Submitted to IEEE Transactions on Robotics and Automation

BasedonthePrincipalStiffnessParameters

QiaoLinandJoelW.Burdick

DivisionofEngineeringandAppliedScience

CaliforniaInstituteofTechnologyPasadena,California91125ElonRimon

DepartmentofMechanicalEngineeringTechnion,IsraelInstituteofTechnology

Haifa32000,Israel

Submittedto

IEEETransactionsonRoboticsandAutomation

Abstract

Thispaperpresentsasystematicapproachtoquantifyingtheeffectivenessofcompliantgraspsandfixturesofanobject.Theapproachisphysicallymo-tivatedandappliestothegraspingof2Dand3Dobjectsbyanynumberoffingers.Wecharacterizetheframe-invariantfeaturesofagrasporfixturestiffnessmatrix.Thenwedefineframe-invariantcharacteristicstiffnesspa-rameters,andprovidephysicalandgeometricinterpretationfortheseparam-eters.Usingaphysicallymeaningfulschemetomakethestiffnessparameterscomparable,wedefineaframe-invariantqualitymeasurecalledthestiffnessqualitymeasure.Thequalitymeasureisthenappliedtotheoptimalgraspingofpolygonalobjectsbythreeandfourfingers.Wedeveloppracticalmethodsforcomputingthegloballyoptimalthreeandfour-fingerarrangement,andprovideexampleswhichshowthattheresultingoptimalgraspsareintuitivelyeffectivegrasps.

1Introduction

Complianceplaysadominantroleinpassivegraspssuchasworkpiecefix-turing,andcanalsobeusedtomodelthefingerforcesinactivegrasps.Thispaperpresentsaframe-invariantqualitymeasureforcompliantgraspsandfixtures,andconsidersitsapplicationtooptimalgraspingandfixturing.Toourknowledge,thequalitymeasurepresentedhereprovidesthefirstsystem-aticapproachtoquantifyingtheeffectivenessofcompliant(asopposedtorigid)graspsandfixtures.Theapproachisframe-invariantandappliestothegraspingorfixturingof2Dand3Dobjectsbyanynumberoffingers.Forthesakeofconvenience,thetermgraspingwillhereafteralsoapplytofixturing.

Compliantgraspshavereceivedmuchattentionintheroboticgraspingliterature.HanafusaandAsada[1]usedalinearspringmodeltofindsta-ble3-fingerplanargrasps.TheirworkwasextendedbyNguyen[2],whousedalinearspringmodeltocomputethestiffnessmatrixofmoregeneralgrasps.HowardandKumar[3]employedamoresophisticatedlinearspringcompliancemodeltostudygraspstability,andshowedhowthecontactge-ometryinfluencesthegraspstability.Instudyingcomplianceinthepresenceoffriction,CutkoskyandWright[4]notedthatstabilityisinfluencedbyinitialloadingaswellascontactgeometry.Whilethelinear-springcom-1

pliancemodelhasbeenwidelyusedbyroboticists,itisnotsupportedbyexperimentsorresultsfromelasticitytheory.RimonandBurdick[5]usedoverlapfunctionstomodelnonlinearcomplianceeffects.Lin,BurdickandRimon[6]usedtheseoverlapfunctionstocomputeandanalyzethegraspstiffnessmatrixforvariouscontactmodels,includingtheexperimentallyandtheoreticallyjustifiedHertzmodel.Whiletheoverlapmodelisusedheretocomputethestiffnessmatrix,ourqualitymeasureisvalidforanycompliancemodel.

Nearlyallpriorworkonquantifyinggraspeffectivenesshasfocusedontherigid-bodymechanicsofthegrasp,whileignoringcomplianceeffects[7].Letthewrench(i.e.forceandtorque)duetoaunitforceappliedbyacontactingfingerbetermedageneratingwrench.LiandSastry[8]suggestedaqualitymeasurebasedonthesmallestsingularvalueofthegraspmatrix,whosecolumnsconsistofthegeneratingwrenches.Kirkpatrick,Mishra,andYap[9]definetheradiusofthemaximalballinscribedintheconvexhullofthegeneratingwrenchesasaqualitymeasure.ThisideaisalsofollowedbyFerrariandCanny[10].However,thesequalitycriteriadependonthechoiceofcoordinateframes—agraspwhichisoptimalunderonechoiceofreferenceframemayfailtobeoptimalunderanother.Severalauthorshavedevisedschemestoavoidthisproblem.MarkenscoffandPapadimitriou[11]minimizetheworst-casefingerforcesneededtobalanceanyexternalunitforceactingatagivenpointoftheobject.MirtichandCanny[12]firstcomputethegraspsthatbestcounteractpureforces,andthenselectamongthesegraspstheonewhichbestresistspuretorques.Someofthecharacteristicsofoptimalfingerarrangementsintheirworkarealsoexhibited,undercertaincircumstances,byouroptimalfingerarrangements.However,theirworkconsidersonlyrigidgrasps,whileweexplicitlyincludecomplianceinouranalysis.Furthermore,weuseacompletelydifferentapproachtodefiningthequalitymeasure—whiletheydefineaqualitymeasureinalexicographicalmannertoavoidcomparingforceswithtorques,wedirectlyaddressthecomparisonofrotationalandtranslationalstiffnesses(explainedbelow).Teichmann[13]suggestsasaqualitymeasurethelargestinscribedball(asdefinedinRef.[9])withrespecttoallchoicesofcoordinateframes.However,hisapproachappliesonlytorigidgrasps,andappearstolackadequatephysicalinterpretation.

Thepotentiallyimportantroleplayedbycomplianceinmanygraspingandfixturingoperationscallsforthedevelopmentofqualitymeasuresthattaketheseeffectsintoconsideration.Todeveloppracticallyusefulandwell-definedqualitymeasures,theeffectsofcompliancemustbeproperlyincor-2

porated,andthedependenceonthechoiceofcoordinateframesmustbeeliminated.Weaddresstheseissuesbyfirstidentifyingtheframe-invariantfeaturesofagrasp’sstiffnessmatrix,intermsofinvariantscalarscalledtheprincipaltranslationalandrotationalstiffnessparameters.TheseparameterswerefirstidentifiedbyPattersonandLipkin[14]usingscrewtheory.Thesameparameterswerediscoveredbyususingadifferentapproach.Wedis-cusstherelationbetweenthetwoapproaches,andprovideanovelgeometricalinterpretationoftheprincipalstiffnessparameters.Oncetheinvariantstiff-nessparametersareidentified,weturntotheissueofhowtoincorporatethetranslationalandrotationalstiffnessparametersinaphysicallymeaningfulwayintoasinglequalitymeasure.Thekeytoourapproachistheconversionoftherotationalstiffnessparametersintoequivalenttranslationalstiffnessesaccordingtoconsiderationsoftheobject’sdeflectionandequivalenceofelas-ticenergy.Basedonthismethod,wedefineastiffnessqualitymeasureintermsoftheprincipalstiffnessparameters.Todemonstratetheusefulnessofthestiffnessqualitymeasure,weconsidertheoptimalgraspingofpolyg-onalobjectsbythreeandfourfingers.Inbothcaseswedeveloppracticalmethodsforcomputingthegloballyoptimalfingerarrangement,andprovideexampleswhichshowthattheresultingoptimalgraspsareindeedintuitivelyeffectivegrasps.

Thepaperisorganizedasfollows.Thenextsectionbrieflyreviewstheconfiguration-spacebasedmodelingofcompliance.Thenweconsidertheframe-invariantfeaturesofagrasp’sstiffnessmatrix,whichleadtothedefini-tionoftheprincipalstiffnessparameters.InSection4wediscussaphysicallymotivatedmethodforcomparingthetranslationalandrotationalstiffnessparameters,andthendefinethestiffnessqualitymeasure.InSection5weconsiderthecomputationofoptimalthreeandfour-fingergraspsofpolyg-onalobjects,andprovideseveralexamplesofthesecomputations.Finally,intheconcludingsectionwesummarizethecontributionsofthisworkanddiscussseveralrelevantissues.

2ModelingComplianceinC-Space

Inthissectionweintroducerelevantterminologyandreviewaconfiguration-spacebasedapproachtomodelingcompliantgrasps.MoredetaileddiscussionofthisapproachcanbefoundinRef.[5].Thiscompliancemodelisusedfortheexamplespresentedinthispaper.However,thequalitymeasurecanbe

3

usedwithanycompliancemodelthatyieldsastiffnessmatrixforthegrasp.AgrasporfixturearrangementconsistsofanobjectBcontactedbykfingersA1,...,Ak.Wemakethefollowingassumptionsonthegraspscon-sideredinthispaper.First,thecontactsarefrictionless,whichallowsforconservativegraspanalysissinceagraspthatisstableintheabsenceoffric-tionwillremainstablewhenfrictionisincluded1Second,thebodieshaveasmoothboundarynearthecontactpoints,asisthecasewithmanyrealisticgrasps.Third,thebodiesareassumedtobequasi-rigid.Thatis,deforma-tionsduetocomplianceeffectsarelocalizedtothevicinityofthecontacts,sothattheoverallmotionofBrelativetoAicanbedescribedusingrigid-bodykinematics.Elasticitytheory[15]suggeststhatthisisanexcellentassump-tionprovidedthebodiesdonotinvolveslenderstructures.Finally,sinceweareinterestedinthemotionofBrelativetoAi,thefingersareassumedtobestationary.

