k e n .T h e p u rp ose of t h e commu n i cat i on l i n k i s t ot r a n s m i t in t e l l ig e n c e . In t h e i o n o s p h e r eex p e r i m en t t h e d es i re d i n t e l l i g en ce i s t h eshift in received signal f requency as a funct i on of t o t a l i on osp h ere i on con t en t an d l ocali on i zat i on d en si t y a t t h e sa t e l l i t e . F o r ah o m o g e ne o u s t r a n s m i s s i o n m e d i u m , t he D o p pl er shift in f requency a s a function of po lare l ev at i on an g l e , fo r a sa t e l l i t e p ass i n gd i r e c t l y o v e r t he r e c e i v i n g s i t e , c a n b ed es cr i b ed a s sh own in F i g . 3 . T h e max i m u mfrequency is see n to be about 1800 cp s. The

act u al r ecei v ed s i g n al sh i ft ma y b e g r ea t e rthan 1800 cps due to instabil i ty of the signals o u r c e i n t h e m i s s i l e . It h a s be e n e s t i m a t e dthat such effects might ca use f requencych an g e s as g r eat as ±1 2 00 cp s . T h e s i g n of

t h e Dop p l er sh i f t may a l so b e e i t h er p os i t i v eor n eg at i v e d ep en d i n g on wh et h er t h e mi ss i l ei s t r a v e l i n g t o w a r d o r a w ay f r o m t h eo b ser v i n g s i t e . T h u s t h e t o t a l f req u en cysh if t may b eco me as l a r g e as ± 3 00 0 cp s .T h e m a x i m u m f i r s t d e r i v a t i v e o r r a t e o fch an g e of Do p p l er f req u en cy ( f) ap p r o x i mat el y60 cps" 2 and the second derivat ive (f) isab ou t 1 . 5 cp s" s .

T h e m a x i m u m f i r s t d e r i v a t i v e o c c u r s a ta p o l a r e l ev at i on an g l e of 9 0 d eg re es wh il et h e m a x i m u m s e c o n d d e r i v a t i v e o c c u r s w h e nt h e p ol a r e l ev at i on an g l e i s 8 8 d e g r ee s1 5 m i n u t e s . T h i s secon d d er i v at i v e d efi n est h e l eas t p o s t -d et ec t i o n b and wi dt h whi ch mayb e u sed t o p ass t h e i n t e l l i g en ce i n real t i me.

T h e t i me-b an d wi d t h con cep t of i n fo rmat ion t h eo ry p e r m i t s t h e su b st i t u t io n of t i m efor b an d wi d t h , o r v i ce v ersa , i n t ran smi t t i n ga given amount of intel l igence. The minimumbandwidth of the t ransmission l ink is f ixed

b y t h e amo u n t of i n fo rm at i on t o b e t r an smi t t ed i n a g i ven t i m e. If i n fo rm at i on s t o ra g ecan n ot be acco mp l i sh ed , i n fo r mat i o n m u st b et r an s m i t t ed as rap i d l y as i t i s b ei n gg en er at ed . Ho wev er , i t i s o f t en p o ss i b l e t op r o v i d e s t o r a g e f o r t h e i n fo r m a t i o n i n w h ic hc a s e t h e r a t e o f t r a n s m i s s i o n c a n b e r e d u c e dand the bandwidth of the sy ste m ca n be r edu cedp ro p o r t i o n al l y . In an y ev en t t h e t i m e-b an d wi d t h p rod u ct wi l l b e a con st an t fo r ag i v en amou n t of i n fo rmat i on .

I t wil l be shown that in this p ar t ic ul arproblem the intel l igence can be handled byt h e sy s t em i n real t i me, i n sp i t e of t h en a r r o w b a n d w i d t h s r e q u i r e d t o a c h i e v ea d e q u a t e s i g n a l - t o - n o i s e r a t i o s .

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The ma ximum f o c c u r s a t a n e le va t io n a sm e a s u r e d a t t h e e a r t h ' s s u r f a c e o f 57 d e g r e e sand hence the system bandwidth must begr e a t e s t a t th is a ngle . The r e c e ive d po w e r ,taking into account the pat ter n facto r of

t r a ns mit t ing a nd r e c e iving a nte nna s i s2.82 x 10"17 w a t t s . F o r a o n e k i l o c y c l ebandwidth this results in a SP /NP r a t i o o f0.71 o r a Sv/Nv « 0.84 . Data of high qualityf r o m a Do pple r s ys te m o f th is type r e qu i r ethat the o utput Sv/Nv be about 10 /1 . High erSv/Nv r a t io s a r e de s i r a ble but no t a bs o lute lyne c e s s a r y . To a c hie ve a Sv/Nv • 10, thebandwidth of the system must be reduced bya facto r of (10/.8)2= 156 or to 6.4 cps. Sincethe minimum ba ndwidth r e qu i r e d fo rma ximum f i s a ppr o xima te ly 3 c ps the 6 .4

c ps ba ndwidth p r o vide s a n a de qua te ma r ginof safety.

At the lowest elevation at which it isdesired to detect the s ignal the value of tis about 0.033 cps" 3 and a bandwidth of0.07 cps will be r eq uir ed . The 1-kc Sv/Nvra tio at this elevatio n angle was shownpr e vio us ly to be 0 .2 . Thus to a c hie ve aSv/Nv • 10, the bandwidth must be reducedby (10/0 .2) 2 = 2500 o r to 0.4 cp s. Again an

adequate margin of bandwidth is available.

W hen the s a te l l i te i s d i r e c t ly o ve r he a dat the lowest al t i tude the Sv/Nv ratio wasshown to be about 11/1 for an assum ed1 kcp s bandwidth. Since thi s is adeq uatefur the r f i l te r ing i s no t r e qui r e d . The ba ndwidth p r io r to the t r a c king f i l te r , ho we ve r ,is gr ea ter than 1 kcps s o the fi l ter will bedepended upon to re duc e the bandwidth toat least 1 kcp s.

T h e c r i t i c a l c h a r a c t e r i s t i c of t h e r e c e i v i n gsys tem has now been specified. Ce rta ino t h e r c h a r a c t e r i s t i c s a r e d e s i r a b l e t o e n a b lethe sy stem to yield data of the highest quality

a nd to ma ke i t a ppl ic a ble to o the r s im i la ri n st r u m e n t at io n p r o b l e m s . T h e s e c h a r a c te r is t ic s wi l l no t be d is c us s e d in de ta i l butwill be presented br iefly to show how theyaffect the system design.

The r e c e iving s ys te m s ho uld be c a pa bleo f o pe r a t ing a s a pa r t o f e i the r a no nc o he r e nto r a c o h e r e n t D o p p l e r s y s t e m . F o r t h ec o he r e nt s ys te m two e s s e nt ia l ly ide nt ic a lr e c e i v i n g c h a n n e l s a r e r e q u i r e d ; o n e f o rthe missile s ignal frequency and one forthe g r o und r e fe r e nc e s igna l . The pha s ec h a r a c t e r i s t i c s o f t h e t w o c h a n n e l s i n p a r t ic ula r mus t be e s s e nt ia l ly the s a m e ,especially over the input s ignal level rangeo f 0 .1 to 10 ,000 mic r o vo l ts .

The output s ign al should be in a fo rms u i t a b l e f o r t r a n s m i s s i o n v i a e i t h e r w i r el ine s o r r a d io l inks to a c e nt r a l m a gne t ict a pe r e c o r d i n g f a c il i ty . T h e s e r e q u i r e m e n t spro hibit use of a syste m in which the s ignalf r e que nc ie s a r e at- o r ne a r DC .

~he system should yield the s ign of theDo ppler shift : i . e . , whe ther the m iss ilei s a p p r o a c h i n g o r r e c e d i n g .

Fig . 4 is a block diag ra m of a rec eivin ga nd r e c o r d i n g s y s t e m c u r r e n t l y b e in gdeveloped for application in the sate ll i teio no s phe r e e xpe r im e nt . At f i r s t g la nc e thes ys te m do e s no t a ppe a r to d i f fe r g r e a t lyf r o m a c o n v en ti o n a l s u p e r h e t e r o d y n e r e c e iving s ys te m . Fu r t he r e xa mina t io n o ft h e c h a r a c t e r i s t i c s o f t he v a r i o u s c o m p o n e n t swi l l , ho we ve r , r e v e a l s e ve r a l d is t inc tde pa r tu r e s f r o m c o nve nt io na l te c hnique s o fr e c e i v e r d e s i g n .

C o ns ide r ing f i r s t the ge ne r a l de s ign i t i sto be no ted that the r ec ei ve r is in fact adua l c ha nne l de vic e e mplo ying t r ip le he te r o d y ne c o n v e r s i o n . T h e r e c e i v e d r a d i o f r e que nc y s igna l , 7 4 mc ps , i s r e duc e d s uc c e s s ive ly to in te r me dia te f r e que nc ie s o f

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a p p r o x i m a t e l y 1 0 m c p s , 2 60 k c p s . a n d 5 k c p s .A h i gh l y s t ab i l i ze d r efe re n ce f req u en cy , a l soat 74 mcp s, i s s i m i l a r l y t rea t e d i n t h er efe re n ce ch an n el of t h e r ec ei v er . It i s t ob e n o t ed t h at t h e d e s i r ed i n t e l l i g en ce, t h e

d i f fe ren ce b et ween t h e referen ce an d s i g n alf re q u en ci es , h as not y et b een ex t r act e d o r" d e t e c t e d . "

T h e b an d wi d t h of t h e sy s t em h as a l sob e e n r e d u c e d i n s u c c e s s i v e s t a g e s . S i nc ea rea so n ab l y h ig h d eg re e of p h as e f i d eli tyi s d e s i r ed an d p h ase t r an s i e n t s in t h e i np u tt o t h e t r a c k i n g f i l t e r a r e u n d e s i r a b l e , t h e5-kc IF amp lif ier wil l mo st l ikely be adjustedt o h av e p assb a n d h al f -p ow er p o i n t s a t ab ou t

1 kcp s and 9 kcp s giving a bandw idth of8 kc ps. Fina l re ductio n of bandwidth to ther e q u i r e d f r a c t io n o f a c y c l e p e r s e c o n dsh ou l d b e accom p l i sh ed p r i o r t o f in al Do p p l ers i g n a l d e t e c t io n f o r tw o r e a s o n s . F i r s t , th ec r o s s m o d u la t io n p r o d u c t s t h a t m i g h t b ei n t rod u ced b y t h e d et ect i on p rocess wi l l n o ti n crease t h e n oi se comp on en t s i n t h e f i l t e rinput . Second, the f i l ter w il l not be re qu ir edt o h a n d l e f r e q u e n c y c o m p o n e n t s a t o r n e a rDC wh i ch wou ld b e p r ese n t af t e r d et ect i o n .Since the signa l f r equency may have any valuebetween 2000 and 8000 cps (5 kcps ± 3 kcps)an d t h e f i l t e r p assb an d mu st be cen t er ed ont h i s f req u en cy , t h e f i l t e r i n g d ev i ce wi l l b er e q u i r e d t o " t r a c k " th e i nc o m i n g s i g n al , i . e . ,t h e cen t er f req u en cy of t h e p assb an d mu stb e ad j u s t ed con t i n u ou sl y t o t h e i n s t an t an eou sv al u e of t h e s i g n al f req u en cy .

S uc h f i l t e r s a r e g e n e r a l l y r e f e r r e d to a sa c t i v e o r t r a c k i n g f i l t e r s . In p r a c t i c e , f i l t e r sof t h i s t y p e u su al l y con si s t o f a l ocal osci l

l a t o r w h o s e f r e q u en c y i s c o m p a r e d t o t h ei n comi n g s i g n al f req u en cy an d i s con t i n u ou sl yadjusted to equal the signal f reque ncy. Fig. 5i s a b l ock d i ag ram of su ch an e l ect ron i cal l ys y n ch r o n i z e d o r " l o c k e d " o s c i l l a t o r . If t hes i g n al co mp ar i n g d ev i ce i s a f req u en cy

sen s i n g t y p e, t h e o sci l l a t o r i s sa i d t o b ef req u en cy - l o ck ed . If a p h ase d et ec t o r i se m p lo y e d , a p h a s e -l o c k e d o s c i l l a t o r r e s u l t s .By ap p ro p r i a t e ch o i ce of t h e t r an sfe r fun cti onY p t h e b an d wi d t h ch aract er i s t i cs of t h e f i l t e r

may b e v ar i ed . It i s p ro p o sed t o u se su cha p h a s e - l o c k e d n a r r o w b a n d t r a c k i n g f i l t e ra t t h e o u t pu t of t h e 5 k cp s i n t e rm ed i at e f r eq u en cy amp l i f i e r . Not e t h at by acc o m p l i sh ing f inal bandwidth reduction at this pointt h e ful l ad v an t ag e of p re d et ect i o n f i l t e r i n gi s r e a l i z e d .

I t was p oi n t ed ou t ear l i e r t h at t h e f i l t e rbandw idths, as de ter m ined by f and Sv/Nvc o n s i d e r a t i o n s , r a n g e b e tw e e n 1 k c p s a nd0 .4 c p s . T h e c h a r a c t e r i s t i c s of t h e t r a c k i n g

f i l t e r fo r t h i s ap p l i cat i on a re b ei n g d es i g n edt o p ro v i d e fo r a ch o i ce of t wel v e b an d wid t hsra nging fro m 0.1 cp s to 128 cp s in binarys t e p s .

T h e t r an s fer fun cti on i s a l so b ei n gd e s i g ne d t o p r o v i d e e s s e n t i a l l y z e r o p h a s et r a c k i n g e r r o r f o r a c o n s t an t r a t e o f c h an g eof s i g n al f req u en cy (f = o ) . F o r v ar i ab l er a t e s o f f r e q u e nc y c h a n ge t h e p h a s e e r r o rb et ween t h e s i g n al an d t h e l ock ed os ci l l a t o rwil l var y but in no ca se ma y i t excee d 90

d e g r e e s o r t he o s c i l l a t o r w i ll f a ll o u t o flock and the f i l ter wil l cease to functionp r o p e r l y . P r o v i s i o n s a r e b e in g m a d e to i n d i c a t e , b y me an s of an al og v o l t ag es , th e f i r s t ,secon d , an d t h i rd d er i v at i v es of s i g n al p h asea nd a c o r r e l a t i o n c o e f fi ci en t p r o p o r t i o n a lto the phase difference between the input andou t p u t s i g n a l s of t h e f i l t e r . By o b ser v at i o nof t h ese i n d i cati n g d ev i ces an o p er a t o r mayman u al l y se l ec t a b an dwi d th ap p r o p r i a t e fo rt h e ex i s t i n g con d i t i on s .

