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RoboticsandComputerIntegratedManufacturing212005368–378KeywordsFixturedesignGeometryconstraintDeterministiclocatingUnderconstrainedOverconstrainedconstraintstatus,aworkpieceunderanylocatingschemefallsintooneofthefollowingthreecategorieslocatingproblemusingscrewtheoryin1989.Itisconcludedthatthelocatingwrenchesmatrixneedstobefullranktoachievedeterministiclocation.Thismethodhasbeenadoptedbynumerousstudiesaswell.Wangetal.3consideredARTICLEINPRESS07365845/seefrontmatterr2005ElsevierLtd.Allrightsreserved.doi10.1016/j.rcim.2004.11.012C3Correspondingauthor.Tel.15088316092fax15088316412.Emailaddresshsongwpi.eduH.Song.1.WellconstraineddeterministicTheworkpieceismatedatauniquepositionwhensixlocatorsaremadetocontacttheworkpiecesurface.2.UnderconstrainedThesixdegreesoffreedomofworkpiecearenotfullyconstrained.3.OverconstrainedThesixdegreesoffreedomofworkpieceareconstrainedbymorethansixlocators.In1985,AsadaandBy1proposedfullrankJacobianmatrixofconstraintequationsasacriterionandformedthebasisofanalyticalinvestigationsfordeterministiclocatingthatfollowed.Chouetal.2formulatedthedeterministic1.IntroductionAfixtureisamechanismusedinmanufacturingoperationstoholdaworkpiecefirmlyinposition.Beingacrucialstepinprocessplanningformachiningparts,fixturedesignneedstoensurethepositionalaccuracyanddimensionalaccuracyofaworkpiece.Ingeneral,321principleisthemostwidelyusedguidingprinciplefordevelopingalocationscheme.Vblockandpinholelocatingprinciplesarealsocommonlyused.Alocationschemeforamachiningfixturemustsatisfyanumberofrequirements.Themostbasicrequirementisthatitmustprovidedeterministiclocationfortheworkpiece1.Thisnotionstatesthatalocatorschemeproducesdeterministiclocationwhentheworkpiececannotmovewithoutlosingcontactwithatleastonelocator.Thishasbeenoneofthemostfundamentalguidelinesforfixturedesignandstudiedbymanyresearchers.ConcerninggeometryAbstractGeometryconstraintisoneofthemostimportantconsiderationsinfixturedesign.Analyticalformulationofdeterministiclocationhasbeenwelldeveloped.However,howtoanalyzeandreviseanondeterministiclocatingschemeduringtheprocessofactualfixturedesignpracticehasnotbeenthoroughlystudied.Inthispaper,amethodologytocharacterizefixturingsystemsgeometryconstraintstatuswithfocusonunderconstraintisproposed.Anunderconstraintstatus,ifitexists,canberecognizedwithgivenlocatingscheme.Allunconstrainedmotionsofaworkpieceinanunderconstraintstatuscanbeautomaticallyidentified.Thisassiststhedesignertoimprovedeficitlocatingschemeandprovidesguidelinesforrevisiontoeventuallyachievedeterministiclocating.r2005ElsevierLtd.Allrightsreserved.CAMLab,DepartmentofMechanicalEngineering,WorcesterPolytechnicInstitute,100InstituteRd,Worcester,MA01609,USAReceived14September2004receivedinrevisedform9November2004accepted10November2004LocatingcompletenessevaluationandrevisioninfixtureplanH.SongC3,Y.Rongwww.elsevier.com/locate/rcimlocator–workpiececontactareaeffectsinsteadofapplyingpointcontact.Theyintroducedacontactmatrixandpointedoutthattwocontactbodiesshouldnothaveequalbutoppositecurvatureatcontactingpoint.Carlson4suggestedthatalinearapproximationmaynotbesufficientforsomeapplicationssuchasnonprismaticsurfacesornonsmallrelativeerrors.HeproposedasecondorderTaylorexpansionwhichalsotakeslocatorerrorinteractionintoaccount.MarinandFerreira5appliedChousformulationon321locationandformulatedseveraleasytofollowplanningrules.Despitethenumerousanalyticalstudiesondeterministiclocation,lessattentionwaspaidtoanalyzenondeterministiclocation.IntheAsadaandBysformulation,theyassumedfrictionlessandpointcontactbetweenfixturingelementsandworkpiece.Thedesiredlocationisq,atwhichaworkpieceistobepositionedandpiecewiselydifferentiablesurfacefunctionisgiasshowninFig.1.ThesurfacefunctionisdefinedasgiðqC3Þ¼0Tobedeterministic,thereshouldbeauniquesolutionforthefollowingequationsetforalllocators.giðqÞ¼0i¼12...n,1wherenisthenumberoflocatorsandq¼½x0y0z0y0f0c0C138representsthepositionandorientationoftheworkpiece.OnlyconsideringthevicinityofdesiredlocationqC3whereq¼qC3þDqAsadaandByshowedthatARTICLEINPRESSH.Song,Y.Rong/RoboticsandComputerIntegratedManufacturing212005368–378369giðqÞ¼giðqC3ÞþhiDq,2wherehiistheJacobianmatrixofgeometryfunctions,asshownbythematrixinEq.3.ThedeterministiclocatingrequirementcanbesatisfiediftheJacobianmatrixhasfullrank,whichmakestheEq.2tohaveonlyonesolutionq¼qC3rankqg1qx0qg1qy0qg1qz0qg1qy0qg1qf0qg1qc0qgiqx0qgiqy0qgiqz0qgiqy0qgiqf0qgiqc0qgnqx0qgnqy0qgnqz0qgnqy0qgnqf0qgnqc0266666666643777777777589¼6.3Upongivena321locatingscheme,therankofaJacobianmatrixforconstraintequationstellstheconstraintstatusasshowninTable1.Iftherankislessthansix,theworkpieceisunderconstrained,i.e.,thereexistsatleastonefreemotionoftheworkpiecethatisnotconstrainedbylocators.Ifthematrixhasfullrankbutthelocatingschemehasmorethansixlocators,theworkpieceisoverconstrained,whichindicatesthereexistsatleastonelocatorsuchthatitcanberemovedwithoutaffectingthegeometryconstrainstatusoftheworkpiece.Forlocatingamodelotherthan321,datumframecanbeestablishedtoextractequivalentlocatingpoints.Hu6hasdevelopedasystematicapproachforthispurpose.Hence,thiscriterioncanbeappliedtoalllocatingschemes.XYZOXYZOx0,y0,z0giUCSWCSWorkpieceFig.1.Fixturingsystemmodel.Theyfurtherintroducedseveralindexesderivedfromthosematrixestoevaluatelocatorconfigurations,followedbyoptimizationthroughconstrainednonlinearprogramming.Theiranalyticalstudy,however,doesnotconcerntheARTICLEINPRESSrevisionofnondeterministiclocating.Currently,thereisnosystematicstudyonhowtodealwithafixturedesignthatfailedtoprovidedeterministiclocation.2.LocatingcompletenessevaluationIfdeterministiclocationisnotachievedbydesignedfixturingsystem,itisasimportantfordesignerstoknowwhattheconstraintstatusisandhowtoimprovethedesign.Ifthefixturingsystemisoverconstrained,informationabouttheunnecessarylocatorsisdesired.Whileunderconstrainedoccurs,theknowledgeaboutalltheunconstrainedmotionsofaworkpiecemayguidedesignerstoselectadditionallocatorsand/orrevisethelocatingschememoreefficiently.AgeneralstrategytocharacterizegeometryconstraintstatusofalocatingschemeisdescribedinFig.2.Inthispaper,therankoflocatingmatrixisexertedtoevaluategeometryconstraintstatusseeAppendixforderivationoflocatingmatrix.ThedeterministiclocatingrequiressixlocatorsthatprovidefullranklocatingmatrixWLAsshowninFig.3,forgivenlocatornumbernlocatingnormalvector½aibiciC138andlocatingposition½xiyiziC138foreachlocator,i¼12...nthenC26locatingmatrixcanbedeterminedasfollowsa1b1c1c1y1C0b1z1a1z1C0c1x1b1x1C0a1y12637Kangetal.7followedthesemethodsandimplementedthemtodevelopageometryconstraintanalysismoduleintheirautomatedcomputeraidedfixturedesignverificationsystem.TheirCAFDVsystemcancalculatetheJacobianmatrixanditsranktodeterminelocatingcompleteness.Itcanalsoanalyzetheworkpiecedisplacementandsensitivitytolocatingerror.Xiongetal.8presentedanapproachtochecktherankoflocatingmatrixWLseeAppendix.Theyalsointroducedleft/rightgeneralizedinverseofthelocatingmatrixtoanalyzethegeometricerrorsofworkpiece.IthasbeenshownthatthepositionandorientationerrorsDXoftheworkpieceandthepositionerrorsDroflocatorsarerelatedasfollowsWellconstrainedDX¼WLDr,4OverconstrainedDX¼ðWTLWLÞC01WTLDr,5UnderconstrainedDX¼WTLðWLWTLÞC01DrþðI6C26C0WTLðWLWTLÞC01WLÞl,6wherelisanarbitraryvector.Table1RankNumberoflocatorsStatuso6Underconstrained¼6¼6Wellconstrained¼646OverconstrainedH.Song,Y.Rong/RoboticsandComputerIntegratedManufacturing212005368–378370WL¼aibiciciyiC0biziaiziC0cixibixiC0aiyianbncncnynC0bnznanznC0cnxnbnxnC0anyn666664777775.7WhenrankðWLÞ¼6andn¼6theworkpieceiswellconstrained.WhenrankðWLÞ¼6andn46theworkpieceisoverconstrained.ThismeansthereareðnC06Þunnecessarylocatorsinthelocatingscheme.TheworkpiecewillbewellconstrainedwithoutthepresenceofthoseðnC06Þlocators.ThemathematicalrepresentationforthisstatusisthatthereareðnC06Þrowvectorsinlocatingmatrixthatcanbeexpressedaslinearcombinationsoftheothersixrowvectors.ThelocatorscorrespondingtothatsixrowvectorsconsistoneARTICLEINPRESSlocatdeterm1.2.3.4.be3.workpiH.Song,Y.Rong/RoboticsandComputerIntegratedManufacturing212005368–378371ingschemethatprovidesdeterministiclocation.ThedevelopedalgorithmusesthefollowingapproachtoinetheunnecessarylocatorsFindallthecombinationofðnC06Þlocators.Foreachcombination,removethatðnC06Þlocatorsfromlocatingscheme.Recalculatetherankoflocatingmatrixfortheleftsixlocators.Iftherankremainsunchanged,theremovedðnC06Þlocatorsareresponsibleforoverconstrainedstatus.Thismethodmayyieldmultisolutionsandrequiredesignertodeterminewhichsetofunnecessarylocatorsshouldremovedforthebestlocatingperformance.WhenrankðWLÞo6theworkpieceisunderconstrained.AlgorithmdevelopmentandimplementationThealgorithmtobedevelopedherewilldedicatetoprovideinformationonunconstrainedmotionsoftheeceinunderconstrainedstatus.Supposetherearenlocators,therelationshipbetweenaworkpiecesposition/Fig.2.Geometryconstraintstatuscharacterization.XZYa1,b1,c12,b2,c2x1,y1,z1x2,y2,z2ai,bi,cixi,yi,ziaFig.3.Asimplifiedlocatingscheme.
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