外文文献翻译五轴机床刀具路径控制的一种方案

附录，文献原文及翻译（1）原文Atool-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.2001ElsevierScienceLtd.Allrightsreserved.Keywords:Five-axismachinetool;Servocontroller;Tool-pathtrackingcontrol1.IntroductionToachievehighprecisioninmodernCNC(computernumericalcontrol)machiningapplications,designofservocontrolsystemsthatgenerateaccuratecoordinatedmulti-axismotionisofgreatimportance.Tosynchronizethemotionsofthedifferentaxeswhenmachiningacomplexsurface,aconventionalmulti-axisservocontrolsystemconsistsofaninterpolatorandseveralaxialcontrollers.Theinterpolatorgeneratesthedesiredtoolmotionthatisrelativetotheworkpiece,andthen,decomposesthedesiredmotionintothereferencepositioncommandsfortheseparatedrivingaxes[1–4].Thepracticalmotionisrealizedbythedrivingaxes.Eachaxisiscontrolledbyanaxialcontroller,whoseobjectiveistotracktheaxialpositioncommand(i.e.,toeliminatethepositionerroralongeachdrivingaxis).Manyresearchershavedevelopedcontrolalgorithmsthatimprovethetrackingaccuracyforanindividualaxis.Traditionalalgorithmsarebasedonthefeedbackprin-ciple[5,6].Inaddition,feedforwardcontrolalgorithmscanbeimplementedtoaugmentthetrackingperformance.Currently,asignificantcontributionhasbeenmadebyTomizuka[7],whoproposedazerophaseerrortrackingcontroller(ZPETC).OnthebasisoftheZPETCmethod,somevariationaloraccessoryalgorithms(e.g.,adaptiveZPETC)havebeenproposed[8,9].Althoughthetrackingperformanceforeachindividualaxiscanbesignificantlyimprovedbytheabovemethods,theoverallcontrolperformanceforthemulti-axismachinetoolisnotalwaysguaranteed[6].Atypicalperformanceindexforevaluationofthemulti-axisservocontrolisthecontourerror,whichdenotesthedeviationfromthedesiredtoolpath.Toconductaneffectivereductionofthecontourerror,Koren[10]proposedacross-couplingcontroller(CCC)thatisconstructedbetweenandparalleltotheaxialcontrollers.Atypicalcross-couplingcontrollerconsistsofareal-timecalculationofthecontourerrorandacontrollawtoeliminatethecontourerror.BasedontheconceptoftheCCCmethod,numerouscross-couplingcontrollers(withdifferentcontour-errormodelsand/orcontrollerlaws)havebeenproposed[11–13].Thecontourerror,however,isnottheonlyconcernformulti-axistool-pathtrackingcontrol.Forinstance,thepositionlagalongthetrackingdirectionisanotherconcern[6].Besides,theorientationerror,whichdenotesthedeviationanglebetweenthepracticaltoolaxisandthedesiredtooldirection,isalsoanimportantconcerninfour-orfive-axismachine-toolcontrol[14].Inthispaper,themainconcernsforafive-axistool-pathcontrolarediscussedfirst.Then,aconventionalfive-axiscontrolsystemisdiscussed,anditsdrawbackisaddressed.Finally,afive-axiscontrolsystemthatconductsadirecteliminationoftheseconcernsisproposedandcomparedwiththeconventionalone.Fig.1.ThetoolpathalongthesculpturedsurfaceFig.2. Machininginaccuracyduetoimperfecttool-pathtrackingcontrol.2.Mainconcernsinfive-axistool-pathcontrolLet’sconsiderthefollowingfive-axismachiningcase(asreferredtoinFig.1):utilizingacylindricaltooltocutasurface.Attheplanningstage,thetoolpaththatcomprisesthetool-centerlocation(L)andthetoolorien-tation(O)isscheduledsothatthecutteredge(S)canpassoverthesculpturedsurface[15,16].Here,welet