CONTROLOFHYBRIDDYNAMICSWITHAPPLICATIONTOAHOPPINGROBOTByFrankBentonMathisADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfullmentoftherequirementsforthedegreeofMechanicalEngineering-DoctorofPhilosophy2016ABSTRACTCONTROLOFHYBRIDDYNAMICSWITHAPPLICATIONTOAHOPPINGROBOTByFrankBentonMathisControlofdynamicmotionisanimportantsubjectofstudyinroboticsasitisdesirableforrobotstohaveaspeccmotionpatternratherthenmovingtoasetpoint.Themotionsofrobotsalsoinvolvechangingdynamicbehaviorsduetointeractionwiththeenvironment,suchasduringcontact,andthisleadstohybridsystemdynamics.Apopularexampleofahybriddynamicalsystemisaleggedrobot;thehybriddynamicsisduetotheperiodicswitchingofswingandstancelegsandimpulsivedynamicsduetogroundcontacts.Leggedrobotsrequirecontrolofadynamictrajectorydenedbythewalkinggaitorrunningmotion.Forleggedrobots,thespringloadedinvertedpendulum(SLIP)modeliscommonlyusedtodescribethedynamicmotioninasimpedmanner.TheSLIPmodelhasalsobeenusedforcontrolofhoppingrobotsandafundamentallimitationofthemodelisthatitfailstoaccountforimpactwiththeground;thisisduetoitssingledegree-of-freedomintheverticaldirection.Weinvestigatethecontrolofahoppingrobotstartingfromamoregeneraltwo-massmodelandthenexpandthetheorytoplanarmulti-linkrobotsystems.Theinvestigationinvolvestwogroundcontactmodels,rigidandelastic,fortheobjectiveofapexheightcontrol.Intherigidcase,thegroundisassumedtoprovideanimpulsiveforcetothehoppingrobotresultinginaninelasticcollision.Ahybridcontrolstrategyisdesignedtodealwiththehybriddynamicalsystem:acontinuouscontrollerbasedonpartialfeedbacklinearizationisusedinconjunctionwithadiscretecontrollerthatupdatesacontrolparameterateachhoptoachievethecontrolobjective.Intheelasticcase,thegroundactsasamasslessspring,whichdeectsastherobotexertsaforceuponcontact.Inthiscase,weshowthatacontinuouscontrollerbasedonthebacksteppingalgorithmcanensureasymptoticconvergencetothedesiredapexheight.Severalrobotcongurationsareconsidered,andforeachcongurationthecompletehybriddynamicsistakenintoaccountwhiledesigningthecontroller.ThecontrollerscompensatefortheimpulsivedynamicsaswellashigherorderdynamicsthatareignoredinsimpedmodelssuchastheSLIPmodel.Experimentalvalidationofapexheightcontrolofatwo-masshoppingrobotonarigidfoundationisprovided.TABLEOFCONTENTSLISTOFFIGURES...................................viChapter1Introduction...............................1Chapter2Two-MassHoppingRobotonaRigidFoundation........52.1Dynamics.....................................62.1.1CoordinateSystemDescription.....................6
2.1.2FlightPhase................................7
2.1.3InelasticImpact..............................7
2.1.4ContactPhase...............................82.2ContinuousControloftheHeightoftheCenter-of-Mass..................................10
2.2.1FeedbackLinearization..........................10
2.2.2FlightPhase................................11
2.2.3ContactPhase...............................11
2.2.4HybridDynamicsofClosed-LoopSystem................122.3PoincareMap...................................132.3.1ConstructionofFirstReturnMap....................13
2.3.2PeriodOneOrbits............................16
2.3.3StabilityAnalysisofFirstReturnMap.................172.4DiscreteControl..................................182.4.1MappingBetweenApexHeightandLiftoConguration.......182.4.2StabilizingtheLiftoConguration...................192.5Simulations....................................21
2.6Two-MassHopperExperiments.........................242.6.1DescriptionofExperimentalHardware.................252.6.2ControlImplementationinExperiments................25
2.6.3PoincareMapReduction.........................27
2.6.4SimulationResultswithHardwareParameters.............282.6.5ComparisonofExperimentalResultswithSimulations........33Chapter3Four-LinkHoppingRobotonaRigidFoundation........353.1Dynamics.....................................353.1.1FlightPhase................................37
3.1.2Impact...................................37
3.1.3ContactPhase...............................393.2ContinuousControl................................403.2.1ContactPhase...............................41
3.2.2FlightPhase................................45
3.2.3HybridDynamicsofClosed-LoopSystem................47iv3.3PeriodicBehavior.................................493.3.1PeriodOneOrbits............................49
3.3.2ChaosControl...............................503.4Simulations....................................52Chapter4Two-MassHoppingRobotonanElasticFoundation......574.1Dynamics.....................................584.1.1SystemDescription............................58
4.1.2FlightPhase................................58
4.1.3ContactPhase...............................59
4.1.4ApexHeight................................604.2MotivationforDerentControlStrategy....................604.3HybridControlStrategy.............................624.3.1ControlProblemDenition.......................62
4.3.2ContinuousControllerDesignforStabilizationinFlightandContactPhases...................................634.3.3FeedbackLinearization..........................63
4.3.4Backstepping...............................64
4.3.5StabilityAnalysis.............................67
4.3.6DiscreteControllerforStabilizationofHybridDynamics.......684.4SimulationResults................................72Chapter5Three-linkHoppingRobotonanElasticFoundation......755.1Dynamics.....................................755.1.1SystemDescription............................75
5.1.2FlightPhase................................78
5.1.3Impact...................................78
5.1.4ContactPhase...............................80
5.1.5ApexHeight................................815.2HybridControlStrategy.............................835.2.1FeedbackLinearization..........................83
5.2.2ControllerDesigninXDirection....................845.2.3ControlProblemDenitionforYDirection...............845.2.4ContinuousControllerDesignforYDirection,Backstepping.....855.2.5StabilityAnalysis.............................89
5.2.6DiscreteControllerforStabilizationofHybridDynamics.......905.3SimulationResults................................93Chapter6Conclusions................................98BIBLIOGRAPHY....................................100vLISTOFFIGURESFigure2.1:Two-masshoppingrobot........................6Figure2.2:Apexheighthcorrespondingtoperiodicorbitsofthesystemforagivenvalue..............................22Figure2.3:Errorbetweendesiredlift-ostates˜andactuallift-ostatesatthekth-hop˜(k)...............................23Figure2.4:Absoluteheightofthetwomassesm1andm2,andcenter-of-massheightareplottedasafunctionoftime................24Figure2.5:Schematic(left)andphotograph(right)oftheexperimentalhardware.26Figure2.6:Apexheighthcorrespondingtoperiodicorbitsofthesystemobtainedwithderentvaluesof.........................29Figure2.7:Simulationresults:Errorbetweentheactuallift-ostate˘=_yandthedesiredlift-ostate˘=_yatthebeginningofthek-thhop,k=1;2;;9...............................30Figure2.8:Simulationresults:Plotoftheheightoftheuppermassx1,thelowermassx2,andthecenter-of-massz,asafunctionoftime........31Figure2.9:SimulationResults:PlotoftheforceFappliedbytheactuator...31Figure2.10:Experimentalresults:Plotoftheheightoftheuppermassx1,thelowermassx2,andthecenter-of-massz,asafunctionoftime....32Figure2.11:ExperimentalResults:PlotoftheforceFappliedbytheactuator..33Figure3.1:Four-linkhoppingrobot.........................37Figure3.2:Centerofmassheightabovethegroundforthefour-linkhopper..55Figure3.3:Inputtorquesforthefour-linkhopper.................56Figure4.1:(a)Two-massrobothoppingonanelasticfoundation(b)free-bodydiagramsofthetwomassesataninstantwhenthelowermassisin
contactwiththeelasticfoundation...................59viFigure4.2:Simulationresults:Plotoftheheightoftheuppermassx1,thelowermassx2,andthecenter-of-massz,asafunctionoftime........73Figure4.3:Simulationresults:Errorsinthediscretestates(EEdes),q1andq2atthebeginningofthek-thhop,k=1;2;;7..........74Figure5.1:Three-linkhoppingrobotonanelasticfoundation..........77Figure5.2:Centerofmassheightandrelativecenterofmassforthethree-linkhopper..................................96Figure5.3:Inputtorquesforthethree-linkhopper................97viiChapter1
Introduction
Ahybriddynamicsystemisonewhosedynamicschangesaccordingtodiscreteevents.Suchsystemsareubiquitousinroboticssincearobotisfrequentlyexpectedtointeractwithitsenvironmentcausingeitheradiscretechangeinmassduetoitpickingupordroppinganobject,orachangeinkinematicsduetocontactwithasurfacesuchastheground.Moreover,thedynamicmotionofrobotsinspiredbybiologicaldesign,suchasswimmingwithatailorwalkingwithlegs,isnotonlyhybridinnaturebutalsorequirescontroltoaperiodicoperatingpoint.Herewewillinvestigatethehoppingmotionofarobotwhichisinspiredbyrunningandwakinginleggedroboticsystems.TheearliestworkonhoppingrobotscanbecreditedtoRaibert[26],[14].Inhiswork,Raibertusedasinglemassonahydraulicpistonandexperimentallydemonstratedahoppinggait;thecontroldesignutilizedaspringlikemotioninthepistonandastabilizingtorqueatthehipjoint.SchwindandKoditchek[30]proposedalosslessmodelofRaibert'splanarhopperforthepurposeofcontrollingitsforwardvelocity.AclosedformofthereturnmapofahopwasderivedandacontrollerwasproposedforimprovinguponthegaitofRaibert'shopper.Saranlietal.[28]introducedtheSLIP(springloadedinvertedpendulum)modelofthehopperandargueditsequivalencetothehigherdegree-of-freedomAKH(ankle-knee-hip)legmodel.Adeadbeatcontroller,whichvariedthespringparametersoftheSLIPmodelateachhop,wasdesignedtoproducestableperiodicmotion.Thespringloadedinvertedpendulum(SLIP)modelwaslatershowntobeanaccuraterepresentationofrunningand1hoppinginbiologicalsystems[5],[8].FollowingtheworkbySchwindandKoditchek[30]andSaranlietal.[28],theSLIPmodelbecameverypopularintheresearchcommunity.HyonusedacombinationofhydraulicactuatorsandspringstoapplyRaibert'scontrolmethodtoakneedrobot[16].Kajitausedaresolvedmomentumapproachtocontrolaleggedrobottoadesiredtrajectory[18].Ghigliazzaetal.[12],[11]andHolmesetal.[15]investigatedpassivestabilityoftheSLIPmodelforvariousparameters.PoulakakisandGrizzle[24],[25]investigatedtheSLIPmodelwithanasymmetricmassandcompareditsbehaviorwiththatofathree-linkhopper.Altendorferetal.[6],[7]studiedthereturnmapofthenon-integrableSLIPdynamicsandinvestigatedthestabilityofthemap.SeipelandHolmes[31],[32]investigatedthestabilityoftheSLIPmodelforathree-dimensionalsystem.HamedandGrizzle[13]laterproposedarobustevent-basedcontrolmethodtoimprovestabilityofthecontrolledleggedsystem.AlthoughtheanalysisoftheSLIPmodelisusefulandprovidesthebasisforthedesignofeectivecontrollers,itisincompletesinceitdoesnotaccountfortheimpulsivedynamicsassociatedwithfoot-groundinteraction.TheSLIPmodelisnotcapableofaccountingfortheimpulsiveforcesthataregeneratedatthetimeofimpactwiththeground.Toaccountfortheseimpulsiveforcesandmodelthedynamicsofthehoppermoreaccurately,itisnecessarytomodelthelegasamassratherthanamasslessspring.Saitouetal.[27]andIshikawaetal.[17]proposedatwo-masssystemforahoppingrobotinaneorttoobtainamoreaccuratemodeloftherobotintheightphase.Saitouetal.[27]usedoptimalcontrolmethodstomaximizedthejumpingheightoftherobotinthepresenceofcontrolconstraints.Ishikawaetal.[17]usedaport-controlledHamiltonianmethodtocontroltheenergyofthetwomasssystemtoadesiredleveltomaintainamaximumjumpingheight.Inbothinstances,thegroundismodeledas2aspringandtheimpulsiveeectofgroundimpactisneglected.Hereweinvestigateaclassofhoppingrobotswithtwomasses,threelinks,andfourlinks.Thecontrolobjectiveforeachhoppingrobotistheachieveadesiredmaximumvalueofthecenterofmassduringeachhop,orapexheight.Allofwhichutilizeaformachaoticcontrolstrategiesasthemethodforstabilizingtheapexheight.WebegininChapter2withatwo-masshoppingrobotundergoinginelasticcollisionwitharigidgroundateachhop.Weusefeedbacklinearizationmethodstostabilizetheinternaldynamicswithderentcontrolparameters.Theperiodicnatureandstabilitycharacteristicsoftheresultinghybriddynamicsystemwasanalyzed[9]and\chaoscontrol"[29]wasusedtodiscretelyadjustasystemparameterandconvergetheapexheighttoitsdesiredvalue.ExperimentalresultsarepresentedinChapter2.6usingavoicecoilactuatortoprovetheecacyofthecontrolmethod.InChapter3,weextendthecontrolmethodutilizedonthetwo-masshoppingrobotonanrigidgroundtocontoltheapexheightofafour-linkhoppingrobotwhichundergoesinelasticcollisionwitharigidground.Generalizingthecontrolmethodfromatwo-massrobottoafour-linkrobotshowstheapplicabilityofthecontroltoamorehumanoidinspiredsystem,aswellasprovidesinsightsastothemethodofextendingthecontrolmethodtosystemwithhigherdegreesoffreedom.Thecontrolmethodforthefour-linkhoppingrobot,aswiththetwomasshoppingrobot,utilizesfeedbacklinearizationtostabilizethesystem,andthenadiscretechangeinaparametereachhoptoconvergetheapexheighttoachosendesiredvalue.Chapter4investigatestheissuesofapplyingthecontrolstrategy,whichwasdevelopedforhoppingonarigidground,tothecasewiththeelasticground.Wethendevelopanewcontrolmethodforcontrollingthetwo-masshopperontheelasticground.Thecontrol3methodutilizesacontinuouscontrolmethodbasedonbacksetpping.Thebacksteppingmethodcontrolstheenergyofthesystemtoadesiredlevelwhilesimultaneouslycontrollingthedistancebetweenthetwomassestoaconstantvalue.Thehybriddynamicsarethenstabilizedutilizingadiscretechangeinthecontrolparameterseachhop.Finally,inChapter5weextendthecontrolmethodusedforthetwo-massrobottoathree-linkrobothoppingonanelasticground.Similartothetwo-masshoppingrobotthehybridcontrollerutilizesacontinuouscontrolstrategybasedonbacksteppingtocontroltheenergyoftherobot.Thenthecontrolparametersarediscretelychangedateachhoptostabilizethesystemtothedesiredapexheight.Theresultsofsimulationsshowthatthepresentedcontrolstrategiesarecapableofachiev-ingarepeatedapexheightforeachhoppingrobot.Furthermore,theexperimentalresultsshowthecontrolstrategyiscapableofachievingthedesiredapexheighteveninthepre-scienceofuncertainty.InChapter6wepresenttheconclusions.4Chapter2
Two-MassHoppingRobotonaRigid
Foundation
Duringnormalwalkingandrunning,ahumanoidrobotmustinteractwiththeground.Themostcommongroundisconsideredtoberigid,i.e.itdoesnotdeformwhentherobotmakescontactwithit.Therigidgroundassumptionalsoimpliesthattherobotmakesinelasticcontactwiththeground,i.e.thevelocityofthepointofcontactoftherobotimmediatelymatchesthatofthegroundandgoestozerouponcontactWebeginwithasimpedmodelofahoppingleggedrobot,atwo-linkprismaticjointrobot.Thetwo-linkroboteectivelyactsastwomassesconnectedthroughanappliedfor.Theadditionofthelowermassallowsyoutoaccountforthelossofenergyduetoimpact.Thecontrolobjectiveistoachieveamotionwhichproducesadesiredvalueofthecenterofmassatthehighestpointofthehop,orapexheight.Ahybridcontrolstrategyconsistingofacontinuousanddiscretecontrollerisimplementedinordertoachievethetheobjectiveofadesiredapexheight.Experimentsarepresentedtovalidatethecontrolstrategy.Thecontrolmethodforthetwo-massrobotisthenextendedforusewithafourlinkrobot.Thetwomasshoppingrobotutilizestwomasseswithanactuatedforcebetweenthem.Thisisslightlymorecomplicatedthantheinvertedpendulummodelswhichinsteadallowforapplicationoftheforcedirectlyatthepointofcontactatthebaseratherthanthebase5mass.ThecontrolmethodpresentedfollowstheresultsofMathisandMukherjee[21]and[22].
