Algebraic estimation and flatness based lateral longitudinal control for automotive vehicle

论文

2011 14th International IEEE Conference onIntelligent Transportation Systems

Washington, DC, USA. October 5-7, 2011

Algebraicnonlinearestimationand?atness-basedlateral/longitudinalcontrolforautomotivevehicles

LghaniMenhour,Brigitted’Andr´ea-Novel,Cl´ementBoussard,MichelFliessandHuguesMounier

Abstract—Acombinedlongitudinalandlateralvehiclecon-trolispresented.Itemploys?atness-basedcontrolandnewalgebraicestimationtechniquesforthenumericaldifferen-tiationofnoisysignals.Thisnonlinearcontrolisdesignedforautomaticpath-trackingviavehiclesteeringangleanddriving/brakingwheeltorque.Itcombinesthecontrolofthelateralandlongitudinalmovementsinordertoensureanaccuratetrackingofstraightorcurvedtrajectories.Itcanalsobeusedtoperformacombinedlane-keepingandsteeringcontrolduringcriticaldrivingsituationssuchasobstacleavoidance,stop-and-gocontrol,lane-changemaneuversoranyothermaneuvers.Promisingresultshavebeenobtainedusingthenoisyexperimentaldataacquiredbyalaboratoryvehicleunderhighdynamicloadsandcharacterizedbyhighlateralaccelerations.

I.INTRODUCTION

Theresearchonintelligenttransportationsystems,theaimofwhichistoimproveroadsafety,isgettingmoreandmoreactive.Someassistancesystemshavebeendevelopedinordertohelpthedriverincriticalsituations,andtoimprovethevehiclestabilityandsteerability.Amongthose,wecan?ndforexampleAnti-LockBrakingSystem,ElectronicStabilityprogram,AdaptiveCruiseControlandrecentlyfourwheelsteeringsystems.

Todevelopautonomousvehicles,thedesignofcombinedlateralandlongitudinalcontrolisrequiredtoperformsomecoupledmaneuversasstop-and-gocontrolwithobstacleavoidance,lane-changemaneuvers,andtheappropriatesteer-ingangletoensurelanekeeping.Amajorityofthestrategiesproposedinthepreviouspublishedworksdealwithlongitu-dinalorlateralcontrolseparately,andrarelywithacombinedimplementation,forexampletoperformabrakingactionincurve.In[22],asurveyofdrivermodelsandtheirapplicationinautomotivedynamicsispresented,andpapersreferredinthislargeinventorydealonlywithsteeringvehiclecontrol.In[25]anintegrationoflongitudinalandlateralcontrolfor

ThisworkwassupportedbytheFrenchnationalprojectINOVE/ANR2010BLANC308.

L.MenhourandB.d’Andr´ea-NovelarewithMines-ParisTech,CAOR-CentredeRobotique,Math´ematiquesetsyst`emes,60boulevardSaint-Michel,75272Pariscedex06,France.

lghani.menhour@mines-paristech.fr

brigitte.dandrea-novel@mines-paristech.frclement.boussard@inria.fr

Cl´ementBoussardiswithIMARA,INRIA,DomainedeVoluceau-Rocquencourt,B.P.105,78153LeChesnay,France.

´MichelFliessiswithLIX(CNRS,UMR7161)Ecolepolytechnique,91128Palaiseau,France.

platooninginhighwaysispresentedinadecoupledway.Thesteeringcontrolstrategyfortheautonomouslanechangeinhighwayorforurbansteeringdrivingarealsopresentedinseveralworks,like[2],[4],[12],[13],[16].

Longitudinalcontrolisconsideredinsomepapers,forex-amplein[17],[20],[27],[28]withsomestop-and-gocontrolstrategiestokeepasafetydistanceandavoidcollisions.Ontheotherhand,someworkshavedealtwiththeproblemofcomfortandimprovementsofagroundvehiclehandlingthroughthejointbrakingandsuspensioncontrol,seeforexample[23].In[24]thisapproachisextendedbytheintroductionofthesteeringcontrolinordertoimprovethevehiclestability,thelongitudinalmotionisnotconsideredasastatecomponentbutasavariableparameter.1

In[19],[14],theauthorsproposeaninventoryconcerninglongitudinalandlateralcontrol,andpointoutthatintheliteraturethelongitudinalcontrolandthesteeringcontrolaretreatedseparatelyinmostofthecases.However,in[3],[5],anonlinearglobalchassiscontrollawisproposedtofollowdesiredtrajectoriesinyawrateandinlongitudinalaccelerationviabrakingtorquesonlywithoutactivesteeringcontrol.

