Analysis and design of parallel mechanisms with flexure joints

Abstract — Flexure joints are frequently used in precision motion stages and micro-robotic mechanisms due to their monolithic construction. The joint compliance, however, can affect the static and dynamic performance of the overall mechanism. In this pape

AnalysisandDesignofParallelMechanismswith

FlexureJoints

ByoungHunKang,JohnT.Wen

CenterforAutomationTechnologiesRensselaerPolytechnicInstitute

Troy,NY12180.

Emails:{kangb,wen}@cat.rpi.edu

Abstract—Flexurejointsarefrequentlyusedinprecisionmo-tionstagesandmicro-roboticmechanismsduetotheirmonolithicconstruction.Thejointcompliance,however,canaffectthestaticanddynamicperformanceoftheoverallmechanism.Inthispaper,weconsidertheanalysisanddesignofgeneralplatformtypeparallelmechanismscontaining exurejoints.Weconsiderstaticperformancemeasuressuchastaskspacestiffnessandmanipulability,whilesubjecttoconstraintssuchasjointstress,mechanismsize,workspacevolume,anddynamiccharacteristics.Basedontheseperformancemeasuresandconstraints,weadoptthemulti-objectiveoptimizationapproach.We rstobtaintheParetofrontier,whichcanthenbeusedtoselectthedesireddesignparametersbasedonsecondarycriteriasuchasperformancesensitivity.Tosimplifypresentation,weconsideronlylumpedap-proximationof exurejointsinthepseudo-rigid-bodyapproach.Aplanarmechanismisincludedtoillustratetheanalysisanddesigntechniques.Toolspresentedinthispapercanalsobeappliedtoabroaderclassofcompliantmechanisms,includingrobotswithinherentjoint exibilityaswellascompliantrobotsforcontacttasks.

I.INTRODUCTION

Flexurejointshavebeenusedinprecisioninstrumentssuchaswatchesandclocksforhundredsofyears,andcontinuetobeusedtodayinapplicationssuchasopticalsystems,micro-robots,andcleanroomequipment.Flexurejointsoffersigni cantadvantagesoverconventionaljoints[1],[2]intermsofbothmanufacturingandoperationalcharacteristics.Mechanicallyassembledjointsinevitablyreduceaccuracyduetomanufacturingtolerances.Flexurejointsaretypicallyman-ufacturedmonolithicallyandthereforeavoidassemblyerrors.Themonolithicconstructionalsoimpliesarelativelyeasymanufacturingprocessandpotentiallyverycompactdesign.Intermsofoperation, exurejointshavelittlefrictionlossesanddonotrequirelubrication.Theygeneratesmoothandcontinuousdisplacementwithoutbacklash.Withasuitablechoiceofmaterial, exurejointsexhibitapredictableandrepeatablerelationshipbetweenforceanddisplacement.Theseattributeshaveendeared exuremechanismstomeso-andmicro-scaleprecisionmotionapplications,fromopticalstagestomicro-electro-mechanical-systems(MEMS).

Thoroughtreatmentsonthecharacterizationanddesignof exurejointsandmechanismsmaybefoundin[1],[3].Flexuremechanismdesignisusuallyaddressedeitherfromakinematicsynthesispointofviewwiththeoverallmechanism

NicholasG.Dagalakis,JasonJ.Gorman

IntelligentSystemsDivision

NationalInstituteofStandardsandTechnology

Gaithersburg,MD20899,USA

Emails:{dagalaki,gorman}@cme.nist.gov

complianceasasecondarycriterion,orfromthecompliancepointofview[1]withtheemphasisonsynthesizingdesiredcompliancecharacteristicsusing,forexample,topologicalop-timization[4],[5]or niteelementanalysis[6],[7].Thegen-eralproblemofcompliancesynthesishasbeenaddressedusingsimplesprings[8]withspeci csolutionsproposedfortor-sionalandlinespringsin[9]–[11].However,suchanapproachhasseveraldrawbacks:thedesigncriteriononlyinvolvesthedesiredcompliance;constraintsarenottakenintoaccount;andtheoverallmechanismispassivewithoutconsiderationofactuators.Thespeci cproblemofsynthesizingadesiredgraspcompliancebychoosingappropriate ngercomplianceisusedin[12].Independentofjointcompliance,optimizationbaseddesignmethodshavealsobeendevelopedforparallelmechanisms[13],[14],butthejointcomplianceisnottakenintoaccount.Thegoalofthispaperistopresentanalysisanddesigntoolsforparallelmechanismscontaining exurejointsbasedonthepseudo-rigid-bodymodel.Ourapproachistobalancethemotionandcomplianceconsiderationthroughamulti-objectiveoptimization.

Awellestablishedcriterionforassessingthebehaviorofaserialorparallelmanipulatoristhemanipulabilityellipsoidwhichisthetaskspaceimageofaballintheactivejointvelocityspace.Thisconceptwas rstproposedforserialmanipulators[15]andlaterextendedtoparallelrobots[16],[17].Weposethedesignproblemasamulti-objectiveop-timizationproblemwiththeperformancemetricsbasedonmanipulabilityandstiffnesssubjecttoconstraints(suchasthemaximumjointstress,workspace,mechanismsize,etc.)andboundsonthedesignparameters.TheParetofrontier[18]isthencalculatedandthe naldesigndeterminedbasedonsecondaryconsiderationssuchasdynamiccharacteristicsandperformancesensitivity.Asanexample,weincludea1-DstagedesignedbytheNationalInstituteofStandardsandTechnology(NIST)toillustratethemodelinganddesignapproachdescribedinthepaper.

II.DIFFERENTIALKINEMATICS

Consideraparallelmechanismwithactivejointsdenotedbythevectorqaandpassivejointsdenotedbyqp.Thedifferential

Abstract — Flexure joints are frequently used in precision motion stages and micro-robotic mechanisms due to their monolithic construction. The joint compliance, however, can affect the static and dynamic performance of the overall mechanism. In this pape

kinematicsmaybedescribedas

xTJTaJTp qa

=0JCaJCp qp

qJT

J:=

JC

(1)

Forparallelmechanismswithconventionalpassivejoints,JCpistypicallysquare(samenumberofpassivejointsasconstraints)sothattherearenoundesirableinternalconstraintforces.ItisalsoessentialtoensurethatJCpisinvertiblesothattherewouldnotbeundesiredmotion(thisisthekinematicstabilitycondition).IfJCpisatallmatrix,themechanismisoverconstrainedanditcannotmoveunlesssomeoftheconstraintsareredundant.Ifthisisthecaseforaworkingmechanism,therigidbodykinematicdescriptionisnotadequate,andeithermorelumpedjointsneedtobeaddedoradistributeddescriptionshouldbeused.IfJCpisafatmatrix,themechanismisunderconstrained.Forconventionalparallelmechanisms,thisisnotdesirable,sincetherecouldbeuncontrolledmotionresultingfromdisturbances.However,weshallseethatfor exuremechanisms,thismaybeacceptableprovidedthatthestiffnessinthedirectionofunwantedmotionissuf cientlylarge.

Wenowconsiderafullyconstrainedmechanism(whenactivejointsarelocked)orunderconstrainedmechanism,i.e.,JCpissquareorfat,andfullrank.IfJCpisfat, qpcannotbeuniquelysolvedsinceanyvectorinthenullspaceofJCpmaybeaddedtothesolution.Inthiscase,weassumethatthesolution qpminimizesthestrainenergyinthepassivejoints,i.e., qpisfoundfrom

1T

Kqp qp,subjectto0=JCa qa+JCp qp,(2)min qp

qp2

wherewehaveassumedlinearspringcharacteristicswithspringconstantKqp.Wetreat qpastheactualjointdis-placementsinceweassumethatthejointdisplacementfromtheequilibriumissmall.Thesolutionof(2)maybereadilyfound:

qp= JCJ qa(3)pCawhere

1/2 1/2 JC:=Kq(JCpKq)ppp

wherefTistheexternallyappliedspatialforce,fCisthe

constraintspatialforce(toenforcethekinematicconstraint,thebottomportionof(1)),τaandτparethetorquevectorsappliedattheactiveandpassivejoints,respectively.Whenthepassivejointsarefree(e.g.,pin,spherical,etc.),τp=0.However,for exurejoints,τpisrelatedto qp.

