AStudyofLateralVehicleControlUnderaVirtualForce Framework Eric J
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AStudyofLateralVehicleControlUnderaVirtualForce Framework Eric J

Rossetter Design Division Department of Mechanical Engineering Stanford University Stanford California 943054021 Email ejrosscdrstanfordedu J Christian Gerdes Design Division Department of Mechanical Engineering Stanford University Stanford Californ

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AStudyofLateralVehicleControlUnderaVirtualForce Framework Eric J




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AStudyofLateralVehicleControlUndera‘Virtual’Force Framework Eric J. Rossetter Design Division Department of Mechanical Engineering Stanford University Stanford, California 94305-4021 Email: ejross@cdr.stanford.edu J. Christian Gerdes Design Division Department of Mechanical Engineering Stanford University Stanford, California 94305-4021 Email: gerdes@cdr.stanford.edu Abstract Futureautomotivesafetyfunctions,suchaslanekeepingandcollisionavoidance,linkthe vehicledynamicallytoitsenvironment. Asaresult,acombinationofthemechanicalsystem

dynamicsandthevirtuallinktotheroadwayorobstaclesdeterminesthevehiclemotion.This paperlooksatthecombinedinfluenceofdecouplinglateralandyawmodes,previewdistance, andcontrollerdampingonthestabilityandperformanceoflateralcontrollers. Theeffectsof thesecharacteristicsarestudiedusinganintuitive‘virtual’forcesanalogywherethecontrol inputsareviewedasasingleforceactingonthevehicle. Withtheabilitytoinfluencethe couplingbetweenlateralandyawmodes,the‘virtual’controlforcecanbeshiftedalongthe lengthofthevehicle.Forstability,thisforcemustbeappliedinfrontoftheneutralsteerpoint.

Inaddition,shiftingthiscontrolforcecangivesatisfactorysystembehaviorwithaminimal previewdistance.Thebenefitsoflookaheadarealsoexploredunderthisframework,givingwell behavedresponseswithouttheadditionofcontrollerdamping. Thecombinationoflookahead anddampingonthevelocitystatesallowscompletefreedomindesigningthesystemresponse. 1 Introduction Asautomobilesincorporatenewdrive-by-wiretechnologiesitwillbepossibletoimplementlateral controllersonpassengervehiclesforcollisionavoidanceandlanekeeping[1][(].)esearchinlateral controlhasfocusedonfourmainfactorsthatinfluencesystemperformance:

vehiclehandling characteristics,previewdistance,actuatorcapability,andcontrollerdesign. +nderstandingthe interactionofallthesefactorsiscrucialinthedesignoflateralcontrolsystems. ,ehiclehandlingcharacteristicsisakeycomponentincontrolsystemdesign.-pen-loopvehicle stabilityiswellunderstoodandresearchinthisareadatesbacktotheearly1./0’s.Atthistime, researchersat1eneral2otorsfirstnoticedthatsomesteeringgeometriescreatedatendencyfor thevehicleto‘oversteer’thedesiredtrajectory.4hortlyafterthisdiscovery,thesame‘oversteering

behaviorwasnoticedwithvehiclesthathadunderinflatedreartiresorarearwardweightbias.This nurturedthestudyoftiredynamicsandtheirinfluenceonvehiclehandling.4tonex[/]wasoneof thefirsttostudysteady-statevehiclestabilityandexploreundersteeringandoversteeringbehavior. Thisledtothedevelopmentandanalysisofdynamicvehiclemodels,resultinginamathematical understandingofundersteeringandoversteeringbehavior[5].Theresultsshowedthatundersteering vehiclesarealwaysstablebutdampingdecreaseswithincreasingspeed. -versteeringvehicles,

however,arestablewithincreasingdampinguptoacriticalspeedwherethevehiclebecomes
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unstable[6][7]. Thesimplemodelsthatyieldtheseresultsarestillusedformostvehiclecontrol systemsdesign. Withsensorandcomputeradvances,thecapabilityforvehiclecontrolbecamepossible.Fenton [8]didinitialworkinvehiclelateralcontrolbyusinganelectricwireasareference. Inthis work,thevehicleonlyhadknowledgeofitscurrentlateralposition,whichcreatedinstabilityat highspeeds. Theselook-downreferencesystemswerealsostudiedbyresearchersat9AT:[8] andlaterby1uldneretal. [.]butstillhadinstabilitiesabove/0m/s.

