Journal of Beijing Institute of Technology, 2011, Vol. 20, No. 3

Trajectory shaping guidance law with terminalimpact angle constraint

LIU Da鄄wei(刘大卫)1,苣, 摇 XIA Qun鄄li(夏群利)1, 摇 WU Tao(武涛)2,摇 QI Zai鄄kang(祁载康)1

(1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;2. China Research and Development Academy of Machinery Equipment, Beijing 100089, China)

Abstract: In order to strike hard targets underground or warships and tanks with expected impactangle by missiles or guided bombs, trajectory shaping guidance law with terminal position and impactangle constraints is derived based on linear quadratic optimal control theory. The required accelera鄄tion expressed by impact angle and heading error is obtained in lag鄄free guidance system in order tofind the optimal relationship of those angles in terminal phase. The adjoint systems of miss distanceand impact angle error of first鄄order guidance system are established based on statistical linearizationadjoint method (SLAM) in order to study the impact performances of the guidance law. Simulationresults show that the miss distance and impact angle error of trajectory shaping guidance law areboth according with the impact position and angle constraint and the required acceleration at impactcan be decreased by an optimal relationship of impact angle and heading error.

Key words: trajectory shaping; required acceleration; miss distance; impact angle errorCLC number: V 448郾 133摇 摇 Document code: A摇 摇 Article ID: 1004鄄0579(2011)03鄄0345鄄06

Received摇 2010鄄10鄄14Supported by the Aeronautical Science Foundation of China(20060112123)苣 Author for correspondenceE鄄mail: 20031145@ bit. edu. cn

摇 摇 In order to attack hard target effectively,guided weapons have to impact the target not onlywith high accuracy in impact position but alsowith specific terminal impact angle constraint.Therefore, optimal terminal guidance law be鄄comes an important field of homing guidancenowadays. Kim and Grider have proposed an op鄄timal impact angle control guidance law for thereentry vehicle in the vertical plane[1] . Ryoo andCho have investigated an optimal guidance law forconstant speed missiles with constraints on theimpact angle and control input[2] . Ernest andCraig have proposed and evaluated a new guid鄄ance law termed generalized vector explicit guid鄄ance[3] . Liu Dan and Chang Chao have proposedan optimal guidance law for constant speed mis鄄

siles with constraints on small impact angle andextended to big impact angle constraint in nonlin鄄ear simulation[4 -5] . However, none of those pa鄄pers give us the overload characteristic, and wedon爷 t know the miss distance and impact angleerror of those guidance laws. In this paper, first鄄ly, a simple geometry is given and an optimalguidance law based on linear quadratic optimalcontrol theory is obtained with impact positionand angle. Then, required acceleration with lag鄄free system is given to disclose the overload char鄄acters with initial conditions and impact con鄄straints. Finally, miss distance and impact angleerror of trajectory shaping guidance law are stud鄄ied by SLAM. Simulation results show that trajec鄄tory shaping guidance law accords to the terminalguidance problem with impact position and angleconstraints perfectly.

1摇 Trajectory shaping guidancelaw

摇 摇 Considering the homing guidance geometry

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Journal of Beijing Institute of Technology, 2011, Vol. 20, No. 3

for a stationary or a slowly moving target, three鄄dimensional guidance problem can be analysed astwo orthogonal plane guidance[6] . The planar ge鄄ometry of guidance is shown in Fig. 1. Here, Rand z denote the relative distance between missileand target and the distance to initial line of sight(LOS), respectively. V, Vr and v are the missiletotal velocity, the relative velocity in initial LOSand the normal velocity component, respectively.q,兹,qf and 着 are the angle of line of missile鄄targetand initial LOS, the angle of flight path, the im鄄pact angle constraint, and the heading error, re鄄spectively, and am is the acceleration applied nor鄄mal to the velocity vector.

Fig. 1摇 Terminal guidance geometry

The velocity of missile is not rapidly changedin terminal phase, we can suppose that Vr is con鄄stant, 兹 is small, and the target is stationary ormoving slowly. So that, impact constraints of thisoptimal guidance law can be described as the nor鄄mal velocity arriving at ve and the position comingto ze at the final guide time.