Byfarthemostcommoncon gurationofparallelmech-anismisaplatformsupportedbymultiplelegs.ForanM-legplatformmechanism,thedifferentialkinematicsmaybewrittenas

xT=JT1 qa1+JC1 qp1=...=JTM qaM+JCM qpM.

(7)

Wecanrewritethisrelationas

Ja10

qa1 .. ... .0

qJaaMM

qp Jp11

... ... 0

qpM

JpM

J

x1I

= ... = ... xT.(8)

xMI

A

SinceAisoffullcolumnrank,wecanimmediatelytransformthistotheform(1):

xT=A J q

q0=AJ

(9)

(10)

isafullrowrankwhereA isthepseudo-inverseofAandA

matrixwhosenullspacecoincideswiththecolumnspaceofA.

III.PERFORMANCEMEASURES

A.Manipulability

ManipulabilityischaracterizedbyJTcompasin(5).Depend-ingonthedesignobjective,differentmetricsmaybeimposed.Ifitisdesirabletohaveanisotropicmechanism(thetaskframeisequallyeasytomoveinalldirections,foractivejointmotionconstrainedinaunitball),thenthemetrictominimizemaybe

smax(JTcomp)

µM(JTcomp)=( 1)2(11)

smin(JTcomp)wheresminandsmaxdenotetheminimumandmaximumsingularvalues,respectively(equivalently,thelengthsoftheprincipalmajorandminoraxesofthemanipulabilityellipsoid).Itmayalsobedesirabletomaximizetheoverallworkspace.Inthiscase,wecanchoosetomaximizethevolumeofthemanipulabilityellipsoidbyminimizingthemetric

1

µM(JTcomp)= sj(JTcomp) .(12)

j

(4)

and denotestheMoore-Penrosepseudo-inverse.IfJCpis

1

squareinvertible,thenJC=JC.pp

Therelationshipbetweenactivejointdisplacementandtaskdisplacementisthen:

xT=(JTa JTpJCJ) qa.pCa

:=JTcomp

(5)

Byapplyingtheprincipleofvirtualwork,weobtainthe

dualrelationship:

T T JTaJC

a

τafT = ,(6)

fCτpTT

JTpJCp

τ

2

Abstract — Flexure joints are frequently used in precision motion stages and micro-robotic mechanisms due to their monolithic construction. The joint compliance, however, can affect the static and dynamic performance of the overall mechanism. In this pape

Ifitisdesiredtoincreasemanipulabilityindirectionsgivenbytheunitvectors{ui}anddecreasemanipulabilityindirectionsgivenby{vi},thenapossiblemetrictominimizeisthefollowingweightedsum:

1

TT

µM(JTcomp)=αiuiJTcompJTcompui

i

Assumethejointtorqueisrelatedtothejointdisplacement

throughalinearspringrelationship:

τa qaKqa0

=,(21)τp qp0Kqp

Kq

+

B.MaximumJointStress

i

TT

βiviJTcompJTcompvi.(13)

whereKqadenotestheactivejointstiffnessandKqpdenotes

thepassivejointstiffness.Ifproportional-derivativetypeoffeedbackisusedfortheactivejoints,then

(p)(a)

Kqa=Kq+Kqaa

(p)

(a)

(22)

Themaximumstressesinthe exurejointsareapproxi-matelyproportionaltothemaximumde ectionsofthesejoints.

Forexample,foracircularnotchhingejointwithradiusR,hingewidtht,andYoung’sModulusE,themaximumstress,σmax,isrelatedtotheangularde ection,θmax,by[3]

3πR

σmax.(14)θmax=

4EtForacantileveredjointwithlengthLandwidtht,the

relationshipisapproximately

θmax=

0.148L

σmax.Et

(15)

whereKqadenotesthepassiveportionandKqadenotestheproportionalfeedbackgain.

Substituting(21)into(20),weget

JCKq q

T

T =JCJTKT xTT =JCJTKTJT q.

TT

(23)

Fromthekinematicconstraint(bottomportionof(1)),we

know qmaybeexpressedas

q=JCφ

forsomevectorφ.Substitutinginto(23),weget

T JCKqJCφ=JCJTKTJTJCφ.

