DOI10.1007/s00170-003-1860-2ORIGINALARTICLEIntJAdvManufTechnol(2005)25:130139J.SongJ.MouAnear-optimalpartsetupalgorithmfor5-axismachiningusingaparallelkinematicmachineReceived:19February2003/Accepted:4July2003/Publishedonline:28July2004Springer-VerlagLondonLimited2004AbstractAnear-optimalpartsetupmodel(NOPSM)isde-veloped.Thepurposeofthismodelistofindthenear-optimalpartsetuppositionandorientationbasedontheworkspace,stiff-nessandaccuracycapabilityofaparallelkinematicmachinethatcanbeusedfor5-axismachining.TobuildtheproposedNOPSM,theknowledgeonthehexapodkinematics,workspace,stiffness,structuralimperfection,nonuniformthermalgradientandaccuracyisrequired.Thus,itisacomprehensiveperform-ancecapabilitystudyforaparallelkinematicmachine.Thepro-posedmodelisasoftwaresolutionconcepttoimprovethema-chinesperformance.Itisverycosteffectiveandcanalsobemodifiedforother5-axismachinetoolapplications.Keywords5-axismachiningParallelkinematicPartsetupPerformanceenhancement1IntroductionManyCAD/CAMalgorithmscansimulatethemachiningpro-cessforslantedandfree-formsurfacesaswellasgeneratethecorrespondingNCcodes；however,thetraditionalthree-degree-of-freedomCNCmachinerestrictsthepotencyofthesealgo-rithms.Oneoftheadvantagesofparallelkinematic(hexapod)machineoverthetraditional3-axismachinecentreisitsdex-terityandflexibility17.Theoretically,thehexapodmachineunderstudypossessessixdegreesoffreedom.Actually,theplat-formorientationaroundthez-axiscoincideswiththemachinespindlerotation；thus,thishexapodmachinehasfiveeffectivede-greesoffreedomformachining.Manypartsthathaveslantedorfree-formsurfacescanbegeneratedonahexapodmachinewithonesetup.Anotheradvantageofthehexapodmachineisitshigherstiffnesscomparedtotheseriallinkedstructuralma-J.SongJ.Mou(a117)DepartmentofIndustrialEngineering,ArizonaStateUniversity,Tempe,AZ85287-5906,USAE-mail:chinesothatthehigh-speedoperationcanbecarriedoutonthismachine8,9.However,adrawbackthatcomeswiththedexterityofthehexapodmachineisitsrelativelysmallworkspace.Likeallmanufacturingequipment,imperfectstructureandnon-uniformthermal-gradient-relatederrorsalwaysexisttodegradethema-chinesperformanceinproducingqualityproducts1014.Duetotheuniquecharacteristicsofparallelkinematicstructures,themachineinaccuracydistributionwithinitsworkspacewillchangeastheplatformpositionandorientationchange.Mean-while,itsstructuralstiffnessvariesatdifferentplatformpositionsandorientations.Therefore,basedontheinformationonthehexapodsnom-inalkinematicstructure,structuralerrors,thermalerrors,andworkspaceandstiffnessanalyses,anear-optimalpartsetupmodel(NOPSM)couldbedevelopedtosub-optimallysetupapartwithintheworkspaceofahexapodmachine.Thecon-ceptofthisalgorithmisgenericandcanbeeasilyintegratedwithexistingkinematicandthermalmodelsofanyotherparallelkine-maticmachineswithsimplemodifications.Thisapproachcouldalsobeemployedintheapplicationofseriallylinkedrobotsandmachinetools.ForNOPSM,thefirstconstraintisthatallsurfacestobema-chinedneedtobelocatedwithinthehexapodsworkspace.Oncetheworkspaceconditionissatisfied,thenextcriterionappliedtofindthenear-optimalpartsetupisthehexapodmachinesstiff-nessanalysis.Togeneratehigh-qualityproduct,thepartneedstobeplacedatthemostdesirablepositionsothatthemachinecanpossessthehigheststiffnessandaccuracywhilegeneratingthepart.Thealgorithmsderivedin15tofindtherelationshipbetweenthemachinesstructural/thermalerrorsanditsaccuracydistributionbasedonthemachinesstructuralcharacteristicsandmachinestemperaturegradientprofilesareadoptedforsearch-ingthenear-optimalpartpositioningandorientation.Inpractice,thehexapodmachinesdynamicsandcontrolsys-temshouldalsobeconsideredfornear-optimalpartpositioningsearches.However,duetotheproblemscomplexityandtolim-itationsonthescopeofthisresearch,wewillnotdiscussthosetopicshere.1312WorkspaceanalysisTheworkspaceistheworkingvolumeofamachinewithspe-cifictoolsandfeasiblespindlepositionandorientation.Inordertodeterminetheusableworkspaceofthehexapodmachine,ade-rivedkinematicmodel15canbeappliedtodeterminethestrutlength,thejointrotationangleandmobileplatformpositionandorientation.Twoconstraintsaretakenintoaccountinthisworkspaceanalysis.First,themachinesstrutlengthlimitations(maximumlength)definethelowerboundoftheworkspace.Fig.1.HexapodmachineworkspaceanalysisflowchartSecond,themachinessphericaljointrotationallimitationsde-finetheupperboundoftheworkspace.Althoughthemachinesminimumstrutlengthlimitationshouldalsobetakenintoconsid-eration,thisconstraintisoverriddenbythesphericaljointrota-tionallimitationindeterminingthemachinesupperworkspace.AnalgorithmforthedeterminationofhexapodworkspaceforaspecificplatformorientationisshowninFig.1.Differentplatformorientationshavediversemachinework-spaceenvelopes.Byupdatingtheorientationinformation,theworkspaceenvelopfordifferentmachineplatformorientationscanbedetermined.InFigs.2and3,theorientationsaroundthe132Fig.2.Workspaceenvelopewithspindleorientationangle000Fig.3.Workspaceenvelopewithspindleorientationangle3000y-andz-axess,and,arekeptconstant；onlytheorientationanglearoundthex-axis,ischanged.Astheorientationanglearoundthex-axisincreases,theworkspaceistiltedandthez-dimensionoftheworkspaceenvelopeisdecreased.Thelargertheorientationangle,themoreseveretheworkspacetilting.Theworkspaceanalysisresultsshowthatasimilarphenomenonoc-curswhentheorientationanglearoundthey-axis,ischanged,butwithdifferenttiltingdirection.Sincetheplatformorientationanglearoundthez-axis,coincideswiththespindlerotatingdi-rection,theeffectofissuperimposedonspindlerotationandthusnottakenintoconsiderationinworkspaceenvelopeanalysis.TheNOPSMadoptstheworkspaceanalysistodeterminewhetherornotthemachiningsurfacesarewithinthehexapodworkspace.Toensuretheefficiencyofthealgorithm,thefollow-ingtwoconstraintsaretestedforalltheselectedpointsonthesurfacetobemachined:1.Thehexapodmaximumstrutlengthconstraint.2.Thehexapodmaximumjointanglerotationconstraint.Ifalltheselectedpointsonthemachiningsurfacessatisfytheabovetwoconstraints,structuralstiffnessandmachineaccu-racywillthenbeanalysedtoidentifythenear-optimalpartsetuplocationandorientation.3StiffnessanalysisForaparallelmechanism,thereusuallyisaclosed-formsolutionfortheinversekinematics.Theinversekinematicsforthehexa-podmachinecouldbeusedtocalculatethesix-strutlengthbasedontheplatformpositionandorientationinformation16.Thiscanbeexpressedasfollows:Li=fi(x,y,z,).(1)Theapplicationofthechainruleyieldsdifferentialsofli(i=1,2,.,6)asfunctionsofthedifferentialsofx,y,z,.li=fixx+fiyy+fizz+fi+fi+fi.(2)DividingbothsidesofEq.1bythedifferentialtimeelementtandexpressingitinmatrixformatyieldsl1l2l3l4l5l6=f1xf1yf1zf1f1f1f2xf2yf2zf2f2f2.f6xf6yf6zf6f6f6xyz.