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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,Z0GIUCSWCSWORKPIECEFIG1FIXTURINGSYSTEMMODELTHEYFURTHERINTRODUCEDSEVERALINDEXESDERIVEDFROMTHOSEMATRIXESTOEVALUATELOCATORCONGURATIONS,FOLLOWEDBYOPTIMIZATIONTHROUGHCONSTRAINEDNONLINEARPROGRAMMINGTHEIRANALYTICALSTUDY,HOWEVER,DOESNOTCONCERNTHEARTICLEINPRESSREVISIONOFNONDETERMINISTICLOCATINGCURRENTLY,THEREISNOSYSTEMATICSTUDYONHOWTODEALWITHAXTUREDESIGNTHATFAILEDTOPROVIDEDETERMINISTICLOCATION2LOCATINGCOMPLETENESSEVALUATIONIFDETERMINISTICLOCATIONISNOTACHIEVEDBYDESIGNEDXTURINGSYSTEM,ITISASIMPORTANTFORDESIGNERSTOKNOWWHATTHECONSTRAINTSTATUSISANDHOWTOIMPROVETHEDESIGNIFTHEXTURINGSYSTEMISOVERCONSTRAINED,INFORMATIONABOUTTHEUNNECESSARYLOCATORSISDESIREDWHILEUNDERCONSTRAINEDOCCURS,THEKNOWLEDGEABOUTALLTHEUNCONSTRAINEDMOTIONSOFAWORKPIECEMAYGUIDEDESIGNERSTOSELECTADDITIONALLOCATORSAND/ORREVISETHELOCATINGSCHEMEMOREEFCIENTLYAGENERALSTRATEGYTOCHARACTERIZEGEOMETRYCONSTRAINTSTATUSOFALOCATINGSCHEMEISDESCRIBEDINFIG2INTHISPAPER,THERANKOFLOCATINGMATRIXISEXERTEDTOEVALUATEGEOMETRYCONSTRAINTSTATUSSEEAPPENDIXFORDERIVATIONOFLOCATINGMATRIXTHEDETERMINISTICLOCATINGREQUIRESSIXLOCATORSTHATPROVIDEFULLRANKLOCATINGMATRIXWLASSHOWNINFIG3,FORGIVENLOCATORNUMBERNLOCATINGNORMALVECTORAIBICIC138ANDLOCATINGPOSITIONXIYIZIC138FOREACHLOCATOR,I12NTHENC26LOCATINGMATRIXCANBEDETERMINEDASFOLLOWSA1B1C1C1Y1C0B1Z1A1Z1C0C1X1B1X1C0A1Y12637KANGETAL7FOLLOWEDTHESEMETHODSANDIMPLEMENTEDTHEMTODEVELOPAGEOMETRYCONSTRAINTANALYSISMODULEINTHEIRAUTOMATEDCOMPUTERAIDEDXTUREDESIGNVERICATIONSYSTEMTHEIRCAFDVSYSTEMCANCALCULATETHEJACOBIANMATRIXANDITSRANKTODETERMINELOCATINGCOMPLETENESSITCANALSOANALYZETHEWORKPIECEDISPLACEMENTANDSENSITIVITYTOLOCATINGERRORXIONGETAL8PRESENTEDANAPPROACHTOCHECKTHERANKOFLOCATINGMATRIXWLSEEAPPENDIXTHEYALSOINTRODUCEDLEFT/RIGHTGENERALIZEDINVERSEOFTHELOCATINGMATRIXTOANALYZETHEGEOMETRICERRORSOFWORKPIECEITHASBEENSHOWNTHATTHEPOSITIONANDORIENTATIONERRORSDXOFTHEWORKPIECEANDTHEPOSITIONERRORSDROFLOCATORSARERELATEDASFOLLOWSWELLCONSTRAINEDDXWLDR,4OVERCONSTRAINEDDXWTLWLC01WTLDR,5UNDERCONSTRAINEDDXWTLWLWTLC01DRI6C26C0WTLWLWTLC01WLL,6WHERELISANARBITRARYVECTORTABLE1RANKNUMBEROFLOCATORSSTATUSO6UNDERCONSTRAINED66WELLCONSTRAINED646OVERCONSTRAINEDHSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378370WLAIBICICIYIC0BIZIAIZIC0CIXIBIXIC0AIYIANBNCNCNYNC0BNZNANZNC0CNXNBNXNC0ANYN6666647777757WHENRANKWL6ANDN6THEWORKPIECEISWELLCONSTRAINEDWHENRANKWL6ANDN46THEWORKPIECEISOVERCONSTRAINEDTHISMEANSTHEREARENC06UNNECESSARYLOCATORSINTHELOCATINGSCHEMETHEWORKPIECEWILLBEWELLCONSTRAINEDWITHOUTTHEPRESENCEOFTHOSENC06LOCATORSTHEMATHEMATICALREPRESENTATIONFORTHISSTATUSISTHATTHEREARENC06ROWVECTORSINLOCATINGMATRIXTHATCANBEEXPRESSEDASLINEARCOMBINATIONSOFTHEOTHERSIXROWVECTORSTHELOCATORSCORRESPONDINGTOTHATSIXROWVECTORSCONSISTONEARTICLEINPRESSLOCATDETERM1234BE3WORKPIHSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378371INGSCHEMETHATPROVIDESDETERMINISTICLOCATIONTHEDEVELOPEDALGORITHMUSESTHEFOLLOWINGAPPROACHTOINETHEUNNECESSARYLOCATORSFINDALLTHECOMBINATIONOFNC06LOCATORSFOREACHCOMBINATION,REMOVETHATNC06LOCATORSFROMLOCATINGSCHEMERECALCULATETHERANKOFLOCATINGMATRIXFORTHELEFTSIXLOCATORSIFTHERANKREMAINSUNCHANGED,THEREMOVEDNC06LOCATORSARERESPONSIBLEFOROVERCONSTRAINEDSTATUSTHISMETHODMAYYIELDMULTISOLUTIONSANDREQUIREDESIGNERTODETERMINEWHICHSETOFUNNECESSARYLOCATORSSHOULDREMOVEDFORTHEBESTLOCATINGPERFORMANCEWHENRANKWLO6THEWORKPIECEISUNDERCONSTRAINEDALGORITHMDEVELOPMENTANDIMPLEMENTATIONTHEALGORITHMTOBEDEVELOPEDHEREWILLDEDICATETOPROVIDEINFORMATIONONUNCONSTRAINEDMOTIONSOFTHEECEINUNDERCONSTRAINEDSTATUSSUPPOSETHEREARENLOCATORS,THERELATIONSHIPBETWEENAWORKPIECESPOSITION/FIG2GEOMETRYCONSTRAINTSTATUSCHARACTERIZATIONXZYA1,B1,C12,B2,C2X1,Y1,Z1X2,Y2,Z2AI,BI,CIXI,YI,ZIAFIG3ASIMPLIEDLOCATINGSCHEMEORIENTIJLLLARTICLEINPRESS3725TOIDENTIFYALLTHEUNCONSTRAINEDMOTIONSOFTHEWORKPIECE,VDXIDYIDZIDAXIDAYIDAZIC138ISINTRODUCEDSUCHTHATVDX09SINCERANKDXO6THEREMUSTEXISTNONZEROVTHATSATISESEQ9EACHNONZEROSOLUTIONOFVREPRESENTSANUNCONSTRAINEDMOTIONEACHTERMOFVREPRESENTSACOMPONENTOFTHATMOTIONFOREXAMPLE,000300C138SAYSTHATTHEROTATIONABOUTXAXISISNOTCONSTRAINED011000C138MEANSTHATTHEWORKPIECECANMOVEALONGTHEDIRECTIONGIVENBYVECTOR011C138THERECOULDBEINNITESOLUTIONSTHESOLUTIONSPACE,HOWEVER,CANBECONSTRUCTEDBY6C0RANKWLBASICSOLUTIONSFOLLOWINGANALYSISISDEDICATEDTONDOUTTHEBASICSOLUTIONSFROMEQS8AND9VXDXDXDYDYDZDZDAXDAXDAYDAYDAZDAZDXXNI1W1IDRIDYXNI1W2IDRIDZXNI1W3IDRIDAXXNI1W4IDRIDAYXNI1W5IDRIDAZXNI1W6IDRIXNI1VW1IW2IW3IW4IW5IW6IC138TDRI010EQ10HOLDSFOR8DRIIFANDONLYIFEQ11ISTRUEFOR8I1PIPNVW1IW2IW3IW4IW5IW6IC138T011EQ11ILLUSTRATESTHEDEPENDENCYRELATIONSHIPSAMONGROWVECTORSOFWRINSPECIALCASES,SAY,ALLW1JEQUALTOZERO,VHASANOBVIOUSSOLUTION1,0,0,0,0,0,INDICATINGDISPLACEMENTALONGTHEXAXISISNOTCONSTRAINEDTHISISEASYTOUNDERSTANDBECAUSEDX0INTHISCASE,IMPLYINGTHATTHECORRESPONDINGPOSITIONERROROFTHEWORKPIECEISNOTDEPENDENTOFANYLOCATORERRORSHENCE,THEASSOCIATEDMOTIONISNOTCONSTRAINEDBYLOCATORSMOREOVER,ACOMBINEDMOTIONISNOTCONSTRAINEDIFONEOFTHEELEMENTSINDXCANBEEXPRESSEDASLINEARCOMBINATIONOFOTHERELEMENTSFORINSTANCE,9W1JA0W2JA0W1JC0W2JFOR8JINTHISSCENARIO,THEWORKPIECECANNOTMOVEALONGXORYAXISHOWEVER,ITCANMOVEALONGTHEDIAGONALLINEBETWEENXANDYAXISDENEDBYVECTOR1,1,0TONDSOLUTIONSFORGENERALCASES,THEFOLLOWINGSTRATEGYWASDEVELOPED1ELIMINATEDEPENDENTROWSFROMLOCATINGMATRIXLETRRANKWLNNUMBEROFLOCATORIFRONCREATEAVECTORINNC0RDIMENSIONSPACEUU1UJUNC0RHI1PJPNC0R1PUJPNSELECTUJINTHEWAYTHATRANKWLRSTILLHOLDSAFTERSETTINGALLTHETERMSOFALLTHEUJTHROWSEQUALTOZEROSETRC26MODIEDLOCATINGMATRIXWLMA1B1C1C1Y1C0B1Z1A1Z1C0C1X1B1X1C0A1Y1AIBICICIYIC0BIZIAIZIC0CIXIBIXIC0AIYIANBNCNCNYNC0BNZNANZNC0CNXNBNXNC0ANYN2666666437777775RC26,WHERGEOMETATIONERRORSANDLOCATORERRORSCANBEEXPRESSEDASFOLLOWSDXDXDYDZAXAYAZ2666666666437777777775W11W1IW1NW21W2IW2NW31W3IW3NW41W4IW4NW51W5IW5NW61W6IW6N2666666666437777777775C1DR1DRIDRN2666666437777775,8EDXDYDZAXAYAZAREDISPLACEMENTALONGX,Y,ZAXISANDROTATIONABOUTX,Y,ZAXIS,RESPECTIVELYDRIISRICERROROFTHEITHLOCATORWISDENEDBYRIGHTGENERALIZEDINVERSEOFTHELOCATINGMATRIXWRWTWWTC01HSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378WHEREI12NIAUJ46CONSTREXAMPLVECTORARTICLEINPRESSL30,0,10,2,1,00,L40,1,00,3,0,20,L50,1,00,1,0,20CONSEQUENTLY,THELOCATINGMATRIXISDETERMINEDWL0013C0100013C0300011C020010C02032666666437777775LLVSV666647775WQKIWQKR66647775C1WL1WLIWLRW61W6IW6R66647775,WHERES126SAQJSAQKL126LAQJREPEATSTEP4SELECTANOTHERTERMFROMQANDSTEP5UNTILALL6C0RBASICSOLUTIONSHAVEBEENDETERMINEDBASEDONTHISALGORITHM,ACPROGRAMWASDEVELOPEDTOIDENTIFYTHEUNDERCONSTRAINEDSTATUSANDUNAINEDMOTIONSE1INASURFACEGRINDINGOPERATION,AWORKPIECEISLOCATEDONAXTURESYSTEMASSHOWNINFIG4THENORMALANDPOSITIONOFEACHLOCATORAREASFOLLOWS10,0,10,1,3,00,20,0,10,3,3,00,CALCULATEDUNDETERMINEDTERMSOFVVISALSOASOLUTIONOFEQ11THERUNDETERMINEDTERMSCANBEFOUNDASFOLLOWSV126663777WQK126663777W11W1IW1R26663777C015WRMWL1WLIWLRW61W6IW6R666477756C26,WHEREL126LAQJNORMALIZETHEFREEMOTIONSPACESUPPOSEVV1V2V3V4V5V6C138ISONEOFTHEBASICSOLUTIONSOFEQ10WITHALLSIXTERMSUNDETERMINEDSELECTATERMQKFROMVECTORQ1PKP6C0RSETVQKC01VQJ0J126C0RJAK2COMPUTETHE6C2NRIGHTGENERALIZEDINVERSEOFTHEMODIEDLOCATINGMATRIXWRWTLMWLMWTLMC01W11W1IW1RW21W2IW2RW31W3IW3RW41W4IW4RW51W5IW5RW61W6IW6R26666666664377777777756C2R3TRIMWRDOWNTOARC2RFULLRANKMATRIXWRMRRANKWLO6CONSTRUCTA6C0RDIMENSIONVECTORQQ1QJQ6C0RHI1PJP6C0R1PQJPNSELECTQJINTHEWAYTHATRANKWRRSTILLHOLDSAFTERSETTINGALLTHETERMSOFALLTHEQJTHROWSEQUALTOZEROSETRC2RMODIEDINVERSEMATRIXW11W1IW1R26663777HSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378373010C0201ARTICLEINPRESSTHISLOCATINGSYSTEMPROVIDESUNDERCONSTRAINEDPOSITIONINGSINCERANKWL5O6THEPROGRAMTHENCALCULATESTHERIGHTGENERALIZEDINVERSEOFTHELOCATINGMATRIXWR000000505C01C00515075C01251500025025C0050005