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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,CHINAABSTRACTCLOSEDFORMSOLUTIONSWEREDEVELOPEDTOOPTIMIZEKINEMATICSDESIGNOFA2DEGREEOFFREEDOM2DOFPLANARPARALLELMANIPULATORTHEOPTIMUMDESIGNBASEDONTHEWORKSPACEWASPRESENTEDMEANWHILE,AGLOBAL,COMPREHENSIVECONDITIONINGINDEXWASINTRODUCEDTOEVALUATETHEKINEMATICDESIGNSTHEOPTIMALPARALLELMANIPULATORISINCORPORATEDINTOA5DOFHYBRIDMACHINETOOLWHICHINCLUDESA2DOFROTATIONALMILLINGHEADANDALONGMOVEMENTWORKTABLETHERESULTSSHOWTHATTHEPLANARPARALLELMANIPULATORBASEDMACHINETOOLCANBESUCCESSFULLYUSEDTOMACHINEBLADESANDGUIDEVANESFORAHYDRAULICTURBINEKEYWORDSPLANARPARALLELMANIPULATOR;GLOBALCONDITIONINGINDEX;HYBRIDMACHINETOOLINTRODUCTIONPARALLELMECHANISMSARECAPABLEOFVERYFASTANDACCURATEMOTION,POSSESSHIGHERAVERAGESTIFFNESSCHARACTERISTICSTHROUGHOUTTHEIRWORKSPACE,HAVELOWERINERTIA,ANDCANMANIPULATEHEAVIERPAYLOADSTHANTHEIRSERIALCOUNTERPARTSTHEREFORE,PARALLELMECHANISMSHAVEBEENSTUDIEDEXTENSIVELYBYNUMEROUSRESEARCHERSINMANUFACTURING,PRIMARILYASPLATFORMSFORCOMPUTERNUMERICALCONTROLMACHININGTHEGOUGHSTEWARTPLATFORMWASTHEMOSTPOPULARPARALLELKINEMATICMACHINECONFIGURATIONWHENPARALLELKINEMATICMACHINEWEREFIRSTDEVELOPEDAPPLICATIONSWITHINTHEROBOTICSCOMMUNITYRANGEFROMHIGHSPEEDMANIPULATION1TOFORCETORQUESENSING2BUTTHEGOUGHSTEWARTPLATFORMHASSOMEDISADVANTAGESFORMANUFACTURINGAPPLICATIONSSUCHASITSRELATIVELYSMALLUSEFULWORKSPACE,COMPLEXDIRECTKINEMATICS,ANDDESIGNDIFFICULTIESPARALLELMANIPULATORSWITHLESSTHAN6DOFSARERELATIVELYEASYTODESIGNANDTHEIRKINEMATICSCANBEDESCRIBEDINCLOSEDFORMTHEREFORE,PARALLELMANIPULATORSWITHLESSTHAN6DOFS,ESPECIALLY2OR3DOFS,HAVEINCREASINGLYATTRACTEDATTENTION35PARALLELMANIPULATORSWITH2OR3TRANSLATIONALDOFSPLAYIMPORTANTROLESININDUSTRYANDCANBEAPPLIEDINPARALLELKINEMATICSMACHINES6,PICKANDPLACEAPPLICATIONS7,ANDOTHERFIELDSMOSTEXISTING2DOFPLANARPARALLELMANIPULATORSARETHEWELLKNOWNFIVEBARMECHANISMWITHPRISMATICACTUATORSORREVOLUTEACTUATORS8,9THEMANIPULATOROUTPUTISTHETRANSLATIONALMOTIONOFAPOINTONTHEENDEFFECTORANDTHEENDEFFECTORORIENTATIONCANNOTREMAINCONSTANTTHEKINEMATICDESIGNMETHODOLOGYISONEOFTHEKEYPARTSOFKINEMATICDESIGNTHEORYFORPARALLELMECHANISMSTHEOPTIMUMDESIGNCANBEBASEDONVARIOUSEVALUATIONCRITERIAINVOLVINGSTIFFNESS10,DEXTERITY,ORAGLOBALCONDITIONINGINDEX11,12HENCE,THEREHASNOTBEENONEWIDELYACCEPTEDDESIGNMETHODHUANGETAL13PRESENTEDAHYBRIDMETHODBASEDONACONDITIONINGINDEX,WHILELIUETAL14GAVEAGLOBALSTIFFNESS﹡﹡﹡RECEIVED20060602;REVISED20060821SUPPORTEDBYTHENATIONALNATURALSCIENCEFOUNDATIONOFCHINANO50305016ANDTHENATIONALHIGHTECHRESEARCHANDDEVELOPMENT863PROGRAMOFCHINANOS2004AA424120AND2005AA424223TOWHOMCORRESPONDENCESHOULDBEADDRESSEDEMAILLITMMAILTSINGHUAEDUCN;TEL861062792792TSINGHUASCIENCEANDTECHNOLOGY,JUNE2007,123269275270INDEXSIMILARTOTHEGLOBALCONDITIONINGINDEXTHISPAPERDESCRIBESTHEANALYSISOFAPLANARPARALLELMANIPULATORWITHTWOTRANSLATIONALDOFSWHICHDIFFERSFROMTHECONVENTIONALFIVEBARMECHANISMINTHATAPARALLELOGRAMSTRUCTUREISUSEDINEACHCHAINTHEKINEMATICSAREANALYZEDTOGETANOPTIMUMKINEMATICDESIGNBYMINIMIZINGAGLOBAL,COMPREHENSIVECONDITIONINGINDEXTHERESULTSGIVETHEOPTIMALLINKLENGTHSA5DOFHYBRIDMACHINETOOLWASDEVELOPEDWITHASERIALPARALLELARCHITECTUREWITHAPARALLELMANIPULATORCOMBINEDWITHA1DOFTRANSLATIONALWORKTABLEANDA2DOFROTATIONALMILLINGHEADTHEHYBRIDMACHINETOOLUSINGTHEPLANARPARALLELMANIPULATORHASBEENUSEDTOMILLTHEBLADESANDGUIDEVANESOFAHYDRAULICTURBINETOSHOWTHATTHEHYBRIDMACHINETOOLISSUITABLEFORTHEMANUFACTURINGINDUSTRY1STRUCTUREDESCRIPTIONTHE2DOFPARALLELMANIPULATORISSHOWNINFIG1THEMECHANISMISCOMPOSEDOFAGANTRYFRAME,AMOVINGPLATFORM,TWOACTIVESLIDERS,ANDTWOKINEMATICCHAINSEACHCHAINISBUILTASAPARALLELOGRAMASDESIGNED,THEMANIPULATORISOVERCONSTRAINEDSINCEONEPARALLELOGRAMLINKANDANOTHERSINGLELINKAREENOUGHFORTHEMOVINGPLATFORMTOPOSSES2TRANSLATIONALDOFSTWOPARALLELOGRAMCHAINSWEREUSEDTOINCREASETHESTIFFNESSANDMAKETHESTRUCTURESYMMETRICFIG1KINEMATICMODELCOUNTERWEIGHTS1PAND2PSEEFIG1WEREADDEDTOTHEMECHANISMTOIMPROVETHELOADCAPACITYANDACCELERATIONOFTHEACTUATORTHESLIDERSAREDRIVENINDEPENDENTLYBYTWOSERVOMOTORSONTHECOLUMNSTOSLIDEALONGTHEGUIDEWAYSMOUNTEDONTHECOLUMNS,THUSMOVINGPLATFORMWITHA2DOFPURELYTRANSLATIONALMOTIONINAPLANE2KINEMATICSANDSINGULARITIES21INVERSEKINEMATICSSINCETHEMOTIONSOFTWOLINKSOFEACHKINEMATICCHAINAREIDENTICALDUETOTHEPARALLELOGRAMSTRUCTURE,THECHAINMODELCANBESIMPLIFIEDASALINKIIAB1,2IASILLUSTRATEDINFIG1THEBASECOORDINATESYSTEMOXYISATTACHEDTOTHEBASEWITHITSYAXISVERTICALTHROUGHTHEMIDPOINTOF12BBAMOVINGCOORDINATESYSTEMOXY′′′ISFIXEDONTHEMOVINGPLATFORMIARANDIBRARETHEPOSITIONVECTORSOFTHEJOINTPOSITIONSIAANDIB,RESPECTIVELY2RISTHEMOVINGPLATFORMWIDTHAND2RISTHEWIDTHBETWEENTHETWOCOLUMNSTHEPOSITIONVECTOROFTHEORIGINO′WITHRESPECTTOTHECOORDINATESYSTEMOXYISDEFINEDASTOXY′R1THEPOSITIONVECTOROFJOINTPOSITIONIAINOXY′′′IS1T0AR′−R22T0AR′R3THENTHEPOSITIONVECTOROFIAINTHEBASECOORDINATESYSTEMOXYCANBEEXPRESSEDASIIAOA′′RRR4THEPOSITIONVECTOROFEACHJOINTPOSITIONIBINOXYIS1T1BRY−R52T2BRYR6THUS,THECONSTRAINTEQUATIONASSOCIATEDWITHTHEITHKINEMATICCHAINCANBEWRITTENASIIABIIL−RRN,1,2I7WHEREILANDINDENOTETHELENGTHANDTHEUNITVECTOROFTHEITHLINK,RESPECTIVELYTAKINGTHE2NORMOFBOTHSIDESOFEQ7GIVES2211YYLXRR−−82222YYLXRR−−9FORTHECONFIGURATIONSHOWNINFIG1,THEINVERSESOLUTIONSOFTHEKINEMATICSARE2211YYLXRR−−10WUJUN吴军ETALOPTIMALKINEMATICDESIGNOFA2DOFPLANAR2712222YYLXRR−−11FROMEQS10AND11,THESOLUTIONSFORTHEDIRECTKINEMATICSOFTHEMANIPULATORCANBEEXPRESSEDAS222221112224121AFYFALYFYA−−−−12WHERE12,2YYARR−−2221,4YYBRR−−2212,4LLCRR−−,FRRBC−XAYBC13FORTHECONFIGURATIONASSHOWNINFIG1,THE“”OFEQ12SHOULDBEONLY“-”EQUATIONS713SHOWTHATTHEDIRECTANDINVERSEKINEMATICSOFTHEMANIPULATORCANBEDESCRIBEDINCLOSEDFORM22SINGULARITYANALYSISTAKINGTHEDERIVATIVESOFEQ10ANDEQ11WITHRESPECTTOTIMEGIVES1221XRRYYXLXRR−−−−142222XRRYYXLXRR−−−−15EQUATIONS14AND15CANBEREARRANGEDINMATRIXFORMAS12YXYY⎡⎤⎡⎤⎢⎥⎢⎥⎣⎦⎣⎦J16WHEREJISTHEJACOBIANEXPRESSEDAS22122211XRRLXRRXRRLXRR−⎡⎤−⎢⎥−−−−⎢⎥−−⎣⎦J17BECAUSESINGULARITIESLEADTOALOSSOFCONTROLLABILITYANDDEGRADATIONOFTHENATURALSTIFFNESSOFTHEMANIPULATORS,THEYMUSTBEAVOIDEDINTHETASKWORKSPACESINGULARITIESCANBECLASSIFIEDASDIRECTKINEMATICSINGULARITIES,INVERSEKINEMATICSINGULARITIES,ANDCOMBINEDSINGULARITIES15,ANDCANBEDISTINGUISHEDBYTHEMANIPULATORJACOBIANWHENONEOFTHELINKSISHORIZONTAL,THEMANIPULATOREXPERIENCESANINVERSEKINEMATICSINGULARITYDIRECTKINEMATICSINGULARITIESOCCURWHENONELINKOFACHAINANDALINKOFTHEOTHERCHAINARECOLLINEARSINCE122LLR,COMBINEDSINGULARITIESCANNOTOCCURINTHISMANIPULATORFIGURE2SHOWSONEEXAMPLEOFEACHKINDOFSINGULARITYINPRACTICALAPPLICATIONS,SINGULARITIESAREAVOIDEDBYLIMITINGTHETASKWORKSPACEFIG2SINGULARCONFIGURATIONS3WORKSPACEANALYSISTHEWORKSPACEFORTHE2DOFPLANARPARALLELMANIPULATORISAREGIONOFTHEPLANEDERIVEDBYTHEWORKSPACEOFTHEREFERENCEPOINTO′OFTHEMOVINGPLATFORMEQUATIONS10AND11CANBEREWRITTENAS22211XRRYYL−−18222XRRYYL−−19THEREFORE,THEREACHABLEWORKSPACEOFTHEREFERENCEPOINTO′ISTHEINTERSECTIONOFTHESUBWORKSPACESASSOCIATEDWITHTHETWOKINEMATICCHAINSASSHOWNINFIG3FIG3MANIPULATORWORKSPACETHETASKWORKSPACEISAPARTOFTHEREACHABLEWORKSPACEINPRACTICALAPPLICATIONS,THETASKWORKSPACEISUSUALLYDEFINEDASARECTANGULARAREAINTHEREACHABLEWORKSPACETSINGHUASCIENCEANDTECHNOLOGY,JUNE2007,123269