Th e sig n of the Do ppler f requen cy i ses t ab l i sh ed b y d et ermi n g i f t h e i n s t an t an eou sf req u en cy i s h i g h er o r l ower t h an t h e ref

er en ce f r eq u en cy . T h i s can mo st co n v en ien t lyb e a c c o m p l i s h e d b e f o r e t h e r e f e r e n c e a nds i g n a l c a r r i e r s a r e c o m b i ne d t o y ie l d t h e

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Do pple r f r e que nc y. The s ign indic a to r i sshown in Fig. 4 as being in parallel with thepha s e de te c to r in the ma gne t ic ta pe r e c o r de routput cir cu it . I t could be used with equale ffe c tive ne s s a he a d of the r e c o r de r . The s ign

indicator is of conventional design, makingus e o f a qua d r a tur e r e la t io ns hip be twe e n ther e fe r e n c e a nd s igna l f r e que nc ie s to de te r m inephas e lead o r lag of one with re spe ct to theo t h e r .

The fi l tered s ignal voltage and the ref

e r e nc e vo l ta ge ma y be c o mpa r e d in a pha s ede te c to r to de r iv e the de s i r e d in te ll ige nc e -the Do ppler f r e que nc y. Sinc e the r e fe r e nc evoltage is essential ly uncontaminated byn o i s e , t h e p h a s e c o m p a r i s o n p r o c e s s y i e l d s

n e a r l y o p t i m u m c r o s s c o r r e l a t i o n d e t e c t i o n .

The s igna l fr e que ncy c a n a ls o be r e c o r d e do n ma gne t ic ta pe o r c o uld be t r a n s m it te dto a c e nt r a l r e c o r ding s ta t io n be fo r e be ingr e c o r d e d. Sinc e the s igna l info r m a t io n i ss t i l l contained within the band of frequenciesf r o m 2 .0 to 8 .0 kc ps a t r a ns m is s io n s y s te mc a pa ble o f t r a n s m it t ing DC is no t r e q ui r e d .I t wi l l o f c o ur s e a ls o be ne c e s s a r y to r e c o r dthe r e fe r e nc e f r e que nc y s o tha t the r e fe r e nc ea nd s igna l f r e que nc ie s ma y la te r be c o mbine d in a pha s e de te c to r to e xt r a c t theDo pple r f r e que nc y.

An a uxi l ia r y ma gne t ic ta pe r e c o r d ingc ha nne l i s us ed to r e c o r d the s igna l f r e quency befo re i t has been passe d thro ughthe na r r o w ba nd t r a c king f i l te r . The s igna lat this point is s t i l l heavily contaminated byno ise. Since the pr efi l ter bandwidth is about8 kcps the Sv/Nv r ati o w ill be 0.05 o r -26 db.The no is e ge n e r a te d by ma gne t ic ta pe r e c o r de r s , ne gle c t ing fo r the mo me nt wo w a ndflutter , is gener ally abo ut 40 db belowma ximum r e c o r d ing le ve l . Thu s , ta pe no is ew i l l n o t s e r i o u s l y d e t e r i o r a t e t h e s t o r e ds igna l info r ma t io n .

The pur po s e o f r e c o r din g th is no is y s igna lis to guar d against co mp lete lo ss of data inthe event the t racking fi l ter should fail tolock on the s ig nal init ial ly o r if due to anyr e a s o n i t s ho uld d r o p o ut o f lo c k dur ing

pa s s a ge o f the s a te l l i te . The f i r s t o f the s eeve nts is m o r e like ly. In any event if theno is y s igna l a nd r e fe r e nc e s igna ls a r e s to r e don magnetic tape they may be played backat a later t ime, passing the noisy s ignalth r o ugh the t r a c king f i l te r a s ma ny t ime s a sma y be ne c e s s a r y to a c hie ve lo c k-o n a nd the nc o mbining the m in the pha s e de te c to r too bta in the Do pple r s igna l info r m a t io n . Thiste c hnique no t o nly pe r m its o ne to t r y avar iety of adjustmen ts of the t ra ckin g fi l terc h a r a c t e r is t i c s but fu r th e r , if lo c k-o n c a nno tyet be achieved, due to very high rates ofchange of s ignal frequency dur ing the ea r l yp o r t i o n s o f t h e r e c o r d , t he r e c o r d c an b epla ye d ba c k in the r e ve r s e d i r e c t io n s o tha tlo c k-o n c a n be a c hie ve d unde r c o ndi t io ns o fbe s t s ign a l - to -no is e a nd lo we s t s igna l a c c e l e r a t i o n .

It is ha rd to believe that o ne of these po st-r e a l t i m e f i l t e r i n g a n d d e t e c t i o n p r o c e s s e swill not work under the condit ions to bee xpe c te d. Ho we ve r , s ho uld th is o c c ur s t i l la no the r te c hnique is a va i la ble to pe r mitr e a c hing do wn in to the no is e to e xt r a c t thes ig n a l i n f o r m a t i o n . S o m e t im e s r e f e r r e d t oa s "de a d s ig na l" de te c t io n , the me tho d ma ke suse of the fact that o ne can substi tu te t imefor bandwidth and by the use of auto co rr elati o nte c hnique s c a n ve r y e ffe c t ive ly s e pa r a tesigna ls deeply imbedded in no ise. A s tudy(Ref. 3) of such techniques has been car r iedo ut by the Sta nfo r d Unive r s i ty R e s e a r c hIns t i tu te unde r s po ns o r s hip o f the B a l l i s t icR e s e a r c h L a b o r a t o r i e s a nd t e ch n i q u es a nde quipme nt fo r a c c o m pl is hing suc h de a d signa l

de te c t io n ha ve be e n pr o po s e d.

W hile a l l o f the e quipme nt r e qui r e d fo rth is p r o po s e d s ys te m ha s no t ye t r e a c he dthe s ta ge whe r e te s t r e s ul ts c a n be quo te d

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there is sufficient evidence at hand fromp rev i ou sl y comp l et ed d ev el op men t s i n t h esef i e ld s t o o f fe r a h ig h d eg re e of cer t a i n t yt h at v ery l ow l ev el s i g n al en erg i es of t h et yp e a s s u m e d h e r e c a n b e s a t i s f a c t o r i l y

d et ec t ed . W i t hi n t h e p as t few y e ar s a s i m i l a rt y p e of rece i v i n g sy s t em was d ev el op ed fo ra p p l ic a t io n t o a p r o g r a m i nv o l v in g a p r o j e c t i l et o o s m a l l t o c a r r y a s o u r c e of r a d i o f r e q u en c y e n e r g y . T h e i n s t r u m e n t a t i o n s y s t e mthat was developed depended upon energyref l ect ed f rom t h e p r o j ec t i l e . In t h e t es t st ha t w e r e c o n d u c te d a p r o j e c t i l e i l l u m i n a t in g so u r ce of on l y 35 wat t s o f p ow er a t af req u en cy of ap p r o x i ma t el y 1 25 mc p s wasu s e d .

F i g . 6 sh ow s b ot h t h e t h eo re t i cal an dm eas u r ed s i g n al en er g i e s as a fun cti on ofg ro u n d ra n g e f ro m l au n ch . It i s t o b e n ot edt h at t h e measu red en erg y l ev el i s s l i g h t l yh i g h er t h an t h e t h eo re t i cal v al u e b u t i s a lway swi t h in 6 d b of th e l a t t e r . T h e t rac k i n g f i l t e r ,with an effect ive no ise bandwidth of 40 cp s,f i r st locked on the sig nal at a level of -177dbw and co ntinued to t ra ck while the signali n cr eas ed t o a h i g h er l ev el an d t h en d ec re ase dto a level of -180 dbw. Re call ing that thep r o p o s e d t r a c k i n g f i l t e r f o r t he i o n o s p h e r eexperiment wil l have a bandwidth of 0 .4 cpso r o n e - h u n d r e d t h o f 4 0 c p s , t h e n o i s e p o w e r

lev el sho uld be 20 db le ss , enabling this f i l terto w o rk down to a leve l of -197 dbw. Sincet h e t h e o r e t i c a l m i n im u m r e c e i v e d p o w e r i s-1 78 d b w t h e p rop osed sy s t em sh ou l d y i e l da Sp /Np ma rg i n of 1 9 d b o r a S v / Nv ra t i o

of about 10/1 which che cks with the valueo f 1 0 / 1 p r e v i o u s l y p r e d i c t e d .

F i g . 7 i s a r e p r o d u c t i o n o f a n o s c i l l o g r a mshow ing the no isy sig nal a t the input to thet r ac kin g f i l ter having a bandwidth of 40 cpsand the f i l tered output signal .

T h e c o r r e l a t i o n f un c ti o n , a n a n a lo gv o l t ag e, g i v es a co n t i nu ou s i n d icat i on ofh ow wel l t h e ou t p u t s i g n al f rom t h e osci l lato r is be ing held in pha se-lo ck with the

i n pu t s i g n al and may b e u sed as a me as u r eof confidence in the val idi ty of the f i l teredDop p l er s i g n al ou t p u t .

T h e ex p er i en c e g ai ned t h rou g h t h e d e velo pm ent and field te st of this pro to typep h ase - l o ck ed t ra ck i n g f i l t e r l eav es l i t t l ed o u b t t h a t t h e c h a r a c t e r i s t i c s r e q u i r e d i n a ni m p r o v e d f i l t e r f o r t h e s a t e l l i t e i o n o s p h e r ee x p e r i m e n t c a n b e a c h i e v e d . T h e a c h i e v e men t wi l l con st i t u t e an ad v an ce of t h e s t a t e-o f - t h e - a r t a nd w i ll b r i n g u s o n e s t e p c l o s e rt o w a r d t h e a t t a i n m e n t o f e l e c t r o n i c i n s t r u

m e n t a t i o n s y s t e m s c a p a b l e o f t r a c k i n g v e r yl o n g r a n g e m i s s i l e s - a nd s p a c e s h i p s .

REFERENCES

1 . Ber n i n g , W . W ., " T h e De t erm i n at i o n of C h ar g e Den si t i es in t h e Ion o sp h ere b y Rad i oD o p p le r T e c h n i q u e s , " B a l l i s t i c R e s e a r c h L a b o r a t o r i e s , .A b er d ee n P r o v i n g G r o u n d ,M a r y l a n d . P u b l i sh e d i n " R o c k e t E x p l o r a t i o n o f t h e U p p e r A t m o s p h e r e , " P e r g a m o nP r e s s L t d . , L o n d o n .

2 . G o e r k e , V . H . , U n p ub li sh e d c o r r e s p o n d e n c e .

3 . H o n e y , J o h n F . , " D e t e c t i o n T e c h n iq u e s in D o p p l er T r a c k i n g S y s t e m s , " F i n a l R e p o r t ,C o n t r a c t D A - 0 4 - 4 9 5- O R D - 2 7 8 f o r B a l l i s t i c R e s e a r c h L a b o r a t o r i e s , A b e r d e e n P r o v i n gG r o u n d , M a r y l a n d .

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ON THE WAY TO AUTOMATED PROCESSING OF FLIGHT MEASUREMENTSD r . W . E , K l e m p e r e r *

SUMMARY

This pa pe r p r e s e n ts a s ur v e y o f the me tho ds a va i la ble fo r a uto m a te d me a s ur ing a ndp r o c e s s i n g o f f li gh t d a t a . T h e m e a s u r e m e n t s n e c e s s a r y t o d e s c r i b e t h e b e h a v io r o f a t e s tm is s i le a r e d iv ide d in to th r e e g r o ups , na me ly , tho s e tha t c a n be o bta ine d f r o m a g r o undstatio n, th o se that can be taken only by dev ices located on the tes t m iss ile itself, and thosetha t c a n be o bta ine d c o nve nie nt ly f r o m a no the r a i r c r a f t . The me tho ds a va i la ble fo r e a c hg r o u p a r e th e n d i s c u s s e d s e p a r a t e l y .

SOMMAIRE

Dans cette note 1'auteur do nne un ta ble a u ge ne r a l d e s me tho de s u t i l i s a b le s po ur lam e s u r e e t 1'evaluation a u t o m a t i q ue d e 1 'i nf o r m a ti o n e n v o l . L e s m e s u r e s n e c e s s a i r e s p o u rd e c r i r e l e c o m p o r t e m e n t d ' u n m i s s i l e e n e s s a i s o n t d i v i s e e s e n t r o i s g r o u p e s , c e l l e s p o u v a n te t r e o b t e n u e s a p a r t i r d ' un p o s t e a u s o l , c e l l e s p o u v an t e t r e e n r e g i s t r e e s s e u l e m e n t a 1'aide

d ' o r g a ne s p la c e s da ns le m is s i le lu i -m e m e , e t c e l le s qui pe uvent e t r e a is e m e nt o b te nue s apa r t i r d ' un a ut r e e ngin vo la nt . Le s me tho de s u t i l i s a ble s da ns c ha que gr o upe s o nt e ns ui tet r a i t £ e s s e p a r e m e n t .

W hile the s c o pe o f th is s ympo s ium isGuidance and Control and a major i ty of thepa pe r s p r e s e nte d de a l wi th the the o r y o fclosed loop guidance and control , the subjecto f th is pa pe r c o nc e r ns tho s e te s t f light m e a s ur e m e nts f r o m which the r e i s no fe e dba c kto the s a me f ly ing ve hic le . Ho we ve r , the r eshould be feedback of sophist icated intell ige nc e f r o m o ne te s t m is s io n to a s ubs e que ntmis s io n in o r d e r tha t a s e r ie s o f f light te s tsm a y r e f l ec t s y s t e m a t i c p r o g r e s s . O b v io u s lythe n, the r e mus t be s o me time a l lo we d fo r theana lysi s and diges t of the data ga the re d onany flight and for the application of thele s s o ns le a r n e d. Eve n s o , the pa c e o f de ve lo pme nt o f ne w pr o je c ts i s s uc h a s todemand that this digest t ime be minimizeda nd tha t a l l tho s e c ho r e s be me c ha niz e d

which do no t r e qu i r e huma n judgme nt . Theva lue o f e la bo r a te f l ight te s t ins t r ume nta t io nha s o nly g r a dua l ly be e n r e a l iz e d.

