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TSINGHUASCIENCEANDTECHNOLOGYISSN1007021406/18pp269275Volume12,Number3,June2007OptimalKinematicDesignofa2DOFPlanarParallelManipulatorWUJun吴军,LITiemin李铁民∗∗,LIUXinjun刘辛军,WANGLiping王立平InstituteofManufacturingEngineering,DepartmentofPrecisionInstrumentsandMechanology,TsinghuaUniversity,Beijing100084,ChinaAbstractClosedformsolutionsweredevelopedtooptimizekinematicsdesignofa2degreeoffreedom2DOFplanarparallelmanipulator.Theoptimumdesignbasedontheworkspacewaspresented.Meanwhile,aglobal,comprehensiveconditioningindexwasintroducedtoevaluatethekinematicdesigns.Theoptimalparallelmanipulatorisincorporatedintoa5DOFhybridmachinetoolwhichincludesa2DOFrotationalmillingheadandalongmovementworktable.Theresultsshowthattheplanarparallelmanipulatorbasedmachinetoolcanbesuccessfullyusedtomachinebladesandguidevanesforahydraulicturbine.KeywordsplanarparallelmanipulatorglobalconditioningindexhybridmachinetoolIntroductionParallelmechanismsarecapableofveryfastandaccuratemotion,possesshigheraveragestiffnesscharacteristicsthroughouttheirworkspace,havelowerinertia,andcanmanipulateheavierpayloadsthantheirserialcounterparts.Therefore,parallelmechanismshavebeenstudiedextensivelybynumerousresearchersinmanufacturing,primarilyasplatformsforcomputernumericalcontrolmachining.TheGoughStewartplatformwasthemostpopularparallelkinematicmachineconfigurationwhenparallelkinematicmachinewerefirstdeveloped.Applicationswithintheroboticscommunityrangefromhighspeedmanipulation1toforcetorquesensing2.ButtheGoughStewartplatformhassomedisadvantagesformanufacturingapplicationssuchasitsrelativelysmallusefulworkspace,complexdirectkinematics,anddesigndifficulties.Parallelmanipulatorswithlessthan6DOFsarerelativelyeasytodesignandtheirkinematicscanbedescribedinclosedform.Therefore,parallelmanipulatorswithlessthan6DOFs,especially2or3DOFs,haveincreasinglyattractedattention35.Parallelmanipulatorswith2or3translationalDOFsplayimportantrolesinindustryandcanbeappliedinparallelkinematicsmachines6,pickandplaceapplications7,andotherfields.Mostexisting2DOFplanarparallelmanipulatorsarethewellknownfivebarmechanismwithprismaticactuatorsorrevoluteactuators8,9.Themanipulatoroutputisthetranslationalmotionofapointontheendeffectorandtheendeffectororientationcannotremainconstant.Thekinematicdesignmethodologyisoneofthekeypartsofkinematicdesigntheoryforparallelmechanisms.Theoptimumdesigncanbebasedonvariousevaluationcriteriainvolvingstiffness10,dexterity,oraglobalconditioningindex11,12.Hence,therehasnotbeenonewidelyaccepteddesignmethod.Huangetal.13presentedahybridmethodbasedonaconditioningindex,whileLiuetal.14gaveaglobalstiffness﹡﹡﹡Received20060602revised20060821SupportedbytheNationalNaturalScienceFoundationofChinaNo.50305016andtheNationalHighTechResearchandDevelopment863ProgramofChinaNos.2004AA424120and2005AA424223Towhomcorrespondenceshouldbeaddressed.Emaillitmmail.tsinghua.edu.cnTel861062792792TsinghuaScienceandTechnology,June2007,123269275270indexsimilartotheglobalconditioningindex.ThispaperdescribestheanalysisofaplanarparallelmanipulatorwithtwotranslationalDOFswhichdiffersfromtheconventionalfivebarmechanisminthataparallelogramstructureisusedineachchain.Thekinematicsareanalyzedtogetanoptimumkinematicdesignbyminimizingaglobal,comprehensiveconditioningindex.Theresultsgivetheoptimallinklengths.A5DOFhybridmachinetoolwasdevelopedwithaserialparallelarchitecturewithaparallelmanipulatorcombinedwitha1DOFtranslationalworktableanda2DOFrotationalmillinghead.Thehybridmachinetoolusingtheplanarparallelmanipulatorhasbeenusedtomillthebladesandguidevanesofahydraulicturbinetoshowthatthehybridmachinetoolissuitableforthemanufacturingindustry1StructureDescriptionThe2DOFparallelmanipulatorisshowninFig.1.Themechanismiscomposedofagantryframe,amovingplatform,twoactivesliders,andtwokinematicchains.Eachchainisbuiltasaparallelogram.Asdesigned,themanipulatorisoverconstrainedsinceoneparallelogramlinkandanothersinglelinkareenoughforthemovingplatformtoposses2translationalDOFs.Twoparallelogramchainswereusedtoincreasethestiffnessandmakethestructuresymmetric.Fig.1KinematicmodelCounterweights1Pand2PseeFig.1wereaddedtothemechanismtoimprovetheloadcapacityandaccelerationoftheactuator.Theslidersaredrivenindependentlybytwoservomotorsonthecolumnstoslidealongtheguidewaysmountedonthecolumns,thusmovingplatformwitha2DOFpurelytranslationalmotioninaplane.2KinematicsandSingularities2.1InversekinematicsSincethemotionsoftwolinksofeachkinematicchainareidenticalduetotheparallelogramstructure,thechainmodelcanbesimplifiedasalinkiiAB1,2iasillustratedinFig.1.ThebasecoordinatesystemOxyisattachedtothebasewithitsyaxisverticalthroughthemidpointof12BB.AmovingcoordinatesystemOxy′′′isfixedonthemovingplatform.iArandiBrarethepositionvectorsofthejointpositionsiAandiB,respectively.2risthemovingplatformwidthand2Risthewidthbetweenthetwocolumns.ThepositionvectoroftheoriginO′withrespecttothecoordinatesystemOxyisdefinedasTOxy′r1ThepositionvectorofjointpositioniAinOxy′′′is1T0Ar′−r22T0Ar′r3ThenthepositionvectorofiAinthebasecoordinatesystemOxycanbeexpressedasiiAOA′′rrr4ThepositionvectorofeachjointpositioniBinOxyis1T1BRy−r52T2BRyr6Thus,theconstraintequationassociatedwiththeithkinematicchaincanbewrittenasiiABiil−rrn,1,2i7whereilandindenotethelengthandtheunitvectoroftheithlink,respectively.Takingthe2normofbothsidesofEq.7gives2211yylxrR±−−82222yylxrR±−−9FortheconfigurationshowninFig.1,theinversesolutionsofthekinematicsare2211yylxrR−−10WUJun吴军etalOptimalKinematicDesignofa2DOFPlanar2712222yylxrR−−11FromEqs.10and11,thesolutionsforthedirectkinematicsofthemanipulatorcanbeexpressedas222221112224121afyfalyfya−−±−−12where12,2yyaRr−−2221,4yybRr−−2212,4llcRr−−,fRrbc−xaybc13FortheconfigurationasshowninFig.1,the±ofEq.12shouldbeonly-.Equations713showthatthedirectandinversekinematicsofthemanipulatorcanbedescribedinclosedform.2.2SingularityanalysisTakingthederivativesofEq.10andEq.11withrespecttotimegives1221xrRyyxlxrR−−−−142222xrRyyxlxrR−−−−15Equations14and15canberearrangedinmatrixformas12yxyy⎡⎤⎡⎤⎢⎥⎢⎥⎣⎦⎣⎦J16whereJistheJacobianexpressedas22122211xrRlxrRxrRlxrR−⎡⎤−⎢⎥−−−−⎢⎥−−⎣⎦J17Becausesingularitiesleadtoalossofcontrollabilityanddegradationofthenaturalstiffnessofthemanipulators,theymustbeavoidedinthetaskworkspace.Singularitiescanbeclassifiedasdirectkinematicsingularities,inversekinematicsingularities,andcombinedsingularities15,andcanbedistinguishedbythemanipulatorJacobian.Whenoneofthelinksishorizontal,themanipulatorexperiencesaninversekinematicsingularity.Directkinematicsingularitiesoccurwhenonelinkofachainandalinkoftheotherchainarecollinear.Since122llR,combinedsingularitiescannotoccurinthismanipulator.Figure2showsoneexampleofeachkindofsingularity.Inpracticalapplications,singularitiesareavoidedbylimitingthetaskworkspace.Fig.2Singularconfigurations3WorkspaceAnalysisTheworkspaceforthe2DOFplanarparallelmanipulatorisaregionoftheplanederivedbytheworkspaceofthereferencepointO′ofthemovingplatform.Equations10and11canberewrittenas22211xrRyyl−−18222xrRyyl−−19Therefore,thereachableworkspaceofthereferencepointO′istheintersectionofthesubworkspacesassociatedwiththetwokinematicchainsasshowninFig.3.Fig.3ManipulatorworkspaceThetaskworkspaceisapartofthereachableworkspace.Inpracticalapplications,thetaskworkspaceisusuallydefinedasarectangularareainthereachableworkspace.TsinghuaScienceandTechnology,June2007,123269275272Letthemaximumlimitoftheanglesαandβ,whicharetheanglesbetweenlinkiiAB1,2iandtheverticalaxis,bedenotedbymaxαandmaxβ.Let,maxiyand,miniyrepresentthemaximumandminimumpositionsoftheithslider.O′reachespoint1Qwhenslider1Breachesitslowerlimitandthevalueofαisthemaximum,namely11,minyyandmaxαα.Similarly,O′reachespoint4Qwhen22,minyyandmaxββ.Averticallinethrough1Qintersectswiththeupperboundofthereachableworkspaceatpoint2Q.3Qisdirectlyabove4QseeFig.3.Theregion1234QQQQthenmakesupthetaskworkspace,asarectangleofwidthbandheighth,denotedbytW.4OptimalKinematicDesign4.1OptimaldesignbasedontheworkspaceTheobjectiveofthissectionistodeterminethemanipulatorparametersforadesiredworkspace.Thescopeofoptimaldesigncanbestatedasgivenr,b,andhoftW,determine,R1,l2l,andthetotaljourney,max,miniiyy−oftheslider.FromEqs.10,11,and17,themanipulatorperformanceisrelatedtoRr−butnottororRalone.Practically,rshouldbeassmallaspossiblesincesmallervaluesofrleadtosmallermanipulatorvolumes.Usually,rdependsontheshaft,bearing,andtooldimensionsonthemovingplatform.Therefore,rshouldbegivenbythedesigner.Whenthemovingplatformreachesthelowerlimit,asshowninFig.4,thefollowingparametricrelationshipscanbeobtainedmax1sindbrlα−20min2sindrlβ−21wheredisthedistancefromtheleftcolumntotheleftlimitofthetaskworkspace.Inpracticalapplications,1lshouldequal2ltoimprovethesystemperformanceandstiffness.Thereforemaxminmaxminsinsinsinsinrbrdαβαβ−−22Fig.4OptimaldesignofthemanipulatorThus,2bRd23For12ll,whenthemovingplatformmovesfrompoint1Qto4Qalongthexaxis,theslidingdistanceofthesliderintheguidewayshouldbe1222211BddylRrlRr⎛⎞⎛⎞−−−−−−⎜⎟⎜⎟⎝⎠⎝⎠24Whenthemovingplatformtravelsfrompoint4Qto3Qalongtheydirection,theslidingdistanceoftheslideris2Byh25Hence,thetotaljourney,max,miniiyy−oftheslideris12,max,miniiBByyyy−26Becausetheoptimumdesignbasedonthetaskworkspacedoesnotconsiderthedexterityandstiffnessofthemanipulator,thelinklengthsarenotoptimal.Theoptimallengthsofthelinksand,max,miniiyy−fortheslideraredeterminedinthenextsubsection.4.2Global,comprehensiveindexTheconditionnumberoftheJacobianisusedasthelocalperformanceindexforevaluatingthevelocity,accuracy,andrigiditymappingcharacteristicsbetweenthejointvariablesandthemovingplatform.Theconditionnumberκisdefinedas211σκσ∞≤≤27where1σand2σaretheminimumandmaximumsingularvaluesoftheJacobianassociatedwithagiven
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