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外文翻译---瑞典轮式全方位移动机器人 英文版.pdf外文翻译---瑞典轮式全方位移动机器人 英文版.pdf -- 5 元

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164IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY200940C.GallettiandP.Fanghella,Singleloopkinematotropicmechanisms,Mech.Mach.Theory,vol.36,no.3,pp.437–450,2009.41P.Fanghella,C.Galletti,andE.Giannotti,Parallelrobotsthatchangetheirgroupofmotion,inAdvancesinRobotKinematics.TheNetherlandsSpringer,2006,pp.49–56.42S.Refaat,J.M.Herv´e,S.Nahavandi,andH.Trinh,Twomodeoverconstrainedthreedofsrotationaltranslationallinearmotorbasedparallelkinematicsmechanismformachinetoolapplications,Robotica,vol.25,no.4,pp.461–466,2007.43X.W.Kong,C.M.Gosselin,andP.L.Richard,Typesynthesisofparallelmechanismswithmultipleoperationmodes,ASMEJ.Mech.Des.,vol.129,no.6,pp.595–601,2007.44D.Zlatanov,I.A.Bonev,andC.MGosselin,Constraintsingularitiesofparallelmechanisms,inProc.IEEEInt.Conf.Robot.Autom.,Washington,D.C.,2002,pp.496–502.45D.Zlatanov,I.A.Bonev,andC.MGosselin,Constraintsingularitiesascspacesingularities,inAdvancesinRobotKinematics,J.LenarcicandF.Thomas,Eds.Dordrecht,TheNetherlandsKluwer,2002,pp.183–192.46K.H.Hunt,ConstantvelocityshaftcouplingsAgeneraltheory,J.Eng.Ind.,vol.95B,no.2,pp.455–464,1973.SwedishWheeledOmnidirectionalMobileRobotsKinematicsAnalysisandControlGiovanniIndiveriAbstractSwedishwheeledrobotshavereceivedgrowingattentionoverthelastfewyears.Theirkinematicmodelshaveinterestingpropertiesintermsofmobilityandpossiblesingularities.Thispaperaddressestheissueofkinematicmodeling,singularityanalysis,andmotioncontrolforagenericvehicleequippedwithNSwedishwheels.IndexTermsMobilerobotkinematics,mobilerobots,motioncontrol,wheeledrobots.I.INTRODUCTIONInthelastfewyears,Swedishwheeledomnidirectionalmobilerobotshavereceivedgrowingattentionamongthemobileroboticsresearchcommunity.ASwedishwheeldiffersfromacommonwheelinthefactthatrollersaremountedonitsperimeterseeFig.1.Ifalltherollersareparalleltoeachotherandmisalignedwithrespecttothewheelhubaxis,theywillprovideanextradegreeofmobilitywithrespecttoatraditionalperfectlyrollingwheel.ThewheelsdepictedinFig.1areoftencalledmecanumorSwedishwheelsoneoftheirdesignparametersistheangleγbetweentherollersrollingdirectiongandthewheelhubaxisdirectionh.Typicalvaluesareγ45◦andγ0◦,asshowninFig.1leftandrightcases,respectively.Notethatthedegeneratecaseγ90◦hasnopracticalinterestasitwouldallowthesamemobilityoftraditionalwheels.IntheManuscriptreceivedJune27,2008revisedOctober6,2008andNovember19,2008.FirstpublishedJanuary21,2009currentversionpublishedFebruary4,2009.ThispaperwasrecommendedforpublicationbyAssociateEditorF.LamirauxandEditorW.K.Chunguponevaluationofthereviewerscomments.G.IndiveriiswiththeDipartimentoIngegneriaInnovazione,UniversityofSalento,73100Lecce,Italyemailgiovanni.indiveriunile.it.Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineathttp//ieeexplore.ieee.org.DigitalObjectIdentifier10.1109/TRO.2008.2010360Fig.1.Mecanumwheelγ45◦leftandγ0◦right.literature,wheelswithγ45◦aremoreoftencalledmecanumwheelswhereastheoneswithγ0◦aregenerallycalledSwedishwheels.Inthefollowing,thetermSwedishwheelwilldenotethegeneralcaseforsomefixedγ.Asreportedin1,themecanumwheelwasinventedin1973byBengtIlon,anengineerworkingfortheSwedishcompanyMecanumAB.Sincethen,thiswheeldesignhasattractedtheattentionofthemobileroboticsresearchcommunity.Theinterestinsuchkindofwheelsisrelatedtothepossibilityofdevelopingomnidirectionalrobotsinthesenseof2,i.e.,robotsthathaveafullmobilityintheplanemeaningtherebythattheycanmoveateachinstantinanydirectionwithoutanyreorientation2.Theneedtoreorientthewheelsornotpriortoimplementinganydesiredlinearvelocityisrelatedtothepresenceornotofnonholonomicconstraints3,4.SincetheearlystudyofAgull´oetal.5,thekinematicsanalysisofSwedishormecanumwheelrobotshasbeenaddressedinseveralpapers6–12.AlltheseexceptthestudyofAgull´oetal.5andSahaetal.9considereitherthreeorfourSwedishwheeledrobotsforsomevalueofγ.Oneoftheobjectivesofthispaperistoderive,inthemostgeneralsettingofNSwedishwheelswithfixedbutarbitraryrollerwheelangle,necessaryandsufficientgeometricalconditionsonrelativewheelarrangementthatguarantee1theabsenceofsingularitiesand2thepossibilityofdecouplingcommandedlinearandangularrobotvelocities.Thiskindofinformationcanbemostvaluabletoguidethedesignoftherobotapriori.TheresultsobtainedaddressingtheNwheelcase,besidesrealizingatonceaunifiedanalysisofthemostcommonthreeandfourwheeldesigns,allowtoeasilyexplore,forexample,possiblesixwheeldesignsthathavealargeinterestinthefieldofoutdoorandroughterrainapplications14.Asasideresultoftheproposedanalysis,allpossiblekinematicssingularitiesoccurringasafunctionoftheroller–wheelangleandwheelarrangementcanbeidentified.Moreover,buildingonwellknownmethods,thetrajectorytrackingandposeregulationproblemsaresolvedforthesesystemstakingexplicitlyintoaccountactuatorvelocitysaturation.ThepaperisorganizedasfollowsthegeneralkinematicsmodelforanNSwedishwheeledvehicleisderivedinSectionII.TheguidancetrajectorytrackingandposeregulationcontrolproblemsinthepresenceofactuatorvelocitysaturationareaddressedinSectionIII.ExperimentalresultsarereportedinSectionIV.Atlast,someconcludingremarksareaddressedinSectionV.II.KINEMATICSMODELWithreferencetoFig.2,forthesakeofintroducingthenotation,athreewheelomnidrivemobilerobotisconsidered.Allwheelmainaxes,i.e.,hubaxes,areassumedtoalwayslieparalleltothefixedgroundplanePhavingunitvectork⊥P.EachoftheNSwedishwheelsisindexedfrom1toNforthehthwheel,therollerincontactwiththegroundplanePisdepictedasanellipsewithmainaxesorientedalongtheunitvectorsnγhbardblnγhbardbl1anduγhbardbluγhbardbl1.15523098/25.00©2009IEEEAuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2015fromIEEEXplore.Restrictionsapply.IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY2009165Fig.2.Threewheelomnidriverobotgeometricalmodel.Theunitvectornγhisalignedwiththerollersrollingaxisonthemainwheelperimeteranduγhnγhkindicatestheinstantaneoustangentvelocitydirectionoftherollerassociatedwithitsrotationaroundnγh.Allwheelsareassumedtobeidenticalandhavethesameradiusρ.Thepositionofthehthwheelinthebodyfixedframeisdenotedbybh.Theunitvectorofeachwheelhubaxis,i.e.,theunitvectorofthewheelsmainrotationaxis,isdenotedbynhbardblnhbardbl1.Atlastforeachwheel,theunitvectoruhnhkisdefinedindicatingtheinstantaneoustangentvelocitydirectionofthewheelasaconsequenceofitsrotationaroundnh.NotethatwithreferencetoFig.1,theunitvectorsuγhandnhwouldcorrespondtogandh,respectively.Inthegivenhypothesis,alltheintroducedvectorsexceptkareparalleltothegroundplaneP.Giventhecomponentsofanytwo3Dvectorsaandbonacommonorthonormalframe,theirvectorproductwillbecomputedusingtheskewsymmetricmatrixSsuchthatabSab.1CallingvcthelinearvelocityoftherobotscenterindicatedaspointcinFig.2andωkitsangularvelocityvector,thevelocityvectorvhofthecenterofeachomnidirectionalwheelhubwillbegivenbyvhvcωkbh,h1,2,3,...,N.2Consideringthegenerichthwheelanddroppingforthetimebeingtheindexhforthesakeofnotationalclarity,inthecaseofperfectrolling,thevelocityvvhgivenby2willbephysicallyrealizedbytherollerrotationaroundnγandthewheelrotationaroundn.Namely,assumingthatnγandnarenotaligned,i.e.,γnegationslash2ν190◦,whereνisanintegervαuγβuimplyingnTvαparenleftbignTuγparenrightbignTγvβparenleftbignTγuparenrightbigandconsequentlyvnTvnTuγuγnTγvnTγuu.3Notethattherollersrotationaroundnγgivingrisetothefirsttermontherighthandsideof3iscompletelypassive,whereasthewheelrotationaroundngivingrisetothesecondtermontherighthandsideof3isassumedtobeactivelyproducedbyamotor.Callingnh˙qhtheangularspeedassociatedwiththehthmotorinthebodyfixedframe,incaseofperfectrolling,themappingbetweenthejointspeed˙qandthecorrespondingvelocityvofthehubofanygivenwheelwillbegivenbynTγuTnγvρ˙q4wherethecontributionnTvuTγnuTγuofvindirectionofuhasbeenexplicitlyassumednottocontributetoρ˙q,asinthegivenhypothesisofperfectrolling,itisfullygeneratedbythepassiverotationoftheroller.Substituting2backinto4,giventhatuThnγh−cosγforanyh,onegets−1cosγnTγhvc1cosγnTγhSbhkωρ˙qh.5Byprojectingallthevectorsin5onacommonbodyfixedframewithitsthirdaxisequaltok⊥P,thetermnTγhSbhkresultsinnTγhSbhknγxhbyh−nγyhbxh−bThuγh.6Summarizing,5canbeinterpretedasagenericcomponentoftheinversedifferentialkinematicsequationinmatrixformMparenleftbiggvcωparenrightbiggρ˙qcosγ7whereM−nγx1nγy1bT1uγ1nγx2nγy2bT2uγ2.........nγxNnγyNbTNuγN∈IRN38and˙q∈IRN1isthevectorofjointvelocities.Equations7and8representthegeneralkinematicsmodel9ofaSwedishwheeledvehiclewithNwheels.Contrarytomostoftheothermodelspresentedintheliteraturethatarerelativetothreeorfourwheelsinfixedconfigurations,theyallowafullydetailedanalysisofthevehiclekinematicspropertiesasafunctionofnotonlytheroller–wheelhuborientationγ,butalsotherelativewheelposition.Suchananalysismaybeextremelyusefulforthemechanicalandcontrolssystemdesignofthevehicle.Assumingcosγnegationslash0andthatMhasrank3,equation7canbeusedtocomputejointvelocitycommandsforadesiredvehiclespeedvTcd,ωdT.Inparticular,thefollowinglemmaholds.Lemma1GivenaSwedishwheeledmobilerobotwithNidenticalwheelsofradiusρsatisfying7,anydesiredvehiclevelocityvTcd,ωdTcanbeimplementedbyusingproperjointvelocities˙qdifandonlyifallofthefollowingconditionsholdc.1cosγnegationslash0c.2rankM3.Inparticular,˙qd1ρcosγMparenleftbiggvcdωdparenrightbigg.9Theviolationofanyofthepreviousconditionsc.1orc.2correspondstokinematicssingularitiesofadifferentkindtheviolationofconditionc.1wouldcorrespondtoatotallossofcontrolauthority,whiletheviolationofconditionc.2wouldcorrespondtoalossofcontrollability,asforanychoiceoftheinput˙q,thestatederivativevTc,ωTwouldnotbeuniquelydefined.MatrixMdefinedin8canbedecomposedasMMlMa,whereMl∈IRN2andMa∈IRN1suchthatMlvcMaωρ˙qcosγ.10AuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2015fromIEEEXplore.Restrictionsapply.166IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY2009III.GUIDANCECONTROLOFSWEDISHWHEELEDOMNIDIRECTIONALROBOTSBasedonthederivedkinematicalmodel,thetrajectorytrackingandposeregulationmotioncontrolproblemsforagenericrobotequippedwithN≥3Swedishwheelsaresolved.Theproposedsolution,besidesbeinggeneralintermsofnumberofwheels,valueofγ,andwheelconfiguration,explicitlyaccountsforjointvelocitysaturation.Thecontrolproblemisformulatedasaguidancecontrolproblem,namelythejointvelocities˙qhareassumedtobecontrolinputsandtherobotslinearandangularvelocitiesvcandωsatisfying7aretheoutputs.Ofcourse,inpractice,suchanapproachrequirestheimplementationofalowerlevelcontrolloopmappingthejointreferencespeeds˙qdinactuatorcommands.A.LowLevelControlDesignAsinstandardrobotmotioncontrolarchitectures22,thelowlevelcontrolsystemcanbedesignedeitherinadecentralizedorinacentralizedfashion.Intheformercase,eachactuatoriscontrolledseparately,typicallywithavelocityPIDloopasfortheexperimentalvalidationreportedinthispaperinthelattercase,acentralizedcontrolsolutioncanbederivedbasedoncomputedtorquei.e.,feedbacklinearizationmethods.Thedecentralizedcontrolorindependentjointcontrolmethodissimplereachcomponentof˙qdisusedasareferencesignalforthecorrespondingPIDactuatorvelocityloop,anddynamiccouplingsamongtheactuatorsareneglected.Ifsuchlowlevelcontrolsystemi.e.,eachactuatorvelocityservoloopisfastwithrespecttotherobotdynamicnavigationone,thelagbetweenthedesiredjointvelocitiesandtherealjointvelocitiesisnegligibleinsuchacase,aslongastheperfectrollingconstraintissatisfied,themappingbetweenthevehiclesvelocityvTc,ωTandthejointspeedswouldbeapproximatedbythesystemskinematicalmodel7havingthedesiredorcommandedjointvelocities˙qdontherighthandsideinplaceoftherealjointvelocities˙q.Centralizedcontrolsolutionsaregenerallybasedonthedynamicrobotmodel17,Ch.12,pp.493–502,andthedynamicequationofSwedishwheeledrobotswithageometryastheonedescribedinSectionIIhasthefollowingstructureI¨qCωF˙qτ11withI∈IRNNbeingthepositivedefiniteinertiamatrix,Cω∈IRNNtheskewsymmetricCoriolisandcentrifugalforcesmatrixωkistherobotsangularvelocity,F∈IRNNthediagonalfrictionmatrix,andτ∈IRN1theactuatortorquesvector.GiventhenondiagonalnatureofmatricesIandCωandthedependency7ofωfrom˙q,11isnonlinearandcoupled.Yet,ascommonlydoneforroboticmanipulators22,thecontrolinputvectorτcanbecomputedbaseduponanonlinearstatefeedbacklinearizationorcomputedtorquesolution,namelyτCωF˙qIy12givingriserecallthatIisfullranktoalinearanddecoupledmodel¨qythatcanthenbeusedtodesignaclosedloopsolutionforyinordertotrackthereferencesignal˙qd.Giventhatthecomputedtorquesolutionexplicitlyaccountsforthesystemsdynamiccouplingterms,itisexpectedtoexhibitabettertrackingperformance,inparticular,forhighspeedandaccelerationreferences.Nevertheless,asalsodemonstratedbythereportedexperimentalresults,theindependentjointsolutionappearstobesufficientlypreciseandaccuratefortrackingsimpletrajectoriesatconstantspeed.Moreover,giventhatthemainobjectiveofthisresearchwastovalidateanactuatorvelocitymanagementstrategyattheguidancelevel,onlytheindependentjointlowlevelcontrolsolutioni.e.,actuatorPIDvelocityservoloopwasimplemented.Inthefollowing,theSwedishwheeledrobotwillbeassumedtohavesuchkindoflowlevelindependentjointcontrolssystem,andthetrajectorytrackingandposeregulationproblemswillbesolvedattheguidancelevel,i.e.,consideringthecommandedjointspeeds˙qdascontrolinputsandvTc,ωTasoutputsaccordingtothepurelykinematicalmodel10.Inordertoformulatethetrajectorytrackingandposeregulationproblems,thefollowingnotationwillbeusedgivenaninertialglobalframe〈G〉i,j,kwithkij⊥P,wherePisthefloorplane,areferenceplanartrajectoryisadifferentiablecurveinPrdtiparenleftbigrTdtiparenrightbigjparenleftbigrTdtjparenrightbig13withcurvilinearabscissastintegraldisplaytt0vextenddoublevextenddoublevextenddoublevextenddoubledrdτdτvextenddoublevextenddoublevextenddoublevextenddoubledτ14andunittangentvectortddrdds.15Thekinematicstrajectorytrackingproblemconsistsinfindingacontrollawforthesystemsinput˙qdsuchthatthepositionandheadingtrackingerrorsertrdt−rct16eϕtϕdt−ϕt17convergetozero,withrctbeingthepositionin〈G〉ofareferencepointe.g.,thegeometricalcenterorthecenterofmassoftherobot,ϕtitsheading,andϕdtthedesiredreferenceheading.Notethatfornonholonomicvehicleshavingaunicycleorcarlikekinematicsmodel,thereferenceheadingϕdtisnotarbitrary,butneedstocoincidewiththeheadingofthetrajectoriesunittangentvectortd.Conversely,givenanypositionreferencetrajectoryrdt,aSwedishwheeledvehiclewillbefreetotrackanyarbitraryheadingϕdtthatdoesnotnecessarilyneedtocoincidewiththeheadingoftd.Theposeregulationproblemisaspecialcaseofthetrajectorytrackingoneoccurringwhenthepositionandorientationreferencesareconstant,i.e.,when˙rd0and˙ϕd0.B.TrajectoryTrackingControllerDesignInaccordancewiththenotationpreviouslyintroduced,considerthesystemin10with˙rctvcand˙ϕtωbeingtherobotslinearandangularvelocities.Assumingthatconditionsc.1andc.2ofLemma1aresatisfied,N≥3,andMa⊥spanMlthiscanbealwaysguaranteedbyaproperdesignoftherobotgeometry,thenanydesiredrobotlinearvelocityvc,,0Tandangularvelocityωkareuniquelymappedtothecontrolinput˙qdasfollows˙qd˙qdl˙qda18˙qdl1ρcosγMlvc19˙qda1ρcosγMaω.20AuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