164IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY200940C.GallettiandP.Fanghella,“Single-loopkinematotropicmechanisms,”Mech.Mach.Theory,vol.36,no.3,pp.437450,2009.41P.Fanghella,C.Galletti,andE.Giannotti,“Parallelrobotsthatchangetheirgroupofmotion,”inAdvancesinRobotKinematics.TheNetherlands:Springer,2006,pp.4956.42S.Refaat,J.M.Herve,S.Nahavandi,andH.Trinh,“Two-modeover-constrainedthree-dofsrotational-translationallinear-motor-basedparallel-kinematicsmechanismformachinetoolapplications,”Robot-ica,vol.25,no.4,pp.461466,2007.43X.W.Kong,C.M.Gosselin,andP.L.Richard,“Typesynthesisofpar-allelmechanismswithmultipleoperationmodes,”ASMEJ.Mech.Des.,vol.129,no.6,pp.595601,2007.44D.Zlatanov,I.A.Bonev,andC.MGosselin,“Constraintsingularitiesofparallelmechanisms,”inProc.IEEEInt.Conf.Robot.Autom.,Washing-ton,D.C.,2002,pp.496502.45D.Zlatanov,I.A.Bonev,andC.MGosselin,“Constraintsingularitiesasc-spacesingularities,”inAdvancesinRobotKinematics,J.LenarcicandF.Thomas,Eds.Dordrecht,TheNetherlands:Kluwer,2002,pp.183192.46K.H.Hunt,“Constant-velocityshaftcouplings:Ageneraltheory,”J.Eng.Ind.,vol.95B,no.2,pp.455464,1973.SwedishWheeledOmnidirectionalMobileRobots:KinematicsAnalysisandControlGiovanniIndiveriAbstractSwedishwheeledrobotshavereceivedgrowingattentionoverthelastfewyears.Theirkinematicmodelshaveinterestingpropertiesintermsofmobilityandpossiblesingularities.Thispaperaddressestheis-sueofkinematicmodeling,singularityanalysis,andmotioncontrolforagenericvehicleequippedwithNSwedishwheels.IndexTermsMobilerobotkinematics,mobilerobots,motioncontrol,wheeledrobots.I.INTRODUCTIONInthelastfewyears,Swedishwheeledomnidirectionalmobilerobotshavereceivedgrowingattentionamongthemobileroboticsresearchcommunity.ASwedishwheeldiffersfromacommonwheelinthefactthatrollersaremountedonitsperimeter(seeFig.1).Ifalltherollersareparalleltoeachotherandmisalignedwithrespecttothewheelhubaxis,theywillprovideanextradegreeofmobilitywithrespecttoatraditionalperfectlyrollingwheel.ThewheelsdepictedinFig.1areoftencalledmecanumorSwedishwheels:oneoftheirdesignparametersistheanglebetweentherollersrollingdirectiongandthewheelhubaxisdirectionh.Typicalvaluesare=45and=0,asshowninFig.1(leftandrightcases,respectively).Notethatthedegeneratecase=90hasnopracticalinterestasitwouldallowthesamemobilityoftraditionalwheels.IntheManuscriptreceivedJune27,2008；revisedOctober6,2008andNovember19,2008.FirstpublishedJanuary21,2009；currentversionpublishedFebruary4,2009.ThispaperwasrecommendedforpublicationbyAssociateEditorF.LamirauxandEditorW.K.Chunguponevaluationofthereviewerscomments.G.IndiveriiswiththeDipartimentoIngegneriaInnovazione,UniversityofSalento,73100Lecce,Italy(e-mail:giovanni.indiveriunile.it).Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineat.DigitalObjectIdentifier10.1109/TRO.2008.2010360Fig.1.Mecanumwheel:=45(left)and=0(right).literature,wheelswith=45aremoreoftencalledmecanumwheelswhereastheoneswith=0aregenerallycalledSwedishwheels.Inthefollowing,thetermSwedishwheelwilldenotethegeneralcaseforsomefixed.Asreportedin1,themecanumwheelwasinventedin1973byBengtIlon,anengineerworkingfortheSwedishcompanyMecanumAB.Sincethen,thiswheeldesignhasattractedtheattentionofthemobileroboticsresearchcommunity.Theinterestinsuchkindofwheelsisrelatedtothepossibilityofdevelopingomnidirectionalrobotsinthesenseof2,i.e.,robotsthat“haveafullmobilityintheplanemeaningtherebythattheycanmoveateachinstantinanydirectionwithoutanyreorientation”2.Theneedtoreorientthewheelsornotpriortoimplementinganydesiredlinearvelocityisrelatedtothepresenceornotofnonholonomicconstraints3,4.SincetheearlystudyofAgulloetal.5,thekinematicsanalysisofSwedishormecanumwheelrobotshasbeenaddressedinseveralpapers612.AlltheseexceptthestudyofAgulloetal.5andSahaetal.9considereitherthreeorfourSwedishwheeledrobotsforsomevalueof.Oneoftheobjectivesofthispaperistoderive,inthemostgeneralsettingofNSwedishwheelswithfixed(butarbitrary)rollerwheelangle,necessaryandsufficientgeometricalconditionsonrelativewheelarrangementthatguarantee:1)theabsenceofsingularitiesand2)thepossibilityofdecouplingcommandedlinearandangularrobotvelocities.Thiskindofinformationcanbemostvaluabletoguidethedesignoftherobotapriori.TheresultsobtainedaddressingtheN-wheelcase,besidesrealizingatonceaunifiedanalysisofthemostcommonthree-andfour-wheeldesigns,allowtoeasilyexplore,forexample,possiblesix-wheeldesignsthathavealargeinterestinthefieldofoutdoorandroughterrainapplications14.Asasideresultoftheproposedanalysis,allpossiblekinematicssingularitiesoccurringasafunctionoftherollerwheelangleandwheelarrangementcanbeidentified.Moreover,buildingonwell-knownmethods,thetrajectorytrackingandposeregulationproblemsaresolvedforthesesystemstakingexplicitlyintoaccountactuatorvelocitysaturation.Thepaperisorganizedasfollows:thegeneralkinematicsmodelforanNSwedishwheeledvehicleisderivedinSectionII.TheguidancetrajectorytrackingandposeregulationcontrolproblemsinthepresenceofactuatorvelocitysaturationareaddressedinSectionIII.ExperimentalresultsarereportedinSectionIV.Atlast,someconcludingremarksareaddressedinSectionV.II.KINEMATICSMODELWithreferencetoFig.2,forthesakeofintroducingthenotation,athree-wheelomnidrivemobilerobotisconsidered.Allwheelmainaxes,i.e.,hubaxes,areassumedtoalwayslieparalleltothefixedgroundplanePhavingunitvectorkP.EachoftheNSwedishwheelsisindexedfrom1toN:forthehthwheel,therollerincontactwiththegroundplanePisdepictedasanellipsewithmainaxesori-entedalongtheunitvectorsnh:bardblnhbardbl=1anduh:bardbluhbardbl=1.1552-3098/$25.002009IEEEAuthorizedlicenseduselimitedto:NanchangUniversity.DownloadedonJanuary12,2010at20:15fromIEEEXplore.Restrictionsapply.IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY2009165Fig.2.Three-wheelomnidriverobot:geometricalmodel.Theunitvectornhisalignedwiththerollersrollingaxisonthemainwheelperimeteranduh:=nhkindicatestheinstantaneoustan-gentvelocitydirectionoftherollerassociatedwithitsrotationaroundnh.Allwheelsareassumedtobeidenticalandhavethesameradius.Thepositionofthehthwheelinthebody-fixedframeisdenotedbybh.Theunitvectorofeachwheelhubaxis,i.e.,theunitvectorofthewheelsmainrotationaxis,isdenotedbynh:bardblnhbardbl=1.Atlastforeachwheel,theunitvectoruh:=nhkisdefinedindicatingtheinstantaneoustangentvelocitydirectionofthewheelasaconsequenceofitsrotationaroundnh.NotethatwithreferencetoFig.1,theunitvectorsuhandnhwouldcorrespondtogandh,respectively.Inthegivenhypothesis,alltheintroducedvectorsexceptkareparalleltothegroundplaneP.Giventhecomponentsofanytwo3-Dvectorsaandbonacommonorthonormalframe,theirvectorproductwillbecomputedusingtheskew-symmetricmatrixS()suchthatab=S(a)b.