ORIGINALARTICLEFunctionalsimulatorof3-axisparallelkinematicmillingmachineMilosGlavonjic&DraganMilutinovic&SasaZivanovicReceived:12July2007/Accepted:27June2008/Publishedonline:24July2008#Springer-VerlagLondonLimited2008AbstractParallelkinematicmachines(PKM)areresearch-and-developmenttopicinmanylaboratoriesalthoughmanyofthem,unfortunately,havenoPKMatall.Therefore,theuseoflowcostbutfunctionalsimulatorofa3-AxisparallelkinematicmillingmachineissuggestedasahelptoacquirethebasicexperiencesinthePKMfield.Theideaisbasedonthepossibilitythatthesimulatorcouldbedrivenandcontrolledbyaconventional3-AxisComputerNumericalControlmachinetool(CNC).Thepaperdescribesthedevelopmentprocedureofasimulatorincludingtheselec-tionofacorrespondingparallelmechanism,kinematicmodelling,andtheprogrammingalgorithm.Thefunctionalsimulatorideahasbeenverifiedbysuccessfulmakingofsomestandardizedtestpiecesofsoftmaterial,underfulloperationalconditions.KeywordsParallelkinematicmachines.Functionalsimulator.Modellingandtesting1IntroductionThestrategicimportanceofeducationandtraining,espe-ciallyintechnologyandscientificsubjects,isgrowingthroughouttheworld.Thisalsoappliestotheparallelkinematicmachines(PKMs)whicharetodayresearch-and-development(R&D)andeducationalworldwidetopic.BasicknowledgeaboutdiverseaspectsofPKMhasbeenpublished1.Manydifferenttopologiesofparallelmechanismswith3to6dof,includinga3-doftranslationalorthogonalparallelmechanism,havebeenused15.Today,unfortunately,thegreatmajorityofresearchinsti-tutes,universitylaboratories,andcompanieshavenoPKM.Thereason,obviously,isthehighcostofeducationandtrainingforanewtechnology,suchasPKM.Inordertocontributetowardstheacquisitionofpracticalexperiencesinmodelling,design,control,programming,andtheuseofPKM,alowcostbutfunctionalsimulatorof3-Axisparallelkinematicmillingmachineisproposed2.Theideaisbasedonthepossibilitythatthesimulatorcouldbedrivenandcontrolledbyaconventional3-AxisCNCmachinetool.Astheaxesoftheconventional3-AxisCNCmachinearemutuallyorthogonal,different3-dofspatialparallelmechanismswithorthogonaltranslatoryjointsmaybeusedtobuildthesimulator2,7.Thepaperdescribestheprocedureforsimulatordevel-opmentincludingtheselectionofacorrespondingparallelmechanism,kinematicmodelling,andtheprogrammingalgorithm.Theideaaboutthefunctionalsimulatorwasverifiedbysuccessfulmakingofsomestandardizedtestpiecesoutofsoftmaterials,madeunderfulloperationalconditions.2SimulatorsconceptItwouldbepossible,thankstothepreviousknowledgeaboutserialkinematicmachinesandavailableresourcesfortheirprogramming,tomakethesimulatorasahybridstructureconsistingofdrivingconventional3-AxisCNCmillingmachineanddriven3-dofspatialparallelmecha-nism.OneofthepossibleconceptsofafunctionalsimulatorIntJAdvManufTechnol(2009)42:813821DOI10.1007/s00170-008-1643-xM.Glavonjic(*):D.Milutinovic:S.ZivanovicMechanicalEngineeringFaculty,UniversityofBelgrade,KraljiceMarije16,11120Belgrade,Serbiae-mail:mglavonjicmas.bg.ac.yufor3Dmillingofsoftermaterials,showninFig.1,consistsof:Fullyparallel3-dofmechanismwithconstantstrutlengthsandlinearjointsactuatedandcontrolledbytheconventional3-AxisCNCmachine.ThemechanismisbasedonlinearDELTAmechanism6butwithorthogonallinearactuatedjointstofacilitateitsconnectionwithXM,YM,andZMaxesofhorizontalorverticalserialkinematicmachines.Theuniversalplatform,whichalwaysremainsparallelwiththebase,enablestheplacementofthespindleinthreedifferentorthogonalXP,YP,ZPdirectionsasshowninFig.1.Outofseveralpossibleconfigurationsofthemecha-nism,theonewiththeplatforminsidethetrihedron(XB,YB,ZB)hasbeenselectedsinceitenableseasymountingoftheparallelmechanismontheserialmachineXMaxisguideways.Serial2-dofpassivemechanismfordecouplingserialmachinesYMandZMaxes.Inadditiontotheselectionandadjustmentofthesimulatorsmechanismwiththechosenserialmachine,thefollowingprocedures,models,algorithms,andsoftwarehavetobedefinedanddeveloped:kinematicmodellingofparallelmechanism,i.e.,in-verseanddirectkinematics,Jacobianmatrices,andsingularityanalysis,workspaceanalysisandselectionofsimulatorproperdesignparameters,simulatordesignandmanufacturing,Fig.1FunctionalsimulatorconceptFig.2Thebasicconceptsofsimulatorsparallelmechanism814IntJAdvManufTechnol(2009)42:813821theprocedureandaccessoriesforadjustmentofparallelmechanismsreferentpointstosimplifytheprogramming,algorithmsandsoftwareforsimulatorprogramming,theprocedurefortestingofsimulatorunderworkingconditionsbymachiningofvarioustestpiecesfromsoftermaterials.3OnsimulatormechanismsAstheaxesoftheverticalandhorizontal3-AxisCNCserialmachinesareorthogonalandactuatingsimulatorsaxesatthesametime,itwouldbethebestif3-dofspatialparallelsimulatorsmechanismhasorthogonaltranslatoryjointsaswell.AsinserialCNCmachinestheaxesarecoupled,itwouldbeessential,inageneralcase,tohaveatleastone2-dofpassiveserialmechanismfortheirdecou-pling.ThemostconvenientCNCmachinetoolsforthesimulatorarethosewithmovabletoolholderandworkingtable.Insuchconceptstwooutofthreeaxesarecoupledsothatone2-dofserialpassivemechanismsufficesfortheirdecouplingandtheactuationofthesimulator.Withoutclassificationofkinematicstructuresofhori-zontalandvertical3-AxisCNCmachines,someexamplesof3-dofspatialparallelmechanismswithorthogonaltranslatoryjoints,whichhavebeenconsideredandusedforthesimulator,arepresentedinFig.2.Theshapesoftheirworkspacesareshownaswellinthefigure.TheaboveandsimilarexamplesofthemechanismaretheresultofthesolutionvariancesoftheinverseanddirectkinematicproblemofthebasicconceptillustratedinFig.1.Theexamplesof2-dofpassiveserialmechanismsusedtodecouplethemotionoftheaxesofdrivingserialCNCmachineareshowninFig.3.InsomeserialCNCmachineconcepts,theiraxesmaybedirectlyusedassimulatorsparallelmechanismtranslatoryjoints.Insuchcases,thegeneralconceptofthesimulatorbasedonmechanismsshowninFig.2maybesimplified.Figure4showsanexampleofthesimplifiedsimulatorwithparallelmechanismwithoutitsownjoints.ThedrivingserialCNCmachineisahorizontalmachiningcenter.Thecorrespondingmechanicalinterfacesconnectjointparalle-logramswithdecoupledaxesofthemachiningcentre.2-dofserialmechanismdecouplesmachiningcentersYandZaxes.Figure5showsthedesignofasimplifiedsimulatorforaverticalCNCmillingmachinewithtwocoupledaxes.Simulatorsmechanismhasoneowntranslatoryjointwhile2-dofserialmechanismisalsousedfordecouplingoftheverticalCNCmillingmachineaxes.4SimulatormodellingexampleDetailedkinematicanalysisofthesimulatorfromFig.1,isbasedonitsgeometricmodel,Fig.6.Astheplatform,bymechanismsnature,remainsparallelwiththebase,eachspatialparallelogram,Fig.1,isrepresentedbyonestrut.Thefactthatthecoordinateframes,BandP,connectedtothebaseandtheplatformareparallelandthattheyare,atthesametime,parallelwiththereferentserialmachinecoordinateframeMenablesgeneralizationofthemodellingoftheentiresimulator.Thismeansthatitisfeasibletoseparatethemodellingoftheparallelmechanismitself,regardlessofitsmountingonthehorizontalorverticalserialmachineandthepositionofthespindleonFig.3TheexamplesofserialmechanismsfordecouplingofdrivingmachinesaxesFig.4TheexampleofthesimulatorwithoutowntranslatoryjointsIntJAdvManufTechnol(2009)42:813821815itsplatform.VectorsvreferencedinframesBandParedenotedbyBvandPv.Vectorsdefinedbysimulatorparameters:ThepositionvectorsofthemidpointsCibetweenjointcentersatmobileplatformaredefinedintheframePas,PPCi；i1；2；3.ThepositionvectorofthetooltipisdefinedintheframePasPPT；xTPyTPzTPC138T,wherezTPC0h.ThepositionvectorsofsimulatorsdrivingaxesreferencepointsRiaredefinedas,BPRi；i1；2；3.Jointcoordinatesvector:ll1l2l3C138T,l1,l2,andl3arescalarvariablespoweredandcontrolledbyserialCNCmachinewithintherangeoflminC20liC20lmax,whileBaiareunitvectors,Ba1100C138T；Ba2010C138TandBa300C01C138T.Worldcoordinatesvector:BPTxTyTzTC138Trepresentstheprogrammedpositionvectorofthetooltip,whilexBPOPxpypzpC2C3Trepresentsthelocationoftheplatform,i.e.,theoriginOpofthecoordinateframePattachedtoit.Therelationshipbet