(3)NotethestandardJocobianexpression,v=Jl.Bylettingl=J1v,theinverseJocobianmatrix,J1,facilitatesthemappingoftheCartesianspacevelocityvectorvintothestrutdisplace-mentratevector.Applyingtheprincipleofvirtualworktoanarbitrarymech-anismallowsonetoequateworkdoneinCartesianspacetermstoworkdoneinconfigurationspaceterms.Specifically,workinCartesiantermsisassociatedwithaCartesianforce/torquevec-tor,F,appliedatamechanismstoolframeandactingthroughaninfinitesimalCartesiandisplacement,v.Workinconfig-urationspacetermsisassociatedwithaconfigurationspaceforce/torquevector,f,appliedatamechanismsjointsandact-ingthroughinfinitesimaljointdisplacements,l.Thestiffnessofthehexapodcanbedeterminedusingmatrixstructuralanalysis,wherethestructureisconsideredtobeacom-binationofelementsandnodes.Thederivationofthehexapodstiffnessmodelisbasedonthefollowingassumptions:1.Theonlydeformationofthelinksisintheaxialdirection.2.Thereisnobendingortwistingofthelinks.3.Thereisnodeformationofthejoints.Workiscalculatedasthedotproductofaforce/torquevectorwithadisplacementvector,FTv=fTl,wherefT=f1,f2,f3,f4,f5,f6aretheforcesexertedoneachofthesix133strutsandFT=Fx,Fy,Fz,Mx,My,Mzaretheforcesandmomentsactingatthecentreofgravityoftheplatform.Notethatv=Jl,soFTJl=fTlFTJ=fT.Transform-ingbothsidesoftheequationyields(FTJ)T=ff=JTF.Onecouldconcludethatactuatingamechanismwithaforce/torquevector,F,appliedatthetoolisequivalenttoac-tuatingthatmechanismwithaforce/torquevector,f,appliedatthejoints,whenthesameamountofvirtualworkisdoneineithercase.TherelationshipbetweenanappliedforceFatthetoolandtheresultingaxialforcesinthestrutfcanbedefinedasF=JTf.Givenpureaxialloading,=li/li=/E=fi/AEfi=(AE/li)li,whereEistheelasticmodulusofthestrutmaterialandAisthecross-sectionalareaofthestrut.Inmatrixformat,f=AE/l1000000AE/l2000000AE/l3000000AE/l4000000AE/l5000000AE/l6.l.(4)Orf=KSl,wherethematrixisidenticaltothestrutspacestiffnessmatrix,Ks.Notethatl=J1v,sof=KSJ1vandF=JTKSJ1v.LetKC=JTKSJ1；thenF=KCv,whereKcistheCartesianspacestiffnessmatrix.Byset-tingupaneigenvalueproblem,theprinciplestiffnessaxes,i,andprinciplestiffness,i,canbefoundasfollows:F=KCv=iv(5)(KCiI6)v=0(6|KCiI6|=0.(7)Here,iisinthedirectionofvwheretheaboveconditionholds.Theprinciplestiffnessiwillchangeasplatformorien-tationandpositionchange.Thehigherthemachinesstructuralstiffness,thebetterthepartsqualityandaccuracy.4StructuralerrordetectionmodelAsmentionedearlier,thehexapodmachinestructureisnotper-fect,andstructuralimperfectionandassemblyerrorsexist.Thestructuralandassemblyerrorsarenotdistributedevenlyamongthehexapodjointsandstruts.Thisunevennesscausesdiverseaccuracylevelsatdifferentplatformpositionsandorientations.Afteramachineisassembled,itisdifficulttomeasurethema-chinestructuralandassemblyerrorbyusinginstrumentsorsen-sorsdirectly.However,themachineplatformsorientationandpositioncanbepreciselymeasuredbyusinganexternalinstru-mentsuchasa5Dlaserinterferometersystemoralasertrackersystem.Amodelisthenneededtoreverseidentifythemachinestructuralerrorsbasedonthemeasuredplatformpositionandorientationerrors.Thehexapodnominalinversekinematicsisderivedas15Tmlm=Tp+TRPPbnTsm；(m=1,2,.,6；n=int(m+1)/2).(8)Differentiatingtheequation,sinceallthevectorsarewithre-specttothetablecoordinatesystem,thesuperscriptofTcanbeomitted:mlm+mlm=p+RPPbn+RPPbnm.(9)Tosimplifythecalculation,therotationerrormatrixcanbewrit-tenasRP=RP,where=,Tistheorientationerrorvector,RPisthenominalorientationmatrix,andisde-finedas=111.Equation9cannowbeexpressedasmlm+mlm=p+RPPbn+RPPbnsm.(10)Sincelmisaunitvector,lmTlm=1；lmTlm=0.MultiplyingEq.10bylmTresultsinm=lTmp+lTmRPPbn+lTmRPPbnlTmsm；(m=1,2,.,6；n=int(m+1)/2),(11)wheremisthestrutlengtherror,smisthetopplatformspher-icaljointpositionalerror,bnisthemobileplatformballjointpositionalerror,pandarethemobileplatformpositionandorientationerrors,respectively,lTmisthestrutvector,andRPisthetransformationmatrixbetweenthemobileplatformand