FTpp点点通2004

Automatica40(2004)1647–1664www.
elsevier.
com/locate/automaticaModeling,stabilityandcontrolofbipedrobots—ageneralframeworkYildirimHurmuzlua;,FrankGenotb,BernardBrogliatocaMechanicalEngineeringDepartment,SouthernMethodistUniversity,Dallas,TX75252,USAbINRIARocquencourt,DomainedeVoluceau—BP105,78153LeChesnay,Cedex,FrancecINRIARhone-Alpes,ZIRSTMontbonnot,655Avenuedel'Europe,38334Saint-Ismier,FranceReceived15October1999;receivedinrevisedform15September2003;accepted7January2004AbstractThefocusofthissurveyisthemodelingandcontrolofbipedallocomotionsystems.
Morespecically,weseektoreviewthedevelopmentsintheeldwithintheframeworkofstabilityandcontrolofsystemssubjecttounilateralconstraints.
Weplaceparticularemphasisonthreemainissuesthat,inourview,formtheunderlyingtheoryinthestudyofbipedallocomotionsystems.
Impactofthelowerlimbswiththewalkingsurfaceanditseectonthewalkingdynamicswasconsideredrst.
Thekeyissueofmultipleimpactsisreviewedindetail.
Next,weconsiderthedynamicstabilityofbipedalgait.
Wereviewtheuseofdiscretemapsinstudyingthestabilityoftheclosedorbitsthatrepresentthedynamicsofabiped,whichcanbecharacterizedasahybridsystem.
Last,weconsiderthecontrolschemesthathavebeenusedinregulatingthemotionofbipedalsystems.
Wepresentanoverviewoftheexistingworkandseektoidentifytheneededfuturedevelopments.
Duetotheverylargenumberofpublicationsintheeld,wemadethechoicetomainlyfocusonjournalpapers.
2004PublishedbyElsevierLtd.
Keywords:Bipedrobots;Non-smoothmechanics;Unilateralconstraints;Complementarityconditions;Multipleimpactlaws;Hybridsystem;Gaitstability;Controlsynthesis1.
IntroductionIngeneral,abipedallocomotionsystemconsistsofsev-eralmembersthatareinterconnectedwithactuatedjoints.
Inessence,aman-madewalkingrobotisnothingmorethanaroboticmanipulatorwithadetachableandmovingbase.
Designofbipedalrobotshasbeenlargelyinuencedbythemostsophisticatedandversatilebipedknowntoman,themanhimself.
Therefore,mostofthemodels/machinesdevel-opedbearastrongresemblancetothehumanbody.
Almostanymodelormachinecanbecharacterizedashavingtwolowerlimbsthatareconnectedthroughacentralmember.
Althoughthecomplexityofthesystemdependsonthenum-berofdegreesoffreedom,theexistenceoffeetstructures,ThispaperwasnotpresentedatanyIFACmeeting.
ThispaperwasrecommendedforpublicationinrevisedformbyManfredMorariEditor.
Inthissurvey,wereviewresearcheortsindevelopingcontrolalgorithmstoregulatethedynamicsofbipedalgait.
Wefocusonissuesthatarerelatedtomodeling,stability,andcontroloftwoleggedlocomotionsystems.
Correspondingauthor.
E-mailaddress:hurmuzlu@seas.
smu.
edu(Y.
Hurmuzlu).
upperlimbs,etc.
,itiswidelyknownthatevenextremelysimpleunactuatedsystemscangenerateambulatorymotion.
Abipedallocomotionsystemcanhaveaverysimplestruc-turewiththreepointmassesconnectedwithmasslesslinks(Garcia,Chatterjee,Ruina,&Coleman,1997)orverycom-plexstructurethatmimicsthehumanbody(Vukobratovic,Borovac,Surla,&Stokic,1990).
Inbothcases,thesystemcanwalkseveralsteps.
Theroboticscommunityhasbeeninvolvedintheeldofmodelingandcontrolofbipedsformanyyears.
Thebooks(Vukobratovic,1976;Vukobratovicetal.
,1990;Raibert,1986;Todd,1985)areworthreadingasanintroductiontotheeld.
Theinterestedreadermayalsorefertothefollowingwebpages:http://www.
androidworld.
com/prod28.
htm,http://robby.
caltech.
edu/kajita/bipedsite.
html,http://www.
fzi.
de/divisions/ipt/WMC/preface/preface.
html,http://www.
kimura.
is.
uec.
ac.
jp/faculties/legged-robots.
html.
Nevertheless,anddespitethetechnologicalexploitachievedbyHonda'sengineers(Japaniscertainlythecountrywherebipedallocomotionhasreceivedthemostattentionandhasthelongesthistory),somefundamental0005-1098/$-seefrontmatter2004PublishedbyElsevierLtd.
doi:10.
1016/j.
automatica.
2004.
01.
0311648Y.
Hurmuzluetal.
/Automatica40(2004)1647–1664modelingandcontrolproblemshavestillnotbeenaddressednorsolvedintherelatedliterature.
Onemaynotice,inpar-ticular,thatthelocomotionofHonda'sP3prototypere-mainsfarfromclassicalhumanwalkingpatternsatthesamespeeds.
AlthoughHonda(HONDA)didnotpublishmanydetailseitheronthemechanicalpartorontheimplementedcontrolheuristic,itiseasytoseeontheavailablevideosthatP3'sfootstrikedoesnotlooknaturalandleadstosometran-sientinstability(http://www.
honda-p3.
com).
ThenumberoffootdesignpatentstakenoutbyHonda(uptoanair-bag-likeplanterarch)revealsagainthatfoot–groundimpactremainsoneofthemaindicultiesonehastofaceinthedesignofrobustcontrollawsforwalkingrobots.
Thiswillbecomethekeyissuewithincreasinghorizontalvelocityrequirement.
Thisproblem,however,ismoresensitivefortwo-leggedrobotsthanformulti-leggedonesduetothealmoststraightlegcongurationandthebiggerloadatimpacttimefortheformer,leadingtostrongervelocityjumpsofthecenterofmass.
WhileHonda'sengineersseemtoconsidertheseve-locityjumpsasunwantedperturbationsandthusappealtomechanicalastutenesstosmooththetrajectory,wearguethatimpactisanintrinsicfeatureofmechanicalsystemslikebipedrobotsandshouldbetakenassuchinthecontrollerdesign.
Otherbipedalrobotshavebeendesigned.
Amongthemostadvancedprojects,wecitetheWasedaUniversityHumanoidRoboticsInstitutebiped,theMITLegLabora-toryrobots,theLMS-INRIABIPsystem(Sardain,Rostami,&Bessonnet,1998;Sardain,Rostami,Thomas,&Besson-net,1999),theCNRS-Rabbitproject(Chevallereauetal.
,2003),andtheGermanAutonomousWalkingprogramme(Gienger,Loer,&Pfeier,2003),whichcanbefoundathttp://www.
humanoid.
rise.
waseda.
ac.
jp/booklet/kato4.
html,http://www.
ai.
mit.
edu/projects/leglab/robots/robots.
html,http://www.
inrialpes.
fr/bip,http://www-lag.
ensieg.
inpg.
fr/PRC-Bipedes/,http://www.
fzi.
de/ids/dfgschwerpunktlaufen/startpage.
html.
respectively.
Amongalltheseexistingbipeds,theHondarobotsseemtobethemostadvancedatthetimeofwritingofthispaperaccordingtotheinformationmadeavailablebytheowners.
However,thesolutionforcontroldesignedbyHondadoesnotexplainwhyagiventrajectoryworksnordoesitgiveanyinsightastohowtoselect,chaintogether,andblendvariousbehaviorstoeectlocomotionthroughdicultterrain(Pratt,2000).
Itisthefeelingoftheauthorsthattheproblemoffeedbackcontrolofbipedalrobotswillnotbesolvedproperlyaslongasthedynamicsofsuchsys-temsisnotthoroughlyunderstood.
Infact,themainmoti-vationforthewritingofthispaperhasbeenthefollowingobservationaboutwalking:thereisnoanalyticalstudyofastablecontrollerwithacompletestabilityproofavailableintherelatedliterature.
Itisourbeliefthatthemainrea-sonforthisisthelackofasuitablemodel.
Weproposeaframeworkthatisnotonlysimpleenoughtoallowsubse-quentstabilityandcontrolstudiesbutalsorealisticassomeexperimentalvalidationsprove.
Inaddition,theframeworkprovidesauniedmodelingapproachformathematical,nu-merical,andcontrolproblems,whichhasbeenmissing.
ItisforinstancesignicantthatthemaineortsoftheMITLegLab(Pratt,2000)havebeendirectedtowardtechnological(actuators)improvementandtestingofheuristiccontrolal-gorithmssimilartoHonda'sworks.
Weshouldemphasizethatthemainthrustofthissur-veydoesoverlookseveralpracticalaspectsthatmayariseduringthedesignanddevelopmentofwalkingmachines.
Admittedly,awalkingmachinecanbebuiltwithoutpayingattentiontomanyofthemainideasofthissurvey.
Therearenumeroustoysthatwalkinacertainfashion.
Therearequiteafewbipedalrobotsthataredesignedtoavoidimpactsal-togetherduringwalking.
Thefactremainsthatthestability,agility,andversatilityofanyexistingbipedalmachinedoesnotevencomeclosetothatofthehumanbiped.
Thesur-veyedconceptswillbetterenablethedesignandevaluationofsuchmachinesthroughmoresuitablecontrolalgorithmsthattakeintoaccountimpactmechanicsandstability.
Thepracticalissuesthatariseinthedesignanddevelopmentac-tualmachinesdeserveanothersurveyarticle.
Intheensuingpartofthissurveywe,therefore,willmainlyfocusonathe-oreticalframework.
2.
GeneraldescriptionofabipedalwalkerAbipedcanberepresentedbyaninvertedpendulumsystemthathasaconstrainedmotionduetothefor-wardandbackwardimpactsoftheswinglimbwiththeground(Cavagna,Heglund,&Taylor,1977;Hurmuzlu&Moskowitz,1986;Full&Koditschek,1999).
Althoughsim-ilartothestructureofvibrationdampersinmanyaspects(Shaw&Shaw,1989),whicharerelativelywellstudied,structureofbipedalsystemshaveafundamentaldierencearisingfromtheunconstrainedcontactofthelimbswiththeground(seeFig.