(1)Callingvcthelinearvelocityoftherobotscenter(indicatedaspointcinFig.2)andkitsangularvelocityvector,thevelocityvectorvhofthecenterofeachomnidirectionalwheelhubwillbegivenbyvh=vc+kbh,h=1,2,3,.,N.(2)Consideringthegenerichthwheelanddroppingforthetimebeingtheindexhforthesakeofnotationalclarity,inthecaseofperfectrolling,thevelocityv=vhgivenby(2)willbephysicallyrealizedbytherollerrotationaroundnandthewheelrotationaroundn.Namely,assumingthatnandnarenotaligned,i.e.,negationslash=(2+1)90,whereisanintegerv=u+uimplyingnTv=parenleftbignTuparenrightbignTv=parenleftbignTuparenrightbigandconsequentlyv=nTvnTuu+nTvnTuu.(3)Notethattherollersrotationaroundngivingrisetothefirsttermontheright-handsideof(3)iscompletelypassive,whereasthewheelrotationaroundngivingrisetothesecondtermontheright-handsideof(3)isassumedtobeactivelyproducedbyamotor.Callingnhqhtheangularspeedassociatedwiththehthmotorinthebody-fixedframe,incaseofperfectrolling,themappingbetweenthe“joint”speedqandthecorrespondingvelocityvofthehubofanygivenwheelwillbegivenbynTuTnv=q(4)wherethecontributionnTvuTnuTuofvindirectionofuhasbeenexplicitlyassumednottocontributetoq,asinthegivenhypothesisofperfectrolling,itisfullygeneratedbythepassiverotationoftheroller.Substituting(2)backinto(4),giventhatuThnh=cosforanyh,onegets1cosnThvc+1cosnThS(bh)k=qh.(5)Byprojectingallthevectorsin(5)onacommonbody-fixedframewithitsthirdaxisequaltokP,thetermnThS(bh)kresultsinnThS(bh)k=nxhbyhnyhbxh=bThuh.(6)Summarizing,(5)canbeinterpretedasagenericcomponentoftheinversedifferentialkinematicsequationinmatrixformMparenleftbiggvcparenrightbigg=qcos(7)whereM=nx1ny1bT1u1nx2ny2bT2u2.nxNnyNbTNuNIRN3(8)andqIRN1isthevectorofjointvelocities.Equations(7)and(8)representthegeneralkinematicsmodel9ofaSwedishwheeledve-hiclewithNwheels.Contrarytomostoftheothermodelspresentedintheliteraturethatarerelativetothreeorfourwheelsinfixedconfig-urations,theyallowafullydetailedanalysisofthevehiclekinematicspropertiesasafunctionofnotonlytherollerwheelhuborientation,butalsotherelativewheelposition.Suchananalysismaybeextremelyusefulforthemechanicalandcontrolssystemdesignofthevehicle.Assumingcosnegationslash=0andthatMhasrank3,equation(7)canbeusedtocomputejointvelocitycommandsforadesiredvehiclespeed(vTcd,d)T.Inparticular,thefollowinglemmaholds.Lemma1:GivenaSwedishwheeledmobilerobotwithNiden-ticalwheelsofradiussatisfying(7),anydesiredvehiclevelocity(vTcd,d)Tcanbeimplementedbyusingproperjointvelocitiesqdifandonlyifallofthefollowingconditionshold:c.1cosnegationslash=0；c.2rankM=3.Inparticular,qd=1cosMparenleftbiggvcddparenrightbigg.(9)Theviolationofanyofthepreviousconditionsc.1orc.2correspondstokinematicssingularitiesofadifferentkind:theviolationofconditionc.1wouldcorrespondtoatotallossofcontrolauthority,whiletheviolationofconditionc.2wouldcorrespondtoalossofcontrollability,asforanychoiceoftheinputq,thestatederivative(vTc,)Twouldnotbeuniquelydefined.MatrixMdefinedin(8)canbedecomposedasM=MlMa,whereMlIRN2andMaIRN1suchthatMlvc+Ma=qcos.