A Robot Joint With Variable Stiffness Using

机器人,刚度,柔性

IEEETRANSACTIONSONROBOTICS,VOL.27,NO.2,APRIL2011229

ARobotJointWithVariableStiffnessUsing

LeafSprings

JunhoChoi,Member,IEEE,SeonghunHong,StudentMember,IEEE,WoosubLee,SungchulKang,Member,IEEE,

andMunsangKim,Member,IEEE

Abstract—Interactionwithhumansisinevitableforservicerobots,http://pliantcomponentisananswertothesafetyissueatthecostofperformancedegradation.Inordertoreducetheperformancedegradation,manipulatorsequippedwithvariablestiffnesshavebeenstudiedbymanyresearchers.Thispaperpresentsavariablestiffnessjoint(VSJ)designedforarobotmanipulator,http://plianceisgeneratedbyleafspringsandtwoactuatorsareusedtocontrolthepositionandstiff-nessofthejointusingfour-barlinkages.Twoactuatorsinparallelcon gurationareconnectedtothespring.Changingtheeffectivelengthofthespringresultsinachangeinstiffness.Thepositionofthejointiscontrolledviatwoactuatorsrotatingatthesamespeedinthesamedirection.AnonlinearcontrollerisusedtocontroltheVSJ,andasingularperturbationmodelisadoptedtoprovethestabilityoftheclosed-loopsystem.Experimentsareconductedtoshowthatthepositionandstiffnessarecontrolledindependentofeachother,andhavinglessstiffnessatthejointhelpsinmakinganunexpectedcollisionwithanobjectsafer.

IndexTerms—Actuators,servicerobots,variablestiffness.

I.INTRODUCTION

ERVICErobotsdesignedfordailychoresathomerequiredifferentdesignapproachesthanindustrialrobotstoen-surephysicalsafety.Fortraditionalindustrialrobots,safetyofahumanoperatorisensuredbysegregatingworkspacesofthehumanoperatorandtherobots,whereasitisdif culttosegre-gatetheworkspaceoftheservicerobotsandhumanssincethetaskrequirestherobotstosharetheworkspacewithhumans.Inordertoaddressthesafetyproblemofservicerobots,intro-ducingcompliancetothejointsoftherobotshasbeenstudied.Activecomplianceismimickingmechanicalcomplianceusingactuatorsandsensordata.In[1],arobotwithrigidjointswas

S

ManuscriptreceivedFebruary23,2010;revisedAugust24,2010;acceptedDecember13,2010.DateofpublicationJanuary20,2011;dateofcurrentver-sionApril7,2011.ThispaperwasrecommendedforpublicationbyAssociateEditorE.GuglielmelliandEditorK.Lynchuponevaluationofthereviewers’comments.ThisworkwassupportedbytheIntelligentRoboticsDevelopmentProgram,whichisoneofthe21stCenturyFrontierR&DProgramsfundedbytheMinistryofKnowledgeEconomyofKorea.

J.Choi,W.Lee,S.Kang,andM.KimarewiththeCenterforCognitiveRoboticsResearch,KoreaInstituteofScienceandTechnology,136-791Seoul,Korea(e-mail:junhochoi@kist.re.kr;robot@kist.re.kr;kasch@kist.re.kr;munsang@kist.re.kr).

S.HongiswiththeCenterforCognitiveRoboticsResearch,KoreaInstituteofScienceandTechnology,136-791Seoul,Korea,andalsowithHanyangUni-versity,133-791Seoul,Korea(e-mail:weluxmea@kist.re.kr).

Colorversionsofoneormoreofthe guresinthispaperareavailableonlineathttp://.

DigitalObjectIdenti er10.1109/TRO.2010.2100450

controlledtoexhibitcompliantmotionusingforcefeedback.De ectionsinCartesiancoordinateandorientationoftheend-effectorweredeterminedbytheforceandtorquesensordataandcompliancegain.In[2],distributedmacro–miniactuationwasusedwithdifferentactuatorcharacteristics.Alargeactuatorwasusedforhigh-torqueandlow-frequencymovement,whilelow-torqueandhigh-frequencymovementwascoveredbyasmallactuator.Inordertoreduceinertiaofthemanipulator,thelargemotorwaslocatedatthebaseofthemanipulator,andthesmallmotorwasplacedatthejoint.Formorecompactdesign,apneumaticactuatorwasusedforlowfrequencybandwidthin[3].

Whenthesensorsfailedorsamplingfrequencywaslow,activecompliancebecomesunreliabletoensuresafetyofthehumans,whichwaspointedoutbyWangetal.[4],whereacomparisonstudywasconducted.Moreover,withoutanysen-sorfailureorwithhighsamplingfrequencies,mechanicalorelectricalbandwidthofthesystemmightpreventsecuringthesafetyofhumansfromunexpectedcollisions.Sinceactivecom-plianceislimitedinensuringthemanipulatortoremainsafe,passivecompliancehasbeenpaidmuchattentionduetoitsre-liabilitywhensafetyisaprimaryconcern.Forvariousdesignsofactuatorswithpassivecompliance,see[5]andreferencestherein.