Sincethefingersarestationary,wecanfocusonB’sconfigurationspace.Chooseafixedworldreferenceframe,FW,andanobjectreferenceframe,FB,fixedtoB.AconfigurationofBisspecifiedbytheposition,d∈3,andorientation,R∈SO(3),ofFBrelativetoFW.Theconfigurationspace(orc-space),denotedC,isthesetofallconfigurationsofB.ToparametrizeC,

󰀄),wherethe“hat”weparametrizetheorientationmatricesbyR(θ)=exp(θ

󰀄suchthatθ󰀄x=θ×xoperatormapsθ∈3totheskew-symmetricmatrixθ

foranyx∈3.Thus,B’sc-spaceisa6-dimensionalsmoothmanifoldandcanbegivenhybridcoordinatesq=(d,θ)∈3×3,whichmapto(d,R(θ)).ThetangentspacetoCataconfigurationq,denotedTqC,isthesetofalltangentvectors,orvelocitiesofB,atq.Inhybridcoordinates,tangentvectorscanbewrittenasvectorsq˙=(v,ω),wherev∈3isthevelocityoftheoriginofFB,andω∈3istheangularvelocityofFB.Thewrenchspaceatq,denotedTq∗C,isthesetofallwrenches(orcovectors)thatcanpossiblyactonBatq.Awrenchtakestheformw=(f,τ)inhybridcoordinates,wheref∈3istheforceactingonBandτ∈3isthetorque.InthecasewhereBisplanar,c-spacecanbeparameterizedwithq=(d,θ)∈2×,bychoosingthez-axesofFWandFBorthogonaltotheplane.Accordingly,thetangentandwrenchspacesare3-dimensionalwithhybridcoordinatesq˙=(v,ω)∈2×andw=(f,τ)∈2×.

Thehybridparametrizationofc-space(asanyotherparametrization)

dependsonthechoiceofcoordinateframes.Consideranalternativechoice

¯W,displacedfromFWby(dw,Rw),andanotherobjectofworldframe,F

¯B,displacedfromFBby(db,Rb).Aconfigurationwithcoordinatesframe,F

¯WandF¯B.Correspondingly,qwillhavenewcoordinatesq¯withrespecttoF

thetangentandcotangentvectorstransformasfollows[16]:

˙=T−1q,q¯˙

¯=TTw,w

(1)

wherethetransformationmatrix2forthe3Dand2Dcasesaregivenby󰀈󰀉󰀈󰀉

󰀁RwJR0dbRwR0dbRw

T6×6=andT3×3=,(2)

010Rw

01

respectively.HereJ=(−10),andR0istheorientationofFBrelativetoFWatq0.SincedwandRbdonotappearinT,atranslationofFWorarotationofFBdoesnotaffectthetransformation.

InRef.[5]RimonandBurdickproposedaschemeformodelingnonlin-earcontactcompliancebasedonoverlapfunctionsparametrizedinc-space.Thismodelallowsonetoignorethespecificdetailsofthecontactingsur-facedeformationswhenBandAiarequasi-rigid.Rather,continuouselasticdeformationsarerepresentedbyalumpedparameterthatcharacterizestherelativeapproachofthetwobodies.Abriefreviewofthismodelfollows.Intheabsenceofdeformation,thetwobodiesBandAicontactatasinglepoint.Whenpressedtogether,thecontactingsurfacesdeform.Insteadofcomputingthesesurfacedeformations,considerthebodiesasrigidvolumesthatareallowedtointepenetrate,oroverlap,whenpressedtogether.LetBbeataconfigurationq.ThentheoverlapbetweenBandAi,denotedδi(q),istheminimumamountoftranslationofBwhichwouldseparateitfromAi.Bydefinition,δi(q)=0whenBisdisjointfromAi.Forsufficientlysmallpositiveoverlap,thereisauniquesegment,calledtheoverlapsegment,whoseendpointslieontheboundaryofBandAi,suchthatthelengthofthesegmentequalsδianditsorientationgivesthedirectionoftheseparatingtranslation.ThenetcontactforceisassumedtoactonB’sendpointoftheoverlapsegment,inthedirectionoftheseparatingtranslation.Theforce’smagnitudeisassumedtodependontheoverlapintermsofadifferentiable,monotonicallyincreasingfunctionfi(δi).Thesimplestmodelassumesthatfiisalinearfunctionoftheoverlap:fi(δi)=kiδi,wherethecoefficientkiis

determinedbythematerialandsurfacepropertiesofBandAi.Whilethismodelislinearinδi,itistypicallynotlinearinq,sinceδi(q)isingeneralnonlinearinq.Moresophisticatedcontactmodels,includingthewell-knownHertzmodel,canbeformulatedbychoosingappropriatefunctionsfi(δi).SeeRef.[6]fordetails.

ConsidernowagraspofBataconfigurationq0.Thefingersformanequilibriumgraspif(intheabsenceofanyexternalwrench)thefingerforcesproduceazeronetwrenchonB.Whensubjectedtoanarbitraryexternaldisturbance,Bmaybedisplacedfromq0.ThegraspissaidtobestableifBreturnstoq0aftertheexternaldisturbanceisremoved.AmoreformaldiscussionofstabilitycanbefoundinRef.[5].

TheelasticpotentialenergyofthesystemconsistingoftheobjectBandfingersA1,...,Akis:

Π(q)=

k󰀂󰀑i=1

δi(q)

fi(δ)dδ.(3)

0

Itcanbeverifiedthatδi(q)isdifferentiablealmosteverywhere,andsmoothat

pointswhereitispositive.Hence,providedthatfiisdifferentiable,Π(q)isadifferentiablefunction.Intheabsenceofadisturbingwrench,anequilibriumgraspatq0ischaracterizedbythecondition:

∇Π(q0)=

k󰀑i=1

fi(δi(q0))∇δi(q0)=󰀹0,

(4)

wherethegradientvector∇Π(q0)representsthederivativeDΠ(q0).Inthis

equation,thegradientofδiatq0isgivenby[5],

󰀉󰀈

ni

,(5)∇δi(q0)=−

R0ri×niwhereriistheendpointoftheoverlapsegmentontheboundaryofB,expressedinFB;niistheunitnormaltotheboundaryofB,pointingintoB;andR0istheorientationofFBrelativetoFWatq0.Ingeneral,thewrenchgeneratedbyaforceFiactingonBatapointriisgivenbywi=(Fi,(R0ri)×Fi).InourcaseFi=fi(δi(q0))ni,anditfollowsfrom(5)that−fi(δi(q0))∇δi(q0)isthewrenchgeneratedbytheithfinger.Thus,condition(4)assertsthatatanequilibriumgraspthenetwrenchexertedbythefingersmustvanish.

6

WhenBisatanequilibriumgraspconfigurationq0,thegrasp’sstiffnessmatrixisdefinedastheHessian,K=D2Π(q0),oftheelasticpotentialenergyΠ(q)atq0.Since∇Π(q0)=0atanequilibriumgrasp,thebehaviorofΠinthevicinityofq0isdeterminedbyK.IfKispositivedefinite,thenq0isalocalminimumpointofΠ,andthegraspisstable[5].Whilethereexiststablegraspswhosestiffnessmatrixisonlypositivesemi-definite,suchgraspsarenotgenericandcannotadequatelyresistexternaldisturbances.Therefore,inthispaperweonlyconsiderstableequilibriumgraspswithpositivedefinitestiffnessmatrices,andrefertothemsimplyasstablegrasps.

Thestiffnessmatrixofastablegrasphasthefollowingwell-knowninter-pretation.WhentheobjectB,atanequilibriumconfigurationq0,issubjectedtoadisturbingwrenchw∈Tq∗0C,itundergoesadisplacement.Providedthatthedisplacementissufficientlysmall,itcanbeapproximatedbyatangentvectorq˙∈Tq0C.(Inthiscasewemayinterchangeablyusethetermstangentvectoranddisplacement.)Thestiffnessmatrixrelatesanexternalwrenchtotheresultingdisplacementaccordingtotheformulaw=Kq˙.Itsinverse,C=K−1,iscalledthecompliancematrixofthegrasp.Thecompliancematrixisalsopositivedefiniteandestablishestherelationshipq˙=Cw.Tocomputethestiffnessmatrix,wetakethederivativeof∇Π(q)givenin(3)and(4),andobtaintheformula:

k󰀑󰀁′󰀃K=fi(δi(q0))∇δi(q0)∇δi(q0)T+fi(δi(q0))D2δi(q0),

i=1

(6)

wherefi′=dfi/dδi.Inthisformula,whiletheoverlapgradients∇δi(q0)

aregivenby(5),generalformulasfortheoverlapsδi(q0)andtheHessiansD2δi(q0)arederivedinRef.[6].Weobservethatthetwosummandsineq.(6)generallydependontheinitialdeformationsδi(q0).ItisshowninRef.[6]thatwhilebothsummandsdependonthefingerpositionsandcontactnormaldirections,thesecondsummandadditionallydependsonthesurfacecurva-tureatthecontacts3.Wesaythatthefirstsummandaccountsforfirst-ordergeometricaleffects,whilethesecondsummandaccountsforsecond-order,orsurfacecurvature,effects.Ifthefirsttermaloneispositivedefinite,thegraspissaidtobestabletofirst-order.Otherwise,iftheentirematrixKispositivedefinite,thegraspisstabletosecond-order.Therelativecontributionsofthe

firstandsecond-ordereffectstograspstabilityandstiffnessareanalyzedinRef.[6].

Finally,wederivethechange-of-frameformulaforthestiffnessmatrixK.Thequadraticform1

whereq0istheequilibriumgraspconfigurationofB.Inwords,thesubspaceVconsistsofsmalldisplacementsofBwhichcausethefingerstoreactinsuchawaysastogenerateapurenettorqueonthebody.Usingthepartition

−1

ofK,weobtainV={(v,ω):v=−K11K12ω},fromwhichitfollowsthatVcanbeparametrizedintermsofω∈3as4

󰀉󰀈−1󰀃󰀁−KK1211

.(9)whereP=V=q˙=Pω:ω∈3

I

LetKVdenotetherestrictionofthestiffnessmatrixKtothesubspaceV.WenowderiveanexpressionforKV.ThestiffnessmatrixKisalinearoperatorfromTq0CtoTq∗0C,andq˙TKq˙isconsequentlyasymmetricbilinearoperatoronTq0C.SincethevectorsinVareparametrizedbyω,wehavefrom(9)thatωTKVω=ωTPTKPωforarbitraryω.ThusKVhastherepresentationKV=

−1T

PTKP=K22−K12K11K12.Inaddition,thepuretorquecorrespondingtoq˙∈Visgivenby(Kq˙)2=KVω.

¯WandF¯B,withoverbarsdenotingobjectsConsidernowtwonewframesF

associatedwiththeseframes.ThelinearoperatorKVhasthefollowinginvarianceproperty.

¯bethesubspacesparametrizedby(9)intheqProposition3.1.LetVandV

¯Vbetherestrictionoftherespectivestiffnessandq¯coordinates.LetKVandK

¯.ThenKVandK¯VobeytheorthogonaltransformationmatrixtoVandV

¯V=RTKVRw,whereRwistherotationmatrixfromFWtoF¯W.Hence,Kw

theeigenvaluesofKVareframe-invariant.

Proof.Usingthestiffnessmatrixtransformationrule(7)andformula(2)

T¯are:K¯11=RwforT,thecomponentsofthestiffnessmatrixKK11Rw,

TT󰀁¯12=RT(K12+K11R¯󰀁󰀁K0db)Rw,andK22=Rw(K22+K12R0db−R0dbK12−w

¯¯¯T¯−1¯󰀁󰀁R0dbK11R0db)Rw.SubstitutingtheseexpressionsintoKV=K22−K12K11K12

T¯V=RwKVRw.givesK

Dually,therealsoexistsasubspaceofwrenchesonwhichthecompliance

matrixChasaframe-invariantstructure.Thissubspaceisgivenby:

󰀁󰀃W=w∈Tq∗0C:ω=(Cw)2=0.

Usingthisparametrization,therestrictionofCtothesubspaceW,denoted

−1T

CW,takestheformCW=QTCQ=C11−C12C22C12.Moreover,foranywrenchw∈W,theresultingpure-translationisgivenbyv=(Cw)1=CWf.

¯WandF¯B,wecanshowthefollow-ByagainconsideringtwonewframesF

inginvariancepropertyofCWinawaysimilartotheproofofProposition3.1.¯bethesubspacesparametrizedby(10)inProposition3.2.LetWandW

¯Wbetherestrictionoftherespectivetheqandq¯coordinates.LetCWandC

¯.ThenCWandC¯WobeytheorthogonalcompliancematrixtoWandW

¯W=RTCWRw,whereRwistherotationmatrixfromFWtransformationCw

¯toFW.Hence,theeigenvaluesofCWareframe-invariant.

Propositions3.1and3.2leadtothefollowingobservations.Thebehav-iorofKonthetangentsubspaceVcharacterizestherotationalstiffnessofthegrasp.InresponsetoaninstantaneousdisplacementinV,thereactionwrenchisapuretorque.Inaddition,thereactiontorquevariesbyatmostapurerotationcorrespondingtodifferentchoicesofframes.Similarly,thebehaviorofConthewrenchsubspaceWcharacterizesthetranslationalcom-plianceofthegrasp.AwrenchinWgeneratesapuretranslationofB,whichisthesameuptorotationwithrespecttodifferentframes.Wealsonote

−1

thatbytheinversionformulaofapartitionedmatrix5,CW=K11.Basedontheseobservations,wedefinetheframe-invariantprincipalparametersofKasfollows.

Definition1.LetKbethegraspstiffnessmatrix,andC=K−1thecom-pliancematrix.LetKVandCWbetherestrictionofKandCtothesub-spacesVandW.Thentheprincipalrotationalstiffnessesofthegraspare

−1T

K12),theeigenvaluesµi(i=1,2,3)ofKV(whereKV=K22−K12K11

andtheprincipaltranslationalstiffnessesofthegrasparetheeigenvaluesσi

−1

(i=1,2,3)ofCW=K11.

ThesubspaceWthusconsistsofexternalwrencheswhoseactiononBcausesittomovewithpuretranslation.ThesubspaceWcanbeparametrizedintermsoff∈3as

󰀉󰀈

󰀁󰀃IW=w=Qf:f∈3.(10)whereQ=−1T

−C22C12

Forplanargraspstheprincipalstiffnessparametershavethefollowingphys-icalinterpretation.Itcanbeshownthateveryplanargrasphasauniquelocationofanobjectframeorigin,givenby

T−1

db=R0JK11K12,

(11)

¯Bisplacedatthislocation,K¯3×3takestheblock-diagonalsuchthatwhenF

¯=diag(RTK11Rw,µ).Thatis,forplanargraspsthetranslationalformKw

androtationaleffectsaredecoupledaboutthisspecialpoint,calledthecen-terofcompliance[2].Theprincipaltranslationalandrotationalstiffnessesofaplanargrasparephysicallythetranslationalandrotationalstiffnessesaboutthecenterofcompliance.For3Dgrasps,thereisgenerallynosuchcenterofcompliance,andthestiffnessmatrixingeneralcannotbemadeblock-diagonal.However,itisimportanttonotethattheprincipalstiffnessparametersarestillwell-definedinthe3Dcase.

3.2ScrewCoordinatesInterpretation

Whilesearchingfora3Danalogofthecenterofcompliance,PattersonandLipkin[14]werethefirsttorecognizetheexistenceoftheprincipalstiffnessparameters.Theyusedscrewcoordinates,andwenowshowthatourprinci-palparametersareequivalenttotheonesderivedbyPattersonandLipkin.Firstwebrieflyreviewthenotionofscrewcoordinates.Thescrewcoordinatesofatangentvectorq˙=(v,ω)consistofascrewaxisandtwoscalarscalledpitchandmagnitude.Ifω=0,theinstantaneousscrewaxisofq˙isthelineparalleltoωwhichpassesthroughthepointv×ω.Thepitchofq˙is(v·ω)/󰀚ω󰀚2,anditsmagnitudeis󰀚ω󰀚.Ifω=0,thetangentvectorhasinfinitepitch,itsmagnitudeis󰀚v󰀚,anditsaxisisthelineparalleltovwhichpassesthroughtheorigin.Similarly,thescrewcoordinatesofawrenchw=(f,τ)alsoconsistofanaxis,pitch,andmagnitude.Iff=0,thescrewaxisofthewrenchisthelineparalleltofwhichpassesthroughthepointf×τ.Thepitchofwis(f·τ)/󰀚f󰀚2,anditsmagnitudeis󰀚f󰀚.Whenf=0,thewrenchhasinfinitepitch,itsmagnitudeis󰀚τ󰀚,anditsaxisisthelineparalleltoτwhichpassesthroughtheorigin.Considernowatangentvectorq˙i=Pωi∈V,whereωiisauniteigenvec-torofKVassociatedwiththeeigenvalueµi.Correspondingly,wehaveapuretorquegivenbyτ=(Kq˙i)2=µiωi.Itfollowsfromthisformulathatthetan-gentvectorq˙iinducesapure-torquewrenchofmagnitudeµiaboutthescrew

11

axisofq˙i.Ontheotherhand,forwi=Qfi∈Wwherefiisauniteigenvec-−1−1torofCWassociatedwiththeeigenvalueσi,wehavev=(Cwi)1=σifi.Hence,thewrenchwigeneratesapure-translationdisplacementofmagnitude−1σialongthescrewaxisofwi.Wecannowinterprettheprincipalstiffnessparametersintermsofscrewcoordinates.EverystiffnessmatrixKhassixframe-invariantscrewaxes.AdisplacementofBalongthefirstthreeaxesresultsinapuretorquewhichactsonBalongthesameaxis,ofmagnitudewhichisdeterminedbytherotationalstiffnessµi(i=1,2,3).AwrenchappliedtoBalongtheotherthreeaxesresultsinapuretranslationofBalongthesameaxis,andthemagnitudeofthetranslationisdeterminedbythetranslationalstiffnessσi(i=1,2,3).