Lord Kelvin has said that we can beginto unde r s ta nd o nly tho s e phe no me na whic hwe ha ve le a r ne d to me a s ur e qua nt i ta t ive ly .This c e r ta in ly a ppl ie s to the de ve lo pme nt ofg u id ed m i s s i l e s . T h e f a s t e r o u r v e h i c l e sha ve be c o me , a nd the mo r e de ta c he d f r o mM o t h er E a r t h , t h e m o r e w e m u s t r e l y o ni n s t r u m e n t s t h a t a r e q u ic k e r a nd m o r e p r e cis e than the five se ns es w ith which natur ee ndo we d us . In the e a r l ie s t a i r c r a f t whichflew at s pe ed s le ss tha n 100 mp h, the pilo twa s a ble to o bs e r ve a nd jo t do wn o n a kne e -pad the few m e a s ur e m e n ts tha t we r e ne c e s s a r y t o d e t e r m i n e t h e b a s i c p e r f o r m a n c e a nd

"Douglas Air craft Co mpany, Inc. , Santa Mo nica, Calif o rnia .

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c o n t r o l b e h a v i o r of his c r a f t . H o w e v e r , w i thn ew su p e rs o n i c a i r c r af t t h i n g s h ap p en too

quickly, and in u n m a n n e d m i s s i l e s w h i c h are

g u i d e d r e m o t e l y or a u t o m a t i c a l l y t h e r e is

n ob od y a l on g to r e a d g a g e s . Y e t t h e i r g r e a t e r

c o m p l e x i t y and p r e c a r i o u s n e s s c a l l s for inc r e a s i n g l y i n t r i c a t e and e l a b o r a t e m e a s u r e m e n t s to d e v e l o p new s y s t e m s , t o p r o v e t h e irp r o p e r f u n c t i o n and the r e l i a b i l i t y of all

c o m p o n e n t s , or to d i s c o v e r u n f o r e s e e n dif

f i cu l t i es and f i n d r e m e d i e s for t h e m .

D u r i n g the d e v e l o p m e n t of the G e r m a nV - l , it was s t a t ed t h at n ear l y 500 w e r elaunched in i n s t r u m e n t e d p r a c t i c e and t e s tflights until an a c c e p t a b l e d e g r e e of r e l i a b i l i t y ag ai n s t all h a z a r d s was a c h i e v e d . In

g e n e r a l it is now p ret t y wel l accep t ed t h atany new f l ight vehicle, manned or u n man n ed ,m u s t go t h r o u g h an i n t en si v e d ev el op men ts t a g e d u r i n g w h i c h e l a b o r a t e i n s t r u m e n t a t i o nm u s t be p r o v i d e d to b r i n g h o m e a c o m p l e t er e c o r d of the function of all its s y s t e m c o m p o n e n t s u n d e r s y s t e m a t i c a l l y v a r i e d c o n d i t i o n s so t h at the m o s t a p p r o p r i a t e b a l a n c eof all s i g n i f i c a n t p a r a m e t e r s can be e s t a b l i s h e d .

T h e m e a s u r e m e n t s w h i c h are needed to

d escr i b e fu l l y the b e h a v i o r of a v eh i cl e in

flight can be d i v i d e d i n t o t h r e e g r o u p s f r o mthe viewpoint of i n s t r u m e n t a t i o n l o g i s t i c s :

(1) Tho se that can be o b s e r v e d and

m e a s u r e d f r o m a g r o u n d s t a t i o nd i r e c t l y .

(2) T h o se t h at can be t ak en on l y by

s e n s i n g d e v i c e s l o c a t e d on the

f lying test vehicle itself.

( 3 ) T h o s e t h a t ( s o m e t i m e s ) can be

a d v a n t a g e o u s l y m e a s u r e d f r o ma n o t h e r a i r c r a f t .

T h e f ir s t g r o u p c o v e r s the d e t e r m i n a t i o nof the t r a j e c t o r y of the v e h i c l e w i t h r e s p e c tto the e a r t h , by m e a n s of o p t i c a l m e t h o d ssu ch as p h o t o t h e o d o l i t e s or c i n e t h e o d o l i t e so r by r a d i o or r a d a r . G r o u n d v e l o c i t i e s can

b e d er i v e d (wi th i n li mi t ed acc u r acy ) by diff e r e n t i a t i o n f r o m p o s i t i o n c o o r d i n a t e h i s t o r i e s or d i r e c t l y d e t e r m i n e d by D o p p l e rt e c h n i q u e s .

T h e s ec o n d g r o u p c o m p r i s e s a ll m e a s u r e m e n t s of the o p e r a t i o n of c o m p o n e n t s , ( c o n t r o l s u r f a c e e x c u r s i o n s , t o r q u e s or m o m e n t s ,f o r c e s , t e m p e r a t u r e s , p r e s s u r e s , f u e l f l o w ,a i r s p e e d s , a n g l e s of at t ack and yaw, a c c e l e r a t i o n s , a nd r o t a t io n and att i tude) as well as

amb i en t con d i t i on s (su ch as e x t e r n a l air

p r e s s u r e , t e m p e r a t u r e , a nd o t h e r a e r o l o g i c a lp r o p e r t i e s ) .

T h e t h i r d g r o u p is p a r t i c u l a r l y a p p l i c a b l eto the g u i d an ce of a n t i a i r c r a f t t e s t m i s s i l e sto an air i n t e r c e p t of a rea l f l y i n g t a rg etw h e r e p r o o f of the end g a m e m u s t be b a s e do n m e a s u r e m e n t s of the r e l a t i v e m o t i o n of

th e two f lying objects in c l o s e d e t a i l .

L e t us c o n s i d e r f i r s t the t r a j e c t o r y de

t e r m i n a t i o n f r o m g r o u n d s t a t i o n s . For

a r t i l l e r y p u r p o s e s t he t i m e - i n t e r r u p t e d t r a c in g of b a l l i s t i c s h e l l t r a j e c t o r i e s by m e a n so f f i x e d - p l a t e c a m e r a s and p h o t o g r a m m e t r i cev al u at i on of the p i c t u r e s t ak e n f r o m two

con j u g at ed s t a t i on s at f i rst sufficed; it has

b e e n p r a c t i c e d for m a n y d e c a d e s . The

g r e a t e r r e s o l u t i o n and a c c u r a c y d e s i r e d for

t h e r e c o r d i n g of a i r c r a f t m a n e u v e r s and for

a n t i a i r c r a f t w o r k led to the d e v e l o p m e n t of

b a l l is t i c s t r i p c a m e r a s for the takeoff phasean d c i n et h eod ol i t es (n ot ab l y by Askania in

G e r m a n y and by M i t ch el l in USA and l a t e rb y C o n t r a v e s in S wi t zer l an d ) for the flight

p h a s e . W ith t h e s e i n s t r u m e n t s , the flyingob j ect is man u al l y t rack ed by watching it

t h r o u g h a t e l esco p e w h i le a i mi n g a t e l esco p i cc a m e r a a p p r o x i m a t e l y at it. At r e g u l a r in

t e r v a l s , p i c t u r e s are taken which furnish a

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te le pho to ima ge o f the o bje c t a ga ins t a c r o s sha i r a nd a ls o a pho to gr a phic impr in t o f thesca le r eading s of azimuth and elevatio na ngle s o f the te le s c o pe a xis .

F r o m t h e s e p i c t u r e s , t h e t i m e h i s t o r i e sof the t rue azimuth and elevation of the l ineso f sight to the o bje ct f r o m two o r m o r es ur ve ye d the o do l ite s ta t io ns a r e de te r min e dand the t rue posit ion his tory of the object isr e c o ns t r uc te d a t s u i ta ble in te r va ls by t r ia ngula tio n r o ut ine s f r o m s imul ta ne o u s s e ts o fs uc h a ngle s . R e dunda nc ie s a r e r e s o lve d byle a s t s qua r e s o r we ighte d a ve r a ging me tho ds .Lo ngha nd, th is wa s a ve r y te dio us p r o c e d ur e .P u n c h c a r d m a c h i n e s a n d l a t e r e l e c t r o n i ccalculating machines reduced the t ime and

te dium o f the t r igo no me t r ic c a lc ula t io n to ave r y a c c e pta ble minimu m. In o r d e r a ls o tofacil i tate and expedite the reading of manyindividua l c ine the o do l i te f r a me s , s e ve r a ls te ps w e r e in t r o duc e d a t the va r io us te s tr a nge s , be ginning a bo ut s e ve n ye a r s a go .The s e inno va t io ns ve r y ma te r ia l ly r e duc e dt h e t i m e r e q u i r e d t o r e c o n s t r u c t a t r a j e c t o r y .

The fi r s t advance was acco mp lished bya n ins t r ume nt c a l le d the Ic o no lo g , a p r o je c

to r wi th me c ha nic a l ly ma nipula te d c r o s sha i r s a nd a pa i r of a na lo g- to -digi ta l c o nve r te r s , the x and y c o o r dina te o utput o fwhic h wa s a uto ma t ic a l ly t r a ns fe r r e d to ar e la y me m o r y a nd the nc e to a n e le c t r i ctype wr i te r o r a n a uto ma t ic c a r d punc h o rbo th . The c r o s s ha i r wa s po s i t io ne d o ve rthe ima ge of the o bject by hand and ey e,deliberately, to employ judgment in al lowingfo r the d i ffe r e nt a ppe a r a n c e unde r d i f fe r e ntcondit ions of range aspect, i l lumination, andc o nt r a s t a nd to ins ur e t r a c king the de s i r e dpa r t o f s e pa r a te s ta g e s . The ma c hine s we r ealso equipped with keyboards on which ther e a d e r c o uld ke y in a l l num e r ic a l info r m a t io nrea d off the pic tur e such as azim uth, ele va t io n , a t t i tude , r un numbe r , and p ic tu r e numbe r (o r t ime ) . Th is info r m a t io n wa s r e ta ine d

o n a s ho r t m e mo r y a nd then e nte r e d a uto ma ti c a l l y o n t h e t y p e w r i t e r r e c o r d a n d / o r o nthe punche d c a r ds . In la te r ma c hine s thereading of azimuth and elevation scales wasr e duc e d to ma nua l s c a le ma tc hing whic h r e

l ie ve d the o pe r a to r o f the ta s k o f in te r po la t inga nd ke ying numbe r s .

M a c hine s o f th is ge ne r a l type we r e de ve lo pe d by Do ugla s, Ge nis c o , B e n s o n-Le hne r ,Coleman, Telecomputing Inc. , e tc . , and thes a m e ide a ha s be e n a ppl ie d to mic r o s c o picc o mpa r a to r s fo r the r e a ding o f the f i lmsta ken with ba l l i s t ic c a m e r a s . The ma c hine sdiffer in var ious mechanical and optical feat u r e s . Most of them display the magnifiedp i c t u r e o n a t r a n s l u c e n t o r o p a q ue s c r e e n

(which is les s t i r ing than looking thro ugh am i c r o s c o p e ) ; s o m e m o v e t h e fi lm o n am i c r o s c o p e s t a g e , o t h e r s m o v e a c r o s s h a i rbe hind the s c r e e n; a nd s t i l l o the r s p r o je c ta lumino us index spot upon the pic tur e. Somema nipula te th is m o t io n by ha nd; o the r s mo to r iz e it a t va r ia ble s pe e d. W he r e i t i sde s i r e d to r e c o r d the a ppa r e nt a t t itude o fthe o bje c t in the p ic tur e , a n a ngle -me a s ur ingfeatur e is added, ei th er in the fo rm of at u r n t a b l e w h i c h r o t a t e s t h e f i l m p r o j e c t o r o rthe f i lm ga te , o r a c r o s s ha i r o r lumino us

l i n e . This r o ta t io n i s the n d igi t iz e d, c o mm i t t e d t o a r e l a y m e m o r y , a n d r e c o r d e da u t o m a t i c a l l y .

Of c o ur se, i t is temp ting to do away withthe human link in the reading of the film.This is readily feasible only when the objecta ppe a r s a s a we l l -de f ine d po int o f c o nt r a s ta ga ins t i t s ba c kgr o und. So me type pho to e le c t r ic s e ns ing de vic e a nd a te le vis io n- l ikes c a nning pr o c e dur e c a n be e mplo ye d, but thepr o pe r a djus tme nt to the d ime ns io ns o f thes po t a nd i t s c o nt r a s t i s a de l ic a te p r o ble m a tb e s t a n d r e q u i r e s s o m e h u m a n s u p e r v i s i o no r mo ni to r ing. It i s e ve n mo r e te mpt ingto do a way wi th the t r a c king e r r o r a l to g e the r ,and thus el iminate the necessity of reading

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or g ag i n g t h e ap p aren t d i sp l acemen t of t h eo b j ect i m ag e f ro m t h e op t i cal ax i s of t h et ra ck i n g t h eod o l i t e so t h at t h e wh ol e p h o t og rap h i c s t ep can b e d i sp en sed wi t h .

T h e i n t ro d u ct i o n of v el oci t y feed b ack fo r" a i d e d t r a c k i n g " w e nt a lo n g w ay t o a s s i s td i e t r a c k e r s i n f o l lo w i n g t h e i r o b j e c ti v et h ro u g h t h e sk y . Al th ou g h so m e d es i g n er sof eq u i p men t feel con f i d en t t h at p erfect man u a l t r a c k i n g c a n t h u s b e a c c o m p l i s h e d , o t h e r sr e l y o n w h o l l y a u t o m a t i c t r a c k i n g w hi chp o s i ti v e l y e l i m i n a t e s h u m a n e r r o r s a nd f a t i g u e . J u s t a s a u t o m a t i c s t a r t r a c k e r s h a v eb e e n m a d e s u c c e s s f u l l y , s o a f l a r e c a r r i e do n a m i s s i l e c an b e t r a c k e d a u t o m a t i c a l lya n d a c c u r a t e l y b y m e a n s o f a p h o t o e l e c t r i c

se ek er a t n i g h t. It i s on l y a m at t e r o f p r o v i d i n g a g ood b al an ce b et ween reso l u t i on ,s l ewi n g sp eed cap ab i l i t y , an d d amp i n g .

