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April WSPCIJHR International Journal of Humanoid Robotics Vol

1 No 1 2004 157173 World Scienti64257c Publishing Company ZEROMOMENT POINT THIRTY FIVE YEARS OF ITS LIFE MIOMIR VUKOBRATOVI Institute Mihajlo PupinVolgina 15 11000Belgrade Serbia and Montenegro vukrobotimpbgacyu BRANISLAV BOROVAC University of Novi

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April WSPCIJHR International Journal of Humanoid Robotics Vol






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April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.Borovacedgescausedbystrongdisturbances,whichisequivalenttotheappearanceofanunpowered(passive)DOF,(ii)gaitrepeatability(symmetry),whichisrelatedtoregulargaitonly,and(iii)regularinterchangeabilityofsingle-anddouble-supportphases.Duringwalking,twodierentsituationsariseinsequence:thestaticallystabledouble-supportphaseinwhichthemechanismissupportedonbothfeetsimultaneously,andstaticallyunstablesingle-supportphase,whenonlyonefootofthemechanismisincontactwiththegroundwhiletheotherisbeingtransferredfromthebacktofrontpositions.Thus,thelocomotionmechanismchangesitsstruc-tureduringasinglewalkingcyclefromanopentoaclosedkinematicchain.Allthesecircumstanceshavetobetakenintoaccountinarti“cialgaitsynthesis.Allofthebipedmechanismjointsarepoweredanddirectlycontrollableexceptforthecontactbetweenthefootandtheground(whichcanbeconsideredasanadditionalpassiveDOF),wheretheinteractionofthemechanismandenvironmentonlytakesplace.Thiscontactisessentialforthewalkrealizationbecausethemech-anismspositionwithrespecttotheenvironmentdependsontherelativepositionofthefoot/feetwithrespecttotheground.Thefootcannotbecontrolleddirectlybutinanindirectway,byensuringtheappropriatedynamicsofthemechanismabovethefoot.Thus,theoverallindica-torofthemechanismbehavioristhepointwherethein”uenceofallforcesactingonthemechanismcanbereplacedbyonesingleforce.ThispointwastermedtheZero-MomentPointZMPRecognitionofthesigni“canceandroleofZMPinthebipedarti“cialwalkwasaturningpointingaitplanningandcontrol.Theseminalmethodforgaitsynthesis(semi-inversemethod)wasproposedbyVuko-bratovi´candJurici´Itshouldbenotedthatdespiteofthelimitationthatthemotioncanbesynthesizedonlyforasmanyjointsasthezero-momentconditionscanbepreset,thismethodhasremainedforalongtimetheonlyprocedureforbipedgaitsynthesis.TheZMPisalsoindispensableinbipedcontrol,forestablish-ingthepracticallyunavoidablefeedbackwithrespecttodynamicgroundreactionforces.Inthispaperwereviewsomebasicissuesrelatedtobipedlocomotionwithpar-ticularattentionpaidtotheZMPbecauseofitscrucialimportanceforgaitanalysis,synthesisandcontrol.DespitethefactthatthenotionofZMPhasneverbeenintro-ducedintheformofaformalde“nition,inthecourseofalmostthreeandahalfItshouldbenotedthatin“rsttwopapersneitherthetermZMP(themechanismhadapin-pointfootandnosupportareawasemployed)norsemi-inversemethodwereexplicitelymentioned.However,thecompensationaldynamicswasobtainedonthebasisofthesemi-inversemethodandtheZMPconcept,althoughthepossiblepositionsofZMPinthiscasewerereducedtothetipofthepin-pointfoot.Acoupleofyearslater,whenaspatiallinkwasusedinsteadofthepin-pointfoot,thenotionofZMPwasformallyintroduced.Actually,wecansetupzero-momentconditionsforanypassive(unpowered)DOFofthemecha-nism.Forexample,apartfromthefoot-groundcontactwecansetupzero-momentconditionsfortheshoulderjointforfreelyswingingarms(passiveDOFs),whilethemotionatallotherjointshastobeprescribed. April10,200422:17WSPC/191-IJHR00008 