T

T

(24)

Ifthemaximumjointstressisgiven(e.g.,fromtheyieldstress

ofthematerial),itcanbeconvertedtoanequivalentmaximum

(max)

byusingtheaboveformulas.Thejointdisplacement, qp

maxjointstressconstraintcanthenbestatedasamaximumde ectionconstraint:

(max)

| qp|≤ qp,

(25)

Sincethisholdsforanyφ,weobtaintheexpressionforthe

taskspacestiffness

T JCKqJC=JCJTKTJTJC.

T

T

(26)

(16)

where|·|and≤aretreatedinthecomponentwisesense.C.TaskSpaceStiffness

Thetaskspacestiffnessisde nedfromtheforcebalance

betweentheappliedexternalspatialforcefTandthecorre-spondingtaskframedisplacement xT:

fT=KT xT.

Rewritetheforcebalanceequation(6)as

TT

τ=JTfT+JCfC.

Ifthemechanismiskinematicallystable,i.e.,JCpissquare

invertible,then

I J,(27) 1C= JCJCapand(26)becomes

1TT

Kqa+JCJ TKqpJCJ=JKTJTcomp,CTaaCpcompp

(28)

(17)

(18)

Byassumption,JCpisfullrowrank,therefore,JCisfullrow

rank.LetJCbethefullcolumnrankmatrixwhosecolumn

spacecoincideswiththenullspaceofJC.Then

TJCτ=JCJTfT.

T

T

(19)

Substitutingin(17)andusingthedifferentialkinematics(top

portionof(1)),weget

JCτ

T

T =JCJTKT xTT =JCJTKTJT q.

TT

whichisthesameexpressionasobtainedin[19].

Foracircularnotchhingetypeof exurejoint(seeFig.1(a)),thejointstiffnessmodeledasapurerotationisgivenby[3]

2Ept5

,(29)K≈

9πR

whereEistheYoung’sModulusofthehingematerial,pisthedepthofthejoint,tthethicknessofthethinnestportionofthejoint,andRistheradiusofthecircle.Afull3D(planartranslationandrotation)jointstiffnessmodelisalsogivenin[3].

Foracantileveredjoint(seeFig.1(b)),thejointstiffnessmaybeapproximatelymodeledas

K≈2γKθ

EI,L

3

(30)

(20)

3

whereEistheYoung’sModulus,I=ptisthemomentofinertiaabouttheaxisperpendiculartothejoint,Listhe

Abstract — Flexure joints are frequently used in precision motion stages and micro-robotic mechanisms due to their monolithic construction. The joint compliance, however, can affect the static and dynamic performance of the overall mechanism. In this pape

lengthofthejointandγandKθareexperimentallydeterminedconstants:

γ=0.8517,Kθ=2.6762

.

versionhasalsobeendesignedandbuilt.Suchstagesarecurrentlybeingconsideredforsatelliteopticalcommunication

[23].

(a)CircularNotchHingeJoint

Fig.1.

(b)CantileveredJoint

FlexureJointModeling

Similartothemanipulabilitymatrix,differentmetricsmaybeuseddependingontheapplication.Forexample,in[19],thegoalistoensurethestiffnessmatrixisdecoupled.Inthatcase,themetricmaybechosentobe

µK(KT)= KT diag(KT) .

(31)

Fig.2.SchematicofNIST1-D

Mechanism

Ifmaximumstiffnessisdesired,themetricmayinvolvemaximizingthevolumeofKTorminimizingitsreciprocal:

1 µK(KT)=sj(KT) .(32)

j

Fig.3.FlexureJointinNIST1-DMechanism

Ifitisdesiredtoincreasestiffnessindirectionsgivenbythe

unitvectors{ui}anddecreasestiffnessindirectionsgivenby{vi},apossiblemetrictominimizeisthefollowingweightedsum:

1 T

µK(JTcomp)=αiuTKu+βvKvTiiTi.ii

i

i

(33)