Inordertoachieve speedsappropriateforhighwaytravel,itisnecessarytoincorporatealook-aheadreference. The advantageofincorporatingapreviewdistancewasshowntheoreticallyby9engandTomizuka[10] andexperimentallybyAlleyneandDe9oorter[11].1uldneretal.[1(]createda‘virtual’preview distanceforhighspeedstabilityusingalook-downreferenceatthefrontandrearofthevehicle. Withadditionalactuatorssuchasfourwheelsteeringordifferentialbraking,itispossibleto changethecouplingbetweenlateralandyawmodes.Ackermann[1/]developedcontrolalgorithms

thatdecouplethelateralandyawdynamicstomakethedrivingexperiencesaferandmorecom- fortable. Alleyne[15]lookedatlateralcontrolofavehicleusingvariouscombinationsofsteering anddifferentialbrakingforimprovedlanekeeping. Theabilitytoalterthecouplingbetweenthe yawandlateralmodesaddsanotherdegreeoffreedominthecontrolsystemdesign. Inorder tostudytheinteractionofallthesedifferentvariablesitisconvenienttolookatcontrolfrom a‘virtual’forcestandpoint. Inessence,mostcontrollawscanbeviewedasthecombinationof externalforcesandtorquesappliedatthecenterofgravity. Thiscanbecombinedintoasingle

virtualcontrolforceactingatsomepointalongthelongitudinalaxisofthevehicle. Thisvirtual forceconceptgivesphysicalinsightintocontrollerdesignthatisextremelyusefulformanytypes ofcontrolapplications. Thispaperpresentsastabilityanalysisillustratingtheinfluenceofthefollowingattributeson thestabilityandperformanceoflateralvehiclecontrol. ,ehicle:andling9roperties(Amountof+ndersteer/-versteerA ,irtualForceApplication9oint(Boordinationof4teeringandCrakingA 4ensingDocation(AmountofDookaheadA BontrollerDamping Theresultsclearlyillustratethatthevirtualcontrolforcemustbeappliedinfrontoftheneutral

steerpointforstability. Adequatesystemresponsecanthenbeachievedbyincorporatingan appropriateamountoflookahead. Thecombinationoflookaheadandcontrollerdampingallows completecontroloverthesystemresponse. Theseresultsprovideinsightintotheproperwayof applyingvirtualforcestocontrolhighspeedvehicles. 2 Vehicle Dynamics Thevehiclemodelusedintheanalysisisasimplethreedegreeoffreedomyawplanerepresentation withdifferentialbraking,showninFigure1. xr xf cos yf sin mrU (1A yr xf sin yf cos mrU ((A aF xf sin aF yf cos bF yr (/A ( xr G xf cos
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xlf ylf yrf xrf xrr yrr ylr xlr

Figure1:,ehicle2odel where xf xrf xlf (5A xr xrr xlr (6A xf xrf xlf (7A xr xrr xlr (8A Assumingsmallanglesandequalslipanglesontheleftandrightwheels, ra (8A rb (.A +singalineartiremodel,thelateralforcesaregivenas yf (10A yr (11A where and arethefrontandrearcorneringstiffnesses,respectively. 4ubstitutingtheex- pressionsforthelateralforcesintoIquations1through/andmakingsmallangleapproximations yields, mrU xr xf ra (1(A rb ra mrU (1/A xf aF xf aC ra AG bC rb A (15A aC ( xr G xf
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Assumingavehiclethathasthrottle,brake,andsteer-by-wirecapabilitiessothatsteering,braking

andtwodrivewheelscanbecontrolled,theequationscanberewrittenas (E AG (E q,u A (16A whereE F[ andthecontrolvector F[ δF xrf xlf xrr xlr .2isthepositivedefinite massmatrix, (E Acontainsthetermsthatarenotinfluencedbythecontrolvectorand (E q,u Ahas theremainingcontrolledterms. 00 00 (E AF mrU rb ra mrU aC ra AG bC rb (E q,u AF xr xf ra xf aF xf aC ( xr G xf Thecontrolledterms, (E q,u A,arethensetequaltothedesiredcontrolforces.Thissetofequations canthenbeusedtosolveforthecontrolvector, [17]. 3 Virtual Force Analogy for Control

Inthispaper,thevirtualcontrolforceisrealizedthroughacombinationofdifferentialbrakingand frontsteering.Figure(showshowforcesderivedfromthecontrolinputs(differentialbrakingand frontsteeringAcanbethoughtofasavirtualcontrolforce. Thisisaccomplishedbycreatingan equivalentforcesystemconsistingofalongitudinalandlateralforceonthevehicle. Forlateral control,thelongitudinalforceissmallanddisappearsinthelinearizationofthesystemdynamics (4ection5A.Theabilitytomanipulatethecouplingbetweenlateralandyawmodesallowsmovement ofthevirtualcontrolforcealongthelongitudinalaxisofthevehicle.

Forexample,ifdifferential brakingisunavailable( xf F xf F0A,thevirtualcontrolforcefromthesteeringangleis constrainedatthefrontaxle. Withdifferentialbraking,themomentequationcanbeinfluenced independentlyfromsteering,effectivelyshiftingthelocationofthevirtualforcefromthefrontaxle. Forgeneralvehiclecontrol,thevirtualforceisdescribedinglobalcoordinates F[ se where isthedistancealongtheroadway, isthedistanceofthevehicle’scenterofgravityfrom thelanecenterand istheheadingangle(Figure/A.Transformationbetweentheglobalandbody fixedcoordinatesisachievedwith ∂w ∂q

cos sin sin cos 001 (17A Inthispaper,thevirtualforceisbasedonasimpleproportionalderivativecontrollaw. For generalvehiclecontrol virtual K (18A where istheproportionalgainmatrixand isthedampingmatrix. Thesecontrolforcesare orientedinglobalcoordinatesandmustbetransformedtovehiclefixedcoordinatestosolveforthe
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xlf ylf yrf xrf xrr xlr Figure(:,irtualForceAnalogy actualcontrolinputs(steeringangleanddifferentialbrakingA. (E q,u AF K ∂q (18A Withthiscontroller,thevehicledynamicscanberewritten,replacingthecontrolinputvectorwith thevirtualcontrolforce: (E AG K ∂q

(1.A Thesystemdynamicsnowconsistofvirtualforcesandthedrifttermcontainingthevehicledynam- icsnotdirectlyinfluencedbycontrolinputs.Thiscontrollawisanalogoustohavingvirtualforces fromlateral,longitudinal,andtorsionalspringsanddampers.Thisspringanddamperanalogyis similartoworkdoneby:ennesseyetal.[18]and4chilleretal.[18][1.]inthedesignofa‘virtual bumper’. Ifanobstaclepenetratesthebumper,imaginaryspringsanddampersarecompressed applyingavirtualforcetothevehicle.Inthispaper,however,onlythelateraldirectionisusedfor lanekeeping(Figure5A.

Thissetofnon-lineardifferentialequationsisthefocusoftheanalysis.Thebehaviorinquestion isthestabilityofthevehiclewithrespecttoadesiredtrajectoryconsistingofaconstantlongitudinal velocityinthecenteroftheroad.4incetheinterestingbehaviorisinglobalcoordinates,itmakes sensetotransformtheequationsofmotionintoglobalstatesandthenlinearizethesystemabout thedesiredtrajectory. 4 Virtual Force at C.G. Thegoalofthestabilityanalysisistostudythevehicle’sresponsetosmallincursionsfromthe lanecenter.Thebasicideaistotransformtheequationsofmotionintotheglobalreferenceframe