(10)Authorizedlicenseduselimitedto:NanchangUniversity.DownloadedonJanuary12,2010at20:15fromIEEEXplore.Restrictionsapply.166IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY2009III.GUIDANCECONTROLOFSWEDISHWHEELEDOMNIDIRECTIONALROBOTSBasedonthederivedkinematicalmodel,thetrajectorytrackingandposeregulationmotioncontrolproblemsforagenericrobotequippedwithN3Swedishwheelsaresolved.Theproposedsolution,besidesbeinggeneralintermsofnumberofwheels,valueof,andwheelcon-figuration,explicitlyaccountsforjointvelocitysaturation.Thecontrolproblemisformulatedasaguidancecontrolproblem,namelythejointvelocitiesqhareassumedtobecontrolinputsandtherobotslinearandangularvelocitiesvcandsatisfying(7)aretheoutputs.Ofcourse,inpractice,suchanapproachrequirestheimplementationofalowerlevelcontrolloopmappingthejointreferencespeedsqdinactuatorcommands.A.Low-LevelControlDesignAsinstandardrobotmotioncontrolarchitectures22,thelow-levelcontrolsystemcanbedesignedeitherinadecentralizedorinacentral-izedfashion.Intheformercase,eachactuatoriscontrolledseparately,typicallywithavelocityPIDloop(asfortheexperimentalvalidationreportedinthispaper)；inthelattercase,acentralizedcontrolsolutioncanbederivedbasedoncomputedtorque(i.e.,feedbacklinearization)methods.Thedecentralizedcontrol(orindependentjointcontrol)methodissimpler:eachcomponentofqdisusedasareferencesignalforthecorrespondingPIDactuatorvelocityloop,anddynamiccouplingsamongtheactuatorsareneglected.Ifsuchlow-levelcontrolsystem(i.e.,eachactuatorvelocityservoloop)isfastwithrespecttotherobotdynamic-navigationone,thelagbetweenthedesiredjointvelocitiesandtherealjointvelocitiesisnegligible:insuchacase,aslongastheperfectrollingconstraintissatisfied,themappingbetweenthevehiclesvelocity(vTc,)Tandthejointspeedswouldbeapproximatedbythesystemskinematicalmodel(7)havingthedesired(orcommanded)jointvelocitiesqdontheright-handsideinplaceoftherealjointvelocitiesq.Centralizedcontrolsolutionsaregenerallybasedonthedynamicrobotmodel17,Ch.12,pp.493502,andthedynamicequationofSwedishwheeledrobotswithageometryastheonedescribedinSectionIIhasthefollowingstructure:Iq+(C()+F)q=(11)withIIRNNbeingthepositive-definiteinertiamatrix,C()IRNNtheskew-symmetricCoriolisandcentrifugalforcesmatrix(kistherobotsangularvelocity),FIRNNthediagonalfric-tionmatrix,andIRN1theactuatortorquesvector.GiventhenondiagonalnatureofmatricesIandC()andthedependency(7)offromq,(11)isnonlinearandcoupled.Yet,ascommonlydoneforroboticmanipulators22,thecontrolinputvectorcanbecom-putedbaseduponanonlinearstatefeedbacklinearization(orcomputedtorque)solution,namely=(C()+F)q+Iy(12)givingrise(recallthatIisfullrank)toalinearanddecoupledmodelq=ythatcanthenbeusedtodesignaclosed-loopsolutionforyinordertotrackthereferencesignalqd.Giventhatthecomputedtorquesolutionexplicitlyaccountsforthesystemsdynamiccouplingterms,itisexpectedtoexhibitabettertrackingperformance,inparticular,forhighspeedandaccelerationreferences.Nevertheless,asalsodemonstratedbythereportedexperi-mentalresults,theindependentjointsolutionappearstobesufficientlypreciseandaccuratefortrackingsimpletrajectoriesatconstantspeed.Moreover,giventhatthemainobjectiveofthisresearchwastovali-dateanactuatorvelocitymanagementstrategyattheguidancelevel,onlytheindependentjointlow-levelcontrolsolution(i.e.,actuatorPIDvelocityservoloop)wasimplemented.Inthefollowing,theSwedishwheeledrobotwillbeassumedtohavesuchkindoflow-levelindependentjointcontrolssystem,andthetrajectorytrackingandposeregulationproblemswillbesolvedattheguidancelevel,i.e.,consideringthecommandedjointspeedsqdascontrolinputsand(vTc,)Tasoutputsaccordingtothepurelykinematicalmodel(10).Inordertoformulatethetrajectorytrackingandposeregulationproblems,thefollowingnotationwillbeused:givenaninertial(global)frameG=(i,j,k)withk:=(ij)P,wherePisthefloorplane,areference(planar)trajectoryisadifferentiablecurveinPrd(t)=iparenleftbigrTd(t)iparenrightbig+jparenleftbigrTd(t)jparenrightbig(13)withcurvilinearabscissas(t):=integraldisplaytt0vextenddoublevextenddoublevextenddoublevextenddoubledrd()dvextenddoublevextenddoublevextenddoublevextenddoubled(14)andunittangentvectortd=drdds.(15)Thekinematicstrajectorytrackingproblemconsistsinfindingacontrollawforthesystemsinputqdsuchthatthepositionandheadingtrackingerrorser(t):=rd(t)rc(t)(16)e(t):=d(t)(t)(17)convergetozero,withrc(t)beingthepositioninGofareferencepoint(e.g.,thegeometricalcenterorthecenterofmass)oftherobot,(t)itsheading,andd(t)thedesiredreferenceheading.Notethatfornonholonomicvehicleshavingaunicycleorcar-likekinematicsmodel,thereferenceheadingd(t)isnotarbitrary,butneedstocoincidewiththeheadingofthetrajectoriesunittangentvectortd.Conversely,givenanypositionreferencetrajectoryrd(t),aSwedishwheeledvehiclewillbefreetotrackanyarbitraryheadingd(t)thatdoesnotnecessarilyneedtocoincidewiththeheadingoftd.Theposeregulationproblemisaspecialcaseofthetrajectorytrack-ingoneoccurringwhenthepositionandorientationreferencesareconstant,i.e.,whenrd=0andd=0.B.TrajectoryTrackingControllerDesignInaccordancewiththenotationpreviouslyintroduced,considerthesystemin(10)withrc(t)=vcand(t)=beingtherobotslin-earandangularvelocities.Assumingthatconditionsc.1andc.2ofLemma1aresatisfied,N3,andMaspan(Ml)(thiscanbeal-waysguaranteedbyaproperdesignoftherobotgeometry),thenanydesiredrobotlinearvelocityvc=(,0)Tandangularvelocitykareuniquelymappedtothecontrolinputqdasfollows:qd=qdl+qda(18)qdl=1cosMlvc(19)qda=1cosMa.(20)Authorizedlicenseduselimitedto:NanchangUniversity.DownloadedonJanuary12,2010at20:15fromIEEEXplore.Restrictionsapply.IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY2009167Followingastandardapproach,tosolvethetrajectorytrackingproblem,considertheLyapunovcandidatefunctionV=12eTrKrer+12eTKe(21)withKrIR22beingasymmetricpositivedefinite(Kr0)matrixandKapositiveconstant.ThetimederivativeofVresultsinV=eTrKr(rd(t)vc)+eTK(d(t).(22)Ifvcandsatisfyvc=rd(t)+Kr(rd(t)rc(t)(23)=d(t)+K(d(t)(t)(24)thenthetimederivativeofVwillbenegativedefinite,i.e.V=eTrKrKrer(Ke)20,theabsolutevalueofthemaximumpossiblevelocityofactuatorj,athresholdforbardblqdbardblcanbeselectedasqmax=minjqmaxj,wherej=1,2,.,N.WhateverthegainsKrandK,dependingond(t),rd(t),er(t),ore(t),thesaturationconditionbardblqdbardblqmax(29)mayalwaysbeviolated.Notethatwhilethefeedforwardsignalsd(t)andrd(t)caneventuallyalwaysbebounded,thetrackingerrorsinitialconditionsarenotdesignparameters.Hence,acommandedqdwithanexceedinginfinitynormduetooddinitialconditionscannotbeaprioriexcluded.D.TrackinginPresenceofActuatorVelocitySaturationThepresenceofactuatorvelocitysaturationhasasevereimpactonperformance:inparticular,giventheadditivestructureof(26),actuatorvelocitysaturationcanaffectthedecouplingbetweencommandedan-gularandlinearve