3.3GeometricInterpretation

Wenowinterprettheprincipalstiffnessparametersintermsofthegeometryoftwolevel-sets.Thefirstisalevel-setinthetangentspace,definedbyS={q˙∈Tq0C:Φ(q˙)=1},whereΦ(q˙)=1wTCw.Theselevelsetsconsistoftangentvectorsorwrenchesthatinduceunitelasticenergy,andgeometricallyrepresentafive-dimensionalellipsoidalsurfaceinthesix-dimensionaltangentorwrenchspace.Theshapeoftheseellipsoidalsurfacesvariesasdifferentcoordinateframesareused.However,thesesurfacespossessframe-invariantfeatureswhichcorrespondtotheprincipalstiffnessparameters.

Firstconsiderthethelevel-setS.Foreachfixedω,thesubsetofSwiththisparticularvalueofωisdenotedSω.EachsubsetSωisalevel-setofthefunctionΦω(v)󰀁Φ(v,ω),inwhichωisafixedparameterandonlyvisavariable.RewritingΦ(v,ω)asafunctionofvonly,gives:

2

Φω(v)=

1

2

ωTKVω.

Henceforeachfixedω,thelevel-setSω={v:Φω(v)=1}isatwo-dimensionalellipsoidalsurfacewithprincipalsemi-axesoflengths((2−ωTKVω)/σi)1/2(i=1,2,3).SincethequadraticformωTKVωisframe-invariant,theselengthsareframe-invariant.Inparticular,whenω=0,theselengthsaresim-󰀆plygivenby

withthepure-translationsubspacegivenbyω=0.Thisfeaturecanbeeasilyvisualizedin2Dgraspsaswillbedescribedshortly.

Thelevel-setSpossessesanotherframe-invariantgeometricalfeature.Considertheprojection,denotedSv=0,ofthesetSontothepure-rotationsubspacegivenbyv=0.ItcanbeverifiedthattheboundaryofSv=0(calledthesilhouetteofSalongthedirectionofprojection)istheprojectionofthepointsonSatwhichthevectornormaltoShaszerov-components.ThelattersetisdenotedSn.SinceSisalevel-setofthefunctionΦ(v,ω),Snisdeterminedbythecondition(∇Φ(q˙))1=0.Thisconditionimpliesthat

−1

Sn={(v,ω)∈S:v=−K11K12ω}={(v,ω)∈S:1

ωTKVω󰀂1}.

2

Theprojectionsetisathree-dimensionalellipsoidwithprincipalsemi-axes󰀆oflength

ττf2hτf2f1(a)

(b)

Figure2:TheelasticenergyellipsoidTinTq∗0C.(a)Tintersectstheτ-axisatthesamepointsτ=±(2µ)1/2.(b)Tisinscribedinthesameellipticcylinder.

¯WandFWandFB,whiletheslantedellipsoidcorrespondstotheframesF

¯B.TheframesFWandFB,aswellasF¯WandF¯B,arecoincident.AsF

canbeobservedfromthefigure,thelengthsoftheprincipalsemi-axesofeachhorizontalcrosssectionofSareframeinvariant.Similarly,theprojec-tion󰀆ofSontotheω-axisisboundedbytwopoints,whoseω-coordinatesare±

2/µ.

Theframe-invariantfeaturesofthelevelsetT,whichisafive-dimensionalellipsoidalsurfaceinthesix-dimensionalwrenchspace,canbeanalogouslyidentifiedandaresummarizedasfollows.EachsubsetofTwithafixedvalueoff,denotedTf,isatwo-dimensionalellipsoidalsurfacewhoseprin-−1

cipalsemi-axesareequalto(µi(2−fTK11f))1/2(i=1,2,3)andareframe-invariant.Inparticular,whenf=0theprincipalsemi-axesofTfaregiven√by

−1fTK11f󰀂1},

2

whichisathree-dimensionalellipsoidwhoseprincipalsemi-axeshaveframe-√invariantlengthsof

inFigure2,wheretheuprightandslantedellipsoidsagaincorrespondto

¯WandF¯B,respectively.WecanseethatthetheframesFWandFW,andF

ellipsoidTintersectstheτ-axis,which√ischosentobetheverticalaxis,attwopointswhoseτ-coordinatesare±

−1T

fKf󰀂1.Hence,withrespecttoarbitrarilychosencoordinateframes,112

Tisalwaysinscribedintheverticalcylinderwhosebasesetisthisellipse,asshowninFigure2(b).

Wealsonotethatthevolumeoftheellipsoidsisframe-invariant,sincethevolumeisdeterminedbydet(K)whichisframe-invariant[8].However,wemakenouseofthevolumeinthestiffnessqualitymeasure.

4AFrame-InvariantQualityMeasure

Inthissectionwedefineaframe-invariantqualitymeasureforcompliantgraspsbasedontheprincipalstiffnessparameters.Firstwemustfindawaytomeaningfullycomparethetranslationalandrotationalstiffnessesofagrasp.Ourapproachisbasedonthenotionofobjectdeflectionandtheelasticenergyassociatedwiththisdeflection.Letq˙=α(v,ω)beaninfinitesimaldisplacementofB,whereαisascalar,󰀚ω󰀚=1ifω=0,and󰀚v󰀚=1ifω=0.WedefinethedeflectionofBduetothedisplacementq˙asthemaximaldisplacementofanypointinB.SinceBhasboundeddimension,suchamaximaldisplacementalwaysexistsandisindependentofframechoice.Ifω=0,thedeflectionofBissimply|α|.Ifω=0,letρmax(q˙)bethegreatestdistancefromtheinstantaneousscrewaxisassociatedwithq˙toB’sboundarypoints.ThenB’sdeflectionis|α|(ρmax(q˙)2+(v·ω)2)1/2,wherev·ωisthepitchofq˙.Inthecaseofplanargraspsthevectorωisperpendiculartov,andthedeflectionofBis|α|ρmax(q˙),whereρmax(q˙)isthemaximaldistancefromB’sinstantaneouscenterofrotationtoB’sboundarypoints.Wenowconverttherotationalstiffnessestoequivalenttranslationalstiffnessesusingthenotionofobjectdeflection.

Wefirstconsiderplanargrasps,wherethereisonlyasingleprincipalrotationalstiffnessparameter,denotedbyµ.Tocompareµwiththetransla-tionalstiffnessparametersσ1,σ2,wedefineaparameterwhichhastheunitsoftranslationalstiffnessandwhoseequivalencewiththeprincipalrotationalstiffnessµisdeterminedasfollows.AsdiscussedinSection3,µisassociated

15

withrotationsofBaboutthegrasp’scenterofcompliance.CorrespondingtoarotationofBwithmagnitudeαaboutthecenterofcompliance,B’sdeflectionisgivenby|α|ρmax,andtheamountofelasticenergyinducedthedeflectionis1

µα2.Thus,fromtheequation1

obtainthefollowingexpressionfortheequivalentstiffnessµeq:

µ

µeq=

2

2

µα2,we

µiα2,andthedeflectionof

21/2

Bisgivenbyα(ρmax2,whereρmaxi=ρmax(q˙i).Nowimaginei+(vi·ωi))

thesituationwheretheobject,whileattachedtoalinearspring,undergoesapuretranslationbytheamountofthisdeflectioninthedirectionofthespring.Wedefinethestiffnesscoefficientofthelinearspring,denotedµeqi,asequivalenttotheprincipalrotationalstiffnessµi,iftheelasticenergyofthespringduetothetranslationequals1

󰀎󰀆µαµα2,weobtainthefollowingformulafor2eqi2iµeqi:

µi

µeqi=

2

Definition2.Thestiffnessqualitymeasurefor2Dand3Dcompliantgraspsis:

󰀌

min{σ1,σ2,µeq}(2Dcase)Q=

min{σmin,µeqmin}(3Dcase)whereσmin=min{σ1,σ2,σ3}andµeqmin=min{µeq1,µeq2,µeq3}.