Introducingcompliance,however,posesacontrolproblem,whichhasbeenstudiedbymanyresearchers.Bicchietal.in-vestigatedarobotjointwithnonlinearspringsandthecon-trolproblemassociatedwiththemechanicalcompliancein[6].Feedbacklinearizationwasusedtocontrolajointwithcon-stantcompliancein[7]andjointswithvariablestiffnessin[8]and[9].Tomeiusedaproportional–derivative(PD)controllerandprovedthestabilityofthejointwithacompliancein[10].Spong[11]presentedtwodifferentmethodsincontrollingajointwithelasticity,whichwereusingfeedback-linearizationmethodandformingthedynamicsasasingularlyperturbedsystem.Inordertorealizeanactuatorwithpassivecompliance,ase-rieselasticactuator(SEA)wasintroducedin[12].TheSEAhadadvantagesoflow-pass- lteringshocksfromandtotheactuatorandchangingtheforce-controlproblemtotheposition-controlproblem.However,complianceduetothespringcausesposi-tioninaccuracyandlowerbandwidth.Adjustablecompliancealleviatestheseproblems.Onewaytorealizeadjustablecom-plianceistochangethepretensionofthespringconnectedtothemotor.In[13],anadditionalmomentarmandamotorwereusedtogenerateandcontrolcompliance.In[14],anonlinearspringwasrealizedwithspiralpulleysandlinearsprings.Theactuatorwasdesignedforajointofarobottohelptherobotto

1552-3098/$26.00©2011IEEE

机器人,刚度,柔性

230walk.In[15],acamdiskandrollerspressedbylinearspringswereused.Bycontrollingthepressingforcefromthespring,thestiffnesswasvaried.Inthesedesigns,onecompliantcomponentandtwomotorswererequired.Thestiffnessandpositionofthejointwerecontrolledindependentlybyeachmotor.However,itwasnotpossibleforthejointtobecomecompletelystiff.

Anotherwayistochangetheeffectivelengthofthespring.Hollanderetal.[16]usedahelicalspringandcontrolledthenumberofeffectivecoiltochangestiffness.Sincechangingthenumberofeffectivecoilchangedtheeffectivelengthofthespringaswellasthestiffness,itwasnotpossibletocontrolthestiffnessandpositionofthejointindependently.In[17],http://inglead–screwmechanism,theeffectivelengthofthespringwaschanged,whichcausedthestiffnessofthejointchange.Inthesedesigns,itwaspossibleforthejointtoberigid.However,themotorandthedevicetochangethestiffnessbe-cameanadditionalloadtothemotortocontrolthepositionsincetheywereconnectedinseries.

Actuatorsinparallelcon gurationhaveadvantageoveractua-torsinseriessincetheloadofthejointissharedbytheactuators.Twomotorswithnonlinearspringsantagonisticallyconnectedresultinajointwithadjustablecompliance.Torealizeanon-linearspring,rolamitespringswereusedin[18].Theshapesofthespringsweredesignedsothatthestiffnessofthespringbecamenonlinear.Withnonlinearityofthespring,stiffnessofthejointwasvariable.In[19],http://edatimingbeltandlinearspringstorealizenonlinearspringsin[20].Sincethepowerofthemotorswastransmittedthroughthetimingbelt,thestrengthofthebeltlimitedthemaximumallowablepayload.Theexistenceoftheidlepulleymadethejointbulky.Toovercomesuchdisadvantages,anotherdesignwaspresentedin[21].ThepresentedVSA-IIusedafour-barlinkagesystemtotransmitthepower,whichshowedmorerobustnessandlargerload-bearingcapability.Intheseantagonisticdesigns,twomotorsandtwospringswereusedtocontrolthestiffnessandposition.Rotationofthemotorswithrelativeangleofthemotorsunchangedmadethelinkrotate.Whentherelativepositionofthemotorsvariedwiththemeanpositionofmotorsunchanged,thestiffnessoftheactuatorwaschanged.Inordertoholdaconstantstiffness,themotorsneedtoapplytorque,whichimpliespoorerenergyef cient.

Preliminaryconceptofthevariablestiffnessjoint(VSJ)waspresentedin[22]withoutcontrolmethodandproofofthesta-bility.Inthispaper,thecontrolmethod,aswellasthedesignoftheVSJ,ispresented.InSectionII,http://plianceisgeneratedbytheleafspringswhoseef-fectivelengthiscontrolledbytheactuatorsviafour-barlinkage.Insteadofoneactuatorbecominganadditionalloadtotheotheractuator,twoactuatorsevenlysharetheloadofthejoint,whichresultsinbetterenergyconsumption.Inaddition,thefour-barlinkagemakestheVSJmoreenergyef cientsinceitisnotre-quiredtoapplyadditionaltorqueinordertoholdstiffness,asdiscussedin[23],whereenergyconsumptionofvarious

actua-

IEEETRANSACTIONSONROBOTICS,VOL.27,NO.2,APRIL2011

Fig.1.SchematicsoftheVSJ.Leafspringsareattachedtotheaxis,andtheeffectivelengthofthespringisvariedbyfour-barlinkages,whichareactuatedbytwoidenticalactuators.

torswithadjustablepassivecompliancewerecompared.Itwasshownthathavingtoapplytorquetoholdaconstantstiffnesswasareasonofhigherenergyconsumption.InSectionIII,acontrolmethodisintroduced.Eachactuatoriscontrolledbyavelocitycontroller,andthelinkiscontrolledviafullstatefeedback.ThedynamicsoftheVSJisformedasasingularlyperturbedsys-tem.ThestabilityoftheVSJwiththecontrollerisshownusingaLyapunovfunction.InSectionIV,theresultoftheexperimentispresented.ToshowtheeffectofthevariablestiffnessoftheVSJ,anexperimentofhittingballswithdifferentstiffnessarepresented.Whenthestiffnessismaximum,ballsthatarehitbythelinkoftheVSJ yfartherthanminimumstiffness.Theexperimentshowsthattheaccelerationoftheballsaftercolli-sionwiththelinkissmallerwhenthestiffnessissmall,whichimpliesthatthecollisionissafer.TheconclusionismadeinSectionV.