2015fromIEEEXplore.Restrictionsapply.IEEETRANSACTIONSONROBOTICS,VOL.25,NO.1,FEBRUARY2009167Followingastandardapproach,tosolvethetrajectorytrackingproblem,considertheLyapunovcandidatefunctionV12eTrKrer12eTϕKϕeϕ21withKr∈IR22beingasymmetricpositivedefiniteKr0matrixandKϕapositiveconstant.ThetimederivativeofVresultsin˙VeTrKr˙rdt−vceTϕKϕ˙ϕdt−ω.22Ifvcandωsatisfyvc˙rdtKrrdt−rct23ω˙ϕdtKϕϕdt−ϕt24thenthetimederivativeofVwillbenegativedefinite,i.e.˙V−eTrKrKrer−Kϕeϕ20,theabsolutevalueofthemaximumpossiblevelocityofactuatorj,athresholdforbardbl˙qdbardbl∞canbeselectedas˙qmaxminj{˙qmaxj},wherej1,2,...,N.WhateverthegainsKrandKϕ,dependingon˙ϕdt,˙rdt,ert,oreϕt,thesaturationconditionbardbl˙qdbardbl∞≤˙qmax29mayalwaysbeviolated.Notethatwhilethefeedforwardsignals˙ϕdtand˙rdtcaneventuallyalwaysbebounded,thetrackingerrorsinitialconditionsarenotdesignparameters.Hence,acommanded˙qdwithanexceedinginfinitynormduetooddinitialconditionscannotbeaprioriexcluded.D.TrackinginPresenceofActuatorVelocitySaturationThepresenceofactuatorvelocitysaturationhasasevereimpactonperformanceinparticular,giventheadditivestructureof26,actuatorvelocitysaturationcanaffectthedecouplingbetweencommandedangularandlinearvehiclevelocitiesinspiteofthefactthatMadoesnotbelongtothespanofMl.Withreferenceto26,suppose,forexample,thatallthecomponentsof˙qdltarewithintheactuatorlimits,butthatasaconsequenceofadding˙qdat,someoftheoverallcommandedjointspeeds˙qdtexceedtheactuatorlimits.Inthiscase,boththerobotcommandedlinearandangularvelocitieswillbecorruptedinanunpredictablefashion.Aknownquickanddirtywayoutofthisproblemcouldbetosimplyscaledown˙qdtsuchthatallitscomponentsarewithinacceptablebounds.This,ofcourse,willalwaysguaranteethatthecommandedjointspeedsarewithintheproperbounds,butonewouldneedtoprovethatsuchastrategydoesnotjeopardizetheasymptoticstabilityofthetrackingerrortozero.Moreover,inmanyapplications,itmayhappenthateitherlinearorangularvelocitycommandsmayhavehighestpriority.Insuchacase,scalingdowntheoveralljointcommandswillhaveanegativeeffectofslowingdowntheconvergencespeedofthehighestprioritytaskduetothepresenceofthelowerpriorityone.Inordertocopewiththisandtoguaranteeaprioritizedexecutionofthepositionandheadingtrackingtasks,thefollowingmodificationoftheproposedcontrollawissuggestedthesumin26shouldbeweightedwitherrorandreferencedependentweightssuchthat1theresulting˙qdcommandhasnormwithintheactuatorlimits2thetaskspositionandheadingtrackinginthepresentcaseareexecutedwithaprioritybasedtimeorderhigherprioritytasksfirstand3thetrackingerrorconvergestozero.ConsiderthesaturationfunctionσIR0,∞−→IRσx,c0,ifx01,if0|x|cc/|x|,otherwise.30Inthefollowing,thenonnegativesecondargumentcofσx,cwillbecalledthecapacityofx.Notethatbydefinition,σx,cissimplyanonnegativescalarscalingfactorsuchthatxσx,cisclippedtocsignxwhenever|x|shouldexceedthecapacitycandisequaltoAuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2015fromIEEEXplore.Restrictionsapply.
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