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ORIGINALARTICLEFunctionalsimulatorof3axisparallelkinematicmillingmachineMilosGlavonjicDraganMilutinovicSasaZivanovicReceived12July2007/Accepted27June2008/Publishedonline24July2008SpringerVerlagLondonLimited2008AbstractParallelkinematicmachinesPKMareresearchanddevelopmenttopicinmanylaboratoriesalthoughmanyofthem,unfortunately,havenoPKMatall.Therefore,theuseoflowcostbutfunctionalsimulatorofa3AxisparallelkinematicmillingmachineissuggestedasahelptoacquirethebasicexperiencesinthePKMfield.Theideaisbasedonthepossibilitythatthesimulatorcouldbedrivenandcontrolledbyaconventional3AxisComputerNumericalControlmachinetoolCNC.Thepaperdescribesthedevelopmentprocedureofasimulatorincludingtheselectionofacorrespondingparallelmechanism,kinematicmodelling,andtheprogrammingalgorithm.Thefunctionalsimulatorideahasbeenverifiedbysuccessfulmakingofsomestandardizedtestpiecesofsoftmaterial,underfulloperationalconditions.KeywordsParallelkinematicmachines.Functionalsimulator.Modellingandtesting1IntroductionThestrategicimportanceofeducationandtraining,especiallyintechnologyandscientificsubjects,isgrowingthroughouttheworld.ThisalsoappliestotheparallelkinematicmachinesPKMswhicharetodayresearchanddevelopmentRDandeducationalworldwidetopic.BasicknowledgeaboutdiverseaspectsofPKMhasbeenpublished1.Manydifferenttopologiesofparallelmechanismswith3to6dof,includinga3doftranslationalorthogonalparallelmechanism,havebeenused1–5.Today,unfortunately,thegreatmajorityofresearchinstitutes,universitylaboratories,andcompanieshavenoPKM.Thereason,obviously,isthehighcostofeducationandtrainingforanewtechnology,suchasPKM.Inordertocontributetowardstheacquisitionofpracticalexperiencesinmodelling,design,control,programming,andtheuseofPKM,alowcostbutfunctionalsimulatorof3Axisparallelkinematicmillingmachineisproposed2.Theideaisbasedonthepossibilitythatthesimulatorcouldbedrivenandcontrolledbyaconventional3AxisCNCmachinetool.Astheaxesoftheconventional3AxisCNCmachinearemutuallyorthogonal,different3dofspatialparallelmechanismswithorthogonaltranslatoryjointsmaybeusedtobuildthesimulator2,7.Thepaperdescribestheprocedureforsimulatordevelopmentincludingtheselectionofacorrespondingparallelmechanism,kinematicmodelling,andtheprogrammingalgorithm.Theideaaboutthefunctionalsimulatorwasverifiedbysuccessfulmakingofsomestandardizedtestpiecesoutofsoftmaterials,madeunderfulloperationalconditions.2SimulatorsconceptItwouldbepossible,thankstothepreviousknowledgeaboutserialkinematicmachinesandavailableresourcesfortheirprogramming,tomakethesimulatorasahybridstructureconsistingofdrivingconventional3AxisCNCmillingmachineanddriven3dofspatialparallelmechanism.OneofthepossibleconceptsofafunctionalsimulatorIntJAdvManufTechnol200942813–821DOI10.1007/s001700081643xM.GlavonjicD.MilutinovicS.ZivanovicMechanicalEngineeringFaculty,UniversityofBelgrade,KraljiceMarije16,11120Belgrade,Serbiaemailmglavonjicmas.bg.ac.yufor3Dmillingofsoftermaterials,showninFig.1,consistsof▪Fullyparallel3dofmechanismwithconstantstrutlengthsandlinearjointsactuatedandcontrolledbytheconventional3AxisCNCmachine.ThemechanismisbasedonlinearDELTAmechanism6butwithorthogonallinearactuatedjointstofacilitateitsconnectionwithXM,YM,andZMaxesofhorizontalorverticalserialkinematicmachines.Theuniversalplatform,whichalwaysremainsparallelwiththebase,enablestheplacementofthespindleinthreedifferentorthogonalXP,YP,−ZPdirectionsasshowninFig.1.Outofseveralpossibleconfigurationsofthemechanism,theonewiththeplatforminsidethetrihedronXB,YB,−ZBhasbeenselectedsinceitenableseasymountingoftheparallelmechanismontheserialmachineXMaxisguideways.▪Serial2dofpassivemechanismfordecouplingserialmachinesYMandZMaxes.Inadditiontotheselectionandadjustmentofthesimulatorsmechanismwiththechosenserialmachine,thefollowingprocedures,models,algorithms,andsoftwarehavetobedefinedanddeveloped–kinematicmodellingofparallelmechanism,i.e.,inverseanddirectkinematics,Jacobianmatrices,andsingularityanalysis,–workspaceanalysisandselectionofsimulatorproperdesignparameters,–simulatordesignandmanufacturing,Fig.1FunctionalsimulatorconceptFig.2Thebasicconceptsofsimulatorsparallelmechanism814IntJAdvManufTechnol200942813–821–theprocedureandaccessoriesforadjustmentofparallelmechanismsreferentpointstosimplifytheprogramming,–algorithmsandsoftwareforsimulatorprogramming,–theprocedurefortestingofsimulatorunderworkingconditionsbymachiningofvarioustestpiecesfromsoftermaterials.3OnsimulatormechanismsAstheaxesoftheverticalandhorizontal3AxisCNCserialmachinesareorthogonalandactuatingsimulatorsaxesatthesametime,itwouldbethebestif3dofspatialparallelsimulatorsmechanismhasorthogonaltranslatoryjointsaswell.AsinserialCNCmachinestheaxesarecoupled,itwouldbeessential,inageneralcase,tohaveatleastone2dofpassiveserialmechanismfortheirdecoupling.ThemostconvenientCNCmachinetoolsforthesimulatorarethosewithmovabletoolholderandworkingtable.Insuchconceptstwooutofthreeaxesarecoupledsothatone2dofserialpassivemechanismsufficesfortheirdecouplingandtheactuationofthesimulator.Withoutclassificationofkinematicstructuresofhorizontalandvertical3AxisCNCmachines,someexamplesof3dofspatialparallelmechanismswithorthogonaltranslatoryjoints,whichhavebeenconsideredandusedforthesimulator,arepresentedinFig.2.Theshapesoftheirworkspacesareshownaswellinthefigure.TheaboveandsimilarexamplesofthemechanismaretheresultofthesolutionvariancesoftheinverseanddirectkinematicproblemofthebasicconceptillustratedinFig.1.Theexamplesof2dofpassiveserialmechanismsusedtodecouplethemotionoftheaxesofdrivingserialCNCmachineareshowninFig.3.InsomeserialCNCmachineconcepts,theiraxesmaybedirectlyusedassimulatorsparallelmechanismtranslatoryjoints.Insuchcases,thegeneralconceptofthesimulatorbasedonmechanismsshowninFig.2maybesimplified.Figure4showsanexampleofthesimplifiedsimulatorwithparallelmechanismwithoutitsownjoints.ThedrivingserialCNCmachineisahorizontalmachiningcenter.Thecorrespondingmechanicalinterfacesconnectjointparallelogramswithdecoupledaxesofthemachiningcentre.2dofserialmechanismdecouplesmachiningcentersYandZaxes.Figure5showsthedesignofasimplifiedsimulatorforaverticalCNCmillingmachinewithtwocoupledaxes.Simulatorsmechanismhasoneowntranslatoryjointwhile2dofserialmechanismisalsousedfordecouplingoftheverticalCNCmillingmachineaxes.4SimulatormodellingexampleDetailedkinematicanalysisofthesimulatorfromFig.1,isbasedonitsgeometricmodel,Fig.6.Astheplatform,bymechanismsnature,remainsparallelwiththebase,eachspatialparallelogram,Fig.1,isrepresentedbyonestrut.Thefactthatthecoordinateframes,{B}and{P},connectedtothebaseandtheplatformareparallelandthattheyare,atthesametime,parallelwiththereferentserialmachinecoordinateframe{M}enablesgeneralizationofthemodellingoftheentiresimulator.Thismeansthatitisfeasibletoseparatethemodellingoftheparallelmechanismitself,regardlessofitsmountingonthehorizontalorverticalserialmachineandthepositionofthespindleonFig.3TheexamplesofserialmechanismsfordecouplingofdrivingmachinesaxesFig.4TheexampleofthesimulatorwithoutowntranslatoryjointsIntJAdvManufTechnol200942813–821815itsplatform.Vectorsvreferencedinframes{B}and{P}aredenotedbyBvandPv.Vectorsdefinedbysimulatorparameters–ThepositionvectorsofthemidpointsCibetweenjointcentersatmobileplatformaredefinedintheframe{P}as,PPCii¼123ðÞ.–Thepositionvectorofthetooltipisdefinedintheframe{P}asPPTxTPyTPzTP½C138T,wherezTP¼C0h.–ThepositionvectorsofsimulatorsdrivingaxesreferencepointsRiaredefinedas,BPRii¼123ðÞ.Jointcoordinatesvector–l¼l1l2l3½C138T,l1,l2,andl3arescalarvariablespoweredandcontrolledbyserialCNCmachinewithintherangeoflminC20liC20lmax,whileBaiareunitvectors,Ba1¼100½C138TBa2¼010½C138TandBa3¼00C01½C138T.WorldcoordinatesvectorBPT¼xTyTzT½C138Trepresentstheprogrammedpositionvectorofthetooltip,whilex¼BPOP¼xpypzpC2C3Trepresentsthelocationoftheplatform,i.e.,theoriginOpofthecoordinateframe{P}attachedtoit.Therelationshipbetweenthesetwovectorsisobvioussincecoordinateframes{B}and{P}arealwaysparallel,i.e.,BPT¼BPOPþPPTð1ÞOthervectorsandparametersaredefinedasshowninFig.6,whereBwiandBqiareunitvectorswhilecisfixedlengthofjointparallelograms.Therelationshipsbetweenthesimulatorsjointcoordinatesvectorl¼l1l2l3½C138Tandtheserialmachinejointcoordinatesm¼x0MyMzMC2C3T,asshowninFig.6,arex0M¼l3yM¼l2zM¼C011ð2ÞOnthebasisofgeometricrelationsshownintheFig.6,thefollowingequationsarederivedkBiwi¼BPOPþP¼BPCiC0BPRið3ÞkBiwi¼lBiaiþcBqið4ÞBytakingthesquareofbothsidesinEq.4thefollowingrelationisderivedc2¼k2iþl2iC02liBaikBiwiC0C1ð5ÞByadoptingP¼BPCiC0BPRi¼0ð6ÞinEq.3,kinematicmodellingisverysimplified.Inordertofulfillthisrequirement,specificcalibrationmethod,i.e.,settingofreferencepointsRihasbeendeveloped.ByFig.6GeometricmodelofthesimulatorFig.5ExampleofasimulatorontheverticalCNCmillingmachine816IntJAdvManufTechnol200942813–821
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