II.DESIGNOFTHEVARIABLESTIFFNESSJOINT

ThedesignoftheVSJispresentedinthissection.ThemainconceptoftheVSJistohavetwoidentical“actuators”http://plianceisgeneratedbyleafsprings.Changingtheeffectivelengthofthespringsresultsinstiffnesschange.Inordertochangetheeffectivelengthofthesprings,afour-barlinkagesystemisused(seeFig.1).

Notethatthecompliancecomponentislocatedinserieswiththeactuators,whereastheactuatorsareinparallelcon guration.Duetotheparallelcon gurationoftheactuators,theloadtothejointisevenlydistributedtoeachactuator,whereasoneactuatorbecomesanadditionalloadtotheotherincaseofseriesactuation.Furthermore,sincecomplianceisgeneratedbytheleafspringandtheactuatorscontroltheeffectivelengthoftheleafsprings,nopowerisnecessarytoholdthestiffness,whichimpliesthattheVSJismoreenergyef cient.Sinceitisnotnecessarytouseanadditionalmechanismtochangestiffness(i.e.,leadscrew),asimplerdesignispossible.Astheslidermovestowardtotheaxisofrotation,theeffectivelengthofthespringbecomeszero,whichimpliesthattheVSJbecomesarigidactuator.

机器人,刚度,柔性

CHOIetal.:ROBOTJOINTWITHVARIABLESTIFFNESSUSINGLEAFSPRINGS

Fig.2.Axiswithfouridenticalspringsattached.Theslidersmovealongwiththespringswiththerollersinsideofthe

sliders.

Fig.3.Linkaisattachedtotheactuatorsandrotatesabouttheaxis.Thesliderisconnectedtothelinkb,whoseotherendisconnectedwithlinka.Thespringsareattachedtotheaxissothattheaxisrotateswhentheslidersrotate.

A.GeneratingComplianceWithLeafSpring

ComplianceoftheVSJisgeneratedbyNleafsprings,whichareattachedtotheaxis.Thespringsareseparatedby360/Ndegrees.Oneendofeachspringisattachedtotheaxisandtheotherendremainsfreesothatthespringbendswithanexternalforceappliedtothespring.Tworollers,whicharecontainedinaframecalled“slider,”rolloneachsideofthespring(seeFig.2).Effectivelengthofeachspringisdeterminedbythepositionofthesliderintheradialdirection.Theupsideanddownsideofeachsliderareheldbytwolinks,whicharecalled“outputlinks”ofafour-barlinkages.Alinkthatisconnectedtoeachactuatoriscalleda“Inputlink.”Theinputlinksaredesignedtobeacircularframe,whichareconnectedtotheharmonicgearsoftheactuators(seeFig.3).

Thefour-barlinkagetransformsrotationintotranslationalmovementanddeterminedthepositionoftheslider.Withtherelativepositionoftheactuatorsbeingunchanged,theeffectivelengthofthespringremainunchanged.Then,theVSJactsasanactuatorwithconstantcomplianceconnectedinseries.Fig.4showstheassembledVSJ.Theguidespreventtheslidersfrombeingdistortedduetotheappliedtorqueattheaxis.

Stiffnessofeachspringiscalculatedusingthepositionoftheslideranddimensionofthespring(seeFig.5).

LetDbethediameteroftheaxisandletlbethedistancetothesliderfromtheaxis.Whenanexternaltorqueisapplied,thespringsbendduetothetorqueandangulardisplacementoftheaxis.LettingdFibetheforceappliedtoithsliderdueto

the

231

Fig.4.AssembledVSJwithfoursprings.Theguidepreventsthepivotfrombeingtwistedbyapplied

torque.

Fig.5.Schematicofithleafspringwithrollers.Whentherollersslideontheleafspring,theeffectivelengthofthespringchanges,whichisthedistancebetweentheaxisandtherollers.Changeoftheeffectivelengthofthespringresultsinchangingstiffness.

externaltorque,thentheforceisgivenasfollows:

dFdθ

i=

σil+(D/2)

,

i=1,...,N

(1)

whereσiistorsionalstiffnessofeachspring,dFiistheforceateachsliderduetothetorque,anddθistheangulardisplacementoftheaxis.Notethattheamountofbentisthesameforeachspringsinceallthespringsareattachedtotheaxisandthatthepositionsoftheslidersareidentical.Then,thedisplacementofthesliderinangulardirectionisapproximatedas

dδ=(l+D/2)dθ.