2.1Dynamics
2.1.1CoordinateSystemDescription
ConsiderthehoppingrobotshowninFig.2.1.Itiscomprisedoftwomasses,m1andm2(rstandsecondmass,respectively),whichareconnectedbyalinkage.Theforceappliedbythelinkageonthetwomasses,denotedbyF,canbeactivelycontrolled.Theheightofsecondmasswithrespecttothegroundisdenotedbyx,andtheheightoftherstmasswithrespecttothesecondmassisdenotedbyy.Bothxandyareassumedpositiveintheverticallyupwarddirection.TheforceappliedbythegroundonthemassisdenotedbyFext;datumdatumm1m1m2m2gxx0yFFextFigure2.1:Two-masshoppingrobot6intheightphaseFext=0.Theequationsofmotionofthehoppingrobotarex=1m2Fg+1m2Fexty=m1+m2m1m2FFextm2(2.1)Thevaluesxand_xaremeasuredwithrespecttothegroundmakingitimpracticalformeasurementduringight.Thus,Itisassumedthatonlythevariablesy,_y,andFextcanbemeasured,andourobjectiveistocontroltheheightofthecenterofmasstoadesiredvalue.Thehybriddynamicsofthehoppingrobotiscomprisedofthreedistinctphases:theightphase,theinelasticimpact,andthecontactphase.
2.1.2FlightPhase
Duringtheightphase,thesystemisnotincontactwiththegroundandthereforex>x0;Fext=0(2.2)2.1.3InelasticImpact
Attheinstantwhenthesecondmasscomesincontactwiththegroundthesystemmovesfromx>0tox=0.Atthisinstantweassumethefollowingtoholdtrue:Assumption1.Theimpactbetweenthesecondmasswiththegroundresultsinaninstan-taneouschangeinthevelocitiesofthesystem,_xand_y.TheimpulsiveforceappliedbythegroundonthesecondmassisdenotedbyFext=Fimp(2.3)7Assumption2.ThecontrolforceFisnotimpulsiveinnature.Assumption3.Thecollisionofthesecondmasswiththegroundisinelastic.Iftcdenotesthetimeinwhichthesecondmasscomesincontactwiththeground,assumption3implieslimt!t+
c_x(t)=0(2.4)UsingthedynamicsgiveninEq.(2.1)andassumptions1and2,wegetFimp=limt!t
cm2_x(t)(2.5)andlimt!t+
c_y=limt!t
c[_y+_x](2.6)fromEq.(2.4)and(2.6)Theimpulsivedynamicscanbedescribedbythemappinglimt!t+
c2
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4y_yx_x3
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5=Slimt!t
c2
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41000
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5(2.7)2.1.4ContactPhase
Thecontactphasedenestheperiodoftimeduringwhichthesecondmassremainsincontactwiththeground.ForthecontactphasewemakethefollowingassumptionsAssumption4.Theexternalforceexertedonthesystemactsinthepositivedirectiononly8(thegroundisnotsticky),andthereforeFext0(2.8)Assumption5.Theenvironmentisrigidandtherefore,xx0(2.9)Usingassumptions4and5,FextcanbecomputedasFext=m2g+F0(2.10)SubstitutingEq.(2.10)into(2.1)weobtainthedynamicsofcontactphase:y=g+1m1Fx=0(2.11)Attheinstantwhenthesystemswitchesfromthecontactphasetotheightphase,FextcrosseszeroandthereforeFext=F+m2g=0(2.12)92.2ContinuousControloftheHeightoftheCenter-of-Mass2.2.1FeedbackLinearization
Ifzdenotestheheightofthecenterofmassofthehoppingrobot,wecanwritez=x+x0+mfy;mf,m1m1+m2(2.13)Sincethemeasurementofxisnotavailable,wedenetheheightofthecenterofmassrelativetothesecondmassbyrwhichcanbewrittenasr=(zxx0)=mfy(2.14)DerentiatingEq.(2.14)twice,wegetr=1m2[FmfFext](2.15)TocontrolthevalueofrwechoosethecontrolinputFasfollowsF=mfFext+m2v(2.16)wherevwillbechosenlater.SubstitutionofEq.(2.16)intoEq.(2.15)resultsinr=v(2.17)10Thechoiceofvfortheightphaseandthecontactphaseisdiscussednext.2.2.2FlightPhase
Thedynamicsofthehoppingrobotduringightphaseisgivenbytherelationsz=gr=v(2.18)WeseenfromEq.(2.18)thatthedynamicsofzisindependentofthecontrolforceandisthereforeuncontrollable.Tocontrolthepositionoftherelativecenterofmass,wechoosevasfollowsv=K(rrd)D_r(2.19)whererd>0issomeconstant,andKandDareconstantproportionalandderivativegains.Thechoiceofvin(2.19)resultsinthedynamicsr+D_r+K(rrd)=0(2.20)whichimpliesthattheequilibrium(r;_r)=(rd;0)isasymptoticallystable.2.2.3ContactPhase
Duringthecontactphase,xx0andthereforez=rr=v(2.21)11Sincex0,thedynamicsofthesystemisreducedfromorder4toorder2.Toemulateanaturalhoppingmotionwedesignvasfollowsv=8
>
<
>
:K(rrd)D_r_r0K(rrd)D_r_r>0(2.22)whereDchosenasfollows0<D<2pK(2.23)SuchachoiceofDallowstherstmasstobehaveasanunder-dampedoscillatorduringitsdownwardmotion.Bychoosing<0weensurethattherstmassbehaveslikeanegativelydampedoscillatorandenergycanbeaddedtothesystemduringitsupwardmotion.Theamountofenergyaddedwilldependonthechoiceofandthiswillbedeterminedbasedonthedesiredmaximumheightofzanditscurrentvalue.Thiswillbediscussedinthesectionondiscretecontrol.
2.2.4HybridDynamicsofClosed-LoopSystem
Thehybriddynamicsoftheclosed-loopsystemcanbesummarizedasfollows:FlightPhase:SubstitutingEqs.(2.16),(2.18),and(2.19)into(2.1)andsettingFext=0,wehavex=Kmfyrd+Dmf_ygy=K"y1mfrd#D_y(2.24)Impact:TheimpulsivedynamicsisdescribedbythemappinginEq.(2.7)12Contactphase:SubstitutingEqs.(2.10),(2.16),(2.21),and(2.22)into(2.1),wehavexx0y=K(y1mfrd)D_y(2.25)where=8
>
<
>
:1_r0_r>0(2.26)2.3PoincareMap
2.3.1ConstructionofFirstReturnMap
Wedenetherstreturnasthemappingoverasinglehopstartingfromliftoandendingatthenextlifto.Withoutlossofgenerality,weconsiderthetimeofrstliftotobet=0,andt1,t2,andt3tobethedurationsoftheightphase,contactphasewithdownwardvelocityoftherstmass,andcontactphaseupwardvelocityoftherstmassrespectively.Forconvenience,wedeneT1=t1T2=t1+t2T3=t1+t2+t3(2.27)andthestatevectorasfollowsX(t)=[y(t);_y(t);x(t);_x(t)]T(2.28)13UsingEq.(2.24),thestatesofthesystemattheendoftheightphasecanbeobtainedasfollowsX(T1)=2
6
4eAft10mfeAft103
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5X(0)+Bf(2.29)whereAfandBfaregivenbytherelationsAf=2
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401KD3
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5Bf=2
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4Bf1Bf1+Bf23
7
5(2.30)andBf1andBf2aregivenbyBf1=Zt10eAf(t1˝)K1mfRd˝Bf2=2
6
4mf[y(0)+_y(0)t1]12g(t1)2mf_y(0)t1gt13
7
5(2.31)andRisgivenbyR=[0rd00]T(2.32)ImmediatelyfollowingtheightphasethesystemimpactsthegroundandthestateschangeaccordingtoEq.(2.7).Followingtheimpacttherstmassdescendsdownwardinthecontactphase.Thestatesofthesystem,whentherstmasshasreachedthelowestverticalposition,areobtainedfromEqs.(2.25)and(2.26)asfollowsX(T2)=eAct2SX(T1)+Bc(2.33)14whereAcandBcaregivenbytherelationsAc=2
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5(2.34)Bc=ZT2T1eAc(T2˝)K1mfRd˝(2.35)UsingEqs.(2.25)and(2.26)again,thestatesatliftoareobtainedasX(T3)=eApt3X(T2)+Bp(2.36)whereApandBparegivenbytherelationsAp=2
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5(2.37)Bp=ZT3T2eAp(T3˝)K1mfRd˝(2.38)15SubstitutingEqs.(2.29)and(2.33)intoEq.(2.36)givesthemappingforthecongurationoftherstmassoveronehop.Thismappingisgivenbellow2
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4y(T3)_y(T3)3
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5=P[X(0);t1;t2;t3](2.39)2.3.2PeriodOneOrbits
Equation(2.39)givestherstreturnmapforonehop.Indexingeachhopbyanintegerwemaywritewithoutlossofgenerality˜(0)=2
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4y(0)_y(0)3
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5(2.40)and˜(1)=2
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4y(T3)_y(T3)3
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5=P[X(0);t1;t2;t3](2.41)Thediscretedynamicsofthehoppingmotionistherefore˜(k+1)=P[˜(k)](2.42)16wherethedependenceon(t1;t2;t3)hasbeendroppedsincetheyalldependon˜(k).Inparticular,forthersthop,t1,t2,andt3canbesolvedusingthefollowingequations:x(T1)=x0_y(T2)=0K[y(T3)1mfrd]D_y(T3)+gmf=0(2.43)Themeaningofthersttwoequationsaboveareclear.ThethirdequationisobtainedbysettingFext=0atthetimeoflifto.Aperiod-onehopofthesystemcorrespondtoequilibriumpointofthediscretesysteminequation(2.42)whicharethepoints˜whichsatisfy˜(k+1)=˜(k)=P[˜(k)](2.44)Equation(2.43)and(2.44)maythenbesolvednumericallytondalltheperiodoneorbitsofthesystemforagivenvalue.2.3.3StabilityAnalysisofFirstReturnMap