In daylight , i t is m o r e difficul t to supplysu ff i c i en t l i g h t i n t en si t y an d con t ras t esp eci a l l y fo r h i gh a l t i t u d e o r l on g ra n g e wo rk .In f ra re d se ek er s can a l so b e b u i lt t o l ock onm i s s i l e s c a r r y i n g a h o t f l a r e . H o w e v e r ,o n c e t h e t a r g e t p h o t o g r a p h y i s r e n d e r e d u n n e c e s s a r y , i t b e c o m e s d e s i r a b l e t o e l i m i n a t e

t h e p h o t o g r a p h i c p r o c e s s e n t i r e l y a n d s u b s t i t u t e o t h e r m e a n s f o r t h e r e t e n t i o n o f t h et h eod o l i te an g l e i n fo r mat i o n , t o d i g i t i ze i t ,a n d a u t o m a t i c a l l y r e c o r d i t a t s u i t a b l e i n t e r v a l s i n rea l t i me on th e f ly . T h i s i s ana m b i ti o u s e n t e r p r i s e b e c a u s e t he r e s o l u ti o nr e q u i r e d i s v e r y h i gh ; t o o b t a i n r e s o l u t i o nof 1/10 mil would require 64,000 digi t izeds t a t i o n s a l o n g o n e c o m p l e t e c i r c l e , a n d e v e nt h i s fu r n i sh es an accu ra cy of n o b et t e r t h anabout 10 to 50 feet at 20 to 100 m ile s r an ge .

T h e u l t i mat e i n t rack i n g d ev i ces sh ou l db e t h e m o d e r n a n t i a i r c r a f t r a d a r . It f u r n i s h e s i t s o w n r a n g e in f o r m a t io n a n d r e q u i r e sn o con j u g ati on of sev e r a l s t a t i o n s ; b u t t oa c h i e v e a n g u l a r a c c u r a c i e s e x c e e d i ng t h o s eo f o p t i c a l t e l e s c o p e s it r e q u i r e s r a t h e r

e l ab ora t e i n s t a l l a t i on s on t h e g rou n d , an d , o fc o u r s e , t r a n s p o n d e r s c a r r i e d o n th e m i s s i l e .T h e m o d e r n a l l - e l e c t r o n i c a u t o m a t i c t r a c k i ng r a d a r f u r n i s h e s i t s o w n g r a p h i c r e c o r dof t h e com p u t ed f li g ht t ra j ect o ry of t h e

t ra ck e d ob j ect i mm ed i at e l y , an d i t can b eeq u ip p ed t o p lo t v el oci t y co mp o n en t s as wel l .W h e r e a h i g h e r a c c u r a c y o r r e s o l u t io n t ha nt h at g iv en b y a l a r g e au t o mat i c p l o t t i n g b o ardi s r eq u i r ed , t h e t rack i n g an g l e and ra n g ei n f o r m a t io n i s t r a n s f o r m e d i n t o d i g i ta l f o r ma n d r e c o r d e d a s s u c h a t s u i t a b l e i n t e r v a l s .

Ho we ver , i t of ten p ays to equip even afu l l y au t omat i c t rack i n g rad ar wi t h t e l e-p h o t o g r a p h i c b a c ku p i n s t r u m e n t a t i o n . A

p i c t u r e , w h ic h a c c o r d i n g to a C h i n e s e p r o v e r bi s w o r t h 1 0 ,0 0 0 wo rd s , an d acco rd i n g t o t h et e l ev i s i on en g i n eer ab ou t 3 0 0 , 0 0 0 p oi n t s , cant e l l t h e s t o ry of a d ramat i c ev en t su ch ast h e s e p a r a t i o n o f a m i s s i l e f r o m a b o o s t e rs t a g e , o r a n a c c i d e n t a l b r e a k u p m o r e v i v i d l yt h an a cod ed s i g n al . T h e p i c t u r es can a l soserv e t o v er i fy t h e t ru e t rack i n g ev en i f t h eyn e e d n o t b e p h o t o g r a m m e t r i c a l l y e v a l u a t e d .O t h e r w i s e , t he y c a n b e e v a l u a te d a s " b o r e s i g h t " p i c t u r e s i n t h e s a m e m a n n e r a s c i n e t h eod o l i te p i c t u r es t o t e l l t h e s t o r y of t h et r a c k i n g e r r o r a n d t o f u r n is h c l u e s f o r i m p r o v e m e n t s o f th e t r a c k i n g s y s t e m . A t a n yr at e, i t can be ant ic ipated that within a fewy e a r s , fu lly a u t o m a t i c g r o u n d - t o - a i r t r a j e c t o r y a n d v e l o c i t y m e a s u r e m e n t e q u i p m e n twi l l b e av ai l ab l e a t a l l maj o r t es t ran g es fo ra i r c r af t an d mi s s i l es ; ev en t u all y i t wi l l b e c o m e s t a n d a r d i z e d . In t h e m e a n t i m e , m u c hwo rk w i l l con t i n u e t o b e d on e wi th p r es en t -d ay s em i au t o ma t i c eq u i p men t i n wh ich t h eh u m a n o b s e r v e r , a i d e d by d e v i c e s th a t r e d u ce b ored om an d fa t i g u e, su p p l i es j u d g men tan d i n t e l l i g en ce.

R a d i a l c o m p o n e n t s g r o u n d - t o - a i r v e l o c i t ym e a s u r e m e n t s by D o p pl er o r o t h e r s i gn a lp h a s e c o m p a r i s o n t e c h n i q u e s s u c h a s R a y d is t

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c an b e r e c o r d e d in t r u e t i m e . T h e r e c o r d so f pha s e d i f fe r e nc e m e a s u r e me nts , ho we ve r ,which m us t be r e fe r r e d to hype r bo lo ida l c o o r d i n a t e s in o r d e r t o r e c o n s t r u c t t r a j e c t o r i e s , r e qui r e c o unt ing a nd c o ns ide r a blepr o c e s s ing a nd c o m put ing. One mus t e xe r c is e c a r e ful judgme nt a nd/o r ta ke a dva nta ge o f r e dunda nc ie s whe ne ve r a ny r e c o r d i si n t e r r u p t e d . E l e c t r o n i c c a l cu l a ti n g m a c h i n e sa r e i n d i s p e n s a b l e h e r e ; s e m i a u t o m a t i c t r a c er e a ding a ids a r e in us e but c o m ple te a uto ma t iz a t io n wi l l de pe nd o n pe r fe c t r e l ia bi l i ty .

Let us now turn to the second category ofm e a s u r e me nts ; na me ly tho s e whic h mu s t beg a t h e r e d b y i n s t r u m e n t s c a r r i e d a b o a r d t h e

test vehicle. Such instru me ntatio n mu st bede s igne d to be a c c o m mo da te d o n the ve hic lewithout inte rfe re nce with i ts function. Hence,a i r b o r n e ins t r um e nta t io n c a nno t s imply bebought and put aboard; il must be plannedfo r in the e a r ly de s ign s ta ge . As a m a t te rof fact , during i ts br ief his tory, the philosophyo f a i r bo r ne f l ight te s t ins t r ume nta t io n ha sunde r go ne dr a s t i c c ha n ge s . In the be ginning,pe o ple we r e r e luc ta nt to lo a d a i r p la ne s o rm i s s i l e s w i t h i n s t r u m e n t s .

When the necessity was recognized, testengineers were at f i rs t satisfied with putt ingr e la t ive ly p r im i t ive a nd ine xpe nsive ins t r ume nts a bo a r d c o ns ide r ing tha t the inve s tme ntwould be of uncertain value at best; l i t t le o rno at tention was paid to the cost in money,t i m e , and effort which la te r would be spentto e va lua te the r e s ul ts . Gr a dua l ly , the p r o ble ms o f ma king a nd ins ta l l ing r e l ia ble t r a ns ducers were solved and efficient methods fort h e t r a n s m i s s i o n , r e c e p t i o n , a n d p r e s e r v a t i o n

o f r e c o r d s we r e de ve lo pe d. This s o o n ledto the a c c umula t io n o f g r e a t ma s s e s o f da tawhic h e ve ntual ly g r e w to s uc h pr o po r t io nsthat i t became impossible to keep up withthem, to read all of them, to o rganize andplo t the r e s ul ts , a nd to in te r pr e t a nd e va lua te

the m in t ime to s e r ve a us eful pur po s e be fo r ee ngine e r ing de c is io ns r e ga r ding a djus tme ntso r cha nges for the next test s had to be ma de.

The ma gni tude o f the jo b i s r e a di ly r e a l

ized when i t is considered that in one nows ta nda r diz e d me tho d o f te le me t r y a bo ut 1000da ta a r e c o l le c te d pe r s e c o nd s o tha t in asingle flight of two to ten minutes, 100,000t o 5 0 0 , 0 0 0 m e a s u r e m e n t s f r o m t h i s s o u r c ealon e ar e co llected. They would have to ber e a d , s i f t e d , d e c o m m u t a t e d , p r o c e s s e d a c c o r ding to c a l ib r a t io n , p lo t te d , inde xe d, c r o s splo t te d , a nd c o r r e la te d wi th info r m a t io ng a t h e r e d by o t h e r i n s t r u m e n t a t i o n i n c o m p u t a t i o n s o f r e s p o n s e c h a r a c t e r i s t i c s a n dper fo rm anc e to exploit fully the wealth ofinfo r m a t io n ga r n e r e d. A who le new indus t r yha s de velo pe d a s the r e s ul t o f va r io us e ffo r tsto fa c i l i ta te the s e ta s ks a nd to o ve r c o medela ys by red ucing the amo unt of human effo r t invo lve d in the s e p r o c e s s e s th r o ughp r o g r e s s i v e m e c h a n i z a t i o n a n d a u t o m a t i o n .This de ve lo pme nt a nd i t s s igna l s uc c e s s no wbegins to exert i ts own influence upon thepla nning o f ins t r ume nta t io n fo r ne w pr o je c tsand new fac il i t ie s . Obvio usly, I t now paysto s o de s ign the ins t r um e nta t io n in the f i r s tplace that it lends i tself b ett er to expedit io usauto matic evaluatio n even if the installat io n

in the individua l ve hic le c o s ts mo r e .

The e a r l ie s t te c hnique pe r fe c te d fo r o bta in ing f r e que nt r e c o r d s o f ins t r um e nt indicatio ns ab o ard a flying v ehicle was to takem o t i o n p i c t u r e s o f i n d i c a t o r s a r r a y e d o n a ninstrument panel and suitably i l luminated.This me tho d, s t i l l s ta nd a r d in ma ny pla c e s ,wa s de ve lo pe d to r o ut ine pe r fe c t io n fo rma nne d a i r c r a f t . Fr o m the r e i t wa s a da pte dto unma nne d mis s i le s , but he r e the p r o ble mo f s a lva ge o f the f i lm r e c o r ds a r o s e whe r e

no pr o visio n for an intact landing could bem a d e . F i l m s w e r e p r o t e c te d b y a r m o r e dc a s e s a nd r e t r ie ve d f r o m the impa c t wr e c kag e, i n s o m e i n s t a n c e s by p a r a c h u t e r e co ver y, but, in any ca se, a tedio us and cum b e r s o m e p r o c e s s . O n ce t h e f i lm s w e r e

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d e l i v e r e d a nd p r o c e s s e d , t he y h ad t o b e r e a d ,wh en mag n i f ied b y a p ro j ect o r su ch as aReco rd ak , on a sc re en con v en i en tl y p os i t io n edi n f r o n t o f t h e r e a d e r .

H e r e , t o o , m e c h a n i c a l a i d s s u c h a s al u mi n ou s p oi n t er man i p u l at ed i n t o co i n ci d en ce wi t h th e sc r ee n i mag e of an y d i a l o rscal e i n s t ru men t an d t h en au t omat i cal l y d i g i t iz i n g an d rec o rd i n g t he r ead i n g (as in t h eB e n s o n - L e h n e r B o s c a r ) o n t h e b a s i s of p r e s e t z e r o a nd s c a l e c a l i b r a t i o n h e lp e d s i g n i f ican t ly t o ex p ed i t e t h e read i n g p r o ce ss an dr e d u c e h um a n e r r o r s ; it w a s a n im p o r t a n ts t ep t oward au t omat i zat i on a l t h ou g h s t i l lq u i t e s h o r t o f c o m p l e t e a u t o m a t i o n .

I n s t e a d o f p h o t o g r a p h i c r e g i s t r a t i o n , t h ei n s t r u m e n t i n d i c a t i o n s w e r e s o m e t i m e s c o n v e r t e d o n b o a r d i nt o f r e q u en c y m o d u l at e d o rd i g i t i zed s i g n al s an d record ed on mag n et i ct ap e. T h e r e t r i e v i n g c h o r e r e m a i n e d , b u tt h e d an g er of l os i n g t h e record b y acci d en t alexposure of the f i lm was avoided and the t imereq u i red fo r d ev el op i n g f i l m was sav ed . Ar a d i o a c t i v e s o u r c e c a r r i e d o n t h e r e c o r d e rca se was found useful to fac i l i tate lo cat ingt h e i mp ac t p l ace on a v as t t es t ran g e. On ce

m a g n e t i c t a p e w a s s a l v a g e d , t h e r e c o r d o ni t cou l d b e p l ay ed b ack , p r o ces sed , and d i sp l ay ed on a l l -au t o m at i c d ev i ce s wi th ou tfu r t h er h u man ef fo r t . T h i s met h o d i s s t i l ls t a n d a r d w i t h s o m e e x p e r i m e n t a l m i s s i l e s .

