RoboticsandComputer-IntegratedManufacturing21(2005)368378Keywords:Fixturedesign；Geometryconstraint；Deterministiclocating；Under-constrained；Over-constrainedconstraintstatus,aworkpieceunderanylocatingschemefallsintooneofthefollowingthreecategories:locatingproblemusingscrewtheoryin1989.Itisconcludedthatthelocatingwrenchesmatrixneedstobefullranktoachievedeterministiclocation.Thismethodhasbeenadoptedbynumerousstudiesaswell.Wangetal.3consideredARTICLEINPRESS0736-5845/$-seefrontmatterr2005ElsevierLtd.Allrightsreserved.doi:10.1016/j.rcim.2004.11.012C3Correspondingauthor.Tel.:fax:E-mailaddress:(H.Song).1.Well-constrained(deterministic):Theworkpieceismatedatauniquepositionwhensixlocatorsaremadetocontacttheworkpiecesurface.2.Under-constrained:Thesixdegreesoffreedomofworkpiecearenotfullyconstrained.3.Over-constrained:Thesixdegreesoffreedomofworkpieceareconstrainedbymorethansixlocators.In1985,AsadaandBy1proposedfullrankJacobianmatrixofconstraintequationsasacriterionandformedthebasisofanalyticalinvestigationsfordeterministiclocatingthatfollowed.Chouetal.2formulatedthedeterministic1.IntroductionAxtureisamechanismusedinmanufacturingoperationstoholdaworkpiecermlyinposition.Beingacrucialstepinprocessplanningformachiningparts,xturedesignneedstoensurethepositionalaccuracyanddimensionalaccuracyofaworkpiece.Ingeneral,3-2-1principleisthemostwidelyusedguidingprinciplefordevelopingalocationscheme.V-blockandpin-holelocatingprinciplesarealsocommonlyused.Alocationschemeforamachiningxturemustsatisfyanumberofrequirements.Themostbasicrequirementisthatitmustprovidedeterministiclocationfortheworkpiece1.Thisnotionstatesthatalocatorschemeproducesdeterministiclocationwhentheworkpiececannotmovewithoutlosingcontactwithatleastonelocator.Thishasbeenoneofthemostfundamentalguidelinesforxturedesignandstudiedbymanyresearchers.ConcerninggeometryAbstractGeometryconstraintisoneofthemostimportantconsiderationsinxturedesign.Analyticalformulationofdeterministiclocationhasbeenwelldeveloped.However,howtoanalyzeandreviseanon-deterministiclocatingschemeduringtheprocessofactualxturedesignpracticehasnotbeenthoroughlystudied.Inthispaper,amethodologytocharacterizexturingsystemsgeometryconstraintstatuswithfocusonunder-constraintisproposed.Anunder-constraintstatus,ifitexists,canberecognizedwithgivenlocatingscheme.Allun-constrainedmotionsofaworkpieceinanunder-constraintstatuscanbeautomaticallyidentied.Thisassiststhedesignertoimprovedecitlocatingschemeandprovidesguidelinesforrevisiontoeventuallyachievedeterministiclocating.r2005ElsevierLtd.Allrightsreserved.CAMLab,DepartmentofMechanicalEngineering,WorcesterPolytechnicInstitute,100InstituteRd,Worcester,MA01609,USAReceived14September2004；receivedinrevisedform9November2004；accepted10November2004LocatingcompletenessevaluationandrevisioninxtureplanH.SongC3,Y.R/locate/rcimlocatorworkpiececontactareaeffectsinsteadofapplyingpointcontact.Theyintroducedacontactmatrixandpointedoutthattwocontactbodiesshouldnothaveequalbutoppositecurvatureatcontactingpoint.Carlson4suggestedthatalinearapproximationmaynotbesufcientforsomeapplicationssuchasnon-prismaticsurfacesornon-smallrelativeerrors.Heproposedasecond-orderTaylorexpansionwhichalsotakeslocatorerrorinteractionintoaccount.MarinandFerreira5appliedChousformulationon3-2-1locationandformulatedseveraleasy-to-followplanningrules.Despitethenumerousanalyticalstudiesondeterministiclocation,lessattentionwaspaidtoanalyzenon-deterministiclocation.IntheAsadaandBysformulation,theyassumedfrictionlessandpointcontactbetweenxturingelementsandworkpiece.Thedesiredlocationisq*,atwhichaworkpieceistobepositionedandpiecewiselydifferentiablesurfacefunctionisgi(asshowninFig.1).ThesurfacefunctionisdenedasgiqC30:Tobedeterministic,thereshouldbeauniquesolutionforthefollowingequationsetforalllocators.giq0；i1；2；.；n,(1)wherenisthenumberoflocatorsandqx0；y0；z0；y0；f0；c0C138representsthepositionandorientationoftheworkpiece.OnlyconsideringthevicinityofdesiredlocationqC3；whereqqC3Dq；AsadaandByshowedthatARTICLEINPRESSH.