AsdiscussedinSection3.1,theprincipalstiffnessparameterscharacterizethestiffnessofagivengrasp,andthequalitymeasureQcharacterizestheworst-casestiffnessofthegrasp.Theworst-casestiffnessisdeterminedbyatrade-offbetweentheworst-casetranslationalandrotationalstiffnesses.Theworst-casetranslationalstiffnessischaracterizedbythesmallestprin-cipaltranslationalstiffnessσmin,andtheworst-caserotationalstiffnessbythesmallestequivalentrotationalstiffnessµeqmin(orµeqinthe2Dcase).Incharacterizingtheworst-caserotationalstiffnessbyµeqmin(orµeq),theprincipalrotationalstiffnessparametersaremeaningfullycomparedwiththetranslationalstiffnessparametersbyconsideringequivalenceofelasticenergybasedontheobject’sdeflection.WenotethatQhasthefollowingproperties.First,Qisvalidforgraspsof2Dand3Dobjectsbyanynumberoffingers.Second,thegraspscanbemodeledbyanycompliancemodel,sinceQde-pendsonlyonthestiffnessmatrixofthegrasp.Third,Qisinvariantwithrespecttochangeofworldandobjectreferenceframes.Last,theoptimalgraspofanobjectistheonewhichmaximizesQ,sincethisgrasphasthehighestworst-casestiffness.

5OptimalGraspingofPolygons

Toillustrateourmethodologyanditspossibleutility,weapplythestiffnessqualitymeasuretotheoptimalgraspingofapolygonalobjectBbythreeandfourfingers.Tocomputethestiffnessqualitymeasure,weemploytheoverlapcontactmodelforthecomputationofthegraspstiffnessmatrix.Weusethesimplestmodelwheretheithcontactforceisalinearfunctionoftheoverlap:fi(δi)=kiδi,wherethecoefficientkiisdeterminedbythematerialproperties.Tofurthersimplifythecomputation,weassumediscfingersofradiusr,andallowthefingerstotouchtheobjectonlyalongitsedges.(However,thestiffnessqualitymeasureisvalidforarbitraryfingersatanycontactlocation.)Inthecasesdiscussedbelowwefirstcharacterizethe

17

k-fingerstableequilibriumgrasps,thendiscusstheoptimizationofQoverthesegrasps.

5.1OptimalThree-FingerGrasps

Firstwecharacterize3-fingerstableequilibriumgraspsbythefollowingtwoproperties.Thecontactnormalsmustpositivelyspantheorigin,andthelinescollinearwiththecontactnormalsmustintersectatacommonpoint.Wediscard3-fingerequilibriumgraspswherethethreefingerstouchonlytwoedgesofB,sincewithoutfrictionthesegraspsareonlyneutrallystablewithrespecttotranslationalongtheedges.Thusweconsidertripletsofedges,andfocusonlyonthosetriplets,whicharecalledadmissibleedge-tripletsandsatisfythetwoconditionsforanequilibriumgrasp.Theadmissibleedge-tripletswhichgivestableequilibriumgraspshaveapositivedefinitestiffnessmatrix,andaformulaforthestiffnessmatrixisgiveninthefollowinglemma.Inthelemma,nidenotestheunitnormaltoanedgeofBattheithcontact,pointingintoB.Further,thecircumscribingcircleofatriangleisthecirclewhichpassesthroughthetriangle’svertices(Figure3).

Lemma5.1.LetthreediscfingersofradiusrholdapolygonalobjectBonanedge-tripletinafrictionlessequilibriumgrasp.ChoosetheframesFWandFBtobecoincident,withtheoriginattheconcurrencypointofthelinesofthecontactnormals.Thenthegraspstiffnessmatrixisgivenby

3󰀑

K=diag(kininiT,µ)whereµ=fT(2aζ+r).

i=1

(14)

Inthe󰀍expressionforµ,fTisthetotalpreloadingfingerforce,givenby3

kiδi(q0);aisradiusofthetriangle’scircumscribingcircle;fT=i=1󰀏󰀍the33

andζ=(i=1sinαi)/(i=1sinαi)isdeterminedfromthetriangle’sthreeinteriorangles,denotedαi(i=1,2,3).

Thelemma,whoseproofappearsinAppendixA,assertsthatKisblock-diagonalwhenFB’soriginisattheconcurrencypointofthecontactnormals.Sincethispropertyuniquelycharacterizesthecenterofcompliance(eq.(11)),theconcurrencypointisatthegrasp’scenterThus,thetwo󰀍3ofcompliance.Teigenvaluesofthe2×2matrixK11=i=1kininiarethetranslationalstiffnessesσ1,σ2,andµistherotationalstiffnessofthegrasp.ForKtobepositivedefinitethethreeparametersmustbepositive.ThesubmatrixK11

18

edge of Bα3edge of BA2circumscribing circleof radius aA3n3n2α1n1A1Sedge of Bα2Figure3:Threefingersonanedgetriplet.

ispositivedefiniteandconstantonagivenadmissibleedge-triplet.Henceσ1andσ2arepositiveconstantsonagivenedge-triplet.Intheparameterµ=fT(2aζ+r),aandrarepositiveconstants,whileζisapositivecon-stantincompressivegraspswherethefingerspushtowardstheconcurrencypoint6.Assumingtheusualcaseofacompressivegrasp,µispositivewhenfTisstrictlypositive.TheconditionfT>0impliesthatthegraspmustbepreloadedforstability.WethereforeassumethatfThasaspecifiedpositivevalueforallpossiblefingerplacements.Thisisareasonableassumption,sinceinpracticeoneoftenwishestocomparedifferentgraspscorrespondingtoacommonpreloadingleveldeterminedbythetaskspecificationsandma-terialstrengthrequirements.Thusµisalsoapositiveconstantonagivenedge-triplet,andallpreloadedequilibriumgraspsonanedge-tripletaresta-ble.

WenowshowthatQ=min{σ1,σ2,µeq}isdeterminedbytheequiva-lentrotationalstiffnessµeq=µ/ρ2max,whereρmaxisthedistancefromtheconcurrencypointtothefarthestvertexofB.Firstweexcludedegenerateedge-tripletsinwhichthethreeedgesarealmostparalleltoeachother.Typi-caledge-tripletsarenon-degenerate,andinspectionofthematrixK11revealsthatitseigenvaluesσ1andσ2areofthesameorderofmagnitudeasthemate-rialconstantki.Wewritethisconditionasσi∼=ki.Accordingto(14),µeq=

󰀍3

SubstitutingforfTandtaking(fT(2aζ+r))/ρ2,wheref=Tmaxi=1kiδi(q0).󰀍3

thequotientµeq/σi∼=µeq/kigives:µeq/σi∼=(i=1δi(q0))(2aζ+r)/ρ2max.Theparameterζinthequotientsatisfiesζ󰀂1/4.Moreover,ρmax,a,andrareofthesameorderofmagnitudeasB’scharacteristicdimension.Hencethevalue󰀍3ofthequotientisdominatedbytheratio:µeq/σi∼=(i=1δi(q0))/ρmax≪1,sincethepenetrationδi(q0)isalwaysmuchsmallerthanB’scharacteristicdimension.Itfollowsthatmin{σ1,σ2,µeq}=µeq,andtherefore

Q=fT

2aζ+r

thecenterofthesmallestdiscwhichcontainsB,suchthatthedisc’scenterliesinS.Itcanbeshownthatthisisthepointwherethehalf-linewhichstartsatp0alongtheperpendicularbisectorofthelongestedgefirstintersectstheregionS.ThisscenarioisshowninFigure4(b).

optimalconcurrency pointp0S(a)p0optimalconcurrencypointS(b)Figure4:Optimal3-fingergraspsoftwotriangularobjects,inwhichp0lies(a)insideSand(b)outsideS.

Tocomputethegloballyoptimalgrasp,wehavetoevaluate(15)ontheadmissibleedge-tripletsofB.Inspectionof(15)revealsthefollowingchar-acteristicsofthegloballyoptimalgrasp.In(15),whilethetotalpreloadingfTistakentobethesameforalledgetriplets,thequantitiesaandζaredifferentfordifferentedgetriplets.Therefore,whethertheoptimalgrasponagivenedge-tripletistheglobaloptimumoveralledge-tripletsdependsonthedistanceρmax,aswellastheshape(characterizedbyζ)andthesize(characterizedbya)ofthetriangledeterminedbythegivenedgetriplet.Forthequalitymeasuretoassumealargevalue,ρmaxispreferredtobesmall,whileaandζarepreferredtobelarge.Itcanbeverifiedthattheshapeparameterζisboundedbyζ󰀂1/4,withequalityholdingforanequilateraltriangle.Thus,theedgesinthetripletarepreferredtobeorientedevenly.Intheidealcase,theedgesare60◦apartandformanequilateral.Itisimportanttonotethattheparametersρmax,a,andζcombinetodeterminethegraspquality;asingleparameteraloneisnotsufficientforthispurpose.Weillustratetheseobservationsintwoexamples.Intheexamples,thefin-gershavezeroradius,andthematerialconstantskiaretakentobeunitywithoutlossofgenerality.Further,thecenterofthesmallestdisccontainingBiscalledthegeometriccenterofB,andtheradiusofthedisciscalledtheradiusρ0ofB.

21

b15e72be8e1e615e5be4e2e3Figure5:3-fingergraspsofanoctagon.

