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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,COMPREHENSIVECONDITIONINGINDEXWASINTRODUCEDTOEVALUATETHEKINEMATICDESIGNSTHEOPTIMALPARALLELMANIPULATORISINCORPORATEDINTOA5DOFHYBRIDMACHINETOOLWHICHINCLUDESA2DOFROTATIONALMILLINGHEADANDALONGMOVEMENTWORKTABLETHERESULTSSHOWTHATTHEPLANARPARALLELMANIPULATORBASEDMACHINETOOLCANBESUCCESSFULLYUSEDTOMACHINEBLADESANDGUIDEVANESFORAHYDRAULICTURBINEKEYWORDSPLANARPARALLELMANIPULATORGLOBALCONDITIONINGINDEXHYBRIDMACHINETOOLINTRODUCTIONPARALLELMECHANISMSARECAPABLEOFVERYFASTANDACCURATEMOTION,POSSESSHIGHERAVERAGESTIFFNESSCHARACTERISTICSTHROUGHOUTTHEIRWORKSPACE,HAVELOWERINERTIA,ANDCANMANIPULATEHEAVIERPAYLOADSTHANTHEIRSERIALCOUNTERPARTSTHEREFORE,PARALLELMECHANISMSHAVEBEENSTUDIEDEXTENSIVELYBYNUMEROUSRESEARCHERSINMANUFACTURING,PRIMARILYASPLATFORMSFORCOMPUTERNUMERICALCONTROLMACHININGTHEGOUGHSTEWARTPLATFORMWASTHEMOSTPOPULARPARALLELKINEMATICMACHINECONFIGURATIONWHENPARALLELKINEMATICMACHINEWEREFIRSTDEVELOPEDAPPLICATIONSWITHINTHEROBOTICSCOMMUNITYRANGEFROMHIGHSPEEDMANIPULATION1TOFORCETORQUESENSING2BUTTHEGOUGHSTEWARTPLATFORMHASSOMEDISADVANTAGESFORMANUFACTURINGAPPLICATIONSSUCHASITSRELATIVELYSMALLUSEFULWORKSPACE,COMPLEXDIRECTKINEMATICS,ANDDESIGNDIFFICULTIESPARALLELMANIPULATORSWITHLESSTHAN6DOFSARERELATIVELYEASYTODESIGNANDTHEIRKINEMATICSCANBEDESCRIBEDINCLOSEDFORMTHEREFORE,PARALLELMANIPULATORSWITHLESSTHAN6DOFS,ESPECIALLY2OR3DOFS,HAVEINCREASINGLYATTRACTEDATTENTION35PARALLELMANIPULATORSWITH2OR3TRANSLATIONALDOFSPLAYIMPORTANTROLESININDUSTRYANDCANBEAPPLIEDINPARALLELKINEMATICSMACHINES6,PICKANDPLACEAPPLICATIONS7,ANDOTHERFIELDSMOSTEXISTING2DOFPLANARPARALLELMANIPULATORSARETHEWELLKNOWNFIVEBARMECHANISMWITHPRISMATICACTUATORSORREVOLUTEACTUATORS8,9THEMANIPULATOROUTPUTISTHETRANSLATIONALMOTIONOFAPOINTONTHEENDEFFECTORANDTHEENDEFFECTORORIENTATIONCANNOTREMAINCONSTANTTHEKINEMATICDESIGNMETHODOLOGYISONEOFTHEKEYPARTSOFKINEMATICDESIGNTHEORYFORPARALLELMECHANISMSTHEOPTIMUMDESIGNCANBEBASEDONVARIOUSEVALUATIONCRITERIAINVOLVINGSTIFFNESS10,DEXTERITY,ORAGLOBALCONDITIONINGINDEX11,12HENCE,THEREHASNOTBEENONEWIDELYACCEPTEDDESIGNMETHODHUANGETAL13PRESENTEDAHYBRIDMETHODBASEDONACONDITIONINGINDEX,WHILELIUETAL14GAVEAGLOBALSTIFFNESSRECEIVED20060602REVISED20060821SUPPORTEDBYTHENATIONALNATURALSCIENCEFOUNDATIONOFCHINANO50305016ANDTHENATIONALHIGHTECHRESEARCHANDDEVELOPMENT863PROGRAMOFCHINANOS2004AA424120AND2005AA424223TOWHOMCORRESPONDENCESHOULDBEADDRESSEDEMAILLITMMAILTSINGHUAEDUCNTEL861062792792TSINGHUASCIENCEANDTECHNOLOGY,JUNE2007,123269275270INDEXSIMILARTOTHEGLOBALCONDITIONINGINDEXTHISPAPERDESCRIBESTHEANALYSISOFAPLANARPARALLELMANIPULATORWITHTWOTRANSLATIONALDOFSWHICHDIFFERSFROMTHECONVENTIONALFIVEBARMECHANISMINTHATAPARALLELOGRAMSTRUCTUREISUSEDINEACHCHAINTHEKINEMATICSAREANALYZEDTOGETANOPTIMUMKINEMATICDESIGNBYMINIMIZINGAGLOBAL,COMPREHENSIVECONDITIONINGINDEXTHERESULTSGIVETHEOPTIMALLINKLENGTHSA5DOFHYBRIDMACHINETOOLWASDEVELOPEDWITHASERIALPARALLELARCHITECTUREWITHAPARALLELMANIPULATORCOMBINEDWITHA1DOFTRANSLATIONALWORKTABLEANDA2DOFROTATIONALMILLINGHEADTHEHYBRIDMACHINETOOLUSINGTHEPLANARPARALLELMANIPULATORHASBEENUSEDTOMILLTHEBLADESANDGUIDEVANESOFAHYDRAULICTURBINETOSHOWTHATTHEHYBRIDMACHINETOOLISSUITABLEFORTHEMANUFACTURINGINDUSTRY1STRUCTUREDESCRIPTIONTHE2DOFPARALLELMANIPULATORISSHOWNINFIG1THEMECHANISMISCOMPOSEDOFAGANTRYFRAME,AMOVINGPLATFORM,TWOACTIVESLIDERS,ANDTWOKINEMATICCHAINSEACHCHAINISBUILTASAPARALLELOGRAMASDESIGNED,THEMANIPULATORISOVERCONSTRAINEDSINCEONEPARALLELOGRAMLINKANDANOTHERSINGLELINKAREENOUGHFORTHEMOVINGPLATFORMTOPOSSES2TRANSLATIONALDOFSTWOPARALLELOGRAMCHAINSWEREUSEDTOINCREASETHESTIFFNESSANDMAKETHESTRUCTURESYMMETRICFIG1KINEMATICMODELCOUNTERWEIGHTS1PAND2PSEEFIG1WEREADDEDTOTHEMECHANISMTOIMPROVETHELOADCAPACITYANDACCELERATIONOFTHEACTUATORTHESLIDERSAREDRIVENINDEPENDENTLYBYTWOSERVOMOTORSONTHECOLUMNSTOSLIDEALONGTHEGUIDEWAYSMOUNTEDONTHECOLUMNS,THUSMOVINGPLATFORMWITHA2DOFPURELYTRANSLATIONALMOTIONINAPLANE2KINEMATICSANDSINGULARITIES21INVERSEKINEMATICSSINCETHEMOTIONSOFTWOLINKSOFEACHKINEMATICCHAINAREIDENTICALDUETOTHEPARALLELOGRAMSTRUCTURE,THECHAINMODELCANBESIMPLIFIEDASALINKIIAB1,2IASILLUSTRATEDINFIG1THEBASECOORDINATESYSTEMOXYISATTACHEDTOTHEBASEWITHITSYAXISVERTICALTHROUGHTHEMIDPOINTOF12BBAMOVINGCOORDINATESYSTEMOXYISFIXEDONTHEMOVINGPLATFORMIARANDIBRARETHEPOSITIONVECTORSOFTHEJOINTPOSITIONSIAANDIB,RESPECTIVELY2RISTHEMOVINGPLATFORMWIDTHAND2RISTHEWIDTHBETWEENTHETWOCOLUMNSTHEPOSITIONVECTOROFTHEORIGINOWITHRESPECTTOTHECOORDINATESYSTEMOXYISDEFINEDASTOXYR1THEPOSITIONVECTOROFJOINTPOSITIONIAINOXYIS1T0ARR22T0ARR3THENTHEPOSITIONVECTOROFIAINTHEBASECOORDINATESYSTEMOXYCANBEEXPRESSEDASIIAOARRR4THEPOSITIONVECTOROFEACHJOINTPOSITIONIBINOXYIS1T1BRYR52T2BRYR6THUS,THECONSTRAINTEQUATIONASSOCIATEDWITHTHEITHKINEMATICCHAINCANBEWRITTENASIIABIILRRN,1,2I7WHEREILANDINDENOTETHELENGTHANDTHEUNITVECTOROFTHEITHLINK,RESPECTIVELYTAKINGTHE2NORMOFBOTHSIDESOFEQ7GIVES2211YYLXRR82222YYLXRR9FORTHECONFIGURATIONSHOWNINFIG1,THEINVERSESOLUTIONSOFTHEKINEMATICSARE2211YYLXRR10WUJUN吴军ETALOPTIMALKINEMATICDESIGNOFA2DOFPLANAR2712222YYLXRR11FROMEQS10AND11,THESOLUTIONSFORTHEDIRECTKINEMATICSOFTHEMANIPULATORCANBEEXPRESSEDAS222221112224121AFYFALYFYA12WHERE12,2YYARR2221,4YYBRR2212,4LLCRR,FRRBCXAYBC13FORTHECONFIGURATIONASSHOWNINFIG1,THE“”OFEQ12SHOULDBEONLY“”EQUATIONS713SHOWTHATTHEDIRECTANDINVERSEKINEMATICSOFTHEMANIPULATORCANBEDESCRIBEDINCLOSEDFORM22SINGULARITYANALYSISTAKINGTHEDERIVATIVESOFEQ10ANDEQ11WITHRESPECTTOTIMEGIVES1221XRRYYXLXRR142222XRRYYXLXRR15EQUATIONS14AND15CANBEREARRANGEDINMATRIXFORMAS12YXYYJ16WHEREJISTHEJACOBIANEXPRESSEDAS22122211XRRLXRRXRRLXRRJ17BECAUSESINGULARITIESLEADTOALOSSOFCONTROLLABILITYANDDEGRADATIONOFTHENATURALSTIFFNESSOFTHEMANIPULATORS,THEYMUSTBEAVOIDEDINTHETASKWORKSPACESINGULARITIESCANBECLASSIFIEDASDIRECTKINEMATICSINGULARITIES,INVERSEKINEMATICSINGULARITIES,ANDCOMBINEDSINGULARITIES15,ANDCANBEDISTINGUISHEDBYTHEMANIPULATORJACOBIANWHENONEOFTHELINKSISHORIZONTAL,THEMANIPULATOREXPERIENCESANINVERSEKINEMATICSINGULARITYDIRECTKINEMATICSINGULARITIESOCCURWHENONELINKOFACHAINANDALINKOFTHEOTHERCHAINARECOLLINEARSINCE122LLR,COMBINEDSINGULARITIESCANNOTOCCURINTHISMANIPULATORFIGURE2SHOWSONEEXAMPLEOFEACHKINDOFSINGULARITYINPRACTICALAPPLICATIONS,SINGULARITIESAREAVOIDEDBYLIMITINGTHETASKWORKSPACEFIG2SINGULARCONFIGURATIONS3WORKSPACEANALYSISTHEWORKSPACEFORTHE2DOFPLANARPARALLELMANIPULATORISAREGIONOFTHEPLANEDERIVEDBYTHEWORKSPACEOFTHEREFERENCEPOINTOOFTHEMOVINGPLATFORMEQUATIONS10AND11CANBEREWRITTENAS22211XRRYYL18222XRRYYL19THEREFORE,THEREACHABLEWORKSPACEOFTHEREFERENCEPOINTOISTHEINTERSECTIONOFTHESUBWORKSPACESASSOCIATEDWITHTHETWOKINEMATICCHAINSASSHOWNINFIG3FIG3MANIPULATORWORKSPACETHETASKWORKSPACEISAPARTOFTHEREACHABLEWORKSPACEINPRACTICALAPPLICATIONS,THETASKWORKSPACEISUSUALLYDEFINEDASARECTANGULARAREAINTHEREACHABLEWORKSPACETSINGHUASCIENCEANDTECHNOLOGY,JUNE2007,123269275272LETTHEMAXIMUMLIMITOFTHEANGLESAND,WHICHARETHEANGLESBETWEENLINKIIAB1,2IANDTHEVERTICALAXIS,BEDENOTEDBYMAXANDMAXLET,MAXIYAND,MINIYREPRESENTTHEMAXIMUMANDMINIMUMPOSITIONSOFTHEITHSLIDEROREACHESPOINT1QWHENSLIDER1BREACHESITSLOWERLIMITANDTHEVALUEOFISTHEMAXIMUM,NAMELY11,MINYYANDMAXSIMILARLY,OREACHESPOINT4QWHEN22,MINYYANDMAXAVERTICALLINETHROUGH1QINTERSECTSWITHTHEUPPERBOUNDOFTHEREACHABLEWORKSPACEATPOINT2Q3QISDIRECTLYABOVE4QSEEFIG3THEREGION1234QQQQTHENMAKESUPTHETASKWORKSPACE,ASARECTANGLEOFWIDTHBANDHEIGHTH,DENOTEDBYTW4OPTIMALKINEMATICDESIGN41OPTIMALDESIGNBASEDONTHEWORKSPACETHEOBJECTIVEOFTHISSECTIONISTODETERMINETHEMANIPULATORPARAMETERSFORADESIREDWORKSPACETHESCOPEOFOPTIMALDESIGNCANBESTATEDASGIVENR,B,ANDHOFTW,DETERMINE,R1,L2L,ANDTHETOTALJOURNEY,MAX,MINIIYYOFTHESLIDERFROMEQS10,11,AND17,THEMANIPULATORPERFORMANCEISRELATEDTORRBUTNOTTORORRALONEPRACTICALLY,RSHOULDBEASSMALLASPOSSIBLESINCESMALLERVALUESOFRLEADTOSMALLERMANIPULATORVOLUMESUSUALLY,RDEPENDSONTHESHAFT,BEARING,ANDTOOLDIMENSIONSONTHEMOVINGPLATFORMTHEREFORE,RSHOULDBEGIVENBYTHEDESIGNERWHENTHEMOVINGPLATFORMREACHESTHELOWERLIMIT,ASSHOWNINFIG4,THEFOLLOWINGPARAMETRICRELATIONSHIPSCANBEOBTAINEDMAX1SINDBRL20MIN2SINDRL21WHEREDISTHEDISTANCEFROMTHELEFTCOLUMNTOTHELEFTLIMITOFTHETASKWORKSPACEINPRACTICALAPPLICATIONS,1LSHOULDEQUAL2LTOIMPROVETHESYSTEMPERFORMANCEANDSTIFFNESSTHEREFOREMAXMINMAXMINSINSINSINSINRBRD22FIG4OPTIMALDESIGNOFTHEMANIPULATORTHUS,2BRD23FOR12LL,WHENTHEMOVINGPLATFORMMOVESFROMPOINT1QTO4QALONGTHEXAXIS,THESLIDINGDISTANCEOFTHESLIDERINTHEGUIDEWAYSHOULDBE1222211BDDYLRRLRR24WHENTHEMOVINGPLATFORMTRAVELSFROMPOINT4QTO3QALONGTHEYDIRECTION,THESLIDINGDISTANCEOFTHESLIDERIS2BYH25HENCE,THETOTALJOURNEY,MAX,MINIIYYOFTHESLIDERIS12,MAX,MINIIBBYYYY26BECAUSETHEOPTIMUMDESIGNBASEDONTHETASKWORKSPACEDOESNOTCONSIDERTHEDEXTERITYANDSTIFFNESSOFTHEMANIPULATOR,THELINKLENGTHSARENOTOPTIMALTHEOPTIMALLENGTHSOFTHELINKSAND,MAX,MINIIYYFORTHESLIDERAREDETERMINEDINTHENEXTSUBSECTION42GLOBAL,COMPREHENSIVEINDEXTHECONDITIONNUMBEROFTHEJACOBIANISUSEDASTHELOCALPERFORMANCEINDEXFOREVALUATINGTHEVELOCITY,ACCURACY,ANDRIGIDITYMAPPINGCHARACTERISTICSBETWEENTHEJOINTVARIABLESANDTHEMOVINGPLATFORMTHECONDITIONNUMBERISDEFINEDAS21127WHERE1AND2ARETHEMINIMUMANDMAXIMUMSINGULARVALUESOFTHEJACOBIANASSOCIATEDWITHAGIVENWUJUN吴军ETALOPTIMALKINEMATICDESIGNOFA2DOFPLANAR273POSTURE1AND2CANBEDETERMINEDBYSOLVINGTHECHARACTERISTICEQUATION2TDET0IJJ28WHEREIDENOTESAUNITMATRIXOFORDER2FORTHEPARALLELMANIPULATORSTUDIEDHERE,2222222222284STSTSTSTSTST29WHERE1XRRSYYAND2XRRTYYSINCEVARIESWITHTHEMECHANISMCONFIGURATION,AGLOBALPERFORMANCEINDEXISUSEDASTHEPERFORMANCEMEASUREINTHEOPTIMALKINEMATICDESIGN,TTTTDDWWWW30NOTETHATITSELFCANNOTGIVEAFULLDESCRIPTIONOFTHEOVERALLGLOBALKINEMATICPERFORMANCEDUETOITSINABILITYTODESCRIBETHEDIFFERENCEBETWEENTHEMAXIMUMANDMINIMUMVALUESOFHUANGETAL7PROPOSEDAGLOBAL,COMPREHENSIVECONDITIONINDEXASTHEOBJECTIVEFUNCTIONINKINEMATICDESIGNSTHEOBJECTIVEFUNCTIONCANBEEXPRESSEDAS22W31WHEREMAXMINWITHMAXANDMINREPRESENTINGTHEMAXIMUMANDMINIMUMVALUESOFINTHETASKWORKSPACEANDWREPRESENTINGTHEWEIGHTPLACEDUPONTHERATIOOFTOANDCANBENUMERICALLYCALCULATEDBY111MNMNMNMN,MAXMINMNMN32WHEREMNISTHEVALUEOFEVALUATEDATNODE,MNOF1M1NEQUALLYMESHEDTW5APPLICATIONS51XNZD2415HYBRIDMACHINETOOLA2DOFPLANARPARALLELMANIPULATORWASCOMBINEDWITHAWORKTABLEHAVINGATRANSLATIONALDOFINTHEZAXISANDAMILLINGHEADWITHTWOROTATIONALDOFSABOUTTHEXANDZAXESTOCREATEA5DOFHYBRIDSERIALPARALLELMACHINETOOLXNZD2415WITHTHETWOAXISMILLINGHEAD,THETOOLCANOBTAINANYDESIREDORIENTATIONINTHETASKWORKSPACE,WHICHPROVIDESHIGHDEXTERITYTHEWORKTABLECANMOVEFREELYALONGTHEZAXIS,WHICHENABLESTHEMACHINETOOLTOMANUFACTURELONGWORKPIECESTHETASKWORKSPACETWOFTHEPARALLELMANIPULATORISDESIGNEDASARECTANGLEB1600MMINWIDTHANDH1000MMINHEIGHTTHEMANIPULATORALSOHASMAX80D,MIN10,DAND75RMMTHESINGULARITYANALYSISSHOWSTHATSINGULARITIESCANNOTOCCURINTHETASKWORKSPACEOFTHEPARALLELMANIPULATORTHEOPTIMALDESIGNBASEDONTHETASKWORKSPACERESULTSIN12174RMMTHEOTHERDESIGNPARAMETERSARETHENCALCULATEDBYMINIMIZINGTHEOBJECTIVEFUNCTIONFIGURE5SHOWSTHATHASAMINIMUMVALUEOF1916WHEN1L2060MMFOR01WBASEDONEQ26,ITCANBEOBTAINEDTHAT,MAX,MIN|IIYYIS23452MMFIG5MINIMIZATIONOFFORVARIOUSLINKLENGTHSTHECONDITIONNUMBEROFTHEJACOBIANOFTHEPARALLELMANIPULATORINTHEMACHINETOOLISSHOWNINFIG6THEDISTRIBUTIONOFTHECONDITIONNUMBERISSYMMETRICALABOUTTHEXAXISTHEGOODKINEMATICPERFORMANCEISACHIEVEDBYBALANCINGTHECONFLICTBETWEENANDFIG6CONDITIONNUMBEROFTHEJACOBIANOFXNZD2415MACHINETOOLTHEMACHINETOOLWASBUILTBYTSINGHUAUNIVERSITYANDWASUSEDINTHEHARBINELECTRICMACHINERYCO,LTDINCHINATOMILLASERIESOFBLADESANDGUIDEVANESFORHYDRAULICTURBINEGENERATORSETSTHEPROTOTYPEISTSINGHUASCIENCEANDTECHNOLOGY,JUNE2007,123269275274SHOWNINFIG7FIG7XNZD2415HYBRIDMACHINETOOL52XNZD2430HEAVYDUTYMACHINETOOLAFTERTHESUCCESSFULAPPLICATIONOFXNZD2415FORVARIOUSMACHININGTASKS,HARBINELECTRICMACHINERYCONEEDEDANOTHERHYBRIDMACHINETOOLTOMACHINEVERYLARGEBLADESANDGUIDEVANESFORHYDRAULICTURBINEGENERATORSETSANOTHERHEAVYDUTYMACHINETOOLXNZD2430WASTHENBUILTBASEDONTHE2DOFPARALLELMANIPULATORTHENEWISSIMILARTOTHEFIRSTMACHINEINSTRUCTUREWITHAMUCHLARGERNEWWORKSPACEOFXNZD2430ANDSOMEOTHERMODIFICATIONSINCLUDINGTWOADDITIONALCOVERSHEETSFIXEDONTHECOLUMNSANDANAUXILIARYTRAYFIXEDONTOPOFTHEMILLINGHEADTHESIDESOFTHECOVERSHEETSCONTACTTHEAUXILIARYTRAYWHICHISCOVEREDWITHCOPPERSHEETTHEAUXILIARYTRAYCONTACTSTHETWOCOVERSHEETANDSLIDESBETWEENTHEMASTHEMILLINGHEADMOVESTHEREFORE,THECOVERSHEETSLIMITTHEDEFORMATIONOFTHEMILLINGHEADANDIMPROVETHEMACHINETOOLSTIFFNESSNORMALTOTHEPLANEOFTHEMANIPULATORMOTIONFOURBRACKETSAREALSOADDEDTOTHENEWMACHINETOOLTOIMPROVETHESTIFFNESSTWOADDITIONALNONROTATINGMILLINGHEADSWEREDESIGNEDTOREPLACETHE2DOFMILLINGHEADFORMOUNTINGONTHESIDEORUNDERSIDEOFTHEMOVINGPLATFORMTHISHEAVYDUTYMACHINETOOLISTHENA3DOFMACHINEWHICHCANMACHINEVERTICALLYORHORIZONTALLYTHETASKWORKSPACETWOFTHENEWPARALLELMANIPULATORWASDESIGNEDASARECTANGLEB3000MMINWIDTHANDH1800MMINHEIGHTWITHMAX79D,MIN5DTHEMANIPULATORCANNOTREACHSINGULARCONFIGURATIONSINTHETASKWORKSPACETHEOPTIMALDESIGNBASEDONTHETASKWORKSPACERESULTSIN23425RMMHASAMINIMUMVALUEWHEN1L3550MMAND01W,MAX,MIN|IIYYCANTHENBEDETERMINEDFROMEQ26THEMAINDESIGNPARAMETERSARELISTEDINTABLE1TABLE1RESULTSOFOPTIMIZATIONFORXNZD2430MACHINETOOLPARAMETERSVALUESL1L2R,MAX,MIN|IIYY3550MM3550MM550MM40107MM199THEDISTRIBUTIONOFFORTHEOPTIMALDESIGNISSHOWNINFIG8THEDISTRIBUTIONSHOWSTHATTHECONDITIONNUMBERISALSOSYMMETRICALWITHRESPECTTOTHEXAXISANDTHEKINEMATICPERFORMANCEISOPTIMALFIG8CONDITIONNUMBEROFTHEJACOBIANOFXNZD2430MACHINETOOLTHEXNZD2430MACHINETOOLSHOWNINFIG9ISCURRENTLYBEINGBUILTBYTSINGHUAUNIVERSITYWITHTHEMACHINESDYNAMICSANDCONTROLSTOBETESTEDSOONFIG9PROTOTYPEOFXNZD2430MACHINETOOLWUJUN吴军ETALOPTIMALKINEMATICDESIGNOFA2DOFPLANAR2756CONCLUSIONSTHEKINEMATICDESIGNOFA2DOFPLANARPARALLELMANIPULATORHASBEENINVESTIGATEDTHEINVESTIGATIONISSHOWNASFOLLOWS1THEDISTANCEBETWEENTHETWOCOLUMNSISDETERMINEDBYTHEOPTIMUMDESIGNBASEDONTHEWORKSPACE2AGLOBAL,COMPREHENSIVEPERFORMANCEINDEX,WHICHINCLUDESBOTHTHEMEANANDTHERANGEOFTHECONDITIONNUMBEROFTHEJACOBIANINTHETASKWORKSPACE,CANBEUSEDTOEFFECTIVELYOPTIMIZETHEKINEMATICDESIGN3THEPLANARPARALLELMANIPULATORBASEDMACHINETOOLWASSUCCESSFULLYUSEDTOMACHINEBLADESANDGUIDEVANESFORAHYDRAULICTURBINEANOTHERHEAVYDUTYMACHINETOOLBASEDONTHEPARALLELMANIPULATORISBEINGBUILTFORLARGERMACHININGTASKSREFERENCES1BEQONP,PIERROTF,DAUCHEZPFUZZYSLIDINGMODECONTROLOFAFASTPARALLELROBOTINIEEEINTERNATIONALCONFERENCEONROBOTICSANDAUTOMATION,1995117811832FERRARESIC,PASTORELLIS,SORLIM,ZHMUDNSTATICANDDYNAMICBEHAVIOROFAHIGHSTIFFNESSSTEWARTPLATFORMBASEDFORCE/TORQUESENSORJOURNALOFROBOTICSYSTEMS,1995,12128838933WUJ,LITM,TANGXQROBUSTTRAJECTORYTRACKINGCONTROLOFAPLANARPARALLELMECHANISMJTSINGHUAUNIVERSITYSCITECHNOL,2005,455642646INCHINESE4KOCKS,SCHUMACHERWPARALLELXYMECHANISMWITHACTUATIONREDUNDANCYFORHIGHSPEEDANDACTIVESTIFFNESSAPPLICATIONSINIEEEINTERNATIONALCONFER