IV.EXAMPLE:NIST1-DSTAGE

A.MechanismArchitecture

A1-degree-of-freedom(DOF)macro-scaleprecisionmo-tionstageusing exurejointswasdesignedandfabricatedbyNIST[20],[21].Severalmeso-scale(aboutthesizeofacreditcard)modelshavealsobeenbuilt[22].AschematicofthemechanismisshowninFig.2.Apiezoelectricactuatortransmitsthey-axismotionthroughjoints1and4tothetwolowerarms.Thesearmspivotaboutjoints2and5andmovetheoutputstagethroughjoints3and6.Tosupporttheoutputstage(andtoreducetheangularcrosstalk,i.e.,undesirableangularmotion),twoadditionalarmsalsosupporttheplatformthroughjoints7-10.Thegoalofthedesignistoachievedesiredmanipulability(puretranslationiny)andstiffness(largestiffnessintheangularandxdirections).Thejointsareconstructedascircularnotchjoints(seeFig.3from[20]).However,dependingontheexactjointmodelused,thedesignresultwouldbedifferent.Thisisdiscussedinthenextsection.Byreplicatingthedesignalongtheorthogonalaxis,a2-DOF

4

B.KinematicModels

Themechanismconsistsof6kinematicchainsconstrainedattheplatform.Thismeansthatthereare15totalconstraints(5loopsinvolving(x,y,θ)).

IfallthejointsinFig.2arechosentobeidealized1Drotationaljoints,thenthereare10passiveDOF’sandthemechanismisoverconstrained(JCpis15×10).Indeed,inthiscase,themechanismcannotmovefromtheequilibriumpositionshown.Thismeansthatthe1Djointapproximationisnotadequatetodescribethismechanism.

Thereareanumberofpossiblemodi cationsthatwecouldintroduce.Atthispoint,wedonotknowwhichmodelmatchesclosestwiththephysicalmechanism;weintendtoconductcalibrationtestsatNISTinthenearfuture.Toillustratethedesignprocedure,wehaveconsideredthefollowingtwojointmodels:

A:Replacejoints1,3,4,6,8,10bytworotationaljointsconnectedbyashortrigidsegment(cantileveredjointmodel).Themotivationofthisassumptionistoallowrotationaswellassheartypeoftranslationatthesejoints.Joints2,5,7,9serveaspivotsandareretainedaspurerotationaljoints(circularnotchjointmodel).Inthiscase,

thereare16passivejointsand15constraints,i.e.,JC isrank2(includingoneactivejoint).SinceJTJC(in

(26))isrank2,onlythex-ycomponentsofKTcanbedetermined.

Abstract — Flexure joints are frequently used in precision motion stages and micro-robotic mechanisms due to their monolithic construction. The joint compliance, however, can affect the static and dynamic performance of the overall mechanism. In this pape

B:Replacealljointsbydoublejoints(cantileveredjointmodel).Nowthereare20passivejointsand15con-straints,andKTmaybefullydetermined.

The exuremechanismismadefrom6061-T6Aluminumalloy.TheYoung’sModulusisE=70GPa.Thefourquadrantsofthestagearenominallyallsymmetric.Weusethefollowingdimensions:

a=1.5mm,b=15mm,L=0.425mm,d=8.625mm.Forcircularnotchjoints,thepassivejointstiffnessiscalculatedusing(29)withr=L=0.2125mm,t=0.1mm,p=6.35mm:

Kp=0.2140N-m/rad.Forcantileveredjoints,thepassivejointstiffnessiscalculatedusing(30)withL=0.25mm,t=0.1mm,andp=6.35mm:

Kp=0.6701N-m/rad.Theactuatorstiffnessisobtainedfrom[22]:

Ka=2.1569×107N/m.

InbothcasesAandB,thetaskspaceJacobianis(thetaskcoordinateisarrangedas(θ,x,y)):

,0JTcomp=

9.9998showingonlyy-directionmotionofthetaskframe.

