附錄ControlofaNon-OrthogonalReconfigurableMachineToolReuvenKatzJohnYookYoramKorenReceived:January3,2003；revised:September16,2003AbstractComputerizedcontrolsystemsformachinetoolsmustgeneratecoordinatedmovementsoftheseparatelydrivenaxesofmotioninordertotraceaccuratelyapredeterminedpathofthecuttingtoolrelativetotheworkpiece.However,sincethedynamicpropertiesoftheindividualmachineaxesarenotexactlyequal,undesiredcontourerrorsaregenerated.Thecontourerrorisdefinedasthedistancebetweenthepredeterminedandactualpathofthecuttingtool.Thecross-couplingcontroller(CCC)strategywasintroducedtoeffectivelydecreasethecontourerrorsinconventional,orthogonalmachinetools.Thispaper,however,dealswithanewclassofmachinesthathavenon-orthogonalaxesofmotionandcalledreconfigurablemachinetools(RMTs).Thesemachinesmaybeincludedinlarge-scalereconfigurablemachiningsystems(RMSs).Whentheaxesofthemachinearenon-orthogonal,themovementbetweentheaxesistightlycoupledandtheimportanceofcoordinatedmovementamongtheaxesbecomesevengreater.Inthecaseofanon-orthogonalRMT,inadditiontothecontourerror,anothermachiningerrorcalledin-deptherrorisalsogeneratedduetothenon-orthogonalnatureofthemachine.Thefocusofthisstudyisontheconceptualdesignofanewtypeofcross-couplingcontrollerforanon-orthogonalmachinetoolthatdecreasesboththecontourandthein-depthmachiningerrors.Varioustypesofcross-couplingcontrollers,symmetricandnon-symmetric,withandwithoutfeedforward,aresuggestedandstudied.Thestabilityofthecontrolsystemisinvestigated,andsimulationisusedtocomparethedifferenttypesofcontrollers.Weshowthatbyusingcross-couplingcontrollersthereductionofmachiningerrorsaresignificantlyreducedincomparisonwiththeconventionalde-coupledcontroller.Furthermore,itisshownthatthenon-symmetriccross-couplingfeedforward(NS-CC-FF)controllerdemonstratesthebestresultsandistheleadingconceptfornon-orthogonalmachinetools.2004ASMEContributedbytheDynamicSystems,Measurement,andControlDivisionofTHEAMERICANSOCIETYOFMECHANICALENGINEERSforpublicationintheASMEJOURNALOFDYNAMICSYSTEMS,MEASUREMENT,ANDCONTROL.ManuscriptreceivedbytheASMEDynamicSystemsandControlDivisionJanuary3,2003；finalrevisionSeptember16,2003.AssociateEditor:J.Tu.Keywords:machinetool,cross-couplingcontroller,non-orthogonal,RMT1IntroductionCurrentlymanufacturingindustrieshavetwoprimarymethodsforproducingmediumandhighvolumemachinedparts:dedicatedmachiningsystems(DMSs)andflexiblemanufacturingsystems(FMSs)thatincludeCNCmachines.TheDMSisanidealsolutionwhenthepartdesignisfixedandmassproductionisrequiredtoreducecost.Ontheotherhand,theFMSisidealwhentherequiredquantitiesarenotsohighandmanymodificationsinthepartdesignareforeseen.Incontrasttothesetwoextremes,Korendescribesaninnovativeapproachofcustomizedmanufacturingcalledreconfigurablemanufacturingsystems(RMS).ThemainadvantageofthisnewapproachisthecustomizedflexibilityinthesystemtoproduceapartfamilywithlowerinvestmentcostthanFMS.AtypicalRMSincludesbothconventionalCNCmachinesandanewtypeofmachinecalledtheReconfigurableMachineTool.TheEngineeringResearchCenter(ERC)forReconfigurableMachiningSystems(RMS)attheUniversityofMichiganwithitsindustrialpartnershasdesignedanexperimentalReconfigurableMachineTool(RMT)。ThismachineallowsERCresearcherstovalidatemanyofthenewconceptsandmachinetooldesignmethodologiesthathavebeenalreadydevelopedinthecenter.TherearemanytypesofRMTs.Thispaperdescribesanarch-typenon-orthogonalmulti-axisRMTmachine。