Zero-MomentPoint—ThirtyFiveYearsofItsLifedecadesthisconcepthasbeeninvolvedinverydiverseapplicationsrelatedtonume-rousanthropomorphiclocomotionmechanismsofdierentdegreesofcomplexity.TheaimofthisworkisprimarilytoremindthereaderoftheseminalresultsrelatedtoZMP,whose“rstpracticalapplicationwasintherealizationofthedynam-icallybalancedbipedgaitin1984(performedbytheWL-10RDrobot,developedinIchiroKatoslaboratoryatWasedaUniversity),andwhichwasreported16yearsaftertheappearanceoftheZMPconcept.Besidesthis,afterinspectingnumer-ouspapers,publishedespeciallyintheproceedingsofinternationalconferencesdevotedtohumanoidrobots,wehavefoundsomeinsucientlyprecise,andsome-timesincomplete,de“nitionsofZMPthatmightpotentiallyleadtoaninappropri-ateunderstandingofthisconcept,especiallybyyoungerresearchers,thoughthisconcepthasgainedunequivocalcon“rmationthroughagreatnumberofsophisti-catedrealizationsofhumanoidrobots.Hence,thispaperaimsatrefreshingtheZMPnotion,re-stressingitsbasicmeaning,andmentioningsomenew,butveryessential,phenomenathathavestillremainedfromthefocusofthestudiesongaitdynamicsandcontrol.Finally,wetouchuponsomeotherformsoflocomotion-manipulationactivitiesconsideredasextremelycomplexcontacttasks.2.TheZMPNotionApartfromtherealizationoftherelativemotionofthemechanismslinks,themostimportanttaskofalocomotionmechanismduringthegaitistopreserveitsdynamicbalance(somenewŽauthorsusethetermstabilityŽ!),whichisachievedbyensuringthefootswholearea,andnotonlytheedge,isincontactwiththeground.Thefootreliesfreelyonthesupportandtheonlycontactwiththeenvi-ronmentisrealizedviathefrictionforceandverticalforceofthegroundreaction.Letusconsiderthelocomotionmechanisminthesingle-supportphase[Fig.1(a)],withthewholefootbeingontheground.Tofacilitatetheanalysiswecanneglectthepartofthemechanismabovetheankleofthesupportfoot(pointA)andreplaceitsin”uencebytheforceandmoment[Fig.1(b)],wherebytheweightofthefootitselfactsatitsgravitycenter(pointG).ThefootalsoexperiencesthegroundreactionatpointP,whoseactionkeepsthewholemechanisminequilibrium.Ingeneral,thetotalgroundreactionconsistsofthreecomponentsoftheforce)andmomentand).Sincethefrictionforceactsatthepointofcontactofthefootwiththeground,andthefootonthegroundisatrest,thosecomponentsoftheforceandmomentthatactinthehori-zontalplanewillbebalancedbyfriction.Therefore,thehorizontalreactionforce)representsthefrictionforcethatisbalancingthehorizontalcomponentoftheforce,whereastheverticalreactionmomentrepresentsthemomentoffrictionreactionforces[Fig.1(c)]thatbalancestheverticalcomponentofthemomentandthemomentinducedbytheforce.Thus,ifweassumethefoot-”oorcontactiswithoutsliding,thestaticfrictionwillcompensateforthehor-izontalforcecomponents()andverticalreactiontorque().Thevertical April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.Borovac R A MAZMZ XYZ0 AA G M RPmgsMFAMAM z yZ Y PR Fig.1.Bipedmechanismandforcesactingonitssole.reactionforcerepresentsthegroundreactionthatbalancesverticalforces.Itremainstoconsiderthebalancingofthehorizontalcomponentofthefootloadmoment.However,duetoaunidirectionalnatureoftheconnectionbetweenthefootandtheground(itisobviousthatthegroundreactionforceinducedbyfootactionisalwaysorientedupwards)horizontalcomponentsofallactivemomentscanbecompensatedforonlybychangingpositionofthereactionforcewithinthesupportpolygon.Therefore,thehorizontalcomponentofthemomentwillshiftthereactionforcetothecorrespondingposition,tobalancetheadditionalload.ThisisillustratedinFig.1(d),where,forthesakeofsimplicity,wepresentasimpleplanarcaseintheplane.Themomentisbalancedbyshiftingtheactingpointoftheforce,whoseintensityisdeterminedfromtheequationofbalanceofalltheforcesactingonthefoot,bythecorrespondingdistance.Itisnecessarytoemphasizethatallthetimethereactionforceiswithintheareacoveredbythefoot,theincreaseintheanklemomentwillbecompensatedforbychangingthepositionofthisforce,andnohorizontalcomponentsofthemomentsandwillexist.ThisisthereasonwhyinFig.1(b)atpointPonlythecomponentHowever,iftherealsupportpolygonisnotlargeenoughtoencompasstheappropriatepositionoftheforcetobalancetheactionofexternalmoments,theforcewillactatthefootedgeandtheuncompensatedpartofthehorizontal April10,200422:17WSPC/191-IJHR00008 