Example5.2.Considertheoptimal3-fingergraspingoftheoctagonshowninFigure5.Thegeometriccenterandradiusoftheoctagonaregivenbythecenterandradiusoftheoctagon’scircumscribingcircle.Hence,ρ0=2b.Wefirstcomparetheoptimalgraspsontheedge-triplets(e1,e4,e7)and(e3,e5,e8).Thesetwoedgecombinationsdeterminetwocongruenttriangles,forwhichthecombinedeffectonQofshapeandsizeisgivenbyaζ=0.3836.Theoptimalconcurrencypointofthetriplet(e1,e4,e7)coincideswiththegeometriccenter.Thus,forthisgraspwehaveρmax=ρ0andQ=0.1918fT/b.Ontheotherhand,theoptimalconcurrencypointof√(e3,e5,e8)liesonthelineofsymmetryofB,atadistance(1−1/

ye3e2e4e1b/2geometric centerxb/2Figure6:3-fingergraspsofaquadrilateral.

centerislocatedat(0.5b,0.125b).Fortheadmissibleedge-triplets(e1,e2,e3)and(e1,e2,e4),theoptimalfingerlocationsareshowninthefigurebyreg-ularandsolidcircles,respectively.Theconcurrencypointsofthesegraspsbothcoincidewiththegeometriccenter.However,theedgetripletscorre-spondtotrianglesofdifferentshapeandsize.Consequently,thesegraspshavedifferentqualitymeasurevalues.Fortheoptimalgraspon(e1,e2,e3)wehaveQ=1.0359fT/b,whilefortheoptimalgraspon(e1,e2,e4)wehaveQ=1.2526fT/b.Theoptimalfingerarrangementon(e1,e2,e4)givesthegloballyoptimalgraspthisobject.

5.2OptimalFour-FingerGrasps

A4-fingergraspofapolygonalobjectinvolvesthreeorfouredges,andwehavetoconsiderall4-fingerplacementsontripletsandquadrupletsofedges.The4-fingergraspsonaparticularedgecombinationareparametrizedasfollows.LetObetheoriginofFBandletEibetheedgecontainingtheithcontact.ThentheithcontactisparametrizedbythesigneddistancesioftheithcontactfromthepointwhereEiintersectsitsperpendicularlinethroughO.(TheparametersiisthetorquegeneratedbyaunitforceniactingonBatthepointri,sincethewrenchisw=(ni,(Rri)×ni)=(ni,(Rri)·ti)=(ni,si),wheretiistheunittangenttoBattheithcontact.)Apoints=(s1,s2,s3,s4)specifiesaparticular4-fingergrasp,andthecollectionofall4-fingergraspsonagivenedgecombinationisarectangularparallelepipedPin4.

Tocharacterizethe4-fingerequilibriumgraspsinP,lethi=(ni,si)denotethewrenchgeneratedbytheunitforceni,andletthefunctionsdi(s)=det([hi+1hi+2hi+3])(mod4)becalledthedeterminantfunctionsas-sociatedwiththeedgecombination.Thedeterminantfunctionscharacterize

23

theequilibriumgraspsasfollows(seeAppendixAforaproof).

Lemma5.2.Anecessaryandsufficientconditionfora4-fingergraspwithacontactconfigurations∈Ptobeanequilibriumgraspisthatd1(s),−d2(s),d3(s)and−d4(s)areallnonzeroandhavethesamesign.

Wewishtoshowthatalmostallthe4-fingerequilibriumgraspsinPhaveapositive-definitestiffnessmatrixandarethereforestable.Accordingtoformula(6),thestiffnessmatrixconsistsoftwosummands,K=K1+K2,whereK1correspondstofirst-ordereffectsandK2tosecond-ordereffects.ItisshowninRef.[6]thatwhenapolygonisgraspedbyfourormorediscfingers,thecontributionsfromK2areoftheorderδ/LcomparedtoK1,whereδisthecharacteristicvalueoftheinitialdeformationsδi(q0)andLisacharacteristicobjectlength.Theratioδ/LisextremelysmallandwemaythereforeuseK1asanexcellentapproximationforK.Substituting∇δi(q0)=−hiin(6)givesthefollowingapproximateformulaforK.Lemma5.3.Leta4-fingerequilibriumgrasphavecontactparameterssifori=1,...,4.Then,withtheframesFWandFBchosentobecoincident,thestiffnessmatrixtakestheapproximateform:

K=

4󰀑i=1

wherekiisthematerialconstantandnitheinwardunitnormalattheith

contact.

󰀍k

TTT

whereIngeneral,i=1vivi=[v1···vk][v1···vk].HenceK=HH√

k4h4]3×4.WeseethatKispositive-definitewhenHhasH=[

fullrowrank,whichholdstrueatall4-fingerequilibriumgraspsexceptinthespecialcasewherethelinesofthefourcontactnormalscoincide.Thus,exceptforonespecialcase,allthe4-fingerequilibriumgraspsarestable.UsingLemma5.2,thecollectionofstableequilibriumgraspsistheunion7T=T1∪T2,where

󰀁󰀃

T1=P∩s∈4:d1(s),−d2(s),d3(s),−d4(s)<0,

󰀃󰀁

T2=P∩s∈4:d1(s),−d2(s),d3(s),−d4(s)>0.

󰀈󰀍4󰀉󰀍4

T

knnii=1kisini,󰀍󰀍i4=1iiTkihihT4i=2ksniiii=1kisii=1

(16)

Weobservethateachfunctiondiislinearins,henceeachTiisaboundedconvexpolytopein4.Thus,foragivenedgecombinationwemayseparatelysearchtheconvexpolytopesT1andT2fortheoptimalfingerarrangement.NextwederiveaformulaforQonaparticularedgecombination.Bydefinition,thetranslationalstiffnessparametersaretheeigenvaluesofthesubmatrixK11,whiletherotationalstiffnessparameterisgivenbyµ=K22−󰀍4−1TK12K11K12.Using(16),thesubmatrixK11=i=1kininT

iisconstanton

agivenedgecombination,andσ1,σ2arepositiveconstantsonagivenedgecombination.Asforµ,substitutionofthesubmatricesKijaccordingto(16)gives:

µ(s)=

whichcanbeshowntobeanon-negativequadraticfunctionofs.Notethatwhileµisaconstantforall3-fingergraspsonagivenedge-triplet,itisaquadraticfunctionofsin4-fingergraspsonagivenedgecombination.TocomputeQ,wealsoneedaformulafortheequivalentrotationalstiffnessµeq=µ/ρ2max,whereρmaxisthedistancefromthegrasp’scenterofcompliancetothefarthestvertexofB.Letpdenotethegrasp’scenterofcompliance.

−1

Thenaccordingto(11),p=JK11K12,whereweassumethattheframeFBisalignedwiththeforKijaccordingtoLemmaW.Substituting󰀍4frameFT󰀍4

−1

5.3gives:p(s)=J[i=1kinini]i=1kisini,whichislinearins.Thus,2ρ2max(p(s))=max{󰀚vi−p(s)󰀚}overtheverticesv1,...,vnofB.Sincep(s)islinearins,ρ2max(p(s))isthemaximumofnpositivedefinitequadraticfunctionsins.ThemaximumvalueofthequalitymeasureQisgivenonaparticularedgecombinationby

󰀊

µ(s)

Q=minσ1,σ2,max{

s∈T1∪T2

4󰀑i=1

kis2i−

4

󰀎󰀑i=144󰀑󰀑󰀐󰀎󰀐−1

kisinTkininTkisini,ii

i=1

i=1

min{σ1,σ2}isconstantonagivenedgecombination,andtheformulaQ=min{σ1,σ2,µeq}indicatesthatQ󰀂σminonagivenedgecombination.Henceifwefindinthecourseofmaximizingµeq(s)somes∗suchthatµeq(s∗)󰀃σmin,thiss∗isnecessarilytheoptimalfingerarrangementonthegivenedgecombination.Second,σministhesmallesteigenvalueofthe󰀍matrix󰀍441TK11=knn.HenceQ󰀂σ󰀂iiminii=1i=1ki,2

wheretr(·)isthetraceoperator.Toimprovethisbound,thecontactnor-malsniarepreferredtobeevenlyoriented.Inparticular,ifthematerialconstantsareuniformwithki=k,thenσmin󰀂2k,andequalityholdswhenthecontactnormalsare90◦apart,namely,theedgecombinationformsarectangle.Nextwediscusstheparametersthatinfluenceµeq(s).Sinceµeq(s)=µ(s)/ρ2max(p(s)),theparameterρmaxisdesiredtobesmallwhileµispreferredtobelarge.Butµisthegrasp’sstiffnessabout󰀍4rotationalthecenterofcompliance,andisgivenbyµ=i=1kis¯2¯iistheithi,wheres

contact’smomentaboutthecenterofcompliance.Thus,forµtoassumealargevalue,each|s¯i|isdesiredtobelarge.Thisindicatesthatthefingersshouldspreadapartasmuchaspossiblewithrespecttothecenterofcompli-ance.Tosummarize,foragrasptohavegoodstiffnessquality,itispreferredthattheedgesbeevenlyorientedtomakeσminlarge;thatthefingersspreadapartwithrespecttothecenterofcompliancetomakeµlarge;andthatthedistancefromthegrasp’scenterofcompliancetoB’farthestvertexbesmall,tomakeρmaxsmall.Theseparameterscombinetodeterminetheoptimalgrasp.

grasp rectangle2b2s22s12acenter ofcomplianceFigure7:4-fingergraspsofarectangularobject,withtheoptimalgraspshowninblackdots.