Thex-yportionofKTinCaseAisalmostdiagonal:

1.4100×109 6.7705×10 5

KT(x,y)=N/m

6.7705×10 57.2841×105witheigenvalues(1.4100×109,7.2841×105).ThefullKT

inCaseBis

6.9911×10253.1518×10162.4655×1015

KT= 3.1518×10168.3672×1081.1115×106 ,

2.4655×10151.1115×1066.4209×105whichshowsveryhighstiffnessintherotationaldirection.Theeigenvaluesare(6.9911×1025,8.2251×108,5.5514×105)witheigenvectorsalmostperfectlyalignedwiththeunitvectors.TheorderofstiffnessvaluesinthexandydirectionsforCasesAandBareveryclose,demonstratingconsistencybetweenthetwoapproaches.C.DesignOptimization

Toillustratethedesignoptimizationprocedure,wechoosetomaximizethemanipulability(alongy)andtherelativestiffnessbetweenthexandydirections:

1

(34)Manipulability:µ1=JTcomp10

Stiffness:µ2=.(35)

xyNotethatthescalingconstantsareaddedtonormalizebetweenthetwomeasures.

5

Forthemaximumjointstress,weusetheyieldstressforAL6061-T6:σmax=220MPa.Themaximumstressconstraintisimposedwhentheactivejointisatitsmaximumextension qamax=9.1µm.Eqs.(14)and(15)areusedtocalculatethejointstressforcircularnotchjointsandcantileveredjoints,respectively.Inthiscase,themaximumallowedjointde ectionforthecircularnotchjointis0.6731 andforthecantileveredjoint0.1003 .Thedesignparametersarechosentobe(a,b,c)withthebounds:

0.5mm≤a≤4.5mm,5mm≤b≤45mm,0.08mm≤L≤0.75mm.

TheParetofrontiersforcasesAandBareshowninFigures4–5.Whenthetwoperformanceindicesarecombinedwithequalweights:

µ=0.5µ1+0.5µ2,theoptimalsolutionsareshowninTableIwiththeoptimalperformanceindicesCaseA:CaseB:

Manipulability=15.9717,

Kx

=680.1686KyKx

Manipulability=13.9626,=915.0178.

Ky

Comparedwiththenominalmanipulabilityof9.9998and

x

relativestiffnessKy=458.8294,forbothcases,theoptimalsolutionsinbothcasesimprovebothµ1andµ2,withCaseBshowingthegreaterimprovement.However,theabsolutestiffnessinthexdirectionisreducedbyafactorof10.Ifthisisnotacceptable,Kxmaybeaddedasanadditionalperformancemeasureoradesignconstraint.Themaximumpassivejointde ectionsrangefrom0.1880 to0.6731 forCaseAandCaseB,wellwithinthemaximumde ectionconstraintforcircularnotchjoint.

valueTABLEI

OPTIMALDESIGNVALUESFORCASEA

AND

CASEB

V.CONCLUSION

Inthispaper,wehavepresentedanalysisanddesigntoolsforparallelmechanismswithlumped exurejoints.Thekeydifferencebetween exuremechanismandparallelmecha-nismswithconventionaljointsisthatkinematicstabilityisnolongeradesignconsideration.Instead,taskspacestiffnessneedstobecarefullydesignedtoavoidundesiredmotioninthepresenceofexternalloads.Weposethedesignproblemasamulti-objectiveoptimizationwithmanipulabilityandstiffnessasperformancemeasuresandmaximumjointstressanddesign

Abstract — Flexure joints are frequently used in precision motion stages and micro-robotic mechanisms due to their monolithic construction. The joint compliance, however, can affect the static and dynamic performance of the overall mechanism. In this pape

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Fig.4.

ParetoFrontierforCase

A

Fig.5.ParetoFrontierforCaseB

parameterboundsasconstraints.A1-DstagedesignedbyNISTisusedasanexampletoillustratethemodelinganddesignapproach.Itisalsoshownthatbymakingdifferentjointassumptions,e.g.,asinglecircularnotchjointvs.abendingbeamjointmodeledasadouble exure,theoptimalsolutioncouldbequitedifferent.WearecurrentlyplanningtoconductexperimentaltrialsatNISTtodeterminethevalidityofthemodelsandalsofabricatenewstagesbasedontheoptimizationresults.

ACKNOWLEDGMENT

ThisresearchisconductedbasedinpartonthesupportbytheNationalScienceFoundationunderGrantIIS-9820709.ThisworkisalsosupportedinpartbytheCenterforAutoma-tionTechnologiesunderablockgrantfromtheNewYorkStateOf ceofScience,Technology,andAcademicResearch.

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