1).
Whilethevibrationdamperremainsincontactwiththereferenceframeatalltimesbecauseofthehingethatislocatedbetweentheinvertedpendulumandthevibratingmass,thelimbsofthebipedarealwaysfreetodetachfromthewalkingsurface.
Detachmentsoccurfre-quentlyandleadtovarioustypesofmotionsuchaswalking,running,jumping,etc.
Asamatteroffact,onecanclassifybipedallocomotionsystemsascomplementaritysystems(Lotstedt,1984;Brogliato,2003).
Suchamodelingframe-workdoesnotatallprecludetheintroductionofexibilitiesatthecontactpoints.
Alsoitallowsonetoincludeotheref-fectslikeCoulombfrictioninasingleframework,whichcanbequiteusefulfornumericalsimulationsinordertovalidatethecontrollers.
Fig.
1depictsothersystemsthatfallintothesamecategory.
Inthelatterpartofthearticle,wewillshowthatsuchsystemscanbeanalyzedbytheuseofPoincaremaps(Hurmuzlu&Moskowitz,1987),andpossesscom-monfeaturesintermsofmotioncontrol.
WewouldliketostressthatthefocusofthisarticleisonmotionsthatincludeY.
Hurmuzluetal.
/Automatica40(2004)1647–16641649(a)(b)(c)(d)(e)Fig.
1.
Bipedsascomplementaritysystems:(a)biped,(b)pendulum,impactdamper,(c)massimpactdamper,(d)juggler,and(e)manipulatorincontactwitharigidwall.
contactandimpacts.
Forexample,abipedcanrockbackandforthwhiletheswinglimbremainsabovethewalkingsurfaceforalltimes.
Suchmotionswillbeoutsidethefocusofthediscussionspresentedhere.
Atypicalwalkingcyclemayincludetwophases:thesinglesupportphase,whenonelimbispivotedtothegroundwhiletheotherisswingingintheforwarddirec-tion(openkinematicchainconguration),andthedoublesupportphase,whenbothlimbsremainincontactwiththegroundwhiletheentiresystemisswingingintheforwarddirection(closedkinematicchainconguration).
Whenbothlimbsaredetached,thebipedisinthe"ight"phaseandtheresultingmotionisrunningorsomeothertypeofnon-walkingmotion(Kar,Kurien,&Jayarajan,2003).
Anyeortthatinvolvesanalyticalstudyofthedynamicsofgaitnecessitatesathoroughknowledgeoftheinternalstructureofthelocomotionsystem.
Whenthesystemishumanoranimal,thisstructureisextremelycomplicatedandlittleis(a)(b)Fig.
2.
(a,b)Typesofoscillationofathree-elementbiped.
knownaboutthecontrolstrategythatisusedbyhumanbe-ingsandanimalstorealizeaparticularmotionandachievestablegait(Full&Koditschek,1999;Vaughan,2003).
Ifthesystemrepresentsaman-mademachineoranumericalmodelrepresentingsuchamachine,onehastosynthesizecontrolstrategiesandperformancecriteriathattransformmulti-bodysystemstowalkingautomata.
Devisingprac-ticalcontrolarchitecturesforbipedalrobotsremainstobeachallengingproblem.
Theproblemistightlycoupledwiththecontrolstudiesintheareaofroboticmanipulators.
Unlikemanipulators,however,bipedalmachinescanhavemanytypesofmotion.
Thecontrolobjectivesshouldbecarefullyselectedtoconformwithaspecictypeofmotion.
Acontrolstrategythatisselectedforhigh-speedwalkingmaycausethesystemtotransfertorunning,duringwhichanentirelydierentcontrolstrategyshouldbeused,similartojugglerscontrol(Brogliato&Zavala-Rio,2000).
ConsiderthebipeddepictedduringthesinglesupportphaseinFig.
1(a).
Thisbipedisequivalenttoapendulumattachedtothefootcontactpointwithamassandalengththatarecongurationdependent.
Invertedpendulummod-elsofvariouscomplexities,therefore,havebeenextensivelyusedinthemodelingofgaitofhumansandbipedalwalk-ingmachines.
Thedynamicsofbipedallocomotionisintu-itivelysimilartothatofaninvertedpendulumandhasbeenshowntobeclosetoitaccordingtoenergeticalcriteria(Karetal.
,2003;Cavagnaetal.
,1977;Full&Koditschek,1999;Blickman&Full,1987).
Asimpletwo-elementbiped(Hurmuzlu&Moskowitz,1987)mayoperateintwomodes(seeFig.
2):(a)impactlessoscillations,(b)progressionwithgroundcontacts.
Theim-portanceofthecontacteventcanbebetterunderstoodifthemotionisdepictedinthephasespaceofthestatevariables.
Wesimplifythepresentdiscussionbydescribingtheeventsthatleadtostableprogressionofabipedforasingle-degree-of-freedomsystem,however,thisapproachcanbegeneralizedtohigher-ordermodels.
ThephaseplaneportraitcorrespondingtothepreviouslydescribeddynamicbehaviorisdepictedinFig.
3(a).
Thesampletrajectoriescorrespondingtoeachmodeofbehaviorarelabelledaccordingly.
Theverticaldashedlinesrepresentthevaluesofthecoordinatedepictedinthephaseplaneforwhichthecontactoccurs.
ForthemotionsdepictedinFig.
3(a)or(b),theonlytrajectorythatleadstocontactisC.
Thecontacteventforthissimplemodelproducestwo1650Y.
Hurmuzluetal.
/Automatica40(2004)1647–1664(a)12BackwardContactLimitForwardContactLimitCABABIMPACTSWITCHING12C(b)(c)45oAccumulationpoint(d)pe1n+1n.
1.
.
.
.
11Fig.
3.
Impactonthephaseplaneportrait.
simultaneousevents:(1)impact,whichisrepresentedbyasuddenchangeingeneralizedvelocities,(2)switchingduetothetransferofpivottothepointofcontact.
ThecombinedeectofimpactandswitchingonthephaseplaneportraitisdepictedinFig.
3(b).
Asshownintheg-ure,theeectofthecontacteventwillbeasuddentransferinthephasefrompoints1to2,whichisgenerallylocatedonadierentdynamictrajectorythantheoriginalone.
Ifthedestinationofthistransferisontheoriginaltrajectory,thentheresultingmotionbecomesperiodic(Fig.
3(c)).
Thistypeofperiodicityhasuniqueadvantagewhentheinvertedpen-dulumsystemrepresentsabiped.
Actually,thisistheonlymodeofbehaviorthatthisbipedcanachieveprogression.
Themoststrikingaspectofthisparticularmodeofbehav-ioristhatthebipedachievesperiodicitybyutilizingonlyaportionofadynamictrajectory.
Theimpactandswitchingmodesprovidetheconnectionbetweenthecyclicmotionsofthekinematicchainandthewalkingaction.
Wecanclearlyobservefromtheprecedingdiscussionthatthemotionofabipedinvolvescontinuousphasessep-aratedbyabruptchangesresultingfromimpactofthefeetwiththewalkingsurface.
Duringthecontinuousphase,wemayhavenone,one,ortwofeetinsimultaneouscontactwiththeground.
Inthecaseofoneormorefeetcontacts,thebipedisadynamicalsystemthatissubjecttounilateralconstraints.
Whenafootimpactsthegroundsurface,wefacetheimpactproblemofamulti-linkchainwithunilat-eralconstraints.
Infact,theoverallmotionofthebipedmayincludeaverycomplexsequenceofcontinuousanddiscon-tinuousphases.
Thisposesaverychallengingcontrolprob-lem,withanaddedcomplicationofcontinuouslychangingmotionconstraintsandlargevelocityperturbationsresultingfromgroundimpacts.
3.
Mathematicaldescriptionofabipedasasystemsubjecttounilateralconstraints3.
1.
DynamicsofthecomplementaritymodelBipedallocomotionsystemsareunilaterallyconstraineddynamicalsystems.
Awaytomodelsuchsystemsistoin-troduceasetofunilateralconstraintsinthefollowingform:F(q)0;q∈Rp;F:Rp→Rm;whereqrepresentsthecompletevectorofindependentgen-eralizedcoordinates.
Inotherwords,pdenotesthenumberofdegreesoffreedomofthesystemwithoutconstraints,i.
e.
whenF(q)0.
Theconstraintsmeanthatthebodiesthatconstitutethesystemcannotinterpenetrate(irrespectiveofthefactthattheyarerigidorexible).
Thedynamicsofap-degree-of-freedommechanicalsystemsubjecttomuni-lateralconstraintsmaybewrittenasthefollowingsystem(S)ofequations,namedacomplementaritydynamicalsys-tem(Brogliato,2003;Heemels&Brogliato,2003):M(q)q+N(q;˙q)=Tu+F(q)n+Pt(q;˙q);(1)TnF(q)=0;n0;F(q)0;(2)Restitutionlaw+shockdynamics;(3)DryFrictionAmontons–Coulombsmodel;(4)whereM(q)istheinertiamatrix,N(q;˙q)includesCoriolis,centrifugal,gravitational,andotherterms,uisanexternalinput,n∈RmistheLagrangemultipliercorrespondingtothenormalcontactforce.
TheorthogonalityTnF(q)=0meansthatifF(q)0thenn=0,whereasanon-zerocon-tactforcen0ispossibleonlyifthereiscontactF(q)=0.
Suchacontactmodelthereforeexcludesgluing,magneticforces.
ComplementarityLagrangiansystemsasin(1)–(4)havebeenintroducedbyMoreau(1963,1966),andMoreauandPanagiotopoulos(1988).
Therestofthetermsandvari-ablesaredenednext.
ForthebipedstheLagrangedynamicscanberewritteninaspecicwaythatcorrespondstocon-trolobjectivesandallowsthedesignertogetabetterunder-standingoftheirdynamicalfeatures(Wieber,2000;Grizzle,Abba,&Plestan,2001;Werstervelt,Grizzle,&Koditschek,2003).
Inotherwords,thechoiceofthegeneralizedcoor-dinatesqiscrucialforcontrolpurpose,andcertainlymuchlessobviousthanitisforserialmanipulators.
ForexampleinFig.
4,qcanbesplitintwosubsetsq1andq2.
