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CHINESEJOURNALOFMECHANICALENGINEERINGVol.24,aNo. 3,a2011 1

DOI:10.3901/CJME.2011.03.***,availableonlineatwww.cjmenet.comwww.cjmenet.com.cn

AdaptiveTrajectoryTrackingControlforaNonholonomicMobileRobot

CAOZhengcai1,ZHAOYingtao1,andWUQidi2

1College ofInformationScienceandTechnology,BeijingUniversityofChemicalTechnology,Beijing100029,China2CIMSResearchCenterTongjiUniversity, Shanghai200092, China

ReceivedJune17,2010revisedMarch 8,2011accepted March 11,2011publishedelectronically March 18,2011

Abstract:Asoneofthecoreissuesofthemobilerobotmotioncontrol,trajectorytrackinghasreceivedextensiveattention.Atpresent,thesolutionoftheproblemonlytakeskinematicordynamicmodelintoaccountseparately,sothatthepresentedstrategyisdifficultto

realizesatisfactorytrackingqualityinpracticalapplication.Consideringtheunknownparametersoftwomodels,thispaperpresentsanadaptive controller for solving the trajectorytrackingproblemofamobilerobot.Firstly, an adaptivekinematic controllerutilized togeneratethecommandofvelocityisdesignedbasedonBacksteppingmethod.Then,inordertomaketherealvelocityofmobilerobotreachthedesiredvelocityasymptotically,adynamicadaptivecontrollerisproposedadoptingreferencemodelandLyapunovstabilitytheory.Finally,throughsimulatingtypicaltrajectoriesincludingcirculartrajectory,foldlineandparabolatrajectoryinnormalandperturbedcases,the resultsillustrate thatthecontrolschemecansolvethetrackingproblemeffectively.Theproposedcontrollaw,whichcantunethekinematicanddynamicmodelparametersonlineandovercomeexternaldisturbances,providesanovelmethodforimprovingtrajectorytracking performanceofthemobilerobot.

Keywords: nonholonomicmobilerobot,trajectorytracking,modelreferenceadaptive

1 Introduction *

In recent years, therehas beenan increasingamount ofresearch on the subject of mobile robotics. As a part ofresearch interests, the trajectory trackingproblem, indeed,is particularly relevant in practical applications. Manyresearchers have worked in this field for a long period.Suchstudieshavebeendividedintotwomainportions:oneutilizes the kinematic trajectory tracking controller toachieve only tracking issue, while the other one, whichcombineskinematicanddynamiccontroller, isbeingmorefrequentlyadopted.Thekinematictrackingcontrolproblemofmobilerobot,

which iscontrolledby thevelocity input,hasbeenwidelystudied. In Ref. [1], a sliding mode control scheme isproposedtosolvethetrajectorytrackingproblembasedonthe exact discrete time model of the robot. A fuzzycontroller[23] is designed to realize tracking control formobile robots. In Ref. [4], a modelpredictive trackingcontrol applied to a mobile robot is presented, where thecontrol law primarily minimized quadratic cost functionconsistingof trackingerrorsand thecontroleffort.InRef.[5], an adaptive tracking controller is proposed for the

* Correspondingauthor.Email:[email protected] project is supported by State Key Laboratory of Robotics and

System ofChina(GrantNo.SKLR2010MS 14)and StateKeyLabofEmbeddedSystemandServiceComputing ofChina(GrantNo. 201011)

mobile robot based on integrating the analog neuralnetworkintothe Backsteppingtechnique.Someresearchersfocusedonsolvingthetrackingcontrol

problemconsideringthedynamicmodelofmobilerobot.InRef. [6], a state feedback control law based on dynamicmodel is proposed forwheeledmobile robot to control itstwo driving motors synchronously. A dynamic controlalgorithm[7] is presented for realizing the tracking controlbasedonmodeling.Adaptive control has been applied to solving tracking

control problem popularly because of the ability ofresolving uncertainties. In Refs. [811], an adaptivedynamic controller is proposed for mobile robot toattenuate the effects of the uncertainties and disturbances.The adaptive control law is obtained by analyzingLyapunov function. A discretetime sliding modeapproach[12]ispresentedforamobilerobot,consideringtheuncertaintiesinthedynamicalmodel.Above methods have played an important theoretical

guiding role in trajectory tracking of mobile robots.However, these control schemes do not consider thekinematicanddynamicmodeloftherobotsimultaneously,sothatthetrackingperformancemaybenotsatisfactoryinpractice.Inthispaper,anadaptivetrajectorytrackingcontrolleris

proposedwithkinematicanddynamicuncertainties,anditsstability is proved by using the Lyapunov theory. Thedesign of the controller is divided in two parts, each partbeing a controller itself. The first one is an adaptive