andthenrewritethethreesecondorderdifferentialequationsassixfirstorderequationsthat arefunctionsoftheglobalcoordinatesandtheirderivatives. Kacobianlinearizationcanthenbe performedaboutastraighttrajectorywherethelongitudinalvelocityisaconstant andallother statesarezero.
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Road Centerline Figure/:1lobalBoordinates Withoutavirtualforce,thisanalysisreturnswell-knownstabilityresultsforvehiclehandling, asshouldbeexpected. Theinteractionbetweenvehiclehandling,virtualforceapplicationpoint, andsensorlocationisshownwithoutaddinganyartificaldamping( F0A. Thiscontrolforceis

onlydependentonthelateralpositionofthevehiclefromthelanecenter. virtual ke ((0A Thebasicanalysisraisesstabilityconcernsaboutthecontrolledsystemthatcanberesolvedby shiftingtheapplicationofthevirtualforceandsensinglocation. Inthefinalsection,artificial dampingisadded,creatingavirtualforcebasedonaproportionalderivative(9DAcontrollaw. 4.1 Linearization Without a Virtual Force Thelinearizationofavehicleaboutaconstantlongitudinalvelocitygives, Aδx ((1A where δx F[ δe e areperturbationsawayfromthenominalstatesand 01 0 0 mS aC bC mS 00 0 1 aC bC aC bC (((A

Takingthedeterminantof( λI Ayieldsthecharacteristicequationofthesystem. λa F0 ((/A where, G( mS G( bC aC mS mS
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Figure5:DanekeepingAnalogy Fromtheaboveequation,therearefoureigenvalues.Twooftheseeigenvaluesdeterminethevehicle handlingandthetwozeroeigenvaluesarisefromtheintegratorsforthepositionalstates. Inan understeeringcar(where b>aC Abothcoefficients and arepositivewhich,forasecond ordersystem,issufficienttoprovestability.Inanoversteeringcase( aC >bC A,thecoefficient willbenegativewhenthespeed S> aC bC ((5A

Thisisthewellknowncriticalspeedforanoversteeringvehicle(1illespie(1..(AA.Asexpected,the transformationtoglobalcoordinatesdidnotchangethebasicstabilitypropertiesofthevehicle:the understeeringvehicleisalwaysstableandtheoversteeringvehicleisstablebelowacriticalspeed. 4.2 Linearization With a Virtual Force AddingavirtualcontrolforceintheformofIquation1.yieldsalinearsystemwhichhasthe controlforceappearinginthelateralequationofmotion.Thisaddsoneextratermtothematrix inIquation((. 01 0 0 mS aC bC mS 00 0 1 aC bC aC bC ((6A Thecharacteristicequationisnow λb F0 ((7A where G( mS G( bC aC mS kI mS


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+ndersteer -versteer m(kgA 1750 1750 N/m /600 /600 (M/radA 100000 100000 (M/radA 170000 80000 6000 6000 a(mA 1./ 1./ b(mA 1.6 1.6 Table1:,ehicle9arameters mS bC aC )egardlessofthevehicle’sspeed,thelastterm, ,isalwaysnegativewhenoversteeringparameters areused. 4ignchangesinthecharacteristicequationcoefficientsfailthenecessaryconditionsfor stability. Theadditionofthevirtualforcealterstheoversteeringvehicledynamicstoproduce instabilitybyloweringthecriticalspeedtozero. Thisinstabilityoccursbecausethevirtualforce

isappliedatthecenterofgravity,causingthevehicletoturnintotheappliedforce.Thistypeof responseiswellknownforanoversteeringcarwithaphysicalforce,suchasasidewind,applied atthec.g. Ifthevehicleisundersteering,allthecoefficientsarepositive. Althoughthisisanecessary conditionforstability,itisnotsufficient.+singthe)outharray,itcanbeshownthatthesystem isstableuptoacriticalvelocity. cr ((8A whereMandDare A( AG( F( (( A( aC bC G( G( aC bC A( +singtheundersteeringparametersinTable1thecriticalspeedis(8 07 m/s .Figure6showsthe

understeeringandoversteeringresponsewiththeproportionalcontroller.Theinitialconditionsfor thesimulationare F0 and F(6 m/s ,representingatypicaldrivingspeedandaninitial offsetfromthelanecenter.AllotherstatesarezeroandtheparametersusedareinTable1. TheresultsshowninFigure6showtheexactbehaviorpredictedbythelinearanalysis. The understeeringvehicleisstablebutunderdamped,oscillatingaboutthelanecenter.Anoversteering vehicleexhibitsdrasticallydifferentresults.Thecontrolforceinitiallypushesthecartowardsthe centerofthelane,butthechangeinthevehicledynamicscausesarotationintotheappliedforce.