(2)

LettingLi0bethelengthofithspring,wibethewidthofthespring,andtibethethicknessofthespring,thenfollowingthedevelopmentin[24],theforcetothesliderisgivenasfollows:dFEi=iwit3i

4l

3dδ(3)whereEiistheYoung’smodulusofeachspring.Therefore,with(1)–(3),thestiffnessofthejointisgivenas

σD/2)dFi=(l+idθ=(l+D/2)2Eiwit3

i

4l3

.(4)

Thestiffnessofthejointisgivenas

σ=

Nσ Ni=

(l+D/2)2

Eiwit3i

(5)

i=1

i=1

4l

3.NotethatstiffnessoftheVSJisdeterminedbytheshapeand

materialoftheleafspring.Sincestiffnessdependsontheshapeofthespring,itispossibletodesignstiffnessoftheVSJbydesigningtheleafspring,i.e.,rectangularshape,taperedshape,

机器人,刚度,柔性

232Fig.6.Schematicsoflinkagesconnectedattheslider.Thelinkaanddareattachedtobothactuators.Allthejointsofthefour-barsystemarerevolutejoints.Thesliderisconnectedtothelinkbandcsothatitispossiblefortheslidertorotatefreely.

etc.However,therangeofstiffnessfunctionobtainedbyde-signingthespringislimited.Forsimplicity,arectangularshape

ischoseninthispaper.B.ChangingtheStiffness

Inordertochangethestiffness,itisnecessarytochangethepositionsoftheslidersinradialdirection.Thepositionsoftheslidersarechangedbychangingtherelativepositionsoftwoinputlinksofthefour-barlinkages.Allthejointofthefour-barsystemarerevolutejoints(seeFig.6).Twooutputlinksthatareconnectedtoasliderhavethesamelength.Theothertwolinksareinputlinksandhavethesamelength.Eachinputlinkisattachedtoeachactuator.

ThepointOislocatedatthecenteroftherotationalaxis.Theinputlinksaredenotedbyaandd,respectively.Theoutputlinksaredenotedbybandc.Letr1bethelengthofthelinksaandd,andletr2bethelengthofthelinksbandc.Letting2ζ1betheanglebetweenthelinksaandd,then,thedistancetothesliderfromthepointO,whichisdenoted

byr,isgivenasfollows:

r=r1cosζ1+r22 r21sin2ζ1(6)wherer>0,and0<ζ1<π/2.Sincetheinputlinksrotate

aroundtheaxis,thedistancetothesliderbecomesr=l+D/2(seeFig.5).Thetorsionalstiffnessiscalculatedusing(5)and(6)asfollows:

σ(ζ(r1cosζ1+r22 r21sin2ζ1)2 NE1)=

iwit3i

(r1

cosζ1

+r22

r21

sin2ζ1

D2

)3i=1

4(7)

whereσ(ζ1)isaninvertiblefunctionfor0<ζ1<π/2.Notethatasζ1increases,theeffectivelengthofthespringbecomeszero.Then,theVSJbecomesarigidjoint.

Letq1andq2bethepositionofeachactuator.Sincethelinkaanddareattachedtoeachactuator,ζ1isgivenasfollows:ζq1=

1 q2

2

.(8)SincestiffnessoftheVSJdependsontheeffectivelengthoftheleafspring,whichisdeterminedbythelocationofthesliderintheradialdirection,controllingthepositionofthesliderintheradialdirectionisequivalenttocontrollingthestiffness.

Equa-

IEEETRANSACTIONSONROBOTICS,VOL.27,NO.2,APRIL

2011

Fig.7.Controllerofeachactuator.ThevelocityofeachcontrolleriscontrolledusingthePIcontroller.

tion(6)showsthatthelocationofthesliderinradialdirectionisafunctionoftherelativeanglebetweentheactuators,whichisdenoted2ζ1inFig.6.Thelocationofthesliderinangulardirection,however,istherestpositionofthespring.Sincethefour-barlinkagesystemissymmetric,therestpositionofthesliderisgivenasfollows:

ζq2=

1+q2

2

(9)

whereq1andq2arethepositionsoftheactuators.

III.CONTROLLINGTHEVARIABLESTIFFNESSJOINTThevelocityofeachactuatoriscontrolledbyaproportional-integral(PI)controller(seeFig.7).

TheresponseofthePIcontrollerisfasterthanothercontrollersothatitisassumedthatthevelocityoftheactuatorreachesthedesiredvelocityinstantaneously.Then,thedynamicsoftheVSJ,includingthe

velocitycontrollers,isgivenasfollows:

q˙1=v1 q˙2=v2q¨3= σ((q(10)1 q2)/2)q1+q2

Jl(q3 2)whereq3isthepositionofthelink,Jlisthemomentofinertia

ofthelink,σ((q1 q2)/2)isthestiffnessgeneratedbytheleafspring,andv1andv2denotethevelocitycommandtoeachactuator,respectively.Since(q1+q2)/2istherestpositionoftheleafspring,andq3isthepositionofthelink,thedifferenceisthede ectionoftheleafspring.

Let usde neacoordinatetransform ζ 1/20 Tq asfollows: 11/2 1q1ζ2 = 1/21/20 q2 =T q2 .(11)

ζ3001q3q3Then,thedynamics oftheVSJinζcoordinatesbecomes ζ˙

ζ˙1=u1

ζ¨2=u23= σ(ζ(12)1)

Jl(ζ3 ζ2)whereu1=(LettingKpv1 v2)/2,andu2=(v1+v2)/2.