RandPdenotethereferencepositionvectorandthepracticalpositionvector,respectively.BothRandParepositionvectorswithfivecomponents(threeforthetool-centerlocationL andtwoforthetoolorientation O).Thedifference(orerror)betweenthereferenceandthepracticalpositionvectors(i.e., ER-P)isaconcerninfive-axismachining.However, Eisnotthemainconcern,becauseasmall Edoesnotnecessarilyguaranteeanegligiblemachininginaccuracy.AsillustratedinFig.2,although P(2)ismuchcloserto RthanP(1)(i.e.,|E(2)| |E(1)|),itresultsinmoremachininginaccuracy.TwomaincausesforpartinaccuracyareillustratedinFig.3.AsshowninFig.3(a),thedeviationerror(ed),whichdenotesthedistancebetweenthepracticaltoollocation(P)andtheclosestlocation(C)onthedesiredtoolpath(ratherthantheinstantaneousreference R),isanimportantconcern.AsshowninFig.3(b),theorien-tationerror( ),whichdenotestheanglebetweenthepracticaltoolaxisandthedesiredtooldirectioncorre-spondingtoC,isanotherimportantconcern.AscanbeseeninFig.3,thedeviationerrorandtheorientationerroraremaincausesformachininginaccuracy.Inadditiontothedeviationerrorandtheorientationerror,atracking-lagFig.3.ThedeviationerrorandtheorientationerrorFig4Twoconsecutivepathsthattheinfive-axismachiningcontroltoolpathpassesover.(a)deviationerror;(b)orientationerror.error(d)thatdenotesthecomponentof E alongthetrackingdirectionisalsoanimportantconcern.AsshowninFig.4,asignificanttrackinglagwillalsocauseanunacceptablemachininginaccuracybetweentwoconsecutivesurfaces.3.Conventionalfive-axiscontrolsystemTypically,afive-axismachinetoolconsistsofthreetranslationalaxes(x,y,z)andtworotationalaxes(a,b).Theblockdiagramforaconventionalcontrolsystemforfive-axismachinetoolsisshowninFig.5.Inthsystemtheinterpolator,whichconsistsofapath-plan-ningmoduleandaninverse-kinematicstransformation[2,14],generatesinrealtimethedesiredreferencepositioncommandstothefiveseparatecontrolloops(respectivelyforthex-,y-,z-,a-,andb-axis).Thepath-planningmodulegeneratesthedesiredtoolmotionrela-tivelytotheworkpiece.Inotherwords,thetoolpathisdefinedintheworkpiece-coordinatebasis(WCB)forwhichtheaxesarefixedontheworkpiece.Inthefollowing,thedesired,practical,anderrorpositionvectorsthataredefinedintheWCBaredenotedasRw,Pw,andEw,respectively.Incontrasttothepath-planningmodule,theaxialcontrolloopsfocustheireffortontrackingtheindi-vidualmotionsalongthefivedrivingaxes.Thesemotions,however,aredefinedinthe drive-coordinatebasis(DCB).Inthefollowing,thedesired,practical,anderrorpositionvectorsthataredefinedintheDCBaredenotedasRd,Pd,andEd,respectively.TotransformthereferencepositionvectorfromWCBtoDCB,aninverse-kinematicstransformationalgorithmisrequiredtobeimplementedintheinterpolator.Notethatinpracticethemechanicalstructureofthefive-axismachinetoolplaysaroleasadirect-kinematicstransformationthatconvertsPdto Pw.Let K(·)and K1(·)representthedirect-andinverse-kinematics transformations, respectively. Inotherwords,wehaveRwK(Rd),PwK(Pd),RdK−1(Rw)andPdK−1(Pw).