Zero-MomentPoint—ThirtyFiveYearsofItsLifecomponentofthereactionmomentwillcausethemechanismsrotationaboutthefootedge,whichcanresultinthemechanismsoverturning.Therefore,wecansaythatthenecessaryandsucientconditionforthelocomotionmechanismtobeindynamicequilibriumisthatforthepointPonthesolewherethegroundreactionforceisacting,(1)Sincebothcomponentsrelevanttotherealizationofdynamicbalanceareequaltozero,anaturalchoicetonamethispointwasZero-MomentPoint.Or,inotherwords,allthetimethereactionofthegroundduetothefootrestingonitcanbereducedtotheforceandverticalcomponentofthemoment;thepointPatwhichthereactionforceisactingrepresentsZMP.Now,alogicalquestioncanbeposed:giventhemechanismdynamics,whatshouldtheZMPpositionbethatwouldensuredynamicequilibrium?Itshouldbenotedthatinviewofthefactthattheentiremechanismissupportedonthefoot,aprerequisiteforthemechanismsdynamicequilibriumisthatthefootrestsfullyonthe”oor.Thus,toanswerthepreviousquestionletusstatethestaticequilibriumequationsforthesupportingfoot[Fig.1(b)]:(2)(3)whereandareradiusvectorsfromtheoriginofthecoordinatesystemxyztothegroundreactionforceactingpoint(P),footmasscenter(G),andanklejoint(A),respectively,whilethefootmassis.IfweplacetheoriginofthecoordinatesystematthepointPandprojectEq.(3)ontothe-axis,thentheverticalcomponentofthegroundreactionmoment(actually,itisthegroundfrictionmoment)willbeInageneralcase,thismomentisdierentfromzeroandcanbereducedtozeroonlybytheappropriatedynamicsoftheoverallmechanism.However,theprojectionofAlthoughnorotationalslippageofthefootoverthegroundsurfacewilloccurinanormalwalk,tocoverthis(veryhypothetical)possibilityitcanberequestedthatduringthemechanismmotioneventhethirdmomentcomponentisequaltozero(=0).Toachievethis,themechanismshouldperformsomeadditionalmovements,forexample,bythetrunkabouttheverticalaxistoensurethat=0,wherethesuperscriptVstandsfortheverticalcomponentofthemomentattheanklejont.However,foraregularmotionandanormalfrictioncoecientbetweenfootandgroundtherequirement=0isnotnecessarybecausethismomentisintrinsicallycompensatedbythefrictionforce. April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.BorovacEq.(3)ontothehorizontalplanegives(4)Thisequationisabasisforcomputingthepositionofthegroundreactionforceactingpoint(P).Equation(4),representingtheequationofthefootequilibrium,answerstheabovequestionconcerningtheZMPpositionthatwillensuredynamicequilibriumfortheoverallmechanismdynamics,butitdoesnotanswertheinversequestion:whetherforthegivenmotionthemechanismisindynamicequilibrium?Toanswerthisquestionwehavetoconsidertherelationshipbetweenthecom-putedpositionofPandthesupportpolygon.IfthepositionofpointP,computedfromEq.(4),iswithinthesupportpolygon,thesystemisindynamicequilibrium.However,inreality,thepointPcannotexistoutsidethesupportpolygon,asinthatcasethereactionforcecannotactonthesystematall.Fromthisfollowsastraightforwardbutveryimportantconclusion:inreality,inordertoensuredynamicequilibrium,apointPthatsatis“esEq.(4)mustbewithinthesupportpolygon.IfwesupposeforamomentthatthepointPisoutsidethesupportpolygon,letusconsiderwhatwouldthenbethemeaningofthispoint.InviewofthefactthatthispositionofPwasobtainedfromthecondition=0,wecanconsideritasa“ctitiousZMP(FZMP).Therefore,inreality,ZMPcanexistonlywithinthesupportpolygon,andthispointwecantermregularZMP,orZMPforshort,andallthecalculatedpositionsofthepointPoutsidethesupportpolygonrepresent“ctitiouslocations.Letusexplainthisinmoredetail.ItisclearfromEqs.(2)and(3)thattheZMPpositiondependsonthemechanismdynamics(i.e.onand).InthesituationwhenthemechanismdynamicschangessothattheZMPapproachesthesupportpolygonedge(ineithersingle-supportordouble-supportphases)letusfocusourattentiononthemomentwhentheZMPisjustreachingthesupportpolygonedge.ThecorrespondingpointwillremaintheZMPonlyifnoadditionalmomentsareactingatthispoint.However,ifanadditionalmomentappeared,thelocomotionmechanismwouldstarttorotateaboutthefootedgeandthemechanismwouldcollapse.Insuchasituation,theactingpointofgroundreactionforcewouldbeonThetermFoot-RotationIndicator(FRI)Pointhasbeensuggested.Obviously,inaregulargait,itiswhollyundesirabletohavetheZMPonthesupportpolygonedge(orclosetoit),asanadditionalmomentthatwouldcausethemechanismtooverturneasily.InthatcaseanurgentactionofthebipedcontrolsystemwouldberequiredtobringtheZMPbacktothesafetyzone.