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Disturbance

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Disturbancebance

    Disturbance

Wangetal/JZhejiangUnivSciA20089(11):15311538

    ;JournalofZhejiangUniversitySCIENCEA

    ;ISSN1673565X(Print);ISSN1862-1775(Online)

    ;www.zjueduc@zus;wwwspringerlinkcom

    ;E-mail:jZUS@zjuedu.cn

    ;1531

    ;Disturbancerejectioncontrolbasedonaccelerationprojection

    ;methodforwalkingrobots

    ;XuyangWANG,ZhaohongXU,TianshengLO

    ;(SchoolofMechanicalEngineering,ShanghaiJiaoTongUniversity,Shanghai200240,China)

    ;E-mail:wangxuyang@sjtu.edu.cn

    ;ReceivedApr.1,2008;revisionacceptedJune10,2008

    ;Abstract:Thispaperpresentsadisturbancerejectionschemeforwalkingrobotsunderunknownextemalforcesandmoments.

    ;Thedisturbancereiectionstrategy.whichcombinestheinversedynamicscontro1withtheaccelerationprojectionontotheZMP

    ;(zeromomentpoint)

    plane,canensuretheoveralldynamicstabilityoftherobotduringtrackingtheprecomputed

    trajectories.

    ;Undernorma1conditions.i.e..thesystemisdynamicallybalanced,aprimaryinversedynamicscontrolisutilized.Inthecasethat

    ;thesystembecomesunbalancedduetoextemaldisturbances,theaccelerationprojectioncontro1fAPC)loop,wil1beactivatedto

    ;keepthedynamicstabilityofthewalkingrobotthroughmodifyingtheinputtorques.Thepreliminaryexperimentalresultsona

    ;robot1egdemonstratethattheproposedmethodcanactuallymaketherobotkeepastablemotionunderunknownextemalper.

    ;tufbations.

    ;Keywords:Inversedynamics,Disturbancerejection,ZMP(zeromomentpoint)

    plane,O~hogonalprojection,Walkingrobot

    ;doi:10.163l~zus.A0820242Documentcode:ACLCnumber:TP242.1

    ;INTR0DUCTION

    ;Stabilitymaintenanceisalwaysregardedasone

    ;ofthemostimportanttopicsinthemotioncontrolfor

    ;walkingrobots.Incontrasttootherindustrialma

    ;nipulators,theinteractionbetweenthewalkingrobots

    ;andthegroundsubjectstounilatera1constraints.

    ;whichmakestherobotseasilyoverturnduringthe

    ;motion.Furthermore.therobotshavetodealwithall ;kindsofdisturbantesintherealenvironment.e.g.the ;interactionswiththeextemalenvironmentorwith ;humans.A1OSSofstabilitymightresultinapoten

    ;tiallydisastrousconsequenceforbothrobotsand ;humanbeings.Inthissense,thedesignedcontroller ;shouldpossessthecapabilityfordisturbance ;rejection.

    ;Researchesrelatedtobiped1ocomotionandsta

    ;blewalkingcontrolhavebeendevelopedrapidlyin ;recentyears.Severalresearchersproposedbalancing ;compensatorsandcompliancecontrollersforhu

    ;manoidwalkingbasedonzeromomentpoint(ZMP) ;criteriafTakanishieta1.,l990;Hirai,1998; ;Kajitaeta1.,20011.Takanishietf_f1990)describeda ;controlmethodofdynamicbipedwalkingtocom

    ;pensateforthemomentonanarbitraryplannedZMP ;bytrunkmotionandstudiedthecorrespondingmo

    ;tioncontrolunderunknownexternalforces.Accord

    ;ingtothepresetZMPtrajectory,thetrunkmotion ;patternwascalculatedtocompensatethedeviations ;causedbytheexternalforces.Theenvironment ;adaptabilityoftherobotwasimprovedwiththisZMP ;compensativemotioncontro1.However,thismethod ;isbasedonthepresetZMPtrajectoriesbutnotthe ;actualones.whichmightresultinlargeerrorsforthe ;deflectionsbetweenthepresetandtheactualZMP. ;Hiraieta1.(1998)putforwardtheconceptoftipping ;moment,whichcanbecalculatedaccordingtothe ;distancebetweenthedesiredZMPandthecenterof ;theactualgroundreactionforce.Torecovertherobot ;posture.thegroundreactionforcecontrolandmodel ;ZMPcontrolwereusedintheirposturerecovery ;contrO1.Thecenteroftheactualgroundreactionforce ;orZMPwasshiftedtoanappropriatepositionby ;adiustingeachfoot’sdesiredpositionandattitude,

    ;1532Wangota1./dZhejiangUnivSciA20089(11):15311538

    ;wherebytherobotcouldberecoveredtoabalanced ;posture.