The impact constraint equations arez(T) = ze, (1)v(T) = ve . (2)

The state equations of the system arez· = v, (3)v· =az . (4)

The control variable of the optimal problem isthe normal acceleration az( t) . Now, let us con鄄sider the following optimal control problem: findaz( t), which minimizes J, defined by

J( t0) =准(x(T),T) + 乙Tt0

12 a2

z( t)dt, (5)

where 准(x(T),T) = cz(z(T) - ze)2 / 2 + cv(v(T) -ve) 2 / 2.

The physical significance of the object func鄄tion is the control energy and the effect of the in鄄duced drag to the velocity become minimum.Here, cz,cv are the parameters which can be de鄄signed to meet Eqs. (1) (2) . The bigger cz and cv

are, the better constraints are met, and the con鄄straints equations will be met strictly if cz寅肄 andcv寅肄 .

The Hamilton is obtained as

H = 12 a2

z + v姿z +az姿v . (6)

The adjoint equations therefore are

-姿·

z = 鄣H鄣z = 鄣

鄣 (z12 a2

z + v姿z +az姿 )v =0, (7)

-姿·

v = 鄣H鄣v = 鄣

鄣 (v12 a2

z + v姿z +az )姿 =姿z . (8)

Thus, the optimal control is

0 = 鄣H鄣az

= 鄣鄣a (

z

12 a2

z + v姿z +az姿 )v =az +姿v .

(9)That is,

az( t) = -姿v( t) . (10)The constraints of impact can be described as

鄣准(T)鄣z

鄣准(T)鄣

é

ë

êêêê

ù

û

úúúúv

-姿z(T)姿v(T

é

ëêê

ù

ûúú)=0郾 (11)

We can obtain the analytic expression of theoptimal problem as

ac( t) = 6(T - t) 2(ze - z( t)) -

4(T - t)v( t) - 2

T - tve . (12)

Eq. (12 ) shows the optimal terminal guid鄄ance law with impact position and angle con鄄straints by the position of missile z( t), the veloc鄄ity of missile v ( t), the desired point of impactze, the predetermined impact normal velocity ve

and time to go tgo =T - t.We can get the relationship between the im鄄

pact angle and the final normal velocity as the fol鄄lowing.

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LIU Da鄄wei(刘大卫) et al. / Trajectory shaping guidance law with terminal impact angle constraint

When zt = 0, the angle of LOS can be ex鄄pressed as the following:

tan q =zt - zR =

zt - zVrtgo

= - zVrtgo

. (13)

When the angle of LOS is small, the expres鄄sion above can be represented as

q抑tan q =zt - zR =

zt - zVrtgo

= - zVrtgo

. (14)

Then, the rotational velocity of the angle ofLOS is

q·( t) = - z·( t)tgoVr

- z( t)t2goVr

. (15)

The position can be replaced by the angle ofLOS, time to go tgo and the relative velocity of theLOS Vr form Eq. (14) .

z( t) = - q( t) tgoVr . (16)From Eq. (15), the derivative of the position

can also be replaced asz·( t) = - q·( t) tgoVr - z( t) / tgo . (17)

As define in Eq. (3), v( t) = z·( t), we canget the normal velocity as

v( t) = - q·( t) tgoVr + q( t)Vr . (18)

Then,substitute Eq. (18) in Eq. (12), thereis

ac( t) =4 q·( t)Vr +

2tgo

q( t)Vr -2tgo

ve + 6t2go

ze .

(19)At the impact point, ze = zt =0,tgo =0,q( t) =

qf, therefore Eq. (18) becomesve = v(T) = qfVr . (20)

Substitute Eq. (20) in Eq. (19), we can getac( t) =4Vrq

·( t) +2Vr(q( t) - qf) / tgo . (21)It is not difficult to find that the optimal ter鄄

minal guidance law is made up by two parts, oneis the proportional navigation to guarantee highposition accuracy, another is the impact angleconstraint to ensure the impact angle. We can callthis optimal terminal guidance law as trajectoryshaping guidance law because it can guide missilein an optimal trajectory to guarantee both the im鄄pact position and angle.

Consider the terminal phase of the air鄄to鄄ground missile starts at the altitude of 1 km with

the velocity of 250 m / s in horizontal plane, andthe target is stationary at 5 km far from the mis鄄sile on the ground. In order to penetrate the tar鄄get, we require a vertical top attack, that is, thedesired falling angle is 90毅. Trajectory and flightpath angle are shown in Fig. 2, considering thedynamics of guidance system with a time constantof 0郾 5 s. Simulation results show that the trajec鄄tory is smooth and resulting in a miss distance of0郾 03 m and impact angle error of - 0郾 67毅. Thehigh accuracy of guidance and satisfaction of theterminal constrains reveals that the guidance lawis suitable for air鄄to鄄ground guided weapons withlarge falling angle.

