外文翻译--五轴联动机床的刀具轨迹控制方案 英文版.pdf

InternationalJournalofMachineTools&Manufacture42(2002)7988Atool-pathcontrolschemeforfive-axismachinetoolsChih-ChingLo*DepartmentofMechanicalEngineering,FengChiaUniversity,Taichung407,TaiwanReceived16November2000；accepted8May2001AbstractThispaperpresentsanewservocontrolmethodforfive-axismachiningapplications.Theproposedmethodconductsadirecteliminationofthedeviationerror,theorientationerror,andthetracking-lagerrorthatarethemainconcernsforfive-axistool-pathcontrol.Toachievethispurpose,theproposedfive-axiscontrolsystemisbasedonareal-timetransformationbetweenthedrive-coordinatebasis,inwhichthefivedrivesareoperated,andtheworkpiece-coordinatebasis,inwhichthedeviationerroretc.,aredefined.Keywords:Five-axismachinetool；Servocontroller；Tool-pathtrackingcontrol1.IntroductionToachievehighprecisioninmodernCNC(computernumericalcontrol)machiningapplications,designofservocontrolsystemsthatgenerateaccuratecoordinatedmulti-axismotionisofgreatimportance.Tosynchronizethemotionsofthedifferentaxeswhenmachiningacom-plexsurface,aconventionalmulti-axisservocontrolsys-temconsistsofaninterpolatorandseveralaxialcontrol-lers.Theinterpolatorgeneratesthedesiredtoolmotionthatisrelativetotheworkpiece,andthen,decomposesthedesiredmotionintothereferencepositioncommandsfortheseparatedrivingaxes14.Thepracticalmotionisrealizedbythedrivingaxes.Eachaxisiscontrolledbyanaxialcontroller,whoseobjectiveistotracktheaxialpositioncommand(i.e.,toeliminatethepositionerroralongeachdrivingaxis).Manyresearchershavedevelopedcontrolalgorithmsthatimprovethetrackingaccuracyforanindividualaxis.Traditionalalgorithmsarebasedonthefeedbackprin-ciple5,6.Inaddition,feedforwardcontrolalgorithmscanbeimplementedtoaugmentthetrackingperform-ance.Currently,asignificantcontributionhasbeenmadebyTomizuka7,whoproposedazerophaseerrortrack-ingcontroller(ZPETC).OnthebasisoftheZPETC*Tel.:+886-445-17250x3504；fax:+886-445-16545.E-mailaddress:cclocc-sun.fcu.edu.tw(C.-C.Lo).method,somevariationaloraccessoryalgorithms(e.g.,adaptiveZPETC)havebeenproposed8,9.Althoughthetrackingperformanceforeachindividualaxiscanbesignificantlyimprovedbytheabovemethods,theoverallcontrolperformanceforthemulti-axismachinetoolisnotalwaysguaranteed6.Atypicalperformanceindexforevaluationofthemulti-axisservocontrolisthecontourerror,whichdenotesthedeviationfromthedesiredtoolpath.Toconductaneffectivereductionofthecontourerror,Koren10proposedacross-couplingcontroller(CCC)thatisconstructedbetweenandparalleltotheaxialcontrollers.Atypicalcross-couplingcontrollerconsistsofareal-timecalcu-lationofthecontourerrorandacontrollawtoeliminatethecontourerror.BasedontheconceptoftheCCCmethod,numerouscross-couplingcontrollers(withdif-ferentcontour-errormodelsand/orcontrollerlaws)havebeenproposed1113.Thecontourerror,however,isnottheonlyconcernformulti-axistool-pathtrackingcontrol.Forinstance,thepositionlagalongthetrackingdirectionisanotherconcern6.Besides,theorientationerror,whichdenotesthedeviationanglebetweenthepracticaltoolaxisandthedesiredtooldirection,isalsoanimportantconcerninfour-orfive-axismachine-toolcontrol14.Inthispaper,themainconcernsforafive-axistool-pathcontrolarediscussedfirst.Then,aconventionalfive-axiscontrolsystemisdiscussed,anditsdrawbackisaddressed.Finally,afive-axiscontrolsystemthatcon-80C.