Workspace Analysis of the Parallel Module of the V

WorkspaceAnalysisoftheParallelModuleoftheVERNEMachine,draftpaperproposedtotheJournalIFToMMProblemsofAppliedMechanics,D.Kanaan,P.WengerandD.Chablat,November2006.p1WorkspaceAnalysisoftheParallelModuleoftheVERNEMachineDanielKanaan,PhilippeWenger,andDamienChablatInstitutdeRechercheenCommunicationsetCybernétiquedeNantes,U.M.R.C.N.R.S.65971,ruedelaNoë,BP92101,44321NantesCedex03FranceAbstract—Thepaperaddressesgeometricaspectsofaspatialthree-degree-of-freedomparallelmodule,whichistheparallelmoduleofahybridserial-parallel5-axismachinetool.Thisparallelmoduleconsistsofamovingplatformthatisconnectedtoafixedbasebythreenon-identicallegs.Eachlegismadeupofoneprismaticandtwopairsofsphericaljoint,whichareconnectedinawaythatthecombinedeffectsofthethreelegsleadtoanover-constrainedmechanismwithcomplexmotion.Thismotionisdefinedasasimultaneouscombinationofrotationandtranslation.AmethodforcomputingthecompleteworkspaceoftheVERNEparallelmoduleforvarioustoollengthsispresented.Analgorithmdescribingthismethodisalsointroduced.Keywords—Parallelmanipulators,parallelkinematicmachines,workspace,hybridmachinetools,complexmotion,mobilityanalysis,inversekinematics,singularity.I.INTRODUCTIONTheworkspacecalculationofaparallelmanipulatorisveryimportantforthedesignerandfortheend-user.Ifweconsideraserialrobot,therepresentationoftheworkspaceisgenerallybasedontheillustrationin3dimensionsofthespacereachablebythecenterofitswrist(characterizingtranslations)andbythespacereachablebytheextremityoftheterminallink(characterizingorientations),thesetwozonesbeinguncoupled.Unfortunately,itisnotthecaseforparallelrobots:thezonereachablebythecenterofthemovingplatformisdependentontheorientationofitsplatform.Thusagraphicalrepresentationoftheworkspaceofparallelmanipulatorswithmorethanthreedegreesoffreedomisonlypossibleifwefixparametersrepresentingtheexceededdegreesoffreedom.Asconsequence,differenttypesofworkspacewereusedintheliterature,accordingtothechoiceofthepresentedparameters[1].Severalmethodsmaybeusedtocalculatetheworkspaceofaparallelmanipulator.Onecanmostlydistinguishbetweendiscretizationmethods,geometricalmethods,andanalyticalmethods.Asimplewayfordeterminingtheworkspaceofaparallelmanipulatoristouseadiscretizationmethod.Inthismethod,agridofnodeswithpositionandorientationisdefined.Theneachnodeistestedtoseewhetheritbelongstotheworkspaceornot[2,3].Thediscretizationalgorithmtakesintoaccountallconstraintsanditissimpletoimplementbutishassomeseriousdrawbacks.Itisexpensiveincomputationaltimeandtheaccuracydependsonthesamplingstepthatisusedtocreatethegrid[4].Geometricalmethodsaremostlyusedtodeterminetheboundaryoftheworkspace.Theprincipleistodefinegeometricalmodelsfortheconstraintsthatlimittheworkspaceoftheparallelmanipulator[5].ThesemodelsareobtainedWorkspaceAnalysisoftheParallelModuleoftheVERNEMachine,draftpaperproposedtotheJournalIFToMMProblemsofAppliedMechanics,D.Kanaan,P.WengerandD.Chablat,November2006.p2foreachlegseparatelyandtheworkspaceistheintersectionbetweenthesemodels[1].Analyticalmethodsaremoredifficulttoapplybecausetheyincreasethedimensionoftheproblembyintroducingsupplementaryvariables.Theyconsistinsolvinganoptimizationproblemwithpenaltiesattheborders[6].Figure1:OverallViewoftheVERNEMachineParallelkinematicmachines(PKM)arecommonlyclaimedtoofferseveraladvantagesovertheirserialcounterparts[7],suchashighstructuralrigidity,betterpayload-to-weightratio,highdynamiccapacitiesandhighaccuracy[1,8].Thus,theyareprudentlyconsideredaspromisingalternativesforhigh-speedmachiningandhavegainedessentialattentionofanumberofcompaniesandresearchers.Sincethefirstprototypepresentedin1994duringtheIMTSinChicagobyGiddingandLewis(theVARIAX)[9],manyotherparallelmanipulatorshaveappeared.However,mostoftheexistingPKMstillsufferfromalimitedrangeofmotion[10].ThisdrawbackcanbediminishedbydesigningahybridmanipulatorasfortheVERNEmachine,whichisa5-axismachine-toolbuiltbyFatronikforIRCCyN[11].Thismachine-toolconsistsofaparallelmoduleandatiltingtableasshowninFig.1.Theparallelmodulemovesthespindlemostlyintranslationwhilethetiltingtableisusedtorotatetheworkpieceabouttwoorthogonalaxes.WorkspaceAnalysisoftheParallelModuleoftheVERNEMachine,draftpaperproposedtotheJournalIFToMMProblemsofAppliedMechanics,D.Kanaan,P.WengerandD.Chablat,November2006.p3AsimplifiedworkspacemodeloftheVernemachineisusedcurrently,butthismodelisreducedwithrespecttotherealone.ThepurposeofthispaperistocalculatetherealworkspacetoenhancetheworkingcapabilityandtoimprovetheproductivityoftheVERNEmachine.Inthefollowingsection,wepresenttheVERNEparallelmoduleandweformulateitsgeometricequations.SectionΙΙΙisdevotedtothecalculationofthecompleteworkspaceforvarioustoollengthsoftheVERNEparallelmodule.Inthissectionwedefinegeometricmodelsforconstraintslimitingtheworkspace.Thenweapplyacombinationofgeometricanddiscretizationmethodsinordertocalculatethecompleteworkspace.FinallyaconclusionisgiveninsectionIV.II.DESCRIPTIONOFTHEVERNEPARALLELMODULEA.ParallelmanipulatorstructureFigure2showsaschemeoftheparallelmoduleoftheVERNEmachine.Theverticesofthemovingpl