275272LETTHEMAXIMUMLIMITOFTHEANGLESΑANDΒ,WHICHARETHEANGLESBETWEENLINKIIAB1,2IANDTHEVERTICALAXIS,BEDENOTEDBYMAXΑANDMAXΒLET,MAXIYAND,MINIYREPRESENTTHEMAXIMUMANDMINIMUMPOSITIONSOFTHEITHSLIDERO′REACHESPOINT1QWHENSLIDER1BREACHESITSLOWERLIMITANDTHEVALUEOFΑISTHEMAXIMUM,NAMELY11,MINYYANDMAXΑΑSIMILARLY,O′REACHESPOINT4QWHEN22,MINYYANDMAXΒΒAVERTICALLINETHROUGH1QINTERSECTSWITHTHEUPPERBOUNDOFTHEREACHABLEWORKSPACEATPOINT2Q3QISDIRECTLYABOVE4QSEEFIG3THEREGION1234QQQQTHENMAKESUPTHETASKWORKSPACE,ASARECTANGLEOFWIDTHBANDHEIGHTH,DENOTEDBYTW4OPTIMALKINEMATICDESIGN41OPTIMALDESIGNBASEDONTHEWORKSPACETHEOBJECTIVEOFTHISSECTIONISTODETERMINETHEMANIPULATORPARAMETERSFORADESIREDWORKSPACETHESCOPEOFOPTIMALDESIGNCANBESTATEDASGIVENR,B,ANDHOFTW,DETERMINE,R1,L2L,ANDTHETOTALJOURNEY,MAX,MINIIYY−OFTHESLIDERFROMEQS10,11,AND17,THEMANIPULATORPERFORMANCEISRELATEDTORR−BUTNOTTORORRALONEPRACTICALLY,RSHOULDBEASSMALLASPOSSIBLESINCESMALLERVALUESOFRLEADTOSMALLERMANIPULATORVOLUMESUSUALLY,RDEPENDSONTHESHAFT,BEARING,ANDTOOLDIMENSIONSONTHEMOVINGPLATFORMTHEREFORE,RSHOULDBEGIVENBYTHEDESIGNERWHENTHEMOVINGPLATFORMREACHESTHELOWERLIMIT,ASSHOWNINFIG4,THEFOLLOWINGPARAMETRICRELATIONSHIPSCANBEOBTAINEDMAX1SINDBRLΑ−20MIN2SINDRLΒ−21WHEREDISTHEDISTANCEFROMTHELEFTCOLUMNTOTHELEFTLIMITOFTHETASKWORKSPACEINPRACTICALAPPLICATIONS,1LSHOULDEQUAL2LTOIMPROVETHESYSTEMPERFORMANCEANDSTIFFNESSTHEREFOREMAXMINMAXMINSINSINSINSINRBRDΑΒΑΒ−−22FIG4OPTIMALDESIGNOFTHEMANIPULATORTHUS,2BRD23FOR12LL,WHENTHEMOVINGPLATFORMMOVESFROMPOINT1QTO4QALONGTHEXAXIS,THESLIDINGDISTANCEOFTHESLIDERINTHEGUIDEWAYSHOULDBE1222211BDDYLRRLRR⎛⎞⎛⎞−−−−−−⎜⎟⎜⎟⎝⎠⎝⎠24WHENTHEMOVINGPLATFORMTRAVELSFROMPOINT4QTO3QALONGTHEYDIRECTION,THESLIDINGDISTANCEOFTHESLIDERIS2BYH25HENCE,THETOTALJOURNEY,MAX,MINIIYY−OFTHESLIDERIS12,MAX,MINIIBBYYYY−26BECAUSETHEOPTIMUMDESIGNBASEDONTHETASKWORKSPACEDOESNOTCONSIDERTHEDEXTERITYANDSTIFFNESSOFTHEMANIPULATOR,THELINKLENGTHSARENOTOPTIMALTHEOPTIMALLENGTHSOFTHELINKSAND,MAX,MINIIYY−FORTHESLIDERAREDETERMINEDINTHENEXTSUBSECTION42GLOBAL,COMPREHENSIVEINDEXTHECONDITIONNUMBEROFTHEJACOBIANISUSEDASTHELOCALPERFORMANCEINDEXFOREVALUATINGTHEVELOCITY,ACCURACY,ANDRIGIDITYMAPPINGCHARACTERISTICSBETWEENTHEJOINTVARIABLESANDTHEMOVINGPLATFORMTHECONDITIONNUMBERΚISDEFINEDAS211ΣΚΣ∞≤≤27WHERE1ΣAND2ΣARETHEMINIMUMANDMAXIMUMSINGULARVALUESOFTHEJACOBIANASSOCIATEDWITHAGIVEN
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