Biologically motivated neuro-mechanical stepping model in the frontal plane with integration of sensor driven balance control [Elektronische Ressource] / Sonja Andrea Karg

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Biologically motivated neuro-mechanical stepping model in the frontal plane with integration of sensor driven balance control [Elektronische Ressource] / Sonja Andrea Karg

Sonja Andrea Karg - technische_universitat_munchen

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Biologically Motivated Neuro-Mechanical SteppingModel in the Frontal Plane with Integration ofSensor-Driven Balance ControlSonja Andrea Karg¨ ¨TECHNISCHE UNIVERSITAT MUNCHENLehrstuhl fu¨r Realzeit-ComputersystemeBiologically Motivated Neuro-Mechanical SteppingModel in the Frontal Plane with Integration ofSensor-Driven Balance ControlSonja Andrea KargVollst¨andiger Abdruck der von der Fakult¨at fu¨r Elektrotechnik und Informationstechnik derTechnischen Universit¨at Mun¨ chen zur Erlangung des akademischen Grades einesDoktor-Ingenieurs (Dr.-Ing.)genehmigten Dissertation.Vorsitzender: Univ.- Prof. Dr. rer. nat. habil. B. WolfPru¨fer der Dissertation: 1. Univ.- Prof. Dr.-Ing. G. F¨arber2. Priv.-Doz. Dr.-Ing. St. GlasauerLudwig-Maximilians-Universit¨at Munc¨ henDie Dissertation wurde am 23.06.2008 bei der Technischen Universit¨at Munc¨ hen eingereichtund durch die Fakult¨at fur¨ Elektrotechnik und Informationstechnik am24.11.2008 angenom-men.AcknowledgementsThis work was i.a. supported by the Deutschen Forschungsgemeinschaft (DFG) start-ing with the SFB-462 “Sensomotorische Analyse biologischer Systeme, Modellierung undmedizinisch-technische Nutzung” and later with the program “Vestibul¨are Funktion undOkulomotorik: Stand- und Gangregulation” STR-384/1,2.First of all, I want to thank my advisor Prof. Dr. Georg F¨arber who gave me theopportunity for this very interesting and thrilling work.

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Published by	technische_universitat_munchen
Published	01 January 2008
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TECHNISCHEUNIVERSIT¨ATM¨UNCHEN

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BiologicallyMotivatedNeuro-MechanicalStepping
ModelintheFrontalPlanewithIntegrationof
Sensor-DrivenBalanceControl

SonjaAndreaKarg

TVoecllsthni¨andischegnerUniAbdverrusitck¨atdMer¨unvonchenderzurFEakulrlatn¨atgufu¨ngrEldesektakroteademichnischkuenndGradesInformateinioesnstechnikder

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Univ.-Prof.Dr.rer.nat.habil.B.Wolf

1.Univ.-Prof.Dr.-Ing.G.F¨arber

2.PLudrivw.-Doz.ig-MaximiDr.-Inglians.St.-UniversGlasauerit¨atM¨unchen

DieDissertationwurdeam23.06.2008beiderTechnischenUniversit¨atM¨uncheneingereicht
unddurchdieFakult¨atf¨urElektrotechnikundInformationstechnikam24.11.2008angenom-
n.me

Acknowledgements

ingThiswithworkthewasSFBi.a.-462supp“SensomoortedbytoristhecheDeutschenAnalysebioFlogischerorschungsgemeinschaftSysteme,Mode(DFllierunG)gstartund-
medizinisch-technischeNutzung”andlaterwiththeprogram“Vestibul¨areFunktionund
Okulomotorik:Stand-undGangregulation”STR-384/1,2.
Firstofall,IwanttothankmyadvisorProf.Dr.GeorgF¨arberwhogavemethe
opportunityforthisveryinterestingandthrillingwork.Withhismannerofgivingme
plentyofspacetodevelopownideasheenabledmetoﬁndmyownwayandtolearnfrom
manyvaluableexperienceswhichﬁnallyleadtothiswork.
IalsowanttothankPD.Dr.StefanGlasauerwithwhomIhadsomeveryhelpfuland
informativediscussions.Withhisexperienceonbothsidesofthetopic,thetechnicaland
thebiological/medicalside,hesupportedmetoaccomplishthesplitbetweenthediﬀerent
s.disciplineAttheNeurologicalClinicofGrosshadernIwanttothankmycolleaguesforthegoodco-
operation,especiallymymedicalcolleaguePD.Dr.KlausJahn,whojoinedmeformany
experimentsandforthelivelydiscussionsaboutnewideasandconcepts.Thanksso
muchtoDr.ErichSchneiderwhohelpedmewithmanytechnicaldetailsforexperimental
setupsandhadalwaysgoodadviseformyquestions.AlsoIwanttothankMarkusHuber
whogavemeahelpfulhandformyexperiments.
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Contents