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378369giqgiqC3hiDq,(2)wherehiistheJacobianmatrixofgeometryfunctions,asshownbythematrixinEq.(3).ThedeterministiclocatingrequirementcanbesatisediftheJacobianmatrixhasfullrank,whichmakestheEq.(2)tohaveonlyonesolutionqqC3:rankqg1qx0qg1qy0qg1qz0qg1qy0qg1qf0qg1qc0:qgiqx0qgiqy0qgiqz0qgiqy0qgiqf0qgiqc0:qgnqx0qgnqy0qgnqz0qgnqy0qgnqf0qgnqc026666666664377777777758:9=；6.(3)Upongivena3-2-1locatingscheme,therankofaJacobianmatrixforconstraintequationstellstheconstraintstatusasshowninTable1.Iftherankislessthansix,theworkpieceisunder-constrained,i.e.,thereexistsatleastonefreemotionoftheworkpiecethatisnotconstrainedbylocators.Ifthematrixhasfullrankbutthelocatingschemehasmorethansixlocators,theworkpieceisover-constrained,whichindicatesthereexistsatleastonelocatorsuchthatitcanberemovedwithoutaffectingthegeometryconstrainstatusoftheworkpiece.Forlocatingamodelotherthan3-2-1,datumframecanbeestablishedtoextractequivalentlocatingpoints.Hu6hasdevelopedasystematicapproachforthispurpose.Hence,thiscriterioncanbeappliedtoalllocatingschemes.XYZOXYZO(x0,y0,z0)giUCSWCSWorkpieceFig.1.Fixturingsystemmodel.Theyfurtherintroducedseveralindexesderivedfromthosematrixestoevaluatelocatorcongurations,followedbyoptimizationthroughconstrainednonlinearprogramming.Theiranalyticalstudy,however,doesnotconcerntheARTICLEINPRESSrevisionofnon-deterministiclocating.Currently,thereisnosystematicstudyonhowtodealwithaxturedesignthatfailedtoprovidedeterministiclocation.2.LocatingcompletenessevaluationIfdeterministiclocationisnotachievedbydesignedxturingsystem,itisasimportantfordesignerstoknowwhattheconstraintstatusisandhowtoimprovethedesign.Ifthexturingsystemisover-constrained,informa-tionabouttheunnecessarylocatorsisdesired.Whileunder-constrainedoccurs,theknowledgeaboutalltheun-constrainedmotionsofaworkpiecemayguidedesignerstoselectadditionallocatorsand/orrevisethelocatingschememoreefciently.AgeneralstrategytocharacterizegeometryconstraintstatusofalocatingschemeisdescribedinFig.2.Inthispaper,therankoflocatingmatrixisexertedtoevaluategeometryconstraintstatus(seeAppendixforderivationoflocatingmatrix).ThedeterministiclocatingrequiressixlocatorsthatprovidefullranklocatingmatrixWL:AsshowninFig.3,forgivenlocatornumbern；locatingnormalvectorai；bi；ciC138andlocatingpositionxi；yi；ziC138foreachlocator,i1；2；.；n；thenC26locatingmatrixcanbedeterminedasfollows:a1b1c1c1y1C0b1z1a1z1C0c1x1b1x1C0a1y1:2637Kangetal.7followedthesemethodsandimplementedthemtodevelopageometryconstraintanalysismoduleintheirautomatedcomputer-aidedxturedesignvericationsystem.TheirCAFDVsystemcancalculatetheJacobianmatrixanditsranktodeterminelocatingcompleteness.Itcanalsoanalyzetheworkpiecedisplacementandsensitivitytolocatingerror.Xiongetal.8presentedanapproachtochecktherankoflocatingmatrixWL(seeAppendix).Theyalsointro-ducedleft/rightgeneralizedinverseofthelocatingmatrixtoanalyzethegeometricerrorsofworkpiece.IthasbeenshownthatthepositionandorientationerrorsDXoftheworkpieceandthepositionerrorsDroflocatorsarerelatedasfollows:Well-constrained:DXWLDr,(4)Over-constrained:DXWTLWLC01WTLDr,(5)Under-constrained:DXWTLWLWTLC01DrI6C26C0WTLWLWTLC01WLl,(6)wherelisanarbitraryvector.Table1RankNumberoflocatorsStatuso6Under-constrained66Well-constrained646Over-constrainedH.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378370WLaibiciciyiC0biziaiziC0cixibixiC0aiyi:anbncncnynC0bnznanznC0cnxnbnxnC0anyn666664777775.(7)WhenrankWL6andn6；theworkpieceiswell-constrained.WhenrankWL6andn46；theworkpieceisover-constrained.ThismeanstherearenC06unnecessarylocatorsinthelocatingscheme.Theworkpiecewillbewell-constrainedwithoutthepresenceofthosenC06locators.ThemathematicalrepresentationforthisstatusisthattherearenC06rowvectorsinlocatingmatrixthatcanbeexpressedaslinearcombinationsoftheothersixrowvectors.ThelocatorscorrespondingtothatsixrowvectorsconsistoneARTICLEINPRESSlocatdeterm.be3.workpiH.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378371ingsch