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ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378KEYWORDSFIXTUREDESIGNGEOMETRYCONSTRAINTDETERMINISTICLOCATINGUNDERCONSTRAINEDOVERCONSTRAINEDCONSTRAINTSTATUS,AWORKPIECEUNDERANYLOCATINGSCHEMEFALLSINTOONEOFTHEFOLLOWINGTHREECATEGORIESLOCATINGPROBLEMUSINGSCREWTHEORYIN1989ITISCONCLUDEDTHATTHELOCATINGWRENCHESMATRIXNEEDSTOBEFULLRANKTOACHIEVEDETERMINISTICLOCATIONTHISMETHODHASBEENADOPTEDBYNUMEROUSSTUDIESASWELLWANGETAL3CONSIDEREDARTICLEINPRESS07365845/SEEFRONTMATTERR2005ELSEVIERLTDALLRIGHTSRESERVEDDOI101016/JRCIM200411012C3CORRESPONDINGAUTHORTEAMAILADDRESSHSONGWPIEDUHSONG1WELLCONSTRAINEDDETERMINISTICTHEWORKPIECEISMATEDATAUNIQUEPOSITIONWHENSIXLOCATORSAREMADETOCONTACTTHEWORKPIECESURFACE2UNDERCONSTRAINEDTHESIXDEGREESOFFREEDOMOFWORKPIECEARENOTFULLYCONSTRAINED3OVERCONSTRAINEDTHESIXDEGREESOFFREEDOMOFWORKPIECEARECONSTRAINEDBYMORETHANSIXLOCATORSIN1985,ASADAANDBY1PROPOSEDFULLRANKJACOBIANMATRIXOFCONSTRAINTEQUATIONSASACRITERIONANDFORMEDTHEBASISOFANALYTICALINVESTIGATIONSFORDETERMINISTICLOCATINGTHATFOLLOWEDCHOUETAL2FORMULATEDTHEDETERMINISTIC1INTRODUCTIONAXTUREISAMECHANISMUSEDINMANUFACTURINGOPERATIONSTOHOLDAWORKPIECERMLYINPOSITIONBEINGACRUCIALSTEPINPROCESSPLANNINGFORMACHININGPARTS,XTUREDESIGNNEEDSTOENSURETHEPOSITIONALACCURACYANDDIMENSIONALACCURACYOFAWORKPIECEINGENERAL,321PRINCIPLEISTHEMOSTWIDELYUSEDGUIDINGPRINCIPLEFORDEVELOPINGALOCATIONSCHEMEVBLOCKANDPINHOLELOCATINGPRINCIPLESAREALSOCOMMONLYUSEDALOCATIONSCHEMEFORAMACHININGXTUREMUSTSATISFYANUMBEROFREQUIREMENTSTHEMOSTBASICREQUIREMENTISTHATITMUSTPROVIDEDETERMINISTICLOCATIONFORTHEWORKPIECE1THISNOTIONSTATESTHATALOCATORSCHEMEPRODUCESDETERMINISTICLOCATIONWHENTHEWORKPIECECANNOTMOVEWITHOUTLOSINGCONTACTWITHATLEASTONELOCATORTHISHASBEENONEOFTHEMOSTFUNDAMENTALGUIDELINESFORXTUREDESIGNANDSTUDIEDBYMANYRESEARCHERSCONCERNINGGEOMETRYABSTRACTGEOMETRYCONSTRAINTISONEOFTHEMOSTIMPORTANTCONSIDERATIONSINXTUREDESIGNANALYTICALFORMULATIONOFDETERMINISTICLOCATIONHASBEENWELLDEVELOPEDHOWEVER,HOWTOANALYZEANDREVISEANONDETERMINISTICLOCATINGSCHEMEDURINGTHEPROCESSOFACTUALXTUREDESIGNPRACTICEHASNOTBEENTHOROUGHLYSTUDIEDINTHISPAPER,AMETHODOLOGYTOCHARACTERIZEXTURINGSYSTEMSGEOMETRYCONSTRAINTSTATUSWITHFOCUSONUNDERCONSTRAINTISPROPOSEDANUNDERCONSTRAINTSTATUS,IFITEXISTS,CANBERECOGNIZEDWITHGIVENLOCATINGSCHEMEALLUNCONSTRAINEDMOTIONSOFAWORKPIECEINANUNDERCONSTRAINTSTATUSCANBEAUTOMATICALLYIDENTIEDTHISASSISTSTHEDESIGNERTOIMPROVEDECITLOCATINGSCHEMEANDPROVIDESGUIDELINESFORREVISIONTOEVENTUALLYACHIEVEDETERMINISTICLOCATINGR2005ELSEVIERLTDALLRIGHTSRESERVEDCAMLAB,DEPARTMENTOFMECHANICALENGINEERING,WORCESTERPOLYTECHNICINSTITUTE,100INSTITUTERD,WORCESTER,MA01609,USARECEIVED14SEPTEMBER2004RECEIVEDINREVISEDFORM9NOVEMBER2004ACCEPTED10NOVEMBER2004LOCATINGCOMPLETENESSEVALUATIONANDREVISIONINXTUREPLANHSONGC3,YRONGWWWELSEVIERCOM/LOCATE/RCIMLOCATORWORKPIECECONTACTAREAEFFECTSINSTEADOFAPPLYINGPOINTCONTACTTHEYINTRODUCEDACONTACTMATRIXANDPOINTEDOUTTHATTWOCONTACTBODIESSHOULDNOTHAVEEQUALBUTOPPOSITECURVATUREATCONTACTINGPOINTCARLSON4SUGGESTEDTHATALINEARAPPROXIMATIONMAYNOTBESUFCIENTFORSOMEAPPLICATIONSSUCHASNONPRISMATICSURFACESORNONSMALLRELATIVEERRORSHEPROPOSEDASECONDORDERTAYLOREXPANSIONWHICHALSOTAKESLOCATORERRORINTERACTIONINTOACCOUNTMARINANDFERREIRA5APPLIEDCHOUSFORMULATIONON321LOCATIONANDFORMULATEDSEVERALEASYTOFOLLOWPLANNINGRULESDESPITETHENUMEROUSANALYTICALSTUDIESONDETERMINISTICLOCATION,LESSATTENTIONWASPAIDTOANALYZENONDETERMINISTICLOCATIONINTHEASADAANDBYSFORMULATION,THEYASSUMEDFRICTIONLESSANDPOINTCONTACTBETWEENXTURINGELEMENTSANDWORKPIECETHEDESIREDLOCATIONISQ,ATWHICHAWORKPIECEISTOBEPOSITIONEDANDPIECEWISELYDIFFERENTIABLESURFACEFUNCTIONISGIASSHOWNINFIG1THESURFACEFUNCTIONISDENEDASGIQC30TOBEDETERMINISTIC,THERESHOULDBEAUNIQUESOLUTIONFORTHEFOLLOWINGEQUATIONSETFORALLLOCATORSGIQ0I12N,1WHERENISTHENUMBEROFLOCATORSANDQX0Y0Z0Y0F0C0C138REPRESENTSTHEPOSITIONANDORIENTATIONOFTHEWORKPIECEONLYCONSIDERINGTHEVICINITYOFDESIREDLOCATIONQC3WHEREQQC3DQASADAANDBYSHOWEDTHATARTICLEINPRESSHSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378369GIQGIQC3HIDQ,2WHEREHIISTHEJACOBIANMATRIXOFGEOMETRYFUNCTIONS,ASSHOWNBYTHEMATRIXINEQ3THEDETERMINISTICLOCATINGREQUIREMENTCANBESATISEDIFTHEJACOBIANMATRIXHASFULLRANK,WHICHMAKESTHEEQ2TOHAVEONLYONESOLUTIONQQC3RANKQG1QX0QG1QY0QG1QZ0QG1QY0QG1QF0QG1QC0QGIQX0QGIQY0QGIQZ0QGIQY0QGIQF0QGIQC0QGNQX0QGNQY0QGNQZ0QGNQY0QGNQF0QGNQC026666666664377777777758963UPONGIVENA321LOCATINGSCHEME,THERANKOFAJACOBIANMATRIXFORCONSTRAINTEQUATIONSTELLSTHECONSTRAINTSTATUSASSHOWNINTABLE1IFTHERANKISLESSTHANSIX,THEWORKPIECEISUNDERCONSTRAINED,IE,THEREEXISTSATLEASTONEFREEMOTIONOFTHEWORKPIECETHATISNOTCONSTRAINEDBYLOCATORSIFTHEMATRIXHASFULLRANKBUTTHELOCATINGSCHEMEHASMORETHANSIXLOCATORS,THEWORKPIECEISOVERCONSTRAINED,WHICHINDICATESTHEREEXISTSATLEASTONELOCATORSUCHTHATITCANBEREMOVEDWITHOUTAFFECTINGTHEGEOMETRYCONSTRAINSTATUSOFTHEWORKPIECEFORLOCATINGAMODELOTHERTHAN321,DATUMFRAMECANBEESTABLISHEDTOEXTRACTEQUIVALENTLOCATINGPOINTSHU6HASDEVELOPEDASYSTEMATICAPPROACHFORTHISPURPOSEHENCE,THISCRITERIONCANBEAPPLIEDTOALLLOCATINGSCHEMESXYZOXYZOX0,Y0,Z0GIUCSWCSWORKPIECEFIG1FIXTURINGSYSTEMMODELTHEYFURT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