ListofSymbols

ivii

1Introduction1
1.1Motivation....................................1
1.2StateoftheArtContext............................2
1.3Outline......................................3

2PassiveMechanicalModels5
2.1StateoftheArtofPassiveMechanicalModels................5
2.2SteppingModelintheSagittalPlane.....................7
2.2.1LagrangianPrincipleofaPendulum.................7
2.2.2Mechanicsofthe2DSteppingModelintheSagittalPlane.....9
2.2.3GroundContactModel.........................10
2.3SteppingModelintheFrontalPlane.....................11
2.3.1Mechanicsofthe2DSteppingModelintheFrontalPlane.....13
2.3.2GroundContact.............................18
2.3.3ExtendedGroundContactModelforActuatedMechanics.....21
2.4SimulationResults...............................22
2.4.1BallisticPeriodicMovementsintheSagittalPlane..........23
2.5Conclusion....................................28

3ActuationofPassiveMechanicalModels29
3.1StateoftheArtofActuationMechanismsforWalkingModels.......30
3.2ExamplesofOscillator-DrivenMovementsinBiology............30
3.2.1Lamprey.................................31
3.2.2Cat....................................32
3.3NeuralOscillatorModel............................33
3.3.1TheMatsuokaOscillator........................34
3.3.2ConstraintsforOscillation.......................36
3.3.3BasicNetworkTypes..........................36
3.3.4NeuronalOscillatorNetworksforWalking..............38
3.4ActivationofMechanicswithOscillators...................43
3.4.1JointTorqueGeneration........................43
3.4.2MuscleFeedbackAppliedtotheOscillator..............44
3.5Stability.....................................46

Contsnet

3.5.1Poincare´Sections............................46
3.5.2StabilityProofAppliedtoPeriodicWalking.............48
3.5.3FindingConﬁgurationsforStablePeriodicMovements.......49
3.6SimulationofSteppingMovementsandVisualization............50
3.7SimulatedSteppingMovementsintheSagittalPlane............51
3.7.1WalkingMovements..........................51
3.7.2VariationofParameters........................55
3.8SimulatedSteppingMovementsintheFrontalPlane.............57
3.8.1SimulationofDiﬀerentMovementPatterns..............58
3.8.2InﬂuenceofParameterChangesonMovementPatterns.......67
3.8.3DiﬀerentFeedbackGains........................70
3.8.4DiﬀerentOscillatorPatterns......................75
3.8.5StabilityofMovementswithExternalPerturbations.........80
3.8.6ComparisonofSimulationDatawithRealSteppingData......86
3.9Discussion....................................90
3.10Conclusion....................................92

High-LevelPostureControl94
4.1StateoftheArtofSensorimotorPostureModels...............96
4.2SensoryModels.................................98
4.2.1VestibularSense.............................98
4.2.2Proprioception.............................100
4.2.3VisualSense...............................102
4.3EstimationforPostureControl........................106
4.3.1TheKalman-FilterTheory.......................106
4.3.2ApplicationoftheKalmanFiltertotheStanceModel........109
4.3.3ExtendedKalmanforNonlinearSensoryModels...........114
4.3.4OptimalLinearQuadraticRegulator.................115
4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl...117
4.4.1PlotsandPresentations........................119
4.4.2InﬂuenceofVisualPerceptionwithEyeMovementsonPostureCon-
trol....................................120
4.4.3InﬂuenceofVestibularPerceptiononStanceControl........132
4.5SimulatedSwayResponsesforVisualandVestibularStimulation......136
4.5.1ParametersofthePostureControlModel...............137
4.5.2VestibularStimulation.........................138
4.5.3RetinalStimulation...........................141
4.5.4EyeMovementStimulation.......................144
4.5.5CombinedRetinalandEyeMovementStimulation..........148
4.6Discussion....................................151
4.7Conclusion....................................153

IntegrationofHigh-LevelandLow-LevelModels154
5.1StateoftheArtofIntegrationModels....................155

Contsnet

5.2ControlStrategyfortheSteppingModel...................157
5.2.1FeedbackLinearizationTheory....................157
5.2.2AppliedFeedbackLinearizationforHipMovements.........159
5.2.3AppliedPreviewControlandOptimizationCriteria.........160
5.3AppliedIntegrationModel...........................161
5.4SimulationofLow-LevelSteppingMovementswithHigh-LevelPosture
Control.....................................164
5.5Discussion....................................167
5.6Conclusion....................................170