ŽThiscanbeachievedbyappropriateinterventionmovements.Inreality,thefootisnotideallyrigidbutdeformable,andinthecaseofitsinitialrotationinclina-tion,theedgewilltransformintoanewsurface,e.g.intoanarrowstrip.Withincreasinginclinationangle,thesizeandpositionofthecontactsurfacewillchange,andconsequently,anewcontactarea(strip)willbeestablished.IftheZMPiswithinthenewcontactarea(strip),themechanismsdynamicequilibriummightbepreservedevenifthecontactareaisoutsidetheprevioussupportpolygon.Inotherwords,thecondition=0willbeful“lleduntiltheZMPiswithintheinstantenouscontactarea,irrespectiveofwhetheritiswithinoroutsidetheprevioussupportpolygonthatexistedbeforetheinclination.Inthisway,i.e.byfootinclination,itispossibletocompensateforawiderspanofdisturbance.Hence,theelucidation,modelingandrealizationof April10,200422:17WSPC/191-IJHR00008 Zero-MomentPoint—ThirtyFiveYearsofItsLife x Fig.2.Examplesofthedispositionofforcesensorsonthemechanismssole.thefootedge(thereactionforcemustopposetheactionforceatthesamepoint!),butthispointwouldnotbeZMPanymore,sincebothconditions=0and=0wouldnotbeful“lledsimultaneously.TofurtherclarifythemeaningoftheZMPoutsidethesupportpolygon(FZMP)letusberemindedthattherearetwodierentcasesinwhichtheZMPplaysakeyrole:(i)indeterminingtheproperdynamicsofthemechanismabovethefoottoensureadesiredZMPposition,(ii)indeterminingtheZMPpositionforthegivenmechanismmotion.Case(i)belongstothetaskofgaitsynthesisandwillnotbefurtherelabo-ratedhere,whereasCase(ii)referstothegaitcontrol,wheretheZMPpositionisakeyindicatorofthemechanismdynamicequilibrium.Thus,acrucialquestionishowtodeterminetheZMPposition.Inthecaseofarealwalkingmechanism,informationaboutZMPpositioncanbeobtainedbymeasuringforcesactingatthecontactofthegroundandthemechanism,withtheaidofforcesensorsonthemechanismssole.Itshouldbenoticedthatmeasurementcouldbeperformedonlyifallforcesensors(seeFig.2)areincontactwiththeground.Ifsomeofthesensorsdeployedfromthegroundsurface,themechanismasawholewouldrotateaboutthefootedgeandoverturn.Toovercomesuchasituationitisnecessarytochangethecontrolstrategy.However,ifthebipedgaitisinvestigatedusingadynamicmodel,theZMPpositionmustbecomputed.Foragivenmechanismmotion,theforceandmomentattheanklejoint(and)canbeobtainedfromthemodelofthemechanismdynamics,andallelementsinEq.(4)exceptforwillbeknown.TheprocedurefordeterminingZMPpositionconsistsoftwosteps.Step1.ComputefromEq.(4)(seeFig.1).LetuscalltheobtainedpositionofthepointPcomputedZMPposition.Noteagainthatatthismomentweactuallythefootasa”exiblestructure,havingasoftcontactsurface,isanimportantandcomplextaskthatremainstobeproperlyresolved.Theonlysituationwhenadynamicallybalancedgaitisperformedwhilethegroundreactionforceisintentionallykeptwithinaverynarrowarea(thetiptoe)occursinaballeticmotion,butthisdoesnotbelongtoaregularbipedgait. April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.Borovac YZ0 A MFAA PR XZ0 PY A MAFA r FZMP Fig.3.IllustrationofthedeterminationofZMPposition:(a)Step1,and(b)Step2.donotknowwhetherthispositionofpointP[seeFig.3(a)]willbewithintherealsupportpolygonoroutsideit.Step2.ThecomputedZMPpositionisjustacandidatetobearegularZMPanditspositionshouldbecomparedwiththerealsupportpolygonsize.Ifthecom-putedZMPisoutsidethesupportpolygon,thismeansthatthegroundreactionforceactingpoint(P)isactuallyontheedgeofthesupportpolygonandthemech-anismrotationaboutthesupportpolygonedgewillbeinitiatedbytheunbalancedmoment,whoseintensitydependsonthedistancefromthesupportpolygonedgetothecomputedpositionofZMP,i.e.totheFZMPposition.TheaboveprocedureisillustratedinFig.3.InStep1,weobtainananswertothequestionconcerningtheZMPlocationforthegivendynamicsnottakingintoaccounttherealfootsize[seeFig.3(a)],whereasinStep2,weobtaintheanswerwhether,regardingthefootsize(moreprecisely,thesupportpolygonsize),themechanismisreallybalancedornot,andwheretheregularZMP(provideditexists)islocated.Ifthecomputedactingpointofthegroundreactionforceiswithintherealsupportpolygon,thispointisZMPandthemechanismisinequilibrium.Ifthisisnotthecase,thegroundreactionforceactingpointwillbeonthesupportpolygonborder(thegroundreactionforcecannotexitthesupportpolygon!)andthedistancefromittothecomputedZMPpositionisproportionaltotheintensityoftheperturbationmomentthatactsonthefoot[Fig.3(b)].TheZMPconcepthasbeenproperlycomprehendedbyresearchers,widelyused,andveryfrequentlycited.Itcanbenotedthat,althoughbeingessentiallycorrect,alltheZMPde“nitionsdiersigni“cantlyintheextentoftheirdetail.Toillustratethiswegivejusttwointerpretations.The“rstinterpretationisbasicallythesameintwopapers:ZMPinterpretation1.ZMPisdeÞnedasthatpointonthegroundatwhichthenetmomentoftheinertialforcesandthegravityforceshasnocomponentalongthehorizontalaxes