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ORIGINALARTICLEFUNCTIONALSIMULATOROF3AXISPARALLELKINEMATICMILLINGMACHINEMILOSGLAVONJICI123THEPOSITIONVECTOROFTHETOOLTIPISDEFINEDINTHEFRAMEPASPPTXTPYTPZTPC138T,WHEREZTPC0HTHEPOSITIONVECTORSOFSIMULATORSDRIVINGAXESREFERENCEPOINTSRIAREDEFINEDAS,BPRII123JOINTCOORDINATESVECTORLL1L2L3C138T,L1,L2,ANDL3ARESCALARVARIABLESPOWEREDANDCONTROLLEDBYSERIALCNCMACHINEWITHINTHERANGEOFLMINC20LIC20LMAX,WHILEBAIAREUNITVECTORS,BA1100C138TBA2010C138TANDBA300C01C138TWORLDCOORDINATESVECTORBPTXTYTZTC138TREPRESENTSTHEPROGRAMMEDPOSITIONVECTOROFTHETOOLTIP,WHILEXBPOPXPYPZPC2C3TREPRESENTSTHELOCATIONOFTHEPLATFORM,IE,THEORIGINOPOFTHECOORDINATEFRAMEPATTACHEDTOITTHERELATIONSHIPBETWEENTHESETWOVECTORSISOBVIOUSSINCECOORDINATEFRAMESBANDPAREALWAYSPARALLEL,IE,BPTBPOPPPT1OTHERVECTORSANDPARAMETERSAREDEFINEDASSHOWNINFIG6,WHEREBWIANDBQIAREUNITVECTORSWHILECISFIXEDLENGTHOFJOINTPARALLELOGRAMSTHERELATIONSHIPSBETWEENTHESIMULATORSJOINTCOORDINATESVECTORLL1L2L3C138TANDTHESERIALMACHINEJOINTCOORDINATESMX0MYMZMC2C3T,ASSHOWNINFIG6,AREX0ML3YML2ZMC0112ONTHEBASISOFGEOMETRICRELATIONSSHOWNINTHEFIG6,THEFOLLOWINGEQUATIONSAREDERIVEDKBIWIBPOPPBPCIC0BPRI3KBIWILBIAICBQI4BYTAKINGTHESQUAREOFBOTHSIDESINEQ4THEFOLLOWINGRELATIONISDERIVEDC2K2IL2IC02LIBAIKBIWIC0C15BYADOPTINGPBPCIC0BPRI06INEQ3,KINEMATICMODELLINGISVERYSIMPLIFIEDINORDERTOFULFILLTHISREQUIREMENT,SPECIFICCALIBRATIONMETHOD,IE,SETTINGOFREFERENCEPOINTSRIHASBEENDEVELOPEDBYFIG6GEOMETRICMODELOFTHESIMULATORFIG5EXAMPLEOFASIMULATORONTHEVERTICALCNCMILLINGMACHINE816INTJADVMANUFTECHNOL200942813821SUBSTITUTINGOTHERMECHANISMSPARAMETERSINEQ5,THESYSTEMOFTHEFOLLOWINGTHREEEQUATIONSISOBTAINEDX2PY2PZ2PL21C02L1XPC0C20X2PY2PZ2PL22C02L2YPC0C20X2PY2PZ2PL232L3ZPC0C2087FROMWHICHAREDERIVEDINVERSEKINEMATICEQUATIONSASL1XPC6C2C0Y2PC0Z2PQL2YPC6C2C0X2PC0Z2PQL3C0ZPC6C2C0X2PC0Y2PQ88ASWELLASDIRECTKINEMATICEQUATIONSASYPC0S6C6S26C04S5S7P2S5XPS1S2YPZPS3S4YP89WHEREARES1L21C0L222L1S2L2L1S3L22C0L232L3S4C0L2L3S51S22S24S62S1S2S3S4C0L1S2S7S21S23C02L1S1C0C2L21LMINC20LIC20LMAXI123ASITWASMENTIONED,BYADJUSTMENTOFSIMULATORSMECHANISMPARAMETERS,EQ6,THESOLUTIONOFINVERSEANDDIRECTKINEMATICSISGREATLYSIMPLIFIEDTOSATISFYTHECONDITIONSFROMEQ6SIXCALIBRATIONSTRUTSOFSELECTEDREFERENTLENGTHWEREUSED,FIG7WITHTHEUSEOFINVERSEANDDIRECTKINEMATICSSOLUTIONSWITHTHECALIBRATEDSTRUTLENGTHS,THEPOSITIONSOFREFERENCEPOINTSRIOFSLIDERSSI,I1,2,3AREDEFINEDANDFIXEDBYCALIBRATIONPLAINRINGS,FIG741THEANALYSISOFINVERSEANDDIRECTKINEMATICSSOLUTIONSWITHTHEANALYSISOFINVERSEKINEMATICSVARIANCESOLUTIONS,EQ8,DIFFERENTCONFIGURATIONSOFPARALLELMECHANISMFORAGIVENPLATFORMPOSITIONMAYBENOTEDTHEBASICCONFIGURATION,FIG2A,WHENINTHEEQ8,ALLSIGNSBEFORETHESQUAREROOTARENEGATIVE,ONEOFALTERNATIVECONFIGURATIONS,FIG2B,WHENINEQ8,ALLSIGNSBEFORESQUAREROOTAREPOSITIVE,OTHERPOSSIBLEMECHANISMCONFIGURATIONS,WHENINTHEEQ8,SIGNSBEFORETHESQUAREROOTARECOMBINEDINASIMILARWAY,THROUGHTHEANALYSISOFDIRECTKINEMATICSOLUTION,EQ9,DIFFERENTCONFIGURATIONSOFPARALLELMECHANISMFORGIVENPOSITIONSOFDRIVINGAXESMAYBEESTABLISHEDTHEBASICCONFIGURATION,FIG2A,CORRESPONDINGTOTHECASE,WHENINEQ9,THEREISAPOSITIVESIGNBEFORESQUAREROOT,ALTERNATIVECONFIGURATIONS,FIGS2CANDD,WHENINEQ9,THEREISANEGATIVESIGNBEFORESQUAREROOTTHEBASICANDALTERNATIVECONFIGURATIONSSHOWNINFIG2MAYBEREALIZEDINDIFFERENTWAYSSUBJECTTOTHESTRUCTUREOFTHEDRIVINGSERIALMACHINE42JACOBIANMATRICESANDSINGULARITYANALYSISINVIEWOFTHESIGNIFICANCEOFPKMSINGULARITY,THISPROBLEMHASBEENANALYZEDINDETAILFORTHEMECHANISMVARIANTSHOWNINFIG2A,USEDFORTHEDEVELOPMENTOFTHEFIG7SETTINGOFSIMULATORSREFERENCEPOINTSINTJADVMANUFTECHNOL200942813821817SIMULATORONHORIZONTALMACHININGCENTER,FIG1DIFFERENTIATINGEQ8WITHRESPECTTOTHETIME,JACOBIANMATRIXISOBTAINEDASJ1YPXPC0L1ZPXPC0L1XPYPC0L21ZPYPC0L2C0XPZPL3C0YPZPL3C0126437510ASTHEEQUATIONSINEQ7AREIMPLICITFUNCTIONSOFJOINTANDWORLDCOORDINATES,JACOBIANMATRIXMAYBEALSOOBTAINEDBYTHEIRDIFFERENTIATIONASJJC01LC1JX11WHEREJC01L121XPC0L1001YPC0L200C01ZPL326437512JX2XPC0L1YPZPXPYPC0L2ZPXPYPZPL3243513AREJACOBIANMATRICESOFINVERSEANDDIRECTKINEMATICSINTHISWAY,THREEDIFFERENTTYPESOFSINGULARITIESCANBEIDENTIFIED,EG,SINGULARITIESOFINVERSEANDDIRECTKINEMATICSASWELLASCOMBINEDSINGULARITIESWITHCAREFULANALYSISOFJACOBIANMATRICESDETERMINANTSDETJXPL2L3YPL1L3C0ZPL1L2C0L1L2L3XPC0L1C0C1YPC0L2C0C1ZPL3C0C114DETJXC08XPL2L3YPL1L3C0ZPL1L2C0L1L2L3C0C115DETJLC08XPC0L1C0C1YPC0L2C0C1ZPL3C0C116THESINGULARITIESOFINVERSEANDDIRECTKINEMATICSASWELLASCOMBINEDSINGULARITYMAYBENOTICEDFIGURE8SHOWSTHESEPOSSIBLESIMULATORSSINGULARITYCONFIGURATIONSWITHCORRESPONDINGDESCRIPTIONSANDEQUATIONSASITCANBESEENFROMFIG8,ALLSINGULARITIESAREONTHEBORDERSOFTHEORETICALLYACHIEVABLEWORKSPACESOTHATITWOULDBEPOSSIBLETOAVOIDTHEMEASILYWITHADEQUATEDESIGNSOLUTIONSAND/ORMECHANICALCONSTRAINSTHISMEANSTHATTHEACHIEVABLESIMULATORSWORKSPACEISSMALLERTHANTHEORETICALWORKSPACETHEBOUNDARIESOFTHEORETICALWORKSPACEAREONCYLINDERSOFRADIUSCWHOSEAXESAREXB,YB,ZBDERIVEDFROMINVERSEKINEMATICEQ8ANDSPHEREOFRADIUSCCENTEREDINOB,FIG85THEEXAMPLESOFSIMULATORSASITISKNOWNINADDITIONTOSELECTINGAPPROPRIATEKINEMATICTOPOLOGY,THESELECTIONOFTHERIGHTGEOMETRICDIMENSIONSISVERYIMPORTANTSINCETHEPERFORMANCEISHIGHLYINFLUENCEDBYPKMGEOMETRICDIMENSIONS1,8TOSELECTTHERIGHTDIMENSIONSWITHRESPECTTOAGIVENAPPLICATIONISADIFFICULTTASK,ANDTHEDEVELOPMENTOFDESIGNTOOLSFORPKMISSTILLOPENRESEARCH1THEDESIGNPARAMETERSOFSIMULATORSSHOWNINFIGS1,4,AND5WEREADJUSTEDINORDERTOGETMOREADEQUATESHAPESANDWORKSPACEDIMENSIONSONTHEBASISOFPERFORMANCESOFAVAILABLECNCMACHINESFORWHICHSIMULATORSWEREPLANNEDTHEPROCEDUREISESSENTIALLYITERATIVEBECAUSEINDETERMINATIONOFTHEBASICDESIGNPARAMETERSTHEATTENTIONISPAIDTOTHEPOSSIBLEINTERFERENCESOFSTRUCTURALELEMENTSANDTHEVALUESOFDETJANDDETJ1DETERMINANTS,EQS14,15,AND16FIG8SINGULARITYTYPES818INTJADVMANUFTECHNOL200942813821INTHEGEOMETRICMODELOFSIMULATORVARIANTFROMFIG6,ITCANBESEENTHATWORKSPACEDIMENSIONSAREPRIMARILYAFFECTEDBYPARALLELOGRAMSLENGTHC,ASWELLASTOTHEADEQUACYOFTHEDISTANCEOFTHEMECHANISMFROMD3,D3I2,ANDD3I1SINGULARITIESSHOWNINFIG8FORAVAILABLECNCMACHINEFORWHICHTHESIMULATORWASPLANNED,PARALLELOGRAMSLENGTHCANDVALUESJOINTOFCOORDINATESL1,2,3MINANDL1,2,3MAXWEREANALYZEDINITERATIVEPROCEDUREINEACHITERATION,ATTENTIONWASPAIDTOTHEPOTENTIALDESIGNLIMITATIONS,INTERFERENCES,ASWELLASTOTHEVALUESOFDETJANDDETJ1,IE,TOTHEDISTANCESFROMSINGULARITIESTHEPARAMETERSDETERMINEDINTHISWAYHAVEBEENSLIGHTLYCORRECTEDINDETAILEDDESIGNOFTHESIMULATORPROTOTYPESHOWNINFIG9SHAPE,VOLUME,ANDPOSITIONOFACHIEVABLEWORKSPACEFORPARALLELOGRAMSLENGTHC850MMANDL1,2,3MIN200MMANDL1,2,3MAX550MMARESHOWNINFIG2AONTHEBASISOFTHEADOPTEDCONCEPTSANDDESIGNPARAMETERS,THEFIRSTTWOSIMULATORSHAVEBEENBUILTFIGS9AND106SIMULATORPROGRAMMINGANDTESTINGTHESIMULATORPROGRAMMINGSYSTEMHASBEENDEVELOPEDINASTANDARDCADCAMENVIRONMENTONPCPLATFORMFIG11ITISPOSSIBLETOEXCHANGEGEOMETRICWORKPIECEMODELSWITHOTHERSYSTEMSANDSIMULATETHETOOLPATHLINEARINTERPOLATEDTOOLPATHISTAKENFROMTHESTANDARDCLFILETHETOOLPATHMAYALSOBEGENERATEDINSOMEOTHERWAYSELECTEDBYTHESIMULATORUSERTHEBASICPARTOFTHESYSTEMCONSISTSOFDEVELOPEDANDIMPLEMENTEDPOSTPROCESSOR,WITHOUTTHEUSEOFPOSTPROCESSORGENERATORTHEPOSTPROCESSORCONTAINSINVERSEANDDIRECTKINEMATICS,SIMULATORDESIGNPARAMETERS,ANDALGORITHMFORSIMULATORSTOOLPATHLINEARIZATIONFIG12SIMULATORSTOOLPATHLINEARIZATIONISESSENTIALBECAUSECNCMACHINESLINEARINTERPOLATIONISUSEDASSIMULATORSJOINTCOORDINATESINTERPOLATIONINTHISWAY,SIMULATORSTOOLPATHREMAINSWITHINTHETOLERANCETUBEOFPREDEFINEDRADIUSBETWEENPOINTSTJ1ANDTJTAKENFROMCLFILETHELONGPROGRAMFORCNCMACHINEOBTAINEDINFIG9COMPLETEDSIMULATORFROMFIG1FIG10COMPLETEDSIMULATORFROMFIG4FIG11SIMULATORPROGRAMMINGSYSTEMINTJADVMANUFTECHNOL200942813821819THISWAYISTRANSFERREDTOCNCMACHINEANDCANBEVERIFIEDDURINGIDLERUNNINGOFTHESIMULATORTHEMOTIONRANGEOFDRIVINGAXESHASBEENALREADYCHECKEDINTHEPOSTPROCESSORTHETESTINGOFTHESIMULATORINTHISPHASEINCLUDEDVERIFICATIONOFTHESYSTEMFORPROGRAMMINGANDCOMMUNICATION,ANDCUTTINGTESTSBYMACHININGVARIOUSTESTPIECESFIG137CONCLUSIONINORDERTOCONTRIBUTETOWARDSTHEACQUISITIONOFPRACTICALEXPERIENCESINMODELLING,DESIGN,CONTROL,PROGRAMMING,ANDTHEUSEOFPKM,ALOWCOSTBUTFUNCTIONALSIMULATOROF3AXISPARALLELKINEMATICMILLINGMACHINEISPROPOSEDTHEDEVELOPEDFUNCTIONALSIMULATOROFTHE3DPARALLELKINEMATICMILLINGMACHINEINTEGRATES,ASAHYBRIDSYSTEM,THEEXISTINGTECHNOLOGICALEQUIPMENTCNCMACHINETOOLS,CADCAMHARDWAREANDSOFTWAREANDPARALLELKINEMATICMECHANISMINTOACOMPREHENSIVEANDSOPHISTICATEDDIDACTICFACILITYTHEIDEAABOUTTHEFUNCTIONALSIMULATORWASVERIFIEDBYSUCCESSFULMAKINGOFSOMESTANDARDIZEDTESTPIECESOUTOFSOFTMATERIALS,MADEUNDERFULLOPERATIONALCONDITIONSITSCAPABILITIESANDCHARACTERISTICSHAVESHOWNTHATTHESIMULATORITSELFWASANINTERESTINGANDVALUABLERDTOPICTHISIDEAMAYBEFURTHERUSEDFORMAKINGOFONESOWNSIMULATORSFIG13TESTPIECESMADEOFFOAMFIG12UNIFORMLINEARIZATIONOFSIMULATORSTOOLPATH820INTJADVMANUFTECHNOL200942813821ACKNOWLEDGEMENTSTHEPRESENTEDWORKWASPARTOFEUREKAPROJECTE3239SUPPORTEDBYTHEMINISTRYOFSCIENCEOFSERBIAREFERENCES1WECKM,STAIMERD2002PARALLELKINEMATICMACHINETOOLSCURRENTSTATEANDFUTUREPOTENTIALSCIRPANNALSMANUFACTURINGTECHNOLOGY512671683DOI101016/S00078506076170652COVICN2000THEDEVELOPMENTOFTHECONCEPTUALDESIGNOFCLASSOFFLEXIBLEMANUFACTURINGSYSTEMSUNIVERSITYOFBELGRADE,BELGRADEFACULTYOFMECHANICALENGINEERING,DISSERTATION,INSERBIAN3CHABLATD,WENGERP2003ARCHITECTUREOPTIMIZATIONOFA3DOFTRA