YCAOZhengcai,etal:AdaptiveTrajectory TrackingControlfor aNonholonomicMobileRobot2

controller,whichispresentedforthekinematicmodelwithuncertainties, the other one is amodel reference adaptivecontroller that isappliedtodynamicmodelwithunknownparameters.Thesimulationresults show thattheproposedcontrolleriscapableofguidingtherobottotrackadesiredtrajectorywithaquitesmallerror.This paper is organized as follows. In section 2, the

model for mobile robot with nonholonomic constrains isintroduced. In section 3, an adaptive kinematic controllerand a model reference adaptive dynamic controller arepresented.Simulationresults for theproposedstrategyaregiveninsection4.Finally section5 givestheconclusions.

2 MobileRobotModel

Themobilerobotwithtwoindependentdrivenwheelsisshown inFig.1.Oxy is theworldcoordinatesystem.ThemasscenterofrobotisCwhichisalsothemiddleofthedrivingwheels. r is the radius of rear wheels and l is thedistanceofrearwheels.mbandmwarethemassofthebodyandwheelwithamotor,Ic,Iw,Imarethemomentofinertiaof the body about the vertical axis through C, the wheelwith amotor about thewheel axis, and thewheelwith amotoraboutthewheeldiameter,respectively.The configuration of themobile robot can be described

byfivegeneralizedcoordinates:

Tr ]l,,,,[

~ f f qyx =q , (1)

where (x, y) are the coordinates ofC, is the orientationangleofrobot,and lr,f f aretheanglesoftherightandleftdrivingwheels.The kinematic model for the mobile robot under the

nonholonomic constraint of pure rolling and nonslippingis[13]

- =

2

1

l

r

10

0122

sin2

sin2

cos2

cos2

v

v

lr

lr

rr

rr

y

x

q q

q q

f f q

&

&

&

&

&

. (2)

Assumingalltheuncertaintiesanddisturbancesarezero,the dynamic equation of the simple model of thenonholonomicmobilerobotis

BvqM = &)~( , (3)

where T21 ][ t t = is the torque applied to the right and

left wheels, T21 ][ vv =v represent the angular velocities ofrightandleftwheels. MandB areselectedas

+ + -

- + + =

w2

2

22

2

2

22

2

w2

2

2

)(4

)(4

)(4

)(4

IImll

rIml

l

r

Imll

rIIml

l

r

M ,

=

10

01B .

where mc2

wwb 22,2 IIlmImmm + + = + = .

w v

C

r2

l2

q

O

Fig.1. Mobilerobotconfigurationanditsmotioncoordination

3 Adaptive Controller Design for MobileRobot

In general, the trajectory tracking problem aims attracking a reference mobile robot with a known postureqr=[xr,yr,r]

T.Thereforetheerrorsbetweentheactualanddesiredposturesare qe=qrq=[xrx, yry, r]

T=[xe, ye, e]T.

'x

'y

O

C

xe

ye

q

y

x

),( yx

),( rr yx

r q

Fig.2. Robotpostureerrorcoordinate

ThepostureerrorepexpressedintheframeCxyoftherealrobot,asshowninFig.2, is:

- = - =

=

e

e

e

rep

100

0cossin

0sincos

)(

y

x

e

e

e

y

x

q q q q

qqTe , (4)

CHINESEJOURNALOFMECHANICALENGINEERING 3

where Teistransformationmatrix.Therefore, the control algorithm should be designed to

forcetherobottotrackthereferencetrajectoryprecisely,i.e.0lim p = et .