Fig. 2摇 Trajectory and history of flight path angle

2摇 Required acceleration with lag鄄free system

摇 摇 Required acceleration is one of the most im鄄portant performance indexes to a terminal guid鄄ance law. In order to study the guidance lawwithout the effect of the control system dynam鄄ics, lag鄄free system is chosen.

Another expression of the closed鄄form of therequired acceleration can be given by the initialcondition and expected constraints as

ac( t) =12z(0)( tgo -0郾 5T) +T z·(0)(6tgo -2T) +T z·(T)(6tgo -4T)

T3 .

(22)According to assumptions above, we have

z(0) =0,z·(0) =Vr着, z·(T) =Vrqf .Take those equations into Eq. (11), then the

required acceleration is

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Journal of Beijing Institute of Technology, 2011, Vol. 20, No. 3

ac( t) = -2Vr着 (T 2 -3 t )T -

2Vrqf (T 1 -3 t )T .

(23)We can see that there are two parts of the re鄄

quired acceleration, one is caused by the impactangle qf, another is caused by the heading error着.

To a certain guidance time T, initial condi鄄tion and impact constraints, acceleration is pro鄄portionable to the flight time t, so the maximalacceleration appears at the initial or final time.So, we take more attention to the following typi鄄cal guidance times from Eq. (23):

ac( t =0) = -2(qf +2着)Vr / T,ac( t =T / 3) = -2着Vr / T,ac( t =T / 2) = (qf - 着)Vr / T,ac( t =2T / 3) =2qfVr / T,ac( t =T) =2(2qf + 着)Vr

ì

î

í

ïïïï

ïïï / T.

(24)

The required accelerations at these typicalguidance times in different relationship betweenheading error and impact angle are given in Fig.3, and we can see that:

Fig. 3摇 Required acceleration at typical guidance times摇

淤 The acceleration commands at t =0,T / 2,Tlie on impact angle qf and heading error 着, how鄄ever, the accelerations at t =T / 3,2T / 3 depend onheading error 着 and impact angle qf, respectively.

于 If we hope the missile maneuver accelera鄄tion minimum at impact, the heading error 着 mustbe twice of the impact angle qf, but in oppositedirection.

3摇 Miss distance and impact angleerror with first鄄order systembased on SLAM

摇 摇 The statistical linearization adjoint method isa new computerized approach for the completestatistical analysis of nonlinear missile guidancesystems through the combination of the CADETmethod and the adjoint technique[7] . SLAM is anexcellent design and analysis tool for missile guid鄄ance systems that provides an error budget formiss distance and contains information concern鄄ing the system behavior[8] .

In order to simplify the homing guidance sys鄄tem, the dynamics of seeker is ignored and thedynamics of autopilot is assumed as first鄄order[7],that is

aL(s)ac(s)

= 1Tgs +1. (22)

The trajectory shaping guidance loop withfirst鄄order lag system is given in Fig. 4. Here, Zt

is the position of target, T is the total terminalguidance time, Tg is the time constant of autopi鄄lot, qf is the impact angle constraint and 着 is theheading error, which represents the initial devia鄄tion of the missile from the LOS.

Fig. 4摇 Trajectory shaping guidance loop

In narrow sense, miss distance is defined fo鄄cusing on the final position only. Trajectory sha鄄ping guidance law pays attention to both impactposition and angle. We defined the generalizedmiss characteristics both including the two factorsabove and study their characteristics based onSLAM.3. 1摇 Miss distance based on SLAM

We can obtain the adjoint system from Fig. 4by adjoint system transform method. Then, T =T / Tg, s = sTg, make the adjoint system output

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LIU Da鄄wei(刘大卫) et al. / Trajectory shaping guidance law with terminal impact angle constraint

non鄄dimension relative to the extraneous parame鄄ters. Finally, the impulse input is replaced bystep input which equal to an impulse pass an inte鄄grator as shown in Fig. 5.

Fig. 5摇 Adjoint loop of miss distance

The adjoint system above is a single鄄inputtwo鄄output system. So, we can assume that theheading error is k times of the size of the impactangle, that is, k = 着 / qf . And the outputs can beexpressed by one input and a coefficient k. Thenon鄄dimension miss distance characteristic withthe non鄄dimension time in different coefficient kis given in Fig. 6.

