An Interval Analysis Based Study for the Design an

arXiv:0708.1049v1[cs.RO]8Aug2007AnIntervalAnalysisBasedStudyfortheDesignandtheComparisonof3-DOFParallelKinematicMachinesD.Chablat1Ph.Wenger1F.Majou1J-P.Merlet21InstitutdeRechercheenCommunicationsetCybern´etiquedeNantes∗1,ruedelaNo¨e,44321Nantes,France2INRIASophia-Antipolis2004RoutedesLucioles,06902SophiaAntipolis,[e-mail]@irccyn.ec-nantes.frF´[e-mail]@sophia.inria.frAbstractThispaperaddressesanintervalanalysisbasedstudythatisappliedtothedesignandthecomparisonof3-DOFparallelkinematicmachines.Twodesigncriteriaareused,(i)aregularworkspaceshapeand,(ii)akinetostaticperformanceindexthatneedstobeashomogeneousaspossiblethroughouttheworkspace.Theintervalanalysisbasedmethodtakesthesetwocriteriaintoaccount:onthebasisofprescribedkinetostaticperformances,theworkspaceisanalysedtofindoutthelargestregulardextrousworkspaceenclosedintheCartesianworkspace.Analgorithmdescribingthismethodisintroduced.Two3-DOFtranslationalparallelmechanismsdesignedformachiningapplicationsarecomparedusingthismethod.Thefirstmachinefeaturesthreefixedlinearjointswhicharemountedorthogonallyandthesecondonefeaturesthreelinearjointswhicharemountedinparallel.Inbothcases,themobileplatformmovesintheCartesianx−y−zspacewithfixedorientation.Keywords:Parallelkinematicmachine,Design,Intervalanalysis,Comparison,Workspace,Transmissionfactors.∗IRCCyN:UMRn◦6597CNRS,EcoleCentraledeNantes,Universit´edeNantes,EcoledesMinesdeNantes11IntroductionParallelkinematicmachines(PKM)areknownfortheirhighdynamicperformancesandlowpositioningerrors.ThekinematicdesignofPKMhasdrawntheinterestofseveralresearchers.Theworkspaceisusuallyconsideredasarelevantdesigncriterion[1,2,3].Parallelsingularities[4]occurintheworkspacewherethemovingplatformcannotresistanyeffort.Thusareveryundesirableandgenerallyeliminatedbydesign.TheJacobianmatrix,whichrelatesthejointratestotheoutputvelocitiesisgenerallynotconstantandnotisotropic.Consequently,theperformances(e.g.maximumspeeds,forces,accuracyandstiffness)varyconsiderablyfordifferentpointsintheCartesianworkspaceandfordifferentdirectionsatonegivenpoint.Thisisaseriousdrawbackformachiningapplications[5,6,7].Fewparallelmechanismsareisotropicthroughouttheworkspace[8,9].Buttheirlowstructuralstiffnessmaketheminadequateformachiningapplicationsbecausetheirlegsaresubjecttobending.Tobeofinterestformachiningapplications,aPKMshouldpreservegoodworkspaceproperties,thatis,regularworkspaceshapeandacceptablekinetostaticperformancesthroughout.Forexampleinmillingappli-cations,themachiningconditionsmustremainconstantalongthewholetoolpath[10,11].Inmanyresearchpapers,thiscriterionisnottakenintoaccountinthealgorithmicmethodsusedtocomputetheworkspacevolume[12,13].Otherpaperspresentmethodsthatcomputethewell-conditionedworkspaceusingdiscretiza-tion[14,15].Thus,theresultstheyprovidecannotbeprovedformally.Conversely,intervalanalysismethodsappliedtowell-conditionedworkspacecomputationprovideguaranteedresults[16,17].ThecomparisonofPKMarchitecturesisadifficultbutrelevantchallenge[7,18].Providingtoolstoallowdesignersorend-userstorigorouslycomparePKMisindeednecessarysincethevarietyofexistingPKMmakesithardtochoosewhichoneisbest-suitedforaspecifictask.Inthispaper,anintervalanalysisbasedmethodisaddressedforthedesignandcomparisonof3-DOFPKM.Thismethodtakesintoaccounttwocriteria,(i)aregularworkspaceshapeand,(ii)akinetostaticperformanceindexthatneedstobeashomogeneousaspossiblethroughouttheworkspace.Twobasictoolsandanalgorithmthatconsidersthesetwocriteriaareintroduced:onthebasisofprescribedkinetostaticperformances,theworkspaceisanalyzedtofindoutthelargestregulardextrousworkspace(square,cube,cylinder,etc...)enclosedintheCartesianworkspace.TwotranslationalparallelmechanismsderivedfromtheDeltarobot[2]arecomparedusingthismethod.Thefirstmachine,calledOrthoglide[19],featuresthreefixedlinearjointswhicharemountedorthogonallyandthesecondone,calledUraneSX(RenaultAutomation)[20],featuresthreelinearjointswhicharemountedinparallel.Inbothcases,themobileplatformmovesintheCartesianx−y−zspacewithfixedorientation.Nextsectionpresentstheintervalanalysisbasedmethodfor3-DOFPKMdesign.Section3presentstheOrthoglideandUraneSXmechanisms,theirkinematicequationsandsingularityanalysis.Section4reportsthecomparisonbetweenthetwomechanismsthroughthedeterminationofthelargestdextrouscubefortheOrthoglideandthelargestdextroussquarefortheUraneSXenclosedintheworkspace.22Descriptionoftheintervalanalysisbasedmethodfor3-DOFtrans-lationalPKMdesign2.1Preliminaries2.1.1DextrousCartesianworkspaceFora3-axisserialmachine-tool,aparallelepiped-shapedCartesianworkspaceallowstheend-usertovisualizeeasilywheretoplacecuttingpaths.ThisconsiderationshouldalsoholdforPKM.Howevertheworkspaceshapeisoftengeometricallycomplexandthushardtovisualize.Therefore,aregular-shapedworkspaceisneededforPKM.Thus,weneedtodefinearegulardextrousworkspacewhichisaregular-shapedworkspaceincludedinthemachineCartesianworkspace.Throughoutthedextrousworkspace,akinetostaticperformanceindex(thatischosenbeforeha