Thevec-torq1=(x;y;)describestheglobalpositionoftherobotinspacewhereasthevectorq2=(1;6)encapsulatesthejointcoordinates.
Thevectorq1couldbeattachedtoanypointofthebiped.
Nevertheless,itisknownfrombiomedi-calstudiesofhumangaitthatoneoftheprimaryobjectivesY.
Hurmuzluetal.
/Automatica40(2004)1647–16641651(x,y,)yB2A2B1A161,2354Fig.
4.
A9-degree-of-freedomplanarbiped.
duringlocomotionisthestabilizationoftheheadwheretheexteroceptivesensors(innerear,sight)arelocated.
Set-tingq=(q1;q2)T,(1)splitsinanupperpartandalowerpartcorrespondingtoheadmotionandjointdynamics,respectively,M1(q)M2(q)q+N1(q;˙q)N2(q;˙q)=0T2u+F(q)n+Pt(q;˙q);(5)F(q)=(yA1;yB1;yA2;yB2)T∈R4;(6)whereyA1denotesthey-coordinateofpointA1,andsoon.
Inthesequelwewillreviewtheavailablemodelingtoolswhichwillallowthedesignertocomplete(5)with(3)and(4).
In(5),T1=0sincethebipedhasonlyjointactuators.
Asamatteroffactheadmotioncanonlybeachievedthankstoacoordinatedactionofjointactuationandcontactforces.
Thisfactisprobablymoreapparentfortheightphases(running)duringwhichtherearenocontactforcesandthetrajectoryofthecenterofmass(attachingthistimeq1tothecenterofmass)yieldsonlytogravity.
AllcomplementaritysystemsdepictedinFig.
1canberepresentedintheformgivenby(1)–(4).
However,bipeddynamicspossessessomespecicfeaturesthatmakethecontrolstudydiersignicantlyfromthecontrolofothercomplementaritysystems.
Bipedssharethefollowingfeatures:(1)withsystems(b),(d)and(e):thecenterofmassisnotcontrolledwhenF(q)0,astheobjectofajuggler(Brogliato&Zavala-Rio,2000),(2)withsystem(e):theirdynamicsisthatofamanipulatorwhenonefootstickstotheground,(3)withsystem(b):theymaybeunderactuated(noankleactuator)andactasaninvertedpendulumwhenonefootstickstotheground.
Thefactthatbipedsmergeallthesecharacteristicsmakestheircontrolanalysiscomplex.
Eq.
(1)representsthedy-namicswheneitherthesystemevolvesinfree-motionorinaphaseofpermanentcontact,i.
e.
Fi(q)≡0forsomei∈I(q){1;m};Fj(q)0forj∈I(q):(7)Then,inEq.
(1)(Fi(q)=(9FTi=9q)(q)∈Rpisthegradientvector)F(q)n=i=mi=1Fi(q)n;i:Noticethatfori∈I(q),onecanexpress(2)as(where˙Fi(q;˙q)=d=dt[Fi(q(t))])n;i˙Fi(q;˙q)=0;n;i0;˙Fi(q;˙q)0(8)or,if˙Fi(q;˙q)≡0fori∈I(q),as(whereFi(q;˙q;q)=(d=dt)[˙Fi(q(t);˙q(t))])n;iFi(q;˙q;q)=0;n;i0;Fi(q;˙q;q)0:(9)Asweshallsee,whentheconstraintsarefrictionlesstheconditionsin(9)denealinearcomplementarityproblem(LCP)withunknownsn;i.
AnLCPisasystemoftheformAx+B0,x0,xT(Ax+B)=0(Cottle,Pang,&Stone,1992).
LCPsareubiquitousinmanyengineeringap-plications(Ferris&Pang,1997),andparticularlyinuni-lateralmechanicsinwhichtheyhavebeenintroducedbyMoreau(1963,1966).
Finally,themodelsin(3)and(4)areneededtocompletethedynamics.
Inparticular,itisnec-essarytorelatethepost-impactvelocitiestothepre-impactdatatobeabletocomputesolutionsthatarecompatiblewiththeconstraints(integratethesystemandrenderthedo-main,{q|F(q)0}invariant).
TheclassicalCoulombfrictionmodelin(4)providestheformofPt(q;˙q)in(1).
Itisapparentthatthesetofequationsin(1)–(9)denesacomplexhybriddynamicalsystem,inthesensethatitmixesbothcontinuousanddiscrete-eventphenomena.
Thestatesofthediscrete-eventsystem(DES)aredenedbythe2mmodesofthecomplementarityconditionsin(2).
Itisalsoimportanttopointoutthatsuchsystemsfundamen-tallydierinnaturefromthosestudiedinBainovandSime-onov(1989),whichconsistofmainlyordinarydierentialequationswithimpulsivedisturbances.
Somediscrepancies1652Y.
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/Automatica40(2004)1647–1664betweenbothmodelsarerecalledinBrogliato(1999,Sec-tions1.
4.
2,7.
1)andBrogliato,tenDam,Paoli,Genot,andAbadie(2002).
Especially,oneshouldalwayskeepinmindthatthecomplementarityconditions(2)playamajorroleinthedynamicsofcomplementaritysystems(forinstancetheyarenecessarytoproperlycharacterizetheequilibriumstates).
Fromamechanicalengineerpointofview,twoques-tionshavetobeansweredto,whenonewantstointegratesystem(1)–(4)(Moreau&Panagiotopoulos,1988):Q1.
AssumethatI(q)in(7)isnon-emptyatatimeinstant,andthatthevelocity˙q()pointsinwardsortangentiallyto9:determinewhichcontactsi∈I(q),willpersistat+.
Inotherwords,determinethesubsequentmode(orDESstate).
Q2.
Atanimpacttimetk,onehas˙q(tk)TFi(q(tk))0forsomei∈{1;m}andFi(q(tk))=0.
Determinetherightvelocity˙q(t+k).
Inotherwords,determineare-initializationofthe(continuous)stateassoonasone(orseveral)oftheunilateralconditionsin(2)isgoingtobeviolated.
Inthesequeltk,k=0;1;2:::willgenericallydenotetheimpactinstants.
Theanswertothosequestionsisfarfrombeingtrivialandhasbeentheobjectofmanyresearches.
TherstoneisrelatedtosolvingtheLCPassociatedtothesystem,i.
e.
beingabletocalculateateachtimeofacollision-freephasetheinteractionforcesPq,F(q)n+Pt(q;˙q);hence,theaccelerationqandthesubsequentmo-tion.
IthasbeenraisedinitiallybyDelassus(1917)(seealsoPfeier&Glocker,(1996)foraveryniceandsim-plerexample).
Thesecondquestionisthatofdeningproperrestitutionrules,orcollisionlaws.
Thisgoesbacktothe17thcenturyandthecelebratedNewton'sconjecture,seeBrogliato(1999)andKozlovandTreshchev(1991)formoredetails.
Inparticular,ifseveralhypersurfacesi={q|Fi(q)=0}areattainedsimultaneously,amultipleimpactoccurs.
Suchaneventoccurstypicallyduringwalk-ingattheendofasinglesupportphasewhentheswinginglegfoothitstheground.
Forexample,fortheplanarbipeddepictedinFig.
4,a3-impacttakesplaceatfootstrike,thatisyA2(t)=yB2(t)=0,yA1(t)0,andyB1(t)0forttk,whileyA2(tk)=yB2(tk)=0,yA1(tk)=0,and˙yA1(tk)0.
Fromourpointofview,onlytoofewattentionwaspayedintheliteraturetotheexistenceofunilateralconstraints.
Infact,mostoftheworksonthecontrolofbipedrobotsmodeltherobotinthesinglesupportphaseasamanipula-torwhosebasecorrespondstothesupportingfootandaddsomeclosedloopconstraintsforthedoublesupportphase.
Evenifthisapproachisveryconvenienttoderivetrajectorytrackinglawsviathecomputedtorquetechnique,itdoesnotaccountforpossibleslippagenordetachmentatgroundcontactsnorsayanythingabouttheinuenceofimpactsonstability.
Noticealsothatthehumanwalkingpatternandtheunderlyingcontrolstrategydiersnotablydependingonthegroundcharacteristics(icearena,basketballplayground,trampoline,etc.
),whichmakesthewalkingpattern(ortra-jectoryplanning)animportanttopicofresearch(ElHa&Gorce,1999;Lum,Zribi,&Soh,1999;Yagi&Lumelsky,2000;Rostami&Bessonnet,2001;Shih,1999;Huangetal.
,2001;Chevallereau&Aoustin,2001;Saidouni&Besson-net,2003).
3.
2.
Frictionlesscontacts(continuousmotion)ThesucceedingsectionsaredevotedtothestudyoftheLCPin(9)(i.
e.
thecalculationofthecontactforces)andmultipleimpactsrespectively.
Inthefrictionlesscase(S)reducesto(1),(2)and(3)withPt(q;˙q)=0.
Moreau(1963,1966)hasbeenthersttoshowthatinthemultiplecon-straintscasem≥1,usingthatF(q;˙q;q)=FT(q)q+f(q;˙q),thecomplementarityrelation(9)combinedwith(1)yieldsanLCPoftheform(assumingherethatI(q)={1;m}in(7))F(q;˙q;q)=A(q)n+b(q;˙q)0;n0andTnF(q;˙q;q)=0;(10)whereA(q)=FT(q)M1(q)F(q)andb(q;˙q)=FT(q)M1(q)h(q;˙q)+f(q;˙q)andsettingh(q;˙q)=TuN(q;˙q).
Iftheactiveconstraints(i∈I(q))areindependent,thenAispositivesymmetricdenite(PSD)anditisknownthattheLCPin(10)pos-sessesauniquesolutionn(Cottleetal.
,1992).
Ifsomecon-straintsaredependent,thenAisonlysemi-PSDandunique-nessonlyholdsfortheaccelerationF(q;˙q;q).
Moreover,F(q)nisalsounique,seeMoreau(1966)andLotstedt(1982),andthusfromdynamics(1)qisuniquetoo.
Asaclassicalexample,onemaythinkofachairwithfourlegsonarigidground:eveniftheinteractionforcescannotbeuniquelydetermined,theaccelerationofthemasscenterisunique(upwards).