Oncetheperiodicpointforeachvaluearedetermined,itisnecessarytodeterminethestabilityofeachpoint.Todosowelinearizethediscretesysteminequation(2.42)aboutagivenperiodicpoint,denotedby˜,giving˜(k+1)ˇP(˜)+dP(˜)d˜j˜=˜(˜(k)˜)(2.45)17wheretheJacobianmatrixdP(˜)d˜isobtainedbyimplicitderentiationofEq.(2.43).BydeningtheerrorbetweenthecurrentpointofliftowiththeperiodicpointasE(k)=˜(k)˜(2.46)wehavethelinearizedequationofthediscretedynamics:E(k+1)ˇdP(˜)d˜E(k)(2.47)TheperiodicpointisasymptoticallystableifandonlyifˆdP(˜)d˜<1(2.48)whereˆ[]isthespectralradius.2.4DiscreteControl
2.4.1MappingBetweenApexHeightandLiftoConguration
Theheighttowhichthehopperwilljump(themaximumvalueofzdenotedherebyh)dependsonthevalueofyand_y,oralternativelyrand_r,atthetimeoflifto,i.e.,h=h(˜).Tondthisfunctionalrepresentationwerealizethatthetotalenergy(kineticpluspotential)ofthecenterofmassremainsconstantduringtheightphase,i.e.E(˜)=12(m1+m2)_z2+(m1+m2)gz=const:(2.49)18Notethatzand_zareidenticaltorand_ratthetimeoflift-o.ThesecondequationneededtodeterminethevaluevalueofhisFext=0atlift-o.DeningthemapT(˜;)asT(˜;)=2
6
41(m1+m2)gE(˜)Fext(˜;)3
7
5=2
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4h03
7
5(2.50)wecansolvefor˜foragivenvalueofhasfollows˜=T1(h;)(2.51)Notethatistheonlyvariableoverwhichwehavecontrol.Therefore,ourobjectiveistodeterminetherightvalueof=thatwillmakethehopperjumptothedesiredheight.2.4.2StabilizingtheLiftoConguration
Equations(2.44)and(2.51)produceasetofpoints(˜;)whichareperiodicandhaveahoppingheighth.Toensureasymptoticconvergenceofthehoppertothedesiredperiodicpoint,orequivalentlytothedesiredhoppingheight,wechangetoaderentconstantvalueforeachhop.LinearizingthediscretedynamicsinEq.(2.44)abouttheperiodicpoint(˜;)gives˜(k+1)=P(˜;)+dP(˜;)d˜j˜=˜=(˜(k)˜)+dP(˜;)dj˜=˜=((k))(2.52)19wheretheJacobianmatricesdP(˜)d˜anddP(˜)daredeterminedbyimplicitderentiationofEq.(2.43).DeningtheerrorandcontrolinputstatesE(k)andU(k)asE(k)=˜(k)˜U(k)=(k)(2.53)andusingequation(2.44)wemaywritethelinearizeddynamicsasE(k+1)=dP(˜;)d˜j˜=˜=E(k)+dP(˜;)dj˜=˜=U(k)(2.54)ChoosingthecontrolactiontobeU(k)=CE(k))(k)=+CE(k)(2.55)whereCisaconstantmatrix.WendthechoiceofthematrixCsuchthatˆdPd˜j˜=˜=+dPdj˜=˜=C<1(2.56)asymptoticallystabilizesthesystemtothedesiredperiodicpoint.202.5Simulations
Thetwomassesofthehopperareassumedtobem1=50kgm2=20kg(2.57)Thedesiredheighthandrdarechosenash=2mrd=1:071m(2.58)Notethatthevalueofr=1:071correspondstoavalueofy=1:5.Weassumethegroundheighttobex0=0:2m(2.59)Thisresultsinrestingheightofthecenter-of-massofthehoppertobe1:271metersabovethedatum.ThecontinuouscontrolgainsKandDarechosentobeK=1500N=(kgm)D=10Ns=(kgm)(2.60)ThevaluesofKandDsatisfyD<2pKdiscussedearlier.Fortheparameterspresented,Fig.2.2providesnumericalsolutionsforthehoppingheightassociatedwithperiodoneorbits,i.e.,solutionstoEqs.(2.44)interms.Notethattherearemultipleheightsassociatedwithagivenvalueof,andthismakesthechoiceofaassociatedwiththedesiredheightdcult.Weseethatthereappearstobeaminimumvalueofhforagiven.Thisisunderstoodasthereisaminimumheightthatthecenterofmassmustreachinorderforthesystemtohopandthusproduceareturnmap.212.252.152.051.951.851.751.6511.522.533.544.5h(m)Figure2.2:ApexheighthcorrespondingtoperiodicorbitsofthesystemforagivenvalueForthedesiredheightofh=2weusedFig.(2.3)todetermine=1:95(2.61)AdiscreteLQRproblemwassolvedtodeterminethecontrolgainsinEq.(2.55).ThesegainsaregivenbelowC=[0:07800:1618](2.62)Thegivenanddesiredheighthdeterminetheliftovaluesx,andthegainsCasymptot-icallystabilizethesystemtothegivenxandthusthedesiredheight.Theinitialconditions,22withoutlossofgenerality,wereassumedtobex(0)=0:2m_x(0)=0m=sy(0)=1:5835m_y(0)=6m=s(2.63)0246810121416182000.0050.010.015024681012141618200.500.51Numberofhops,k(yy)(m)(_y_y)(m/s)Figure2.3:Errorbetweendesiredlift-ostates˜andactuallift-ostatesatthekth-hop˜(k)Figure2.3plotstheerrorbetweenthedesiredperiodicpoint˜,andtheactuallift-opoint˜(k)foreachhop.Thesystemconvergesinapproximately8hops.However,wenotethatthecontrolresultsinsteadystateerror.ThisisduetoinexactestimationoffromFig.2.2.Figure2.4showstheheightoftherstmass,secondmass,andcenterofmassabovetheground.Weseethatthepeakheightofthecenterofmassconvergestoapprox.2metersasprescribedbythecontrolobjective.Fromthesimulationsweseethatecacyofthecontrolalgorithm.However,tofurther23024681012141618200.500.511.522.53time(s)x,(x+y),z(m)(x+y)zgroundxFigure2.4:Absoluteheightofthetwomassesm1andm2,andcenter-of-massheightareplottedasafunctionoftime
investigatetherobustnessandapplicabilityofthecontrolalgorithm,itbehoovesustoexper-imentallytestthecontrolmethod.Therefore,inthenextsectionwewilldiscussthedesign,application,andresultsofatwomasshoppingrobotexperiment.
2.6Two-MassHopperExperiments
InChapter2,section2wepresentedamethodofcontrollingtheapexheightofatwomasshoppingrobot.Toexpandoftheecacyofthepresentedalgorithm,wewillnowinvestigateanexperimentalrobotthatiscontrolledviathepresentedalgorithm.242.6.1DescriptionofExperimentalHardware
Intheexperimentalsetup,thetwo-masshopperiscomprisedofavoice-coilactuatorandtwolinearguides-seeFig.2.5.Onelinearguideisconnectedtothecoilhousingofthevoice-coilactuatorandtogethertheymakeupthelowermassofthehopper.Theotherlinearguideisconnectedtothecylindricalpermanentmagnetofthevoice-coilactuatorandtogethertheymakeuptheuppermassofthehopper.Bothlinearguidesaremountedtotheverticalrail;thisconstrainstheupperandlowermassestomoveintheverticaldirectionandpreventscollisionbetweenthem.Thevoice-coilactuatorisaproductofMoticont[1];itservesasalinearmotor,andforacommandedinputcurrentitoutputsaforcebetweentheupperandlowermasseswithagainof10:6N/A.Ithasastrokelengthof0.1334mandhasthecapabilitytoapplyacontinuousforceof21.5Nandanintermittentforceof68.1N(10%dutycycle).Thepositionofthetwomassesaremeasuredbytwolinearencoderswhosescaleismountedontheotherverticalmember;theencodersareaproductofUSDigital[4]andtheyhavearesolutionof120lines/inch.Thecurrentinthevoice-coilisprovidedbyamotorcontroller[2]poweredbya80voltpowersupply.ADS1104dSpaceboard[3],residinginahostpersonalcomputer,isusedfordataacquisitionandreal-timecontrol.Themassandlengthparametersofthehopperaregivenby.m1=2:668kg;m2=0:808kg;`=0:059m(2.64)2.6.2ControlImplementationinExperiments
Thecontinuouscontrollerdenedby(2.16),(2.24),(2.25)and(2.26)requirestheknowledgeoftheexternalforceFextduringthecontactphase.Theexternalforcecanbemeasured25directlyusingaforcesensorbutsuchasensorisnotpresentinourexperimentalhardware.Toovercomethisproblem,wesubstitute(2.9)and(2.1)into(2.16)torewritethecontrolforceinthecontactphaseasfollowsF=mf(F+m2g)+m2v)F=m2(mfg+v)1mf(2.65)Itcanbeveredfrom(2.13)thatmf<1;theforcein(2.65)isthereforenon-singular.Theparametersofthecontinuouscontrollerwerechosentobethesameasthoseinsimulationwhicharegivenin(2.71).Thechoiceofthevalueofrd=0:0979mcanbeexplainedasencoder for
upper massencoder for
lower massslider forupper massslider forlower massdatumuppermasslowermassvertical railcenter-of-massof lower masscenter-of-massof upper massx1x2`Figure2.5:Schematic(left)andphotograph(right)oftheexperimentalhardware.26follows:Intheshortestlengthconguration,theupperandlowermassesofthevoice-coilareseparatedbyadistanceofymin=0:064m.Themaximumstrokelengthofthevoice-coilis0:1334mbuttheforcedecaysrapidlyattheendofitsstroke.Theactivestrokelengthofthevoice-coilisthelengthoverwhichtheforcecanbeaccuratelycontrolled;thisisequaltosmax=0:127masperthespeccationsofthemanufacturer[1].Bychoosinghalfthestrokelengthasthedesiredneutralpositionofthehopper,thevalueofrdisdeterminedasrd,mfyd=m1(m1+m2)(ymin+12smax)=0:0979Thedesiredapexheightwaschosentobehdes=0:213m(2.66)Thexedpoint(˘;)andgainCforthediscretecontrollerwerechosenfromthesimulation.