    ;TheZMPtrackingmethodprovidesaneffective ;approachforwalkingmotions.However,thestability ;ofthemotionisimprovedatthecostofreducingthe ;motionflexibilities.Presently,moreandmorere

    ;searchersintendtocombinetheZMPcontrolwith

    ;othermode1.basedmethods(Kajitaeta1.,2003a; ;2003b;Sugihareta1.,2002;Kondaketa1.,2003).As ;oneofthem,Kondak(2003)proposedaZMPplane

    ;projectioncontrolmethod,inwhichthealgorithmis ;basedonadecouplingofthenonlinearmodeland

    ;steeringtorquestomaintainthestabilityofthesystem. ;AsanextensionoftheZMPplaneprojectionmethod.

    ;adisturbancerejectionschemeispresentedinthis ;paper.Thisschemebehavesaswitchingmannerbe- ;tweentheinversedynamicscontrolandthe ;ZMPplaneprojectionmethod.Whentherobotbe

    ;comesunbalancedduetoexternalunexpecteddis. ;turbances,theinputtorquescanberegulatedtoac

    ;countforboththetrack.followingsandthestabilityof ;thesystem.

    ;INVERSEDYNAMICSCONTR0L

    ;ByapplyingtheLagrange’sequationofmotion

    ;toaconservativesystem,thedynamicmodelof ;walkingrobotcanbederivedas

    ;()+h(q,)+G(g)=f,

    ;wheregisthevectorofthegeneralizedcoordinates, ;D(q)isaF/x/’/,symmetric,positive—definiteinertia

    ;matrix,h(q,)isanx1,centrifugalvectorwhich

    ;containsthecentrifugalaccelerationandCoriolis ;terms,G(g)isanxlgravitationalvector,andfisa ;×1externaltorquevector.

    ;Givenadesiredreferencetrajectoryqr(‘),acon—

    ;trollercanbedesignedandimplemented.Theoutput ;ofthecontrollerwilldriveasymptoticallytheac

    ;tualtrajectorygandtoand,respectively.

    ;Thedesignofthecontrollerinvolvesdecom- ;posingthecontroldesignproblemintoaninner-loop ;designandanouterloopdesign.Theinner-loopisto ;performfeedbacklinearizationoftheroboticsystem, ;whichdependsontheinversedynamicmodelofthe ;robot.Thecomputedtorquecanbewrittenas ;fg=D(q)il+h(q,)+G(q)

    ;Thenonlinearitiesofthesystemcanbeelimi

    ;natedwithEq.(2).Therefore,bysubstitutingEq.(2) ;intoEq.(1),andassumingthatD(q)isnonsingular,we ;obtain

    ;g

    ;Inthisway,thecomplicatednonlinearsystemis

    ;convertedintoalinearone.Furthermore,tracking

;performancecanbeimprovedbydesigningafeed

    ;backcompensationloop.ThePDcontrolruleisused ;inthispaper,thatis

    ;fg=D(g)(KDKpe)+h(q,)+G(g),(4)

    ;whereP:0disthetrajectorytrackingerrorvector. ;KDandKPareconstantgainmatrixes.Throughse- ;lectingappropriatevaluesforKDandKp,acritically ;dampedclosedloopperformancecanbeacquired.

    ;Thedetaileddiagramoftheinversedynamicscontrol ;isshowninFig.1.

    ;Feedbacklinearizationbyusing

    ;Fig.1Blockdiagramoftheinversedynamicscontrol ;ZMPANDZMP.PLANE

    ;Theinversedynamicscontrolworksquitewell ;intrajectory-followingifweselecttheappropriate ;gainsforthecontroller.However.itdoesnotconsider ;thedynamicalstabilityofthewalkingrobot.Walking ;robotisadynamicallybalancedsystem.anditiseasy ;toovertumduringthemotion.Whenlargedistur. ;banceisappliedtotherobot,theimpactsgeneratedby ;thedisturbanceandthecompensatorytorqueswill ;haveaninfluenceonthestability.Insomecircum. ;stances,therobotwillbecomeunstableand ;Wangota1./dZhejiangUnivSciA20089(11):1531.1538 ;overthrowentirely.Thus,intermsofthecomputed ;torques,theaccelerationshouldberegulatedcor

    ;respondinglytoguaranteethedynamicsstabilityof ;therobotaswellaseliminatetrackingerrors. ;ZMPisoneofthemostfamousstabilitycriteria ;widelyusedinthebipedmotioncontro1.Basedonthe ;ZMPtheory.theZMPplaneconceptisproposedby

    ;Kondak(2003)forthebipedrobottoperfoITnastable ;movement.Inthispaper,thedrivingtorquesconsid. ;eringtheoverallstabilitycanbecomputedaccording ;totheZMP-plane.Theaccelerationisprojected ;ontoadefiniteZMP?-planetoresisttheexternaldis-

    ;turbanceappliedtotherobot.

    ;ZMP

    ;ZMPisdefinedasthepointonthegroundat

    ;whichthenetmomentoftheinertialforcesandthe ;gravityforceshasnocomponentalongthehorizontal ;axes(Vukobratovic,2004).Letusconsiderthelo

    ;comotionmechanisminthesinglesupportphase ;(Fig.2).Thecontributionofthepartabovetheankleis

;replacedbytheforceFAAx,FFandthemo-

    ;mentMA,MAy,MAz).Thefootalsoexperiences ;thegroundreactionatpointPaswellasthegravity ;G=.Thegroundreactiongenerallyconsistsofthe ;groundreactionforceRandthemomentlThe ;horizontalcomponentsofthereactionforceR,i…e

    ;andRv,representthefictionforcethatisbalancingthe ;horizontalcomponentoftheforceFA.whereasthe ;verticalmomentrepresentstheresultantmoment ;atpointPinducedbythefrictionforce. ;Fig.2Forcesactingonthesupportingfoot ;1533

    ;pointOtothefootmasscenterG-OAisavectorfrom ;pointOtotheanklejointA,andmfisthefootmass. ;TheprojectionofEq.(5)ontothehorizontal ;planeyields

    ;(OPx)—(OGx17”1fg)—(M)(OA×)0,

    ;(6)

    ;wherehdenotesthehorizontalyplane.Eq.f61 ;representstheequilibriumequationofthefoot, ;wherebythedynamicstabilityconditionscanbeob. ;tained.Thatis,thepositionofpointPwherethe ;groundreactionisactedcanbesolvedfromEq.f61.In ;lhissense,pointPjscalledZMP.ThesystemisdV_ ;namicallybalancedifthepositionofpointPiswithin ;thesupportpolygon.

    ;ZMP.plane

    ;ZMPplanedescribesthelinearizationrelation

    ;shipbetweentheZMPandthejointaccelerations, ;whichcanbederivedfromtheprimaryZMPequations ;AccordingtotheZMPdefinition.ifthecoordinate ;systemoriginOisshiftedtothegroundprojectionof ;thefootmasscenter,Eq.f6)canbesimplifiedas ;(OPx)()0,(7)

    ;whereM=MA+OAxFAistheresultantmomentof ;thewholemechanismatpoint0.DefineM=[Mx,, ;Mz]T,=,,T,andMP,m0]T,th

    ;Eq.(7)canberewrittenas

    ;IMP=0,

    ;+XZMp=0.(8)

    ;Inaccordancewiththedynamicequationsofthe ;robot,MyandRzcanberepresentedexplicitlyas ;Thestaticequilibriumequationforthesupport’here

    ;ingfootcanbewrittenas

;OP×+OG×mfg+MA+:+OA×=0,(5)

    ;whereOPisavectorfrompointOtogroundreaction ;pointPinCartesiancoordinate,OGisavectorfrom ;M

    ;y

    ;=oc0+alq..l+…+,

    ;Rz=po+p+…+pnn!

    ;=

    ),i:0,?? ;(g,

    ;=

    ;(g,),i:0,??

