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ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368–378KEYWORDSFIXTUREDESIGN;GEOMETRYCONSTRAINT;DETERMINISTICLOCATING;UNDERCONSTRAINED;OVERCONSTRAINEDCONSTRAINTSTATUS,AWORKPIECEUNDERANYLOCATINGSCHEMEFALLSINTOONEOFTHEFOLLOWINGTHREECATEGORIESLOCATINGPROBLEMUSINGSCREWTHEORYIN1989ITISCONCLUDEDTHATTHELOCATINGWRENCHESMATRIXNEEDSTOBEFULLRANKTOACHIEVEDETERMINISTICLOCATIONTHISMETHODHASBEENADOPTEDBYNUMEROUSSTUDIESASWELLWANGETAL3CONSIDEREDARTICLEINPRESS07365845/SEEFRONTMATTERR2005ELSEVIERLTDALLRIGHTSRESERVEDDOI101016/JRCIM200411012C3CORRESPONDINGAUTHORTEL15088316092;FAX15088316412EMAILADDRESSHSONGWPIEDUHSONG1WELLCONSTRAINEDDETERMINISTICTHEWORKPIECEISMATEDATAUNIQUEPOSITIONWHENSIXLOCATORSAREMADETOCONTACTTHEWORKPIECESURFACE2UNDERCONSTRAINEDTHESIXDEGREESOFFREEDOMOFWORKPIECEARENOTFULLYCONSTRAINED3OVERCONSTRAINEDTHESIXDEGREESOFFREEDOMOFWORKPIECEARECONSTRAINEDBYMORETHANSIXLOCATORSIN1985,ASADAANDBY1PROPOSEDFULLRANKJACOBIANMATRIXOFCONSTRAINTEQUATIONSASACRITERIONANDFORMEDTHEBASISOFANALYTICALINVESTIGATIONSFORDETERMINISTICLOCATINGTHATFOLLOWEDCHOUETAL2FORMULATEDTHEDETERMINISTIC1INTRODUCTIONAfiXTUREISAMECHANISMUSEDINMANUFACTURINGOPERATIONSTOHOLDAWORKPIECEfiRMLYINPOSITIONBEINGACRUCIALSTEPINPROCESSPLANNINGFORMACHININGPARTS,fiXTUREDESIGNNEEDSTOENSURETHEPOSITIONALACCURACYANDDIMENSIONALACCURACYOFAWORKPIECEINGENERAL,321PRINCIPLEISTHEMOSTWIDELYUSEDGUIDINGPRINCIPLEFORDEVELOPINGALOCATIONSCHEMEVBLOCKANDPINHOLELOCATINGPRINCIPLESAREALSOCOMMONLYUSEDALOCATIONSCHEMEFORAMACHININGfiXTUREMUSTSATISFYANUMBEROFREQUIREMENTSTHEMOSTBASICREQUIREMENTISTHATITMUSTPROVIDEDETERMINISTICLOCATIONFORTHEWORKPIECE1THISNOTIONSTATESTHATALOCATORSCHEMEPRODUCESDETERMINISTICLOCATIONWHENTHEWORKPIECECANNOTMOVEWITHOUTLOSINGCONTACTWITHATLEASTONELOCATORTHISHASBEENONEOFTHEMOSTFUNDAMENTALGUIDELINESFORfiXTUREDESIGNANDSTUDIEDBYMANYRESEARCHERSCONCERNINGGEOMETRYABSTRACTGEOMETRYCONSTRAINTISONEOFTHEMOSTIMPORTANTCONSIDERATIONSINfiXTUREDESIGNANALYTICALFORMULATIONOFDETERMINISTICLOCATIONHASBEENWELLDEVELOPEDHOWEVER,HOWTOANALYZEANDREVISEANONDETERMINISTICLOCATINGSCHEMEDURINGTHEPROCESSOFACTUALfiXTUREDESIGNPRACTICEHASNOTBEENTHOROUGHLYSTUDIEDINTHISPAPER,AMETHODOLOGYTOCHARACTERIZEfiXTURINGSYSTEM’SGEOMETRYCONSTRAINTSTATUSWITHFOCUSONUNDERCONSTRAINTISPROPOSEDANUNDERCONSTRAINTSTATUS,IFITEXISTS,CANBERECOGNIZEDWITHGIVENLOCATINGSCHEMEALLUNCONSTRAINEDMOTIONSOFAWORKPIECEINANUNDERCONSTRAINTSTATUSCANBEAUTOMATICALLYIDENTIfiEDTHISASSISTSTHEDESIGNERTOIMPROVEDEfiCITLOCATINGSCHEMEANDPROVIDESGUIDELINESFORREVISIONTOEVENTUALLYACHIEVEDETERMINISTICLOCATINGR2005ELSEVIERLTDALLRIGHTSRESERVEDCAMLAB,DEPARTMENTOFMECHANICALENGINEERING,WORCESTERPOLYTECHNICINSTITUTE,100INSTITUTERD,WORCESTER,MA01609,USARECEIVED14SEPTEMBER2004;RECEIVEDINREVISEDFORM9NOVEMBER2004;ACCEPTED10NOVEMBER2004LOCATINGCOMPLETENESSEVALUATIONANDREVISIONINfiXTUREPLANHSONGC3,YRONGWWWELSEVIERCOM/LOCATE/RCIMLOCATOR–WORKPIECECONTACTAREAEFFECTSINSTEADOFAPPLYINGPOINTCONTACTTHEYINTRODUCEDACONTACTMATRIXANDPOINTEDOUTTHATTWOCONTACTBODIESSHOULDNOTHAVEEQUALBUTOPPOSITECURVATUREATCONTACTINGPOINTCARLSON4SUGGESTEDTHATALINEARAPPROXIMATIONMAYNOTBESUFfiCIENTFORSOMEAPPLICATIONSSUCHASNONPRISMATICSURFACESORNONSMALLRELATIVEERRORSHEPROPOSEDASECONDORDERTAYLOREXPANSIONWHICHALSOTAKESLOCATORERRORINTERACTIONINTOACCOUNTMARINANDFERREIRA5APPLIEDCHOU’SFORMULATIONON321LOCATIONANDFORMULATEDSEVERALEASYTOFOLLOWPLANNINGRULESDESPITETHENUMEROUSANALYTICALSTUDIESONDETERMINISTICLOCATION,LESSATTENTIONWASPAIDTOANALYZENONDETERMINISTICLOCATIONINTHEASADAANDBY’SFORMULATION,THEYASSUMEDFRICTIONLESSANDPOINTCONTACTBETWEENfiXTURINGELEMENTSANDWORKPIECETHEDESIREDLOCATIONISQ,ATWHICHAWORKPIECEISTOBEPOSITIONEDANDPIECEWISELYDIFFERENTIABLESURFACEFUNCTIONISGIASSHOWNINFIG1THESURFACEFUNCTIONISDEfiNEDASGIQC30TOBEDETERMINISTIC,THERESHOULDBEAUNIQUESOLUTIONFORTHEFOLLOWINGEQUATIONSETFORALLLOCATORSGIQ0;I1;2;;N,1WHERENISTHENUMBEROFLOCATORSANDQX0;Y0;Z0;Y0;F0;C0C138REPRESENTSTHEPOSITIONANDORIENTATIONOFTHEWORKPIECEONLYCONSIDERINGTHEVICINITYOFDESIREDLOCATIONQC3;WHEREQQC3DQ;ASADAANDBYSHOWEDTHATARTICLEINPRESSHSONG,YRONG/ROBOTICSANDCOMPUTERINTEGRATEDMANUFACTURING212005368–378369GIQGIQC3HIDQ,2WHEREHIISTHEJACOBIANMATRIXOFGEOMETRYFUNCTIONS,ASSHOWNBYTHEMATRIXINEQ3THEDETERMINISTICLOCATINGREQUIREMENTCANBESATISfiEDIFTHEJACOBIANMATRIXHASFULLRANK,WHICHMAKESTHEEQ2TOHAVEONLYONESOLUTIONQQC3RANKQG1QX0QG1QY0QG1QZ0QG1QY0QG1QF0QG1QC0QGIQX0QGIQY0QGIQZ0QGIQY0QGIQF0QGIQC0QGNQX0QGNQY0QGNQZ0QGNQY0QGNQF0QGNQC0266666666643777777777589;63UPONGIVENA321LOCATINGSCHEME,THERANKOFAJACOBIANMATRIXFORCONSTRAINTEQUATIONSTELLSTHECONSTRAINTSTATUSASSHOWNINTABLE1IFTHERANKISLESSTHANSIX,THEWORKPIECEISUNDERCONSTRAINED,IE,THEREEXISTSATLEASTONEFREEMOTIONOFTHEWORKPIECETHATISNOTCONSTRAINEDBYLOCATORSIFTHEMATRIXHASFULLRANKBUTTHELOCATINGSCHEMEHASMORETHANSIXLOCATORS,THEWORKPIECEISOVERCONSTRAINED,WHICHINDICATESTHEREEXISTSATLEASTONELOCATORSUCHTHATITCANBEREMOVEDWITHOUTAFFECTINGTHEGEOMETRYCONSTRAINSTATUSOFTHEWORKPIECEFORLOCATINGAMODELOTHERTHAN321,DATUMFRAMECANBEESTABLISHEDTOEXTRACTEQUIVALENTLOCATINGPOINTSHU6HASDEVELOPEDASYSTEMATICAPPROACHFORTHISPURPOSEHENCE,THISCRITERIONCANBEAPPLIEDTOAL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