6SummaryandFinalConclusion171
6.1Outlook.....................................173

hyograpBibli

175

vii

ListofSymbols

CNSCOMCOPCPGFDOgGVSFFTfpsRLQMSEdeoRMSPZM

viii

centralnervoussystem
assmoftercencenterofpressure
centralpatterngenerator
mdofreeofdegree(s)gravitationalforce
gafastlvaFonicuriervestitrabularnsfostimrmatioulusn
framespersecond
linearquadraticregulator
rerrosquaremeanordinarydiﬀerentialequation
rootmeansquare
zeromomentpoint

SymbolsofEquations
Asystemstatematrixindiscretestatespacedescription
ai,jweightofinhibitingsynapticinputbetweenneuroniandj
αangleofstanceleginfrontal-planemodel
α˙Bangsystemularvinputelocitymaoftrixstainncedislegcreteinfrstateontal-spaceplanemodescriptiodeln
βangleofhipinfrontal-planemodel
˙Cβangularmeasuremenvelotcitymatofrixhipininfrondiscretetstaal-planetespamocedeldescription
GD(q)matrixmeasuremenofgratvitainputtionalmatrfoixrceinsdiscretestatespacedescription
γangleofswingleginfrontal-planemodel
γ˙angularvelocityofswingleginfrontal-planemodel
fiﬁringrateofneuroni
fdfeedbackgainforposition
fHdv,hfeehipwdbaidtckhgainforvelocity

)wj(HlL,mM)q(MN(q,q˙)
niΦ˙ΦΦcom˙ΦcomPiqQRsθ˙θT,Tbauuctauc=(ua,ub,0)
vkwkwijxy

transferfunctioninthefrequencyrange
lengthlegblegodymassmass
matrixofmassorinertia
matrixofcentrifugalandCoriolisterms
neuronactivationwhichisthemembranepotentialofthei-thneuron
angleofstanceleginstancemodel
angularvelocityofstanceleginstancemodel
angleoftheCOMpositioninrelationtostancefootposition
angularvelocityofCOM
networknumberiofoscillatorsforthefrontal-planemodel
generalvectorofsystemstates
noisecovariancematrixofthesystem
noisecovariancematrixofthemeasurements
externalinputtoaneuron
angleofstanceleginsagittal-planemodel
angularvelocityofstanceleginsagittal-planemodel
constantsforoscillatortimeconstraints
generalvectorofinputs
veinputctorvofectosuproerpfosositiocillantorcognetroneraltedinputjoforintjointortquestorques
statisticnoiseofmeasurements
statisticnoiseofthesystem
weightofneuronactivationfortorquegeneration
appliedvectorofsystemstates
vectorofsystemoutputs