(1)Fig.5. Aconventionalcontrolsystemforfive-axismachinetoolsThefunctionofthefivecontrolloopsistotrackthereferencepositioncommandsthataregeneratedbytheinterpolator.Foreachloop,thecontrollerobjectiveistominimizethepositionerroralongthedrivingaxis.Let[Hd]and[Gd]berespectivelythetransferfunctionmatricesforthecontrollersandthedrives.Inthematrices,[Hd]and[Gd],thenon-diagonaltermsarezerosandthediagonaltermsarethetransferfunctionsfortheaxialcontrollersandtheaxialdrives,respectively.Notethatinacomputer-controlledsystem,[Hd]and[Gd]arefunctionsof z-variable(indiscrete-timedomain).Thestrat-egyoftheconventionalfive-axiscontrolsystemistoreducethepositionerrorsalongthedrivingaxes(i.e.,EdRdPd),andthen,toexpectaqualityfive-axistool-pathcontrolthatfocusesontheeliminationofthedeviationerror,theorientationerror,andthetracking-lagerror.However,theexpectationisindoubt.Ashasbeenillustratedintheabovesection(refertoFig.2),thereductionof Eddoesnotnecessarilycorrespondtothereductionofthedeviationerror,etc.4.Proposedfive-axiscontrolsystemTheproposedfive-axiscontrolsystemisdepictedinFig.6.Incontrasttotheconventionalsystemthatconstructsfivelocalandseparatecontrolloops(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., PwK(Pd);(2)anerrormodelforcalculationofthedeviationerrorantheorientationerror(thatarerepresentedbye),andthetracking-lagerror(d);(3)acontrollawthateliminateseandd;(4)aninverse-JacobianmatrixthattransformsthecontrolintheWCB(i.e., Uw)tothatintheDCB(i.e., Ud).Throughtheaboveprocedure,theproposedcontrolsystemfocusesitscontroleffortintheWCBandconductsadirecteliminationofeandd.ThelastthreepartsoftheservocontrolleraredescribedindetailinthefollowingdFig.6. Theproposedcontrolsystemforfive-axismachinetools.4.1.ErrormodelAsstatedabove,thedeviationerror,theorientationerror,andthetracking-lagerrorarethemainconcernsforfive-axistool-pathtrackingcontrol.Therefore,themodelfortheseconcernederrorsisthecoreoftheproposedservocontroller.AschematicillustrationfortheseconcernederrorsisshowninFig.7.InFig.7,Cwdenotesthepositionthatislocatedonthedesiredtoolpathandistheclosesttothepracticaltoolposition,Pw.LetthedifferencebetweenCwandPwbedenotedase,i.e.,eCwPw. (2)Note that e consists of fivecomponents, i.e.,e(ex,ey,ez,ea,eb).CwfromPwtothedesiredtoolpath.Fig.7. ThereferencepositionRw,thepracticalpositionPw,andtheclosestpositionAshasbeendefinedabove,thedeviationerroristhedistancebetweenthepracticaltoollocationandtheclosestpointonthedesiredtoolpath.Consequently,thefirstthreecomponentsofeareinpracticethecomponentsofthedeviationerror(ed),i.e.,ed(ex)2+(ey)2+(ez)2. (3)Theorientationerroristheanglebetweenthetoolaxisandthetoolorientationfortheclosestpointonthedesiredtoolpath.Inthispaper,thetworotationalanglesaredefinedsothatthetoolwithrespecttotheWCBisoriginallyinthez-direction,thenrotateswithaalongthe x-axis,andfinallyrotateswith b alongthey-axis.Basedonthedefinitionsofthetworotationalangles(a,b),theorientationerror(f)canbecalculatedbyf cos−1[cos(Pwa)sin(Pwb),sin(Pwa),cos(Pwa)cos(Pwb)]·cos(Pwa+ea)sin(Pwb+eb)(4),−sin(Pwa+ea)cos(Pwa+ea)cos(Pwb+eb)AsshowninEq.theorientationerrorisdeterminedbyeaandeb,whicharethelasttwocomponentsofe.Therefore,intheproposedcontrolsystem,theeliminationofthedeviationerrorandtheorientationerrorcanbeconductedthroughthecontrolofe.Thetracking-lagerror(d)istheprojectionvectorofEw(RwPw)alongthetoolpath.AccordingtoFig.7,wehaved RwCw.