April10,200422:17WSPC/191-IJHR00008 Zero-MomentPoint—ThirtyFiveYearsofItsLifeTheotherinterpretationZMPinterpretation2.pisthepointthatandrepresentthemomentsaroundx-andy-axisgeneratedbyreactionforceandreactiontorque,respectively.ThepointpisdeÞnedastheZeroMomentPoint(ZMP).WhenZMPexistswithinthedomainofthesupportsurface,thecontactbetweenthegroundandthesupportlegisstable:ZMPZMPZMPwhereZMPdenotesapositionofZMP.Sdenotesadomainofthesupportsurface.ThisconditionindicatesthatnorotationaroundtheedgesofthefootoccursPrimarilybecauseofthoseyoungerresearchersthatarejustbeginningtheirworkinthis“eldandwhooftenhavehadnoinsightintotheoriginalworksinwhichtheZMPnotionwasintroduced,letusnoticethatZMPhasoftenbeeninsucientlypreciselyrelatedtothegroundsurface(asurfaceofpracticallyunlimitedsize),evenwithoutmentioningthesupportpolygon.Also,ithasoftenbeenmissedtostressthataZMPoutsidethesupportpolygonpracticallyhasnosense,asinZMPdefactodoesnotexist,andinrealitythemechanisminsuchsituationsfallsbyrotatingabouttheedgeofthesupportpolygon.Herewehavetopointoutanotherimportantissue,andthisisthedierencebetweenthecenterofpressure(CoP)andZMP,asitisveryimportanttomakeacleardistinctionbetweenthetwonotions,whichmustnotgenerallyberegardedasidentical.Thepressurebetweenthefootandthegroundcanalwaysbereplacedbyaforceactingatthecenterofpressure„CoP.Ifthisforcebalancesallactiveforcesactingonthemechanismduringthemotion(inertia,gravitation,Coriolisandcentrifugalforcesandmoments)itsactingpointisZMP.Thus,inthecaseofadynamicallybalancedgait,CoPandZMPcoincide.Whenthegaitisnotdynamicallybalanced,ZMPdoesnotexistandthemechanismcolapsesaboutthefootedge.TomaketheZMPnotionanditsrelationshipwithCoPperfectlyclearwewillsummarizeourpreviousdiscussioninthreecharacteristiccasesforanon-rigidfootincontactwiththeground,assketchedoutinFig.4.Inabalancedgait,theZMPcoincideswithCoP[Fig.4(a)].Inthecaseofadisturbancethatbringstheactingpointofthegroundreactionforcetothefootedge,theperturbationmomentwillcauserotationofthebipedsystemaboutthefootedge(aswealreadymentioned,thefootedgeisinfactaverynarrowstripastheshoesoleisnottotallyrigid)anditsoverturning.Inthatcasewecanspeakonlyofthe“ctitiousZMP,whosedistancefromthefootedgerepresentstheintensityoftheperturbationmoment[Fig.4(b)].However,itispossibletorealizethebipedmotion,forexample,onthetoetips[Fig.4(c)]withspecialshoeshavingapinpointarea(balleticmotion),whilekeepingtheZMPpositionwithinthepinpointarea. April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.Borovac –R–R–R–M ZMP CoP ZMPFZMPCoP Fig.4.PossiblerelationsbetweenZMPandCoPforanon-rigidfoot:(a)dynamicallybalancedgait,(b)unbalancedgaitwhereZMPdoesnotexistandthegroundreactionforceactingpointisCoPwhilethepointwhere=0and=0isoutsidethesupportpolygon(FZMP).Thesystemasawholerotatesaboutthefootedgeandoverturns,and(c)tiptoedynamicbalance(balleticmotionŽ).AlthoughtheZMPnowcoincideswithCoP,itisnotaregulargait,andthepersonshouldbespeciallytrainedtoperformit.Here,itisnecessarytoberemindedthatthetaskofderivingamodelofnominaldynamicsofahumanoidrobotisconcernedwithsatisfyingacertainnumberofdynamicconnections.Thisisinfacttheso-calledmixedtypeoftask,whenthelinksmotionandthedrivingtorquesarebothpartlyknownandtheircomplementsaresought.Inthecaseofinvestigatingthedynamicsofbipedstructure,themotionofthelinksperformingagiventypeofgaitisknown,whiletheknownmomentsareequaltozero.Thelatterfollowsfromtheequilibriumconditionsholdingforaselectedpointwithinthesupportpolygonandforthejointsofpassivelinks.Therefore,therearetwotypesofzero-momentpoints.Bothofthemservetoformthemodelofnominaldynamicsofthehumanoidrobot,butthosewithinthesupportpolygonarepracticallyunavoidableingaitsynthesisaswellasfortheoverallcontrolofdynamicallybalancedgait.TorelatetheZMPnotiononlytoCoPisnotcorrectastheZMPcanexistatsomeotherpointsinthesystem,e.g.attheshoulderjontsifweconsiderarmsasfreely-swingingpendulumswithnoactuatorsatthejoints.Insummary,theZMPalwayscoincideswiththeCoP(dynamicallybalancedgait),buttheCoPisnotalwaysZMP(dynamicallyunbalancedgait).However,theFZMPnevercoincideswiththeCoPbecauseCoPcannot,naturally,existoutsidethesupportpolygon.3.SomeFurtherNotesZMPandFZMPItisofcrucialimportancetoexplainthesigni“canceandroleoftherealZMPandits“ctitiouspositionoutsidethesupportpolygon„FZMP.Inhumanoidgaitrealization,thetaskofprimaryimportanceisundoubtedlytoconstantlymaintaindynamicequilibrium,i.e.toperformdynamicstabilization.HencethebasictaskofthecontrolsystemistokeeptheZMPwithinthesupport