UsingKuhn–Tucker'stheory(Kuhn&Tucker,1951)itispossibletoshowthatanysolutionntotheLCPin(10)isalsoasolutiontothequadraticproblemminn012TnA(q)n+Tnb(q;˙q);(11)whichisequivalenttoGauss'principleofleastdeviation(Moreau,1966;Lotstedt,1982).
Tosummarize,itisclearthatifoneisabletoobtainattimeavalueforn,thenintroducingthisvalueinto(2)allowsonetodeterminewhichcontactspersistandwhichonesaregoingtobreak(becomeinactive)on0,smallenough.
Remark1.
TheanswertoQ1doesnotnecessarilyre-quiretheexplicitcalculationofthecontactforcesn.
Letusconsiderthesimpleexample(p=1)ofaballrestingontheground(m=1)at0.
Supposethatanexternalforcefisappliedtotheballat0.
Thenthedynamicsisq(t)=max(0;f)(recallthatmaxfunctionscanbewrittenwithcomplementarity).
LetusdiscretizetheunconstrainedY.
Hurmuzluetal.
/Automatica40(2004)1647–16641653ABCR(t)R(t)R(t)(a)(b)(c)Fig.
5.
Possiblemotionsandshockoutcomesinanabstractcongurationspace.
motion(q0)asqi+1=qi+h˙qi;˙qi+1=˙qi+hf:(12)Using(12),thedynamicsisdiscretizedas˙qi+1=prox[R+;˙qi+hf];(13)whereproxdenotestheproximalpointto˙qi+hfinR+.
Thisimplicitschemecanbegeneralizedtomorecomplexsystems(p1,m1),asproposedinMoreau(1986).
Moreover,itcanbegeneralizedtoprovideonesolutiontothemultipleimpactproblemaswillbeshowninSection3.
3.
3.
3.
Frictionlesscontacts(multipleimpacts)Section3.
2wasdevotedtopartiallyanswertoQ1.
WenowfocusonQ2.
Ingeneral,walkingrobotsuseseveralsupportpointsduringlocomotion(bipedrobots;Hurmuzlu&Moskowitz,1986,1987;Hurmuzlu,1993,quadrupedrobots,Chevallereau,Formal'sky,&Perrin,1997;Perrin,Chevallereau,&Formal'sky,1997).
Inotherwordsm2.
Thismeansthattheboundary9oftheadmissibledomainisnotdierentiableeverywhere.
Itssingularitiescorre-spondtosurfaceswithcodimension2.
Thus,theeventu-alitythatthestatecollidesinaneighborhoodofasingularitycannotbeexcludedinmechanicalsystemsas(S).
Intheframeworkofbipedallocomotion,sucheventsintrinsicallybelongtothedynamicsofwalking.
Letusviewforsimplic-itythebipedalrobotasapointR(t)inatwo-dimensionalgeneralizedcongurationspace.
Theclassicalstandardwalkingassumptioncanbeseenasa"bilateralslidingmotion"ofthegeneralizedpointR(t)connedtothetwoconstraintboundaries,seeFig.
5(a).
SinglesupportphasescorrespondtoR(t)∈[A;B)∪(B;C],thedoublesupportphasetoR(t)=B.
HoweveronecannotaprioriexcludereboundingmultipleshocksatB,seeFig.
5(b);detachmentduringsinglesupportphases,seetheforego-ingparagraph,andFig.
5(c).
Themostrecentstudiesintheeldcanbedividedintwodierentapproaches.
Therstone(Hurmuzlu&Marghitu,1994;Marghitu&Hurmuzlu,1995;Han&Gilmore,1993)consistsofanenumerationprocedure.
TheyapplyarestitutionlawforsimpleimpactatAjandsequentiallypropagateitseectsontheothercontactpointsAi,i∈I(q).
Nevertheless,theoverallprocessisassumedtobeinstantaneous,i.
e.
allshocksoccursimultaneouslyatallthecontactpoints:thisisthereforereallyamultipleimpact.
Ateachpointtheyinvestigatewhetherassumingzeroornon-zerolocalper-cussionsyieldconsistentoutcomes.
Themaindrawbackofthesestudiesisthattheydonotruleonwhetherornottheproposedalgorithmsalwaysterminatewithauniquesolution.
Inthecaseofseveraladmissiblesolutions,theydonotgiveacriteriontomakethechoicebetweenthesesolutions.
However,thisshouldnotbeseenasarealdraw-back(exceptforsimulationtooldesignpurpose)sinceanyheuristictomakethe"good"choice,excludinginthesametimeseveraladmissiblesolutions,dropstherigidbodyassumption.
ThesecondapproachcorrespondstothedenitionofacollisionmappingPc:9*{V(q(tk))}→9*{V(q(tk))};(q(tk);˙q(tk))→(q(tk);˙q(t+k));(14)whereV(q)={v∈Rp:i∈I(q),vTFi(q)0}denotesthetangentconeto9atq(t)(Moreau&Panagiotopou-los,1988),i.
e.
thesetofadmissiblepost-impactvelocities.
ItisclearthatthechoiceofPcshouldyieldamathemat-ically,mechanicallyandnumericallycoherentformula-tionofthestudiedphenomenon.
Theso-called"sweepingorMoreau'sprocess"(Brogliato,1999;Moreau,1985;Moreau&Panagiotopoulos,1988)isageneralformula-tionofthedynamicsin(1)–(4)basedonconvexanalysistools.
Itallowsonetowritethedynamicsasaparticu-larMeasureDierentialInclusion,atermcoinedbyJ.
J.
Moreau.
Itimplicitlydenesacollisionmappingbasedonthecomputationofthepost-impactmotionviaaprox-imationprocedureinthekineticmetric˙q(t+k)=proxM(q(tk))[˙q(tk);V(q(tk))]:(15)Itisnoteworthythatthiscanalsobeequivalentlywrit-tenasaquadraticprogrammeunderunilateralconstraints,andconsequentlyunderaLCPformalism,see(10),(11).
Letusnotethatthemappingin(15)appliedtotheshockoftworigidbodiescorrespondtotakingazerorestitu-tioncoecient.
Henceitmaybenameda"generalizeddissipativeimpactrule".
However,itispossibletointro-ducesomerestitution∈[0;1]bysubstituting˙q(t+k)in(15)by12(1+)˙q(t+k)+12(1)˙q(tk)(Moreau&Pana-giotopoulos,1988;Mabrouk,1998).
TheinterestedreadermayhavealookatBrogliato(1999,Section5.
3)foranon-mathematicalintroductiontothismaterial.
SimilarresultshavebeenobtainedinLotstedt(1982).
InPfeierandGlocker(1996),anextensionofPoisson'simpactlawisproposedthattakestheformoftwoLCPs(see,e.
g.
,1654Y.
Hurmuzluetal.
/Automatica40(2004)1647–1664Brogliato,1999,Section6.
5.
6forasimpleexampleofapplicationofthisimpactmapping).
3.
4.
ContactswithAmontons–Coulomb'sfrictionInthefrictionlesscase,wesawthattheLCP(10)waswell-posedsincetheonlypossibleindeterminacies,result-ingfromdependentactiveconstraints,donotinuencetheglobalmotionofthemechanicalsystem(uniquenessoftheaccelerationstillholds).
Theproblembecomesmorecom-plicatedthroughtheintroductionofdryfriction,anessentialparameteroftheleggedlocomotion.
ThenextstepreliesonthefactthattherelaycharacteristicofCoulomb'sfriction,canberepresentedanalyticallyinacomplementarityfor-malismbyintroducingsuitableslackvariables(orLagrangemultipliers).
Thisallowsonetoconstructasetofcomple-mentarityconditionswhichmonitorallthetransitionsfromstickingtoslipping,andfromcontacttodetachment.
Thedy-namicsin(1)–(4)canconsequentlyberewrittenas(Pfeier&Glocker,1996)M(q)q=h(q;˙q)+[W(q)+Nslide(q)];(16)y=A(q)+b(q;˙q);y0;0;yT=0(17)forappropriatematricesA(·)andb(·),WandNslide,wherethecomponentsofaresuitableslackvariables.
Thepow-erfulnessofthecomplementarityformalismclearlyappearsfrom(16),(17)inwhichtheoveralldynamicsiswritteninacompactway:thecontinuousdynamicspluscomple-mentarityconditions.
Thisperfectlytswithinthegeneralcomplementaritydynamicalframework(Brogliato,2003).
4.
ThestabilityframeworkThemostcrucialproblemconcerningthedynamicsofbipedalrobotsistheirstability,seee.
g.
http://www.
ercim.
org/publication/ErcimNews/enw42/espiau.
html.
AshasbeenexplainedinSections2and3,abipedisfarfrombeingasimplesetof(controlled)dierentialequations.
Moreover,theobjectivesofwalkingarequitespecic.
Oneisthereforeledtorstanswerthequestion:whatisastablebipedAnd,consequently,whatmathematicalcharacterizationofthisstabilitycanbeconstructedfromthecomplementaritymodelsAswewillseenext,thisiscloselyrelatedtothefactthatbipedscanbeconsideredashybriddynamicalsystems,thestabilityofwhichcanbeattackedfromvariousangles.
Thegoalofthissectionistopresentsometoolswhichcanserveforthestabilityanalysisofmodelsasin(2)–(5),andwhicharesuitableforbipedsbecausetheyencapsulatetheirmainfeatures.
Firstly,wespendsometimeondescribinginvariantsetsforcomple-mentaritysystems.
Thepointofviewthatisputforthisthatvariousexisting,ortobeinvestigated,stabilityframeworksxumAdmissibleRegionInadmissibleRegionA(a)(b)0FCISCVISUISFig.
6.
Controlledmasssubjecttoaunilateralconstraint.
arebetterunderstoodwheninvariantsetsareclassied.
Sec-ondly,wereviewtheso-calledimpactPoincaremaps,whichhavebeenusedextensivelyintheappliedmathematicsandmechanicalengineeringliteratureforvibro-impactsystems(Masri&Caughey,1966;Shaw&Holmes,1983;Shaw&Shaw,1989).
Thispointofviewseemsnaturalifonecon-sidersbipedsasjugglers(Brogliato&Zavala-Rio,2000;Zavala-Rio&Brogliato,2001).
However,itpresentslimi-tationswhichwepointout.
4.
1.
InvariantsetsofsystemssubjecttounilateralconstraintsApplicationanduseofmappingtechniquesistightlycou-pledwiththestructureoftheinvariantsetthatrepresentsthesteady-statemotion.