-C.Lo/InternationalJournalofMachineTools&Manufacture42(2002)7988Fig.1.Thetoolpathalongthesculpturedsurface.ductsadirecteliminationoftheseconcernsisproposedandcomparedwiththeconventionalone.2.Mainconcernsinfive-axistool-pathcontrolLetsconsiderthefollowingfive-axismachiningcase(asreferredtoinFig.1):utilizingacylindricaltooltocutasurface.Attheplanningstage,thetoolpaththatcomprisesthetool-centerlocation(L)andthetoolorien-tation(O)isscheduledsothatthecutteredge(S)canpassoverthesculpturedsurface15,16.Here,weletRandPdenotethereferencepositionvectorandthepracti-calpositionvector,respectively.BothRandParepos-itionvectorswithfivecomponents(threeforthetool-centerlocationLandtwoforthetoolorientationO).Thedifference(orerror)betweenthereferenceandthepracticalpositionvectors(i.e.,E=R-P)isaconcerninfive-axismachining.However,Eisnotthemaincon-cern,becauseasmallEdoesnotnecessarilyguaranteeanegligiblemachininginaccuracy.AsillustratedinFig.2,althoughP(2)ismuchclosertoRthanP(1)(i.e.,|E(2)|lessmuch|E(1)|),itresultsinmoremachininginaccuracy.TwomaincausesforpartinaccuracyareillustratedinFig.3.AsshowninFig.3(a),thedeviationerror(ed),whichdenotesthedistancebetweenthepracticaltoollocation(P)andtheclosestlocation(C)onthedesiredtoolpath(ratherthantheinstantaneousreferenceR),isanimportantconcern.AsshowninFig.3(b),theorien-tationerror(H9278),whichdenotestheanglebetweentheFig.2.Machininginaccuracyduetoimperfecttool-pathtrackingcontrol.Fig.3.Thedeviationerrorandtheorientationerrorinfive-axismachiningcontrol(a)deviationerror；(b)orientationerror.Fig.4.Twoconsecutivepathsthatthetoolpathpassesover.practicaltoolaxisandthedesiredtooldirectioncorre-spondingtoC,isanotherimportantconcern.AscanbeseeninFig.3,thedeviationerrorandtheorientationerroraremaincausesformachininginaccuracy.Inadditiontothedeviationerrorandtheorientationerror,atracking-lagerror(d)thatdenotesthecomponentofEalongthetrackingdirectionisalsoanimportantconcern.AsshowninFig.4,asignificanttrackinglagwillalsocauseanunacceptablemachininginaccuracybetweentwoconsecutivesurfaces.3.Conventionalfive-axiscontrolsystemTypically,afive-axismachinetoolconsistsofthreetranslationalaxes(x,y,z)andtworotationalaxes(a,b).Theblockdiagramforaconventionalcontrolsystemforfive-axismachinetoolsisshowninFig.5.IntheFig.5.Aconventionalcontrolsystemforfive-axismachinetools.81C.-C.Lo/InternationalJournalofMachineTools&Manufacture42(2002)7988systemtheinterpolator,whichconsistsofapath-plan-ningmoduleandaninverse-kinematicstransformation2,14,generatesinrealtimethedesiredreferencepos-itioncommandstothefiveseparatecontrolloops(respectivelyforthex-,y-,z-,a-,andb-axis).Thepath-planningmodulegeneratesthedesiredtoolmotionrela-tivelytotheworkpiece.Inotherwords,thetoolpathisdefinedintheworkpiece-coordinatebasis(WCB)forwhichtheaxesarefixedontheworkpiece.Inthefollow-ing,thedesired,practical,anderrorpositionvectorsthataredefinedintheWCBaredenotedasRw,Pw,andEw,respectively.Incontrasttothepath-planningmodule,theaxialcontrolloopsfocustheireffortontrackingtheindi-vidualmotionsalongthefivedrivingaxes.Thesemotions,however,aredefinedinthedrive-coordinatebasis(DCB).Inthefollowing,thedesired,practical,anderrorpositionvectorsthataredefinedintheDCBaredenotedasRd,Pd,andEd,respectively.TotransformthereferencepositionvectorfromWCBtoDCB,aninverse-kinematicstransformationalgorithmisrequiredtobeimplementedintheinterpolator.Notethatinpracticethemechanicalstructureofthefive-axismachinetoolplaysaroleasadirect-kinematicstransformationthatconvertsPdtoPw.LetK(·)andKH110021(·)representthedirect-andinverse-kinematicstransformations,respectively.Inotherwords,wehaveRwH11005K(Rd),PwH11005K(Pd),RdH11005K1(Rw)andPd(1)H11005K1(Pw).Thefunctionofthefivecontrolloopsistotrackthereferencepositioncommandsthataregeneratedbytheinterpolator.Foreachloop,thecontrollerobjectiveistominimizethepositionerroralongthedrivingaxis.LetHdandGdberespectivelythetransferfunctionmatr-icesforthecontrollersandthedrives.Inthematrices,HdandGd,thenon-diagonaltermsarezerosandthediagonaltermsarethetransferfunctionsfortheaxialcontrollersandtheaxialdrives,respectively.Notethatinacomputer-controlledsystem,HdandGdarefunc-tionsofz-variable(indiscrete-timedomain).Thestrat-egyoftheconventionalfive-axiscontrolsystemistoreducethepositionerrorsalongthedrivingaxes(i.e.,Ed=RdH11002Pd),andthen,toexpectaqualityfive-axistool-pathcontrolthatfocusesontheeliminationofthedevi-ationerror,theorientationerror,andthetracking-lagerror.However,theexpectationisindoubt.Ashasbeenillustratedintheabovesection(refertoFig.2),thereductionofEddoesnotnecessarilycorrespondtothereductionofthedeviationerror,etc.4.Proposedfive-axiscontrolsystemTheproposedfive-axiscontrolsystemisdepictedinFig.6.Incontrasttotheconventionalsystemthatcon-Fig.6.Theproposedcontrolsystemforfive-axismachinetools.structsfivelocalandseparatecontrolloops(refertoFig.5),theproposedcontrolsystemconstructsaglobalandcoupledlooptoachieveaneffectivecontroloftheover-allperformancethatisintermsofthedeviationerror,theorientationerror,andthetrackinglag.Thedeviationerror,etc.,whichwillbederivedlaterinthefollowing,areerrorcomponentsdefinedintheWCB.Incontrast,thefedbackpositionsignals(Pd)andthecontrolsignals(Ud)senttotheaxialdrivesarebothdefinedintheDCB.Consequently,coordinatetransformationsareintroducedtotheproposedcontrolsystem.AsdepictedinFig.6,theservocontrollerconsistsoffourparts:(1)adirect-kinematicstransformationalgorithmthatcalculatesthepracticaltoolpositioninWCB,i.e.,Pw=K(Pd)；(2)anerrormodelforcalculationofthedeviationerrorandtheorientationerror(thatarerepresentedbye),andthetracking-lagerror(d)；(3)acontrollawthateliminateseandd；(4)aninverse-JacobianmatrixthattransformsthecontrolintheWCB(i.e.,Uw)tothatintheDCB(i.e.,Ud).Throughtheaboveprocedure,theproposedcontrolsystemfocusesitscontroleffortintheWCBandconductsadirecteliminationofeandd.Thelastthreepartsoftheservocontrolleraredescribedindetailinthefollowing.4.1.ErrormodelAsstatedabove,thedeviationerror,theorientationerror,andthetracking-lagerrorarethemainconcernsforfive-axistool-pathtrackingcontrol.Therefore,themodelfortheseconcernederrorsisthecoreofthepro-posedservocontroller.AschematicillustrationfortheseconcernederrorsisshowninFig.7.InFig.7,Cwdenotesthepositionthatislocatedonthedesiredtoolpathandistheclosesttothepracticaltoolposition,Pw.LetthedifferencebetweenCwandPwbedenotedase,i.e.,eH11005CwH11002Pw.