2.6.3PoincareMapReduction
ThePoincaremap,P(˜;)dependsonthestates(˜;)wherethestates˜aregivenby˜2;=f(y;_y)2R2jx=0;_x=0;Fext=0g(2.67)WhichimpliesthatthePoincaremapisahomeomorphismonR3.However,Wenotethatattheinstantoflifto,theforcesFandFextaregivenby27F=mfFext+m2vv=KmfyrdDmf_yFext=F+m2g=0(2.68)CombiningEqns.(2.68)andsolvingforygivesy=gDmf_yKmf+rdmf(2.69)Equation(2.69)meansthatthedimensionalityofthereturnmapmaybereduced.NamelythatwemaywriteP(˘;)where˘2;=f_y2Rjx=0;_x=0;Fext=0;y=gDmf_yKmf+rdmfg(2.70)FromEqn.(2.70)weseethatthePoincaremap,P(˘;)isactuallyahomeomorphismonR2.Thetheexperimentalapplicationwewillusethethemapon(˘;)ratherthanthemapon(˜;).2.6.4SimulationResultswithHardwareParameters
Themassandlengthparametersofthehopperwereassumedtobethesameasthoseinourexperimentalsetup-see(2.64).28TheparametersofthecontinuouscontrollerwerechosenasK=600s2;D=5s1;rd=0:0979m(2.71)Thechoiceofrdisthesameasthatusedinexperimentsandwillbeexplainedlaterinsection2.6.2.-5.5-4.5-3.5-2.5-1.5-0.50.01.02.0h(˘;)(m)(0:213;4:55)Figure2.6:Apexheighthcorrespondingtoperiodicorbitsofthesystemobtainedwithderentvaluesof.Thediscretecontrollerparameterrequiresustochooseanominalvalue,wherecorrespondstothexedpoint(˘;).Anumericalsearchwasusedtondasmanyxedpoints(˘;)aspossible.Eachxedpointcorrespondstoauniquevalueofh=h(˘;)andFig.2.6plotsthesexedpointsintheh-plane.Thisplotisusefulforchoosingforadesiredvalueofh=hdes.Ourdesiredapexheightisgivenin(2.66)29FromFig.2.6,thiscorrespondsto=4:55(2.72)Forthesevaluesofhdesand,thevalueof˘was˘=_y=0:76m=s(2.73)-0.60.0-0.30123456789k(_y_y)(m/s)Figure2.7:Simulationresults:Errorbetweentheactuallift-ostate˘=_yandthedesiredlift-ostate˘=_yatthebeginningofthek-thhop,k=1;2;;9.Thegainofthediscretecontrollerwaschosentoplacethepoleat0:1;thegainwasfoundtobeC=1:08(2.74)Theinitialconditions1wereassumedtobex1(0)=0:149m;_x1(0)=0:0m=sx2(0)=0:059m;_x2(0)=0:0m=s(2.75)andtheresultsareshownhereinFigs.2.7,2.8and2.9.Figure2.7plotstheerrorsbetween1Theseinitialconditionsarethesameasthoseusedinexperiments.Thevalueofx2(0)waschosentobeequalto`=0:059mandthevalueofx1(0)correspondstoy(0)=0:09m.300.100.200.30
0.000.0590.01.02.03.00.213replacementime(s)x1;x2;z(m)x1zx2k=1k=2k=3k=4k=5k=6k=7k=8k=9Figure2.8:Simulationresults:Plotoftheheightoftheuppermassx1,thelowermassx2,andthecenter-of-massz,asafunctionoftime.0.01.02.03.0100200-1000CCCCCCCCupper mass changes direction0.174time(s)F(N)Figure2.9:SimulationResults:PlotoftheforceFappliedbytheactuator.theactuallift-ostate˘=_yanddesiredlift-ostate˘=_yasadiscretefunctionoftime;thehopsaresequentiallynumberedusingtheintegervariablekandtheerrorsatthe310.100.200.30
0.000.0590.01.02.03.00.213replacementime(s)x1;x2;z(m)x1zx2Figure2.10:Experimentalresults:Plotoftheheightoftheuppermassx1,thelowermassx2,andthecenter-of-massz,asafunctionoftime.beginningofeachhopareplottedfork=1;2;;9.Itcanbeseenthattheerrorsbecomenegligibleafterthreehops,i.e.,fork4.Thepositionsoftheuppermass,lowermass,andcenter-of-massoftherobotareplottedasacontinuousfunctionoftimeinFig.2.8.Thetimeintervalsduringwhichx2remainsconstantatitslowestvalueof`=0:059mindicatethecontactphases.Therobotlift-osthegroundattheendofeachcontactphase;thediscreteinstantsoftimecorrespondingtok=1;2;;9inFig.2.7aremarkedby\"inFig.2.8.ItisclearfromFig.2.8thattheapexheightoftheroboth(maximumvalueofzduringightphase)convergestoitsdesiredvaluehdes=0:213withinthreehops.TheforceappliedbytheactuatorF,denedby(2.16),isshowninFig.2.9.Itcanbeseenthattheactuatorforceincreasessigncantlyduringthecontactphase.Eachcontactphaseismarkedby\c"andthemaximumvalueoftheactuatorforcereachesapproximately200N,whichisslightlylargerthan7.5timestheweightoftheuppermassm1.Ineachcontactphase,theuppermasschangesdirectionfromdownwardmotiontoupwardmotion.32Thischangeindirectionresultsinthenon-smoothcontrolactiondescribedby(2.26),whichisdiscerniblefromFig.2.9.Intheightphase,therelativemotionofthemassesisdesignedtoemulatethemotionofanunder-dampedmass-springdampersystem;thisexplainstheoscillatorynatureoftheactuatorforce.Atthebeginningoftheightphase,theactuatorforceisnegativesincetheuppermassliftsthelowermassotheground.2.6.5ComparisonofExperimentalResultswithSimulations0.01.02.03.0100200-1000CCCCCC0.239time(s)F(N)Figure2.11:ExperimentalResults:PlotoftheforceFappliedbytheactuator.TheexperimentalresultsareshowninFigs.2.10and2.11forthesameinitialconditionsasthoseusedinsimulations,namely(2.75).Figure2.10showsthepositionsoftheuppermass,lowermass,andcenter-of-massofthehopperasacontinuousfunctionoftime;itcanbeseenthattheapexheightconvergestoitsdesiredvalueintwohops.Thecontrolobjectiveisachievedbuttherearederencesbetweensimulationandexperimentalresults.AcomparisonofFigs.2.8and2.10indicatederenceintransientbehavior.Theapexheightgraduallyincreasestoitsdesiredvalueinsimulationswhereasitovershootsandconverges33toitsdesiredvalueinexperiments.Thetime-periodforeachhopisalsoderent.Inaperiodof3sec,thehoppercompletessevenhopsinexperimentsbutisonitsninthhopinsimulations.Thisisduetothederenceinthedurationofthecontactphases.Forexample,thecontactphasepriortothesixthhop2isapprox.0.174secinsimulationbutapprox.0.239secinexperiment.Thisisduetothesaturationoftheactuatorforceinexperiments.AcomparisonofFigs.2.9and2.11indicatethatthepeakactuatorforceis200Ninsimulationbutonly150Ninexperiments.Sincethemaximumactuatorforceislowerinexperiments,itisappliedforalongerdurationinthecontactphasesuchthatthehoppercanreachthesameapexheight.FromtheexperimentweseethevalidityofthecontrolalgorithmpresentedinChapter2.2.Utilizingtheseresultswewillnextextendthecontrolalgorithmfromthesimpedtwomassmodeltoa4linkedmodelhoppinginaplane.2Atthetimeofthesixthhop,thetransientphaseisoverandthehopperexhibitssteady-statehoppingbehaviorinbothsimulationsandexperiments.34Chapter3
Four-LinkHoppingRobotonaRigid
Foundation
Fromthesimulationandexperimentalresultsofthetwo-masshopperinChapter2weseethatthecontrolmethodonrigidgroundiscapableofachievingadesiredapexheightforthesimpedtwo-masssystem.Wenowgeneralizethecontrolmethodologytoamulti-doflinkedrobotsystem.Tothisendweconsidertheapexheightcontrolofafour-linkhoppingrobot.Thechoiceofafour-linkhoppingrobotistomorecloselyresembleahumanoidrobot.ThecontrolisdesignedfollowingasimilarstructureinChapter2.Additionalconsider-ationisgivenduringboththecontinuousanddiscretephasestoaccountfortheadditionaldegrees-of-freedominthesystem.TheresultspresentedinthechaterfollowtheworkofMathisandMukherjee[21].
3.1Dynamics
Considerthefour-link,monoped,hoppingrobotshowninFig.3.1.LetxandybetheCartesiancoordinatesofthebaseofthefootoftherobot(pointO)relativetothexedgroundreference.Fori=1;2;3;4,themass,momentofinertia,andlengthofeachlinkaredenotedbymi,Ii,andlirespectively.Theangulardisplacementoftheithlinkisdenotedbyiandthedistancetoitscenterofmassisdenotedbydi-seeFig.3.1.Thestatesare35denedasq=xy1234T(3.1)TheequationsofmotionofthehopperaregivenbyM(q)q+N(q;_q)=AT+Fext(3.2)whereM(q)isthemassmatrix,N(q;_q)isthevectorofCoriolis,centrifugal,andgravitationalforces,A2R63isthematrixgivenbelowA=2
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4000100
000010
0000013
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7
5T;(3.3)TisthevectorofinputtorquesT=˝1˝2˝3T;(3.4)andFextittheforceappliedbythegroundontherobotgivenbyFext=FxFy0000T(3.5)InEq.(3.5),FxandFydenotethexandycomponentsoftheforceappliedtotherobotbythegroundatpointO.Thedynamicsofthehoppermaybeseparatedintothreephases:theightphaseforwhichy>0;theimpactphase,whichoccursattherstinstantwheny=0;andthecontactphase,whichoccursforthedurationinwhichthefootremainsin36xy1234XYOgithlinkdiliFigure3.1:Four-linkhoppingrobotcontactwiththeground,y0.Thephasesmirrorthoseofthetwo-masshopper.3.1.1FlightPhase
DuringtheightphaseFext=0.Furthermore,thedynamicsinEq.(3.2)resultinthenon-holonomicconstraintduetoconservationofangularmomentumaboutthecenterofmassofthehopper.
3.1.2Impact
Atthetimeofimpactweassume:
Assumption1:TheappliedtorquesTarenotimpulsive.Assumption1doesnotimplythatthetorquesTcannotbediscontinuous;itsimply37impliesthatthetorquescannotproducediscretejumpsinthestates.Assumption2:Thehopper'sfootcomesincontactwiththegroundonlyatpointO.Assumption2canbeenforcedthroughproperchoiceofcontrolgains.Assumption3:Attheinstantthefootcontactstheground(y=0),thegroundappliesanimpulsiveforcethatresultsin_x=_y=0instantaneously.Assumption3simplyimpliesinelasticimpact.
TakingtheintegralovertheinnitesimalperiodoftimeinwhichtheimpactoccurswehaveZt0+t0qdt=Zt0+t0M1(q)[ATN(q;_q)+Fext]dt(3.6)_q+=_q+M1(q)Fext(3.7)where_q+and_qaretherightandleftlimitsintimeof_q.Thisfollowsfromourearlierwork[10].Partitioningqaccordingtoq=[xyj]T(3.8)whereisgivenby=[1234]T(3.9)resultsinthecorrespondingpartitionofM1(q)givenbyM1(q)=2
6
6
4(M1)11(M1)12(M1)21(M1)223
7
7
5(3.10)38SolvingEq.(3.7)resultsinthefollowingchangeinthestatevariables:q+=q_x+=0_y+=0_+=_(M1)21[(M1)11]12
6
4_x_y3
7
5(3.11)3.1.3ContactPhase
Duringthecontactphase,Fextissuchthaty=0.Assumption4:ThefrictionforceFxisalwayssucientlylargesuchthatx=0duringthecontactphase.Duringthecontactphase,thedynamicsofthehopperisgivenbyDM(q)DTDq+DN(q;_q)=DAT(3.12)whereDisthematrixD=2
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4001000
000100
000010
0000013
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5(3.13)ThecontactphasetransitionstotheightphasewhenFy=0_Fy<0(3.14)393.2ContinuousControl
Similartothetwo-masscase,inordertoachieveadesiredapexheightwedesignacontinuouscontrollerthatregulatesthecenter-of-massposistionrelativetothatbaseoftherobot,pointO.Tocontrolthecenter-of-mass,wedenertobethevectorfromthebaseofthefoottothecenter-of-massofthehopper.Ifrxandrydenotethehorizontalandverticalcomponentsofr,wecanwriter=2
6
4rxry3
7
5=2
6
4fx(q)fy(q)3
7
5(3.15)wherefx(q)andfy(q)aregivenbyfx(q)=a1cos(1)+a2cos(1+2)+a3cos(1+2+3)+a4cos(1+2+3+4)(3.16)fy(q)=a1sin(1)+a2sin(1+2)+a3sin(1+2+3)+a4sin(1+2+3+4)(3.17)40InEq.(3.16)and(3.17),theconstantshavetheexpressionsa1=m1d1+(m2+m3+m4)l1ma2=m2d2+(m3+m4)l2ma3=m3d3+m3l3ma4=m4d4mm=m1+m2+m3+m4(3.18)Derentiatingwithrespecttotimegives_r=2
6
4_rx_ry3
7
5=2
6
4Jx(q)Jy(q)3
7
5D_q(3.19)whereJx(q)andJy(q)areJacobianmatrices.Inadditiontothecontrolofthecenterofmasspositionr,wewishtocontroltheangleoftherstlink,1.Tothisendwedenethedesiredequilibriumpointofthesystemasfollows:(rx;ry;1;_rx;_ry;_1)=(0;yd;d;0;0;0)(3.20)3.2.1ContactPhase
Thecontinuouscontrollerusedduringthecontactphaseisdenedonthepositionofthecenterofmass,r,theangleofthefoot,andtheangularmomentumaboutthefoot.Thecenterofmasspositionsarechosensinceduringcontactphase,therelativecenterofmasspositionisthetotalcenterofmassposition.Theangularmomentumischosenasitwill41determinethetotalangularmomentumduringtheightphase,andtheangleofthefootischosentohelpensurethatonlythepointOisincontactwiththeground.DuringthecontactphasethesystemdynamicsaredescribedbyEq.(3.12).Totransformthisdynamicstonormalform[20],weusethetransformationsinEqs.(3.15),(3.19),and= 1(q)1= 2(q;_q)=1mgCDM(q)DTD_q(3.21)wherethematrixCisgivenbyC=1000(3.22)Inaddition,i,i2[2;7],aredenedas2=rx3=ryyd4=1d5=_rx6=_ry7=_1(3.23)Itcanbeshown_=@ 1(q)@q_q=f(;)_1=@ 2(q;_q)@q_q+@ 2(q;_q)_qq=CDmghATN(q;_q)+_M(q)DTD_qi=2(3.24)42andh_2;_3;_4iT=[5;6;7]T=J(q)D_q(3.25)h_5;_6;_7iT=J(q)Dq+_J(q)D_q(3.26)whereJ(q)isgivenbyJ(q)=2
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4Jx(q)Jy(q)C3
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5(3.27)Theexpression_1=2followsintuitivelyfrommg1beingtheangularmomentumofthehopperaboutitsfoot,andmg2beingtheresultingtorqueaboutthefootduetogravity.ThedynamicsinEq.(3.12)aredescribedbyEqs.(3.24),and(3.26)intheregionwherethetransformationsinEq.(3.21)and(3.23)aredeomorphic.SubstitutingEq.(3.12)into(3.26)gives2
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4_5_6_73
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5=J(q)(DMDT)1D[ATN(q;_q)]+_J(q)D_q(3.28)DenethevectoroftorquesTtobeT=[J(q)(DM(q)DT)1DA]#[vg+J(q)(DM(q)DT)1DN(q;_q)_J(q)D_q](3.29)where()#istherightpseudo-inverseof().Topreventasingularitycondition,wemake43thefollowingassumption:
Assumption4:TheJacobeanmatrixJ(q)isfullrowrankoverthedurationthatthetorqueTisapplied.Substituting(3.29)into(3.28)resultsinh_5;_6;_7iT=vg(3.30)Wechoosevgtothebegivenbyvg=2
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4K11K22K55K33K66K44K773
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5(3.31)withdenedas=8
>
<
>
:1606>0(3.32)andthegainsKichosensuchthatKi>08iandK6<2pK3(3.33)ThisensuresasymptoticconvergenceoftrajectoriestothemanifoldM=f2R7j(1;2;4;5;7)=(0;0;0;0;0)g(3.34)44OnM,thetrajectoriesofthesystemobey2
6
4_3_63
7
5=2
6
401K3K63
7
52
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4363
7
5(3.35)whichrepresentsa"massspringdamper"whosedampingispositiveornegativebasedonthevalueof.Bymodulatingwewillincreaseordecreasetheenergyofthesystemandachieveapexheightcontrol.