(5)Notethatdconsistsoffive components, i.e.,d(dx,dy,dz,da,db),andthefirstthreecomponentsofdareforthetracking-lagdistance(dd),i.e.,dd(dx)2+(dy)2+(dz)2(6)SubstitutingEq.(5)intoEq.(2)yieldse(Rwd)PwEwd(7)AscanbeseeninEq.(7),ifdisdetermined,eisalsoobtained.However,ananalyticalsolutionof d isnotavailableforgeneraltrajectories.Anumericaliterativemethodistime-consumingandisnotsuggestedforreal-timecontrol.Therefore,anapproximatedmodelforcalculationofdisrecommendedhere.AsshowninFig.7,Cwisapositionvectorthatislocatedonthedesiredtoolpathandlagsbehindtheinstantaneousreferencepositionvector,Rw.Thedistance(dd)thatCwlagsbehindRwcanbeapproximatedbytheprojectionofthepositionerrorvector(Ewx,Ewy,Ewz)onthetangentialdirectionofthepathonRw,i.e.,ddEwxtxEwytyEwztz,(8)where(tx,ty,tz)isthetangentialvectortothetoolpathonRw.Thetracking-lagerror(d)canberegardedasavariationofthereferencepositionvector Rwaccordingtothetracking-lagdistance,dd.Consequently, d canbeapproximatedbyWhereddiscalculatedbyEq.(8);thevariable l isthepathlengthalongthedesiredtoolpath dRw/dl andd2Rw/dl2arethefirstandsecondderivativesofthereferencepositionvectorwithrespecttothepathlength(l).InaCNCsystem,dRw/dlandd2Rw/dl2canbeapproxi-matedbywherefisthefeedrate,Tisthesamplingperiod,andkTdenotesthesamplinginstant.4.2.ControllerlawLet[He]and[Hd]bethetransferfunctionmatricesforthecontrollawsforeandd,respectively.Notethatboth[He]and[Hd]are55diagonalmatrices,forwhicheachdiagonaltermrepresentsthecorrespondingcontrollawforeachcomponentofeandd.Withthecontrollaws,thecontrollercommandsareUw[He]·e[Hd]·d (12)4.3.Inverse-JacobianmatrixInthefinalstep,thecontrolsignalsintheWCBaretransformedtothoseintheDCB,andthen,fedtothefiveaxialdrives.ThistransformationisconductedbymultiplyingthecontrollercommandsbytheinverseoftheJacobianmatrix,i.e.,Ud[J]−1Uw,wheretheinverse-Jacobianmatrixaredefinedas∂Pdj∂Pwi;i,j x,y,z,a,b(14)5.StabilityconsiderationForsimplicity,weadoptthesamecontrollawfor eand d (i.e.,letting[He][Hd][Hw])inthefollowing.Consequently,basedonEqs.(7)and(12),wecanhaveUw[Hw]Ew[Hw](RwPw). (15)Consequently,theoutputsofthedrivesarePd[Gd][J]−1[Hw]Ew.(16)Inpractice,EdandEwareincrementalpositionvectorsforPdandPw,respectively.Accordingtothedefinitionoftheinverse-JacobianmatrixreferredtoEq.(14),wecanhaveEd[J]−1Ew,(17)andconsequently,getPdRdEdK−1(Rw) [J]−1Ew.(18)ThecombinationofEqs.(16)and(18)yieldsEw([J]−1[Gd][J]−1[Hw])−1·K−1(Rw). (19)Consequently,thecharacteristicequationfortheproposedcontrolsystemis(z) ||[J]−1[Gd][J]−1[Hw]|| 0. (20)Itiswellknownthat,tosatisfythestabilityrequirementforacomputer-controlledsystem,therootsoftheaboveequationmustbelocatedinsidetheunitcircle.BecauseEq.(20)includestheinverse-Jacobianmatrix[J]1,theconfigurationofthemachinetoolisintroducedtothedesignofthecontrollaw.Thiswillbringadifficultytothedesignofthecontrollerlaw,becausethemachine-toolconfigurationnotonlydependsonthemachine-toolstructurebutalsoisvaryingduringthemachiningprocess.However,iftheparametersofthecontrollawsarethesameforthedifferentcomponentsof[Hw](i.e.,[He]and[Hd]),wecanhave[Hw] hw[I],（21）where[I]isa55identitymatrixandhwisthetransferfunctionfortheindividualcontrollaw.BasedonEq.(21),thecharacteristicequationcanbesimplifiedas(z) ||[I] hw[Gd]||(1 hwgi) （22）where giisthetransferfunctionforanaxialdrive.AsillustratedbyEq.(22),ifhwcanseparatelystabilizethefivedrives,thesystem’sstabilityisassured.Notethattheabovesimplifiedresultistrueonlywhenadoptingthesamecontrollerparametersforthedifferentaxialcomponents.Ifdifferentcontrollerparametersareutilizedforthedifferentaxialcomponents,weshouldcheckthesystem’sstabilitybasedonEq.(20).6.SimulationexamplesAschematicillustrationforthefive-axismachinetoolusedinthesimulationisshowninFig.8,wherethecorrespondingdrivingaxesaredefined.Themachinetoolincludesthreeslidingaxes(forx,yandz)andtwotiltingaxes(foraandb)sothatthetoolisfixedandtheworkpieceisdriven.InFig.8,L(lxxˆ+lyyˆ+lzzˆ)denotesthecutterlocationandisafixedpointontheFig.8. Themachinetoolstructureusedinthesimulation.machine.Misapointdrivenbythethreeslidingaxesandisthepivotoftherotationalangle a. C isthepivotoftherotationalangle b andistheoriginoftheworkpiececoordinate frame. Besides, we define C Mmxxˆ+myyˆ+mzzˆ.Thecontinuous-timemodel(inLaplace-ors-domain)forthemotor-drivenservomechanismischosenas[5,6]wherekiandti(ix,y,z,a,b)arerespectivelythespeedgainsandthetimeconstants.Precededbyazero-order-hold(ZOH),thedrive’sdigitaltransferfunctionisrepresentedbyInthesimulation,atypicalPIDcontrollawisutilizedforboththeconventionalmethod(forcontrolofEd)andtheproposedcontrolmethod(forcontrolofeandd).ThetransferfunctionofthePIDcontrollerisdescribedbywhere hp, hi,and hdareproportional,integral,andderivativegains,respectively.Notethatinordertogetafaircomparison,thesamesetofPIDgainsarechosenforthetwodifferentmethods.ThemachinegeometryandthesystemparametersusedinthesimulationarelistedinTable1.Notethatweintroducesomedifferences(ormismatches)betweenthedrivedynamicsofthedifferentaxes(5%forthespeedgainsand20%forthetimeconstants).ThePIDgainsarechosensothatthestabilityrequirementissatisfied.Fortheconventionalcontrolmethod,theinverse-kin-ematicsalgorithmthattransforms Rwto RdcanbedescribedbyRdx−Rwxcos(Rwb)−Rwzsin(Rwb)（27）Rdymzsin(Rwa)−Rwxsin(Rwa)sin(Rwb)−Rwycos(Rwa)+Rwzsin(Rwa)cos(Rw)Rdzlz−mzcos(Rwa)−Rwxcos(Rwa)sin(Rwb)−Rwysin(Rwa)−Rwzcos(Rwa)cos(Rwb)RdaRwaRdbRwbFortheproposedcontrolmethod,thedirect-kinemat-icsalgorithmthattransforms Pdto PwcanbedescribedbyPwxmzsin(Pdb)−Pdxcos(Pdb)−Pdysin(Pda)sin(Pdb)−(lz−Pdz)cos(Pda)sin(Pdb)Pwy−Pdycos(Pda)+(lz−Pdz)sin(Pda)Pwx−mzcos(Pdb)−Pdxsin(Pdb)+Pdysin(Pda)cos(Pdb)−(lz−Pdz)cos(Pda)cos(Pdb)PwaPdaPwbPdb(28)BasedonEqs.(14)and(27),theinverse-JacobianmatrixthattransformsUwtoUdcanbeformulatedasInthefirstsimulationexample,acylindricalsurfaceiscutbyacylindricaltool.Thecylindricalsurfaceisexpressedas(x+y)2+2z2800.AsillustratedinFig.9,Sisthecutter–contactlocationthatfollowsacircularpathonthesurface,whileListhecutter–centerlocationthatisobtainedthroughcutter–offsettingFig.