    ;(9)

    ;(10)

    ;(11)

    ;(12)

    ;where….,and….,arenon—linearfunctions

    ;ofthegeneralizedcoordinatesqandgeneralized ;1534Wangetal/dZhejiangUnivSciA20089(11):1531?1538 ;velocities.BysubstitutingEqs.(9)and(10)into ;Eq.(8),weobtain

    ;M

    ;——

    ;0

    ;+1I+.?+

    ;++…+

    ;(13)

    ;Accordingly,theexplicitrepresentationofXZMP ;withrespecttotheaccelerationisacquired.An ;equivalentexpressiontoEq.(13)canbeobtainedas ;here

    ;7,+…+r,4/1+yo=o,

    ;=

    ;(g,,xzMP),i=0,…,

    ;Therefore,Y0,…,arenon—linearfunctions

    ;dependingonthegeneralizedcoordinates,gener

    ;alizedvelocitiesandxcoordinateofZMPXZMp.

    ;Atthemomentt,thesystemaccelerationscanbe ;changedarbitrarilybyapplyingcorresponding ;torquestothejoints.Incontrast,thechangesof ;velocitiesandcoordinatescorrespondtoactual

    ;accelerationsandvelocities(doubleintegratorbe

    ;havior),respectively.Therefore,thecoefficients ;,...,ateachmomentcanbeconsideredascon

    ;stantandtheaccelerationsasarbitraryvalues.

    ;Obviously,thisconclusionisalsoapplicabletoYZMP. ;SimilartoEq.(14),wecanobtain

    ;here

    ;,7+…+771l+q0=0,

    (,,YzM),i=0,…, ;77f=

    ;Foreachmomentf,Eq.(14)orEq.(16)describes ;aplaneintheaccelerationspace.Thisplaneisthe ;ZMP.planewhichwasproposedbyKondakfirstlyin ;2003(Kondakefa1..2003).Thephysicalmeaningof ;theZMP..planecanbedescribedasselectinganac.. ;celerationcombination,…,,andthesystemcan

    ;moveatthismomentinsuchawaythatthexory ;coordinateofZMPwillbeequaltoXZMporYZMP. ;ACCELERATIONpROJECTl0NALG0RITHM

    ;Theaccelerationprojectionalgorithmistopro

    ;jecttheoriginalaccelerationvector=[1,…,]

    ;ontotheZMPplane,wherebytheresultingaccelera-

    ,…,:1thatthetionvectorensuresthatthecoord—l一一 ;tion//=

    ;g,J

    ;nateofZMPwillbeequaltothegivenXZMporYZMP. ;Diversifiedwaysareavailabletoimplementsucha ;projectionoperation,whereastheultimategoalisto ;createtheaccelerationvectorthatliesonthe ;ZMPplane(orintersectiondeterminedbybothXZMp ;andyzMp).AsshowninFig.3,twodifferentprojection ;methodsareillustratedina3Daccelerationspace.In ;thefirstmethod,theoriginalaccelerationvectoris ;changedtoinawaythatitstipliesonthe

    ;ZMPplane.Inthesecondmethod,i.e.ano~hogonal ;projection,theresultingaccelerationvectoris ;determinedbytheperpendicularfoot(pointP)from ;theendoftheoriginalaccelerationvectortothe ;ZMPplane.

    ;Fig.3Projectionoperationsin3Daccelerationspace ;Thestudyhasshownthattheo~hogonalprojec

    ;tionleadstothebestmovementperformanceamong ;alltheothers(Kondaketa1.,2003).Accordingly,itis ;adoptedinourdisturbancerejectioncontrollerto ;acquireastablemovement.Sincetheperpendicular ;footisthenearestpointfromtheoriginalacceleration ;vectortotheZMP.plane.theresultingacceleration ;vectorcanbeobtainedbysolvingtheminimal ;distancebetweentheoriginalvectorandthe ;ZMPplane.Letthedistancebe

;J

    ;

    ;

    ;

    ;

    ;?

    ;=

    ;,J

    ;g

    ;D

    ;Wangotal/JZhefiangUnivSciA20089(11):1531.1538 ;inaddition,shouldsatisfytheconstraints1and ;,

    ;i…e

    ;()=?Yiqi+Yo=0,(19)1

    ;(20)

    ;Theminimaldistancesubjecttotheconstraints ;Eqs.(19)and(20)canbefoundbyusingLagrange ;multiplierprinciple.Correspondingly,theresulting ;accelerationvectorcanbealsoobtainedaccording ;to

    ;++

    ;++dq{oqdql

    ;=0,=0,i=1,?