3.2.2FlightPhase
Duringtheightphase,thepositionofthefootrelativetothecenterofmassiscontrolledinordertoachieveadesiredfootplacementatthetimeoftouchdown.Inthisphasethesystemhastheadditionaldynamicsofx,_x,y,and_y,whichwerenotpresentinthecontactphase.Wedenethestatesas.d=x+rx;h=y+ry_d=_x+_rx;_h=_y+_ry(3.36)wheredandhrepresentthehorizontalandverticalcomponentofthecenterofmassintheinertialframeofreference.UsingEq.(3.2)wecanshowd=0;h=g(3.37)45Additionally,theangularmomentumofthesystemaboutitscenterofmassisconserved.TheangularmomentumaboutthecenterofmassisgivenbyHc=4Xi=12
4rimi_ri+IiiXj=1_j3
5(3.38)SubstitutingEqs.(3.21)and(3.23)into(3.38)givesHc=m[g1+(3+yd)526](3.39)sovlingEq.(3.39)for1gives1=1mg[m26m(3+yd)5Hc](3.40)Thisshows1isrelatedto2,3,5,and6viaanalgebraicrelationship.Thedynamicsof2,3,and4arethesameasinEq.(3.25),whereasthedynamicsof5,6,and7canbeobtainedbysubstitutingEq.(3.2)intoEq.(3.26):2
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5=J(q)DM1[ATN(q;_q)]+_J(q)D_q(3.41)DeningthevectoroftorquesTtobeT=[J(q)DM1(q)A]#[vf+J(q)DM1(q)N(q;_q)_J(q)D_q](3.42)46resultsinh_5;_6;_7iT=vf(3.43)Wechoosevfasfollows:vf=2
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4K22K55K33K66K44K773
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7
7
5(3.44)whereKi>08i2[2;7].Thisguaranteesthatthevariablesii2[2;7]willasymptoticallyconvergetozero.Additionally,ifHc=0,1willasymptoticallyconvergetozero.SincetheangularmomentHccannotbecontrolledintheightphase,ourobjectiveistobringittozeroduringthecontactphase.
3.2.3HybridDynamicsofClosed-LoopSystem
Thehybriddynamicsofthesystemoveronehopissummarizedasfollows:ThesystembeginswiththeFlightPhase.Thedynamicsduringightphaseisdescribedby2
6
6
6
6
6
4dh_3
7
7
7
7
7
5=2
6
6
6
6
6
40gf(;)3
7
7
7
7
7
5;2
6
6
6
6
6
4_2_3_43
7
7
7
7
7
5=2
6
6
6
6
6
45673
7
7
7
7
7
5(3.45)2
6
6
6
6
6
4_5_6_73
7
7
7
7
7
5=2
6
6
6
6
6
4K22K55K3(3yd)K66K4(4d)K773
7
7
7
7
7
5(3.46)andthenon-holonmicconstraintgivenbyEq.(3.40)for1.FollowingtheightphasethesystemundergoesImpact.Thehoppermakescontactwith47thegroundwheny=0orh3yd=0.TheimpulseduetoimpactisgivenbyEq.(3.11).FollowingtheImpact,thesystemisinContactPhase.FromEqs.(3.24),(3.26),(3.30)and(3.31),andx=y=0thedynamicsduringcontactphasearegivenby2
6
6
6
6
6
6
6
6
6
6
6
6
6
4__1_2_3_43
7
7
7
7
7
7
7
7
7
7
7
7
7
5=2
6
6
6
6
6
6
6
6
6
6
6
6
6
4f(;)25673
7
7
7
7
7
7
7
7
7
7
7
7
7
5;2
6
6
6
6
6
4_5_6_73
7
7
7
7
7
5=2
6
6
6
6
6
4K11K22K55K3(3yd)K66K4(4d)K773
7
7
7
7
7
5(3.47)where=8
>
<
>
:1606>0(3.48)andthestateshand_daregivenbyh3+yd;_d5(3.49)ThecontactphaseendsattheinstantFyinEq.(3.5)isequalto0,thatisFy=m_6+mg=0)_6=g(3.50)483.3PeriodicBehavior
Hoppingisdescribedbyconsecutivesequencesofightphase,impact,andcontactphase.Todescribeasinglehop,wedenethestate˜˜2;=f(;_)j_6(;_)+g=0g(3.51)whichdenethecongurationofthehopperatthetimeoftransitionfromthecontacttotheightphase.Theconguration˜doesnotincludedsincetheobjectiveofthispaperistocontrolonlytheheight.Therstreturnmapbetweenthekthhopandthe(k+1)thhopisdenedas˜(k+1)=P(˜(k))(3.52)whereP(˜(k))isthesolutionoftheclosed-loophybriddynamics.3.3.1PeriodOneOrbits
Foraperiod-oneorbit[9]wehave˜(k)P(˜(k))=0(3.53)Let˜tobeanyvalueof˜thatsatisesEq.(3.53).Notethat˜liesina7embeddedmanifoldofR8.LetVtobethematrixoflinearlyindependentunitvectorsV=[v1;v2;:::v7]vi2R8;jvij=1;i2[1;7](3.54)49wherespan(V)=(3.55)LinearizationofEq.(3.52)abouttheperiodicpoint˜gives˜(k+1)ˇP(˜)+7Xi=1@P(˜)@vivTij˜=˜(˜(k)˜)(3.56)where@P(˜)=@viisgivenby@P(˜)@vi=limh!0P(˜+hvi)P(˜)h(3.57)ItfollowsthatEq.(3.56)isasymptoticallystableˆ 7Xi=1@P(˜)@vivTij˜=˜!<1(3.58)whereˆ()isthespectralradius.3.3.2ChaosControl
Aperiodicorbitdenedby˜maynotbestableforagivensetofsystemparameters.However,wenotethatP(˜)isdependentonthevariable.BydeningtobethevalueofthatsatisesEq.(3.53)for˜=˜,wewillvarythevalueoftoensureasymptoticstabilityof˜.50Todesigntheinput(k),rstdenethevectorvasv=2
6
4ab3
7
5jvj=1;a2R8;b2Rf0g(3.59)whereasatisesEq.(3.50).DeningtheerrorEandtheinputuasE(k)=˜(k)˜u(k)=(k)(3.60)wehavethelinearizedequationofEqn.(3.53):E(k+1)=AE(k)+Bu(k)(3.61)whereAandBaregivenbyA=7Xi=1@P(˜)@vivTi+@P(˜(k))@vaTB=@P(˜(k))@vb(3.62)andwherethedirectionalderivatives@P(˜)=@viand@P(˜)=@vareevaluatedat(˜;).Forasymptoticstability,u(k)isdesignedasu(k)=GE(k)(3.63)51whereGischosensuchthatˆ(A+BG)<1(3.64)3.4Simulations
Forthefour-linkhopper,themassesareassumedtobem1=2:5kg;m2=5kg;m3=10kg;m4=20kg(3.65)Thelengthofthelinksofthehopperareassumedtobel1=0:1m;l2=l3=l4=0:3m(3.66)andthedistancetothecenterofmassofeachlink-seeFig.3.1areassumedtobed1=0:05m;d2=d3=d4=0:15m(3.67)ThemomentofinertiaofeachlinkiscomputedasIi=112mil2i8i2[1;4](3.68)ThegainsusedforthecontinuouscontrolareK1=12000K2=8000K3=300K4=300K5=120K6=10K7=8(3.69)52andthesetpointsydanddforthecontinuouscontrolareyd=0:4967md=ˇ20:1(3.70)Thevalueofydis0:14mbelowthemaximumheightofthecenterofmassrelativetothefoot.Wechoosethedesiredapexheightofthecenterofmasstobe0:65metersandcomputetheperiodicpoint,(˜;)tobegivenby˜
1=1:579˜
2=0:407˜
3=1:135˜
4=1:178˜
5=6:696˜
6=11:659˜
7=10:360˜
8=8:960=0:607(3.71)withallvaluesgiveninradandrad=swhereappropriate.Theperiodicpointiscomputednumericallyviaaregressionalgorithm.Thealgorithmusesaninitialguessfortheperiodicvalues(˜;).Thealgorithmrestrictstheguessestothemanifoldwhichsatisfythedesiredapexheight.Itthenusesaregressiveminimizationalgorithmtominimizethederencebetweentheguessedlift-ovaluesandthevaluesfortheresultingvaluesfromlettingthesystemprogressforasinglehop.ThestabilizingcontrolgainsGfortheperiodicpointinEq.(5.73)arebysolvingthediscreteLQRproblem:G1=0:023G2=0:025G3=0:005G4=0:011G5=0:051G6=0:030G7=0:005G8=0:027(3.72)53Wechoosetheinitialcongurationofthesystemisassumedtobe(x(0);_x(0);y(0);_y(0))=(0:00;0:00;0:03;0:00)(0)=[1:65;0:50;1:07;0:94]T_(0)=[0:00;0:00;0:00;0:00](3.73)wheretheunitsareinmeters,rad,andrad/sec.Theseinitialconditionswerechosensuchthatthecenterofmassofthehopperliesverticallyabovethepointofsupport.Theinitialvalueofthediscretecontrolinputischosentobeu(0)=0)(0)=(3.74)Figure3.2showstheheightofthecenterofmassasafunctionoftime.Itcanbeseenthatthecenterofmassconvergestothedesiredheightinapproximately4hops.Figure3.3displaystheinputstorques˝1,˝2,and˝3.Thesharppeaksindicatethediscontinuousjumpsinthetorquesimmediatelyfollowingimpact.Fromthesimulationsweseetheecacyofthecontrolmethodasappliedtoafour-linkedhoppingrobot.Wenotethatallresultsofthischapterareforrobotsthatareinteractingwitharigidground,namelygroundsurfaceswhoserigidityissucientlyhighastoconsiderthemtonotdeformundertheappliedforcesoftherobot.However,itbehoovesustoinvestigatethemotionofhoppingrobotswhenthetheircontactisanelasticsurfaceinsteadofarigidone,namelyasurfacewhichdoesdeformunderthemassoftherobot.Tothisend,wewillinvestigatebothatwomasshoppingrobot,andathree-linkhoppingrobotwheninteractingwithandelasticground.5400.511.522.533.544.550.40.450.50.550.60.65h(m)time(sec)Figure3.2:Centerofmassheightabovethegroundforthefour-linkhopper5500.511.522.533.544.55 10010203000.511.522.533.544.55 60 40 200204000.511.522.533.544.55 50 20104070˝1(Nm)˝2(Nm)˝3(Nm)time(sec)Figure3.3:Inputtorquesforthefour-linkhopper56Chapter4
Two-MassHoppingRobotonan
ElasticFoundation
Unlikewiththerigidgroundinteraction,andelasticgroundinteractionassumesthatthegrounddoesdeformunderthemassoftherobot.Forthispurposewemodeltheinteractionoftheelasticgroundasaninteractionwithamasslessspring.Similarlywiththerigidfoundation,webeginwiththeanalysisofasimpedtwo-masshoppingrobotbeforemovingontotheanalysisofathree-linkhoppingrobot.AswithChapter2,Westarttheanalysisofalinkedhoppingrobotonanelasticfoun-dationwiththeanalysisandcontrolofasimped2massrobothoppingonanelasticfoundation.Webeginwithadiscussionofthedynamicsofthetwomasssystemhoppingonanelasticfoundationandthediscussthenecessityofchangingthecontrolalgorithmfromtherigidfoundationcase.Wethendiscussahybridcontrolalgorithmwhichusesbackstepping[19]tostabilizethedynamicsoftherobotintheightandcontactphases.TheperiodicnatureofthehybriddynamicsystemisthenanalyzedusingaPoincaremap[9]andtheOGY1methodofchaoscontrol[29]isusedtoadjustaparameterdiscretelyandconvergetothedesiredapexheight.TheresultsinthechapterfollowtheresultspresentedbyMathisandMukherjee[23].1AmethodintroducedbyOtt,GrebogiandYorkeforachievingstabilizationofaperiodicorbit[29].574.1Dynamics
4.1.1SystemDescription
Considerthetwo-massrobotinFig.4.1(a),whichishoppingonamasslesselasticfoundationofstnessKext.Itiscomprisedofanuppermassm1andalowermassm2.Theupperandlowermassesareconnectedbyaprismaticjoint.TheforceappliedbytheactuatorintheprismaticjointonthetwomassesisdenotedbyF;itisassumedthatthisforcecanbeactivelycontrolled.Thepositionsofthecenter-of-massofm1andm2fromthedatumaredenotedbyx1andx2.Thepositionofthecenter-of-massofm1relativetothatofm2isdenotedbyyandtheheightofthecenter-of-massofm2fromthebaseofm2isdenotedby`.TheforceofinteractionbetweenthelowermassandtheelasticfoundationisdenotedbyFext.4.1.2FlightPhase
Duringtheightphase,thefollowingconditionshold:x2>`Fext=0(4.1)Theequationsofmotionoftherobotareasfollows:x1=g+Fm1x2=gFm2(4.2)Usingtherelativedisplacementy,(x1x2)toreplacex1,analternateformoftheequations58ofmotionarey=mtm1m2Fx2=gFm2(4.3)wherethetotalmassofthesystemismt,(m1+m2)(4.4)4.1.3ContactPhase
Forthecontactphase,wemakethefollowingassumptions:
Assumption1:Theelasticfoundationhasnomassandbehaveslikeaspring.Assumption2:Theforceexertedbytheelasticfoundationonmassm2isnon-negative,i.e.,(a)(b)undeformed
configuration
of springdatumx1x2m1Fextm2Fg`Kextym1gm2gFigure4.1:(a)Two-massrobothoppingonanelasticfoundation(b)free-bodydiagramsofthetwomassesataninstantwhenthelowermassisincontactwiththeelasticfoundation.59Fext0.Fromthefree-bodydiagrams,theequationsofmotionoftherobotcanbeobtainedasfollows:x1=g+Fm1x2=g+(FextF)m2(4.5)Inyandx2coordinates,theseequationscanbewrittenasy=mtm1m2FFextm2x2=g+(FextF)m2(4.6)In(4.5)and(4.6),FextisgivenbytheexpressionFext=Kext(x2`)(4.7)4.1.4ApexHeight
Ifzdenotestheheightofthecenter-of-massofthehoppingrobot,wehavez=m1x1+m2x2m1+m2=x2+mfy;mf,(m1=mt)(4.8)Foreachightphase,theapexheightisdenedasthemaximumvalueofz,andisdenotedbyh.4.2MotivationforDierentControlStrategy
