1Proceedingsofthe1998IEEEInternationalConferenceonRobotics&AutomationLeuven,BelgiumMay1998ApracticalapproachtofeedbackcontrolforamobilerobotwithtrailerF.LamirauxandJ.P.LaumondLAAS-CNRSToulouse,Franceflorent,jpllaas.frAbstractThispaperpresentsarobustmethodtocontrolamobilerobottowingatrailer.Bothproblemsoftrajectorytrackingandsteeringtoagivenconfigurationareaddressed.Thissecondissueissolvedbyaniterativetrajectorytracking.Perturbationsaretakenintoaccountalongthemotions.ExperimentalresultsonthemobilerobotHilareillustratethevalidityofourapproach.21IntroductionMotioncontrolfornonholonomicsystemshavegivenrisetoalotofworkforthepast8years.Brockettscondition2madestabilizationaboutagivenconfigurationachallengingtaskforsuchsystems,provingthatitcouldnotbeperformedbyasimplecontinuousstatefeedback.Alternativesolutionsastime-varyingfeedbackl0,4,11,13,14,15,18ordiscontinuousfeedback3havebeenthenproposed.See5forasurveyinmobilerobotmotioncontrol.Ontheotherhand,trackingatrajectoryforanonholonomicsystemdoesnotmeetBrockettsconditionandthusitisaneasiertask.Alotofworkhavealsoaddressedthisproblem6,7,8,12,16fortheparticularcaseofmobilerobots.Allthesecontrollawsworkunderthesameassumption:theevolutionofthesystemisexactlyknownandnoperturbationmakesthesystemdeviatefromitstrajectory.Fewpapersdealingwithmobilerobotscontroltakeintoaccountperturbationsinthekinematicsequations.lhoweverproposedamethodtostabilizeacaraboutaconfiguration,robusttocontrolvectorfieldsperturbations,andbasedoniterativetrajectorytracking.Thepresenceofobstaclemakesthetaskofreachingaconfigurationevenmoredifficultandrequireapathplanningtaskbeforeexecutinganymotion.Inthispaper,weproposearobustschemebasedoniterativetrajectorytracking,toleadarobottowingatrailertoaconfiguration.Thetrajectoriesarecomputedbyamotionplannerdescribedin17andthusavoidobstaclesthataregivenininput.Inthefollowing.Wewontgiveanydevelopmentaboutthisplanner,werefertothisreferencefordetails.Moreover,weassumethattheexecutionofagiventrajectoryissubmittedtoperturbations.Themodelwechosefortheseperturbationsisverysimpleandverygeneral.Itpresentssomecommonpointswithl.Thepaperisorganizedasfollows.Section2describesourexperimentalsystemHilareanditstrailer:twohookingsystemswillbeconsidered(Figure1).Section3dealswiththecontrolschemeandtheanalysisofstabilityandrobustness.InSection4,wepresentexperimentalresults.32DescriptionofthesystemHilareisatwodrivingwheelmobilerobot.Atrailerishitchedonthisrobot,definingtwodifferentsystemsdependingonthehookingdevice:onsystemA,thetrailerishitchedabovethewheelaxisoftherobot(Figure1,top),whereasonsystemB,itishitchedbehindthisaxis(Figurel,bottom).AistheparticularcaseofB,forwhichrl=0.Thissystemishoweversingularfromacontrolpointofviewandrequiresmorecomplexcomputations.Forthisreason,wedealseparatelywithbothhookingsystems.Twomotorsenabletocontrolthelinearandangularvelocities（vr，r)oftherobot.Thesevelocitiesaremoreovermeasuredbyodometricsensors,whereastheanglebetweentherobotandthetrailerisgivenbyanopticalencoder.Thepositionandorientation（xr,yr,r）oftherobotarecomputedbyintegratingtheformervelocities.Withthesenotations,thecontrolsystemofBis:cossinsin()cos()rrrrrrrrrrrrttxvyvvlll（1）Figure1:Hilarewithitstrailer43Globalcontrolscheme3.1MotivationWhenconsideringrealsystems,onehastotakeintoaccountperturbationsduringmotionexecution.Thesemayhavemanyoriginsasimperfectionofthemotors,slippageofthewheels,inertiaeffects.Theseperturbationscanbemodeledbyaddingaterminthecontrolsystem(l),leadingtoanewsystemoftheform(,)xfxuwheremaybeeitherdeterministicorarandomvariable.Inthefirstcase,theperturbationisonlyduetoabadknowledgeofthesystemevolution,whereasinthesecondcase,itcomesfromarandombehaviorofthesystem.Wewillseelaterthatthissecondmodelisabetterfitforourexperimentalsystem.Tosteerarobotfromastartconfigurationtoagoal,manyworksconsiderthattheperturbationisonlytheinitialdistancebetweentherobotandthegoal,butthattheevolutionofthesystemisperfectlyknown.Tosolvetheproblem,theydesignaninputasafunctionofthestateandtimethatmakesthegoalanasymptoticallystableequilibriumoftheclosedloopsystem.Now,ifweintroducethepreviouslydefinedterminthisclosedloopsystem,wedontknowwhatwillhappen.Wecanhoweverconjecturethatiftheperturbationissmallanddeterministic,theequilibriumpoint(ifthereisstillone)willbeclosetothegoal,andiftheperturbationisarandomvariable,theequilibriumpointwillbecomeanequilibriumsubset.Butwedontknowanythingaboutthepositionofthesenewequilibriumpointorsubset.Moreover,timevaryingmethodsarenotconvenientwhendealingwithobstacles.Theycanonlybeusedintheneighborhoodofthegoalandthisneighborhoodhastobeproperlydefinedtoensurecollision-freetrajectoriesoftheclosedloopsystem.Letusnoticethatdiscontinuousstatefeedbackcan