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InternationalJournalofMachineToolsManufacture42200279–88AtoolpathcontrolschemeforfiveaxismachinetoolsChihChingLoDepartmentofMechanicalEngineering,FengChiaUniversity,Taichung407,TaiwanReceived16November2000accepted8May2001AbstractThispaperpresentsanewservocontrolmethodforfiveaxismachiningapplications.Theproposedmethodconductsadirecteliminationofthedeviationerror,theorientationerror,andthetrackinglagerrorthatarethemainconcernsforfiveaxistoolpathcontrol.Toachievethispurpose,theproposedfiveaxiscontrolsystemisbasedonarealtimetransformationbetweenthedrivecoordinatebasis,inwhichthefivedrivesareoperated,andtheworkpiececoordinatebasis,inwhichthedeviationerroretc.,aredefined.KeywordsFiveaxismachinetoolServocontrollerToolpathtrackingcontrol1.IntroductionToachievehighprecisioninmodernCNCcomputernumericalcontrolmachiningapplications,designofservocontrolsystemsthatgenerateaccuratecoordinatedmultiaxismotionisofgreatimportance.Tosynchronizethemotionsofthedifferentaxeswhenmachiningacomplexsurface,aconventionalmultiaxisservocontrolsystemconsistsofaninterpolatorandseveralaxialcontrollers.Theinterpolatorgeneratesthedesiredtoolmotionthatisrelativetotheworkpiece,andthen,decomposesthedesiredmotionintothereferencepositioncommandsfortheseparatedrivingaxes1–4.Thepracticalmotionisrealizedbythedrivingaxes.Eachaxisiscontrolledbyanaxialcontroller,whoseobjectiveistotracktheaxialpositioncommandi.e.,toeliminatethepositionerroralongeachdrivingaxis.Manyresearchershavedevelopedcontrolalgorithmsthatimprovethetrackingaccuracyforanindividualaxis.Traditionalalgorithmsarebasedonthefeedbackprinciple5,6.Inaddition,feedforwardcontrolalgorithmscanbeimplementedtoaugmentthetrackingperformance.Currently,asignificantcontributionhasbeenmadebyTomizuka7,whoproposedazerophaseerrortrackingcontrollerZPETC.OnthebasisoftheZPETCTel.88644517250x3504fax88644516545.Emailaddressccloccsun.fcu.edu.twC.C.Lo.method,somevariationaloraccessoryalgorithmse.g.,adaptiveZPETChavebeenproposed8,9.Althoughthetrackingperformanceforeachindividualaxiscanbesignificantlyimprovedbytheabovemethods,theoverallcontrolperformanceforthemultiaxismachinetoolisnotalwaysguaranteed6.Atypicalperformanceindexforevaluationofthemultiaxisservocontrolisthecontourerror,whichdenotesthedeviationfromthedesiredtoolpath.Toconductaneffectivereductionofthecontourerror,Koren10proposedacrosscouplingcontrollerCCCthatisconstructedbetweenandparalleltotheaxialcontrollers.Atypicalcrosscouplingcontrollerconsistsofarealtimecalculationofthecontourerrorandacontrollawtoeliminatethecontourerror.BasedontheconceptoftheCCCmethod,numerouscrosscouplingcontrollerswithdifferentcontourerrormodelsand/orcontrollerlawshavebeenproposed11–13.Thecontourerror,however,isnottheonlyconcernformultiaxistoolpathtrackingcontrol.Forinstance,thepositionlagalongthetrackingdirectionisanotherconcern6.Besides,theorientationerror,whichdenotesthedeviationanglebetweenthepracticaltoolaxisandthedesiredtooldirection,isalsoanimportantconcerninfourorfiveaxismachinetoolcontrol14.Inthispaper,themainconcernsforafiveaxistoolpathcontrolarediscussedfirst.Then,aconventionalfiveaxiscontrolsystemisdiscussed,anditsdrawbackisaddressed.Finally,afiveaxiscontrolsystemthatcon80C.C.Lo/InternationalJournalofMachineToolsManufacture42200279–88Fig.1.Thetoolpathalongthesculpturedsurface.ductsadirecteliminationoftheseconcernsisproposedandcomparedwiththeconventionalone.2.MainconcernsinfiveaxistoolpathcontrolLetsconsiderthefollowingfiveaxismachiningcaseasreferredtoinFig.1utilizingacylindricaltooltocutasurface.Attheplanningstage,thetoolpaththatcomprisesthetoolcenterlocationLandthetoolorientationOisscheduledsothatthecutteredgeScanpassoverthesculpturedsurface15,16.Here,weletRandPdenotethereferencepositionvectorandthepracticalpositionvector,respectively.BothRandParepositionvectorswithfivecomponentsthreeforthetoolcenterlocationLandtwoforthetoolorientationO.Thedifferenceorerrorbetweenthereferenceandthepracticalpositionvectorsi.e.,ERPisaconcerninfiveaxismachining.However,Eisnotthemainconcern,becauseasmallEdoesnotnecessarilyguaranteeanegligiblemachininginaccuracy.AsillustratedinFig.2,althoughP2ismuchclosertoRthanP1i.e.,|E2|lessmuch|E1|,itresultsinmoremachininginaccuracy.TwomaincausesforpartinaccuracyareillustratedinFig.3.AsshowninFig.3a,thedeviationerrored,whichdenotesthedistancebetweenthepracticaltoollocationPandtheclosestlocationConthedesiredtoolpathratherthantheinstantaneousreferenceR,isanimportantconcern.AsshowninFig.3b,theorientationerrorH9278,whichdenotestheanglebetweentheFig.2.Machininginaccuracyduetoimperfecttoolpathtrackingcontrol.Fig.3.Thedeviationerrorandtheorientationerrorinfiveaxismachiningcontroladeviationerrorborientationerror.Fig.4.Twoconsecutivepathsthatthetoolpathpassesover.practicaltoolaxisandthedesiredtooldirectioncorrespondingtoC,isanotherimportantconcern.AscanbeseeninFig.3,thedeviationerrorandtheorientationerroraremaincausesformachininginaccuracy.Inadditiontothedeviationerrorandtheorientationerror,atrackinglagerrordthatdenotesthecomponentofEalongthetrackingdirectionisalsoanimportantconcern.AsshowninFig.4,asignificanttrackinglagwillalsocauseanunacceptablemachininginaccuracybetweentwoconsecutivesurfaces.3.ConventionalfiveaxiscontrolsystemTypically,afiveaxismachinetoolconsistsofthreetranslationalaxesx,y,zandtworotationalaxesa,b.TheblockdiagramforaconventionalcontrolsystemforfiveaxismachinetoolsisshowninFig.5.IntheFig.5.Aconventionalcontrolsystemforfiveaxismachinetools.81C.C.Lo/InternationalJournalofMachineToolsManufacture42200279–88systemtheinterpolator,whichconsistsofapathplanningmoduleandaninversekinematicstransformation2,14,generatesinrealtimethedesiredreferencepositioncommandstothefiveseparatecontrolloopsrespectivelyforthex,y,z,a,andbaxis.Thepathplanningmodulegeneratesthedesiredtoolmotionrelativelytotheworkpiece.Inotherwords,thetoolpathisdefinedintheworkpiececoordinatebasisWCBforwhichtheaxesarefixedontheworkpiece.Inthefollowing,thedesired,practical,anderrorpositionvectorsthataredefinedintheWCBaredenotedasRw,Pw,andEw,respectively.Incontrasttothepathplanningmodule,theaxialcontrolloopsfocustheireffortontrackingtheindividualmotionsalongthefivedrivingaxes.Thesemotions,however,aredefinedinthedrivecoordinatebasisDCB.Inthefollowing,thedesired,practical,anderrorpositionvectorsthataredefinedintheDCBaredenotedasRd,Pd,andEd,respectively.TotransformthereferencepositionvectorfromWCBtoDCB,aninversekinematicstransformationalgorithmisrequiredtobeimplementedintheinterpolator.NotethatinpracticethemechanicalstructureofthefiveaxismachinetoolplaysaroleasadirectkinematicstransformationthatconvertsPdtoPw.LetKandKH110021representthedirectandinversekinematicstransformations,respectively.Inotherwords,wehaveRwH11005KRd,PwH11005KPd,RdH11005K−1RwandPd1H11005K−1Pw.Thefunctionofthefivecontrolloopsistotrackthereferencepositioncommandsthataregeneratedbytheinterpolator.Foreachloop,thecontrollerobjectiveistominimizethepositionerroralongthedrivingaxis.LetHdandGdberespectivelythetransferfunctionmatricesforthecontrollersandthedrives.Inthematrices,HdandGd,thenondiagonaltermsarezerosandthediagonaltermsarethetransferfunctionsfortheaxialcontrollersandtheaxialdrives,respectively.Notethatinacomputercontrolledsystem,HdandGdarefunctionsofzvariableindiscretetimedomain.Thestrategyoftheconventionalfiveaxiscontrolsystemistoreducethepositionerrorsalongthedrivingaxesi.e.,EdRdH11002Pd,andthen,toexpectaqualityfiveaxistoolpathcontrolthatfocusesontheeliminationofthedeviationerror,theorientationerror,andthetrackinglagerror.However,theexpectationisindoubt.AshasbeenillustratedintheabovesectionrefertoFig.2,thereductionofEddoesnotnecessarilycorrespondtothereductionofthedeviationerror,etc.4.ProposedfiveaxiscontrolsystemTheproposedfiveaxiscontrolsystemisdepictedinFig.6.IncontrasttotheconventionalsystemthatconFig.6.Theproposedcontrolsystemforfiveaxismachinetools.structsfivelocalandseparatecontrolloopsrefertoFig.5,theproposedcontrolsystemconstructsaglobalandcoupledlooptoachieveaneffectivecontroloftheoverallperformancethatisintermsofthedeviationerror,theorientationerror,andthetrackinglag.Thedeviationerror,etc.,whichwillbederivedlaterinthefollowing,areerrorcomponentsdefinedintheWCB.Incontrast,thefedbackpositionsignalsPdandthecontrolsignalsUdsenttotheaxialdrivesarebothdefinedintheDCB.Consequently,coordinatetransformationsareintroducedtotheproposedcontrolsystem.AsdepictedinFig.6,theservocontrollerconsistsoffourparts1adirectkinematicstransformationalgorithmthatcalculatesthepracticaltoolpositioninWCB,i.e.,PwKPd2anerrormodelforcalculationofthedeviationerrorandtheorientationerrorthatarerepresentedbye,andthetrackinglagerrord3acontrollawthateliminateseandd4aninverseJacobianmatrixthattransformsthecontrolintheWCBi.e.,UwtothatintheDCBi.e.,Ud.Throughtheaboveprocedure,theproposedcontrolsystemfocusesitscontroleffortintheWCBandconductsadirecteliminationofeandd.Thelastthreepartsoftheservocontrolleraredescribedindetailinthefollowing.4.1.ErrormodelAsstatedabove,thedeviationerror,theorientationerror,andthetrackinglagerrorarethemainconcernsforfiveaxistoolpathtrackingcontrol.Therefore,themodelfortheseconcernederrorsisthecoreoftheproposedservocontroller.AschematicillustrationfortheseconcernederrorsisshowninFig.7.InFig.7,Cwdenotesthepositionthatislocatedonthedesiredtoolpathandistheclosesttothepracticaltoolposition,Pw.LetthedifferencebetweenCwandPwbedenotedase,i.e.,eH11005CwH11002Pw.2Notethateconsistsoffivecomponents,i.e.,eex,ey,ez,ea,eb.82C.C.Lo/InternationalJournalofMachineToolsManufacture42200279–88Fig.7.ThereferencepositionRw,thepracticalpositionPw,andtheclosestpositionCwfromPwtothedesiredtoolpath.Ashasbeendefinedabove,thedeviationerroristhedistancebetweenthepracticaltoollocationandtheclosestpointonthedesiredtoolpath.Consequently,thefirstthreecomponentsofeareinpracticethecomponentsofthedeviationerrored,i.e.,edH11005H20881ex2ey2ez2.3Theorientationerroristheanglebetweenthetoolaxisandthetoolorientationfortheclosestpointonthedesiredtoolpath.Inthispaper,thetworotationalanglesaredefinedsothatthetoolwithrespecttotheWCBisoriginallyinthezdirection,thenrotateswithaalongthexaxis,andfinallyrotateswithbalongtheyaxis.Basedonthedefinitionsofthetworotationalanglesa,b,theorientationerrorfcanbecalculatedbyfH11005cos−1H20898cosPwasinPwb,H11002sinPwa,cosPwacosPwbH20900cosPwaeasinPwbeb−sinPwaeacosPwaeacosPwbebH20901H20899.4AsshowninEq.4,theorientationerrorisdeterminedbyeaandeb,whicharethelasttwocomponentsofe.Therefore,intheproposedcontrolsystem,theeliminationofthedeviationerrorandtheorientationerrorcanbeconductedthroughthecontrolofe.ThetrackinglagerrordistheprojectionvectorofEwRwH11002Pwalongthetoolpath.AccordingtoFig.7,wehavedH11005RwH11002Cw.5Notethatdconsistsoffivecomponents,i.e.,ddx,dy,dz,da,db,andthefirstthreecomponentsofdareforthetrackinglagdistancedd,i.e.,ddH11005H20881dx2dy2dz2.6SubstitutingEq.5intoEq.2yieldseH11005RwH11002dH11002PwH11005EwH11002d.7AscanbeseeninEq.7,ifdisdetermined,eisalsoobtained.However,ananalyticalsolutionofdisnotavailableforgeneraltrajectories.Anumericaliterativemethodistimeconsumingandisnotsuggestedforrealtimecontrol.Therefore,anapproximatedmodelforcalculationofdisrecommendedhere.AsshowninFig.7,Cwisapositionvectorthatislocatedonthedesiredtoolpathandlagsbehindtheinstantaneousreferencepositionvector,Rw.ThedistanceddthatCwlagsbehindRwcanbeapproximatedbytheprojectionofthepositionerrorvectorEwx,Ewy,EwzonthetangentialdirectionofthepathonRw,i.e.,ddH11015EwxtxH11001EwytyH11001Ewztz,8wheretx,ty,tzisthetangentialvectortothetoolpathonRw.ThetrackinglagerrordcanberegardedasavariationofthereferencepositionvectorRwaccordingtothetrackinglagdistance,dd.Consequently,dcanbeapproximatedbydH11005RwH11002CwH11015dddRwdlH11001d2d2d2Rwdl2,9whereddiscalculatedbyEq.8thevariablelisthepathlengthalongthedesiredtoolpathdRw/dlandd2Rw/dl2arethefirstandsecondderivativesofthereferencepositionvectorwithrespecttothepathlengthl.InaCNCsystem,dRw/dlandd2Rw/dl2canbeapproximatedbydRwdlH11015RwkT−RwkT−TfT,10d2Rwdl2H11015RwkT−2RwkT−TRwkT−2TfT2,11wherefisthefeedrate,Tisthesamplingperiod,andkTdenotesthesamplinginstant.4.2.ControllerlawLetHeandHdbethetransferfunctionmatricesforthecontrollawsforeandd,respectively.NotethatbothHeandHdare55diagonalmatrices,forwhicheachdiagonaltermrepresentsthecorrespondingcontrollaw
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