FromChapter2.2wehavethedynamicsoftheclosedloopsystemaregivenby:60x=K[mfyrd]+Dmf_yg+m2Fexty=Ky1mfrdD_y(4.9)Theconvergenceofthealgorithmreliesontheuseofthegradientofthetrajectorieswithrespecttogainaroundtheperiodicpoint.Thisworksinthecaseoftherigidfoundationbecauseofthechangeinmomentumduetotheimpulseduringgroundimpactensuresthattheperiodicorbitdoesnotsatisfy_y0.However,theexternalforcecreatesaconservativesystemintheelasticfoundation.Thisproducestheperiodicorbitx=g+m2Fexty_y0(4.10)Wenotthatthisperiodictrajectorysetstheapexheightbasedontheinitialconditionandisindependentofthevalue.Furthermore,thisFromthisweseethatthegradientoftheperiodictrajectorywithrespecttois0since_y0duringtheseperiodicpoints.Thismeansthatthepreviousmethodwillnotworkforthesepointssincethecontrolhasnoeectatanyperiodicorbit.Therefore,wemustutilizeanewclosedloopmethodwhichutilizesparametervariationswhichhavenon-zerogradientattheperiodicorbit.614.3HybridControlStrategy
4.3.1ControlProblemDenition
Werstdenertobetheheightofthecenter-of-massoftherobotrelativetothatofthelowermassm2.Using(4.8),itcanbeshownr,(zx2)=mfy(4.11)Next,wedeneease=(rrd)(4.12)whererd>0issomedesiredvalueofr.From(4.11)and(4.12)itcanbeveredthate0!_e0!_y0,whichimpliesnorelativemotionbetweenthetwomasses.Ifthereisnorelativemotionofthemasses,thetotalenergyofthesystemisthesumofthekineticandpotentialenergiesofthecenter-of-massplusthepotentialenergystoredintheelasticfoundation.Thepotentialenergyofthecenter-of-massisdenedrelativetothedatumz=zd,wherezd,zj(x2=`;r=rd)=(rd+`)Intheabsenceofrelativemotionofthemasses,thetotalenergycanbewrittenasE=mt12_z2+g(zzd)+12Kext(zzd)2(4.13)62where=8
>
<
>
:0:x2`FlightPhase1:x2<`ContactPhase(4.14)Thesecondtermontheright-handsideof(4.13)representsthepotentialenergystoredinthespringwhenthetwomassesareintheirnominalpositionrelativetooneanother.Fortherobottoreachitsdesiredapexheighthd,thetotalenergyshouldbeequaltoEEdes=mtg(hdzd)(4.15)inadditiontoe0.Thedesiredequilibriumcongurationisthereforegivenby(EEdes;e;_e)=(0;0;0)(4.16)4.3.2ContinuousControllerDesignforStabilizationinFlightandContactPhases4.3.3FeedbackLinearization
Fortheconvenienceofcontroldesign,werewritethedynamicsofthehybridsystemintermsofvariableszande.Using(4.4),(4.6),(4.7),(4.8),(4.11)and(4.12),thehybriddynamicsoftherobotcanbedescribedbytherelationsz=g1mtKext(x2`)(4.17)e=1m2[FmfFext](4.18)63whereisdenedin(4.14).ThefollowingchoiceofthecontrolinputFF=mfFext+m2v(4.19a)=mfKext(x2`)+m2v(4.19b)resultsinthehybriddynamicsz=g1mtKext(x2`)(4.20)e=v(4.21)wherevisthenewcontrolinput.NotethatthecontrolinputFcanbechosenaccordingto(4.19a)or(4.19b)dependingonwhethertheexternalforceFextorthepositionofthelowermassx2isavailableformeasurement.4.3.4Backstepping
Withtheobjectiveofstabilizingtheequilibriumin(4.16),werstdenetheLyapunovfunctioncandidateV1=12ke(EEdes)2(4.22)wherekeisapositiveconstant.ItshouldbenotedthatV1isafunctionof(sinceEisafunctionof)butitiscontinuouslyderentiableinboththeightphaseandcontactphase.TheLyapunovfunctioncandidatesintroducedbelowbeusedforouranalysisofstabilityinbothphasesandthereforewetreatasconstantanddonotmakeanydistinctionbetweenthetwophasesinourderivation.Using(4.12),(4.20),(4.13),and(4.22),_V1canbecomputed64as_V1=ke(EEdes)_E=ke(EEdes)_z[mt(z+g)+Kext(zzd)]=ke(EEdes)Kext_ze(4.23)Bychoosinge=fke(EEdes)_zg,˚1(4.24)wecanmake_V1negativesemi-denite;therefore,integratorbacksteppingisintroducedbydeningthenewvariableq1=e+ke(EEdes)_z=(e˚1)(4.25)andthecompositeLyapunovfunctionV2=V1+12q21=12ke(EEdes)2+12q21(4.26)wherewherek1isapositiveconstant.DerentiatingV2andsubstituting(4.23)and(4.25),weget_V2=ke(EEdes)Kext_ze+q1_q1=ke(EEdes)Kext_z[q1ke(EEdes)_z]+q1_q1=2k2eKext(EEdes)2_z2+q1[_q1+ke(EEdes)Kext_z](4.27)65Bychoosingk1>0and_q1=fkeKext(EEdes)_zk1q1g,˚2(4.28)wecanmake_V2negativesemi-denite.Weintroduceintegratorbacksteppingagainbydeningthenewvariableq2=(_q1˚2)(4.29)andthecompositeLyapunovfunctionV3=V2+12q22=12ke(EEdes)2+12q21+12q22(4.30)DerentiatingV3andsubstituting(4.27)and(4.29),weget_V3=2k2eKext(EEdes)2_z2+q1[_q1+keKext(EEdes)_z]+q2_q2=2k2eKext(EEdes)2_z2+q1[_q1˚2k1q1]+(_q1˚2)(q1_˚2)=2k2eKext(EEdes)2_z2k1q21+q2hq1_˚2+q1i(4.31)66Ourchoiceofq1=_˚2q1k2q2;k2>0(4.32)resultsinanegativesemi-denite_V3andyieldsthecontrollerv=˚1+_˚2q1k2q2(4.33)Theaboveequationwasobtainedfrom(4.32)bysubstituting(4.21)and(4.25).Fromthedenitionof˚1in(4.24),itisclearthat˚1willinvolvethethirdderivativeofz.Thisisnotaproblemsincethethirdderivativeofzcanbecomputedeasilyfrom(4.20)as...z=1mtKext_x2(4.34)Thecompletecontrollawisgivenby(4.33)and(4.19a)or(4.19b).
4.3.5StabilityAnalysis
Using(4.11),(4.12),(4.13),(4.17),(4.25)and(4.29)itcanbeshownthat(EEdes;e;_e)=(0;0;0),(EEdes;q1;q2)=(0;0;0)Therefore,V3in(4.30)isacandidateLyapunovfunctionforinvestigatingthestabilityoftheequilibriumin(4.16).Intheightphase,=0.Forthecontrollawgivenby(4.33),thederivativeofthe67Lyapunovfunctionin(4.31)canbeshowntobe_V3=k1q21k2q220(4.35)Therefore,(EEdes;q1;q2)=(0;0;0)isstable.Inthecontactphase,=1.Forthecontrollawgivenby(4.33),thederivativeoftheLyapunovfunctionin(4.31)is_V3=keKext(EEdes)2_z2k1q21k2q220(4.36)Therefore,(EEdes;q1;q2)=(0;0;0)isstable.Remark1.Thestabilityof(EEdes;q1;q2)=(0;0;0)intheightandcontactphasesdonotguaranteeitsstabilityforthehybriddynamics.
4.3.6DiscreteControllerforStabilizationofHybridDynamicsToinvestigatethestabilityofthehybriddynamicsystem,weuseaPoincaremapwiththePoincaresectiondenedbytheinstantwhenthesystemtransitionsfromthecontactphasetotheightphase.ThePoincaresectionisdenedasZ:=nX2R3jx2=`;_x2>0o)Z:=nX2R3jz=e+zd;_z>_eo(4.37)whereXisdenedasX=_ze_eT68TousethesamesetofvariablesusedintheLyapunovanalysis,namely,(EEdes),q1andq2,wedenethePoincaresectionusingthecoordinates,whereisdenedbythecoordinatetransformationH():R3)R3,asfollows=(EEdes)q1q2T=H(X)(4.38)ItcanbeshownthatthemapH()isalocalhomeomorphismandthereforelocallytopo-logicallyconjugate[9];thisimpliesthatthestabilityofthePoincaremapsinandXcoordinatesareequivalent.ThePoincaremapP()andthesequenceofpointsk2H(Z)satisfyk+1=P(k);P():H(Z)7!H(Z)(4.39)withperiodicpointdenedas=P()(4.40)Fortheelasticfoundation,theperiodicpointwhichachievesthedesiredapexheightisgivenby=000T(4.41)Wedenetheerrorstatekask=(k)=kBylinearizingthePoincaremapabout,wehavetheapproximatediscretedynamicsgivenbyk+1=AkA,dP()d


=(4.42)69Theperiodicpointwillbeasymptoticallystableifandonlyifˆ(A)<1(4.43)whereˆ(A)isthespectralradiusofA.Sincetheconditionin(4.43)maynotbesatised,wedesignadiscretecontrollertostabilizetheclosed-loopsystem;thediscretecontrollerisdiscussednext.Todesignthediscretecontroller,weredeneasfollows=+100Tu(4.44),(EEd)q1q2T;u,(EdEdes)whereEdisdesiredlevelofenergyforagivenhop.ThenewPoincaremapP(;u)andthesequenceofpointsk2H(Z)satisfyk+1=P(k;uk);P(;u):H(Z)R7!H(Z)(4.45)withperiodicpointdenedas=P(;u)(4.46)Fortheelasticfoundation,theperiodicpointwhichachievesthedesiredapexheightisgivenby=000T;u=0(4.47)70Wedenetheerrorstatekask=(k)=kBylinearizingthePoincaremapabout(;u),wehavetheapproximatediscretedynamicsgivenbyk+1=Ak+Buk(4.48)A,dP(;u)d


=;u=u;B,dP(;u)du


=;u=uForourchoiceofinputuk=Kk(4.49)theclosed-loopsystemdynamicstakestheformk+1=(A+BK)kIffA;Bgiscontrollable,wecanchooseKsuchthatˆ(A+BK)<1(4.50)andthehybriddynamicalsystemisasymptoticallystable.
Remark2.Iftheconditionin(4.43)isnotsatisedandthediscretecontrollerin(4.49)isimplemented,thecontinuouscontrollerwillhavetomoded.Inparticular,thexeddesired71valueoftheenergyEdeswillhavetobereplacedbythedesiredvalueofenergyforeachhopEdtoaccountforthechangeinthePoincaremapfromP()toP(;u).4.4SimulationResults
Themassandlengthparametersoftherobotandthestnessoftheelasticfoundationareprovidedbelow:m1=2:668kg;m2=0:808kg`=0:059m;Kext=11560N=mThevalueofrdandthedesiredapexheightwerechosenasrd=0:0979m;hd=0:2m(4.51)Theparametersofthecontinuouscontrollerwerechosenaske=0:001;k1=600;k2=18(4.52)Thevalueofkewaschosentobemuchsmallerthanthoseofk1andk2toreducethedominanceoftermsinvolvingKext,whichisO(104).ThematrixA,denedin(4.42),wasfoundtohaveeigenvalues:0:0003,0:0017and0:3678;theconditionin(4.43)wasthereforesatised.AdiscretecontrollerwasneverthelessdesignedandthegainmatrixKin720.01.02.00.100.200.25
0.000.059time(s)x1;x2;z(m)x1zx2k=1k=2k=3k=4k=5k=6k=7Figure4.2:Simulationresults:Plotoftheheightoftheuppermassx1,thelowermassx2,andthecenter-of-massz,asafunctionoftime.(4.49)waschosenasfollowsK=0:10000:09000:0189(4.53)Thisresultsintheclosed-loopsystemeigenvalues:0:0003,0:0641and0:2000.Thegainsin(4.53)werechosentomovetheeigenvaluewiththelargestmagnitudefrom-0.3678to0.2000.Theinitialconditionswereassumedtobex2(0)=0:056m;y(0)=0:088m_x2(0)=0:0m=s;_y(0)=0:0m=s(4.54)andtheresultsareshowninFigs.4.2and4.3.Thedisplacementsoftheuppermass,lower73mass,andcenter-of-massareplottedinFig.4.2.Theintervalsoftimeduringwhichx2`=0:059mindicatethecontactphases.Thevalueofx2(0)=0:056mindicatesthatthespringisinitiallycompressedduetotheweightoftherobot.ItisclearfromFig.4.2thattheapexheightoftherobotconvergestoitsdesiredvalueintwohops.ThediscretestatesareplottedinFig.4.3;theycorrespondtothevaluesof(EEdes),q1andq2atthebeginningofthek-thhop,k=1;2;;7,andtheyconvergetozeropriortothethirdhop.Thediscreteinstantsoftimecorrespondingtok=1;2;;7inFig.4.3aremarkedby\"inFig.4.2.Fromthesimulationsweseetheecacyofthemodedcontrolmethodfortotwomasshoppingrobotonanelasticfoundation.Wenextinvestigatetheapplicationofthecontrolmethodonelasticgroundtoamulti-linksystem.Themulti-linksystemischosenasamoreaccuraterepresentationofahumanoidsystem.01234567-0.50.0
0.00.5-4.0-5.00.05.0X 10-4k(EEdes)q1q2Figure4.3:Simulationresults:Errorsinthediscretestates(EEdes),q1andq2atthebeginningofthek-thhop,k=1;2;;7.74Chapter5
Three-linkHoppingRobotonan
ElasticFoundation
Fromthesimulationresultsofthetwo-masshopperinChapter4weseethatthecontrolmethodonelasticgroundiscapableofachievingadesiredapexheightforthesimpedtwo-masssystem.Wenowgeneralizethecontrolmethodologytoamulti-doflinkedrobotsystem.Tothisendweconsidertheapexheightcontrolofathree-linkhoppingrobot.Thechoiceofathree-linkedhoppingrobotissuchthatitprovidesthesmallestdimen-sionalsystemtoachievethefullcontroldimensions,asopposedtothefour-link,whichrequiredadditionalobjectivesfortheextradigreesoffreedom.ThecontrolisdesignedfollowingasimilarstructureinChapter3.Additionalconsider-ationisgivenduringboththecontinuousanddiscretephasestoaccountfortheadditionaldegrees-of-freedominthesystem.