[15,16].Toachievepivotoftherotationalangle a. C isthepivotoftherotationalangle b andistheoriginoftheworkpiececoordinate frame. Fig.9. Five-axismachiningofacylindricalsurfaceBesides,we define C Mmxxˆ+myyˆ+mzzˆ.Thecontinuous-timemodel(inLaplace-ors-domain)forthemotor-drivenservomechanismischosenas[5,6]wherekiandti(ix,y,z,a,b)arerespectivelythespeedgainsandthetimeconstants.Precededbyazero-order-hold(ZOH),thedrive’sdigitaltransferfunctionisrepresentedbyInthesimulation,atypicalPIDcontrollawisutilizedforboththeconventionalmethod(forcontrolofEd)andtheproposedcontrolmethod(forcontrolofeandd).ThetransferfunctionofthePIDcontrollerisdescribedbywhere hp, hi,and hdareproportional,integral,andderivativegains,respectively.Notethatinordertogetafaircomparison,thesamesetofPIDgainsarechosenforthetwodifferentmethods.ThemachinegeometryandthesystemparametersusedinthesimulationarelistedinTable1.Notethatweintroducesomedifferences(ormismatches)betweenthedrivedynamicsofthedifferentaxes(5%forthespeedgainsand20%forthetimeconstants).ThePIDgainsarechosensothatthestabilityrequirementissatis-fied.Fortheconventionalcontrolmethod,theinverse-kin-ematicsalgorithmthattransforms。Rdx−Rwxcos(Rwb)−Rwzsin(Rwb)Rdymzsin(Rwa)−Rwxsin(Rwa)sin(Rwb)−Rwycos(Rwa)+Rwzsin(Rwa)cos(Rwb)(27)Rdzlz−mzcos(Rwa)−Rwxcos(Rwa)sin(Rwb)−Rwysin(Rwa)−Rwzcos(Rwa)cos(Rwb)RdaRwaRdbRwbFortheproposedcontrolmethod,thedirect-kinematicsalgorithmthattransformsPdtoPwcanbedescribedbyPwxmzsin(Pdb)−Pdxcos(Pdb)−Pdysin(Pda)sin(Pdb)−(lz−Pdz)cos(Pda)sin(Pdb)Pwy−Pdycos(Pda)+(lz−Pdz)sin(Pda)Pwx−mzcos(Pdb)−Pdxsin(Pdb)+Pdysin(Pda)cos(Pdb)−(lz−Pdz)cos(Pda)cos(Pdb)PwaPdaPwbPdb(28)BasedonEqs.(14)and(27),theinverse-JacobianmatrixthattransformsUwtoUdcanbeformulatedasInthefirstsimulationexample,acylindricalsurfaceiscutbyacylindricaltool.Thecylindricalsurfaceisexpressedas(x+y)2+2z2800.AsillustratedinFig.9,Sisthecutter–contactlocationthatfollowsacircularpathonthesurface,whileListhecutter–centerlocationthatisobtainedthroughcutter–offsetting[15,16].Toachievehighefficientmachiningofconvexsurface[17],wemaysettheinclinationandthetwistinganglesabout S aszero. Consequently, thetool-orientation vector,OO(a,b),isequaltothenormalvectortothesurfaceonS.Basedontheaboveassignment,aspecifictoolpath(Rw)isscheduledasRwz20sin(q)+rtcos(q))Rwa−sin−1cos(q)(31)Rwbtan−12tan(q)Wherert(5mm)isthetoolradiusand q isthepathparameterthatstartsfrom90to0(inCW).Thefive-axismachiningisconductedatafeedrateof450mm/min.Inthesecondexample,aconicalsurfaceiscut.Theconicalsurfaceisexpressedasx2+y2(45 2z)2.Asillus-tratedinFig.10,thecutter–contactFig.10. Five-axismachiningofaconicalsurfacelocation(S)followsacircularpathonthesurfaceandthetoolisneitherinclinednortwistedabout S.Consequently,aspecifictoolpath(Rw)isscheduledasRwx15cos(u)+rtsin(u)(32)where rt(5mm)isthetoolradiusanduisthepara-meteralongthetoolpath.Fig.11. Five-axismachiningofaruledsurface.Inthesimulation,ustartsfrom0to2p(inCCW).Thefive-axismachiningisconductedatafeedrateof600m/min.Inthethirdexample,aruledsurfaceiscut.Theruledsurfaceisexpressedax 2(20u3−30u2+30u+5)+20vy 2(20u3−30u2+30u+5)−20v(33)whereuandvarethesurfaceparameters.AsillustratedinFig.11,thecutter–contactlocation(S)followsacubicsplinepath(intheu-direction).Inordertoavoidthereargaugingattheconcavepart,thecutterisinclinedat13.Notethatananalyticalsolutionforthetoolpath(suchasEqs.