Systemssubjecttounilateralcon-straintsbehaveinamorecomplexmannerthantheonesthatarenot(Budd&Dux,1994).
Forexample,letuscon-siderthesystemgiveninFig.
6(a).
Supposewewouldliketodevelopacontrollertoplacethemassatatimevaryingpositionxd(t)=A+Bsin(!
t)startingfromanarbitraryini-tialconditionintheadmissibleregion.
Onecanuseasimplecontrollerthatyieldsthefollowingclosed-loopdynamics:m(xxd(t))+k1(˙x˙xd(t))+k2(xxd(t))=n;06n⊥x0;˙x(t+k)=e˙x(tk)withe∈[0;1):(18)Ourobjectivehereisnottoexploreallpossibletypesofinvariantsetsthatmaybeattainedbythesystemgivenby(18).
Instead,wewouldliketoshowthattheY.
Hurmuzluetal.
/Automatica40(2004)1647–16641655invariantsetsoftheclosedloopsystemcanbeclassiedunderthreecategories(seeFig.
6(b)):(i)Constraintviolatinginvariantsets(CVIS):Aninvari-antsetthatincludesatleastonecollisionwiththeconstraintsurfacepercycleofmotion(includingorbitsthatstabilizeinnitetimeontheconstraintsurfaceafteraninnitenum-berofcollisions).
Hence,animpactPoincaremappingPiswelldenedthatcapturestheseorbits.
Thistypeofinvari-antsetisauniquefeatureofsystemssubjecttounilateralconstraints.
AllthesystemsshowninFig.
1canbemadetoexhibitthisbehaviorwithasetofproperlyselectedparam-etervalues.
Specically,forlocomotionsystems,thiswillbetheonlymodeofmotionthatcandescriberunningandwalking.
(ii)Unconstrainedinvariantsets(UIS):Aninvariantsetthatdoesnotincludecollisionswiththeconstraintsurface.
Inthesinglemasscase,thiscorrespondstothecyclicmo-tionsofthemassthatoccurtotherightoftheconstraintsurface(x0t6t6∞).
ThistypeofinvariantsetcanbeobservedforallsystemsofFig.
1exceptthejuggler.
Forthebiped,thiscorrespondstorockingwhenoneortwolimbsareincontactwiththeground.
(iii)Fullyconstrainedinvariantsets(FCIS):Anin-variantsetthatneverleavestheconstraintsurface.
Thiscorrespondsforthesystemin(18)toastaticequilibriumwherethesystemrestsontheconstraintsurface(x(t)=0t6t6∞).
AllthesystemsinFig.
1canexhibitthisbe-haviorwithaspecicchoiceofthecontrolparametervaluesandinitialconditions.
ItiseasytoimaginethattrajectoriesofasystemmaybeCVIS,FCISandUISsimultaneously.
Inparticular,no-ticethatalthoughthecomplementarityrelationsin(18),i.
e.
x0,n0,xn=0,aprioridenetwomodesx0andx=0(henceabimodalsystem),forcontrolanddynamicsystemsanalysispurposeoneisledtocon-siderthosephasesthatcorrespondtoCVISasindependentones(Brogliato,Niculescu,&Monteiro-Marques,2000;Bourgeot&Brogliato,2003):theyaredescribedneitherbythefreemotionnorbytheconstrainedmotiondynamicsbutbythewholedynamicsofthehybridsystem.
4.
2.
BipedalrobotsashybriddynamicalsystemsTheaboveclassicationoftheinvariantsetsnaturallyleadsonetoconsidercomplementaritysystemsasinFig.
1ashybriddynamicalsystemswhoseDESstatesaredenedfromthedescribedinvariantsets.
Inthefollowing,weshallgenericallydenotethephasesthatcorrespondtoCVISasIkandthosethatcorrespondtoFCISand/orUISask.
Withsomeabuseofnotation,weshalldenotetheDESstatesandthecorrespondingtimeintervalsinthesamemanner.
Asweshallseefurthersubdivisionswillbeneeded.
AsanexampleletusconsideranimpactdamperasinFig.
1(c)andwithasinusoidalexcitationappliedtothebasismass:foraproperchoiceofthespring-dashpot,theexcitationparametersandoftheinitialconditions,thesystempossessesperiodictra-jectorieswithoneimpactperperiod(henceCVISs)(Masri&Caughey,1966).
ThusonehasR+=I0:(19)ConsidernowamanipulatorasinFig.
1(e)thatperformsacompleterobotictaskwithasuccessionoffree-motionandconstrainedmotionsphases,duringwhichitisexplicitlyrequiredtotrackdesiredmotionand/orcon-tactforce(Bourgeot&Brogliato,2003;Brogliato,1999;Brogliato,Niculsecu,&Orhant,1997;Brogliatoetal.
,2000;Menini&Tornambe,2001).
Duringtheforce/positioncon-trolphases,thetrajectorieswillingeneralbebothUIS(inthetangentdirectiontotheconstraintsurface)andFCIS(inthenormaldirection).
Duringthefree-motionphases,thetrajectoriesareUIS.
Itisthereforenaturaltosplitsuchtaskintothreephases2k,2k+1andIkthatcorrespondtoUIS,FCISandCVIS,respectively,i.
e.
R+=0∪I0∪1∪2∪I120)Considernowthebiped.
Itisclearthatinordertodescribewalking,runningandhoppingmotions,oneneedsmorethantheabovethreetypesofinvariantsets(Karetal.
,2003).
Moreover,oneneedsmorethanthethreephasesproposedforthemanipulatorcase.
Indeed,aswealreadypointedoutconcerningthechoiceofthePoincaresection,describingforinstanceawalkingmotioninvolvestotakecareofthenon-slidingconditions.
Hence,oneisledtodierentiatecontactphases(FCISandUIS)andimpactphases(CVISandFCISand/orUIS)thatslideandthosethatstick.
Noticemoreoverthatthismaybedoneindependentlyofthepres-enceofAmontons–Coulombfrictionatthecontactpoints:frictionaddsmodestotheplantmodel,whereasourdescrip-tionconcernsthenatureofthetrajectoriesandisdirectlyrelatedtostabilityandcontrolobjectives.
Butitisclearthattheplantmodelingwillstronglyinuencetheconditionsunderwhichthosemodeswillbeactivated.
Suchade-nitionyieldsgenerallyalargenumberofDESstates.
Weshalldeneonlythosethatareneededtodescribethethreementionedtypesofmotion:fk:ightphases(bothfeetdetached);slk:leftfootsticks,rightfootdetached;srk:rightfootsticks,leftfootdetached;dssk:doublesupportphase,bothfeetstick;Irsk:impactsontherightfoot,leftfootsticks;Ildk:impactsontheleftfoot,rightfootdetached.
Thenweobtainthefollowing:Walking:R+=sl0∪Irs0∪dss0∪sr0∪Irl0∪dss1∪sl121a)Hopping:R+=f0∪Ild0∪f1∪Ild121b)Running:R+=f0∪Ild0∪sl0∪f1∪Ird1∪sr1∪f2∪Ild121c)1656Y.
Hurmuzluetal.
/Automatica40(2004)1647–1664Theconditionsofactivationofonemodehavetobestud-ied.
Forinstance,conditionssuchthatstickingoccursatanimpactorduringastephavebeenstudiedinRubanovichandFormal'sky(1981),Hurmuzlu(1993),ChangandHur-muzlu(1994)andGenot,Brogliato,andHurmuzlu(1998).
TheyevidentlystronglydependontheprocessmodellikeAmontons–Coulombfrictionandthemultipleimpactresti-tutionlaw.
Theconcatenationofphasesin(21)correspondstodesiredinvariantsetsoftheDES.
Fromageneralviewpoint,anystabilitycriterionshouldtakeintoaccountboththenon-smoothandthehybridnaturesofsuchcomplemen-taritysystems.
Asitisknown,onemayhaveseveralpointofviewsofhybriddynamicalsystems:continuous-time,discrete-event,ormixed,seee.
g.
Automatica(1999).
4.
3.
Constraintsthatguaranteefootsticking,stabilitymarginsThecontroltorquesu(·)havetoobeycertainconditionstoensurestickingofthecontactingfeetatalltimes.
Theseconditionscanbewrittenas(Genot,1998;Genotetal.
,1998)A(q;)u+B(q;˙q;)0(22)duringsmoothmotion,andasA(q(tk);)˙q(tk)0(23)atimpacttimes(andforsomechoiceofthedoubleimpactmodel),whereisAmontons–Coulomb'sfrictioncoe-cientatcontact.
Thedetailedcalculationsofinequalities(22)and(23)canbefoundinGenot,Brogliato,andHurmuzlu(2001).
Bothinequalitiesin(22)and(23)arenecessaryandsucientconditionstobesatisedbythecontrolinputusothatduringthewholemotion(smoothandnon-smooth)stickingismaintainedanddetachmentismonitored.
Thein-equalityin(22)canberewrittenasA()0andcanbeusedtocomputestabilitymargins(Wieber,2002).
Theno-tionofstabilitymarginhasbeenintroducedcorrectlyinSeoandYoon(1995),whoformulatedasetofconstraintsinthespiritof(22),(23).
The"distance"fromthetrajectorytotheconstraintboundaryisdenedasthemaximalmagnitudeofadisturbancethatisappliedonthebiped.
Stabilitymar-ginshelpunderstandingthedierencebetweenstaticgait(centerofgravitylocatedwithinitsbaseofsupport),anddynamicgait(centerofgravitymayfalloutsidethesup-portbase)(Vaughan,2003).
Tothebestofourknowledge,noneofthecontrollawsproposedintheliteratureuntilnowwasshowntobe"stable"withrespecttothesefundamen-talconditions,mainlyduetothefactthattheunderlyingdynamicsoftherobotdoesnotcapturetheunilateralfea-tureofthefeet-groundcontacts.
Tosummarize,acontrollerwhichguaranteesboth(22)and(23)impliesthatthesystemevolvesintheDESpathin(21a).
4.
4.
PoincaremapsandstabilityGenerally,theapproachtothestabilityanalysistakesintoaccounttwofactsaboutbipedallocomotion:themotionisdiscontinuousbecauseoftheimpactofthelimbswiththewalkingsurface(Hurmuzlu&Moskowitz,1987;Hurmuzlu,1993;Katoh&Mori,1994;Zheng,1989;Grizzleetal.