T h e r e t r i e v i n g p r o b l e m w a s e l i m i n a t e d byr a d i o - t e l e m e t r y w h i c h , d u r i n g t h e l a s t 1 0y e a r s , h a s g r a d u a l l y r e p l a c e d a i r b o r n e r e c o r d i n g . S e v e r a l s y s t e m s h a v e e m e r g e d i n top r a c t i c a l s e r v i c e a nd a t ta i n e d a c c e p t a b le

re l i ab i l i t y . T h ey a r e u su al l y d i s t i n g u i sh edby the metho d of m ult iplexing the many ind i v i d u al measu r i n g ch an n el s i n t h e s i g n alt r a n s m i s s i o n o v e r t h e r a d i o c a r r i e r l i n k .T h ese m et h o d s l en d t h em sel v es d i f fe ren t l yt o m e c h a n i z a ti o n o r a u t o m a t i z a ti o n o f t he

d a t a r e d u c t i o n w h ic h c o m p r i s e s t h e s t e p s o fu n s c r a m b l i n g ( d e m u l t i p l e x i n g ) , r e a r r a n g i n g ,l i n ear i z i n g , re ad i n g , p l o t t i n g , and u ti l i z in gt h e d at a .

T h e most wi d el y accep t ed mu l t i p l ex i n gme t h o d s a r e (a) sh ar i n g a f req u en cy ban db y s u b c a r r i e r f r e q u en c y m o d u la t io n , ( b) t i m es h a r i n g b y c o m m u t a t i o n ( s u c c e s s i v e s a m pling), and (c) by combinations of these. Thech o i ce of t h e mu l t i p l ex i n g met h od a l so af fectst h e ch o i ce of t h e t y p es of t r an sd u c er s to b ei n s t a l le d a b o a r d t h e v e h i c l e . T r a n s d u c e r sa r e t h o s e i n s t r u m e n t s w h ic h t r a n s l a t e t heo r i g i n a l r e s p o n s e o f t he m e a s u r i n g s e n s o r( p r e s s u r e s y lp h o n , m o v e m e n t g a ge , t o r q u e -m e t e r , d y n a m o m e t e r . B o u r d o n tu b e , t h e r

m o m e t e r , d i l a t o m e t e r , f lo w g a g e , v a n e , a c c e l e r o m e l e r , g y r o , e t c .) i nt o a n e l e c t r i c a lq u an t it y su ch as a DC v o l t ag e; h en ce, a co n d e n s e r c h a r g e , o r a n i nd u c ta n c e o r r e a c t a n c e ,an d h en ce an e l e ct r i c o sci l l a t i on f r eq u en cy .

F r o m a n o t h e r v ie w p o i n t, t he v a r i o u sm e a s u r e m e n t s m a y b e d is t in g u i s h e d a s t o t h eres o l u t i o n i n q u an ti ty an d f req u en cy re q u i r ed .S om e mea su r ed v al u es ch an g e b ut s lo wl y andmon ot on i cal l y ; e . g . , t h e su p p l y of fu el , p ressu r e med i u m , e t c . , so th at i n freq u en t sa m

p l in g su ff i ces . Ot h er s may v ar y fa i r l yr a p i d l y in c o n t r o l l e d m a n e u v e r s o r in t h ep e r f o r m a n c e o f a u t o m a t i c s t a b i l i z a t i o n o rr e m o t e g u id a n c e; e . g ., c o n t r o l m o v e m e n t s ,t o r q u e s , f o r c e s , a c c e l e r a t i o n s , a n g l e s o fa t t a c k ; a n d , t o a l e s s e r d e g r e e , a i r s p e e d s ,a m b i e n t a i r d e n s i t y , a nd t e m p e r a t u r e . F i n al l y , t h ere a re t h ose t h at a t t a i n h i g h f req u en c i es f ro m 3 0 t o o v er 1 00 0 cp s; v i z . ,f l u t t e r , v i b r a t i o n s , a n d s t r u c t u r a l o s c i l l a t i o n s . T h e s e a r e b e l t e r c a ug h t b y c o n t i nu o u sr e c o r d i n g t h a n b y s a m p l i n g .

F o r c o n t i n uo u s r e c o r d i n g , i t h a s b ee nfound p r ac t ic al to mu lt iplex six and up to12 i n s t r u m e n t c h a n n e l s o v e r o n e c a r r i e r l i nk ,b y t h e F M -F M t ech n i q u e. E ach ch an n el i sa s s i g n e d a s o n i c s u b c a r r i e r b a nd ; t he c e n t e r s

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of the bands are usually s taggered by afa c to r o f 1 .5 , mo r e o r le s s , de pe nding o n theba la nc e o f num be r o f c ha nne ls r e qui r e d a ndthe r e s o lu t io n de s i r e d o f e a c h. To go o d a dva nta ge , the c ha nne ls r e qui r ing the h ighe s t

f r e que nc y r e s po ns e a r e a s s igne d the uppe rf r e quenc y ba nds . Fo r s a fe gua r d a ga ins tc r o s s - t a l k , t h e s u m o f a l l c h a n n e l r e s p o n s e sa va i la ble i s le s s tha n the t r a ns m is s io n linkb a nd w id th . T h e t e l e m e t r y s ig n a l s a r e r e c e ive d o n the g r o und a nd e i the r s to r e d o nma gne t ic ta pe o r imme dia te ly c o nve r te d tot r u e t i m e g r a p h s o n s t r i p r e c o r d e r s o r b o t h .T h e s u b c a r r i e r c o m p o n e n t s a r e s e p a r a t e d bysuitable fi l ter c i rcuits and individually andc o nt inuo us ly r e c o r d e d s o a s to fu r n is h a nimme dia te qua l i ta t ive g r a phic p ic tur e o f the

h i s t o r y o f t h e f l i g h t m e a s u r e m e n t s .

The s ta nda r d iz e d PDM s ys te m , a s de ve lo pe d by GE a nd o the r s , e mplo ye d a c o m mu ta to r which s uc c e s s iv e ly s wi tc he d 28 ins t r um e n ts o n a nd o ff (p lus two pa u s e s fo rcycle identification) in any o ne rev o lutio n,

whi le tu r n ing a t a bo ut 30 r e vo lut io ns pe rs e c o n d , s o t h a t a l m o s t 9 0 0 m e a s u r e m e n t swo uld be t r a ns m it te d pe r s e c o nd, e a c h ta kinga bo u t 1 m i l l i se c o n d t o s a m p l e . In s o m e a i r c r a f t i n s t a l l a t i o n s , m e c h a n i c a l c o m m u t a t i o nwas s tepped up to 88 channels and 1500 readi n gs p e r s e c o n d . ( W ith e l e c t r o n i c c o m m u t a t io n , s e ve r a l tho us a nd r e a dings pe r s e c o ndc a n be o bta ine d, but the r e a r e l imi ta t io nsto the r e s o lu t io n a t ta ina ble o ve r a va i la bler a dio l inks s inc e the puls e dur a t io n i s ine vi ta bly s ho r te ne d wi th h ighe r r e pe t i t io n

r a t e s . )

The s e g r a phs ma y s t i l l be inc o nve nie ntin s c a l e , n o n l i n e a r i t y , z e r o p o i n t s , c r i s s c r o s s in g, a nd le ngth . The y ma y be a na m o r -phic a lly r e pr o duc e d to o the r t ime a nd o r d ina tes c a l e s i n a u t o m a t i c r e p r i n t i n g m a c h i n e s( s u c h a s t h e G e n i s c o - P a n o r a m a g r a p h ) . A b o v ea l l , the o r ig ina l r e c o r ds ma y be ma nua l lyc o r r e c te d a t a s ma ny po ints a s de s i r e d , byt r a ns po s ing the m, l ine a r iz ing the m, a djus t ing them for ze r o po int and sca le value for

in-fl ight c a l ib r a t io n (by ma c hine -a ide d m a nua l o pe r a t io ns , fo r ins ta nc e o n the B e ns o n-L e h n e r " O s c a r " ) a nd by r e p l o t t i n g t h e ma uto m a t ic a l ly to the de s i r e d s c a le s by me a nsof a digit ized output memory and an autom a t i c d a t a p l o t t e r .

Se ve r a l ye a r s a go , i t s e e me d tha t FM -FMte le m e t r y wo uld be c o me s ta nda r d iz e d to thee x c lu s i o n o f a l l o t h e r s y s t e m s , l a r g e l y b e cause of the ease with which i t permittedg r a p h i c a l r e p r e s e n t a t i o n o f r e s u l t s , d e s p i t e

the s e r io us l im i ta t io ns in the numbe r ofins t r ume nt c ha nne ls tha t c o uld be mul t i plexed on any one ra dio l ink chann el. Howe ve r , th is p ic tur e c ha nge d wi th the de ve lo pme nt o f a uto ma t iz e d da ta r e a de r s fo r r e a ding,a na lyz ing, a nd gr a phic p lo t t ing .

T h e m e a s u r e d v a l u e s e n s e d b y a t r a n s duc e r i s us ua l ly fu r nis he d a s a DC po te nt io me te r vo l ta ge . Th is vo l ta ge , when ma tc he db y a g r a d u a l l y c h a r g e d c o n d e n s e r , d e t e r m i n e s t h e d u r a t i o n o f t h e c h a r g i n g p r o c e s sand with i t , the dur atio n of the em issio n ofa m o d u l a te d s i g n a l o v e r a t e l e m e t r y t r a n s m i t t e r . T h e s e p u l s e s o f v a r i o u s l e ng t h s a r ed i s p l a y e d a t t h e t e l e m e t r y r e c e i v e r s t a t i o nby a s we e p o s c i l lo gr a ph a nd pho to gr a phe d

o n a r u n n i ng fi lm . In t h e " H e r m o g r a p h "type ma chin e (d evelo ped about 1948) thef i lms a r e p r o je c te d th r o ugh a n o pt ic a l s y s tem and automatically scanned by a sweepingm ir r o r , pa s t a pho to e le c t r ic c e l l whic h, whenit se ns es the end of a l ine, s to ps the c har gingo f a c o n d e n s e r a n d t h u s r e c o n s t r u c t s t h ea na lo g vo l ta ge o f the t r a ns d uc e r . A pa tc hbo a r d pe r m its s e le c t io n o f up to s ix o f the28 channels to be read in any sequence ofs o r t in g . Ea c h m e a s ur e d va lue i s p lo t te da u t o m a t i c a ll y o n a p e n a n d s t r i p r e c o r d e r ,

d i f fe r e nt po in t s ymbo ls d is t inguis hing diffe r e nt c ha n ne ls . Fi lm a dva nc e a nd t imec o u n t i n g a r e a l s o a u t o m a t i c .

A bank of po te nt io m e te r s i s a va i la ble tope r mit a djus tme nt o f z e r o , s ign , a nd s c a le

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v al u e of each t ra ce t o co n for m t o su i t ab l eu n i t s on t h e g rap h read i l y l eg i b l e i n t e rmsof p ou n d s , p ou n d s p e r sq u ar e i n ch , feet p ers e c o n d s q u a r e d , d e g r e e s , d e g r e e s p e r s e c on d , e t c . , t o ac cu r ac i es of t h e o r d e r o f 1 /2 %

f r o m i np u t, o r 1 t o f r o m t r a n s d u c e r .T h ese mach i n es , p l o t t i n g a t t h e ra t e of ab ou ton e p oi n t p er seco n d , an d a l so fu rn i sh i n gp u n c h e d c a r d s f o r l a t e r c o m p u t a t i o n , w e r ea v a s t i m p r o v e m e n t o v e r p r e v i o u s m e t h o d si nv o l v in g hu m a n o p e r a t o r s a nd th e y d r a s t i cal l y cu t d own t h e b ack l og of record s wh i chr e m a i n e d u n r e a d b e f o r e . Y e t , a s t h e d e v el op men t p ace of mi ss i l es q u i ck en ed , t h i ssp eed n o l o n g er su f fi ced an d t h e d e s i r e t op r o c e s s t h e r e c o r d s a t m o r e n e a r l y r e a lt i m e g r e w . T h i s b e c a m e p o s s i b l e t h r o u g h

t h e su b st i tu t i on of mag n et i c t ap e fo r p h o t o g r a p h i c f i l m a s t h e s t o r a g e m e d i u m .

M ag n et i c t ap e was f i rs t i n t rod u ced as ab a ck u p d e v i c e ; no w th e r o l e s a r e r e v e r s e d .In Dou g l as ' l a t es t sy s t em, (on e u s i n g a con v e r t e r b u il t b y M a gn a v o x R e s e a r c h L a b o r a t o r i es an d an ot h er b u i l t b y Dou g l as) , t h em a i n r e c o r d s a r e r e c e i v e d o n t a p e a n d o n l ya s p a r e r e c o r d i s f il m e d a s a s a f et y m e a s u r ea n d f o r v i s u a l m o n i t o r i n g . P r e s e n t l y , t h ema g n et i c t ap es a r e co l l ect ed a t t h e ran g est a t i on s an d mai l ed o r f l own t o t h e cen t ra l

d a t a p r o c e s s i n g f a c il it y a t t he m a n u f a c t u r e r ' sp l an t , wh ere t h ey a re p l ay ed b ack , so r t ed ,ed i t ed , an d an al y zed b y en g i n ee rs i n c l o setouch with the design staff. F u t u r e p l a n s ,esp eci a l l y i n t h e a i rp l an e t es t f l i g h t d i v i s i on ,e n v is a g e d i r e c t r e t r a n s m i s s i o n o f t h e s i g n a l sas re cei v ed a t t h e ran g e v i a a 1 0 0 -m i l e ch ai no f m i c r o w a v e l in k s t a ti o n s t o t h e c e n t r a lfaci li t y a t t h e fact o ry . He re t h e s i g n al s wi l lb e red i sp l ay ed an d wat ch ed as t h ey come i n .T h i s w i l l b e a n o t h e r s t e p t o w a r d c o m p l e t eautomation, but i t is not yet in the service

s t a g e .

On ce t h e record i s av ai l ab l e i n mag n et i ct ap e fo rm as a s t r i n g of s i g n a l s of v ary i n gl e n gt h , c o n c u r r e n t w i t h s o m e s t a n d a r d i z e d

t i m e r e c o r d , i t r e m a i n s t o s e p a r a t e t he i n d i v i d u al ch an n el s b y a d ecommu t at o r wh i chr ep ea t s o n l y t h e s i g n al s b el on g i n g t o an yo n e s e l e c t e d c h a n n e l a nd r e - r e c o r d s i t w i thp r o p e r t i m e r e f e r e n c e . T h i s i s a c c o m p l i s he d

fu l l y au t omat i cal l y i n a sy n ch ron i zed d e-c o m m u t a t o r w h i le pl a yi n g t he o r i g i n a l r e c o r don t o a n ew t ap e fo r mai n t en an ce of accu racy .C u r r e n t l y a v a i l a b l e m a g n e t ic t a p e c a n s t o r eup to 14 t r ac ks pe r inch width. Up to abo utt h r e e - q u a r t e r s o f a m i l l io n d e c i m a l d i g i t scan b e s t o red on t ap e occu p y i n g on e cu b i cinch when rol led up.