3.1 DesignoftheadaptivekinematiccontrollerThedesignof thekinematiccontroller isbasedon the

kinematic model of the robot. Linear velocity v andangular velocity w are related to v1 and v2 by thetransformation:

- =

w

v

rl

r

rl

rv

v1

1

2

1 . (5)

Substituted Eq. (5) for Eq. (2), the ordinary form of amobilerobotwithtwoactuatedwheelscanbeobtained

=

=

w

vy

x

10

0sin

0cos q q

q & &

&

&q , (6)

where q=[x, y, ]T stands for the pose ofmobile robot ininertialcoordinationO xy.Eq.(4)describesthedifferenceofpositionanddirection

of the reference robot from the real robot. Using theBackstepping method, the input T],[ wv =u , which makesepconvergetozero,isgivenbythefollowing

[8]:

+ + =

+ =

,sin

,cos

r3r2rc

1rc

y

x

evkevkww

ekevv(7)

where k1, k2 and k3 are positive constants, vr, and wraredesiredlinearandangularvelocitiesofrobot.TheobjectiveofsuchacontrolleristogeneratethereferencevelocitiesofwheelsforthedynamiccontrollerbasedonEq.(5).IftheparametersrandlinEq.(2)areunknown,Eq.(7)

cannotbechosenasaninputbecauseoftherelationshipEq.(5).Hence,anadaptivecontroller isdesigned toattain thecontrolobjectivebyusingtheestimatesofrand l.Setting1=1/r and 2=l/r, the velocity control input can beexpressedas

- =

c

c

21

21

c2

c1

w

v

v

v, (8)

where 1 b and 2 b are the estimates of 1 b and 2 b , respectively,

and 111~ b b b + = , 222

~ b b b + = ,noticing 1 b , 2 b >0.Theparameterupdatinglawsarechosenas[13]

=

=

,sin

,

2

c22

c11

kew

ve

x

a b

a b &

&

(9)

where 1 a and 2 a arepositiveconstants.Accordingto Ref.[13],thecontrolsystemisstable.

3.2 Design of the model reference adaptive dynamiccontroller

Inthissection,velocityvector T2c1cc ][ vv =v generatedbyEq.(8)areconsideredasthereferenceangularvelocitiesofwheels.Thesevaluesareregardedasthereferenceinputsforthedynamiccontroller.The following control laws are employed to prepare

trackingof v1c,and v2c[14]:

c1

a~ vMBv & - + =k , (10)

where ka is a positive constant, and vvv - = c~ is angular

velocityerrorvector.It is well known that the parameter variations of the

system, suchasmass and inertia,aredifficult tomeasure.Therefore,thecontrollawsshouldbewrittenasfollows:

ca~ vPv & + =k , (11)

where MBP 1 - = 22 R , P 22 R isestimateof P.Substituting Eq. (3) in Eq. (11), the following closed

loopequationofvelocitiesisobtained:

vvPvP ~ dc k + = & & . (12)

Define '1 MPP = - 22 R , NP = - d1k 22 R , theEq.

(12)canberewrittenas[14]

vNvMv ~c' + = & & . (13)

Referencemodelforangularvelocityerrorisselectedasfollows:

0ee = + vv T& , (14)

where T is the positive time constant of error damping.

mce vvv - = , mv is the velocity of the reference model.ThenEq.(15)canbeobtained:

mccm vvvv T - + = & & , (15)


The adaptation error, which is the difference betweenv and the velocity of the reference model mv , is

mvve - = .

Thederivativeof eis

vENvENvEMe

vvvNvNvvM

vve

)()()( cc'

mcccc'

m

TTT

TT

- - - + - + - =

+ - - - + =

- =

&

& &

& & &

, (16)

where ER22isunitmatrix.Defining

=

=

2221

1211

2

1'

mm

mm

M

MM ,

=

=

2221

1211

2

1

nn

nn

N

NN ,

=

2

1

e

ee ,

=

=

10

01

2

1

E

EE ,

and considering the following Lyapunov function

candidate

T1111

2

T1111

1

211

))((1

))((1

2

1

ENEN

EMEM

TT

eV

- -

+ - - + =

l

l, (17)

with1,2>0,itsderivativeobtainedbyusingEq.(16)

is

.)1

(

)1

)((

)1

(

)1

)(1(

)(1

)(1

212c1222

22

111c1212

21

2c1121

12

1c1111

1121

T111

2

T111

1111

vevemm

vevemTm

vemm

vemmTe

TeeV

- +

+ - + -

+ +

+ + - + - =

- + - + =

&

&

& &

& &

& & & &

l

l

l

l

l lNENMEM

(18)

Choose

- - =

- - =

- =

- =

),(

),(

,

,

22c1221

11c1221

2c1112

1c1111

vvem

vvem

vem

vem

&

&

& &

& &

(19)

thenEq.(18)becomes

211 TeV - = & 0. (20)

Similarlyforthenextpart:

- - =

- - =

- =

- =

,)(

,)(

,

,

22c2421

11c2421

2c2312

1c2311

vven

vven

ven

ven

&

&

& &

& &

(21)

where parameters i (i=1,2,3,4)>0, and will be tuned toreachthedesiredperformance.AccordingtotheLyapunovtheory,thecontrolsystemisasymptoticallystable.