List

bSym

ols

tacAbstr

Anewmodelforfrontal-planesteppingmovementsisdevelopedinordertoevaluate
medio-lateralgaitmovements.Usingthismodelitispossibletostudyindividualstep-
pingparameters,stabilityofmovementsandvarioussteppingpatterns.Gaitresearch
todatehasmainlyfocusedonforwardlocomotion,butasmaintaininglateralbalanceis
criticalforstablegait,theproposedmodelconcentratesonsteppinginthefrontalplane.
Themodelingiscarriedoutonthebasisofbiologicalprinciplesandusingabottom-up
approach.Themodelisaccordinglysplitintoalow-levelandahigh-levelcomponentin
linewithbiologicalprocesses,wherelow-leveltasksarechieﬂyautomaticandhigh-level
tasksareprimarilydirective.Theactuationofthepassivemechanicalmodelisachieved
bycreatinganeuronaloscillatorstructurewithmuscularfeedbackandantagonisticjoint
torquegeneration.Characteristicparametersofthislow-levelmodelareidentiﬁedfor
functionssuchasstepfrequencyorsteppingpatterns.Variousmovementsarepresented
forstablesteppinginplacewithdroppingorliftinghip,steppingtotheside,andstepping
upwards.Thesimulatedsteppingiscomparedwithrealvideotrackingdataandfoundto
beverysimilar.Thesteppingisalsotestedunderdisturbinginﬂuencessuchasslipping
orgettingstuck;themodelshowsrobustreactionsandreturnstoastablesolutionwithin
afewsteps.Thestabilityandperformanceofthelow-levelsteppingmodelhavetheir
limitationsasthissystemlacks”perception”oftheoverallcontext.
Ahigh-levelmodelisthereforedevelopedtorepresentperceptionofthewholebodypo-
sitionandtheenvironmenttoaccomplishposturecontroltasks.Thebasisforthismodel
ismodelknowledgeintheformofstatisticalestimationandsensorymodelsderivedfrom
biology.Thisisintegratedinafeedbackloopwherethetwomainoptimizationcriteria
areuprightpostureandlowactuationinput.Toevaluatetheperformanceofthesensor-
drivenposturecontrolmodel,twoexperimentswithrealsubjectsareperformed,onefor
vestibularstimulationandanotherforvisualpursuitstimulation.Theexperimentaldata
forpostureresponsearereproducedandveriﬁedbythehigh-levelmodel.
Toenhancetheperformanceandabilitiesofthelow-levelmodel,thetwomodelsare
integratedbyasuperpositioncontrolconcept.Superpositioningdoesnotinﬂuencethe
low-levelactuationdirectly,butthetwolevelsaresuperimposedwhereverthewholebody
balanceisconsideredtobeatrisk.Thisintegrationleadstoimprovedstabilityofthe
steppingmovementswithoutreducinglow-levelautonomy.Stabilityofmovementisno
longermainlydependentontheinitialvaluesandthisleadstoanincreasedrangeofstable
solutionsandthepossibilityofinﬂuencingsteppingmovementsbysensorycues.
Thisrelativelysimpleframeworkintegratesthemainlow-levelandhigh-levelmechanisms
ofsteppingmovementsinonemodel.Itisusedtosimulateautonomoussteppingmove-
mentsundertheaspectofsensor-drivenposturecontrol,withthepossibilityofanalyzing
medio-lateralsteppingcharacteristics,andofinﬂuencingthem.

xii

onductiIntro1

cesHumansesstaenablingnceandthegboaitdymectohapenisrfomsrmarelocoacomotiomplen,xtoinmaterainctiontainofbaexlancetremelyandtodiversereactproto-
environmentalinﬂuences.Theseprocessesrelatetothemechanics,themuscles,the
neuronacomplexlstscenarioructures,andstheensorvaycrietuesyofandintegtheratbraionin,mectohanismsnameonlyrequirtehedtomoasctohievbeviotheus.waThislk-
ingtaskarestillnotwellunderstoodandresearchintothemcoversaverywideﬁeld
rangingfrombiologyandmedicinetoengineeringandnaturalsciences.

vationiotM1.1

Theworkpresentedinthefollowingevolvedfromajointprojectwithaneurological
medicalresearchunit.Thekeyideasweretriggeredbyseveralstudiesofsensoryinﬂuence
onlocomotioninalateraldirection.Hence,amodeltoevaluatesidewardsmovements
ofstanceandgaitisofspecialinterest.Forclinicalpurposesthiscouldleadtoabetter
understandingoflocomotionbehaviorsofpatientswithavarietyofdefects,knowledge
whichcansubsequentlybeusedtochangeandimprovetherapy.Tothisendafrontal
planeposturemodelforsteppingmovementswasneeded.
Thisthesispresentsandanalyzesamathematicalmodeloriginatedintheengineering
environmentandbasedonknownandacceptedbiologicalstructures,inordertoanalyze
posturalstanceandsteppingtasks.
Abiologicalmodelsuchasthisstandsincontrasttotoday’sroboticstanceandgait
realizationsbecausetheconceptisdiﬀerentandbecausetheperformanceisoftenworse
atﬁrstsight.Thereasonforcreatingsuchbiologicallymotivatedmodelsistherealization
thatroboticshasitslimitsandthatconventionalsolutionscanpushthoselimitsbutnot
overcomethementirely.Itisnotwellknownhowhumanbeingssolvemovementtasksand
theclassicroboticapproachbringsusnonearertoanexplanation.Biologicallymotivated
modelsareoftenusedtoobtainfurtherunderstandingandinsightintocommoncomplex
movementactionsandinteractions.
Especiallythemedio-lateralsteppingmovements(steppinginthefrontalplane)arenota
wellexploredaspectofmovementingaitresearch.Thisthesisstudiesseveralaspectsof
medio-lateralsteppingmovementsandintroducesageneralmodelwiththepossibilitiesof
researchingthissteppingplanedependingontheindividualcomponentsandonsensory
inﬂuences.Iflocomotionisconsideredasataskofhumansurvival,thestabilityofthis