Thepresentpaperdealswiththeproblemofanonlinearcoupledlongitudinalandlateralcontrolbasedonalgebraicnonlinearestimationtechniques[7],[18]anddifferential?atness[8],[9](seealso[10]forarelatedapproach).Todealwiththismethod,athreedegrees-of-freedomtwowheelsvehiclemodelisconsidered.Theestimationapproachallowstoobtainaverygoodestimationofthederivativesofnoisysignals.Indeedthesensorsusedinarealautomotiveapplica-tionaregenerallylowcostandtheirmeasurementsareverynoisy.Henceitisnoteasytousethesemeasurementsdirectlytoobtainthereferencesignalsforthecontrolapplications.Forthisreason,algebraicestimationisuseful.

Theremainderofthispaperisorganizedasfollows.SectionIIpresentsthe3DoFtwowheelsvehiclemodelusedforthecontrollawdesign.SectionIIIdescribesthedesignstepsofa?atness-basedrobustcombinedlongitudinalandlateralnonlinearcontrolandalgebraicnonlinearestimation.SectionIVpresentsdetailedexperimentalresultsusingthenoisymeasurementsacquiredonrealroadswithhighdy-namicloads.SomeconcludingremarksandfutureworkarepresentedinSectionV.

Michel.Fliess@polytechnique.eduhugues.mounier@lss.supelec.fr

978-1-4577-2197-7/11/$26.00 ©2011 IEEE

HuguesMounieriswithL2S(UMR8506),CNRS–Sup´elec–Universit´eParis-Sud,3rueJoliot-Curie,91192Gif-sur-Yvette,France.

1See

also[30].

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II.VEHICLEMODELSANDCOUPLEDDYNAMICANALYSISTwononlinearvehiclemodelsareused:the?rstoneisa3DoFtwowheelsnonlinearmodel(3DoF-NLTWVM)whichwillbeusedtodesignacombinednonlinearcontrollaw.Thesecondisa10DoFfourwheelsnonlinearmodel(10DoF-NLFWVM)whichwillbeusedasacompletevehiclesimulatortotesttheproposedcontrollawunderhighloadswithcurvedtrajectories.Inthefollowingsubsectionswepresentthereasonsforwhichthecouplednonlinearvehiclemodelsareused.A.Coupleddynamics

Theaimofdesigningacombinedlateralandlongitudinalcontrolleristoenhanceandensureacertainlevelofcontrolperformance,andrealizesomecombinedmaneuvers.There-fore,toensureacombinedrobustcontrol,coupledvehicledynamicsmustbetakenintoaccount.Clearly,tosatisfytheserequirements,theidenti?cationofsomecouplingeffectsisanimportantsteptodesignacombinedcontrollaw:

?The?rstcouplingthatcanbecitedistheonerelatedtothekineticsvehicle.Thelongitudinalmotionisaffectedbythesteeringangle;infactwhenthevehiclewheelsaresteered,thelongitudinalforcesarefunctionsofthewheelforcesandsteeringangle.Thelateraldeviationisalsoafunctionofthelongitudinalspeedandyawangle.Thesamereasoningisdoneforthelateralforcesandthelongitudinaldisplacement.

?Theloadtransferisanimportantcouplingofthevehicledynamics.Thiscouplingistheconsequenceofthebraking/traction(changetheweightdistributionbetweenthefrontandreartires),lateralacceleration(changetheweightdistributionbetweentherightandlefttires),roadbankangle,roadslopeangle,rollandsuspensionmotions.Allthesecouplingareconsideredinthecomputationofthetireverticalforces.