,Ksp,andKdtrollawofthef

VSJisde nedsbenonzeroconstants,thenthecon-asfollows:

u1=Kpf(ζ1 ζ1d

)(13)upK2=Ksζ2

sp

Kp2ζ3 sdd2

ζ3+Ksζ˙3(14)

whereζdandζdandlink1

3arethedesiredrelativepositionoftheactuatorsposition,respectively.IntypicalapplicationoftheVSJ,sinceitistypicalthattheVSJisconnectedtoalink,which

机器人,刚度,柔性

CHOIetal.:ROBOTJOINTWITHVARIABLESTIFFNESSUSINGLEAFSPRINGShasbiggerinertiathanfour-barlinkagesandtheactuatorsaresharedtocontrolthepositionandthestiffnessoftheVSJ,itisinevitabletocontrolthepositionofthelinkslowerthanpositionoftheactuators.ItisassumedthatKprelative

=(1/ )K(13),where isasmallpositiveconstant.Then,f

thedynamicsspinofthe closed-loopsystembecomes

ζ˙1=Ksp(ζd ζ˙1 ζ1)Kp(ζp

2=2 ζ3)+Ks(ζ3 ζd)+Kdζζ¨s3=

σ(ζ23s˙3(15)1)

Jl

(ζ3 ζ2).Thedζ˙equilibriumpointof(15)isζ1=ζζf=ζ1

,ζ2=ζ3=ζ3d

,and3=0.Letting1 ζ1

d,andζs ζ 2 ζζs= ζ2

s3d

ζ˙3 = ζ3

s3 dζ˙ζ3 (16)3thenthedynamicsoftheclosedloopsysteminthestatespace

isgivenas

ζ˙f=Kp s

ζf=:ff(ζf)pssp

s

sd˙s ζ˙s= K(ζ ζ˙s2 ζ3)+Kζ+Ksζ33s23

(17) σ(ζf)

J(ζ3s ζ2sl)

=:fs(ζf,ζs)

(18)

withζf0=0,andζ0s

=[0,0,0]Tbeingtheequilibriumpoint.

Sinceζf=0istheuniquesolutiontoff(ζf)=0whenKp0istheisolatedequilibriumpointof(17)forζss=0,h(ζs0

):=3∈IR.Then,models(17)and(18)areinthestandardformofsingularperturbation.Followingthestandardapproachin[25],theboundary-layersystemisgivenas

ζ˙f=ff(ζf)=Ks

pζf.(19)

Thereducedsystemis givenasfollows:Kps ζ˙s(ζ2 ζ3s)+Kp

sζs+Ksdζ˙ss=fs(ζf0

,ζs)= ζ˙3s23

3 ) .(20)

σ(ζf

0Jl(ζ3

ζ2)

Lemma1stableatζf.Theboundary-layersystem(19)isexponentially

=0,ifKpProof:Letting0

Vs<0.f=(1/2)(ζf)2bethecandidateLyapunovfunction,then,forallζf=0,thederivativeoftheLyapunovfunctionisgivenby

V˙f=ζfζ˙f=Ks

p(ζf)2<0.(21)

Sincethesystemislinear,thesystemisexponentiallystable.

Notethattherelativepositionoftheactuatorsneedstobecon-trolledwithinboundariesduetothelimitedrangeofmovement.Theboundariesaredeterminedbythelengthofthefour-barlinkagesandthelengthoftheleafspring.Whendesigningareferencetrajectoryfortherelativepositionoftheactuators,thereferenceneedstoremainwithintheboundaries.

Lemma2:Letλbeapositiveconstant,andletκ0=σ(ζf

)/Jl.LettingKsp= 2λ,andKsd

= 1 (λ2/κ0

0),thenthecontrollawofthereducedsystemin(14)isastabilizingcontrol

of

233

Fig.8.ImplementedVSJ.Itconsistsoftwoidenticalmotorsandharmonicgearsandfouridenticalleafspringsandsliders.

thereducedsystemin(20).Moreover,thereducedsystemisexponentiallystable.

Proof:ForproofofLemma2,seetheAppendix. Theorem1:Letubeacontrollawforthefullmodelandde nedasfollows:

u=u1= 2 λζf

u2= 2λζ2s+λζ3s 1+J2 lσ(ζλf)ζ˙3s(22)whereλ>0,thenthefullmodel(12)withthecontroller(22)

isexponentiallystable.

Proof:ζf=0isanisolatedrootoftheequation0=ff(ζf).ByLemmas1and2,theboundary-layersystemandthereducedsystemareexponentiallystable.Therefore,by[26,Th.9.3],thesystemisexponentiallystable. NotethattorquecontroloftheVSJispossible.SincetheVSJactsasanSEAin[12]whenthestiffnessisheldconstant,asimpleapproachistochangetheforce-controlproblemtoaposition-controlproblem.Thetorquegeneratedbytheleafspringisgivenas

τ=σ(ζ1)(ζ3 ζ2).

(23)

Whenthelinkpositionis xedatζ3=ζd

ζ3andtherelativepo-sitionis xedatζ1=d

with(23)as1,thedesiredrestpositionoftheVSJiscalculatedfollows:

ζτd

2

d=ζ3d σ(ζd)(24)

1

whereτdisthedesiredtorque,ζ1d

isthedesiredrelativeposition,

andζ3d

isthepositionofthelink.Then,itcontrolstherelativepositionoftheactuatorsandtherestpositionofthespring.Othermethodsarefoundin[27]–[29].

IV.EXPERIMENT

Inthissection,resultsofexperimentstoevaluatetheimple-mentedVSJarepresented.

Fig.8showstheimplementedVSJ.Twoidenticalactuatorsareconnectedtotheinputlinksofthefour-barlinkages.Each

机器人,刚度,柔性

234TABLEI

MODELPARAMETERSFORTHEVSJANDEACHC

OMPONENTFig.9.TorquegeneratedbytheVSJwithdifferentangulardisplacements.

actuatorconsistsofaharmonicgearandanelectricalmotor.Theelectricalmotorandtheharmonicgearisconnectedthroughatimingbeltwitha3:1gearratio.Sincethegearratiooftheharmonicgearis51:1,theoverallgearratiois153:1.Fouridenticalleafspringsareattachedtotheaxis.DimensionsoftheVSJandotherparametersarelistedinTableI.Theparametersaredesignedtomeetthespeci cationofthemanipulatorofaservicerobot.

Aslidermovesalongeachspringandtheoutputlinksofthefour-barlinkageareconnectedtothesliders.Therotationoftheaxisismeasuredbyanencoder,whiletherotationoftheshaftofeachmotorismeasuredbyanotherencoderattachedtotheshaft.Guidesareusedtopreventtheslidersfrombeingdistortedduetotheforceatbothendsofthesliders.Whenthefour-barsystemreachesoutthemost,theeffectivelengthofthespringis26.3mm.TheVSJisdesignedtogenerate30N·m.

Fig.9showsthemeasuredstiffnessoftheVSJ.Inordertomeasurethestiffness,angulardisplacementoftheaxisismeasuredwhenknowntorqueisappliedattheaxis.

Notethattheangulardisplacementisalmostlinearwithre-specttotheappliedtorque.Therefore,itispossibletoapprox-imatethestiffnesswithaconstantwhentherelativeposition

IEEETRANSACTIONSONROBOTICS,VOL.27,NO.2,APRIL2011

TABLEII

TORSIONALSTIFFNESSOFTHE

VSJ

Fig.10.FrequencyresponseoftheVSJ.(a)Frequencyresponseoftheposi-tion.Bandwidthofthepositionis2Hz.(b)Frequencyresponseofthestiffness.Bandwidthofthestiffnessis3.5Hz.

oftheactuatorsisunchanged,whichmakesthecontrollersim-pler.TableIIshowstheapproximatedstiffnesswithdifferentrelativepositionsoftheactuators.Maximumstiffnessisabout3648Nm/rad,andminimumstiffnessisabout252Nm/rad.TheresultoffrequencyanalysisisshowninFig.10.Mag-nitudeandphaseofthepositionandstiffnessoftheVSJaremeasuredwhencommandstothepositionandstiffnessaresinu-soidalfunctionswithvariousfrequencies.Fig.10(a)showsthe

机器人,刚度,柔性

CHOIetal.:ROBOTJOINTWITHVARIABLESTIFFNESSUSINGLEAFSPRINGS

Fig.11.StepresponseoftheVSJwhenthereferenceofthelinkpositionisastepfunction.(a)Positionofthelink,i.e.,ζ3,whenthepositioncommandisastepfunction.(b)Angulardisplacementoftwoactuators,i.e.,ζ1.

frequencyresponseofthepositionoftheVSJ.The3-dBcutofffrequencyofthepositionis2Hz.Fig.10(b)showsthefre-quencyresponseofthestiffness.The3-dBcutofffrequencyofthestiffnessis3.5Hz.

ResponsestopositioncommandswithdifferentstiffnessareshowninFig.11.Therelativepositionoftheactuators,whichisdenotedbyζ1in(12),iscontrolledtobeunchanged.Alink,whoseinertiaisabout0.5kg·m2,isattachedtotheVSJ.Fig.11(a)showsthepositionofthelink,whichisdenotedbyζ3in(12),withdifferentstiffness.Whenthestiffnessismaxi-mum,vibrationdoesnotoccur,therisetimebecomes0.164s,andthesettlingtimeis0.475s.However,whenthestiffnessisminimum,vibrationduetothepositioncommandoccurs.Therisetimeis0.153s,andthesettlingtimeis1.133s.Duetothevibration,thesettlingtimeislongerwithminimumstiff-ness.Modeluncertaintyandfrictioncausesteady-state

error.

235

Fig.12.StepresponseoftheVSJwhenthereferenceoftherelativepositionoftheactuatorisastepfunction.(a)Positionofthelink,i.e.,ζ3,whenthepositioncommandisastepfunction.(b)Relativepositionoftwoactuators,i.e.,ζ1.

Fig.11(b)istherelative-positionerrorwhenthepositionrefer-enceisastepfunction.Therelative-positionerrorremainslessthan0.010rad.