,2001),andthedynamicsishighlynonlinearandnon-smoothandlinearizationaboutverticalstancegenerallyshouldbeavoided(Vukobratovicetal.
,1990;Hurmuzlu,1993;Grizzle,Abba&Plestan,1999).
AclassicaltechniquetoanalyzedynamicalsystemsisthatofPoincaremaps.
Inthethree-linkbipedalmodelofSection2,wehaveshownthatperiodicmotionsofasimplebipedcanberepresentedasclosedorbitsinthephasespace.
Fig.
3(d)depictsarstreturnmapobtainedfromthepointsofthetrajectorythatcoincidewiththeinstantofheelstrike.
APoincaremapforageneralizedcoordinate1(whichistypicallyajointangleinbipedalsystems)attheinstantofheelstrikenowcanbeobtainedbyplottingthevaluesof1atithversusthevaluesat(i+1)thheelstrike.
Onecanchooseaneventsuchasthemereoccurrenceofheelstrike,todenethePoincaresection(Hurmuzlu&Moskowitz,1987;Kuo,1999).
Wecanconstructseveralmappingsdependingonthetypeofmotion.
Ingeneral,how-ever,thesectioncanbewrittenasfollows:+i={(q;˙q)∈R2p|t=tck};i=1;l;(24)wheretheconditionestablishesthePoincaresection.
Thediscretemapobtainedbyfollowingtheprocedurede-scribedabovecanbewritteninthefollowinggeneralform:Qi=P(Qi1);(25)whereQisareduceddimensionstatevector,andthesub-scriptsdenotetheithand(i1)threturnvalues,respec-tively.
PeriodicmotionsofthebipedcorrespondtothexedpointsofPwhereQ=Pk(Q);(26)wherePkisthekthiterate.
ThestabilityofPkreectsthestabilityofthecorrespondingow.
ThexedpointQissaidtobestablewhentheeigenvaluesi,ofthelinearizedmap,Qi=DPk(Q)Qi1(27)havemodulilessthanone.
ThistechniqueemployedinHurmuzluandMoskowitz(1987),Hurmuzlu(1993),FrancoisandSamson(1998),Kuo(1999),Grizzleetal.
(1999,2001),PiiroinenandDankowicz(2002),DankowiczandPiiroinen(2002),Dankowicz,AdolfssonandNordmark(2001),Piiroinen,Dankowicz,andNordmark(2001,2003)andQuintvanderLinde(1999)hasseveraladvantages.
Usingthisapproachthestabilityofgaitconformswiththeformalstabilitydenitionacceptedinnonlinearmechanics.
Theeigenvaluesofthelinearizedmap(Floquetmultipliers)providequantitativemeasuresofthestabilityofbipedalgait.
Finally,toapplytheanalysistolocomotiononeonlyY.
Hurmuzluetal.
/Automatica40(2004)1647–16641657requiresthekinematicdatathatrepresentalltherelevantdegreesoffreedom.
Nospecicknowledgeoftheinternalstructureofthesystemisneeded.
Thisfeaturealsomakesitpossibletoextendtheanalysistothestudyofhumangait.
Usingthisapproachonecandevelopquantitativemeasuresforclinicalevaluationofthehumangait(Hurmuzlu&Bas-dogan,1994;Hurmuzlu,Basdogan,&Stoianovici,1995).
5.
ControlofbipedalrobotsThecontrolproblemofbipedalrobotscanbedenedaschoosingaproperinputuin(S)suchthatthesystembe-havesinadesiredfashion.
Thekeyissueofcontrollingthemotionofbipedsstillhingesonthespecicationofade-siredmotion.
Therearenumerouswaysthatonecanspecifythedesiredbehaviorofabiped,whichinitselfisanopenquestion.
Thecontrolproblemcanbecomeverysimpleorextremelycomplexdependingonthespecieddesiredbe-haviorandthestructureofthesystem.
Typicalbipedalma-chinesaredesignedtoperformtasksthatarenotconnedtosimplewalkingactions.
Suchtasksmayincludemaneuver-ingintightspaces,walkingorjumpingoverobstacles,andrunning.
Inthisarticle,weplaceourmainfocusontasksthatareprimarilyrelatedtowalking.
Therefore,wewillnotbeconcernedwithactionssuchasperformingmanipulationtaskswiththeupperlimbs.
5.
1.
PassivewalkingPossiblytherstbipedalwalkingmachinewasbuiltbyFallis(1888).
TheideaofabipedwalkingwithoutjointactuationduringcertainphasesoflocomotionwasinitiallyproposedinMochonandMcMahon(1980a).
Theauthorstermedthistypeofwalkas"ballisticwalking".
Theinspi-rationofthisideaoriginatedfromevidenceinhumangaitstudies,whichpointedouttorelativelylowlevelsofmus-cleactivityintheswinglimbduringtheswingperiod(seeBasmajian,1976;Zarrugh,1976).
Twoplanarmodelswereused:(1)athree-elementmodelwithasingle-linkstanceandtwo-linkswingleg,(2)afour-elementmodelwithtwo-linkstanceandswinglimbs.
Ineachcase,theauthorssearchedforinitialconditionsattheonsetoftheswingphasesuchthatthesubsequentmotionsatisedasetofkineticandkine-maticconstraints.
Then,theyisolatedtheinitialconditionsthatsatisedtheimposedconstraints.
Thesimulationresultswerecomparedtoexperimentallymeasuredkneeanglesandgroundreactionforces.
Theyconcludedthattheiroutcomesandhumandatahadthesamegeneralshape.
InMochonandMcMahon(1980b),theyimprovedtheirmodelbyaddingthestanceknee(two-elementstancelimb).
Withthismoreso-phisticatedmodel,theauthorsanalyzedthemodelresponsebyusingthreeoutofthevegaitdeterminants(Saunders,Inman,&Eberhart,1953).
Thepremiseofwalkingwith-outjointactuation,promptedMcGeer(1990)toproposethe"passivewalking".
McGeerdevelopednumericalaswellasexperimentalmodelsofbipedsinspiredbyFallis(1888)thathavecompletelyfreejoints.
Hedemonstratedthatthesesim-ple,unactuatedbipedscanambulateondownwardplanesonlywiththeactionofgravity.
Now,returningto(1)–(4),thetypicalpassivecontrolschemeisconcernedwiththefreedynamicsofthesystem(S)giveninSection3.
1(i.
e.
thedynamicsofthesystemsubjecttou=0).
Then,aswehaveshowninSection4.
1,aPoincaresection+canbeselected(see(24))toobtainanonlinearmappingintheformofQi=P(Qi1;');(28)wheretheparametervector'typicallyincludestheslopeofthewalkingsurface,memberlengths,andmemberweights.
Then,theunderlyingquestionbecomestheexistenceandstabilityofthexedpointsofthismap(Kuo,1999)andtheresemblanceoftheresultingmotionstobipedalwalk-ing.
Thistaskisgenerallyverydiculttorealizewiththeexceptionofverysimplesystems.
Forexample,inthecaseofvibro-impactsystems,analyticalexpressionstoshowex-istencecanbefound(Shaw&Shaw,1989).
Inthecaseofslightlymorecomplexsystems,suchexplicitcalculationsbecomeimpossiblesincethefree-motiondynamicsisnolongerintegrable.
ThenonehastorelyonnumericaltoolstoderiveboththePoincaremappinganditslocalstabil-ity(Kuo,1999;Piiroinenetal.
,2001,2003;Piiroinen&Dankowicz,2002;Dankowicz&Piiroinen,2002).
Themainenergylossinthesebipedsisduetotherepeti-tiveimpactsofthefeetwiththegroundsurface.
Thegravi-tationalpotentialenergyprovidesthecompensationforthisloss,thusresultinginsteadyandstablelocomotionforcer-tainslopesandcongurations.
Thegroundimpacts(Hurmu-zlu&Moskowitz,1986)provideauniquemechanismthatleadstostableprogressioninverysimplebipedsashasbeenrecentlydemonstratedbyseveralinvestigators.
InGoswami,Thuilot,andEspiau(1996,1998),theauthorsconsiderasimplemodelthatincludestwovariablelengthmemberswithlumpedmassesrepresentingtheupperbodyandthetwolimbs.
Athirdlumpedmassisattachedtothispoint,whichrepresentstheupperbody.
Theyanalyzethenonlineardynamicsof(28)subjecttoprescribedvariationsintheele-mentsof'.
Theyprimarilyfocusontheeectofthegroundslope,massdistribution,andlimblengths.
Numericalanal-ysisofthenonlinearmap,resultsinthedetectionofstablelimitcyclesaswellaschaotictrajectoriesthatarereachedthroughperioddoublingcascade.
Asimpler,two-linkmodelwasconsideredinGarciaetal.
(1997).
Theyalsodemon-stratedthatthissimplebipedcanproducestablelocomotionaswellasverycomplexchaoticmotionsreachedthroughfrequencydoublingcascade.
ChatterjeeandGarcia(1998)andDasandChatterjee(2002)studiedtheexistenceofperiodicgaitsinthelimitofzeroslope.
Theadditionofpassivearmsservedtoreduceside-to-siderockingina3Dpassivewalker(Collins,Wisse,&Ruina,2001).
QuintvanderLinde(1999)includesphasicmusclecontractionastheenergysource,andvarythemusclemodelparametersto1658Y.
Hurmuzluetal.
/Automatica40(2004)1647–1664createnewperiodicgaits.
Themainchallengeofthestudyofpassivegaitistotranslatetheunderstandinggainedbystudy-ingpassivesystemstoactivesystems.
TheMITLegLabpla-narbipedsarecontrolledthisway(Pratt,2000)forperiodicwalking.
Inotherwords,thedesiredtrajectoriesqd(t)aredesignedfromthestudyofpassivewalking.
ThisisnotthecasefortheHondarobotswhereqd(t)areobtainedfromhu-manrecordings.
Oneofthemainobstaclestobuildbipedalrobotsremainstobetheprohibitivelyhighjointtorquesthatareoftenrequiredtorealizeevenroutinewalkingtasks.
Acomprehensiveinvestigationthatbridgesthepassivestud-iestobetterdesignofactivecontrolschemeswouldbeanaturalextensionofpassivelocomotionresearch.
