(06020137)Locating completeness evaluation and revision in fixture plan.pdf

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)368378371ingschemethatprovidesdeterministiclocation.Thedevelopedalgorithmusesthefollowingapproachtoinetheunnecessarylocators:FindallthecombinationofnC06locators.Foreachcombination,removethatnC06locatorsfromlocatingscheme.Recalculatetherankoflocatingmatrixfortheleftsixlocators.Iftherankremainsunchanged,theremovednC06locatorsareresponsibleforover-constrainedstatus.Thismethodmayyieldmulti-solutionsandrequiredesignertodeterminewhichsetofunnecessarylocatorsshouldremovedforthebestlocatingperformance.WhenrankWLo6;theworkpieceisunder-constrained.AlgorithmdevelopmentandimplementationThealgorithmtobedevelopedherewilldedicatetoprovideinformationonun-constrainedmotionsoftheeceinunder-constrainedstatus.Supposetherearenlocators,therelationshipbetweenaworkpiecesposition/Fig.2.Geometryconstraintstatuscharacterization.XZY(a1,b1,c1)2,b2,c2)(x1,y1,z1)(x2,y2,z2)(ai,bi,ci)(xi,yi,zi)(aFig.3.Asimpliedlocatingscheme.orientijLLLARTICLEINPRESS3725.Toidentifyalltheun-constrainedmotionsoftheworkpiece,Vdxi;dyi;dzi;daxi;dayi;daziC138isintroducedsuchthatVDX0.(9)SincerankDXo6;theremustexistnon-zeroVthatsatisesEq.(9).Eachnon-zerosolutionofVrepresentsanun-constrainedmotion.EachtermofVrepresentsacomponentofthatmotion.Forexample,0;0;0;3;0;0C138saysthattherotationaboutx-axisisnotconstrained.0;1;1;0;0;0C138meansthattheworkpiececanmovealongthedirectiongivenbyvector0;1;1C138:Therecouldbeinnitesolutions.Thesolutionspace,however,canbeconstructedby6C0rankWLbasicsolutions.Followinganalysisisdedicatedtondoutthebasicsolutions.FromEqs.(8)and(9)VXdxDxdyDydzDzdaxDaxdayDaydazDazdxXni1w1iDridyXni1w2iDridzXni1w3iDridaxXni1w4iDridayXni1w5iDridazXni1w6iDriXni1Vw1i;w2i;w3i;w4i;w5i;w6iC138TDri0.10Eq.(10)holdsfor8DriifandonlyifEq.(11)istruefor8i1pipn:Vw1i;w2i;w3i;w4i;w5i;w6iC138T0.(11)Eq.(11)illustratesthedependencyrelationshipsamongrowvectorsofWr:Inspecialcases,say,allw1jequaltozero,Vhasanobvioussolution1,0,0,0,0,0,indicatingdisplacementalongthex-axisisnotconstrained.ThisiseasytounderstandbecauseDx0inthiscase,implyingthatthecorrespondingpositionerroroftheworkpieceisnotdependentofanylocatorerrors.Hence,theassociatedmotionisnotconstrainedbylocators.Moreover,acombinedmotionisnotconstrainedifoneoftheelementsinDXcanbeexpressedaslinearcombinationofotherelements.Forinstance,9w1ja0;w2ja0;w1jC0w2jfor8j:Inthisscenario,theworkpiececannotmovealongx-ory-axis.However,itcanmovealongthediagonallinebetweenx-andy-axisdenedbyvector1,1,0.Tondsolutionsforgeneralcases,thefollowingstrategywasdeveloped:1.Eliminatedependentrow(s)fromlocatingmatrix.LetrrankWL;nnumberoflocator.Ifron;createavectorinnC0rdimensionspaceUu1:uj:unC0rhi1pjpnC0r;1pujpn:SelectujinthewaythatrankWLrstillholdsaftersettingallthetermsofalltheujthrow(s)equaltozero.SetrC26modiedlocatingmatrixWLMa1b1c1c1y1C0b1z1a1z1C0c1x1b1x1C0a1y1:aibiciciyiC0biziaiziC0cixibixiC0aiyi:anbncncnynC0bnznanznC0cnxnbnxnC0anyn2666666437777775rC26,whergeometationerrorsandlocatorerrorscanbeexpressedasfollows:DXDxDyDzaxayaz2666666666437777777775w11:w1i:w1nw21:w2i:w2nw31:w3i:w3nw41:w4i:w4nw51:w5i:w5nw61:w6i:w6n2666666666437777777775C1Dr1:Dri:Drn2666666437777775,(8)eDx;Dy;Dz;ax;ay;azaredisplacementalongx,y,zaxisandrotationaboutx,y,zaxis,respectively.Driisricerroroftheithlocator.wisdenedbyrightgeneralizedinverseofthelocatingmatrixWrWTWWTC01H.