April10,200422:17WSPC/191-IJHR00008 Zero-MomentPoint—ThirtyFiveYearsofItsLifepolygon,topreventitfromcomingtooclosetothesupportpolygonedge,andthusavoidthelossofequilibriumoftheoverallsysteminthecaseofasuddendisturbance.However,thequestionremainswhattodoifsuchasituationstillaroseandwhetherpotentialinformationabouttheFZMPcouldbeofanyhelp.Inthegaitperformedbyawalkingmechanism,atthemomentoftheoccurrenceofanexternaldisturbance,thecontactofthehumanoidmechanismwiththegroundwillbereducedtoanarrowstriponthefootedge,andthatverymomentwillceasethepossibilityofregularmaintainingofthemechanismsdynamicbalance.Namely,bylosingregularcontactofitsfootwiththeground,thehumanoidlosestheforcefeedbackofthegrounddynamicreaction,i.e.thepossibilityofstabilizingitselfasawhole.Suchasituationcanarisebothinthesingle-supportanddouble-supportphasesofthegait.Inthatcaseanemergency-copingstrategycanbeapplied,whichprimarilyassumesthemovementsofthearmsinanattempttodiminishtheperturbationmoment,combinedwithanincreaseinstrideandmovingthelegasidetoenlargethetrace.Thiseventuallycanbringabouttheenlargementofthesupportpolygonwithinwhichistobelocatedanew,emergencyŽZMP.Ifthecriticalsituation(i.e.therobotsoverturning)isthusovercome,furtherrobotmotionmaybeinterruptedandrestartedintheformofaregulargait,or,ifpossible,themotionwillnotbeinterruptedbut,afterseveraltransitionalsteps,continuedinthesamemanneraspriortotheoccurrenceofthedisturbance.Itshouldbeemphasizedthatthisoutlinedemergency-copyingŽstrategyisanextremelydelicatetask,requiringspecialsensorslikegyroscopesandotherhigh-techtransducers,aswellasverypowerfulcontrolunitscapableofupdatingactuatordatainmicroseconds.Insteadofusingspecialhighlysophisticatedsensorsandfastmicroprocessorcontrolunitstostabilizethehumanoidrobotinrealtimeinthecaseofemergency,theproblemofdynamicinterventioncanbesolvedinanother,lesssophisticatedway.Theprocedurewouldconsistofarmmotionbywhichsomeaddi-tionalcontactswouldbemade(themechanismmayleanusingitshandsagainstsomeobjectinitssurroundings),resolvingthustheproblemofthemomentarylossofdynamicbalanceofthepreviousanthropomorphiccon“guration.Preventingtherobotssoverturningcanalsobeachievedbytemporaryrecon“gurationintoaquadrupedusingtheupperextremities,followedbyre-establishingthemotionintheformofregulardynamicallybalancedbipedgait.Namely,byensuringadditionalsupportpointsstaticequilibriummaybere-establishedandthedynamicallybal-ancedgaitcontinued.Thisprocedureofre-establishingdynamicequilibriummightbeconsideredasakindoftotalcomplianceprocedure.SomeprospectivetasksTheexpectationstobemetbyhumanoidrobotsareconstantlygrowingbothinnumberandspeci“city.Alreadytodaywecanenvisagetheambitioususeofserviceroboticsinthewidestsense,fromhelping(orreplacing)humansinhazardoussitu-ationsandhostileenvironmentstoentertainmentandsocializationŽofman-robotcommunication. April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.BorovacHenceitisnecessarytomakecertainimprovementsandre“nementstohumanoidrobots,bothinthedomainofcomplexityoftheirmechanisms(DOFs)andensuringnewperformance,whichontheotherhandwoulddemandtheinclusionofsomenew,previouslyneglectedphenomenainthemodelingandcontrolofhumanoidrobots.Letusmentiononlythosephenomena(limitedtothedynamics-controldomainonly)whosepresencecouldyieldnew,signi“cantlyimproved,performanceandcapabilitiesofhumanoidrobots:Elasto-dynamicsandincreasedcomplexityoftherobotfootformorerealisticdescriptionofthecontacttaskRobot-DynamicEnvironment,toenableappro-priatedynamiccontrolwithrespecttopositionandcontactforceofthedynamicreaction.Softnessofthetwo-linksemi-rigidfootinsteadoftheconventionallytreatedrigidfoot,asthisplaysanimportantroleintheappearanceofunpoweredDOFsbetweenthefootandtheground.Namely,insteadoftheunnaturaledgeaboutwhichthemechanismwouldrotateinthecaseofalargedisturbance,itismorerealistictoconsiderthecontactintheformofsomenarrowerareaofthearti“cialfootappearingasaresultofthementionedfootelasticity.ThisphenomenonisimportantbecauseoftheZMPposition,which,incontrasttotheconservativecaseofsuddenrotationaboutthefootedge(theoreticalline),isfoundonabor-derstripofthefoot,givingthusahigherchanceofusingamoreeectivecontrolstrategyinthecriticalregimesofthesynthesizedgait.Elasticityofhumanoidrobotjoints,especiallyoftheanklejoints,whereappro-priateactiveabsorberscouldbebuiltin,whosedampingwouldchangedependingontheimpactoftherobotsfootagainsttheground.Thisphenomenonhasbeeninitiallyconsideredinsomepapersfromthedomainofhumanoidrobotics.Inadditiontothevariabledampingcoecient,careshouldalsobepaidtothereal-izationofvariableactivestiness,whichrepresentsasomewhatmorecomplexcaseforimplementationinhumanoidrobots.Synthesisofdynamicposition-forcecontrolofarti“cialgaitinthecaseofelasticactivelydampedjointsandtheelastodynamiccharacterofthefoot.Resolvingtheproblemofquasi-continuoustransitionoftheZMPfromthesingle-supporttothedouble-supportgaitphase.Itshouldalsobeborneinmindthequasi-continuityoftheZMPtrajectoriesthatareapproximatelyrealizedattheirdiscretelocations.Whenanalyzingthein”uenceofthecharacterofZMPtra-jectoriesinrespectofthedegreeofanthropomorphismofhumanoidrobots,thelattercharacteristicshouldalsobeconsideredinrelationtotheincreaseinthenumberofDOFs.Smoothtransitionfromonewalkingpatterntoanother(e.g.transitionfromwalk-ingona”atsurfacetowalkingupstairsanddownstairs,avoidingobstacles,walkation,etc.).AspecialchallengerepresentstheindependentŽuseofhandsforanothertaskduringthewalk(e.g.takingobjectsfromthetableinpassingwithoutstoppingandtheirmanipulation,carryingheavyobjects,etc.). April10,200422:17WSPC/191-IJHR00008 Zero-MomentPoint—ThirtyFiveYearsofItsLifeAtthispointwewillalsobrie”yformulateanothermajortopicthatseemsinevitableorprospective.Assomeoftheseproblemshavealreadybeenrecognizedandelab-orated,wewillmentiononlythosetasksthathavenotbeenresolvedyet.ThetermgroundŽusuallymeanssomethingimmobile(perhapsdeformable,butimmobile).However,ageneralapproachrequirestheoptionofwalkingonamobilesupport.Moreover,suchsupportshouldnotbeconsideredasapurenonstationaryconstraintbutratherasadynamicsystemthatinteractswiththewalker.Thus,amobileplatformthathasitsowndynamicshastobeintroduced.TheplatformshouldhaveuptosixDOFs.Itisclearthattheplatformcanbemodeledindierentways.AconstructiveapproachmayrefertotheuseofspecialStewartplatformstructures,asshowninFig.5(a).Indescribingtheeectsthatshouldbetakenintoaccountwhenworkingonhumanorhumanoiddynamicswecometothecon“gurationofthesysteminthedynamicssimulatorGHDS(GeneralHuman/HumanoidDynamicsSimulator).ItisimportanttomakeadistinctionbetweentheGHDSandtestbedthatinvolvestherealdevices:robot,cameras,etc.[sketchedinFig.5(b)].Thefusionofthesetwosystemsmaybeconsideredasanultimategoal.Theabovetopicsconcerninghumanoidrobots,beingstillinamodestinitialstageornotyetformulated,areofcrucialimportancetoachievethosecapabilitiesofhumanoidrobotsthatcouldrealisticallymeethighrequirementsoftheirenvisagedapplications.4.ConclusionTheconceptofZMPhasandwillhaveanessentialroleinboththeoreticalconsid-erationsandthepracticaldevelopmentofhumanoidrobotsandbipedlocomotion.Afterseveraldecadesofitsapplicationitcanbenoticedthatinreferringtoit,probablybecausethenotionhasbecometrulyacceptedandcommonlyknown,theZMPhassometimesbeende“nedinaninsucientlypreciseandover-simpli“edway.HavinginmindthattheoriginalworksinwhichtheZMPconceptwasintroducedarenoteasilyaccessibletoallresearchers,especiallytoyoungerones,wethoughtitusefultorefreshthisnotionandremindreadersofitsoriginalmeaningandthusavoiditssuper“cialunderstandingandpossibleconfusion.Besidesthis,itisevidentthatthedevelopmentofhumanoidroboticsisgoinginthedirectionofincorporat-ingrobotsintointimateŽhumanenvironments,coexistenceandco-operationwithhumans(evenasapartneronthesametask),sothatitisrightlyexpectedthattheperformanceofrobotswillbecomecloserto,andinsomesegmentsevenbetterthan,thoseofhumans.Also,wehavetomentiontheimportantareaofserviceroboticsandtheroleofrobotsinhostileenvironments.Hence,initslastsectionsthispapertouchesuponsomeimportantbutstillunresolvedlocomotion-manipulationissues. April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.Borovac detail“A”: flexiblesole:flexiblesupport: TRUEGROUNDforcesand platform Fig.5.Modelofthegeneraltaskforensuringbipedsdynamicequilibrium. 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April10,200422:17WSPC/191-IJHR00008 M.Vukobratovi´c&B.Borovac MiomirVukobratovi´wasborninBotos,Serbia,1931.HereceivedhisB.Sc.andPh.D.degreesinMechanicalEngineeringfromtheUniversityofBelgradein1957and1964,respectively,andhisD.Sc.degreefromtheInstituteMashinovedeniya,Soviet(nowRussian)AcademyofScience,Moscow,1972.From1968hewasheadoftheBiodynamicsDepartment,thendirectoroftheLaboratoryforRoboticsandFlexibleAutomationandDirectoroftheRoboticsCenterrespectivelyatMihailoPupinInstitute,HehasservedasavisitingprofessorteachinggraduatecoursesinroboticsatseveraluniversitiesintheformerYugoslaviaandabroad.Heistheauthor/co-authorofmorethan200scienti“cpapersinthe“eldofroboticsandsystemtheory,haspublishedinleadinginternationaljournals,andisalsotheauthor/co-authorofabout360papersinproceedingsofinternationalconferencesandcongresses.Hehasalsoauthored/co-authored13researchmonographspublishedinEnglish,Japanese,Russian,ChineseandSerbian,twoadvancedtextbooksinroboticsinEnglish,andtenchaptersininternationalmonographsandhandbooks.Amongothers,heisaholderofJosephEngelbergerŽawardinroboticsforhispioneeringgloballyrecog-nizedresultsinappliedresearchandeducationinrobotics,awardedbytheRoboticIndustriesAssociationintheUSAin1996.Prof.Vukobratovi´cisafullmemberoftheSerbianAcademyforSciencesandArts,aforeignmemberoftheRussian(formerlySoviet)AcademyofSciences,afullmemberoftheInternationalAcademyofNonlinearSciences,andseveralotherforeignacademies,presidentofYugoslavAcademyofEngineering,aforeignfullmemberoftheInternationalEngineeringAcademy,Moscow,aforeignmemberoftheChineseAcademyofEngineering,anhonorarymemberoftheHungarianAcademyofEngineering,andothernationalacademies.HeisdoctorhonoriscausaofMoscowStateUniversitynamedafterM.V.Lomonosovandseveralotheruni-versitiesinEurope.BasedontheCitationIndex,hehasbeencitedabout1,350times.Prof.Vukobratovi´chaspresentedthirtyopeningandplenarylecturesatworldconferences,symposiaandcongresses,andhaslecturedbyinvitationatmorethan150scienti“cseminarsintheUSA,Japan,Russia,ChinaandEurope.Hismajorinterestisinthedevelopmentofecientcomputeraidedmodelingofroboticsystemsdynamics,inparticulardynamicnon-adaptiveandadaptivecontrolofnon-contactandcontacttasksinmanipulationrobotics,aswellasdynamicsmodeling,stabilityandcontrolinleggedlocomotion,especiallyhumanoidrobots. April10,200422:17WSPC/191-IJHR00008 Zero-MomentPoint—ThirtyFiveYearsofItsLife BranislavBorovacwasborninLeskovac,Serbia,1951.HereceivedhisM.Sc.andPh.D.degreesinMechanicalEngineer-ingfromtheUniversityofNoviSadin1982and1986,respec-tively.HebecameAssistantProfessorofEngineeringDesignin1987,AssistantProfessorofRoboticsin1988,AssociateProfessorofRoboticsin1993andsince1998,hehasbeenfullProfessorofRobotics,allattheFacultyofTechnicalSciences,UniversityofNoviSad.Heiscoauthoroftworesearchmonographpub-lishedbySpringer-Verlag(1990)andCRCPress(2001).Heistheauthor/coauthorof15scienti“cpapersinthe“eldofrobotics,publishedininternatinaljournals,aswellastheauthor/coauthorofabout50papersinproceedingsofinternationalconferencesandcongresses.Hisresearchinterestsincludebipedlocomotion,robotmodelingandcontrol,industrialrobotics,sensorsandsensorinformationintegration,forcesensorsandtheiruseincontacttasks,assembly,mechatronics,productdesignand”exiblesystems. April10,200422:17WSPC/191-IJHR00008 April10,200422:17WSPC/191-IJHR00008 InternationalJournalofHumanoidRoboticsVol.1,No.1(2004)157…173WorldScienti“cPublishingCompany ZERO-MOMENTPOINT—THIRTYFIVEYEARSOFITSLIFE