?Anothercouplingeffectcomesfromthebehaviorofthetires;thiscouplingisexpressedbythelongitudinalandlateralforces.Infact,toobtainarealisticrepresentationofthevehiclebehaviorsinextremedrivingsituations,thecouplingoflongitudinalandlateralforcesisre-quired.Toachievethisgoal,thecoupledformulaofPacejka[21]modelisused.Inthismodelthecouplingoflongitudinalslip,lateralslip,verticalforcesandcamberangleistakenintoaccount.B.3DoFNonlinearTwoWheelsVehicleControlModelThe3DoF-NLTWVMshowninFigure1usedtodesignthecombinedcontrollawprovidesasuf?cientapproximationofthelongitudinalandlateraldynamicsofthevehicleinnormaldrivingsituations.Thismodeliscomposedfromthe

˙motions(seetableIlongitudinalVx,lateralVyandyawψ

fornotations).

Thecoupledequationsofthismodelare:

?

˙x?ψ˙Vy)=(Fx1+Fx2)?max=m(V

˙y+ψ˙Vx)=(Fy1+Fy2)may=m(V(1)

?¨

Izψ=Mz1+Mz2

464

xVyaxay˙ψψβωiTωTmTbTbfTbrδ

Cf,CrFxiFyiFxfFyfRRr

Lf,LrIzIrαigmMz

TABLEISOMENOTATIONS

lateralspeed[km.h]

longitudinalacceleration[m/s2]lateralacceleration[m/s2]yawrate[rad/s]yawangle[rad]sideslipangle[rad]

wheelangularspeedofthewheeli[rad/s]wheeltorque[Nm]

wheeltractiontorque[Nm]wheelbrakingtorque[Nm]

frontwheelbrakingtorque[Nm]rearwheelbrakingtorques[Nm]wheelsteerangle[deg]

frontandrearcorneringstiffnesses[N.rad?1]longitudinalforceinthevehiclecoordinate[N]lateralforceinthevehiclecoordinate[N]

frontlongitudinalforceinthewheelcoordinate[N]frontlateralforceinthewheelcoordinate[N]tireradius[m]roadradius[m]

distancesfromtheCoGtothefrontandrear

axles[m]yawmomentofinertia[Kg.m?2]wheelmomentofinertia[kgm2]tireslipangle[rad]

accelerationduetogravity[m/s2]vehiclemass[kgm2]yawmoment[Nm]

TheforcesandmomentsofEq.(1)inthevehiclecoordi-natestakingintoaccountthekineticscouplingare:

?

Fx1=Fxfcos(δ)?Fyfsin(δ)????Fx2=Fxr???

Fy1=Fxfsin(δ)+Fyfcos(δ)

(2)

F=F?yry2???Mz1=Lf(Fyfcos(δ)+Fxfsin(δ))???

Mz2=?LrFyrConsideringthesmallanglesassumption,theforcesandmoments,inEq.(2)becomeasfollows:

?

Fx1=Fxf?Fyfδ????Fx2=Fxr???

Fy1=Fxfδ+Fyf

(3)

F=F?yry2???Mz1=Lf(Fyf+Fxfδ)???

Mz2=?LrFyr

Fig.1.Nonlineartwowheeledvehiclemodel

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InEq.(1)thelongitudinalforcesarecalculatedusingthedynamicalmodelofthewheels.Forthefrontwheel:

˙f+Tm?Tbf)Fxf=(1/R)(?Irω

(4)

?

?

g(x,t)=??0

?g2=?

1

Cf??

˙Vy+Lfψx

??

?

?

˙f)/mR(CfR?Irω˙f)/IzR(LfCfR?LfIrω

?Cf

?

?,g1=???

1

Lfz???,

Consideringthatthevehicleisonlypropelledbythefrontwheel,thedynamicalmodeloftherearwheelisasfollows:

˙r)Fxr=?(1/R)(Tbr+Irω

(5)

?T

?T

????TVx

u1?,x=?Vy?,u=

u2

˙ψ

?