Fig. 6摇 Non鄄dimension miss distance of different k

3. 2摇 Impact angle error based on SLAMIn first鄄order lag system of impact angle er鄄

ror, we can only obtain three results - 仔 / 2, 0and -仔 / 2 because of the impact angle is calculat鄄ed by the missile position z and distance of LOSVrtgo, and the calculation is singularity when z屹0and Vrtgo = 0. However, in missile homing guid鄄ance engineering, people pay more attention tothe velocity vector, so the angle of LOS q is re鄄placed by angle of flight path 兹 near the impactpoint. And the angle of flight path 兹 can be ob鄄tained by

兹抑淄 / Vr . (23)After replace q by 兹, trajectory shaping guid鄄

ance loop with first鄄order lay system in Fig. 4 canbe rewritten, and the adjoint system of impact an鄄gle error can also be obtained by the same princi鄄ple as one of the miss distance in Fig. 7.

Fig. 7摇 Adjoint loop of impact angle error摇

The non鄄dimension impact angle error char鄄acterisitc with the non鄄dimension time in differentcoefficient k is given in Fig. 8.

Fig. 8摇 Non鄄dimension impact angle error of different k摇

Simulation results of terminal errors of trajec鄄tory shaping guidance law due to the first鄄ordermissile guidance model show that:

淤 Trajectory shaping guidance law can in鄄sure both the impact position and angle to meetthe expected value. The convergence times of themiss distance and the impact angle error are a鄄bout 12 and 15 times of the first鄄order time con鄄stant Tg, respectively.

于 The bigger the heading error is, the biggerthe miss distance and impact angle error are forthe same impact angle constraint, and the longerthe converge time is.

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Journal of Beijing Institute of Technology, 2011, Vol. 20, No. 3

4摇 ConclusionsOptimal guidance law both with impact posi鄄

tion and angle based on optimal control problemis deduced and trajectory shaping guidance lawwith lag free system is given upon the assumptionof linear model and small angle. It is concludedthat: 淤 Trajectory shaping guidance law can beused in guided weapon to implement the terminalguide problem both with impact position and an鄄gle; 于In order to obtain minimal required accel鄄eration at impact in lag鄄free system, the headingerror angle must be twice of the impact angle ex鄄pected; 盂 Miss distance and impact angle errorwill go to zero, when terminal guidance times areabove 12 and 15 times of the first鄄order systemtime constant, respectively. All of which give usan important enlightenment and reference in thehoming guidance engineering.

References:[1] 摇 Kim M, Grider K V. Terminal guidance for impact

attitude angle constrained flight trajectories [ J ] .IEEE Transactions on Aerospace and Electronic Sys鄄tems,1973,9(6): 852 -859.

[2] 摇 Ryoo Chang鄄Kyung, Cho Hangju, Tahk Min鄄Jea.

Optimal guidance laws with terminal impact angleconstraint [ J] . Journal of Guidance, Control andDynamics,2005,11(4):724 -732.

[3] 摇 Ohlmeyer Ernest J, Phillips Craig A. Generalizedvector explicit guidance[ J] . Journal of Guidance,Control and Dynamics, 2006, 29(2): 261 -268.

[4] 摇 Liu Dan, Qi Zaikang. Impact angle and final positionconstrained optimal guidance law [ J] . Journal ofBeijing Institute of Technology, 2001, 21(3): 278 -281. ( in Chinese)

[5] 摇 Chang Chao, Lin Defu, Qi Zaikang, et al. Study onthe optimal terminal guidance law with interceptionand impact angle [J] . Journal of Beijing Institute ofTechnology,2009,29(3):233 -236. ( in Chinese)

[6] 摇 Cai Hong, Hu Zhengdong, Cao Yuan. A survey ofguidance law with terminal impact angle constraints[J] . Journal of Astronautics, 2010, 31(2): 315 -332. ( in Chinese)

[7] 摇 Paul Zarchan P. Tactical and strategic missile guid鄄ance[M] . 4th ed. [ S. l. ]: American Institute ofAstronautics and Aeronautics, Inc. , 2002: 541 -548.

[8] 摇 Paul Zarchan P. Complete statistical analysis ofnonlinear missile guidance systems—SLAM [ J ] .Journal of Guidance and Control, 1979, 2(1): 71 -78.

(Edited by Wang Yuxia)

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