4 SimulationResults

Toobserveuniversalityoftheapproach,threekindsoftypicalreferencetrajectoriesforthesimulationarechosen:the firstoneisacircular trajectory, thesecondoneis foldline trajectory and the other one is a parabola trajectory.Systemparametersoftherobotareselectedasfollows

r=0.15 m, l=0.75 m, mb=30 kg , mw=1 kg, Ic=15.625

kgm2, Iw=0.005kgm2, Im=0.0025kgm2,1=2=5, T=10,k1=4.0, k2=6.0, k3=5.5, i(i=1,2,3,4)=5.The resultingmobile robot trajectory tracking, obtained

by the proposed adaptive controller, is shown inFig.3Fig.14 including trajectory tracking and trackingerrors. In these results, the sampling period was set toT0=0.01s,theidealandrealinFigs.3,5,7,9,11standfor ideal trajectory and real trajectory of mobile robot,respectively.

4.1 NominalCaseIn this case, there are no external disturbances.

Fig.3Fig.8 showsthetrackingresults.(1) The circular trajectory was generated form the

desiredvelocity vr=1m/sandangularvelocity wr=1rad/s.

Fig.3. Circulartrajectorytracking


Fig.4. Trackingerrorsincirculartrajectorytracking

The initial posture of the reference trajectory is set atqr(0)=[5, 0, 0]

Twhile theactual initial posture of robot isq(0)=[6,0,/2]T.Circulartrajectorytrackingsimulationanderror curve are seen in Fig .3 and Fig. 4, respectively,whereadjustmenttime t=2.0s.(2) The fold line trajectory was generated form the

desiredvelocityintime:

<

< <

=

sts

sts

st

v

1510,m/s5.1

105,m/s2.1

50,m/s5.1

r ,

andangularvelocityiswr=0rad/s.

Fig.5. Foldlinetrajectorytracking

Fig.6. Trackingerrorsinfoldlinetrajectorytracking

The initial posture of the reference trajectory is set atqr(0)=[0, 0, /4]

T while the actual initial posture of therobot is q(0)=[0, 1, /4]T. Fold line trajectory trackingsimulationanderrorcurveareshown inFig.5andFig.6,respectively. In Fig. 5, the reference orientation alterssuddenly form/4 tozeroat 5s andchanges fromzero to/4at10s,sotwopulsescanbeobservedinerrorcurveasshowninFig.6.Moreover,aftereachreferenceorientationchanging,thetrackingerrorsconvergetozeroquickly.(3)Theparabolatrajectorywasproducedbythedesired

velocity vr=0.8 m/s and angular velocity wr=0.16 rad/s.The initial posture of the reference trajectory is set atqr(0)=[0,0,/4]

Twhiletheactualinitialpostureofrobotisq(0)=[1, 0, /2]T. Parabola trajectory tracking simulationand error curve are depicted in Fig .7 and Fig. 8,respectively,whereadjustmenttime t=2.5s.

Fig.7. Parabolatrajectorytracking

Fig.8. Trackingerrorsinparabolatrajectorytracking

Evidently,thiscontrollerhasaperfecttrackingabilityasshown in above results. It can be seen that the controllercan correct deviations quickly and there are little trackingerrors.

4.2 PerturbedCase:To test capacity of resisting disturbance of controller,

white noise randn(t) is added based on above reference


trajectories in tracking process, respectively. randn(t)standsforwhitenoisewithaveragevalue0andvariance1.

)('r tx , )('r ty and )(

'r t q are respectively measured values

of )(r tx , )(r ty and )(r t q . Fig.9Fig.14 demonstrate thetrackingresults.(1) Supposing )(01.0)()( r

'r trandntxtx + = , the circular

trajectorytrackingprocessandtrackingerrorareshowninFigs. 9, 10,respectively,where adjustmenttime t=2.3s.