Intheaboveanalysis,thevehicleiscontrolledbyavirtualforceappliedatthec.g. ofthe vehiclewithsensingbasedonthelateralpositionofthec.g. Thelocationofthiscontrolforce resultsininstabilityfortheoversteeringvehiclethatcannotberectifiedbyaddinglookaheador controllerdamping.Theonlysolutionistoshiftthecontrolforceforwarduntilstabilityisachieved. Fortheundersteeringvehicle,thiscontrolstrategyresultsinstabilitybelowacriticalspeed.The lowdampingcharacteristicsandacriticalspeedwithintheoperatingrangeofthevehiclecanbe
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10 −0.5 0.5 Time (s) Lateral Position (m) Vehicle Lateral

Position Oversteering Understeering Figure6:4imulation:,irtualForceatB.1. attributedtosensingatthec.g. (i.e. nolookaheadA. Aninterestingresultisthatwithoutany lookaheadthecriticalspeedisfairlylargecomparedwithpreviousresultsforlook-downlateral controllers. Thisisaresultofapplyingthecontrolforceatalocationthatalmostdecouplesthe lateralandyawmodesofthevehicle.Thisabilitypresentstheoptionofproducingadequatevehicle responsewithoutlookahead. 5 Shifting the Virtual Force Withthevirtualcontrolforceactingatadistance, cf ,infrontofthec.g.,theequationsofmotion

haveanextratermappearinginthemomentequation. 01 0 0 mS aC bC mS 00 0 1 kx cf aC bC aC bC ((8A Thecharacteristicequationis λc F0 ((.A where G( mS G( bC aC mS kI mS
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cf bC aC AA mS bC aC cf AA Fromthelastsection,thefinalterminthecharacteristicequationissensitivetochangesinthe handlingpropertiesofthevehicle.Withtheabilitytoshiftthevirtualforceitispossibletonegate thisproblem.Forstability,thefollowinginequalitymustbesatisfied. bC aC G( cf 0 (/0A so, cf aC bC (/1A Therighthandsideoftheaboveexpressionisthedefinitionoftheneutralsteerpoint,whichhas

physicalsignificanceinvehicledynamics.Theneutralsteerpointisthelocationonthecenterline ofavehiclewhereanexternalforcewillproducenosteadystateyawvelocity.Thisconceptisoften usedtodiscusssidewindsensitivityofavehicleandhasanaturalinterpretationwhenconsidering virtualforcesandstability. Forstability,thevirtualforcemustbeappliedinfrontoftheneutral steerpointofthevehicle.Thisensuresthatthevehiclewillyawinthesamedirectionasthevirtual controlforce. Ifthevirtualforceisappliedattheneutralsteerpoint,thesystemismarginallystablewith threenegativeeigenvaluesandoneattheorigin.

Thecoefficientsforthecharacteristicequation become G( mS G( bC aC mS kI mS kC mS F0 Withthevirtualforceattheneutralsteerpointthereisstillthepossibilityofacriticalspeed.The onlyterminthissystemwhichcanpossiblygonegativeforanoversteeringvehicleis .From lookingatthe)outharray,thesystemactuallyhasacriticalspeedthatoccursbefore becomes negative. cr (/(A where A( AG( F(( aC bC A(( bC aC AA A( aC bC A( +singtheoversteeringparametersinTable1,theirhappenstobenocriticalspeedforthesystem (thesolutiontoIquation/(isimaginarybutcertainoversteeringparameterscangiverealcritical speedsA.