Thewheelsrotationsωfandωraresupposedtobe

measuredthroughodometers.Theyareintroducedasexternalinputsintheequationsofthe3DoF-NLTWVM.WeassumethattherearbrakingtorqueTbrisequaltothefrontoneTbf.Thewheelsrotationaccelerationswillbeestimatedusingthealgebraicnonlinearestimationandthemeasuredrotationspeedswhichareavailableonthevehicle’sCANbus.Notethatthelateraltireforces,wehave,modeledasproportionaltotheslipanglesofeachaxle.Thefrontandrearforces(Fyf=Cfαf,Fyr=Crαr)arede?nedasfollows:

?????˙LfVy+ψ

?Fyf=Cfδ?x

????(6)˙Lr?F=?CVy?ψ

yrrxwhereCfandCrarethecorneringstiffnessforthefrontand

reartiresrespectively.Thelongitudinalspeedisconsidereddifferentfromzero(Vx>ε).ReplacingEq.(3),(4),(5)and(6)inEq.(1)yields:

?1r˙˙x=mψ˙Vy?I˙f)+mV(ωr+ω(Tm?Tbf?Tbr)????????˙V+Lψ??δ?Cfδ2+Cfyxf?????????????˙Vy+Lfψ˙LrVy?ψ?˙˙Vx?CfmVy=?mψ?Crx

??xI??

r˙?+(1/R)(Tm?Tbf)δ+Cf?ωfδ?????????????˙Vy+Lfψ˙?Vy?Lrψ?¨=?LfCfIzψ+LrCr?xx???I˙f)δ+(Lf/R)(Tm?Tbf)δ+Lf(Tm?ω

III.FLATNESS-BASEDCONTROL

Inordertoreducethecomplexityofthenonlinearmodelin

2Eq.(8),nonlineartermssuchasu1u2andu22areneglected.

Despitethesesimpli?cationssomecoupledbehaviorsarekeptasshownbythefunctionsf(x,t)andg(x,t).Eq.(8)becomes:

x˙=f(x,t)+g(x,t)u

A.FlatnesspropertyConsiderthesystem

x˙=f(x,u)

wherex=(x,···,xn)∈Rnandu=(u,···,um)∈Rm.Itis

saidtobedifferentially?at(see[8],[9]and[15],[26])if,andonlyif,

?thereexistsavector-valuedfunctionhsuchthat

y=h(x,u,u˙,···,u(r))

?

(9)

(10)

(7)

wherey=(y,···,ym)∈Rm,r∈N;

thecomponentsofx=(x,···,xn)andu=(u,···,um)maybeexpressedas

x=A(y,y˙,···,y(rx)),rx∈Nu=B(y,y˙,···,y(ru)),ru∈N

Thelongitudinalmovementiscontrolledviathetrac-tion/brakingwheeltorqueTω=Tm?TbwithTb=Tbf+Tbr,

andthelateralmovementviathesteeringangleδ.De?nethetwocontrolvariables:

??

u1=Tωu2=δThemodelgivenbyEq.(7)maynowbewrittenasfollows:

x˙=f(x,t)+g(x)u+g1u1u2+g2u22

where

?

??

f(x,t)=?

?

˙Vy?ψ????????????˙Vy+Lfψ˙Vy?Lrψ1˙Vx+?Cf?ψ?Crxx

????????????˙Vy+Lfψ˙Vy?Lrψ1

+LrCrz?LfCfxx

I˙˙(ωr+ωf)

RememberthatyinEq.(10)iscalleda?atoutput.B.Flatness-basedlongitudinalandlateralcontrol1)Aproofof?atness:Wewanttoshowthaty1andy2

givenby??

y1=Vx

˙y2=LfmVy?Izψde?nea?atoutput.Somealgebraicmanipulationsyield?

x=

yf(8)

?????

2The

??

Vx

IzfVy????

˙ψ

Ty1

=A(y1,y2,y˙2)=

????

????

(11)

????,?

465

?

??

?

??

Lfmy1y˙2+Cr(Lf+Lr)y2

Cr(Lf+Lr)(Iz?LrLfm)+(Lfmy1)Lfmy1y˙2+Cr(Lf+Lr)y2

rfrzrff1??

resultsofSectionIVfullyjustifythoseapproximations.

论文

and

????

y˙1y¨2u1u2

????

=?(y1,y2,y˙2)

??

?where

=??1(y1,y2,y˙2)

u1

u2????