In one system in use at Douglas, thei n fo r mat i o n g o es d i r ect l y i n to a d i g i t a l t ab u l arr e c o r d . O t h e r w i s e t h e i n fo r m a t i o n m u s t b e

c o n v e r t e d f r o m a n a lo g f o r m ( p u ls e d u r a ti o n )t o d i g i ta l fo r m a t s o m e s t a g e o r o t h e r . M an ys c h e m e s a r e i n v o g u e f o r t h i s p u r p o s e a ndm a n y e l e c t r o m e c h a n i c a l , m a g n e t i c , a nd e l e c t ron i c d ev i ces h av e b een d ev el op ed an d ap p l ie d t o i m p l e m e n t t h e m . T h e f o r m e r a r ea d e q u a t e f o r r e l a t i v e l y s lo w p r o c e s s e s o rw h e r e a r e a d - o u t i s r e q u i r e d o n ly w h i le t h esy st em i s b ei ng sam p l ed wh i le a t r es t . S uchc o n v e r t e r s a r e m a n u f ac t ur e d b y G e n i s co ,Gi an n i n i , Col eman , an d T el ecomp u t i n g .

W h ere h i g h sp eed s an d read -ou t on t h ef l y a r e r e q u i r e d , e l e c t r o n i c c o n v e r t e r s a r eu sed . S u ch i n s t r u m en t s (q uot i n g 8,000 to1 0 0 , 0 0 0 c o d e t r a n s l a t i o n s p e r s e c o n d ) a r eb ei n g ma d e b y Rad i at i on , In c . , Vi cd ar , Co n s o l i d a t e d E l e c t r o d y n a m i c s , J . B . R e a , an dE p sco . Di g i t iz i n g i s u su al l y acco mp l i sh ed b ycomp ari n g t h e u n k n own v ol t ag e wi t h a b i n arys e r i e s o f kn o w n p r e c i s i o n v o l t a g e s , th e v a l u eof each vol tage being one-half the value ofi t s p r e d e c e s s o r . T h e b i n a r y d i g it a l c o d ere p r ese n t i n g t h e un kn own v o l t ag e i s fo r med

b y ass i g n i n g a b i n ar y v al u e of on e t o t h o ser e f e r e n c e v o l t a g e s w h o s e s u m m o s t n e a r l yap p rox i mat es t h e u n k n own , an d zero t o a l lo t h e r s . T h e r e s i d u a l e r r o r w i ll b e no l a r g e rt h a n t h e s m a l l e s t r e f e r e n c e v o l t a g e .

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PDM type da ta a r e s i m ple r to d ig i t iz e a ndto p r e s e nt tha n FM -FM da ta , be c a us e the ya r e a l r e a d y r e c e i v e d in d i s c r e t e s a m p l e s .One Douglas system for digit izing PDM datae mplo ys a s imple f ixe d-f r e que nc y o s c i l la to r

that is gated with the leading edge of the d atapuls e and c lo s e d a t the t r a i l ing e dge . Thec yc le c o unt i s the n r e a d d i r e c t ly in to r e gis te r sa nd the c o r r e s po nding b ina r y numb e r i sf o r m e d . T h e o t h e r s y s t e m c o n v e r t s t hePDM signal back to analog voltage, appliess c a le fa c to r s , then c o nve r ts d ig it vo l ta ge tod i g i t s .

Onc e digi t iz e d, the da ta ma y be r e c o r de do n ma gne t ic ta pe fo r us e wi th a c o mpute ro r the y may be se nt to a r e c o r de r c o nta in ing,

say, 128 fixed s tyli , each s tylu s re pr ese nti ngo ne o f the 128 po s s ib le qua nt iz ing le ve ls .The rapid ity w ith which the s tyli ar e ac t iva te d s uff ice s to p r e s e n t a p r a c t ic a l ly c o nt i n u o u s t r a c e .

Analog to digital conversion is s t i l l in theflux of develo pm ent. No digital code hasye t e me r ge d a s undis pute d s ta nda r d; s o met r a n s l a t e i n d e c i m a l n u m b e r s , o t h e r s inbina r y , a ga in o the r s in b ina r y c o de d de c ima ld i g i t s ; the choice depends largely on the

language of the computer to be used whent he d a t a a r e f u r t h e r p r o c e s s e d .

Guided m is s i le s a r e us ua l ly me a nt to beguide d to a ta r ge t . The ta r g e t ma y be o nthe g r o und, o n the wa te r , o r in the a i r . Thes uc c e s s of the m is s i le s ys te m is us ua l lyr e f le c te d by the m is s ve c to r . It i s the r e fo r eo f v i ta l i n t e r e s t t o m e a s u r e t h i s m i s s v e c t o r .If th is i s do ne by r a d a r o r c ine the o do l i tef r o m the g r o und, the r e s ul t i s a s m a l l dif

fe r e nc e be twe e n two la r ge ve c t o r s . It i st h e r e f o r e o f te n w o r t h w h il e t o m e a s u r e t hem is s ve c to r a lo f t by d i r e c t c o mm unic a t io nbetween missile and target during the endga me wi th h ighe r po te nt ia l a c c ur a c y . Ina nt ia i r c r a f t wo r k , the o nly r e a l ly c o nvinc ing

pr o o f o f s uc c e s s o f the s ys te m is o bta ine dby f i r ing te s t m is s i le s a t r e a l f ly ing a i r c r a f t . F o r r e a s o n s o f p e r s o n n e l s a fe t y ,t h e s e a r e e i t h e r t o w - t a r g e t s o r r e m o t e l yc o n t r o l l e d d r o n e s .

It has been found expedient to equip thesed r o n e s w i t h h i g h - s p e e d c a m e r a s w h i c h r e c o r d the ne a r pa s s a ge o f a te s t m is s i le a nda pyr o te c hnic a l to ke n de to na t io n . Th e s ec a me r a s ha ve wide a ngle o pt ic s , le ns e s tha te nc o mp a s s 140 to 160 de gr e e s . Fo ur o f the min a t e t r a h e d r a l a r r a y c o v e r t he e n t i r esp he ri ca l f ield and photo graph anything thatgo e s by. The c a m e r a s r un a t f ilm s pe e d sbetween 200 and 500 frames per second.The y do no t r un in e nfo r c e d s ync h r o nis m ,

but the y pho to gr a ph o n e ve r y f r a me a n a r r a yo f s ma l l ne o n l ights whic h indic a te s c o mmo ntim e in an unambiguo us binar y co de to onem i l l i se c o n d r e s o l u t i o n . C l u s t e r s o f f o u rc a m e r a s a r e o f t e n m o u n t e d i n s t r e a m l i n e dpods on each of the wing t ips and on somel a r g e d r o n e s a d d i t i o n a l c a m e r a s a r e i n s t a l l e din the no se and tai l of the fuselage. Thec a m e r a s a r e s t a r t e d a u t o m a t i c a l ly b y r a d i os i g n a l . P a r a c h u t e s b r i n g t h e c a m e r a p o d sto e a r th in c a s e the d r o ne i s h i t a nd de s t r o ye din the a i r .

Eva lua t io n o f the p ic tur e s whic h s ho w them i s s i l e i s d o n e by a s e m i a u t o m a t i c r o u t i n e .The a ngu la r c o o r din a te s o f the ta r ge t ima gea r e ma tc he d by the c r o s s ha i r s o f a n Ic o no lo g ,B o s c a r , o r s i m i l a r m a c h i n e . H e r e h u m a njudgme nt i s ne e de d be c a u s e the m is s i le ima gemay change s ize and aspect rapidly fromfr a m e to f r a m e . The a ngula r po s i t io n o f thel ine o f s ight f r o m the c a m e r a to the mis s i leis the n a uto ma t ic a l ly punche d o n a c a r d to get her w ith the t im e cod e. The r es t of the

c a lc ula t io n i s the n a c c o mpl is he d o n a n e l e c t r o n i c c o m p u t e r in a r o u t i n e p r o c e d u r e .

T h e c o m p u t a t i o n c o m p r i s e s t h e c o r r e c t i o nfor the optical dis tor t ion of the wide anglel e n s , t h e t r i g o n o m e t r i c a l c a lc u l a t io n s to r e f e r

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f r o m c a m e r a to d r o n e c o o r d i n a t e s , th e d e cod i n g of t i me, t h e i n t e rp ol a t i on t o s i mu l t a n e o u s t i m e i n s t a n t s b e tw e e n tw o c a m e r a sat op p osi t e en d s of a b asel i n e su ch as t h ewi n g sp an o r fu sel ag e, t h e t r i an g u l at i on f rom

t wo t h u s con j u g at ed camera s t a t i on s , an d t h ep l o t t i n g of t h e mi ss v ect o r comp on en t s i nt a r g e r c o o r d i n a t e s . T h e D o u g la s a i r b o r n er e c o r d i n g s y s t e m h a s b e c o m e s t a n d a r d i z e du n d e r t h e n a m e K i n e s c o r e .

W ith a i r - l a u n c h e d m i s s i l e s , it m a y s o m e t i m e s b e d e s i r a b l e t o t a k e m o t i o n p i c t u r e so f th e m i s s i l e f r o m t h e m o t h e r a i r c r a f t a ndt o p h o t o g r a p h o n t h e s a m e p i c t u r e s t h e i n s t r u m en t s wh i ch g i v e a r eco rd of t h e g u i d an ces y s t e m p e r f o r m a n c e . S p ec ia l c a m e r a s f o r

t h i s p u r p o s e a r e u n d e r d e v e l o p m e n t .

W h i l e a i rb orn e p h ot og rap h y i s q u i t e ef fect ive, i t is admit tedly expensive and cumb e r s o m e . T h e r e f o r e , e f fo r t i s b e in g d i r e c t e du p o n t h e d e v e l o p m e n t o f a l l - e l e c t r o n i c m i s s -m e t e r s . In t h e s e , s i g n a l s a r e e x c ha n g edb etween t h e t a r g et an d th e m i ss i l e i n r ad arfash i on an d t h ey a re recei v ed an d ev al u at edei t h er on a g ro u n d s t a t i o n o r i n t h e a i r , b eit o n t he t a r g e t , o r o n t he m o t h e r a i r c r a f t .S e v e r a l s u c h s y s t e m s a r e p r e s e n t l y u n d e rd ev el o p men t . T h ey wi l l p ro b ab l y ev en t u al lyco me i n to g en er al u se , bu t s i n ce t h ey can n otg i v e as man y d et a i l s as p h ot og rap h y , t h el a t t e r w i l l p r o b a b l y r e m a i n d e s i r a b l e a s ab ack u p .

Becau se of t h e g reat v ar i e t y of con d i t i on su n d e r w h ic h m i s s i l e - t a r g e t e n c o u n t e r s m a yo ccu r (d i f fe ren t t a r g et s i z e , sp ee d , a l t i t u d e,e v a s i v e m a n e u v e r s , a n d a t d i f f e r e n t r a n g e s ) ,i t wou l d b e v ery ex p en si v e t o cov er a l l o ft h em by d r o n e i n t e r c e p t t e s t s . E c o n o m yd i ct a t es t h at ac t u al d r o n e t es t s b e l i mi t ed

t o a r e a s o n a b l e n u m b e r o f t y p ic a l o r c r i t i c a lco n d i t i on s , an d th at t he r es t b e co v er ed b yf i r i n g t e s t s w i th s y n t h e ti c t a r g e t s p r e s e n t e dby a s i m u l a t o r w ith a g r e a t e r o r l e s s e rd e g r e e o f s o p h i s t i c a t i o n .

The big established f l ight test stat ions andf i r i n g ran g es a re b u si l y p l an n i n g an d p rog r e s s i n g t o w a r d a f u tu r e wh en a l l m e a s u r e men t s p er t a i n i n g t o t h e f l i g h t o f ex p er i men t ala i r c r a f t a n d m i s s i l e s c an b e g a t h e r e d , r e

c o r d e d , s o r t e d , p r e s e n t e d , a n a l y z e d , a n d i n t e r p r e t e d , ful ly au t o ma t i cal l y , in a cen t ra lfaci l i ty , so quickly that information valuablefor the next mission wil l be avai lable in am a t t e r o f h o u r s o r m i n u t e s i n s t ea d o f d a y so r w e e k s . S o m e p l a n s g o s o f a r a s t o c o m bine in this faci l i ty the pr o ce ssi ng of thev ast ou t p u t o f a v ar i e t y of f i e l d ex p er i men t s ,su ch as s t a t i c p ower p l an t t es t s t an d s an dr o c k e t s l e d t e s t s , th e b e t t e r t o u ti l iz e t h ei n e vi ta b l y e x p e n s i v e a n d e l a b o r a t e d a t a p r o c ess i n g an d com p u t in g faci l i ty . Ho wev er ,su ch l on g ra n g e d ev el o p me n t s sh ou l d n ot b ep r e s s e d f o r w a r d t o o h a s t i l y . T h ey m u s t b ep r o j e c t e d w ith c o n s i d e r a b l e f o r e s i g h t a ndmu st re m ai n v er sat i l e an d ad ap t ab l e so th att h ey can t ak e ad v an t ag e of n ew measu remen tt ech n i q u es t h at may b ecome p ract i cal i n t h em e a n t i m e .