TheeconomicjustificationofRMTsisgiveninsection2ofthispaper.Acontouringmotionrequiresthatthecuttingtoolmovesalongadesiredtrajectory.Typically,computerizedcontrolsystemsformachinetoolsgeneratecoordinatedmovementsoftheseparatelydrivenaxesofmotioninordertotraceapredeterminedpathofthecuttingtoolrelativetotheworkpiece.Toreducethecontouringerror,whichisdefinedasthedistancebetweenthepredeterminedandtheactualpath,therehavebeentwomaincontrolstrategies.Thefirstapproachistousefeedforwardcontrolinordertoreduceaxialtrackingerrors.however,theyarelimitedwhennon-linearcutsarerequired.Theotherapproachistousecross-couplingcontrolinwhichaxial-feedbackinformationissharedbetweenthemovingaxes.Thecross-couplingcontrollerisusedinadditiontotheconventionalaxialservocontroller.Ateachsamplingtime,thecross-couplingcontrollercalculatesthecurrentcontourerrorandgeneratesacommandthatmovesthetooltowardtheclosestpointonthedesiredtoolpath.Thiscontrolstrategyofthecross-couplingcontroller(CCC)effectivelydecreasesthecontourerror.Advancedcontrolmethodshavebeenappliedtofurtherimprovethecontrolpropertiesoftheoriginalcross-couplingcontroller(CCC).AnoptimalCCCissuggestedin,toimprovethecontrollerperformancewhenhighcontouringspeedswererequired.Anothermethodtoovercomethesameproblemforhighercontourfeedratesisaddressedin,whichusesadaptivefeederatecontrolstrategytoimprovethecontrollerperformance.Thelatesttrendofcross-couplingcontrollerimprovementistheapplicationoffuzzylogic.Allthesemethods,however,donotworkformachineswithnon-orthogonalaxes.Surfacecut(e.g.,acircularcutintheX-Yplane)ona3-axisorthogonalmillingmachinerequiresamotionoftwoaxes(e.g.,XandY).However,surfacecutsinthenon-orthogonalRMTrequiresimultaneousmotionofallthreeaxes.Therefore,inadditiontothecontourerror,thismotioncreatesanothererror,calledthein-deptherror,whichisintheZdirection.Thiserroraffectsthesurfacefinishqualityoftheworkpiece.Whilecontouring,thetooltipoftheRMThasnotonlytofollowthepredeterminedpath,butalsotocontrolcontinuouslythedepthofcut.Thesimultaneouscontrolofbotherrors,theconventionalcontourerrorandthein-deptherror,requiresanewcontrolstrategysincethestandardCCCalgorithmscannotbedirectlyapplied.Inotherwords,theRMTcontroldesignproblemrequiresanewcontrolapproachthatisabletocorrectsimultaneouslytwotypesofcuttingerrors.Thisproblemhasnotbeenaddressedintheliterature.Inthispaper,wedescribethreetypesofcontrollersaimedatreducingthecontourandin-deptherrorsimultaneously.Firstweinvestigateasymmetricalcross-coupling(S-CC)controller,whichunfortunatelydoesnotshowgoodperformanceinreducingbotherrors.Thepoorperformanceisduetotheconflictingdemandsinreducingthetwoerrorsandthelackofinformationsharingbetweenthetwopairsofaxes(X-YandY-Z),whichareresponsibleforerrorcompensation.Toovercomethisproblem,therequiredmotioninformationofonepairofaxesisfedforwardtotheother.Thisidearesultsintwonewcontrollertypes,symmetricalcross-couplingfeedforward(S-CC-FF)controllerandnon-symmetricalcross-couplingfeedforward(NS-CC-FF)controller.Finally,theinfluenceofthereconfigurableangularpositionofthecuttingtoolonsystemstabilityisinvestigated.2MachineCharacteristicsandtheControlProblemInthissectionweexplaintheeconomicadvantageoftheRMT,anddevelopthemathematicalrepresentationofthecontourerrorandthein-deptherror.aMachineCharacteristicsTypicalCNCmachinetoolsarebuiltasgeneral-purposemachines.Theparttobemachinedhastobeadaptedtoagivenmachinebyutilizingprocessplanningmethodologies.Thisdesignprocessmaycreateacapitalwaste:SincetheCNCmachineisdesignedattheoutsettomachineanypart(withinagivenenvelope),itmustbebuiltwithgeneralflexibility,butnotallthisflexibilityisutilizedformachiningaspecificpart.TheconceptofRMTsreversesthisdesignorder:Themachineisdesignedaroundaknownpartfamily.Thisdesignprocesscreatesalesscomplex,althoughlessflexiblemachine,butamachinethatcontainsallthefunctionalityandflexibilityneededtoproduceacertainpartfamily.TheRMTmaycontain,forexample,asmallernumberofaxes,whichreducescostandenhancesthemachinereliability.Therefore,inprinciple,aRMTwithcustomizedflexibilitywouldbelessexpensivethanacompar