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REVERSALSINPARTICULAR,WESHOWTHATREVERSALSAREPRONOUNCEDINMACHININGEVENPLANESURFACESANDSTRAIGHTLINESWITHPARALLELMACHINETOOLSANDAREGENERALLYDICULTTOAVOIDINMULTIAXISFREEFORMMACHININGWITHMOSTTYPESOFMACHINETOOLSMECHANISMANDMACHINETHEORY392004299322CORRESPONDINGAUTHORTEL16172531925FAX161725375491INTRODUCTIONWHENJOINTSINAMACHINETOOLMECHANISMUNDERGOAREVERSALINTHEIRDIRECTIONOFMOTION,SEVERALANOMALIESOCCUR,WHICHMUSTBETAKENCAREOFTOACHIEVEHIGHPRECISIONTHISPAPERDEALSWITHTHEKINEMATICSOFREVERSALSINMACHINETOOLMECHANISMS,CONCENTRATINGMOSTLYONTHETHEORYANDTHEPROPERTIESOFREVERSALSANDREVERSALFREEPATHSTHESEEECTSTENDTOBEIGNOREDDURINGTHEINITIALSTAGESOFMECHANISMDESIGN,ANDOURGOALISTOPRESENTTHETHEORETICALBASISFORPREDICTINGONACTUATORREVERSALMOTIONSOFMACHINETOOLSTAEJUNGKIMA,SEUNGKILSONB,SANJAYESARMAA,ADEPARTMENTOFMECHANICALENGINEERING,MASSACHUSETTSINSTITUTEOFTECHNOLOGY,35010,MIT,CAMBRIDGE,MA02139,USABENGINEERINGDIVISION,LAWRENCEBERKELEYNATIONALLABORATORY,BERKELEY,CA94720,USARECEIVED30AUGUST2002RECEIVEDINREVISEDFORM16JUNE2003ACCEPTED17SEPTEMBER2003ABSTRACTWHENJOINTSINAMACHINETOOLREVERSETHEDIRECTIONOFTHEIRMOTION,NONLINEARITIESTHATAREIGNOREDINMACHINEDESIGNANDCONTROLAREREFLECTEDNOTICEABLYINTHEACCURACYOFMACHINEDSURFACESFOREXAMPLE,FRICTIONCHARACTERISTICSOFAMACHINETOOLBECOMEHIGHLYNONLINEARATLOWOPERATINGSPEEDS,DEMANDINGSOPHISTICATEDCOMPENSATIONWEPRESENTATHEORETICALTREATMENTOFTHEKINEMATICSOFREVERSALSANDREVERSALFREEPATHSOFMACHINETOOLSWEVISUALIZEANDCOMPAREREVERSALCHARACTERISTICSOFACTIVEJOINTSINSERIAL,PARALLEL,ANDHYBRIDMECHANISMSFORVARIOUSTRAJECTORIESANDSWEEPINGPATTERNSREVERSALCHARACTERISTICSHAVEIMPLICATIONSBOTHINTHEDESIGNOFMACHINETOOLSANDINPATHPLANNINGC2112003ELSEVIERLTDALLRIGHTSRESERVEDKEYWORDSREVERSALFRICTIONMACHINETOOLKINEMATICSWWWELSEVIERCOM/LOCATE/MECHMTEMAILADDRESSSESARMAMITEDUSESARMA0094114X/SEEFRONTMATTERC2112003ELSEVIERLTDALLRIGHTSRESERVEDDOI101016/JMECHMACHTHEORY200309002300TKIMETAL/MECHANISMANDMACHINETHEORY392004299322THECOMPLICATIONSASSOCIATEDWITHREVERSALSARISEFROMTHEFACTTHATTHEYEXACERBATENONLINEARITIESINTHEBEHAVIOROFBEARINGSANDDRIVEMECHANISMSACLASSICALEXAMPLEOFAREVERSALRELATEDPROBLEMISBACKLASHINLEADSCREWSBACKLASHARISESWHENAREVERSALINTHELOADINGOFADRIVEMECHANISMCAUSESTHEPOINTOFCONTACTINTHEBEARINGTOCHANGESUDDENLYINPRACTICE,BACKLASHINPRECISIONMECHANISMSCANBEREDUCEDBYPRELOADINGJOINTSTOREDUCEPLAYAMOREFUNDAMENTALNONLINEARITYWHICHISMAGNIFIEDDURINGMOTIONREVERSALSISTHETRANSITIONBETWEENSTATICANDDYNAMICREGIMESOFFRICTIONINJOINTSFIG1ASHOWSTHETYPICALBEHAVIOROFFRICTIONINALUBRICATEDJOINTASITTRANSITIONSFROMLOWVELOCITYTOHIGHVELOCITYMOTIONSTHEFRICTIONMECHANISMCHANGESFROMSTICTIONEECTSTOTHESTRIBECKEECTTOMORELINEARLYVISCOUSBEHAVIOR2THEMAGNITUDEOFTHEFRICTIONFORCEINCREASESANDSWITCHESDIRECTIONASTHEJOINTPASSESTHROUGHTHEZEROSPEEDREGIMESIMPLELINEARCONTROLSTRATEGIES,DESIGNEDFORLINEARVISCOUSFRICTIONMODELS,COMPENSATETHISEECTPOORLY,ANDCANSUERFROMDELAYS,TRACKINGIRREGULARITIESANDLOSSOFMACHININGPRECISION3QUADRANTGLITCHES,ASSHOWNINFIG1B,ARETYPICALEXAMPLESOFFRICTIONINDUCEDERRORSGENERATEDDURINGCIRCULARENDMILLINGONA3AXISMILLINGMACHINE4FIG1ATYPICALFRICTIONCHARACTERISTICINAJOINTANDITSNEGATIVEEECTINPRECISIONARELATIONBETWEENFRICTIONANDVELOCITYBQUADRANTGLITCHESCOURTESYOFEUGENETUNG4BETTERTRACKINGPRECISIONCANBEATTAINEDINTHISREGIMEBYINCREASINGTHEGAININTHECONTROLLEREVERYTIMETHEANOMALYOCCURS,ORBYADOPTINGMORECOMPLEXCONTROLSTRATEGIESHOWEVER,IMPLEMENTINGTHESESTRATEGIESREQUIRESMOREADVANCEDCONTROLLERS,ANDCAREFULEXPERIMENTATIONFORPARAMETERIDENTIFICATION,WHICHAREBOTHNECESSARILYMOREEXPENSIVEITISBEST,THEREFORE,TOAVOIDTHEOCCURRENCEOFREVERSALSDURINGCRITICALMOTIONSTHATTHEDESIGNERCONSIDERSIMPORTANT11REVERSALCHARACTERISTICSASDESCRIBEDABOVE,ITWOULDBEDESIRABLEFORAMACHINETOOLTOBEDESIGNEDINSUCHAWAYTHATREVERSALSDONOTOCCURINSITUATIONSWHERETHEYWOULDHURTPERFORMANCEINCARTESIANMACHINES,FOREXAMPLE,REVERSALSDONOTOCCURWHILETHEMACHINEISEXECUTINGAUNIDIRECTIONALMOTIONINASTRAIGHTLINETRAJECTORYTHISISVERYIMPORTANTBECAUSETHESTRAIGHTLINEHASASPECIALANDFUNDAMENTALROLEINDESIGNANDMANUFACTURINGASADATUMANDASAMATINGSURFACEREVERSALSDOOCCURINCARTESIAN3AXISMACHINES,BUTONLYINPLACESWHERETHEDIRECTIONOFMOTIONISEXPLICITLYREVERSED,SUCHASINMACHININGACIRCLEASSHOWNINFIG1BTHISCONSTITUTESGOODKINEMATICTKIMETAL/MECHANISMANDMACHINETHEORY392004299322301BEHAVIORFROMTHEPOINTOFVIEWOFREVERSALSALLREVERSALSOCCURINANEASILYDESCRIBABLEWAY,ANDREVERSALCHARACTERISTICSAREPOSITIONINDEPENDENTANDCONFIGURATIONINDEPENDENTUNFORTUNATELY,NOTALLMACHINESHAVETHESEDESIRABLEPROPERTIES,ANDREVERSALSAREESPECIALLYDICULTTOPREDICTINPARALLELMACHINESAPARALLELMACHINEMAYSUERAREVERSALWHILEPRESCRIBINGASTRAIGHTLINEMOTIONINITSWORKSPACETHISBEHAVIORCOULDTURNOUTTOBEASEVEREIMPEDIMENTTOTHEUSEOFPARALLELMACHINESINMACHININGAPPLICATIONSSERIALCARTESIANMACHINETOOLSTOOSUERFROMREVERSALSDURINGFREEFORMMACHININGINMACHININGANAERODYNAMICALLYCRITICALSURFACE,FOREXAMPLE,ACURVEDTRAJECTORYMAYRESULTINUNDESIRABLEREVERSALINDUCEDRIDGESUNDERSTANDINGTHESEPROBLEMSISIMPORTANT,ANDWEEXAMINEANUMBEROFISSUESINTHISPAPERWHERETHEREVERSALSOCCURINAPATH,WHATTHENATUREOFREVERSALFREEPATHSIS,WHEREREVERSALSAREINEVITABLEREGARDLESSOFDIRECTIONOFMOTION,HOWTHEREGIONSOFREVERSALFREEREACHABILITYAREDERIVEDANDSOONOURTREATMENTISMOSTLYTHEORETICAL,ANDBASEDONTHEKINEMATICSOFAGIVENMACHINEMUCHOFTHECONTRIBUTIONOFTHISPAPERLIESINDEVELOPINGTHEMATHEMATICALMACHINERYNECESSARYTODESCRIBEREVERSALSANDRELATEDCONCEPTSPRECISELYANDUNAMBIGUOUSLY12BACKGROUNDTHEPROBLEMOFBACKLASHHASBEENANAGEOLDCHALLENGEINMACHINEDESIGN,ANDHASRECEIVEDAGREATDEALOFATTENTIONINTHEMACHINEELEMENTCOMMUNITYADESCRIPTIONOFIDEASANDTECHNIQUESINMACHINEELEMENTSISAVAILABLEIN1THEREHASBEENAGREATDEALOFLITERATUREONFRICTIONMODELS,ANDACOMPREHENSIVESUMMARYISAVAILABLEIN5ANELABORATEMODELOFFRICTIONPRESENTEDBYCANUDASDEWITETAL6CAPTURESTHEDIERENTEECTSTHATMUSTBECONSIDEREDINREVERSALSITUATIONSINCLUDINGTHEEARLYWORKOFCOULOMB,OFDUDLEY7,WHOSHOWEDTHATSTICKSLIPISNOTCAPTUREDBYCOULOMBANDVISCOUSMODELS,OFRABINOWICZ9,WHOSHOWEDTHATTHEREISATIMELAGBETWEENTHEBEGINNINGOFMOTIONANDTHEAPPEARANCEOFAFRICTIONFORCE,ANDOFDAHL8WHOFURTHERINCORPORATEDHYSTERETICEECTSHOWEVER,EVENCANUDASDEWITC213S6PARAMETERMODELDOESNOTCAPTUREALLTHESUBTLETIESOFFRICTION,SUCHASTHEDEPENDENCEOFFRICTIONONTHETIMEDERIVATIVEOFAPPLIEDFORCEINADDITION,FRICTIONPARAMETERSHAVEBEENSHOWNTOBEHIGHLYVARIABLEWITHINTHEWORKSPACE,ANDOVERTIME10NEVERTHELESS,THISMODELFORMSTHEBASISOFANUMBEROFCONTROLMETHODSTHATHAVEBEENPROPOSEDINTHELASTFEWYEARSRESEARCHERSINTHEFIELDOFMOTIONCONTROLHAVEPROPOSEDARICHVARIETYOFCOMPENSATIONMETHODSTOACHIEVESMOOTHMOTIONWITHHIGHACCURACYUNDERTHEACTIONOFFRICTIONNOTBEINGTHEFOCUSOFTHISPAPER,WEPRESENTONLYAROUGHOVERVIEWHERECOMPREHENSIVESUMMARIESAREAVAILABLEIN11,12THETECHNIQUESFORCONTROLOFSYSTEMSUNDERFRICTIONINCLUDEESTIMATIONANDADAPTIVEMETHODS13,14,ROBUSTCOMPENSATIONSCHEMES15,REPETITIVECONTROL4,MULTILOOPCONTROL16,NONLINEARCOMPENSATIONSIMILARTOSLIDINGMODE17,NONLINEARPIDWHICHCHANGESTHEGAINDEPENDINGONPLANTSTATE18ANDABROADCLASSOFPULSECODEMODULATIONTECHNIQUESWHICHARESUMMARIZEDIN12THEREISAGREATDEALOFLITERATUREONPARALLELMACHINES,ANDTHEREAREDOZENSOFMILLINGMACHINESWHICHUSEPARALLELKINEMATICSTRUCTURESASUMMARYISAVAILABLEIN19THEPUTATIVEADVANTAGEOFPARALLELMACHINESOVERTHEIRSERIALCOUNTERPARTSISTHATTHELOADCANBEDISTRIBUTEDMOREEVENLYTOTHEIRKINEMATICLINKSAND,ASARESULT,AHIGHERSTINESSCANBEACHIEVEDUSINGLIGHTERSTRUCTURALELEMENTSANDLESSEXPENSIVEACTUATORSHOWEVERWEWILLSHOWTHATTHECOUPLEDKINEMATICSOF302TKIMETAL/MECHANISMANDMACHINETHEORY392004299322PARALLELMECHANISMSCANCAUSEREVERSALSINTHEIRJOINTSFORSEEMINGLYSIMPLEANDSTRAIGHTTRAJECTORIESFURTHERMORE,STUDIESHAVESHOWNTHATTHEERRORSINANACTUATORPROPAGATESINTOALLTHECOORDINATEAXES19,ANDTHISCANBEPROBLEMATICWHENITISIMPORTANTTOMAINTAINPLANARITYORSTRAIGHTNESSINADDITION,THEREISLITERATURETHATSHOWSTHATFREQUENTACTUATORREVERSALSCANEXCITEHIGHERHARMONICCOMPONENTSINTHEMACHINETOOLSTRUCTURE20,21THISINCREASESTHENECESSARYSTRUCTURALSTINESSOFPARALLELMACHINESAND,TOSOMEEXTENT,UNDERMINESTHEVERYBASISFORWHICHPARALLELMECHANISMSAREPREFERREDINTHEFIRSTPLACEBEYONDTHIS,HOWEVER,THEREAPPEARSTOBELITTLEPRIORWORKINTHESTUDYOFREVERSALSINSERIALORPARALLELMACHINETOOLS13OUTLINEANABSOLUTELYFAIRCOMPARISONAMONGSTDIERENTMACHINETOOLMECHANISMSISNOTEASYTOESTABLISHFORREVERSALBEHAVIORBECAUSEPERFORMANCEDEPENDSONTHETRAJECTORIESCONSIDEREDANDONTHETASKSATHANDIFFOREXAMPLE,THEMACHINEISNOTREQUIREDTOCUTFLATSURFACES,BUT,SAY,SPHERESFIG2THEMACHINESWECONSIDERAASERIAL,BHYBRIDANDCPARALLELMACHINETOOLINSTEAD,APARALLELMACHINEOFACERTAINCONFIGURATIONMAYBEMORENATURALTHANACARTESIANSERIALMACHINEINSTEADOFMAKINGASSUMPTIONSABOUTTHETASKTHATTHEMACHINEISCONSIDEREDFOR,WEWILLDEFINETHECONCEPTOFATASKSPACEWHICHPROVIDESTHECAPABILITYOFCAPTURINGANYTASKSURFACE,ANDWEWILLASKHOWMANYREVERSALSCANOCCURINATYPICALPATHANDWHETHERITISPOSSIBLETOACHIEVETHESAMEOBJECTIVEWITHAREDUCEDNUMBEROFREVERSALSINSECTION2,WESETAMATHEMATICALFRAMEWORKUPONWHICHTHERESTOFTHISPAPERISBASEDINSECTION3,WEANSWERFUNDAMENTALQUESTIONSABOUTTHEBEHAVIOROFREVERSALSANDDEVELOPMETHODSOFANALYSISINSECTION4,WEPRESENTSOMEEXAMPLESWITHTHREETYPESOFMACHINETOOLSASERIAL,AHYBRIDANDAPARALLELMACHINEASSHOWNINFIG2WECONCLUDEINSECTION52AFRAMEWORKFORANALYSISINTHISSECTION,WELAYDOWNAMATHEMATICALFRAMEWORKUPONWHICHTHERESTOFTHISPAPERISBASEDITISASSUMEDTHATWHATWEWILLCALLARESTRICTEDINVERSEKINEMATICSMAPISKNOWNTHISSECTIONSIMPLYDEFINESTHETERMANDJUSTIFIESTHENEEDFORTHISASSUMPTIONTODESCRIBETHEMOTIONOFACUTTINGTOOL,WEATTACHAFIDUCIALPOINTONTHECENTERLINEOFTHECUTTINGTOOLNEARTHETOOLTIP13TUPLESOFTHECARTESIANCOORDINATESOFTHEFIDUCIALPOINTCONSTITUTEWHATWEREFERTOASTHEWORKSPACEOFTHEMACHINETOOLINGENERAL,ACUTTINGTOOL,ASARIGIDBODY,HAS6DEGREESOFFREEDOMTHREEFORTHETRANSLATIONOFTHEFIDUCIALPOINTANDANOTHERTHREEFORINTERESTIDENTICALTKIMETAL/MECHANISMANDMACHINETHEORY392004299322303WECANPERFORM5AXISMACHININGUSING6AXISMACHINESTHEADDITIONALONEDEGREEOFFREEDOMISREDUNDANTIN3AXISMACHINING,ONLYTRANSLATIONOFACUTTINGTOOLISALLOWEDANDTHETASKSPACEDIMENSIONIS3INTHEPATHPLANNINGSTAGE,THEDIMENSIONCANBEFURTHERREDUCEDBYINTRODUCINGTASKCONSTRAINTSFOREXAMPLEIN3AXISROUGHING,ITISCOMMONTOREMOVEMATERIALFROMTHESTOCKLAYERBYLAYERINTHISCASE,THETASKCONSTRAINTISTHATTHEFIDUCIALPOINTSHOULDSTAYONAPLANEWHILEALAYEROFMATERIALISREMOVED,ANDTHETASKSPACEDIMENSIONOFMACHININGEACHLAYERIS2INFINISHINGTHESURFACEOFAFREEFORMPARTUSING5AXISMACHINING,THETOOLTIPISCONSTRAINEDTOCONTACTTHEREQUIREDSURFACE,ANDTHETASKSPACEDIMENSIONOFFINISHINGISREDUCEDTO4WHENPLANNINGFINISHINGPATHS,COLLISIONFREEORIENTATIONSOFACUTTINGTOOLATEACHPOINTONTHEDESIGNEDSURFACECANBEDETERMINEDBYACOLLISIONDETECTIONPREPROCESSOR,ANDTHEREMAININGPLANNINGTASKCANBEPERFORMEDONA2DIMENSIONALSPACEINTHISCASE,THETASKSPACEDIMENSIONIS2CONSIDERAMACHINETOOLPOSSESSINGNDEGREESOFFREEDOMANDAMACHININGSTRATEGYWHOSETASKSPACEDIMENSIONISM6NTHEMOTIONOFACUTTINGTOOLRESTRICTEDTOTHESTRATEGYCANBECAPTUREDBYAHISTORYOFMTUPLESOFREALNUMBERSWEREFERTOTHETOTALITYOFTHEMTUPLESREPRESENTINGTHERESTRICTEDPOSTURESASTHETASKSPACEUNDERTHESTRATEGYTHETOTALITYOFTHENTUPLESOFDISPLACEMENTSOFACTUATORSINTHEMACHINEISREFERREDTOASTHEACTUATIONSPACEWEASSUMETHATTHEKINEMATICSOFAMACHINETOOLANDTHECONSTRAINTSINTRODUCEDBYAMACHININGSTRATEGYDETERMINEASMOOTHMAPFFROMTHETASKSPACEUC26RMOFTHESTRATEGYTOTHEACTUATIONSPACEHC26RNOFTHEMACHINETOOL,FUHUHFUWHICHWEREFERTOASTHERESTRICTEDINVERSEKINEMATICMAPANYREDUNDANCYMUSTBERESOLVEDINAPOINTWISEMANNERORHOLONOMICALLYBASEDONOURASSUMPTIONWEASSUMETHATTHETASKSPACEUISSIMPLYCONNECTEDANDTHATITAVOIDSSINGULARITIESBYCHOICETHERANGEFUOFTHEMAPISANMDIMENSIONALSUBSPACEOFTHENDIMENSIONALACTUATIONSPACETHEACTUATORDISPLACEMENTHCANBETHOUGHTOFASONEOFTHELOCALCOORDINATESOFTHECONFIGURATIONSPACEOFTHEMACHINETOOLTHISSETTINGISGENERALENOUGHFOROURPURPOSE,EVENTHOUGHCOMPLETEGENERALITYCANBEACHIEVEDBY1SINCEACUTTINGTOOLHASANONZERODIAMETER,THECONTACTPOINTBETWEENTHETOOLANDTHESURFACEISNOTNECESSARILYONTHELINEOFSYMMETRYOFTHETOOL22HOWEVER,ALOOSEDEFINITIONOFTHELOCATIONOFTHEFIDUCIALPOINTSUCESFOROURCURRENTASTHETASKSPACEDIMENSIONOFTHEAPPLICATIONTHETASKSPACEDIMENSIONISNOTNECESSARILYTOTHEDEGREESOFFREEDOMOFTHEMACHINEINVOLVEDINTHEMACHININGTASKFOREXAMPLE,THEROTATIONOFTHECUTTINGTOOLSINCEWEARECONSIDERINGMILLINGMACHINES,ANDBECAUSECUTTINGTOOLSCANBECONSIDEREDTOBEAXISYMMETRIC,THESIXTHFREEDOMOFMOTION,WHICHISTHESPINABOUTTHECENTERLINEOFACUTTINGTOOL,ISTYPICALLYUNNECESSARYANDREDUNDANTTHEREFORE,TWOADDITIONALCOORDINATESAREENOUGHTODESCRIBETHEORIENTATIONOFA5AXISMACHINEINTHECONTEXTOFMILLINGALLTHREEDEGREESOFFREEDOMOFROTATIONMAYBENECESSARYINOTHERAPPLICATIONSWEREFERTOTHENUMBEROFDEGREESOFFREEDOMOFTHECUTTINGTOOLRESTRICTEDTOOURSUBJECTIVEANALYSISDISPLACEMENTSARIGOROUSSTATEMENTOFTHEABOVEASSUMPTIONEXCEEDSTHESCOPEOFOURCURRENTTREATMENT,DESCRIBED3REVERSALANALYSISREVERSINGMOTIONCOMPARE311AREVERSALCONDITIONONATRAJECTORYFORATRACKINGTASKANYI,WEREFERTOTHEPOINTUT0ASAREVERSALPOINTIFTHEREISNOREVERSALPOINTALONGATRAJECTORY,312REVERSALSINGULARPOINTSANDSTRONGREVERSALSINGULARPOINTSATAPOINTU2UINTHETASKSPACEU,IFTHEREISANINTEGERK2F12NGSUCHTHAT304TKIMETAL/MECHANISMANDMACHINETHEORY392004299322RFKC17OFKOU1OFKOU2OFKOUMC20C210WEREFERTOTHEPOINTUASAREVERSALSINGULARPOINTWITHRESPECTTOTHEKTHACTUATOR,WHEREFKISTHEWESAYTHATTHETRAJECTORYISREVERSALFREEFORTHEITHACTUATORTOBEREVERSEDATT20TC138ALONGAREGULARTRAJECTORY,ITISSUCIENTTHAT_HITC17DDTFIUTXMJ1OFIOUJC1DUJDTC18C190ANDHITC17D2DT2FIUT60THISCONDITIONITSELFSUGGESTSASTRAIGHTFORWARDPROCEDUREFORFINDINGTHEREVERSALPOINTSALONGAGIVENREGULARTRAJECTORYINTHETASKSPACEPLOTTINGANDCOUNTINGREVERSALPOINTSALONGSPECIFIEDTRAJECTORIESISAONEWAYTOVISUALIZEORTOCOMPARETHEREVERSALCHARACTERISTICSOFMACHINETOOLSAFUNDAMENTALFUNCTIONOFAMACHINETOOLISTHEABILITYTOTRACKATRAJECTORYSPECIFIEDFORITSENDEECTORWECONSIDERAMACHINETOOLMOVINGALONGAPIECEWISEREGULARTRAJECTORYUT20TC138UT2UINTHETASKSPACEUIFHITC17