Cyshiftingtheforcetotheneutralsteerpoint,theoversteeringvehicleismarginally stable,butexhibitsanicelydampedresponseascomparedwiththeundersteeringvehiclefromthe previoussection(Figure7A. 4incethiscontrolforcelocationcreatesnosteadystateyawvelocity 10
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thereisminimalrotation,yieldinganacceptableresponse.4hiftingthecontrolforcetotheneutral steerpointoftheundersteeringgivessimilarresults(Figure7AA.Theundersteeringvehiclehasan overdampedresponsethatisduetothetransientbehaviorofthevehicleasitreachessteadystate. 10 0.1 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) Lateral Position (m) Vehicle

Lateral Position Oversteering Understeering Figure7:4imulation:,irtualForceatMeutral4teer9oint Figure8andFigure8showhowtheeigenvaluesshiftforanoversteeringandundersteering carasthevirtualforceisshiftedfrom0 behindtheneutralsteerpointto0 infrontofthe neutralsteerpoint. Thesquaredenotestheinitialpositionbehindtheneutralsteerpoint. The eigenvaluesverifythatthesystemisunstablewhentheapplicationpointisbehindtheneutralsteer point.Asthevirtualforceisshiftedforward,thesystembecomesstable,butastheforceismoved furtherforwardthesystembecomesoscillatoryandeventuallyunstable.Thisinstabilityisdueto

thelackoflookaheadinthesystem. Interestingly,inthecasediscussedinthissection,theforceisshiftedtotheneutralsteerpoint butthesensinglocationremainesatthec.g.Therefore,intheoversteeringcasethesensinglocation isactuallybehindthecontrolforcewhileintheundersteeringvehicleitisslightlyinfrontofthis controlforce. Ineithercase,withhardlyanylookaheadthesystemresponseisexcellent. The limitationisthattheneutralsteerpointisonlymarginallystable,yieldinganeigenvalueatzero. Ifthecontrolforceismovedslightlyrearward,thesystemwillbecomeunstable.Itisunlikelythat

thevehicleparameterscanbeknownwiththeaccuracynecessarytopinpointthislocation.Iven variationsinvehicleloadingortirepressurecanshiftthispointandcreateinstability.Toberobust toparameteruncertaintiesthecontrolforceshouldbeshiftedinfrontoftheneutralsteerpoint. Asthisforceismovedtowardsthefrontaxleofthevehicle,thedampinglowersandacritical speedexists. Thesystemresponsecanbeimprovedbyincorporatinganappropriateamountof lookahead.Theroleoflookaheadinvehicleswiththecontrolforceatthefrontaxle(corresponding tohavingonlysteeringAiswellknownbutwiththeabilitytoshifttheforcetherearetwodifferent 11


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Eigenvalues for Oversteering Vehicle Real Imaginary Figure8:-versteering,ehicleIigenvaluesasApplication9oint4hiftsForward variablesthatinfluencestabilityandsystembehavior. 6 Incorporating Lookahead Inordertoincorporatelookahead,thecontrolforcemustdependonalateralpositionthatisin frontoftheapplicationpoint. Defining la tobetheprojectedlateralpositionadistance la in frontofthec.g.,asshowninFigure .,yieldsthefollowingcontrolforce. virtual ke la la sin AA (//A Ifsmallangleapproximationsaremade,thisexpressionforlookaheadcanbeincludedinthe linearizedvehiclemodel. 01 0 0

mS kx la aC bC mS 00 0 1 kx cf aC bC aC bC kx la cf (/5A Thecharacteristicequationis λd F0 (/6A where G( mS 1(
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Eigenvalues for Understeering Vehicle Real Imaginary Figure8:+ndersteering,ehicleIigenvaluesasApplication9oint4hiftsForward G( bC aC mS kS mx cf la mS G( cf la A( bC aC AG( cf la mS bC aC cf AA Mowtherearetwofactorseffectingstability:applicationpointofthevirtualforceandlookahead distance.Fromtheprevioussectionthereareclearlimitationsontheapplicationpointofthecontrol force.