Fig.12showstheresponseoftheVSJwhenthereferencefunctionfortherelativepositionisastepfunction,whilethepositionofthelinkiscontrolledtoremainunchanged.Fig.12(a)showsthepositionerrorofthelink.Thepositionofthelink,i.e.,ζ3,changedlessthan0.015rad.Thechangesinthelinkpositionareduetodiscrepanciesinactuators,whichareassumedtobeidentical.Fig.12(b)showstherelativepositionoftheactuators,i.e.,ζ1.Therisetimeis0.093s,andthesettlingtimeis0.155s.Notethatlargercontrolgainresultsintheriseandsettlingtimeoftherelativepositionbeinglessthanthelinkposition.

InFig.13,theresponseoftheVSJisshownwhenbothposi-tionandstiffnessoftheVSJarechanged.Stiffnessischanged

机器人,刚度,柔性

236Fig.13.StepresponseoftheVSJwhenunitstepfunctionsareappliedasreferencesforthepositionofthelinkandtherelativepositionoftheactuators.(a)Positionofthelink,i.e.,ζ3,whenthepositioncommandisastepfunction.(b)Relativepositionoftheactuators,i.e.,ζ1.

fromitsmaximumvaluetominimumvaluewithinlessthan0.3s.Duetotheinertiaofthelink,positionresponseshowsanovershootsimilartothestepresponseofthelinkwithminimumstiffnessinFig.11(a).

Steady-stateerrorsexistduetothefrictionandmodeluncer-taintiesnotincludedinthemodel.

InordertoshowtheeffectoftheVSJinunexpectedcolli-sion,hittingagolfballandatable-tennisballexperimentwasconducted.Agolfballandatable-tennisballarehitbytheVSJwithdifferentstiffness.A520-mm-longlink,whichismadeofaluminum,isattachedtotheaxisoftheVSJ.Theballsarehitbythelinkattheheightof17mm(seeFig.14).TheVSJrotateswithmaximumandminimumstiffnessat1.2rad/s.Inordertoreducetheeffectofvibrationduetothelink,theVSJrotates1080 beforehittingtheballwhenthestiffnessis

minimum.IEEETRANSACTIONSONROBOTICS,VOL.27,NO.2,APRIL

2011

Fig.14.Experimentalsetupforhitting-ballexperiment.AlinkisattachedtotheVSJandthelinkhitstheballonthestand.

TABLEIII

FLIGHTDISTANCEOFTHEBALLSHITBYAL

INK

The ightdistanceoftheballsaremeasured.Themeanvalueofthedataaretakenafter vemeasurementsforeachstiffnesscondition,whichisshowninTableIII.

Forbothcases,theballs yfartherifthestiffnessoftheVSJismaximum.Thegolfball ies43mmfartherwhenthestiffnessismaximum.Thetable-tennisball ies104mmfartherwhenthestiffnessismaximum.Sincethemassoftheballsdoesnotchangeandtheballrestsbeforehitting,theexperimentshowsthattheaccelerationoftheballsaftercollisionisaffectedbydifferentstiffness.Lessstiffnessresultsinlessaccelerationoftheballs,whichimpliescollisionwithlessstiffnessissafer.

V.CONCLUSION

http://pli-ancewasgeneratedbyleafspringsattachedtotherotationalaxis.Changingtheeffectivelengthofthespringsresultedinchangesinstiffnessofthesprings.Theeffectivelengthofthespringswerechangedviamovingtheslider,whichhadrollerssledonthespring.Theslidersweremovedwithfour-barlink-age.Twoidenticalactuatorswereconnectedtothefour-barlinkageresultinginparallelactuation.Duetoparallelactuation,theloadtotheVSJwassharedbytheactuators.Furthermore,notorquewasrequiredbytheactuatorstomaintainthestiff-nessconstant,whichimpliesbetterenergyef ciency.Whenthebarsrotatedatthesamespeedinthesamedirection,theaxisrotateswithoutchangingthelocationofthesliders,whichresultedinthestiffnessbeingunchanged.Whenthebarsro-tatedintheoppositedirections,theslidersmovedwhiletheaxis

机器人,刚度,柔性

CHOIetal.:ROBOTJOINTWITHVARIABLESTIFFNESSUSINGLEAFSPRINGSremainmotionless,whichresultedinchangeinstiffnessofthejoint.Anonlinearcontrollerwassuggestedandproventobeastabilizingcontrollerusingasingularperturbationmodel.Thecontrollerassumedtheactuatorswereidentical,whichwasnottrueintheimplementedsystem.Asymmetriesanduncertaintiesinmodelparametersresultedinnonzerosteady-stateerrors,whichshouldbefurtherinvestigatedinordertoreducetheeffectsofthem.Experimentswereconductedtoshowthatthepositionandstiffnessofthejointwerecon-trolledindependentlyusingtwoactuators.Thestiffnesswaschangedmuchfasterthanthepositionofthelinkandvibra-tionoccurredwithminimumstiffness.WhenaballwashitbythelinkattachedtotheVSJwithmaximumstiffness,theball ewfartherthanwhenthestiffnesswasminimum.Sincethesafetyoftheobjectdependsontheaccelerationafterthecollision,theexperimentshowedthathavingalessstiffjointwassaferincaseofanunexpectedcollisionwithhumans.AmanipulatorofaservicerobotcanbeequippedwiththeVSJtopreventdamagetohumanswhenanunexpectedcollisionoccurs.