C00500000005C00526666666643777777775THERSTROWISRECOGNIZEDASADEPENDENTROWBECAUSEREMOVALOFTHISROWDOESNOTAFFECTRANKOFTHEMATRIXTHEOTHERVEROWSAREINDEPENDENTROWSALINEARCOMBINATIONOFTHEINDEPENDENTROWSISFOUNDACCORDINGTHEREQUIREMENTINSTEP5OFTHEPROCEDUREFORUNDERCONSTRAINEDSTATUSTHESOLUTIONFORTHISSPECIALCASEISOBVIOUSTHATALLTHECOEFCIENTSAREZEROHENCE,THEUNCONSTRAINEDMOTIONOFWORKPIECECANBEDETERMINEDASVC0100000C138THISINDICATESTHATTHEWORKPIECECANMOVEALONGXDIRECTIONBASEDONTHISRESULT,ANADDITIONALLOCATORSHOULDBEEMPLOYEDTOCONSTRAINTDISPLACEMENTOFWORKPIECEALONGXAXISXZYL3L4L5L2L1FIG4UNDERCONSTRAINEDLOCATINGSCHEMEHSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378374EXAMPLE2FIG5SHOWSAKNUCKLEWITH321LOCATINGSYSTEMTHENORMALVECTORANDPOSITIONOFEACHLOCATORINTHISINITIALDESIGNAREASFOLLOWSL10,1,00,896,C0877,C05150,L20,1,00,1060,C0875,C03780,L30,1,00,1010,C0959,C06120,L409955,C000349,00880,977,C0902,C06240,L509955,C000349,00880,977,C0866,C06240,L60088,0017,C009960,1034,C0864,C03590THELOCATINGMATRIXOFTHISCONGURATIONISWL01051500089600103781060001061200010009955C00034900880C01012445C070726640863809955C00034900880C0980728C07072664082800088000170C009960866625799824660093626666666643777777775,RANKWL5O6REVEALSTHATTHEWORKPIECEISUNDERCONSTRAINEDITISFOUNDTHATONEOFTHERSTVEROWSCANBEREMOVEDWITHOUTVARYINGTHERANKOFLOCATINGMATRIXSUPPOSETHERSTROW,IE,LOCATORL1ISREMOVEDFROMWLTHEARTICLEINPRESSMODIEDLOCATINGMATRIXTURNSINTOWLM0103780010600010612010009955C00034900880C01012445C070726640863809955C00034900880C0980728C07072664082800088000170C0099686662579982466009362666666437777775THERIGHTGENERALIZEDINVERSEOFTHEMODIEDLOCATINGMATRIXISWR18768C018607C02066652137160499530551C020551C03244483244480C01095610862120648C0124764C002916C0000440004400061C000061000025C00002500065C00006900007C0000040000400284C000284026666666643777777775THEPROGRAMCHECKEDTHEDEPENDENTROWANDFOUNDEVERYROWISDEPENDENTONOTHERVEROWSWITHOUTLOSINGGENERALITY,THERSTROWISREGARDEDASDEPENDENTROWTHE5C25MODIEDINVERSEMATRIXIS23FIG5KNUCKLE610MODIEDFROMREALDESIGNHSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368378375WRM30551C020551C03244483244480C01095610862120648C0124764C002916C0000440004400061C000061000025C00002500065C00006900007C0000040000400284C000284066666647777775THEUNDETERMINEDSOLUTIONISVC01V2V3V4V5V6C138TOCALCULATETHEVEUNDETERMINEDTERMSOFVACCORDINGTOSTEP5,18768C018607C02066652137160499526666666643777777775TC130551C020551C03244483244480C01095610862120648C0124764C002916C0000440004400061C000061000025C00002500065C00006900007C000004000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