Moticethattheadditionoflookaheaddoesnoteffectthefinalterminthecharacteristic equation. Therefore,theinstabilitycausedbyapplyingthevirtualforcebehindtheneutralsteer pointcannotberectifiedusinglookahead.-ncestabilityhasbeenachievedbyapplyingtheforce infrontoftheneutralsteerpoint,systemresponseandcriticalspeedarechangedbymanipulation ofthesensinglocation. Figure10showshowthesystemeigenvalueschangeasthelookaheadisvariedfromthec.g.toa previewdistanceof70 foranundersteeringvehicle.Thespeedisheldataconstant/0 m/s andthe applicationpointisapplied0

infrontoftheneutralsteerpoint.Initially,thevehicleisunstable withapairofeigenvaluesintherighthalfplane. Asthelookaheaddistanceisincreased,these unstablepoleseventuallymigrateintothelefthalfplane,stabilizingthesystem. Increasingthe lookaheadbytoogreatamarginhassomenegativesideeffectsonsystemperformance.Iventually, twoeigenvaluesmoveontotherealaxiswhiletheothertwoeigenvaluesmovetowardstheimaginary axis,decreasingthesystemdamping. Figure11depictsthesystemeigenvaluesforanoversteeringvehicleusingthesamerangeof lookaheaddistances.

Asintheundersteeringcase,thesystemisinitiallyunstablewithapairof eigenvaluesintherighthalfplane. Asthelookaheadisincreased,thiseigenvaluepairmovesinto 1/
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RoadCenterline la la F=-Ke la Figure.:Dookahead 10mLookahead 30mLookahead 50mLookahead -0.7858G(.0878i -6.1087 -0.717/ -0.7858-(.0878i -1.1... -8./.(8 -5.5876G6.1.(0i -(.0081G6.8/87i -1.1678G7..661i -5.5876-6.1.(0i -(.0081-6.8/87i -1.1678-7..661i F0./088 F1.0 F1.0 F0.76/8 F0.// F0.175 Table(:IigenvaluesandDamping thelefthalfplanestabilizingthesystem. Withincreasinglookahead,however,theeigenvalues

thatwereinitiallyunstablebecomelessdamped.Aswiththeundersteeringcase,thereisanideal lookaheaddistancethatwillyieldastablesystemwithappropriatedampingcharacteristics. Figure1(showsthesystemstatesforanundersteeringvehiclewithvaryingamountsoflooka- head. Thevirtualforceapplicationpointandspeedarethesameasthoseusedintheeigenvalue figuresandtheinitialconditionisalateraloffsetof0 fromthelanecenter.Withalookahead of10 thesystemisunderdampedwithaslownaturalfrequency. Whenthepreviewdistance isincreasedto/0 theresponseslookmuchbetter. Thedampinghasincreasedandallstates

convergetonominalvaluesinashortertime. Increasingthepreviewdistanceto60 ,however, createsapoorersystemresponsewithhighfrequencyoscillationsinthestates.Table(showsthe eigenvaluesanddampingforthesethreecasesoflookahead. 1ivenacertainapplicationpointforthevirtualforce,alookaheaddistancecanbechosen thatwillcreateastable, adequatelydampedsystemwithoutanyartificialcontrollerdamping (i.e. dampingon E and A. Thefinalwaytoaltertheperformanceofthesystemistoinclude controllerdamping. Addingdampingtothevelocitystatesallowsthedesignertoimplementfull

statefeedbackonthesystem,givingthefreedomtotheoreticallyplacethepolesatanarbitrary location(ofcourse,therearelimitsduetoactuatorcapability,bandwidth,etc.A. 15
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Eigenvalues for Understeering Vehicle with Increasing Lookahead Real Imaginary Figure10:IigenvaluesastheDookaheadDistanceis,ariedfrom0-70m:+ndersteering,ehicle 7 Controller Damping Ifcontrollerdampingisaddedtothesystemthevirtualforceisnowafunctionofthelateral positionatsomelookaheaddistance, la ,thelateralvelocity,E ,andtheyawrate, virtual la sin (/7A where and

arethedampingcoefficientsonthelateralandyawvelocitiesrespectively. Thiscontrollawisessentiallyaproportionalderivativecontrollergivingthefollowinglinear system. 01 0 0 mS kx la aC bC mS 00 0 1 kx cf aC bC cf aC bC kx la cf cf (/8A IftheinputisthevirtualforcefromIquation/7appliedatadistance, cf ,fromthec.g.,the closedloopsystemcanbewrittenas F( OL BG δx (/8A where OL istheopenloopvehicledynamicsgiveninIquation((withCand1definedas cf (/.A kD kx la (50A 16
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10 10 Eigenvalues for Oversteering Vehicle with Increasing Lookahead Real Imaginary