It m u st a l so b e r em em b er ed t h at fulla u t o m a t i o n d o e s n o t m e a n c o m p l e te d i s p e n s a t ion of h u man ju d g m en t . S om e i n t e l l ig en tm o n i t o r i n g a n d m a n a g e m e n t w i l l r e m a i n n e c es sa r y . F o r i n s t an ce, it wou l d b e wast efulof communicat ion bandwidth i f al l avai lablechannels were equipped to handle the highesto c c u r r i n g f r e q u e n c y r e s p o n s e a n d a s s i g n e db l i n d l y t o c h a n n e l s r e g a r d l e s s o f t h e i r c h a r a c t e r i s t i c s . A n u n n e c e s s a r y a m o u n t o f t r i v i a ldata would be col lected and handled for noo n e t o ex am i n e. It i s co n cei v ab l e t o d es i g ni n s t r u m e n t s s o t h at t he t i m e s c a l e i s m a d ev ar i ab l e an d au t om at i cal l y o p en ed u p wh en e v e r a n i m p o r t a n t e v e nt o r m a n e u v e r i s e x p e c te d o r p r o g r a m m e d , o r w h en a n u n f o r e

s e e n d e p a r t u r e f r o m t he a n ti c ip a t e d b e h a v i o ro cc u r s . Ho wev er , ev en t h i s ref i n emen t wi l lnot tak e the p lace of the human inte l l igencen ec ess ar y i n p l ann i n g o r mod i fyi ng t h ee v a lu a ti o n p r o c e d u r e . T h e r e f o r e , s o - c a l l e d

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" q u i c k - l o o k " g r a p h i c a l p r e s e n t a t i o n s a r eins e r te d a t s u i ta ble s te ps o f the p r o c e dur et o e n a b l e a n o b s e r v e r t o s e p a r a t e t h e s i g ni fica nt p o r t io ns o f the r e c o r d f r o m thet r iv ia l a nd the h igh pr io r i ty c r i t ic a l ma te r ia l

f r o m the lo we r .

T h e " q u i c k - l o o k " g r a p h i c a l p r e s e n t a t i o nis usually considered adequate as quali tat iveda ta wi th 5 to 10 pe r c e nt a c c ur a c y. Ho we ve r ,i t is often possible to achieve much highera c c ur a c y, in whic h c a s e the quic k- lo o k pr e s entation can be accepted as final o r quan tit a ti v e d a t a . P r e s e n t l y , r e a l - t i m e o r " o n l i n e " reduction is being accomplished bytaking the output of a receiver and feedingthe input directly to the data system forgr a p hic a l p r e s e nta t io n . This i s the f i r s ts tep toward full automation; the next phasewo uld be d i r e c t input in to la r g e s c a le c o m pu te r s . The c o mple te imple me nta t io n o f thet r e nd to wa r d a uto ma t io n ma y s t i l l ta ke ye a r sand it is po ssib le that it will be followed bya r e gr e s s io n to a llo w mo r e huma n judgme nta g a i n t o e n t e r t h e p r o c e d u r e s .

In a paper presented before the conventionof the Insti tute of Radio Engineers in March1956 and published in the IRE Proceedings,D. G. M a z ur o f the Na val R e s e a r c h La bo r a to r ys u m m a r i z e d t h e p r e p a r a t i o n s be in g m a d efo r te le me t r y f r o m the e a r th s a te l l i te s tobe launched under the auspices of the International Geo physical Ye ar . The fol lowingi s q uo t ed f r o m h is p a p e r : " A p r o g r a m e m bracing the use of t ime divis ion and frequencydivision multiplexing has been planned. Theequipment includes pulse posit ion modulation,frequency mo dulation and pulse width mo dula t io n , c o mm uta t io n de vic e s , a l l p r o ve n in

a c t i v e s e r v i c e in o t h e r r o c k e t p r o g r a m s ,under fi r e , so to speak. Multiple gro undstations will be employed to afford backupprotection in case of any ground equipmentma lfunct io n . Pe r ma ne n t r e c o r din g o f da ta

wil l be ma de in s o me ins ta nc e s o n pho to gra phic fi lm and in o the rs on mag netic tape.R e a l - t i m e p r e s e n t a t i o n o n m e t e r s a nd p e nr e c o r d e r s w i ll b e a v a i la b l e a t s o m e l o c a t i o n s . It is expected that a v ast amo unt of

data will be collected from each rocket. Someda ta wi l l be o f the fo r m tha t me r e v is ua linspection will suffice. Other data will r e qui r e no r m a l iz a t io n a nd wi ll be r e duc e ds e m i-a uto ma t ic a l ly . Us e o f a uto m a t ic da tareduction methods will be made insofar as i ti s p r a c t i c a l . "

The plans for the t ra ckin g of the sat ell i teswe r e de s c r ibe d by J . T. M e nge l , a ls o o f theNa va l R e s e a r c h La bo r a to r y , a t the s a meme e t ing a nd in the s a me Pr o c e e dings o f theIR E. He s a id : "T he M ini t r a c k Sys tem o frad io angle t r ackin g develo ped by the NavalR e s e a r c h L a b o r a t o r y u t i l i z e s a n o s c i l l a t o rof m inimu m siz e and weight within thesatell i te to i l luminate pairs of antennas at agr o und s ta t io n whic h me a s ur e s the a ngula rpo s i t io ns o f the s a te l l i te us ing pha s e -c o mp a r is o n te c hnique s , inde pe ndent o fwe athe r , vis ibil i ty, and t ime of day . . .Pr im a r y da ta p r e s e nta t io n wi l l be by th r e edi r e c t wr i t ing r e c o r ds o f the a na lo gy vo l ta geof the ambiguity resolving antenna phases,

a nd two d i r e c t wr i t ing r e c o r ds in d ig i ta lform of the fine antenna phases, a l l as afunction of tim e . . . Six pi ec es of da ta,the l ime s o f c r o s s ing the -4° , 0° , a nd +4°zenith angles , and the nor th-south angles thatwe r e me a s ur e d a t the s e t im e s , wi ll be s e ntto a central computing facil i ty within 20minutes of a t racking event, to be used indetermining the orbit of the satell i te . . .The central computing facil i ty will receivedata fro m as many a s nine of these grounds ta t io n s . . . C o mple te e phe me r ide s wi l l

p r o vide t r a c king a ngle a nd t ime info r ma t io nfor the pr incipal locations at which opticalt r a c king s ta t io ns a r e lo c a te d , to pe r mit thet r a c k ing e quipme nt to a c qu i r e the s a te l l i te ,as well as for most of the major ci t ies from

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which the sate l l i te could be vis ible . It isn ot t oo u n real i s t i c t o p red i c t t h at d u r i n g asat e l l i t e ev en t t h e ev en i n g n ewsp ap ers wi l lp u b l i sh on t h ei r f ron t p ag es t h ree b ox es , on ef o r t h e b a s e b a l l s c o r e s , o n e f o r t he h o r s e

race resu l t s , an d on e fo r t h e ev en i n g t i meand angles at which the satel l i te can be pickedu p ! "

On e may wel l ask h ow E u rop ean en t erpr is e, ta lent , and ingenuity f it in to thisp i c t u r e . T h e an sw er s can b e sou g h t on t wol e v e l s . F i r s t , i t m a y b e w i s e t o p r o c e e dcautiously and avoid laying out extremely

e l a b o r a t e f a c i l i t i e s u n l e s s t h e r e i s r e a l l y ademand for high test ing r ate and high eva luat io n s p e e d . O t h e r w i s e s e m i - a u t o m a t i c d e v i c e s a n d o p t i c a l - m e c h a n i c a l a n d e l e c t r i c a laids may suffice. One may actua l ly inter cep t

t h e Ameri can d ev el op men t on t h e reb ou n d .S econ d , d esp i t e t h e t ech n i cal p rog ress mad eb y a g rowi n g n ew i n d u st ry t h ere i s s t i l la m p l e r o o m f o r th e i m p r o v e m e n t o f c o m p o n e n ts , s uc h a s t r a n s d u c e r s , c o m m u t a t o r s( e s p e c ia l l y b r u s h l e s s o n e s ) , a n t e n n a s , d e -c o m m u t a t o r s , fi l t e r s , p l ay ba c k s y n c h r o n i z e r s , h i g h - s p e e d d i g i t i z e r s , p l o t t e r s , a n dr e c o r d e r s . T h i s o f f e r s a c h a l le n g e o f c o n s i d e r a b l e m a g n i t u d e .

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PA PE R ON THE GUIDANCE AND CONTROL OF MISSILESStephen Waldron*

SUMMARY

This p a pe r c o nc e r ns the r e l ia bi l i ty o f e le c t r o nic e quipme nt , pa r t ic ula r ly a s it e f fe c tsthe rel iab il i ty of guided m is sil es . Reliabil i ty is her e defined as the pro babil i ty of sur viv alfor a specified t im e. The s t r es se s applied to a guided m iss ile at the var io us s tag es of i tshis to ry and which affect rel ia bil i ty ar e disc usse d and the co nclusion is reac hed that a re alga in in m is s i le r e l ia bi l i ty r e q ui r e s inve s t iga t io n of fa i lu r e s whic h o c c ur e a r ly in the mis s i l eflight.

SOMMAIRE

Dans cet ar t icle i l s'agit de la securi te de fonctionnement de 1'appareil e l e c t r o n i q u e ,

pa r t ic ul ie r e m e nt da ns la me s u r e o u e l le influenc e la s c c ur i te ' de fo nc tio nne me nt de s p r o je c t i le sguide s . No t r e de f in i tio n de "s e c ur i te de fo nc t io n ne me nt" e s t la p r o ba bi l i te de s ur vie dur a n tun temp s sp ecifie . Nous discuto ns les tensio ns qui agissen t su r un pr o jecti le guide auxdi ve r se s etap es de son v ol , e t qui influencent la se cu ri te de leur fonctio nnement. Nousa r r iv o ns a la co nc lus io n qu ' une a m e l io r a t io n r e e l le de la s e c u r i te d' un pr o je c t i le e xige uneetude des fai l l i tes qui surviennent au debut du vol du projecti le .

This pa pe r c o nc e r ns the r e l ia bi l i ty o fe l e c t r o n i c e q u ip m e n t , p a r t i c u l a r l y t he r e l i a bi l i ty o f guide d mis s i le s .

It i s ne c e s s a r y to de fine " r e l i a b i l i t y . "Let us accept a definition often used inm i l i t a r y c i r c l e s . A d e v i ce m u s t o p e r a t e a spla nne d fo r a c e r ta in t ime in o r d e r to beuseful. Rel iability of the dev ice is definedas the pro babil i ty that i t sur viv es for thene e de d t ime .

I n o r d e r t o m e a s u r e r e l i a b i l i t y o n e m u s tbe a ble to de te c t fa i lu r e whe n i t o c c ur s .Assume for the moment that one can identifyfa i lu r e s a s the y a r is e .

Failures in complex equipment often occurin such a way that the ir cau ses ar e difficult

to identify. As a co nseque nce, fai lu re s ar eoften classified with rela tio n to t ime o rdensity of fai lures per unit t ime, rather thanwith r e la t io n to c a us e . Fa i lu r e s a r e us ua l lydivide d in to the th r e e c a te go r ie s : in i t ia l,c ha nc e , a nd we a r o ut fa i lu r e s .

Ea r ly o pe r a t io n ne a r ly a lwa ys e xpo s e scom ponent defects which cau se the equ ipment to fail. Such initial defe cts ar e builtin o r a r e de ve lo pe d by the r ig o r s o f t r a n s po r ta t io n o r r e wo r k a t the c o ns umptio n po int .After fair ly fixed operating t imes the equipment m ay fail due to wearo ut fai lur es in suchc o m po ne nts a s ba t te r ie s a nd r o ta t ing m a chin er y. The equipment may fail , ho wever ,as the r esu lt of com ponent fai lur es which

are due neither to infant mortali ty nor o lda g e . Fo r ma ny c o mpo ne nts the p r o ba bi l ity

*Office of the Chief of Naval Operations, Washington, D. C.

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of fai lure is independent of the cumulat ivet i me t h e com p on en t h as o p er a t ed ; fa i l u re i sa chan ce even t . Th is is typical of vacuumt u b es , fo r i n s t an ce , wh i ch may " g o " a t an yt i m e .

W h en eq u i p men t fa i l u re resu l t s f romch an ce fa i l u re of comp on en t s , t h e eq u i p men tre l i ab i l i t y can b e re l a t ed mat h emat i cal l y t ot h e r e l i a b i l i t i e s o f t h e s e p a r a t e c o m p o n e n t s .T h e m a t h e m a t i c a l r e l a t i o n s h i p e x p r e s s e s t h efact that a device composed of many long-l ived components wil l i tself have a relat ivelysh o r t mea n li fe . T h i s i s n o t su r p r i s i n g . Infact , the equipment mean l i fe is dominatedb y t h e weak est comp on en t s .

M u ch of th e effo r t t o i mp r o v e re l i ab i l i t yof e l ect ron i c eq u i p men t h as g on e i n t o d et e rmi n i n g an d i n creas i n g t h e mean l i v es ofco mp o n en t s . T h i s n eeded ef fo r t h as g en er at edv a s t f a i lu r e - r e p o r t i n g s y s t e m s a nd d a t a -h an d li n g faci l i t i es . As a r esu l t , a t t en t io nh as b een focu sed on t h ose comp on en t s wh i chm o s t n e e d i m p r o v e m e n t .

Now co n si d e r t h e b as i c q u est i o n , why d oco mp o n en t s fa i l . A com p on en t fa i l s b ecau sei t i s su b j ect ed to s t r e ss , t h rou g h a p er i o d oft i m e . In g e n e r a l , f a i lu r e s r e s u l t s o o n e r

f r o m l a r g e s t r e s s e s th a n f r o m s m a l l o n e s .F or a f i x ed s t ress an d t i me, t h e p rob ab i l i t yof fai lure wil l have a dist r ibution of valuesfo r a co l l ect i on of su p p osed l y i d en t i cal comp o n e n t s . S ta te d d i ff e r e n tl y , s t r e s s r e s i s t a n c ev a r i e s a m o n g " i d e n t i c a l " c o m p o n e n t s i n ac o l l e c t i o n , du e t o d i f fe r e n c e s i n m a t e r i a l s ,a s s e m b l y , a nd o t h e r f a c t o r s . B y c o m b i n in gt h e e ff e c ts o f r a n d o m v a r i a t i o n s in s t r e s sre s i s t an ce an d ran d om v ar i a t i o n s in ap p l ieds t r e s s , it i s p o s s i b l e t o a c c o u n t f o r c h a n c efai l u res of comp l ex eq u i p men t .