??y˙1y¨2

C.Algebraicnonlinearestimation

+Φ(y1,y2,y˙2)??

??

?Φ(y1,y2,y˙2)

(12)

Itshouldbepointedoutthatthecontrollawcontains

derivativesofreferencesignalswhichareestimatedfrom

˙refandthederivativesmeasurementssuchasVxref,Vyref,ψ

ofthemeasuredfrontandrearrotationspeedwheelsωfandωr.Inordertoperformthistaskandavoidtheeffectsofthenoisymeasurementsweproposetousethenumericaldifferentiationbasedonanalgebraicnonlinearestimation.3Thisestimationisperformedusingtherecentadvancesin[7],[18],whichyieldef?cientreal-time?lters.Thefollowingformulae(see,e.g.,[11])maybeused:?Denoising:

2!

y?(t)=T

?

1

?11(y1,y2,y˙2)=

mR????

˙CfVy+Lfψ

?12(y1,y2,y˙2)=,

my1

????˙f)LrCr(Lf+Lr)(LfCfR?LfIωω

?22(y1,y2,y˙2)=+?Lfmy1+1z+

˙)?Lfmψ˙y2˙)(Cr(Lf+Lr)(Vy?Lrψ1)Cf(Vy+Lfψ

y11

??t

t?T

(3(t?τ)?T)y(τ)dτ(15)

?

˙fCr(Lf+Lr)RCf?Iωω

1Numericaldifferentiationofanoisysignal:

3!˙y?(t)=?T

??t

t?T

˙)?Lfmψ˙y2(Cr(Lf+Lr)(Vy?Lrψ1)?21(y1,y2,y˙2)=

mRy1

˙Vy?Φ1(y1,y2,y˙2)=ψ

Iω

˙r+ω˙f)(ωmR

Cr(Lf+Lr)

f2(x,t)1

(2T(t?τ)?T)y(τ)dτ(16)

Φ2(y1,y2,y˙2)=?Lfmy1f3(x,t)?+

Notethattheslidingtimewindow[t?T,t]maybequite

short.

Remark3.1:Theaboveestimationmethodsarenotofasymptotictypeanddonotrequireanystatisticalknowledgeofthecorruptingnoises(see[6]fordetails).

http://wendang.chazidian.comPARISONWITHEXPERIMENTALTESTSSeveraltrialsinpresenceofcoupledtraction/brakingandsteeringmaneuvershavebeenrealizedunderhighdynamicloadsusingthelaboratoryvehicle.Duringthesetestsalargesetofdynamicparametershasbeenacquiredsuchaslongitudinalandlateralspeeds,lateralandlongitudinalaccelerations,yawrate,wheelrotationspeeds,allforcesandmomentsonfourwheels,steeringangle,etc.Fortheseteststheacquisitiondeviceoperatesatfrequency400Hz.

Thesedatahavebeenusedinoursimulationprocessasreferencetrajectories,andcomparedtotheobtainedsim-ulationresultsinclosedloopwiththeexperimentalones.Moreprecisely,thecombinedcontrolapproachpresentedhereistestedusingseveraltrialsdataperformedonrealracetrackasshownbycurvatureofFigure4.ThissectionpresentsatestconductedonaracetrackwhichisperformedunderhighdynamicloadsasshowninFigures2,3,4and5whichsummarizesomedynamicparametersoflongitudinalandlateralmotions.Thistestpresentsacoupledsteering,accelerating,asuddenandsharpbrakingmaneuver.

Figures2and3showthedynamicparametersofthetwocontrolledvehiclemodels(3DoF-NLTWVMand10DoF-NLFWVM)whichareclosetothosemeasuredsuchasyawrate,longitudinalspeed,longitudinalandlateralaccelera-tions.Noticealsothatthesteeringangleandbraking/tractionwheeltorqueprovidedbythecombinedcontrollawaresimilartothemeasuredones,this,foranygivenvehiclemodel(3DoFor10DoF)andtheperformedmaneuversasillustratedinFigures2and3betweenthepositionsfrom600mto750mandfrom1000mto1080m.

[1],[28],[29]forprevioussuccessfulapplicationstointelligent

transportationsystems.