Figure11:IigenvaluesastheDookaheadDistanceis,ariedfrom0-70m:-versteering,ehicle Intheory,thereisnowcompletecontrolovertheclosedloopsystemdynamics. Thepolesofthe systemcanbeputatanylocationgiventhecorrectchoiceoftheparametersinthevirtualforce expression. -fcourse,therearepracticallimitsontheseparametersdependingontheactuation andsensingcapabalitiesofaparticularsystem.Ignoringtheseissues,theproblemoflateralvehicle controlnowseemstrivial.Thegain,lookaheaddistance,anddampingvaluescanbechosentogive anydesiredresponse.

Inreality,however,itmightbeundesirabletohavelargedampingvaluesinthecontroller.With largedampingvalues,thesystemisconstantlylosingenergywhenthereisalateraloryawvelocity. Fromanefficiencystandpointthisisapoordesignmove.Forlanekeepingdriverassistancesystems, theabilitytoproduceadequateresponseswithoutaddinganyartificialcontrollerdampingonthe velocitystatesisquitebeneficial. Inthiscase,thecontrolforcehasanaturalassociationwitha positionalstatecorrespondingtotheedgeoftheroadwherecontrolforcesaredesiredtoassistthe driver.

Incaseswheredampingisundesirableorwherelookaheadisunavailablefullstatefeedbackis notpossible.Therefore,thecontrollerwillnotbeabletousecertaininputstoaltertheoutput.In thesecases,theentirecontrolspaceisnotavailableandtheresultsandintuitionfromtheprevious sectionscanbeusedtodesignasuitablelateralcontroller. 8 Concluding Remarks Inthecaseofvehiclecontrol,itiscrucialtopayattentiontotheapplicationpointofvirtualcontrol forces.Bhangesinvehiclepropertiescanhavedeleteriouseffectsonthesystemsperformanceand stability.Theanalysisinthispaperillustratestheinstabilitiesthatcanoccurwhenvirtualcontrol

forcesareappliedtoavehicle. Thispaperlookedatthreemaincontributionstolateralstability 17
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0.2 0.1 0.1 0.2 0.3 0.4 0.5 Time (s) Lateral Position (m) Vehicle Lateral Position 50 m Lookahead 30 m Lookahead 10 m Lookahead 0.8 0.6 0.4 0.2 0.2 0.4 0.6 Time (s) Lateral Velocity (m) Vehicle Lateral Velocity 0.025 0.02 0.015 0.01 0.005 0.005 0.01 Time (s) Yaw Angle (rad) Vehicle Yaw Angle 0.06 0.04 0.02 0.02 0.04 Time (s) Yaw Velocity (rad/s) Vehicle Yaw Velocity Figure1(:+ndersteering,ehicle)esponsewith,aryingDookahead andperformance.

Thevirtualcontrolforcemustbeappliedinfrontoftheneutralsteerpoint Dookaheadprovidesadequateperformancewithoutcontrollerdamping Dampingonthevelocitystatesandlookaheadoffercompletecontrolofthesystemresponse Akeyadvantageofhavingcontrolovertheapplicationpointoftheforceistheabilitytouseless lookahead.Astheapplictionpointapproachestheneutralsteerpoint,lateralsensorsonthefront ofthevehiclecouldbeusedtoachieveanacceptableresponseeliminatingtheneedforprojecting offthefrontofthevehicle. Futureworkincludesimplementationandexperimentalvalidationof

theseresultsandtheirroleindriveracceptanceoflanekeepingassistsystems. 18
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