5.1Dynamics
5.1.1SystemDescription
Considerthethree-linkedrobotinFig.5.1(a),whichishoppingonamasslesselasticfoun-dationofstnessKext.Itiscomprisedofthreelinksoflengthl1,whicheachhavemassmi75fori21;2;3.Acontrollabletorqueisappliedatthejointscomprisingtheintersectionofeachlink.Thedistanceofthecenter-of-massofeachlinkisfromtheendofthepreviouslinkandisgivenbydi.TheCartesianlocationofthebaseoftherstlink,pointo,relativetothegrounddatumisgivenbyxandy.Wedenotetherelativeangleoftheithlinkrelativetothepreviouslinkbyi-seeFig.5.1.Thestatesaredenedasq=xy123T(5.1)TheequationsofmotionofthehopperaregivenbyM(q)q+N(q;_q)=AT+Fext(5.2)whereM(q)isthemassmatrix;andN(q;_q)isthevectorofCoriolis,centrifugal,andgravi-tationalforces;andA2R63isthematrixgivebelowA=2
6
400010
000013
7
5T;(5.3)TisthevectorofinputtorquesT=˝1˝2T;(5.4)andFextittheforceappliedbythegroundontherobotgivenbyFext=FxFy000T(5.5)76InEq.(5.5),FxandFydenotethexandycomponentsoftheforceappliedtotherobotbythegroundatpointO.Thedynamicsofthehoppermaybeseparatedintothreephases:theightphaseforwhichy>0;theimpactphase,whichoccursattheinstanty=0;andthecontactphase,whichoccursforthedurationinwhichthefootremainsincontactwiththeground,y0.Thephasesmirrorthoseofthetwo-masshopper.xy123XYOgKextithlinkdiliFigure5.1:Three-linkhoppingrobotonanelasticfoundation775.1.2FlightPhase
Duringtheightphase,thefollowingconditionshold:y>0Fx=Fy=0(5.6)TheequationsofmotionduringtheightphasearegivenbyM(q)q+N(q;_q)=AT(5.7)wherethematrixM(q)2R55isthemassmatrixandN(q;_q)2R5isthenon-lineargravitationalandCorioliseects.ThematrixArepresentsthemappingoftorquestotherelativestates,andthevectoroftorques,T,itisgivenbyT=˝1;˝2T(5.8)5.1.3Impact
Atthetimeofimpactweassume:
Assumption1:TheappliedvectoroftorquesTarenotimpulsive.Assumption1doesnotimplythatthevectoroftorquesTcannotbediscontinuous;itsimplyimpliesthatthetorquescannotproducediscretejumpsinthestates.Assumption2:Thehopper'sfootcomesincontactwiththegroundonlyatpointO.Assumption2canbeenforcedthroughproperchoiceofcontrolgains.Assumption3:Attheinstantthefootcontactstheground(y=0),thegroundappliesanimpulsiveforcethatresultsin_x=0.78Assumption3simplyimpliesinelasticimpactintheXdirection.Itcanbenotedthatassumption3isnotentirelyphysical.Thisisbecauseinaphysicalsystem,thelateralforceFxisdependentonthefrictionforce;and,thusitsmodelormodelsarechosentorepresentsaidfriction.Toalleviatetheneedofadditionalanalysisofthemodel,theforceisassumedtoholdthepointOinthexdirectionfortheentiredurationofthecontactphase.TakingtheintegralovertheinnitesimalperiodoftimeinwhichtheimpactoccurswehaveZt0+t0qdt=Zt0+t0M1(q)[ATN(q;_q)+Fext]dt(5.9)_q+=_q+M1(q)Fext(5.10)where_q+and_qaretherightandleftlimitsintimeof_q.Thisfollowsfromourearlierwork[10].Partitioningqaccordingtoq=[xjy]T(5.11)whereisgivenby=[123]T(5.12)resultsinthecorrespondingpartitionofM1(q)givenbyM1(q)=2
6
6
4(M1)11(M1)12(M1)21(M1)223
7
7
5(5.13)79SolvingEq.(5.10)resultsinthefollowingchangeinthestatevariables:q+=q_x+=02
6
4_y+_+3
7
5=2
6
4_y_3
7
5(M1)21[(M1)11]1_x(5.14)5.1.4ContactPhase
Forthecontactphase,wemakethefollowingassumptions:
Assumption1:Theelasticfoundationhasnomassandbehaveslikeaspring.Assumption2:OnlythepointOcontactstheelasticsurface.Assumption3:TheverticalforceexertedbytheelasticfoundationonthepointOisnon-negative,i.e.,Fy0.Assumption4:ThehorizontalforceofthegroundonthepointOissuchthatitpreventsmotionintheXdirectionofpointO.Duringthecontactphase,wemaywritethedynamicsofthehopperwhicharegivenbyDM(q)DTDq+DN(q;_q)=DAT+DFext(5.15)whereDisthematrixD=2
6
6
6
6
6
6
6
6
6
401000
00100
00010
000013
7
7
7
7
7
7
7
7
7
5(5.16)80WiththeexternalforceFygivenbyFy=Kexty(5.17)FromEqn.(5.17),thecontactphasetransitionstotheightphasewheny=0_y>0(5.18)Equation(5.15)duringcontactisidenticaltoEqn.(5.2),butispresentedasanalternativetohavingtosolveforFx,ortobeusedforcontrolifFxcannotbemeasured.5.1.5ApexHeight
Similartothetwo-masscase,wewishtocontrolthepositionofthecenterofmassofthehopperrelativetothatbaseoftherobot,pointO.Tocontrolthecenterofmass,wedenertobethevectorfromthebaseofthefoottothecenterofmassofthehopper.Ifrxandrydenotethehorizontalandverticalcomponentsofr,wecanwriter=2
6
4rxry3
7
5=2
6
4fx(q)fy(q)3
7
5(5.19)wherefx(q)andfy(q)aregivenbyfx(q)=a1cos(1)+a2cos(1+2)+a3cos(1+2+3)(5.20)81fy(q)=a1sin(1)+a2sin(1+2)+a3sin(1+2+3)(5.21)inEq.(5.20)and(5.21),theconstantshavetheexpressionsa1=m1d1+(m2+m3)l1ma2=m2d2+m3l2ma3=m3d3mm=m1+m2+m3(5.22)Ifzdenotestheheightofthecenter-of-massofthehoppingrobot,wehavez=y+ry(5.23)Foreachightphase,theapexheightisdenedasthemaximumvalueofz,andisdenotedbyh.Derentiatingrwithrespecttotimegives_r=2
6
4_rx_ry3
7
5=2
6
4Jx(q)Jy(q)3
7
5_q(5.24)whereJx(q)andJy(q)areJacobianmatrices.FromtheequationsofmotioninEqn.(5.2)wendthatdynamicsofzmaybegivenbymz=mg+Fy(5.25)825.2HybridControlStrategy
5.2.1FeedbackLinearization
Thecontrolofthethree-link,hoppingrobotisdenedbasedontherelativecenterofmass,givenbyrinEqn.(5.19).Todothis,wedenetheerrorstatesetobegivenbye=2
6
4exey3
7
5=rrd=2
6
4rxrd;xryrd;y3
7
5(5.26)whererd;x=0andrd;ysaconstantdenedbasedasthedesiredrestingheightofthecenterofmassrelativetothebaseo.DerentiatingEqn.(5.26)andsubstitutingEqn.(5.24)gives_e=J(q)_q=2
6
4Jx(q)Jy(q)3
7
5_q(5.27)Derentiatingasecondtimewendthedynamicsofetobegivenbye=J(q)q+_J(q)_q(5.28)SubstitutingEqn.(5.2)intoEqn.(5.28)givese=J(q)M1(q)[AT+FextN(q;_q)]+_J(q)_q(5.29)DeningthevectoroftorquesTtobegivenbyT=hJ(q)M1(q)i#nvg+J(q)M1(q)[N(q;_q)Fext]_J_qo(5.30)where()#istherightpseudo-inverseof(),resultsin83e=2
6
4exey3
7
5=vg=2
6
4vxvy3
7
5(5.31)5.2.2ControllerDesigninXDirectionForthecontrolofthesystemintheXweseethatwedesiretohaveex!0,whichwillpreventanylateralmotion.Wefurtherseethatwhenex0thattheimpulsedynamicsgiveninEqn.(5.14)resultinanimpulseofmagnitude0asthereisnolinearmomentumtocancelatthetimeofimpact.Designingvxtobegivenbyvx=KxexKd;x_ex(5.32)whereKxandKd;xarepositiveconstants,resultsinasymptoticconvergenceofe!0duringthecontinuousphases.
5.2.3ControlProblemDenitionforYDirectionItcanbeveredthat,aswiththetwo-masssystem,e0!_e0,whichimpliesnorelativemotionofthecenterofmass.Ifthereisnorelativemotionofthecenterofmass,thetotalenergyofthesystemisthesumofthekineticandpotentialenergiesofthecenter-of-massplusthepotentialenergystoredintheelasticfoundation.Thepotentialenergyofthecenter-of-massisdenedrelativetothedatumz=zd,wherezd,zj(rY=rd;y)=rd;y84Intheabsenceofrelativemotionofthecenterofmass,thetotalenergycanbewrittenasE=m12_z2+g(zzd)+12Kext(zzd)2(5.33)where=8
>
<
>
:0:y`FlightPhase1:y<`ContactPhase(5.34)Thesecondtermontheright-handsideof(5.33)representsthepotentialenergystoredinthespringwhenthetwomassesareintheirnominalpositionrelativetooneanother.Fortherobottoreachitsdesiredapexheighthd,thetotalenergyshouldbeequaltoEEdes=mg(hdzd)(5.35)inadditiontoe0.Thedesiredequilibriumcongurationisthereforegivenby(EEdes;e;_e)=(0;0;0)(5.36)5.2.4ContinuousControllerDesignforYDirection,BacksteppingForthecontrollerdesignintheYdirectionwewillassumethatex0.ThisisafairassumptionastheconvergenceofthedynamicsintheXdirectionareindependentofthecontrolintheYdirection.WiththeobjectiveofstabilizingtheequilibriuminEq.(5.36),85werstdenetheLyapunovfunctioncandidateV1=12ke(EEdes)2(5.37)wherekeisapositiveconstant.ItshouldbenotedthatV1isafunctionof(sinceEisafunctionof),butitiscontinuouslyderentiableinboththeightphaseandcontactphase.TheLyapunovfunctioncandidatesintroducedbelowcanbeusedforouranalysisofstabilityinbothphases;and,therefore,wetreatasconstantanddonotmakeanydistinctionbetweenthetwophasesinourderivation.UsingEqs.(5.26),(5.25),(5.33),and(5.37),_V1canbecomputedas_V1=ke(EEdes)_E=ke(EEdes)_z[m(z+g)+Kext(zzd)]=ke(EEdes)Kext_zey(5.38)Bychoosinge=fke(EEdes)_zg,˚1(5.39)wecanmake_V1negativesemi-denite;therefore,integratorbacksteppingisintroducedbydeningthenewvariable1=e+ke(EEdes)_z=(e˚1)(5.40)86andthecompositeLyapunovfunctionV2=V1+12q21=12ke(EEdes)2+1221(5.41)wherewherek1isapositiveconstant.DerentiatingV2andsubstitutingEq.(5.38)andEq.(5.40),weget_V2=ke(EEdes)Kext_zey+1_1=ke(EEdes)Kext_z[1ke(EEdes)_z]+1_1=2k2eKext(EEdes)2_z2+1h_1+ke(EEdes)Kext_zi(5.42)Bychoosingk1>0and_1=fkeKext(EEdes)_zk11g,˚2(5.43)wecanmake_V2negativesemi-denite.Weintroduceintegratorbacksteppingagainbydeningthenewvariable2=(_1˚2)(5.44)andthecompositeLyapunovfunctionV3=V2+1222=12ke(EEdes)2+1221+1222(5.45)87DerentiatingV3andsubstitutingEq.(5.42)andEq.(5.44),weget_V3=2k2eKext(EEdes)2_z2+1h_1+keKext(EEdes)_zi+2_2=2k2eKext(EEdes)2_z2+1h_1˚2k11i+(_1˚2)(1_˚2)=2k2eKext(EEdes)2_z2k121+2h1_˚2+1i(5.46)Ourchoiceof1=_˚21k22;k2>0(5.47)resultsinanegativesemi-denite_V3andyieldsthecontrollervy=˚1+_˚21k22(5.48)TheaboveequationwasobtainedfromEq.(5.47)bysubstitutingEqs.(5.31)and(5.40).Fromthedenitionof˚1inEq.(5.39),itisclearthat˚1willinvolvethethirdderivativeofz.ThisisnotaproblemsincethethirdderivativeofzcanbecomputedeasilyfromEq.(5.25)as...z=1mKext_y(5.49)ThecompletecontrollawisgivenbyEqs.(5.48),(5.32),(5.31),and(5.30).885.2.5StabilityAnalysis
Assumingex0,andusingEqs.(5.19),(5.23),(5.33),(5.25),(5.40)and(5.44)itcanbeshownthat(EEdes;ey;_ey)=(0;0;0),(EEdes;1;2)=(0;0;0)Therefore,V3inEq.(5.45)isacandidateLyapunovfunctionforinvestigatingthestabilityoftheequilibriuminEq.(5.36).Intheightphase,=0.Forthecontrollaw,givenbyEq.(5.48),thederivativeoftheLyapunovfunctioninEq.(5.46)canbeshowntobe_V3=k121k2220(5.50)Therefore,(EEdes;1;2)=(0;0;0)isstable.Inthecontactphase,=1.ForthecontrollawgivenbyEq.(5.48),thederivativeoftheLyapunovfunctioninEq.(5.46)is_V3=keKext(EEdes)2_z2k121k2220(5.51)Therefore,(EEdes;1;2)=(0;0;0)isstable.Remark3.Thestabilityof(EEdes;1;2)=(0;0;0)intheightandcontactphasesdoesnotguaranteeitsstabilityforthehybriddynamics.895.2.6DiscreteControllerforStabilizationofHybridDynamicsToinvestigatethestabilityofthehybriddynamicsystem,weuseaPoincaremapwiththePoincaresectiondenedbytheinstantwhenthesystemtransitionsfromthecontactphasetotheightphase.ThePoincaresectionisdenedasZ:=nX2R8jy=0;_y>0;_x=0o)Z:=nX2R8jz=ey+zd;_z>_ey;_x=0o(5.52)whereXisdenedasX=x_z1_12_23_3TFromthesystemdynamics,wecanndthatthePoincaremapmaybeconsideredinde-pendentofthevariablexasitonlyplacesatranslationontheinitialconditionsofthesectionwithouteectingthedynamics.WethereforedenethePoincaresectionusingthecoordinates,whereisdenedbythereducedcoordinates=_z1_12_23_3T(5.53)DeningthereducedsectionLasL:=nX2R7jy=0;_y>0;_x=0oˆZ(5.54)90ThePoincaremapP()andthesequenceofpointsk2Lsatisfyk+1=P(k)(5.55)withperiodicpointdenedas=P()(5.56)Fortheelasticfoundation,theperiodicpointsofinterestsatisfy2fX2LjE=Edes;e=0;_e=0g(5.57)Wedenetheerrorstatekask=(k)BylinearizingthePoincaremapabout,wehavetheapproximatediscretedynamicsgivenbyk+1=AkA,dP()d


=(5.58)Theperiodicpointwillbeasymptoticallystableifandonlyifˆ(A)<1(5.59)whereˆ(A)isthespectralradiusofA.SincetheconditioninEq.(5.59)maynotbesatised,wedesignadiscretecontrollertostabilizetheclosed-loopsystem;thediscretecontrollerisdiscussednext.Todesignthediscretecontroller,wedenetheadditionalstateEdasthedesiredenergy91inputtobacksteppingcontroller.Tothisendwedenethediscreteinputu(k)tobegivenbyu(k),(EdEdes)(5.60)Weseethatattheperiodicpoint,thatu(k)=u=0(5.61)ThenewPoincaremapP(;u)satisesk+1=P(k;uk)(5.62)withperiodicpointdenedas=P(;u)=P(;0)(5.63)Wedenetheerrorstatekask=(k)andlinearizingthePoincaremapabout(;u),wehavetheapproximatediscretedynamicsgivenbyk+1=Ak+Buk(5.64)A,dP(;u)d


=;u=u;B,dP(;u)du


=;u=u92Forourchoiceofinputuk=Kk(5.65)theclosed-loopsystemdynamicstakestheformk+1=(A+BK)kIffA;Bgiscontrollable,wecanchooseKsuchthatˆ(A+BK)<1(5.66)andthehybriddynamicalsystemisasymptoticallystable.
Remark4.IftheconditioninEq.(5.59)isnotsatisedandthediscretecontrollerinEq.(5.65)isimplemented,thecontinuouscontrollerwillhavetobemoded.Inparticular,thexeddesiredvalueoftheenergyEdeswillhavetobereplacedbythedesiredvalueofenergyforeachhopEdtoaccountforthechangeinthePoincaremapfromP()toP(;u).5.3SimulationResults
Forthethree-linkhopper,themassesareassumedtobem1=2kg;m2=2kg;m3=2kg(5.67)93Thelengthofthelinksofthehopperareassumedtobel1=l2=l3=0:3m(5.68)andthedistancetothecenterofmassofeachlink-seeFig.3.1areassumedtobed1=d2=d3=0:15m(5.69)ThemomentofinertiaofeachlinkiscomputedasIi=112mil2i8i2[1;4](5.70)ThegainsusedforthecontinuouscontrolareKx=8000Kd;x=300ke=0:01k1=300k2=10(5.71)andthesetpointydforthecontinuouscontrolisyd=0:38m(5.72)Wechoosethedesiredapexheightofthecenterofmasstobe0:65metersandcompute94theperiodicpoint,(˜;)tobegivenby˜
1=0:0547˜
2=0:0˜
3=0:4361˜
4=0:0˜
5=1:6828˜
6=0:0Ed=1:5892(5.73)withallvaluesgiveninradandrad=swhereappropriate.ThestabilizingcontrolgainsKfortheperiodicpointinEq.(5.65)arebysolvedbyusngthediscreteLQRproblemK1=0:1K2=0:03K3=0:1K4=0:04K5=0:1K6=0:003(5.74)Wechoosetheinitialcongurationofthesystemisassumedtobe(x(0);_x(0);y(0);_y(0))=(0:00;0:00;0:05;0:00)(0)=[0:2041;0:6979;1:1537]T_(0)=[0:00;0:00;0:00](5.75)wheretheunitsareinmeters,rad,andrad/sec.Theseinitialconditionswerechosensuchthatthecenterofmassofthehopperliesverticallyabovethepointofsupport.Theinitialvalueofthediscretecontrolinputischosentobeu(0)=0)Ed(0)=Ed(5.76)Figure5.2showstheheightofthecenterofmassasafunctionoftime.Itcanbeseenthatthecenterofmassconvergestothedesiredheightinapproximately4hops.Figure5.39500.511.522.533.544.55h0.20.40.60.800.511.522.533.544.55ry0.350.40.45Time (s)00.511.522.533.544.55rx-0.1-0.0500.05Figure5.2:Centerofmassheightandrelativecenterofmassforthethree-linkhopperdisplaystheinputtorques˝1,˝2,and˝3.Thesharppeaksindicatethediscontinuousjumpsinthetorquesimmediatelyfollowingimpact.9600.511.522.533.544.55=1-101234Time (s)00.511.522.533.544.55=2-10-505Figure5.3:Inputtorquesforthethree-linkhopper97Chapter6
Conclusions
Presentedrstwasamethodofcontrollingatwo-masshoppingrobotinteractingwithrigidground.Thecontactbetweenthelowermassandthegroundwasconsideredtobeinelasticwhichresultsinanimpulsivereactionwhenthemasscontactstherigidground.Thecontrolutilizesfeedbacklinearizationtostabilizethesystemduringcontinuousmotion.Adiscretefeedbackcontrolleronthecontrolparametersprovidesstabilitytoadesiredapexheight.Ecacyofthecontrolmethodisinitiallyshownthroughsimulation.Thentherobustnessandapplicabilityaretestedthroughexperiments.Theexperimentsutilizeavoicecoillinearactuatortoprovidetheforcebetweenthetwo-massesandalinearguidetomeasurethedistancebetweenthemassesaswellastheheight.Fromtheexperiments,wesawtheconvergenceofthealgorithm,althoughthemaximumheightwasrestrictedbythemaximumforceabletobeappliedbythevoicecoil.Fromthecontrolresultsofthetwo-masshopperonrigidground,acontrolmethodforafour-linkhoppingrobotinteractingwithrigidgrounditpresented.Followingthestrategyofthetwo-masshopper,feedbacklinearizationisusedincontinuoustimetostabilizethefour-linkhopper.Discretevariationsofacontrolparameterarethenusedtoensureasymptoticstabilitytoadesiredapexheight.Simulationresultsarethenpresentedtoshowtheecacyofthecontrolalgorithm.Followingthecontrolofthefour-linkhopper,investigationofthecontrolmethodonanelasticfoundationwasconducted.Throughasimpleinvestigationwithatwo-mass,98hoppingrobot,itwasshownthatmodcationofthecontrolmethodusedwheninteractingwithrigidgroundwasrequired.Tothisend,amodedcontrolmethodwasproposed.Thecontrolmethodutilizedbacksteppingtoproduceacontinuousstabilizingcontroller.Discretevariationsofthecontrolparameterswereusedwitheachhoptostabilizethesystemtoadesiredapexheight.Simulationresultsarepresentedtoshowtheecacyofthecontrolmethod.Fromthecontrolofatwo-massrobotonelasticground,thecontrolmethodisextendedforthecontrolofathree-linkhopperinteractingwithelasticground.Thecontrolmethodusesbacksteppingforcontinuousstabilization.Thenadiscretevariationofacontrolparameterisusedtostabilizethesystemtoadesiredapexheight.Thecontrolmethodisveredthroughsimulationresults.Fromthisthesisweseetheuseofchaoscontrolintheuseofahybriddynamicsystemtoachieveperiodorbits.Thecontrolreliesontheconstructionofcontinuouscontrollerstoensureinternalstabilityduringthecontinuousportionoftheorbit,andthediscretechaoscontrollerprovidesstabilityofthewholeoftheorbit.Alternatemethodswerepresentedhereforderenthoppingrobotswithderentsurfaceinteractions.Furtherworksincludeinvestigationofasystemwithlateralmotion,aswellassystemwithmultiplelegs.99BIBLIOGRAPHY100BIBLIOGRAPHY[1]LinearVoiceCoilMotorActuator,ModelLCVM-051-165-01.www.moticont.com/lvcm-051-165-01.htm.accessed20-Nov-2014.[2]Servodrives-brushed,analog,panelmount,Model25A8.www.a-m-c.com.accessed20-Nov-2014.[3]Single-boardreal-timehardware,ModelDS1104.www.dspace.com.accessed20-Nov-2014.[4]Transmissiveopticalencodermodule,ModelEM1-0-120-N.www.usdigital.com/products/encoders/incremental/modules/EM1.accessed20-Nov-2014.[5]RAlexander.Threeusesforspringsinleggedlocomotion.TheInternationalJournalofRoboticsResearch,9(2):53{61,1990.[6]R.Altendorfer,D.EKoditschek,andP.Holmes.Towardsafactoredanalysisofleggedlocomotionmodels.InIEEEInternationalConferenceonRoboticsandAutomation,volume1,pages37{44.IEEE,2003.[7]R.Altendorfer,D.EKoditschek,andP.Holmes.Stabilityanalysisofleggedlocomo-tionmodelsbysymmetry-factoredreturnmaps.TheInternationalJournalofRoboticsResearch,23(10-11):979{999,2004.[8]R.Blickhan.Thespring-massmodelforrunningandhopping.Journalofbiomechanics,22(11):1217{1227,1989.[9]N.Boccara.Modelingcomplexsystems.SpringerVerlag,2004.[10]LouisLFlynn,RouhollahJafari,andRanjanMukherjee.Activesynthetic-wheelbipedwithtorso.IEEETransactionsonRobotics,26(5):816{826,2010.[11]RMGhigliazza,R.Altendorfer,P.Holmes,andD.Koditschek.Asimplystabilizedrunningmodel.SIAMreview,2:519{549,2005.101[12]RMGhigliazza,RichardAltendorfer,PhilipHolmes,andDKoditschek.Asimplystabilizedrunningmodel.SIAMJournalonAppliedDynamicalSystems,2(2):187{218,2003.[13]K.A.HamedandJWGrizzle.Robustevent-basedstabilizationofperiodicorbitsforhybridsystems:Applicationtoanunderactuated3dbipedalrobot.InProceedingsofthe2013AmericanControlConference,2013.[14]J.K.HodginsandMNRaibert.Adjustingsteplengthforroughterrainlocomotion.RoboticsandAutomation,IEEETransactionson,7(3):289{298,1991.[15]P.Holmes,R.J.Full,D.Koditschek,andJ.Guckenheimer.Thedynamicsofleggedlocomotion:Models,analyses,andchallenges.SiamReview,48:207{304,2006.[16]SHHyon,T.Emura,andT.Mita.Dynamics-basedcontrolofaone-leggedhoppingrobot.ProceedingsoftheInstitutionofMechanicalEngineers,PartI:JournalofSystemsandControlEngineering,217(2):83{98,2003.[17]M.Ishikawa,A.Neki,J.I.Imura,andS.Hara.Energypreservingcontrolofahoppingrobotbasedonhybridport-controlledHamiltonianmodeling.InControlApplications,2003.CCA2003.Proceedingsof2003IEEEConferenceon,volume2,pages1136{1141.IEEE,2003.[18]S.Kajita,T.Nagasaki,K.Kaneko,K.Yokoi,andK.Tanie.Ahoptowardsrunninghumanoidbiped.InRoboticsandAutomation,2004.Proceedings.ICRA'04.2004IEEEInternationalConferenceo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