Example5.4.Inthefollowingexamplesweassumepointfingersanduni-formelasticityconstantsofki=k.Figure7showsarectangularobjectBofsize2a×2b.WhenBisgraspedbyfourfingers,eachfingermustcontactadifferentedgeofB.Thecontactnormalsare90◦apart,andσminachieves

26

itsupperbound:σmin=2kforallfingerarrangements.Nowconsidertheequivalentrotationalstiffness,µeq=µ/ρ2max.GivenanyequilibriumgraspofB,thelinesofthecontactnormalsformarectangle,whichwecallthegrasprectangletodistinguishitfromtherectangularobject.Itcanbeverifiedthatthegrasp’scenterofcomplianceislocatedatthecenterofthegrasprectangle[2].Moreover,therotationalstiffnessisgivenbyµ=2k(¯s2¯21+s2),wheres¯1ands¯2arethehalf-lengthandhalf-widthofthegrasprectangle.Clearly,whenthefingersareplacedattheendsoftheobject’sedgeswiths¯1=aands¯2=basshowninthefigure,µachievesitsmaximumvalue:µ=2k(a2+b2).Indeed,thisgraspobeystherulethatthefingersshouldspreadapartwithrespecttothecenterofcompliance.Also,thecenterofcomplianceforthisgraspcoincideswithB’scenterofsymmetry,andthedistance√fromthecenterofcompliancetoB’sfarthestvertexisminimized:ρmax=

ofabasefingerataninteriorpointofthebaseedge,thereexistsanalternativeplacementofahigherµeqvalue,suchthatbothbasefingersarelocatedatthebase’sendpoints.Thuswemayrestrictourattentiontofingerarrangementswherethebasefingersareattheendpointsofthebaseedge.

LetAbetheintersectionpointofthesidefingers’forcelines.Bysymme-try,weneedonlyconsiderfingerarrangementsinwhichthepointAliesintherighthalfplaneboundedbythelineofsymmetryℓs(Figure8(a)).LetxdenotethehorizontaldistancebetweenAandℓs.WiththestiffnessmatrixcomputedfromLemma5.3,itcanbeshownthatµ=2k(b2+x2/5),andthatthecenterofcomplianceislocatedonthesamehorizontallineasthepointA,atadistanceofx/5fromℓs(Figure8(a)).Nowconsiderafixedvalueofx,i.e.,thesidefingersmoveinawaysuchthatthepointA,alongwiththecenterofcompliance,isatafixeddistancetothelineℓs.Thenµeqismaximizedasthecenterofcompliancemovesontothebisectoroftheleftsideedge,sincewithµaconstant,thisminimizesρmaxforthegivenx.Thus,wecanfocusonaone-parameterfamilyofgrasps,wherethehorizontallinethroughAintersectstheleftedge’sbisectoratadistancex/5toℓs.Astheparameterxincreases,

22

µ(x)andρ2max(x)=(100b+20bx+4x)/75bothincreaseandcompetetodeterminethevariationofµeq(x)=µ(x)/ρ2max(x).Asimplecalculationshowsthatµeq(x)ismaximizedatx=0,withµeq(0)=1.5k.Itfollowsthattheoptimalgraspcorrespondstox=0,withQ=min{σmin,µeq}=1.5k.AsshowninFigure8(b),intheoptimalgraspthesidefingers’forcelinesintersectattheB’scenterofsymmetry.

ygeometric centerxbFigure9:Globaloptimalgraspofaquadrilateral.

Example5.6.Inthepreviousexamples,thesymmetryoftheobjectsal-lowedanalyticalanalysisoftheoptimalfingerarrangement.Forageneralpolygonalobjectitisnecessarytousethenumericalprocedureoutlinedin

28

AppendixB.Thisexampleconsiderstheoptimal4-fingergraspingofthequadrilateralusedinExample5.3forthe3-fingercase.Recallthattheverticesofthisquadrilateralhavecoordinates(0,0),(b,0),(0.7b,0.6b)and(0.15b,0.45b),andthatthegeometriccenterhascoordinates(0.5b,0.125b).Byconsideringallfeasibleedgecombinationswecanfindtheoptimalgraspassociatedwitheachcombination,andfurtherdeterminethegloballyopti-malgrasp,whichisshowninFigure9.Forthisgrasp,σmin=1.684kandµeq=1.882k,henceQ=1.684k.Inaddition,thecenterofcomplianceofthisgraspcoincideswiththegeometriccenteroftheobject.Thereforetheopti-malfingerarrangementmaximizesµeqbyminimizingρmax,andbyspreadingapartthetwofingersonthebaseedgetoallowµtoassumealargevalue.

6Conclusion

Wedescribedtheframe-invariantparametersofthestiffnessmatrixandusedtheseparameterstodefineastiffnessqualitymeasureforcompliantgraspsorfixtures.Thequalitymeasureisbasedontheprincipaltranslationalstiff-nessesσi(i=1,2,3)andtheprincipalrotationalstiffnessesµi(i=1,2,3)ofagrasp.Themeasurealsodependsonascalingfactor.Thisfactor,basedonequivalenceofelasticenergyandtheobject’sdeflection,convertstherotationalstiffnessesintoequivalentstiffnessesµeqi(i=1,2,3),whichcanbemeaningfullycomparedwiththetranslationalstiffnesses.There-sultingqualitymeasureisgivenbyQ=min{σ1,σ2,µeq}for2DgraspsandbyQ=min{σmin,µeqmin}for3Dgrasps,whereσmin=min{σ1,σ2,σ3}andµeqmin=min{µeq1,µeq2,µeq3}.Thequalitymeasurereflectstheworst-casestiffnessofagrasp,andingeneralthehigherthequalitymeasurethebetterthegrasp.

Thestiffnessqualitymeasurehasseveralimportantproperties.First,themeasureisindependentofthechoiceofobjectandworldframes.Second,themeasureisexplicitlydesignedforcompliantgrasps,andisthefirstsys-tematiceffortinquantifyingtheeffectivenessofcompliantgrasps.Moreover,thequalitymeasureisformulatedintermsofageneralclassofcompliancemodels,whichincludesthewellknownHertzmodelasaspecialcase.Third,thequalitymeasureisvalidforgraspsof2Dand3Dobjectsbyanynumberoffingers.Inparticular,itisknownthatcurvatureeffectscansignificantlyreducethenumberoffrictionlessfingersorfixturesrequiredtostablygraspanobject.(AnadaptationofRef.[18]yieldsthat3convexfingerssuffice

29

tostablygraspalmostany2Dobject,and4convexfingersseemtosufficetostablygraspalmostany3Dobject.)Thestiffnessqualitymeasureauto-maticallyincludesfirst-ordereffects(i.e.fingerpositionsandcontactnormaldirections)withsecond-ordereffects(i.e.surfacecurvatureatthecontacts)inasinglemeasure.Thequalitymeasureisthususefulforassessinginauniformwaytheeffectivenessofgraspswhichinvolvedifferentnumberoffingersanddifferenttypesofgeometricaleffects.

Wealsoconsideredthecomputationoftheoptimalgraspofapolygonalobjectbythreeandfourfingers.Ineachcaseweattemptedtocharacterizethequalitativepropertiesoftheoptimalgrasp.Forexample,wefoundthatQhasahighervalueonevenlyorientededgecombinations.Wealsofoundthatthefingersshouldspreadapartasmuchaspossiblewithrespecttothegrasp’scenterofcompliance,whileattemptingtobringthecenterofcomplianceasclosetotheobject’spointsaspossible.Thesepropertiesallcombinetodeterminetheoptimalgrasp,whichasillustratedintheexamplesisintuitivelyeffective.Thereareseveralissuesthatcallforfurtherresearch.Whilewedevelopedanefficientprocedureforgloballyoptimalgraspingusingthelinearcontactmodel,practicalglobaloptimizationwithnonlinearcontactmodels,suchastheHertzmodel,needstobeaddressed.Wehaveindicatedthatourqualitymeasurecanbeusedtoevaluatetheeffectivenessofanygiventhree-dimensionalgrasp.However,itremainsopentocomputeoptimalthree-dimensionalgraspswiththisqualitymeasure,basedoneitherlinearornonlinearcontactmodels.Finally,thenotionofobjectdeflectionisinvokedtocomparetranslationalandrotationalstiffnesses.However,thequalitymeasureisnotanindicatorthatdirectlyassessestheobject’sdeflection,whichmaybedesirableinapplicationssuchasworkpiecefixturing.Thisissueisbeingaddressedbyourongoingresearch.

Finallywementionpotentialapplicationsofthiswork.Thestiffnessqual-itymeasureisusefulforpassivegraspandfixtureplanning.Animportantapplicationofthistypeisworkpiecefixturing,whereapartisheldbyfix-tureelementsformachiningpurposes.Thefixtureelementshavetoprotecttheworkpiecefromdeflectingundertheloadofthemachiningforces,andthestiffnessqualitymeasurecanindicatethenumberandlocationoffixtureelementsthatbestsuitthegiventask.Thequalitymeasureisalsousefulforactivegraspplanning,wherefingerlinkageshavetostablygraspanob-ject.Asseveralresearchershavesuggested[1,2],wemayperformsuchtasksbysimulatingvirtualspringsatthecontacts.Thestiffnessqualitymeasureisusefulforselectingtheoptimalplacementandstiffnessofsuchsprings,

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basedontheobject’sgeometry.Themeasurealsoallowsaninclusionofthefingers’geometryintotheplanning,therebyprovidingatoolforselectingamongseveralpossiblefingergeometriesinapplicationswheresuchachoiceisavailable.

AProofofLemmas5.1and5.2

ProofofLemma5.1.Denotethefirstandsecondsummandsin(6)byK1andK2,respectively.Bythechoiceoforigintheoverlapgradientfor-mula󰀍(5)simplifiesto∇δi(q0)=(ni,0),fromwhichitfollowsthatK1=

T

diag(3i=1kinini,0).TocomputeK2,itcanbeshown[6]thatforapointfingercontactingastraightedge,theoverlapderiavativeformulaisgivenbyD2δi(q0)=diag(0,0,ρi+r),whereρiisthedistancefromtheconcur-rencypointtotheithcontactpoint(positiveifthefingerandthecon-currencypointlieontheopposite󰀍sidesoftheithedge).weobtain󰀍Thus,33K2=diag(0,0,µ),󰀍whereµ=fTi=1νi(ρi+r)=fT(i=1νiρi+r).Inthisformula,fT=3i=1fiisthetotalpreloadingforce,fiarethepreloadingfingerforces,andtheratiosνiaredefinedbyνi=fi/fT.Whennipositivelyspan2,itcan󰀍3beshownfrom(4)thatthesenormalsuniquelydetermineνibyνi=di/(j=1dj),wheredi=det([ni+1ni+2])(mod3).Usingelemen-tarygeometry,wecanexpressνiintermsofthetriangle’sinteriorangles,andexpressρiintermsofthetriangle’sedgelengthsandinteriorangles,aswellastheoftheconcurrencypoint.Thenwecanfurthershowthat󰀍location3

thesumi=1νiρiisactuallyindependentofthelocationoftheconcurrencypoint,andisgivenby2aζ.

ProofofLemma5.2.Thefingerarrangementcorrespondingtosisanequi-libriumgraspiffthewrenchesh1,h2,h3,h4positivelyspanthezerowrench.Firstsupposethatd1(s),−d2(s),d3(s)and−d4(s)arenonzeroandhavethesamesign.Toprovethatthewrencheshipositivelyspantheorigin,weshowthatthereexistν1,ν2,ν3>0suchthatthevectorν=(ν1,ν2,ν3)solvestheequation[h1h2h3]ν=−h4.Sinced4=det([h1h2h3])=0,wecanuseCramer’sruletoobtainν=−(d1d4,d3

normalsniandni+1intersect,andhencedet([nini+1])=0,forallimod4.Nowsupposethatdi(s)=0forsomei.Withoutlossofgenerality,wemaychoosethelocationoftheoriginsothatitcoincideswiththeintersectionofni+1andni+2.Thussi+1=si+2=0,andthedeterminantfunctionditakestheformdi(s)=si+3det([ni+1ni+2]).Butdi(s)=0byassumption,hencesi+3=0.Thusthenormalsni+1,ni+2andni+3intersectatacommonpoint.Tomaintainequilibrium,nimustpassthroughthisintersection,hencesi=󰀍0.Now,equilibriumalsoimpliesthatthereexistν1,...,ν4>0suchthat4i=1νihi=0,or[h1h2h3]ν=−ν4h4.UseofCramer’sruleagaingivesν=−ν4(d1d4,d3

f(s)

,(17)

whereµandfarepositivesemidefinitequadraticfunctionswithf(s)>0whenevers∈D,andthedomainDisaconvexpolyhedralsubsetof4.Weareinterestedinfindingtheglobalmaximumofµeq(s)overD.How-ever,µeqisanonconvex,stronglynonlinearfunction,anditsglobalmaximummayingeneralbeverydifficulttofind.Fortunately,theapproachtobedis-cussedinthesectionoffersaneffectivealgorithmtofindtheglobaloptimum.Letusfirstdefineafunctionφ:×D→by

φ(t,s)=µ(s)−tf(s).

Foragivent∈,thereexistss∈Dsuchthatt=µeq(s)ifandonlyifφ(t,s)=0.Thus,themaximizationproblem(17)isequivalenttomaximizingt∈suchthat(t,s)isazeroofφforsomes∈D.Toaddressthisequivalentproblem,wefurtherdefineascalarfunctionψ:→by

ψ(t)=maxφ(t,s).

s∈P

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Thisfunctionhassomeinterestingproperties.For∆t>0,sincef(s)isstrictlypositive,wehave

ψ(t+∆t)=max(µ(s)−tf(s)−∆f(s))s∈D

s∈D

Inotherwords,ψisstrictlymonotonicdecreasing.Inaddition,sinceψ(0)=maxs∈Dµ(s)>0,andψ(t)→−∞ast→∞,thereexistsauniquet∗>0suchthatψ(t∗)=0.Thatis,ψhasaunique,positivezero.Thefollowingpropositionindicatesthatthemaximizationofµeq(s)overDisequivalenttothecomputationoftheuniquezeroofthescalarfunctionψ,whoseevaluationisperformedbymaximizingφ(t,s),aquadraticfunctionofs.

PropositionB.1.Maximizingµeq(s)overDisequivalenttofindingtheuniquezeroofψinthefollowingsense.Apositivenumbert∗>0satisfiesψ(t∗)=0ifandonlyift∗=maxs∈Dµeq(s).Inthiscase,acontactconfigu-rations∗∈Dmaximizesφ(t∗,s),regardedasafunctionofs,overDifandonlyifitmaximizesµeq(s)overD.

Proof.Ift∗=µeq(s∗)=maxs∈Dµeq(s),thenφ(t∗,s∗)=0.Foranys∈D,wehave

φ(t∗,s)=µ(s)−t∗f(s)=f(s)(µeq(s)−t∗)󰀂0=φ(t∗,s∗).

Henceψ(t∗)=φ(t∗,s∗)=maxs∈Dφ(t∗,s)=0.Conversely,supposethatψ(t∗)=0forsomet∗>0.Givenany∆t>0,wehave

φ(t∗+∆t,s)󰀂ψ(t∗+∆t)<ψ(t∗)=0,

wherethestrictmonotonicityofψhasbeenused.Thisindicatesthatthereexistnos∈Dsuchthatφ(t∗+∆t,s)=0forany∆t>0.Hencet∗=maxs∈Dµeq(s).Moreover,lets∗∈Dbesuchthatψ(t∗)=φ(t∗,s∗)=maxs∈Dφ(t∗,s).Thenbydefinitionofφ,wehavet∗=µ(s∗)/f(s∗)=µeq(s∗).Hence,s∗maximizesµeq.

Itfollowsfromthispropositionthattheoptimizationproblem(17),whichmaximizesµeqoverD,isequivalenttosolvingfortheuniquerootofthescalarequationψ(t)=0.Notethattoevaluateψatsomet,weneedtomaximizeaquadraticfunctionofs,whichisingeneralindefinite,i.e.,,thematrixas-sociatedwiththequadraticterminthisfunctionhaspositiveandnegative

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eigenvalues.Indefinitequadraticprogramming(IQP)isunfortunatelyNP-hard,andtheknownalgorithmsareexponentialinthenumberofvariables.Forexample,Refs.[19,20]describeanindefinitequadraticminimizational-+1)p

gorithmwhichtakesO(l(m(mǫ))steps,wheremisthenumberofvariablesandpthenumberofnegativeeigenvaluesofthequadraticobjectivefunction.Inthisbound,listhetimeittakestosolveaconvexquadraticoptimiza-tionproblemofthesamesize,whichisO(nlogǫ)wherenisthenumberoflinearconstraintsinthepolytopeD.Sinceinourcasem=4andp󰀂4,thenumberofstepsislinearinthenumberofconstraints,withasomewhatlargeconstantdeterminedbythedimensionm=4.Thus,givenmbeingsmall,ourapproachprovidesapracticalprocedurewhichguaranteestofindtheglobaloptimumatareasonablecomputationalcostdespitethestronglynonlinearandnonconvexnatureofthestiffnessqualitymeasure.

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