3See

˙)?Lfmψ˙y2Cr(Lf+Lr)(Vy?LrψLrCr(Lf+Lr)1

f(x,t)+f3(x,t)11y1

The?atnesspropertyholdsthereforeifthematrix

?(y1,y2,y˙2)isinvertible.Itreads

det(?(y1,y2,y˙2))=?11?22??21?12=

˙f?CfR)(Iωω

????

2m2?Cr(L+L)LLm+CILL2yrrfrzff1

z1=0

Thisdeterminant,whichdependsonlyonthelongitudinal

speedy1=Vx,isindeednonzero:

?ThewheelrotationaccelerationislessthanRCf/Iω:

˙f?CfR=0.RCf/Iωisaround104,thenIωωNoticethatIz>Lfm,then:Cr(Lr+Lf)(Iz?Lfm)+

22L2fmy1=0.

Thus

????Tω

˙1,y2,y˙2,y¨2)=u==B(y1,y

δ????????(13)

y˙1?1?(y1,y2,y˙2)?Φ(y1,y2,y˙2)y¨2

?

withrx=1andru=2.

2)Atrackingfeedbackcontrol:Inordertotrackthe

fref

desiredoutputtrajectoriesyre1andy2,set

??y˙1

y¨2

??

??=

f

y¨re2

f12y˙re1+K1ey1+K1ey1dt1e+K2˙y2

2e+K2y2

3+K2

??

??

(14)

??

ey2dt

frefref

where,ey1=yre1?y1=Vx?Vxandey2=y2?y2.The

1,K2,K1,K2andK3ischoiceofthegainparametersK11222

straightforward.

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DuringthesetwophasesasmentionedonFigures2and3,thecombinedsteeringandbrakingmaneuversareperformedsimultaneouslyinthemostdangerousbendsasillustratedinFigure4.TheperformancesofthecontrollawintermsoftrackingtrajectorieserrorsaredepictedinFigure5.Letusemphasizethattheseerrorsarequitesmallandsatisfactorywhateverthevehiclemodelused,thenatureofthetrajectoryandthedynamicloads(lateralacceleration?5m/s2??ay??5m/s2).

Duringthistest,thecontrollawisef?cientinsimulationwhichisemphasizedbythebehaviorofthecontrolledmodelsbeingclosetothemeasuredone.Thisremarkisveri?edespeciallywhenthe10DoFvehiclemodeliscoupledwiththenonlinear?atcontrollaw,indeed,asshownbyFigure4,thecurvatureof3DoF-NLTWVM+nonlinear?atcontrolisslightlydifferentfromthereferenceroadcurvature.

5

Longitudinal acceleration

[m/s2]

[deg/s][deg]

[m/s2]

Distance [m]

Fig.3.Resultsoflateralbehaviors:lateralacceleration,yawrateandsteeringanglecontrol

?50[km/h]200400600

8001000120014001600

[Nm]

Distance [m]

Distance [m]

Fig.2.Resultsoflongitudinalbehaviors:longitudinalacceleration,longitudinalspeedandtraction/brakingtorquecontrol

Fig.4.Curvatures:referenceandcontrolledvehiclemodelcurvatures

Fromtheresultsobtainedwiththetwotests,thecontrollawprovidedmorerealisticbehaviorsintermsofthetrackingtrajectories,whencouplingthebraking/tractiontorqueandsteeringangleonarealisticsimulator10DoF-NLFWVM.Thisshowstheeffectivenessandtheabilityofthecombinedcontrollawtooperateunderextremeandcouplednonlin-earvehiclebehaviors(forlateralacceleration?5m/s2??ay??5m/s2andlongitudinalacceleration?5m/s2??ax??3.5m/s2)incurvedtrajectories(20m<R<35m).Itshouldbepointedoutthattheemergencysituations,suchascol-lisionsandobstacleavoidance,lane-changemaneuvers,caneasilycausecriticaldynamicaldrivingsituations.Forthisreasontheperformanceofthecontrollawmustbetestedunderhighdynamicsloads.

467

Remark4.1:Letusemphasizethatour?rstresultsshowthattheperformanceofthecontrollawisnotaffectedbythemodelsimpli?cations(nonlineartermsinthecontrolinputs).