    ;=

    ;0,

    ;

    ;,,

    ;where21andA2areLagrangemultipliers ;DISTURBANCEREJECT10NCONTROLLER

    ;AccordingtothedefinitionofZMP.plane.the ;inputaccelerationvectorcanberevisedconsider

    ;ingtheoveralldynamicstabilityaswellasthe ;trackfollowing.AsshowninFig.4anacceleration ;projectioncontroIfAPC)loopiSinsertedjntothe ;inversedynamicscontroller.Theaccelerationpro

    ;jectionalgorithmiSadoptedintheAPCloop.which ;makestheoriginalinputaccelerationvectorbe ;projectedontothezMPplane.WhenthesystemiS

    ;dynamicallybalanced,onlytheprimaryinversedy

    ;namicscontroliSusedintrackingthepredefined ;trajectories.OneelargedisturbanceiSappliedorthe ;systembecomesunbalanced.theAPCloopwillbe ;activatedtokeepthesystemmoveinastablemanner.

    ;Fig.4Blockdiagramofthedisturbancerejection ;controller

    ;1535

    ;ToconstructanappropriateZMP.planeiSim

    ;portantintheaccelerationprojectionalgorithm. ;Whenthesupportingpolygonisrectangleandone

    axis(andtheotherjSparallelto ;edgeiSparalleltoX

    ;Yaxis),theboundaryconditionthatthesystemjS ;dynamicallybalancediS

    ;i<MP<X

    ;max,Y<YzMP<Yax,(22)

    ;whereXmaxandXminaremaximumandminimum ;boundaryofthesupportingpolygonalongthe.axis ;respectively,YmaxandYminaremaximumandmini

    ;mumboundaryofZMPalongtheyaxisrespectively.

    ;ThesystemwillbecomeunbalancedifZMPliesout

    ;sideoftheboundaries.Inthiscase,theZMPplane

    ;shouldbeplacednearesttothestablezoneforthe ;facilitiestocontro1.ThatiS.theplaneiSdetermined ;byXmax,Xminormax,min.Thedetailedcontrolpro

    ;cedureforthedisturbancerejectioniSshowninFig.5. ;g

    ;Fig.5Flowchartofthedisturbancecontrolalgorithm ;SIMULAT10NEXPERIMENTS

    ;Alotofoneleggedsystemshavebeendeveloped

    ;duringthepastfewdecades.Workonthedynamics ;andbalanceofleggedsystemsdatesbacktoMat

    ;souka(1979)andRaibert(1984)makingamajor ;contributiontothefield.Furthermore,aparticular ;0

    ;=

    ;

    ;}f?

    ;?

    ;=

    ;,

    ;g

    ;/l\

    ;1536Wangota1./JZhejiangUnivSciA20089(11):1531-1538 ;interesthasbeenexpressedintheconstructionof ;oneleggedrobottosimulatethesinglesupportphase ;whichappearsfrequentlyinthebipedwalkingmotion ;(Pannuela1.,1995;Mitobeeta1.,2004).Thebalance ;maintenanceofoneleggedsystemisachallengeto

;achieveforitslimitedsupportconvex.whichpro

    ;videsagoodplatforilltotestthecontrolalgorithm. ;Toverifythedisturbancerejectioncontrollaw. ;anexperimentaloneleggedrobotisbuiltasshownin

    ;Fig.6.Therobotconsistsof4links,i…efoot,shank,

    ;thigh,andwaist,andtheparametersofthemodelare ;giveninTable1.Thetrunkismodeledasamasspoint ;(=0.5036kg),wherebytheinfluenceoftheupper ;bodycanbestudied.Thecontrolalgorithmisim. ;plementedinMatlab/Simulinkenvironment.The ;robotisexpectedtoperformasquattingandraising ;motion.Duringthemotion,disturbancesareapplied ;totherobotfromdifferentdirections.Theresulting ;movementisillustratedinFig.7.wherethreepushes ;withthemagnitudeof2NareactedatpointsAandB ;f0r0.125s.respectively.Indetail.thefirsttwopushes ;occuratpointAalongaxisandthelastonehappens

    ;atpointBalongy-axisindependently.

    ;Sincethecontactstateisunchangedinthemo

    ;tion,thestabilityareaisfixed?

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