FIUTHASALOCALEXTREMUMATTT0FORMOSTOFOURPROPOSITIONSCANBEUNDERSTOODINTUITIVELYWEOMITSOMEPROOFSBUTPROVIDETHEIRRATIONALEINTHEMAINBODYOFTEXT31REVERSALCONDITIONSINTHISSECTION,WESHOWVARIOUSWAYSTOINTERPRETREVERSALSOFACTUATORSKTHCOMPONENTPATHSWEALSOASKATWHICHPOINTINATRAJECTORYTHEACTUATORSREVERSETHEIRDIRECTIONOFFINALLY,WEANALYZETHEREVERSALCHARACTERISTICSOFSURFACEMACHININGORSWEEPINGWETHETHREETYPESOFMACHINETOOLSINSECTION4USINGTHECONCEPTSDEVELOPEDINTHISSECTIONINTHISSECTION,WEDEVELOPSEVERALCONCEPTSWHICHAREUSEFULINJUDGINGTHEREVERSALCHARACTERISTICSOFMACHINETOOLSWEWILLASKWHETHERAPOINTTOPOINTTASKCANBECOMPLETEDWITHOUTANYACTUATORREVERSALSORWHETHERTHEREISACONDITIONTHATGUARANTEESREVERSALFREEORCERTAINLYBUTMOREDETAILSAREAVAILABLEIN23,24INTHISPAPER,WEWILLSHOWHOWTHESETTINGABOVECANCAPTURETHEREVERSALCHARACTERISTICSOFMOSTMACHINESCONSIDERINGTHECONFIGURATIONSPACEOFMACHINETOOLSINCLUDINGBOTHACTIVEANDPASSIVEJOINTOFTHEINVERSEKINEMATICSFAREVERSALSINGULARPOINTWITHRESPECTTOTHEKTHOUOUPLACEMENTFKISEITHERALOCALMAXIMUMORMINIMUMTHEREFORE,THEKTHACTUATORMUSTREVERSEITSACTUATORSANDTHEVELOCITY_UINTHETASKSPACEHAVETHEFOLLOWINGLINEARRELATIONTKIMETAL/MECHANISMANDMACHINETHEORY392004299322305WHERECISTHEJACOBIANMATRIXOFTHERESTRICTEDINVERSEKINEMATICSFTHEMTUPLE_UISANELEMENTOFTHETANGENTSPACEOFTHETASKSPACETHETANGENTSPACEATAPOINTU2UISDENOTEDBYTUUTHEUNION_HC_U1DIRECTIONOFMOTIONATTHESTRONGREVERSALSINGULARPOINT,WHICHLEADSTOTHEFOLLOWINGPROPOSITIONPROPOSITION1WHENTHEFIDUCIALPOINTOFAMACHINETOOLPASSESASTRONGREVERSALSINGULARPOINTWITHRESPECTTOTHEKTHACTUATOR,REVERSALOFTHEKTHACTUATORISUNAVOIDABLEMEREREVERSALSINGULARPOINTSWHICH,FOREMPHASIS,WEWILLSOMETIMESREFERTOASNONSTRONGREVERSALSINGULARPOINTS,ALSOIMPLYAHIGHCHANCEOFREVERSALSBECAUSETHETRAJECTORIESPASSINGAREVERSALSINGULARPOINTMUSTSATISFYAVERYSPECIALCONDITIONTOCONSTITUTEREVERSALFREEMOTIONSTHISISESPECIALLYEASYTOUNDERSTANDFORA2DIMENSIONALTASKSPACEIFANONDEGENERATEREVERSALSINGULARPOINTINA2DIMENSIONALTASKSPACEWITHRESPECTTOTHEKTHACTUATORISNOTSTRONG,THEKTHCOMPONENTFKOFTHEINVERSEKINEMATICSHASASADDLEPOINTATTHEREVERSALSINGULARPOINTTHISIMPLIESPROPOSITION2ATANONDEGENERATE,NONSTRONGREVERSALSINGULARPOINTINA2DIMENSIONALTASKSPACE,AREGULARTRAJECTORYMUSTBETANGENTIALTOONEOFTHESEPARATRICESOFTHEFUNCTIONFKIFITISREVERSALFREETHEPROOFCANBEEASILYESTABLISHEDTHROUGHTHEMORSELEMMATHECONDITIONSTATEDABOVEISARATHERSPECIALCONDITIONWHICHISRARELYSATISFIED,ESPECIALLYINSWEEPINGTASKS,WHICHWEDISCUSSINSECTION33APLOTOFTHEREVERSALSINGULARPOINTSISACANONICALWAYTOVISUALIZETHEREVERSALCHARACTERISTICSOFMACHINETOOLSBECAUSEITSAPPEARANCEDOESNOTDEPENDONAPARTICULARTRAJECTORYWEMAYHAVECHOSEN313REVERSALFREEDIRECTIONSINTHETANGENTSPACESOFTHETASKSPACEWECANSPECIFYAREVERSALCONDITIONINTERMSOFTANGENTVECTORSOFTRAJECTORIESWEALSODEFINERELATEDTERMSFORFUTUREUSEINTHISSECTIONIFACUTTINGTOOLMOVESINATRAJECTORYOBEYINGTHEGIVENTASKCONSTRAINTS,THEVELOCITY_HOFTHEIJAREVERSALSINGULARPOINTISSAIDTOBEDEGENERATEIFITSCORRESPONDINGHESSIANMATRIXISSINGULARDEGENERATEREVERSALSINGULARPOINTSAREUNCOMMONINTHATASLIGHTPERTURBATIONOFKINEMATICPARAMETERSCANREMOVETHEMATASTRONGREVERSALSINGULARPOINTWITHRESPECTTOTHEKTHACTUATOR,THEKTHACTUATORDISHKIJC17O2FKACTUATORISSAIDTOBESTRONGIFTHEHESSIANMATRIXHKIJC138OFTHEKTHCOMPONENTFKOFTHEINVERSEKINEMATICSFISDEFINITEATTHEPOINT,WHEREOFTANGENTSPACESISREFERREDTOASTHETANGENTBUNDLE,ANDSYMBOLICALLYASTUATEVERYPOINTINTHETASKSPACEATANINSTANT,EACHACTUATORHASTWOALTERNATIVES,EITHERINCREASINGORDECREASINGITSDISPLACEMENTINOTHERWORDS,EITHER_HIP0OR_HI602INPARTICULAR,IFATRAJECTORYISREVERSALFREE,ONLYONEOFTHETWOINEQUALITIESMUSTHOLDDURINGITSENTIREINTERVALOFMOTION,EXCEPTFOR_HI0INOTHERWORDS,EACHACTUATORDISPLACEMENTMUSTBEMONOTONICALONGA306TKIMETAL/MECHANISMANDMACHINETHEORY392004299322REVERSALFREETRAJECTORYTHEPERMUTATIONSOFTHESEBINARYALTERNATIVESOVERALLTHEACTUATORSCANBEENCODEDUSINGABINARYSYSTEMUSINGTHESYMBOLSIANDDASTHEBASICDIGITS,WHERETHESYMBOLSREPRESENTTHEWORDSINCREASINGANDDECREASINGFOREXAMPLE,INA5AXISMACHINETOOL,IFTHEFIRSTANDTHELASTACTUATORDISPLACEMENTSAREINCREASINGANDTHEOTHERDISPLACEMENTSAREDECREASING,WEDENOTETHECASEBYIDDDI,WHICHWEREFERTOASANACTUATORPERMUTATIONTHENUMBEROFPOSSIBLEACTUATORPERMUTATIONSIS2N,WHERENISTHENUMBEROFACTUATORSCONVERSELY,ANACTUATORPERMUTATIONPIMPOSESNINEQUALITIESINTHETANGENTSPACETUUATAGIVENPOINTU2UEACHINEQUALITY,_HIP0OR_HI60,CANBETHOUGHTOFASAHALFSPACEINTHETANGENTSPACEDUETOTHELINEARRELATIONSHOWNINEQ1THEINTERSECTIONOFSUCHHALFSPACESFORMSASIMPLYCONNECTEDCONEINTHETANGENTSPACEWEREFERTOTHERESULTINGCONEASTHEPTHREVERSALFREECONEOFAGIVENPOINTU2U,ANDSYMBOLICALLYASCPUUWESAYTHATCPUUISVOIDIFCPUUF0G,A