(31)and(32)fortheprevioustwoexamples)isnotavailable.However,wecanobtainthenumericalsolutionthroughthetooloffsetting.Inthesimulationexample,aspecifictoolpathisscheduledalongv0.Theproposedtool-pathcontrolmethodisevaluatedandcomparedwiththeconventionalmethod.Thecomparisonisbasedontheircapabilityineliminationofthedeviationerror(ed),theorientationerror(f),andthetracking-lagerror(dd).ThesimulationresultsforthethreeexamplesareshowninFigs.12–14,andaresum-marizedinTable1.Basedonthesimulationresults,wecanhavethefollowingobservations.1.Thedeviationerror(ed)issignificantlyreducedbytheproposedmethod(forboththemaximumandthemeanvalues).2.Theorientationerror(f)issignificantlyreducedbytheproposedmethodforthefirstexample.Forthesecondandthethirdexamples,onlyalittleimprovementcanbeachieved.AsillustratedinFig.3(a),thetoolorientationerrorcancauseasurfaceerror(overcutorundercut)nearthecuttercontactlocation(S).Geometrically,thiserrorcanbeapproximatedbyrtf where rtisthetoolradius.Fortheabovethreeexamples,themaximumorientationerrorislessthan0.06deg(0.001rad).Accordingly,thesurfaceerrorduetothetoolorientationerrorislessthan0.005mm.Ascomparedwiththetooldeviationerror(ed),thisisnotimportant.3.Thetracking-lagerror(dd)isnotsignificantlyreducedbytheproposedmethod(especiallyatthetransientstate).However,ashasbeenillustratedinFig.4,thetracking-lagerrordoesnotcausemachiningerrorduringthetracking.Itplaysanimportantroleonlyattheendofthepathoronthecorneroftwoconsecutivepaths.AscanbeseeninFigs.12(c),13(c)and14(c),theproposedmethodcandrivethetracking-lagerrortoaverysmallvalueatthesteadystate.Thismeansthatthemachiningerrorattheendofthepath(orthecorner)canbewellcontrolledbytheproposedmethod(ascomparedwiththeconventionalmethod).However,wemaynotreallywantddtogotozero,especiallywhenasignificantovershooterroratthecornerisnotallowed.Toavoidasignificantovershootatthecorner,wecanutilizePDlaw(byremovingtheintegralpart)tocontroldd.However,PDcontrolmayresultinalargedd,andconsequentlycauseanunder-cutatthecorner(pleaserefertoFig.4).Therefore,whenchoosingthePIDgainsfor dd,thereisatrade-offbetweentheovercutandtheundercutatthecorner.Fig.12. SimulationresultsformachiningofFig.13. Simulationresultsformachiningacylindricalsurface;solidline:conventionalofaconicalsurface;solidline:conventionalmethod,dashedline:proposedmethod.method,dashedline:proposedmethod.表14.Inordertoobtainafaircomparisonbetweentheconventionalandtheproposedmethods,weutilizethesamePIDgains.Inpractice,toaugmenttheFig.14. Simulationresultsformachiningofarulesurface;solidline:conventionalmethod,dashedline:proposedmethodperformanceoftheproposedmethod,wemayutilizedifferentPIDgains(orevendifferentcontrollaws)forcontrollinged,fanddd.Inotherwords,[He]isdifferentto[Hd].Forinstance,wecanutilizehigh-gainPIDcontrolfor[He](becauseourpurposeistoeliminateedand f),andutilizemedium-gainPDcontrolfor[Hd](sothatddisneithertoolargenortoosmall).7.ConclusionThemainconcernsforfive-axistool-pathtrackingcontrolarethedeviationerror,theorientationerror,andthetracking-lagerror.Consequently,theconven