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378wherei1;2;:;niauj:4.6.constrExamplvectorARTICLEINPRESSL3:0,0,10,2,1,00,L4:0,1,00,3,0,20,L5:0,1,00,1,0,20.Consequently,thelocatingmatrixisdetermined.WL0013C0100013C0300011C020010C02032666666437777775.LLvs:v666647775wqki:wqkr66647775C1wl1:wli:wlr:w61:w6i:w6r66647775,wheres1;2;:;6saqj;saqk;l1;2;:;6laqj:Repeatstep4(selectanothertermfromQ)andstep5untilall6C0rbasicsolutionshavebeendetermined.Basedonthisalgorithm,aC+programwasdevelopedtoidentifytheunder-constrainedstatusandun-ainedmotions.e1.Inasurfacegrindingoperation,aworkpieceislocatedonaxturesystemasshowninFig.4.Thenormalandpositionofeachlocatorareasfollows:1:0,0,10,1,3,00,2:0,0,10,3,3,00,CalculatedundeterminedtermsofV:VisalsoasolutionofEq.(11).Therundeterminedtermscanbefoundasfollows.v1:26663777wqk1:26663777w11:w1i:w1r:26663777C015.Wrmwl1:wli:wlr:w61:w6i:w6r666477756C26,wherel1;2;:;6laqj:Normalizethefreemotionspace.SupposeVV1;V2;V3;V4;V5;V6C138isoneofthebasicsolutionsofEq.(10)withallsixtermsundetermined.SelectatermqkfromvectorQ1pkp6C0r:SetVqkC01;Vqj0j1;2;:;6C0r;jak;(2.Computethe6C2nrightgeneralizedinverseofthemodiedlocatingmatrixWrWTLMWLMWTLMC01w11:w1i:w1rw21:w2i:w2rw31:w3i:w3rw41:w4i:w4rw51:w5i:w5rw61:w6i:w6r26666666664377777777756C2r3.TrimWrdowntoarC2rfullrankmatrixWrm:rrankWLo6:Constructa6C0rdimensionvectorQq1:qj:q6C0rhi1pjp6C0r;1pqjpn:SelectqjinthewaythatrankWrrstillholdsaftersettingallthetermsofalltheqjthrow(s)equaltozero.SetrC2rmodiedinversematrixw11:w1i:w1r:26663777H.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378373010C0201ARTICLEINPRESSThislocatingsystemprovidesunder-constrainedpositioningsincerankWL5o6:Theprogramthencalculatestherightgeneralizedinverseofthelocatingmatrix.Wr000000:50:5C01C00:51:50:75C01:251:5000:250:25C00:5000:5C00:50000000:5C00:526666666643777777775.Therstrowisrecognizedasadependentrowbecauseremovalofthisrowdoesnotaffectrankofthematrix.Theotherverowsareindependentrows.Alinearcombinationoftheindependentrowsisfoundaccordingtherequirementinstep5oftheprocedureforunder-constrainedstatus.Thesolutionforthisspecialcaseisobviousthatallthecoefcientsarezero.Hence,theun-constrainedmotionofworkpiececanbedeterminedasVC01;0;0;0;0;0C138:Thisindicatesthattheworkpiececanmovealongxdirection.Basedonthisresult,anadditionallocatorshouldbeemployedtoconstraintdisplacementofworkpiecealongx-axis.XZYL3L4L5L2L1Fig.4.Under-constrainedlocatingscheme.H.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378374Example2.Fig.5showsaknucklewith3-2-1locatingsystem.Thenormalvectorandpositionofeachlocatorinthisinitialdesignareasfollows:L1:0,1,00,896,C0877,C05150,L2:0,1,00,1060,C0875,C03780,L3:0,1,00,1010,C0959,C06120,L4:0.9955,C00.0349,0.0880,977,C0902,C06240,L5:0.9955,C00.0349,0.0880,977,C0866,C06240,L6:0.088,0.017,C00.9960,1034,C0864,C03590.ThelocatingmatrixofthiscongurationisWL010515:000:8960010378:1:0600010612:00:01000:9955C00:03490:0880C0101:2445C0707:26640:86380:9955C00:03490:0880C098:0728C0707:26640:82800:08800:0170C00:9960866