ThisisachievedinPiiroinenandDankowicz(2002),DankowiczandPiiroinen(2002),whoproposeacontrolmethodbasedondiscreteadjustmentsoftheswing-footorientationpriortocontact,henceindirectlyaectingthenatureandtimingofthesubsequentimpact.
Thisresultsinthe(local)stabi-lizationofamotionthatisnaturallyoccurringinthesystem.
5.
2.
Walkingwithactivecontrol5.
2.
1.
ControllerdesignThecontrolactionmustassurethatthemotionofamulti-linkkinematicchain,whichcancharacterizeatypicalbiped,isthatofawalker.
Although,thecharacteristicsofthemotionofawalkerisstillopentointerpretation,wemaytranslatethisrequirementtoasetoftarget/objectivefunctionsgivenintheformgi(q(t);˙q(t);qd(t);˙qd(t);n;t;;u)=0i=1;k6p;(29)whereisavectorofparametersthatprescribescertainas-pectsofthewalkingactionsuchasprogressionspeed,steplength,etc.
Notethatforthesakeofsimplicityofthenota-tionsqwilldenoteinthesequelthevectorofgeneralizedcoordinatesofthemodelconsideredbythereferencedau-thors,i.
e.
eitherofthefullordermodel,orofthereducedordermodelwhenassumingthatthegroundcontacts,whenactive,arebilateralcontacts.
Thecontrolproblemcanbedescribedasspecifyingthevectorofjointactuatortorquesuin(1)suchthatthesystembehavesinacertainway.
Thesimplestwaytoproceedistospecifythetimeprolesofthejointtrajectories.
Inves-tigatorsintheeldusedkinematicsofhumangaitasde-siredproles(seeHemami&Farnsworth,1977;Khosravi,Yurkovich,&Hemami,1987;Vukobratovicetal.
,1990).
Onecanalsosimplyspecifycertainaspectsoflocomo-tionsuchaswalkingspeed,steplength,uprighttorso,etc.
(Chudinov,1980,1984;Lavrovskii,1979,1980;Beletskii,1975;Beletskii&Kirsanova,1976;Beletskii&Chudi-nov,1977b,1980;Beletskii,Berbyuk,&Samsonov,1982;Grishin&Formal'sky,1990;Novozhilov,1984;Hurmuzlu,1993;Chang&Hurmuzlu,1994;Yang,1994).
InBeletskiiandChudinov(1980),theauthorsusethecomponentsofthegroundreactionforcesinadditiontokinematicsinordertocompletelyspecifythecontroltorques.
Oncetheobjectivefunctionsarespecied,onehastochooseacon-trolschemeinordertospecifythejointmoments(controltorques)thatdrivethesystemtowardthedesiredbehavior.
Weencounterseveralapproachestothisproblemthatcanbeenumeratedasfollows:(1)Linearcontrol:Theequationsofmotionarelin-earizedabouttheverticalstance,assumingthatthepostureofthebipeddoesnotexcessivelydeviatefromthisposi-tion.
Forexample,inJalics,Hemami,andClymer(1997),KajitaandTani(1996),Kajita,Yamaura,andKobayashi(1992),Grishin,Formal'sky,LenskyandZhitomirsky(1994),ZhengandHemami(1984),Hemami,Zheng,andHines(1982),GollidayandHemami(1977),HemamiandFarnsworth(1977),Gubina,Hemami,andMcGhee(1974),Mitobe,Capi,andNasu(2000),SeoandYoon(1995)andGarcia,Estremera,andGonzalesdeSantos(2002),aPDcontrollerwasusedtotrackjointtrajectories.
Thelinearcontroller,however,cannottracktimefunctions.
Thus,theauthorsdiscretizedthedesiredjointprolesandletthecontrollertrackthetrajectoryinapoint-to-pointfashion.
vanderSoest"Knoek",Heanen,andRozendaal(2003)studytheinuenceofdelaysinthefeedbacklooponstancestability,includingmusclemodel.
(2)Computedtorquecontrol:Thismethodwasapplied(Hemami&Katbab,1982;Lee&Liao,1988;Hurmuzlu,1993;Yang,1994;Jalicsetal.
,1997;Lumetal.
,1999;Song,Low,&Guo,1999;Park,2001;Taga,1995:Chevallereau,2003)tobipedallocomotionmodelswithvariouslevelsofcomplexities.
InChevallereau(2003),thecomputedtorqueiscombinedwithatime-scalingofthedesiredtrajectoriesoptimallydesigned(Chevallereau&Aoustin,2001),whichallowsthenite-timeconvergenceofthesystem'sstateto-wardsthedesiredmotion.
Thenite-timeconvergenceespe-ciallyallowsonetoavoidthetrickyproblemsduetotrack-ingerrorsinducedbyimpacts(Bourgeot&Brogliato,2003;Brogliatoetal.
,1997,2000).
(3)Variablestructurecontrol:Thismethodresultsinafeedbacklawthatensurestrackingdespiteuncertaintiesinsystemparameters.
Inthisapproach,onechoosesthecontrolvectorasui=uikisign(s);(30)whereuiisatrajectorytrackingcontrollerwithxedesti-matedparameters.
Thesecondterm,isthevariablestructurepartofthecontrolinput.
Thefunctionsdenestheslidingsurfacethatrepresentsthedesiredmotion.
Thisisahighgainapproachthatisadvantageousbecauseitensurescon-vergenceinnitetime.
Inlocomotion,thestabilityoftheoverallmotionreliesontheeectivenessofthecontrollerineliminatingtheerrorsinducedbyimpactduringthesub-sequentstep.
ThereadercancheckChangandHurmuzlu(1994)andLumetal.
(1999)toseetheapplicationofsuchacontrollertoave-elementplanarmodel.
Y.
Hurmuzluetal.
/Automatica40(2004)1647–16641659(4)Optimalcontrol:Optimalcontrolmethodshavebeenusedbyresearcherstoregulatethesmoothdynamicphaseofbipedallocomotionsystems.
Twoapproacheshavebeentakentotheoptimizationproblem.
Therstmethodisbasedoncomputingthevaluesofselectedparametersintheob-jectivefunctionsthatminimizeenergy-basedcostfunctions(Frank,1970;Vukobratovic,1976;Beletskii&Chudinov,1977a;Beletskiietal.
,1982;Rutkovskii,1985;Channon,Hopkins,&Pham,1992;Saidouni&Bessonnet,2003).
Wenotethatoptimaltrajectoryplanningincludingthedy-namicsin(1)–(4)isequivalenttosearchingforanoptimalopen-loopcontrolu(t).
Thesecondapproachisbasedonvariationalmethodstoobtaincontrollersthatminimizecostfunctions(Beletskii&Bolotin,1983;Bolotin,1984;Fu-rusho&Sano,1990;Channon,Hopkins,&Pham,1996a,b).
Itisthedirectapplicationofclassicaloptimalcontrolmeth-odstobipedallocomotion,seeChannon,Hopkins,&Pham(1996c)forthemostadvancedworkinthistopic.
Heretheauthorsregulatethemotionofthebipedoverasupportphasewithaquadraticcostfunction.
vanderKooij,Jacobs,Koopman,&vanderHalm(2003)proposeamodelpre-dictivecontrollerdesignedfromatangentlinearizationtoregulategaitdescriptorsformulatedasend-pointconditions.
Themainobstacletowardsrealimplementationisatoolargecomputationtime.
(5)Adaptivecontrol:Theadaptivecontrolapproachhasreceivedverylittleattentioninbipedcontrol.
Perhapsitisdoesnothaverealadvantageincontrollingbipedallocomo-tion.
Nevertheless,Yang(1994)hasappliedadaptivecon-trolapproachtoathreelink,planarrobot.
ExperimentshavebeenledattheMITLegLab(Pratt,2000)usingadaptivecontrol.
(6)Shapingdiscreteeventdynamics:Theabruptnatureofimpactmakesitpracticallyimpossibletodirectlycon-trolitseectonthesystemstate.
EvenanapproximationofanimpulsiveDiracinputwoulddemandactuatorswithtoohighbandwidth(tosaynothingofinducedvibrationsinthemechanicalstructure).
Analternateapproachcanbefoundinshapingthesystemstatepriortotheimpactinstantsuchthatadesiredoutcomeisassured.
SuchanapproachwastakeninHurmuzlu(1993)andChangandHurmuzlu(1994).
Inthesestudies,asetofobjectivefunctionsintheformof(29)wastailored.
Assumingperfecttracking,theauthorsderivedtheexpressionforthesystemstateimmediatelybe-foretheinstantofimpactintermsoftheparametervector.
Subsequently,thepost-impactstatewascomputedforspe-cicvaluesoftheparametervector.
Theparameterspacewaspartitionedintoregionsaccordingtoslippageandcon-tactconditionsthatresultfromthefootimpact.
Then,thispartitioningwasusedtospecifycontrollerparameterssuchthattheresultinggaitpatternhasonlysinglesupportphaseandthefeetwouldnotslipasaresultofthefeetimpact.
DunnandHowe(1994)developedconditionsintermsofmotionandstructuralparameterssuchthattheyminimize/eliminatethevelocityjumpsduetogroundimpactandlimbswitch-ing.
Thus,intheircase,theobjectiveoftheshapingwastoremovetheeectoftheimpactaltogether.
MiuraandShi-moyama(1984)usedafeedforwardinputthatmodiesthemotionattheendofeachstepfrommeasurementsinfor-mations.
Grizzleetal.
(1999,2001)andWersterveltetal.
(2003)havealsousedasimilarapproach.
Theyshapethestatebeforetheimpact,sothatatthenextstepthestatere-sidesinthezerodynamics.
Doingsotheycreateaperiodicgaitthatcorrespondstothezerodynamicsdenedfromasetofoutputfunctions.
PiiroinenandDankowicz(2002)lo-callystabilizeapassivewalkwithaspecicstrategy,seeSection5.
1.
(7)Stabilityandperiodicmotions:Stabilityoftheover-allgaitisoftenoverlookedinlocomotionstudies.
Typically,controllershavebeendeveloped,andfewgaitcycleshavebeenshowntodemonstratethatthebiped"walks"withthegivencontroller.
Athoroughanalysisofthenonlineardy-namicsofaplanar,ve-elementbiped(Hurmuzlu,1993)revealsarichsetofstable,periodicmotionsthatdonotnecessarilyconformtotheclassicalperiodonelocomotion.