APPENDIX

PROOFOFTHEEXPONENTIALSTABILITYOF

THEREDUCEDMODEL

Proof:Sinceσ(ζf)>0,andJl>0,κ0ispositive.Thedy-namicsofthereducedsystemwiththecontrol(14)isgivenby

ζ˙ 2s= 2λζ2s+λζλ2 3s 1+κζ˙s(25)0

3

ζ¨3

s= κ0ζ3+κ0ζ2.(26)

LettingacandidateLyapunovfunctionbede nedas

V=

κ02ξ2+κ0212ζ3+2

ζ˙32

(27)whereξ=ζ2s+(λ/κ0)ζ˙s,thenthecandidateLyapunovfunc-tionispositive,exceptfor3

ζs=ζthecontrolLyapunovfunction2is3s=ζ˙s=0.Thederivativeofgiven3

by

V

˙=κ0ξξ˙+κ0ζ3ζ˙3+ζ˙3ζ¨3=κ0ξξ˙+ζ˙3s(κ0ξ λζ˙3

s)= λ(ζ˙3s)2+ξ(κ0ξ˙+κ0ζ˙3

s)= λ(ζ˙ 3s)2+κ0ξ λζ2

s λ2

sκζ˙03= λ(ζ˙3

2)2 κ0λξ2≤0.(28)

Therefore,thereducedsystemLettingΩ={ζs∈IR3|V

˙isstable.

=0},thenΩ={ζs∈IR3|ζ2s=0,ζ˙3

s=0}.(29)

LettingM ΩbethelargestinvariantsubsetofΩ,then

M={ζs∈Ω|ζs=[ζs,ζ3s,ζ˙3s]T=0}.Therefore,thesystemisasymptoticallystable,2

whichalsoimpliesthatthesystemisexponentiallystablesincethesystemislinear.

237

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3534–3538.

JunhoChoi(M’10)receivedtheB.S.degreeinelectricalengineeringfromtheHanyangUniversity,Seoul,Korea,in2000,andthePh.D.degreeinelec-tricalengineeringsystemsfromtheUniversityofMichigan,AnnArbor,in2005.

HeiscurrentlyaSeniorResearchScientistwiththeKoreaInstituteofScienceandTechnology,Seoul.Hisresearchinterestsincludenonlinearcontrol,bipedalrobotcontrol,manipulatorcontrol,andthedesignandcontrolofasafe-jointforrobot

manipulators.

SeonghunHong(S’10)receivedtheB.S.degreeinelectricalandcomputerengineeringin2007fromHanyangUniversity,Seoul,Korea,whereheiscur-rentlyworkingtowardthePh.D.degreeinelectricalengineering.

HereceivedthelicenseofSecondarySchoolTeacher(Class-II)withinterestinthe eldofEd-ucation.Since2007,hehasbeenwiththeCenterforCognitiveRoboticsResearch,KoreaInstituteofScienceandTechnology,Seoul.Hiscurrentresearchinterestsincludenonlinearsystems,applicationsto

roboticmanipulationand eldrobotsinthe eldofcontrolengineering,cogni-tivescience,andmultipleintelligenceinthe eldof

education.

IEEETRANSACTIONSONROBOTICS,VOL.27,NO.2,APRIL2011

WoosubLeereceivedtheB.S.degreeinmechanicalengineeringfromSogangUniversity,Seoul,Korea,in1999,andtheM.S.degreeinelectronicengineer-ingfromYonseiUniversity,Seoul,in2004.

Since2004,hehasbeenaResearchScientistwiththeKoreaInstituteofScienceandTechnology,Seoul.Hiscurrentresearchinterestsarethedesignandcontrolof eldrobotsystemsanddependable

manipulators.

SungchulKang(M’98)receivedtheB.S.,M.S.,andPh.D.degreesinmechanicaldesignandproductionengineeringfromSeoulNationalUniversity,Seoul,Korea,in1989,1991,and1998,respectively.

Since1991,hehasbeenwithKoreaInstituteofScienceandTechnology,Seoul,whereheiscurrentlyaPrincipalResearchScientist.In2000,hewasaone-yearPostdoctoralResearcherwiththeMechanicalEngineeringLaboratory,Japan.In2006,hewasaVisitingResearcherwiththeArti cialIntelligenceLaboratory,StanfordUniversity,Stanford,CA.His

currentresearchinterestsincluderobotmanipulation,hapticsensinganddisplay,and eldrobot

systems.

MunsangKim(M’96)receivedtheB.S.andM.SdegreesinmechanicalengineeringfromtheSeoulNationalUniversity,Seoul,Korea,in1980and1982,respectively,andtheDr.-Ing.degreeinroboticsfromtheTechnicalUniversityofBerlin,Berlin,Germany,in1987.

Since1987,hehasbeenaResearchScientistwiththeKoreaInstituteofScienceofKorea,Seoul,wherehehasbeenleadingtheAdvancedRoboticsResearchCentersince2000andhasbeentheDirectorofthe“IntelligentRobot—TheFrontier21Program,”since

October2003.Hiscurrentresearchinterestsincludethedesignandcontrolofnovelmobile-manipulationsystems,haptic-devicedesignandcontrol,andsensorapplicationtointelligentrobots.