:6257998:24660:093626666666643777777775,rankWL5o6revealsthattheworkpieceisunder-constrained.Itisfoundthatoneoftherstverowscanberemovedwithoutvaryingtherankoflocatingmatrix.Supposetherstrow,i.e.,locatorL1isremovedfromWL;theARTICLEINPRESSmodiedlocatingmatrixturnsintoWLM010378:001:0600010612:01000:9955C00:03490:0880C0101:2445C0707:26640:86380:9955C00:03490:0880C098:0728C0707:26640:82800:08800:0170C00:996866:6257998:24660:09362666666437777775.TherightgeneralizedinverseofthemodiedlocatingmatrixisWr1:8768C01:8607C020:666521:37160:49953:0551C02:0551C032:444832:44480C01:09561:086212:0648C012:4764C00:2916C00:00440:00440:0061C00:006100:0025C00:00250:0065C00:00690:0007C00:00040:00040:0284C00:0284026666666643777777775.Theprogramcheckedthedependentrowandfoundeveryrowisdependentonotherverows.Withoutlosinggenerality,therstrowisregardedasdependentrow.The5C25modiedinversematrixis23Fig.5.Knuckle610(modiedfromrealdesign).H.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378375Wrm3:0551C02:0551C032:444832:44480C01:09561:086212:0648C012:4764C00:2916C00:00440:00440:0061C00:006100:0025C00:00250:0065C00:00690:0007C00:00040:00040:0284C00:0284066666647777775.TheundeterminedsolutionisVC01;v2;v3;v4;v5;v6C138:TocalculatetheveundeterminedtermsofVaccordingtostep5,1:8768C01:8607C020:666521:37160:499526666666643777777775TC13:0551C02:0551C032:444832:44480C01:09561:086212:0648C012:4764C00:2916C00:00440:00440:0061C00:006100:0025C00:00250:0065C00:00690:0007C00:00040:00040:0284C00:0284026666666643777777775C010;C01:713;C00:0432;C00:0706;0:04C138.SubstitutingthisresultintotheundeterminedsolutionyieldsVC01;0;C01:713;C00:0432;C00:0706;0:04C138ThisvectorrepresentsafreemotiondenedbythecombinationofadisplacementalongC01,0,C01.713directioncombinedandarotationaboutC00.0432,C00.0706,0.04.Torevisethislocatingconguration,anotherlocatorshouldbeaddedtoconstrainthisfreemotionoftheworkpiece,assuminglocatorL1wasremovedinstep1.TheprogramcanalsocalculatethefreemotionsoftheworkpieceifalocatorotherthanL1wasremovedinstep1.Thisprovidesmorerevisionoptionsfordesigner.4.SummaryDeterministiclocationisanimportantrequirementforxturelocatingschemedesign.Analyticalcriterionfordeterministicstatushasbeenwellestablished.Tofurtherstudynon-deterministicstatus,analgorithmforcheckingthegeometryconstraintstatushasbeendeveloped.Thisalgorithmcanidentifyanunder-constrainedstatusandindicateqfiqfiqfiARTICLEINPRESSwXH.Song,Y.Rong/RoboticsandComputer-IntegratedManufacturing21(2005)368378376ithlocatoriniLilrGilXfiXli;Hli;rliqXliDXliqHliDHliqrliDrli,14workpieceWiwrwXwtheun-constrainedmotionsofworkpiece.Itcanalsorecognizeanover-constrainedstatusandunnecessarylocators.Theoutputinformationcanassistdesignertoanalyzeandimproveanexistinglocatingscheme.Appendix.LocatingmatrixConsiderageneralworkpieceasshowninFig.6.ChoosereferenceframefWgxedtotheworkpiece.LetfGgandfLigbetheglobalframeandtheithlocatorframexedrelativetoit.WehaveFiXw;Hw;rwifiXli;Hli;rli,(12)whereXw23C21andHw23C21(Xli23C21andHli23C21)arethepositionandorientationoftheworkpiece(theithlocator)intheglobalframefGg;rwi23C21(rli23C21)isthepositionoftheithcontactpointbetweentheworkpieceandtheithlocatorintheworkpieceframefWg(theithlocatorframefLig).AssumethatDXw2