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AUTOMATICFIXTURESYNTHESISIN3D

KamenPenev

ProgrammableAutomationLaboratory

ComputerScienceDepartmentandInstituteforRoboticsandIntelligentSystems

UniversityofSouthernCalifornia

LosAngeles,CA90089-0781AristidesA.G.Requicha

LosAngeles,CA90089-0781

Abstract

Afixtureisanarrangementoffixturingmodulesthatlocateandholdaworkpartduringamanufacturingoperation.Inthisworkwe.considerfixtureswithfrictionlesspointcontactsandpresentamethodforplacementofcontactpointsonanon-prismatic3Dworkpart.Itisanon-deterministic,potentialfieldalgorithmforcontactpointplacement.Themethodprovidesabasicframeworkfortheintegrationofheterogeneoushigh-levelfixturingagentsthroughaninterfacebasedonzonesofattractionandrepulsionontheworkpartboundary.Thealgorithmmayproduceredundantfixtures,andcanaugmentpartialsolutionstocompleteformclosurefixtures.

1.Introduction

Afixtureisanarrangementoffixturingmodulesthatlocateandholdaworkpartduringamanufacturingoperation,suchasmachining,assemblyandinspection.Fixturingisofessentialimportancetoindustrialmanufacturingandconstitutesasignificantpartofallmanufacturingcosts.Therefore,fixturedesignautomationisveryimportant.Fixturedesigninvolvesagreatvarietyofconsiderations,suchasrestraint,deterministiclocation,loadability,andtoolaccessibility.Efficientalgorithmsthataddressthewholerangeoffixturingissuesforacomprehensivedomainofworkpartsdonotyetexist.Recently,BrostandPeterspublishedanalgorithm[Brost&

Peters1996]thatextendstheearlierclassicworkofBrostandGoldberg[Brost&

Goldberg,1994]tothe3Ddomain.Thisalgorithm,however,requiresverticalandhorizontalplanarsurfacestoconstituteasubstantialpartoftheworkpartboundary.Itgeneratesallpossiblefixturesandthenratesthemaccordinglytocertainmetrics.Thisiscomputationallyexpensive.Wagneretalpresentedanalgorithmthatusessevenmodularstrutsmountedinaboxtofixturepolyhedra[Wagneretal1995].Thisalgorithmisnotcompleteinthesensethatitcannoteffectivelyhandlecertaincases,suchasacubewithfacesparalleltothebox.Italsosuffersfromhighcomputationalcomplexity.WallackandCannysuggestedanothermethodwithan“enumerate-and-rate”flavor[Wallack&

Canny1996].Itcanfixtureprismaticworkpartswithplanarandcylindricalverticalsurfaces.Ponceproposedanalgorithmthatutilizescurvatureeffectstocomputefixtureswithfourfingersforpolyhedralparts[Ponce96].Thereducednumberofcontactsshouldprovideforbettercomplexityofthisalgorithm,butthequalityoftheproducedfixturesseemstobeinferiortotheonesthatutilizemorecontactsandprovideclassicalformclosure.

Inthispaperwepresentanewpotential-fieldalgorithmthatefficientlyproducesqualityfixturedesigns.Ouralgorithmworksforarbitraryworkpartsandprovidesconvenientuniversalmeansforrepresentingvariousfixturingrequirements.Thisalgorithmisadirectgeneralizationofthe2DpotentialfieldfixturingalgorithmofPenevandRequicha[Penev&

Requicha1996].Weconsiderfixtureswithfrictionlesspointcontacts.Ithasbeenproventhatsevencontactsarenecessary.[Somoff,1900]andsufficient[Markenscoffetal,1990]toimmobilizeanyworkpartin3D

Followingaleast-commitmentstrategy,theprocessoffixturesynthesismaybeseparatedintothreestages–fixturingtaskanalysis,contactpointplacement,andfixturelayoutdesign.Inthefixturingtaskanalysisphasetheworkpartgeometryandmanufacturingprocessareanalyzedtoidentifyvariousparametersofthefixturingproblem,suchascuttingforces,inaccessibleorforbiddenareas,andalsotofindfeaturesthatmaybeusefulforapplyingfixturingdevices,suchasmachinedflatsurfaces,horizontalandverticalsurfaces,pairsofparallelsurfaces,pairsofperpendicularsurfaces,etc.

Figure1:

Contactpointplacement

Inthecontactpointplacementphaseanumberofcontactpointsareplacedontheworkpartboundary（Figure1）,sothattheresultingconfigurationofcontactssatisfiestheconstraintsidentifiedintheanalysisphaseaswellascertainkinematicrequirementsthatmustbesatisfiedbyanyfixture,suchastotalrestraint.

Figure2:

Fromcontactpointconfigurationtofixturelayoutdesign

Inthelayoutdesignphase“towers”offixturingcomponentsarebuiltandplacedaroundtheworkpartsoastocontactthepartatthepointlocationscomputedinthecontactpointplacementphase.Forexample,acontactpointonahorizontalworkpartsurface（Figure2a）mayleadtotheinstantiationofanoverheadclampthatcontactstheworkpartatthatparticularpoint（Figure2b）.Thisisadesign-for-functionproblemconstrainedbythesetofavailablefixturingmodulesandtheirparameters.Thesetofcontactpointsarethefunctionalspecificationandthefixturelayoutisaconfigurationofcomponentsthatachievesit.

Inthisresearchwefocusoncontactpointplacementanditsintegrationwithpartandtaskanalysis.Anarrangementofcontactpointsmustsatisfycertainkinematicconditionsinordertobeabasisforagoodfixture.Inparticular,itmustprovideformclosure,deterministiclocation,clampingstability,detachabilityandloadability[Asada&

By].

Thealgorithmusesadiscretizationoftheworkpartboundary,similartothemeshesusedinFEA.However,unlikeFEA,ourattentionisonthemeshnodes,ratherthanonthemeshelements.Discretizationwaschosenforthefollowingreasons:

First,wecanhandleworkpartswitharbitrarygeometry,aslongasthepart’sboundaryisacollectionofsmoothsurfaceswhichweknowhowtomesh.ThisrequirementissatisfiedbyallsurfacesusedinmodernCADsystems.Second,discretizationisnecessaryinordertoavoidanexpensivecomputationofgeodesiccurves.Third,discretizationshouldnotsignificantlyaffecttheresults,aslongasthenumberofdiscretecandidatelocationsontheboundaryismuchlargerthanthenumberofsurfaces.Inourimplementationthediscretizedboundaryconsistsofseveralhundredpointsonly.Experimentalevidenceindicatesthatthisissufficientforrealisticworkparts.

Weintroduceapotentialfieldontheworkpartboundarydefinedbyzonesofattractionandrepulsion,whichwecallP-zones.Thecontactsaremodeledaschargedparticlesthatmoveontheboundarydrivenbythispotentialfield.Thecontactsarealsosubjecttomutualrepulsionbasedonthedistancebetweeneachtwocontactsinthewrenchvectorspace.Thealgorithmexecutesaseriesofsimulationepochs.Eachepochstartswitharandomconfiguration,proceedsthroughacertainnumberofstepstowardlowerpotentialenergyandendswithatestforkinematicconditions（formclosure）.Thealgorithmterminateswhenanepochproducessatisfactoryconfiguration.

Tospreadthecontactpointsontheboundarywesimulaterepulsionbetweeneachpairofthem.Theintensityofrepulsionbetweentwocontactpointsdependsonthedistancebetweentheircorrespondingwrenchesinthewrenchvectorspace.Oursimulationproceedsinalimitednumberofstepsoruntilequilibriumisreached.Theresultingplacementshouldhaveagoodchanceofleadingtoagoodfixture.Sucharandomizedmethodassumesthatthesetofn-tuplesofcontactpoints（forngreaterthanthree）thatsatisfythekinematicrequirementshasmeasuregreaterthanzeroandisrelativelylarge.Thatis,thesolutionspaceislarge.Althoughwehavenotbeenabletoprovethishypothesismathematically,ourexperimentshaveconfirmedit.Moreover,themeasureincreaseswiththenumberofcontactpoints,e.g.itiseasiertofindaformclosurearrangementwitheightpointsthanwithseven.

Thenotionofrepulsionisessentialinourmethodasitallowsotherconsiderationstobeaccommodatedeasily.Wecanputadditionalrepulsionspotsontheworkpartboundarytorepresentforbiddenregions.Wecanalsointroducecentersofattraction.Thesecorrespondtoareasthatwererecommendedbytheanalysisphaseasdesirableforplacingcontactpoints,e.g.datumsurfaces.Thus,weproposeapotentialfieldforuniformlyrepresentingheterogeneousfixturinginformation.Regionsofrepulsioncorrespondtoareaswithpositivepotential.Negativepotentialisassociatedwithattraction.Zeropotentialcorrespondstoneutralareas.Theinitialrandomlyselectedcontactpointsareregardedasparticlesthatarebeingattractedorrepelledbyapotentialfieldthatincludesapairwiserepulsion.Thegoalofthesystemofcontactpointsistominimizeitstotalpotentialenergy.

2TheInput

TheinputtoouralgorithmconsistsofCADmodelsoftheworkpartboundaryandasetofsolidP-zones.EachP-zonedefinesapotential-fieldinfluencingregionwithnon-zerocharge.

3DiscretizingtheWorkpartBoundary

Thefirststepinourmethodistodiscretizetheboundaryoftheworkpart,thuscreatingthecandidatecontactpointlocationswhichwecallnodes.Discretizationisdonebyinvokingastandardfaceterembeddedinthegeometricmodelerweuse.Thediscretizationisstoredinanorientedgraphdatastructure.Eachnodeofthegraphcorrespondstoanodeonthemesh.Theedgesofthegraphcorrespondtoedgesofthemeshconnectingneighboringnodes.Ateachnodethescrewrepresentingthepointcontactiscomputedandstored.Ascrewisaconciseandconvenientrepresentationofthesurfacenormalandthelocationofthenode.Itisusedinallkinematictestsbasedonscrewtheory.

4ComputingthePo