Fig.9. Circulartrajectorytrackingintheperturbedcase

Fig.10. Trackingerrorsincirculartrajectorytracking

(2)Supposing )(005.0)()( r'r trandntyty + = ,thefoldline

trajectorytrackingprocessandtrackingerroraredescribedinFig. 11 andFig. 12,respectively.

Fig. 11. Foldlinetrajectorytrackingintheperturbedcase

Fig.12. Trackingerrorsinfoldlinetrajectorytracking

(3) Supposing )(01.0)()( r'r trandntt + = q q , the parabola

trajectorytrackingprocessandtrackingerrorareshowninFigs. 13, 14,respectively,whereadjustmenttime t=2.4s.

Fig.13. Parabolatrajectorytrackingintheperturbedcase

Fig.14. Trackingerrorsinparabolatrajectorytracking

All the simulation results indicate that posture errorsconverge to a small neighborhood of zero in case ofexternaldisturbances.Inotherwords,theresultsverifythatthe controlmethod is robust enough to resist the externaldisturbances.


5 Conclusions

(1)Considering uncertainties in kinematicmodel of themobile robot,anadaptivekinematiccontroller isproposedbasedonBackstepping method.(2) A model reference adaptive dynamic controller is

presentedfortheunknowndynamicparameters.(3)Thesimulationofthecontrollawis implementedin

nominal and perturbed cases. All the simulations indicatethattheproposedschemeisindeedfeasibleandeffective.

References[1] NINOSUAREZPA,ARANDABRICAIREE,VELASCOVILLA

M. Discretetime sliding mode trajectorytracking control for awheeledmobileRobot[C]//Proceedingsofthe45thIEEEConferenceonDecisionandControl,SanDiego,USA,December13-15,2006:3 025-3 057.

[2] ZOU X Y, XU D, LI Z Y. Piecewise fuzzy control for trajectorytrackingofnonholonomicmobileRobots[J].ControlandDecision,2008,23(6): 655-659.(inChinese)

[3] CHIAVERINI S, FUSCO G. A fuzzy logic based approach formobilerobotpathtracking[J].IEEETransactiononFuzzySystems,2007,15(2):211-221.

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[8] CHENCY,LITHS,YEHYC,etal.Designandimplementationofanadaptive slidingmodedynamiccontroller forwheeledmobile

robots[J].Mechatronics,2009,19(2):156-166.[9] MARTINSFN,CELESTEWC,CARELLIR, et al.Anadaptive

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[10] KIMMS,SHINJH,HONGSG,etal.Designingarobustadaptivedynamiccontrollerfornonholonomicmobilerobotsundermodelinguncertainty and disturbances [J]. Mechatronics, 2003, 13(5):507-519.

[11] POURBOQHRAT F, KARLSSON M P. Adaptive control ofdynamic mobile robots with nonholonomic constraints [J].ComputersandElectricalEngineering,2002,28:241-253.

[12] CORRADINI M L, ORLANDOG. Control of mobile robots withuncertainties in the dynamicalmodel: a discrete time slidingmodeapproachwithexperimentalresults[J]. ControlEngineeringPractice,2002,10(1):23-34.

[13] FUKAOT,NAKAGAWAH,ADACHIN.Adaptivetrackingcontrolofanonholonomicmobilerobot[J].IEEETransactionsonRoboticsandAutomation,2000,16(5):609-614.

[14] GHOLIPOURA,YAZDANPANAHMJ.Dynamictrackingcontrolof nonholonomicmobile robotwithmodel referenceadaptation foruncertainparameters[C]//EuropeanControlConference,Cambridge,UK,September,2003.

BiographicalnotesCAOZhengcai is anassociate professor atBeijingUniversity ofChemicalTechnology,China.Hismajorresearchinterestssensortechnique,intelligentcontrolofrobot.Email: [email protected].

ZHAO Yingtao is a graduate candidate in Beijing University ofChemical Technology, China. His major research interestsintelligentcontrolofrobot.Email:[email protected].

WUQidiisaprofessorinSchoolofElectronicsandInformationEngineering, Tongji University, China. Her research interestcovers complex systems scheduling, system engineering,managementengineering,andintelligentcontrol.

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