(2)Notethateconsistsoffivecomponents,i.e.,e=(ex,ey,ez,ea,eb).82C.-C.Lo/InternationalJournalofMachineTools&Manufacture42(2002)7988Fig.7.ThereferencepositionRw,thepracticalpositionPw,andtheclosestpositionCwfromPwtothedesiredtoolpath.Ashasbeendefinedabove,thedeviationerroristhedistancebetweenthepracticaltoollocationandtheclos-estpointonthedesiredtoolpath.Consequently,thefirstthreecomponentsofeareinpracticethecomponentsofthedeviationerror(ed),i.e.,edH11005H20881(ex)2+(ey)2+(ez)2.(3)Theorientationerroristheanglebetweenthetoolaxisandthetoolorientationfortheclosestpointonthedesiredtoolpath.Inthispaper,thetworotationalanglesaredefinedsothatthetoolwithrespecttotheWCBisoriginallyinthez-direction,thenrotateswithaalongthex-axis,andfinallyrotateswithbalongthey-axis.Basedonthedefinitionsofthetworotationalangles(a,b),theorientationerror(f)canbecalculatedbyfH11005cos1H20898cos(Pwa)sin(Pwb),H11002sin(Pwa),cos(Pwa)cos(Pwb)·H20900cos(Pwa+ea)sin(Pwb+eb)sin(Pwa+ea)cos(Pwa+ea)cos(Pwb+eb)H20901H20899.(4)AsshowninEq.(4),theorientationerrorisdeterminedbyeaandeb,whicharethelasttwocomponentsofe.Therefore,intheproposedcontrolsystem,theelimin-ationofthedeviationerrorandtheorientationerrorcanbeconductedthroughthecontrolofe.Thetracking-lagerror(d)istheprojectionvectorofEw(=RwH11002Pw)alongthetoolpath.AccordingtoFig.7,wehavedH11005RwH11002Cw.(5)Notethatdconsistsoffivecomponents,i.e.,d=(dx,dy,dz,da,db),andthefirstthreecomponentsofdareforthetracking-lagdistance(dd),i.e.,ddH11005H20881(dx)2+(dy)2+(dz)2.(6)SubstitutingEq.(5)intoEq.(2)yieldseH11005(RwH11002d)H11002PwH11005EwH11002d.(7)AscanbeseeninEq.(7),ifdisdetermined,eisalsoobtained.However,ananalyticalsolutionofdisnotavailableforgeneraltrajectories.Anumericaliterativemethodistime-consumingandisnotsuggestedforreal-timecontrol.Therefore,anapproximatedmodelforcal-culationofdisrecommendedhere.AsshowninFig.7,Cwisapositionvectorthatislocatedonthedesiredtoolpathandlagsbehindtheinstantaneousreferencepositionvector,Rw.Thedistance(dd)thatCwlagsbehindRwcanbeapproximatedbytheprojectionofthepositionerrorvector(Ewx,Ewy,Ewz)onthetangentialdirectionofthepathonRw,i.e.,ddH11015EwxtxH11001EwytyH11001Ewztz,(8)where(tx,ty,tz)isthetangentialvectortothetoolpathonRw.Thetracking-lagerror(d)canberegardedasavari-ationofthereferencepositionvectorRwaccordingtothetracking-lagdistance,dd.Consequently,dcanbeapproximatedbydH11005RwH11002CwH11015dd·dRwdlH11001d2d2·d2Rwdl2,(9)whereddiscalculatedbyEq.(8)；thevariablelisthepathlengthalongthedesiredtoolpathdRw/dlandd2Rw/dl2arethefirstandsecondderivativesoftherefer-encepositionvectorwithrespecttothepathlength(l).InaCNCsystem,dRw/dlandd2Rw/dl2canbeapproxi-matedbydRwdlH11015Rw(kT)Rw(kTT)fT,(10)d2Rwdl2H11015Rw(kT)2Rw(kTT)+Rw(kT2T)(fT)2,(11)wherefisthefeedrate,Tisthesamplingperiod,andkTdenotesthesamplinginstant.4.2.ControllerlawLetHeandHdbethetransferfunctionmatricesforthecontrollawsforeandd,respectively.NotethatbothHeandHdare5×5diagonalmatrices,forwhicheachdiagonaltermrepresentsthecorrespondingcontrollaw