MichaelColemanAnindyaChatterjeeDepartmentofTheoreticalandAppliedMechanics212KimballHallCornellUniversityIthacaNewYork14853ph6072557108fax6072552011ruinacornelledugarciatamcornelleduIn ID: 112818
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PASSIVE-DYNAMICMODELSOFHUMANGAITMarianoGarciaAndyRuina MichaelColemanAnindyaChatterjeeDepartmentofTheoreticalandAppliedMechanics212KimballHall,CornellUniversity,Ithaca,NewYork14853ph:607-255-7108,fax:607-255-2011,ruina@cornell.edu,garcia@tam.cornell.eduIntroductionHumanmotioniscontrolledbytheneuro-muscularsystem.Butbipedalwalking,anexampleofabasichumanmotion,mightbelargelyunderstoodasapassivemechanicalprocess[3].Forexample,TadMcGeerdemonstrated,bybothcomputersimulationandphysicalmodelconstruction(bothofwhichwehaverepeated),thatasomewhatanthropomorphicleggedmechanismwithfourmovinglinkscanexhibitstable,human-likewalkingonarangeofshallowslopeswithactuationandcontrol(energylostinfrictionandcollisionsisrecoveredfromgravity)[1].Sofar,ourstudiesfocusontwo2Dmodels:akneedwalker,andasimpler`pointfoot'model.Thesearelinkagesmadeofrigidbarsthatareconnectedwithhinges.Unlikecontrol-basedmodels,wherethecontrollertriestoforceamotiononthesystem,themodels'gaitcycles(exactsequencesofrepeatedsteps)areinherentproductsofthemodelparameters. -qstqthqsh grS ,mScTlTrT ,mTwTwScSlSg eT R b) DYNAMIC VARIABLESa) DIMENSIONAL PARAMETERS T Figure1:McGeer'skneedwalkingmodel.KneedPassiveWalkingModelOurkneedwalkerisessentiallyacopyofMcGeer's[2].Ithasfourlinks,whichrepresenttwothighsandtwoshanks.Kneestops,modeledbyplasticcollisions,preventtheshanksfromhyperextendingrelativetothethighs.Theshanksarerigidlyconnectedtoo®setsemicircularfeet.Masscenters,momentsofinertia,footradiusandfootplacementrelativetotheshankareallmodelparameters.Themechanismisonarampofslope.Heelstrikeismodeledbyaplastic(no-slip,no-bounce)collisionoftheswingingfootwiththerampsurface.Doublesupportisinstantaneous.PointfootPassiveWalkingModelOur2DpointfootwalkerisanextremespecialcaseofMcGeer's2Dstraight-leggedround-footwalker[1].Westudyitbecauseofitssimplicity.Twomasslessrigidrods(legs)oflengthconnectedbyafrictionlesshingeatthehip.Thereisapointmassatthehipandamuchsmallerpointmassateachfoot.Themechanismisonarigidrampofslope.Whenthefoot hitsthegroundatheelstrike,ithasaplasticcollision.Duringwalking,onlyonefootisincontactwiththegroundatanytime,exceptingtheinstantofdoublesupport.ThePoincar¶esectionforthepointfootwalker,usingtheconditionsjustafterheelstrike,turnsouttobejusttwo-dimensional.Sothestanceangleandrate,,justafterheelstrike,determinethesubsequentmotionsofthewalker.ResultsandInsightsfromtheModelsUsinganumericalNewton-Raphsoncalculationwhilevaryingmodelparameters,we¯ndvariousgaitcycles.SomeobservationsthatgobeyondtheworkofMcGeerarelistedbelow.Thesearefoundfromcomputerandperturbationcalculationsand,forthemostpart,havenotyetbeenveri¯edwithphysicalexperiments.1)Suprisingly,solutionsexistallthewaytodownhillslopesofforthepointfootwalker.Thesteplengthsvanishasthedownhillslopegoestozero.Sincedownhillslopeisameasureofine±ciency,thepointfootwalkeriscapableofessentiallyperfectlye±cientlocomotion.Forpassivegaitcyclestoexistatarbitrarilysmallslopes,thecenterofmassofeachlegmustlieonalineconnectingthefootcenterwiththehip,aconditionautomaticallyful¯lledbyourpointfootmodel.Wedonotyetknowwhetherornotthisplacementofthecenterofmassissu±cienttoallowthekneedwalkertowalkatarbitrarilysmallslopes.2)Anotherwaytocharacterizee±ciency,onethatgivescredittospeed,istousetheratio(averageforwardkineticenergy)/(energylossoveronestep).Forthekneedwalker,thissecondmeasurehasamaximumatgaitcycleswhich,althoughtheyseemtolookthemostanthropomorphicwiththeirsmoothheelstrikesandlateknee-strikes,areunstable.Perhapsthehumanbodyselectssomethingclosetoanaturally`e±cient'passivegaitmodeandstabilizesitatasmallcontrolcost.3)Forthepointfootwalker,walkingwithsmallleganglesonsmallslopes,wefoundtwoperiod-onecyclesofinterest:along-periodcycleandashort-periodcycle,asobservedbyMcGeerinhismoregeneralwalkerson¯niteslopes[1].Inthesesolutions,initialstanceangleandstanceanglescaleaccordingto).Thelong-periodcycleisstableondownhillslopeslessthan015radians.4)Asthelong-period-1cyclebecomesunstableataslopeof015,astableperiod-2(limping)cycleappears.Higherordercyclesappearinaperioddoublingcascadeuntilthestablegaitisapparentlychaotic(randomstumbling)atadownhillslopeof019.Wehavenotyetfoundperiod3(waltzing)solutions.Furtherstudiesofthetypeswearepursuingmayrevealthatmanyhumanmotionsarelargelynaturalorquasi-passiveandnotheavilycontrolled.Inthecontextofwalking,suchresultsmightbeusefultothosestudyingtheactivecontrolaspectsofwalking,tothosetryingtodesignanthropo-morphicrobots,andtothosetryingtoimproveprostheticdevicesandrehabilitationprocedures.[1]TadMcGeer.Passivedynamicwalking.InternationalJournalofRoboticsResearch,9:62{82,[2]TadMcGeer.Passivewalkingwithknees.ProceedingsoftheIEEEConferenceonRoboticsandAutomation,2:1640{1645,1990.[3]SimonMochonandThomasMcMahon.Ballisticwalking:Animprovedmodel.MathematicalBiosciences,52:241{260,1980.