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TSINGHUASCIENCEANDTECHNOLOGYISSN1007-021406/18pp269-275Volume12,Number3,June2007OptimalKinematicDesignofa2-DOFPlanarParallelManipulator*WUJun(吳軍),LITiemin(李鐵民),LIUXinjun(劉辛軍),WANGLiping(王立平)InstituteofManufacturingEngineering,DepartmentofPrecisionInstrumentsandMechanology,TsinghuaUniversity,Beijing100084,ChinaAbstract:Closed-formsolutionsweredevelopedtooptimizekinematicsdesignofa2-degree-of-freedom(2-DOF)planarparallelmanipulator.Theoptimumdesignbasedontheworkspacewaspresented.Meanwhile,aglobal,comprehensiveconditioningindexwasintroducedtoevaluatethekinematicdesigns.Theoptimalparallelmanipulatorisincorporatedintoa5-DOFhybridmachinetoolwhichincludesa2-DOFrotationalmill-ingheadandalongmovementworktable.Theresultsshowthattheplanarparallelmanipulator-basedma-chinetoolcanbesuccessfullyusedtomachinebladesandguidevanesforahydraulicturbine.Keywords:planarparallelmanipulator；globalconditioningindex；hybridmachinetoolIntroductionParallelmechanismsarecapableofveryfastandaccu-ratemotion,possesshigheraveragestiffnesscharacter-isticsthroughouttheirworkspace,havelowerinertia,andcanmanipulateheavierpayloadsthantheirserialcounterparts.Therefore,parallelmechanismshavebeenstudiedextensivelybynumerousresearchersinmanufacturing,primarilyasplatformsforcomputernumericalcontrolmachining.TheGough-Stewartplat-formwasthemostpopularparallelkinematicmachineconfigurationwhenparallelkinematicmachinewerefirstdeveloped.Applicationswithintheroboticscom-munityrangefromhigh-speedmanipulation1toforce-torquesensing2.ButtheGough-Stewartplatformhassomedisadvantagesformanufacturingapplicationssuchasitsrelativelysmallusefulworkspace,com-plexdirectkinematics,anddesigndifficulties.Parallelmanipulatorswithlessthan6DOFsarerelativelyeasytodesignandtheirkinematicscanbedescribedinclosedform.Therefore,parallelmanipula-torswithlessthan6DOFs,especially2or3DOFs,haveincreasinglyattractedattention3-5.Parallelma-nipulatorswith2or3translationalDOFsplayimpor-tantrolesinindustryandcanbeappliedinparallelkinematicsmachines6,pickandplaceapplications7,andotherfields.Mostexisting2-DOFplanarparallelmanipulatorsarethewell-knownfive-barmechanismwithprismaticactuatorsorrevoluteactuators8,9.Themanipulatoroutputisthetranslationalmotionofapointontheend-effectorandtheend-effectororienta-tioncannotremainconstant.Thekinematicdesignmethodologyisoneofthekeypartsofkinematicdesigntheoryforparallelmecha-nisms.Theoptimumdesigncanbebasedonvariousevaluationcriteriainvolvingstiffness10,dexterity,oraglobalconditioningindex11,12.Hence,therehasnotbeenonewidelyaccepteddesignmethod.Huangetal.13presentedahybridmethodbasedonacondition-ingindex,whileLiuetal.14gaveaglobalstiffnessReceived:2006-06-02；revised:2006-08-21SupportedbytheNationalNaturalScienceFoundationofChina(No.50305016)andtheNationalHigh-TechResearchandDevel-opment(863)ProgramofChina(Nos.2004AA424120and2005AA424223)Towhomcorrespondenceshouldbeaddressed.E-mail:；Tel:86-10-62792792TsinghuaScienceandTechnology,June2007,12(3):269-275270indexsimilartotheglobalconditioningindex.ThispaperdescribestheanalysisofaplanarparallelmanipulatorwithtwotranslationalDOFswhichdiffersfromtheconventionalfive-barmechanisminthatapar-allelogramstructureisusedineachchain.Thekinemat-icsareanalyzedtogetanoptimumkinematicdesignbyminimizingaglobal,comprehensiveconditioningindex.Theresultsgivetheoptimallinklengths.A5-DOFhybridmachinetoolwasdevelopedwithaserial-parallelarchitecturewithaparallelmanipulatorcombinedwitha1-DOFtranslationalworktableanda2-DOFrotationalmillinghead.Thehybridmachinetoolusingtheplanarparallelmanipulatorhasbeenusedtomillthebladesandguidevanesofahydraulicturbinetoshowthatthehybridmachinetoolissuitableforthemanufacturingindustry1StructureDescriptionThe2-DOFparallelmanipulatorisshowninFig.1.Themechanismiscomposedofagantryframe,amov-ingplatform,twoactivesliders,andtwokinematicchains.Eachchainisbuiltasaparallelogram.Asde-signed,themanipulatorisover-constrainedsinceoneparallelogramlinkandanothersinglelinkareenoughforthemovingplatformtoposses2translationalDOFs.Twoparallelogramchainswereusedtoincreasethestiffnessandmakethestructuresymmetric.Fig.1KinematicmodelCounterweights1Pand2P(seeFig.1)wereaddedtothemechanismtoimprovetheloadcapacityandac-celerationoftheactuator.Theslidersaredriveninde-pendentlybytwoservomotorsonthecolumnstoslidealongtheguidewaysmountedonthecolumns,thusmovingplatformwitha2-DOFpurelytranslationalmotioninaplane.2KinematicsandSingularities2.1InversekinematicsSincethemotionsoftwolinksofeachkinematicchainareidenticalduetotheparallelogramstructure,thechainmodelcanbesimplifiedasalinkiiAB(1,2i=)asillustratedinFig.1.Thebasecoordinatesystem-Oxyisattachedtothebasewithitsyaxisverticalthroughthemidpointof12BB.Amovingcoordinatesystem-Oxyisfixedonthemovingplatform.iArandiBrarethepositionvectorsofthejointpositionsiAandiB,respectively.2risthemovingplatformwidthand2Risthewidthbetweenthetwocolumns.ThepositionvectoroftheoriginOwithrespecttothecoordinatesystem-OxyisdefinedasTOxy=r(1)ThepositionvectorofjointpositioniAin-Oxyis1T0Ar=r(2)2T0Ar=r(3)ThenthepositionvectorofiAinthebasecoordi-natesystem-OxycanbeexpressedasiiAOA=+rrr(4)ThepositionvectorofeachjointpositioniBin-Oxyis1T1BRy=r(5)2T2BRy=r(6)Thus,theconstraintequationassociatedwiththei-thkinematicchaincanbewrittenasiiABiil=rrn,1,2i=(7)whereilandindenotethelengthandtheunitvectorofthei-thlink,respectively.Takingthe2-normofbothsidesofEq.(7)gives()2211yylxrR=+(8)()2222yylxrR=+(9)FortheconfigurationshowninFig.1,theinversesolutionsofthekinematicsare()2211yylxrR=+(10)WUJun(吳軍)etal：OptimalKinematicDesignofa2-DOFPlanar271()2222yylxrR=+(11)FromEqs.(10)and(11),thesolutionsforthedirectkinematicsofthemanipulatorcanbeexpressedas222221112(22)4(1)()2(1)afyfalyfya+=+(12)where12,2()yyaRr=2221,4()yybRr=2212,4()llcRr=,fRrbc=+xaybc=+(13)FortheconfigurationasshowninFig.1,the“”ofEq.(12)shouldbeonly“”.Equations(7)-(13)showthatthedirectandinversekinematicsofthemanipulatorcanbedescribedinclosedform.2.2SingularityanalysisTakingthederivativesofEq.(10)andEq.(11)withre-specttotimegives()1221xrRyyxlxrR+=+(14)()2222xrRyyxlxrR+=+(15)Equations(14)and(15)canberearrangedinmatrixformas12yxyy=J(16)whereJistheJacobianexpressedas()()22122211xrRlxrRxrRlxrR+=+J(17)Becausesingularitiesleadtoalossofcontrollabilityanddegradationofthenaturalstiffnessofthemanipu-lators,theymustbeavoidedinthetaskworkspace.Singularitiescanbeclassifiedasdirectkinematicsin-gularities,inversekinematicsingularities,andcom-binedsingularities15,andcanbedistinguishedbythemanipulatorJacobian.Whenoneofthelinksishori-zontal,themanipulatorexperiencesaninversekine-maticsingularity.Directkinematicsingularitiesoccurwhenonelinkofachainandalinkoftheotherchainarecollinear.Since122llR+,combinedsingularitiescannotoc-curinthismanipulator.Figure2showsoneexampleofeachkindofsingularity.Inpracticalapplications,sin-gularitiesareavoidedbylimitingthetaskworkspace.Fig.2Singularconfigurations3WorkspaceAnalysisTheworkspaceforthe2-DOFplanarparallelmanipu-latorisaregionoftheplanederivedbytheworkspaceofthereferencepointOofthemovingplatform.Equations(10)and(11)canberewrittenas()22211()xrRyyl+=(18)()222()xrRyyl+=(19)Therefore,thereachableworkspaceofthereferencepointOistheintersectionofthesub-workspacesassociatedwiththetwokinematicchainsasshowninFig.3.Fig.3ManipulatorworkspaceThetaskworkspaceisapartofthereachablework-space.Inpracticalapplications,thetaskworkspaceisusuallydefinedasarectangularareainthereachableworkspace.TsinghuaScienceandTechnology,June2007,12(3):269-275272Letthemaximumlimitoftheanglesand,whicharetheanglesbetweenlinkiiAB(1,2i=)andtheverticalaxis,bedenotedbymaxandmax.Let,maxiyand,miniyrepresentthemaximumandmini-mumpositionsofthei-thslider.Oreachespoint1Qwhenslider1Breachesitslowerlimitandthevalueofisthemaximum,namely11,minyy=andmax=.Similarly,Oreachespoint4Qwhen22,minyy=andmax=.Averticallinethrough1Qintersectswiththeupperboundofthereachableworkspaceatpoint2Q.3Qisdirectlyabove4Q(seeFig.3).Theregion1234QQQQthenmakesupthetaskworkspace,asarectangleofwidthbandheighth,denotedbytW.4OptimalKinematicDesign4.1OptimaldesignbasedontheworkspaceTheobjectiveofthissectionistodeterminethema-nipulatorparametersforadesiredworkspace.Thescopeofoptimaldesigncanbestatedas:givenr,b,andhoftW,determine,R1,l2l,andthetotaljour-ney,max,miniiyyoftheslider.FromEqs.(10),(11),and(17),themanipulatorper-formanceisrelatedtoRrbutnottororRalone.Practically,rshouldbeassmallaspossiblesincesmallervaluesofrleadtosmallermanipulatorvolumes.Usually,rdependsontheshaft,bearing,andtooldimensionsonthemovingplatform.Therefore,rshouldbegivenbythedesigner.Whenthemovingplatformreachesthelowerlimit,asshowninFig.4,thefollowingparametricrelation-shipscanbeobtainedmax1sindbrl+=(20)min2sindrl=(21)wheredisthedistancefromtheleftcolumntotheleftlimitofthetaskworkspace.Inpracticalapplications,1lshouldequal2ltoimprovethe