T h e t h eory of ch an ce fa i l u res ou t l i n eda b o v e a p p li e s s a t i s f a c t o r i l y t o m o s t e l e c t r o n i c e q u i p m e n t. E q u ip m e n t s t r e s s e s a r esm all , fai rly unifo rm , and well enough known

s o t ha t c o m p o n e n ts m a y b e t e s t e d a p p r o p r i a t e l y d u r i n g d es ig n and ma n u fact u r e . Asa c o n s e q u e n c e t h e r e a r e n o s p e c t a c u l a r dif

fe r en c es in t h e mean l i v es of co mp o n en t si n vol v ed , an d eq u i p men t fa i l u r es o ccu r r an

d o m l y .T h e t h eory of ch an ce fa i l u res mu st b e

au g men t ed b y o t h er con si d era t i on s wh eng u i d e d m i s s i l e r e l i a b i l i t y i s c o n s i d e r e d .Gu i d ed mi ss i l es d i f fe r f rom more con v en t i on al e l ec t ro n i c d e v i ce s b ot h i n co mp l ex i t yand in the enviro nm ent which they ex pe r ien ce . T h e g u id ed m i ss i l e mu st no t on l yh an d l e i n fo rm at i on , b ut a l so co n t ro l i t s owne n e r g y r e s o u r c e s . T h e c i r c u i t r y i s o f te nwork i n g n ear t h e l i mi t s o f i t s cap ab i l i t i esa n d t h e r e a r e s t r o n g i n t e r a c t i o n s a m o n g

co mp o n en t s . In ad d i t ion , t h e g u id ed m i ss i l een v i ro n men t i s mu ch m o re sev er e t h an t h ati n w h i c h c o n v e n t i o n a l e l e c t r o n i c s o p e r a t e s .

Co n si d er t h e s t r e ss es ap p l ied t o a g u id edm i s s i l e a t v a r i o u s s t a g e s o f i t s h i s t o r y .T a b l e 1 l i s t s t h e s e s t r e s s e s f o r a t y p i c a lr o c k e t - l a u n c h e d m i s s i l e .

U n d e r m e c h a n i c a l s t r e s s , s h o c k ( o r i m p u l se) cou l d resu l t f rom rock et l au n ch ,

d r o p p i n g, c a t a p u lt i ng , o r a r r e s t e d l a n di n g.T h e o t h e r m e c h a n i c a l s t r e s s i n g s h o u ld b es e l f - e x p l a n a t o r y . E l e c t r i c a l s t r e s s i n c lu d e st h e ef fect s o f t ran s i en t o v er l o ad s of v o l t ag eo r c u r r e n t . T h e r m a l s t r e s s i nc lu d es h ig ha m b i e n t t e m p e r a t u r e a n d , p e r h a p s , t h e r m a lg r a d i e n t s . C h e m i c a l s t r e s s i s m e a n t t o c o v e rt h e c o r r o s i v e a n d o t h e r e f f e c t s o f c h e m i c a l sin t he a t m o s p h e r e . P r e s s u r e , r e d u c ed a thigh al t i tude s, is effective in al lowing co ro naan d mak i n g seal ed u n i t s b reat h e i n ch emi cal s .T h e s t r e ss of man wi l l b e b rou g h t ou t sh o r t l y .

L e t u s c o n s i d e r p o s s i b l e c a u s e s a n dr e m e d i e s f o r f a i l u r e s w h ic h o c c u r a t v a r i o u sp o i n t s in a m i s s i l e ' s l i f e . F o l lo w i n g p r o d u c t io n , a m i s s i l e i s t r a n s p o r t e d , r e c e i v e d , a nd

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Tab le 1, St r es se s Applied to a M issile During i ts Co nsum er Life

S t r e s s e s

"rt_o'cr t

r Cocu

2

"rto"S«x:CJ0)

crtx;ucu

s .mm

o

S h o c k *

V i b r a t i o n *

N o r m a l

F u n c t i o n * *

E l e c t r i c a l

T h e r m a l

P r e s s u r e

C h e m i c a l * * *

M a n

T r a n s p o r t

X

x

X

S t o r a g e

X

X

T e s t

X

X

X

X

R e p a i r

X

X

U s e

X

X

X

X

X

X

X

?

*Applied or f l ight-induced.**lnc lude s hydr a ul ic s ys te ms a s we l l a s a e r o dyna mic c o nt r o ls .***C a r r ie d by the a tmo s phe r e .

ins pe c te d. The c o ns um e r e xa m ine s a nd te s t sthe mis s i le a nd r e pa i r s i t s " in i t ia l" de fe c ts .The lat ter are often such mechanical f lawsas co ld-so lder ed jo ints which have beenvibr a te d lo o s e in t r a ns p o r t . A few in i tia lde fe c ts a r e fo und in e le c t r o nic c o m po ne ntswhich may have had the ir e le ctr ica l o rt h e r m a l s t r e s s l i m i t s r ed u c ed by t r a n s p o r t a t ion and handling. The cu re for such t ro uble smay l ie in better mo tivatio n of the fabr icat o rsa nd ins pe c to r s , c o uple d wi th be t te r me c ha nical design of the missile and its shippingc o n t a i n e r .

T h e c o n s u m e r n o w s t o r e s t h e m i s s i l e .

He ho pes that its sto r ag e life will be long.In o r d e r to a s c e r ta in tha t the mis s i le i s s t i l lo pe r a ble a f te r s to r a ge , he te s ts i t . Le t ussuppo se that ther e is low confidence in them is s i le ' s a bi li ty to la s t th r o ugh s to r a ge .

Such is o ften the ca se . Given this low co nf ide nc e , the c o ns ume r te s ts h is mis s i le a f te ronly a few da ys. At this t ime the m iss ileoften fails to check out pr o pe rly . It isther efo re concluded that the m issi le is unableto wi ths tand the r i go r s of s to r a ge . It c a nbe s e e n wha t th is me a ns in te r ms o f ma n-ho ur s s pe nt te s t ing a nd r e pa i r ing mis s i le s .

The mis in te r pr e ta t io n o f s to r a ge e ffe c tsc o me s a bo ut be c a us e we ha ve he r e a no the rexam ple of the uncer tainty pr incip le. A"go o d" m is s i le i s def ine d o pe r a t io na l ly . Amis s i le c a nno t r e a l ly be de f ine d a s "go o d"in s to ra ge , becau se it must function to be

goo d. When it functions, the test pro ced ur emay disab le i t . The consequent fai lure m ustbe examined carefully and should not bea s c r ibe d to s to r a ge unle s s s uc h a fa i lu r ehas never been known to result f rom testing.

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I t has just been implied chat a mi ss i ler e m a i n s o p e r a b l e w h i l e l e f t i n s t o r a g e . T h i si s t r u e f o r l o n g p e r i o d s o f t i m e , p r o v i d e dt ha t a t m o s p h e r i c - b o r n e w a t e r a nd o t h e rch e m i cal s a r e kep t t o low l ev e l s . Gi v en

p r o p e r a t m o s p h e r e , it i s fo u nd t h at s t o r a g el i fe of a mi ss i l e g rows as rap i d l y as on eh as t h e cou rag e t o l en g t h en i t .

L e t u s e x a m i n e t h e t e s t p r o c e s s f u r t h e r .W h e n c h a r a c t e r i s t i c e l e c t r o m e c h a n i c a l a n dt h e r m a l s t r e s s e s a r e a p p li e d, c h a nc e f a i lu r e sre duc e the m ean l ife of a typical mi ss i le bya fa c t o r o f tw o o r t h r e e , c o m p a r e d t o a m o r ec o n v e n t i o n a l d e v i c e m a d e o f t h e s a m e c o m p o n e n t s . T h i s i m p l i e s th a t t h e m i s s i l e t e s te n v i r o n m e n t i s o n e o f i n c r e a s e d s t r e s s e s ,

w h ic h i s no g r e a t s u r p r i s e .

T h e s t r e s s o f m a n i s b r o u g h t t o b e a r d u r i ng t h e t e s t p r o c e d u r e . F i r s t , m a n o f te np r o d s o r g o u g e s i n t h e w r o n g p l a c e s w h i l ea d j u s t i n g t h e m i s s i l e o r p r e p a r i n g i t f o rt e s t . W o r s e , h e m a y i n c o r r e c t l y i n t e r p r e tt h e t es t resu l t s an d s t a r t an u n n eed ed rep ai ref fo r t . T h e l a t t e r , i n vol v i n g d i sa sse m b l yan d h an d l i n g , o f t en p rod u ces d efect s wh eren o n e e x i s t e d b e f o r e .

T h e p r o b l e m o f f a l s e d i a g n o s i s i s o fc o u r s e t w o - p r o n g e d . H o w e v e r , p e r h a p s o u to f r e s p e c t f o r m i s s i l e c o s t s , t h e t e s t m a nw i l l m o r e o f t e n r e p a i r a g o o d m i s s i l e t h a npa ss a dud. The lat te r effect has been foundi n s t u d i es of v acu u m t u b e fa i l u res .

T h e p r o b l e m o f m i s d i a g n o s i s o f m i s s i l et e s t s s e e m s i n s o l u b l e . W i th t h e p r e s e n tco mp l i cat ed t es t eq u i p m en t , t h e on l y way t or e d u c e p o s s i bi li ty o f e r r o r i s to t a ke m o r et i m e. M o re t i me b r i n g s t h e i n ev i t ab l e t r u ed efect c l o se r . A m i ss i l e wh i ch n eed s n ot e s t i n g s e e m s a n u n r e a l i z a b l e g o a l .

T h e d an g ers of c reat i n g n ew d efect s wh i l er e p a i r i n g o l d o n e s h a v e b e e n m e n t i o n e d .

One of the diff icul t ies of the chan ce -f a i l u r e a p p r o a c h to m i s s i l e r e l i a b i l i ty m a ynow b e ap p ar en t . T h at ap p ro ach h as l ed t ot h e c o l l e c t i o n o f m y r i a d d a t a o n f a i l u r e sd u r i n g mi s s i l e l ife b efo r e u se . Un fo r t u n at e ly ,

t h e r e c o r d s o f t r u e c h a n ce f a i l u r e s a r ed i lu t e d by a l a r g e n u m b e r of m a n - c r e a t e dfai l u r es an d of t en by an ev en l a r g e r n u mb e rof fa i l u res wh i ch n ev er ex i s t ed a t a l l .

N ow l e t u s c o n s i d e r t h e u s e o f m i s s i l e s ,i .e. , the actua l f l ight. Fi gu r e 1 sho ws ar e p r e s e n t a t i v e p lo t o f th e p r o b a b i l it y o fsurvival of a missi le, as a function of f l ightt i m e . Z e r o o n th e t i m e s c a l e i n d i c a t e s t h ei n s t an t o f s i g n al t o l au n ch . E l ec t r i ca l an do t h e r s t r e s s e s h a v e u s u a l l y b e e n a p p l i e d

b efor e l au n ch i s d e s i r ed . As a co n seq u en c e,some mi ss i l es d o n ot l au n ch , as y ou can see .T h e m i s s i l e r e p r e s e n t e d h e r e i s r o c k e t -l a u n c h e d , a s m e n t i o n e d e a r l i e r .

Sho rt ly a fter launch, pa r t of the guidancesy st em i s mad e o p er a t i v e . It i s t h en d i sc o v e r e d t h at m o r e m i s s i l e s h a v e s u c c u m b e d ,p r i o r t o t h i s i n s t an t . A wh i l e l a t e r , t h er es t of the guida nce syst em is cut in andt h i s a c t r e v e a l s a n o t h e r s e t o f f a i l u r e s . F r o mh e r e o n t h e f a i l u r e s o c c u r a t a m u c h l o w e rr a t e . In f ac t t h ey o c c u r a t a r a t e a p p r o x i m a t i ng t h e f a i l u r e r a t e d u r i n g t e s t s o n t h e g r o u n d .T h e f a i l u r e r a t e d u r i n g t h e f i r s t p o r t i o n o ff l ig h t , on t h e o t h er h an d , i s o r d e r s of m ag n i t u d e l a r g e r t h a n t h e u l t i m a t e r a t e .

T h e s o l id c u r v e o f t h e f i gu r e is r e p r e s e n t a t i v e of ac t u al d at a and fo rm s t h e u p p erb ou nd on su r v i v a l p ro b ab i l i t y . W h at i sprobably happening is indicated by the dot tedc u r v e . T h e f a i l u r e s c a n no t b e r e c o g n i z e du n t i l t h e re l ev an t c i rcu i t i s p u t t o work .

T h e f i g u re i l l u s t r a t e s t h at t h e b i g g ai n t ob e m a d e i n m i s s i l e r e l i a b i l i t y c o m e s f r o mr e d u c i n g t h e e a r l y , e x t r e m e l y h i g h f a i l u r er a t e . The fact that the r at e is so high ind i c a t e s t h at t h e e n v ir o n m e n t i s m u c h w o r s e

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NETHERLANDS Net h er l an d s Del eg at i on t o AGARDK a n a a l s t r a a t 1 0Delft . Holland

NORWAY

P ORT UGAL

Ch i ef E n g i n eer i n g Di v i s i on

R o y a l N o r w e g i a n A i r F o r c eDep u ty Ch i ef of S t af f / M at er i e lMyntgaten 2O s l o , N o r w a yAttention: Lt . Col . S. Heglund

S u b s e c r e t a r i a d o d a E s t a d o d a A e r o n a u t i c aAv . d a L i b erd ad e 2 5 2L i s b o n , P o r t u g a lA t t en t io n : L t . C o l . J o s e P e r e i r a d o N a s c i m e n t o

TURKEY

UNITED KINGDOM

UNIT E D S T AT E S

M. M. Vekalet i

E r k a n i h a r b i y e i U m u m i y e R i y a s e t il l m i . l s t i s a r e K u r u l u M u d u r l u g uA n k a r a , T u r k e yAttentio n: Co lonel Fuat Ulug

Minist ry of SupplyT.I.L., Room 0 0 9 AF i rs t Av en u e Hou seHi g h Hol b ornL on d on W . C. I , E n g l an d

N a t i o n a l A d v i s o r y C o m m i t t e e f o r A e r o n a u t i c s1512 H Street , N. W.Washington 25 , D. C , U. S. A.

*P r i n t e d by T e c h n i c a l E d i t i n g a nd R e p r o d u c t i o n L t d

9 5 G r e a t P o r t l a n d S t . L o n d o n, U . l .

1057-2200

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