Trackingerrorsinthecontrolactionmayleadtostablegaitpatternsthataredierentthantheonesthatareintendedbytheobjectivefunctions.
Onewaytoovercomethisdi-cultyistopartitiontheparameterspacesuchthatonewouldchoosespecicvaluesthatleadtoadesiredgaitpattern.
ThisapproachistakeninHurmuzlu(1993)andChangandHurmuzlu(1994).
(8)Otherspecializedcontrolschemes:Severalinvesti-gators(Grishin&Formal'sky,1990;Grishinetal.
,1994;Beletskii,1975;Chudinov,1980,1984;Katoh&Mori,1994;Lavrovskii,1979,1980).
usedsimpliedmodelswithoutimpactandconstructedperiodictrajectoriesbyconcatenationoforbitsobtainedfromindividuallycon-trolledsegmentsofthegaitcycle.
ThisapproachisquitesimilarinspirittotheKobrinskii(1965)methodthatisusedtheexistenceoftrajectoriesoftheimpactdamperandtheimpactinginvertedpendulum(seeFig.
1).
BlajerandSchielen(1992)computeanonlinearfeedforwardtorquecorrespondingtoa"non-impacting"referencewalkandusePDmotionandPIforcefeedbacktostabilizearoundthereferencetrajectory.
Fuzzylogiccontrolwasused(Shih,Gruver,&Zhu,1991)todevelopaforcecontrollerthatregulatesgroundreactionforcesinswayingactionsofanexperimentalbiped.
Agroupofinvestigatorschangedtheparametersintheobjectivefunctionssuchthatthedesiredmotionisadaptedtochangingterrainconditions(Igarashi&Nogai,1992;Shih&Klein,1993;Zheng&Sheng,1990).
Zheng(1989)usedanaccelerationcompensationmethodtoeliminateexternaldisturbancesfromthemotionofanexperimentaleightjointrobot.
Kuo(1999)derivesnumer-icallyanimpactPoincaremapthatrepresentsthewalkingcycle,andproposesalinearstatefeedbackthatstabilizesthiscycle.
Clearly,thisisconceptuallycompletelydier-entfromtheworksdescribedabove(seeitem(1)Linearcontrol)sincethedesignisbasedonalinearizationofthePoincaremapitselfandnotofthecontinuousdynamicsononestep.
1660Y.
Hurmuzluetal.
/Automatica40(2004)1647–16645.
2.
2.
DESstabilizationAswehaveshowninSection4.
2,walkingcorrespondstoaparticularsequenceofactivationsofthemodesoftheDESassociatedtothebipedseenasacomplementaritymechani-calsystem.
SuchasequencecanbeseenasaninvariantsetoftheDESdynamics,see(21a).
Thesecontroltechniquesaimatstabilizingthisinvariantsetinthesensethattherobotshouldultimatelybeabletorecoverfromfallsandrestartwalking(Fujiwaraetal.
,2002).
Noticethatthisapproachdoesnotemphasizethelow-leveldetailsofthewalk(walk-ingspeed,stepslength,etc.
).
Aninterestingapproachintheareaisthezeromomentpoint(ZMP)methodproposedrstbyVukobratovicandhisco-workers(Vukobratovic&Juricic,1969;Vukobratovicetal.
,1990).
Thereadercanre-fertoGoswami(1999)andWieber(2002)andreferenceswithinforadetaileddiscussionregardingtherealmeaningofZMP,andtoGarciaetal.
(2002)foracompletede-scriptionofequivalentstabilityconcepts.
Severaldierent,butequivalent,denitionsoftheZMParegiven(Hemami&Farnsworth,1977;Takanishi,Ishida,Yamaziki,&Kato,1985;Arakawa&Fukuda,1997;Hirai,Hirose,&Kenada,1998).
Thesimplestoneis(Hemami&Farnsworth,1977)thepointwheretheverticalreactionforceintersectstheground,i.
e.
thecenterofpressure.
TheZMPstabilitycrite-rionstatesthatthebipedwillnotfalldownaslongastheZMPremainsinsidetheconvexhullofthefoot-support.
Inthesestudies,theauthorsimposethemotionofthelowerlimbkinematicsfromhumankinematicdata,whichtheytermsynergies.
Thisway,theZMPcriterionisusedtoswitchbetweenlow-levelcontrollers(whichsatisfysomeobjec-tivefunctionsliketrajectorytracking),soastostabilizetheDESorbitin(21a)(Park,2001),andpossiblyavoidob-stacles(Yagi&Lumelsky,2000).
TheZMPmethodwasalsoappliedwithothercontrollersthatarenotbasedonprescribinghumandata(Borovac,Vukobratovic,&Surla,1989;Fukuda,Komota,&Arakawa,1997;Shih,Gruver,&Lee,1993;Mitobeetal.
,2000;Park,2001;Huangetal.
,2001;Vanel&Gorce,1997).
OneofthebestexampleofthehighdegreeofeciencythatsuchcontrolapproachesareabletoattainarethebipedsconstructedbyHonda(Hiraietal.
,1998;Hirai,1997;Pratt,2000),whosecontrolmainlyrelyonasuitablecombinationoflocallinearcontrollerwithhigh-level(orlogical)conditions.
InPratt,Chew,Torres,Dilworth,andPratt(2001),anintuitiveapproachformak-ingsomebipedalmachineswalkisproposed.
Itisbasedontheso-calledvirtualmodelcontrol.
Theneedforbothlow-andhigh-levelcontroltogetherwithon-linedesiredtrajec-toriesplanningisexplainedinElHaandGorce(1999)andVanelandGorce(1997)whereonlythesupervisoryaspectsarestudied.
Wieber(2002)proposesaquiteinter-estingstudy,startingfrom(22).
AgeneralcriterionfortheDESpath(21a)stability(equivalently,itsinvariance)ises-tablished,andthelinkwithLyapunovfunctionsismade(exceptingimpacts).
Stabilitymargin(roughly,thedistancefromtheactualtrajectorytotheboundaryofanadmissiblesetoftrajectories,outsideofwhichtherobotfallsdown(Seo&Yoon,1995)canbederived.
Suchstudiesareofprimaryimportanceforcharacterizingthestabilityofreha-bilitatedparaplegics(Popovicetal.
,2000).
6.
ConclusionsanddirectionsforfutureresearchThissurveyisdevotedtotheproblemofmodelingandcontrolofaclassofnon-smoothnonlinearmechanicalsys-tems,namelybipedalrobots.
Itisproposedtorecastthesedynamicalsystemsintheframeworkofmechanicalsys-temssubjecttocomplementarityconditions.
Unilateralcon-straintsthatrepresentpossibledetachmentofthefeetfromthegroundandCoulombfrictionmodelcanbewrittenthisway.
InthelanguageofFullandKoditschek(1999),thisisasuitabletemplate.
Suchapointofviewpossessesseveraladvantages:(1)Itprovidesauniedapproachformathematical,nu-mericalandcontrolinvestigations.
Thisisaquiteim-portantpointsincenumericalstudiesaremandatoryinanymechanicaland/orcontroldesign.
(2)Thisframeworkencompassesallthemodelswhichhavebeenusedtostudylocomotioninthecontrolandroboticsliterature.
(3)Thoughwerestrictourselvestorigidbodycon-tact/impactmodels,lumpedexibilitiescaneasilybeintroduced,bothatthecontactorinthestructureit-self(exiblejoints).
Introducingexibilitiesmaybenecessary(Pratt,2000;Pratt&Williamson,1995),andisphysiologicallysound(Gunther&Blickman,2002).
Thiswill,however,makethecontrolproblemhardertosolveandmaybeineectiveinwalking(Karetal.
,2003).
Itmayalsocreateseriousdicultiesintheanalysis,especiallywithmultiplecontacts(Paoli&Schatzman,2002).
(4)Suchmodelshaveprovedtopredictquitewellthemo-tionasseveralexperimentalvalidationsavailableintheliteratureshow(Abadie,2000).
(5)Asshowninthissurvey,theproposedmodelingap-proachallowsonetoclarifywhichstabilitytoolsonemayusetocharacterizethestabilityofabipedalrobot.
(6)Finally,itistheopinionoftheauthorsthatoneimpor-tantdevelopmentisstillmissingintheeldofbipeddesign:aconciseandsucientlygeneraltheoreticalanalysisframework,basedonrealisticmodels,thatal-lowsthedesignertoderivestablecontrollerstakingintoaccountthehybriddynamicsintheirentirety.
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YildirimHurmuzlureceivedhisPh.
D.
degreeinMechanicalEngineeringfromDrexelUniversity.
Since1987,hehasbeenattheSouthernMethodistUniversity,Dal-las,Texas,whereheisaProfessorandChairmanoftheDepartmentofMechani-calEngineering.
HisresearchfocusesonnonlineardynamicalsystemsandControl,withemphasisonrobotics,biomechanics,andvibrationcontrol.
Hehaspublishedmorethan60articlesintheseareas.
Dr.
HurmuzluistheassociateEditoroftheASMETransactionsonDynamicsSystems,MeasurementandControl.
FrankGenotwasbornin1970inZweibrucken(Germany).
HegraduatedfromtheEcoleNationaleSuperieured'InformatiqueetdeMathematiquesAp-pliqueesdeGrenoble(France)in1993.
HegotthePh.
D.
degreefromtheInsti-tutNationalPolytechniquedeGrenobleinComputerScienceinJanuary1998.
SinceSeptember2000,hehasbeenanINRIAResearcherintheMACSresearchprojectatINRIARocquencourt(France).
Hismainresearchinterestsincludemodellingandsimulationissuesofsystemswithunilateralconstraints,inMechanicsandFinance,andStructuralControl.
BernardBrogliatogothisPh.
D.
fromtheInstitutNationalPolytechniquedeGrenobleinJanuary1991.
HeispresentlyworkingfortheFrenchNationalInstituteinCom-puterScienceandControl(INRIA),intheBipopproject.
Hisscienticinterestsareinnon-smoothdynamicalsystems,modelling,stabilityandcontrol.
HeisamemberoftheEuromechNonLinearOscillationsCon-ferencecommittee(ENOCC),reviewerforMathematicalReviewsandtheASMEAp-pliedMechanicsReviews,andisAssociateEditorforAutomaticasinceOctober1999.