ionducttroIn1

taskisthemostessentialproperty,followedbyrobustnessandﬂexibility.Thisrequiresan
intensivestudyofinﬂuenceparametersonsteppingmovements,alwaysundertheaspect
ofstabilityandrobustness.Theﬂexibilityandvarietyofthesteppingmovementsisso
importantbecausethesteppingtaskisadaptedtosuiteveryreallifesituation.Thisleads
tothecreationofvariationsofthesteppingmovementinthefrontalplane-steppingin
place,steppingtothesideandsteppingupordownunderdiﬀerentsteppingstrategies
-andtosystematicanalysisoftheinﬂuencingfactors.Tostudyhowthevestibularor
visualsensorsinﬂuenceposturecontrol,thoseinﬂuenceshavetobeveriﬁedexperimentally
andexplainedbymathematicalrelations.Thisworkfocusesparticularlyonvestibular
andvisualinﬂuences(througheyemovements)asthesubjectofexperimentsandmodel
evaluationbecausethesearetwoimportantsensorycuesinﬂuencingsteppingtaskand
performance.Theextendtowhichsensor-drivenposturecontrolinﬂuencesthetaskof
steppingisnotclearfromabiologicalpointofview.Forthisreasongeneralposturetasks
andenvironmentalinﬂuencesareconsideredseparatelyandjointly.
Thegeneralprinciplefollowedforcreatingthemodelpresentedinthisworkistolook
fortheabilitiesofthemostsimplemodelandextendthemwherebiologicalconstraints
andfactsfoundexperimentallyrequireit.Thestrategyistobuildthemodelbottomup,
startinglow-levelandextendingitbyahigh-levelcontrol.Themodelconsidersmedio-
lateralsteppingmovementsastherearenoothermodelsknowntotheauthorwhich
analyzelateralstepsinmoredetail.

1.2StateoftheArtContext

EitherwalkingandGaitortheoperationofposturecontrolaregenerallystudiedun-
derdiﬀerentaspectsinliterature:ontheonehandfromthetechnicalangle,whichis
theapplicationoflocomotionmechanismsinrobotics,andontheotherhandfromthe
biologicalormedicalangle,whichfocusesonunderstandinglocomotionmechanismsfor
therapypurposesortoexplainbiologicalstructures.Thesetwoapproachesdiﬀerbothas
regardsthemethodsused,aswellastherequirementtoﬁndaworkingtechnicalsolution
ontheonehand,andtoexplainthebiologicalrealityontheotherhand.Twoexamples
canbegiventodemonstratethediﬀerencesbetweentechnicalsystemsandthehuman
being.Firstly,theenergyconsumptionofawalkingrobotismuchhigherthanthatofa
humanbeing.In[23]itisdeterminedat0.2[cet]forhumansand3.2[cet]forHonda’s
ASIMOrobot,where[cet]isdeﬁnedasthespeciﬁccostoftransport,whichistheenergy
usedperweightanddistancemoved.Becauseofthis,manynewroboticapproachestry
tocompensatethisenergywasteasshownin[182].In[20]arobotrealizedbypassive
mechanicsisactuatedattheanklesforpush-oﬀwithaveryenergy-eﬃcientgait.This
canbeseeninthegraphicspublishedbyCollins[22].
Thesecondexampleisthenaturalappearanceofasteppingmovement.Acharacteristic
ofhumanstepsisthatthestepsareoftensimilarbutneveridentical;thevariability
ofsteptrajectoriesislarge[175,56].Anapproachoftenusedinroboticsiscontrolby
predeﬁneddesiredtrajectories[16,102,11,17].Thisapproachdoesnotresemblethe

1.3utlineO

Figure1.1:Energycomparisonbetweenhumanbeing,theCornellnearlypassiverobot
andtheHondaASIMOrobottakenfrom[22]

naturalvariability.Itisalsonotveryﬂexibleaboutadaptingtochangingconditions
andrequiresagoodaprioriknowledgeofthemovementoralargememoryforsaved
possibletrajectories[103].Noneofthese3pointsarepresentinnaturalsystems,which
neverthelessproducerobustwalkingandposturemovements.
Ontheotherhandtherearetechnicalapproacheswhichtrytoexploitthenaturalresources
fortheirabilitytogeneratewalkingmovements.Therearethedynamicsofmechanics
whichareabletousegravityastheironlyactuation[125,113,21]toproducerealistic
gaits.Groundreactionforcesortrajectorieswhicharecharacteristicforhumansarean-
alyzedandappliedtotechnicalsystemstoevaluatetheireﬃciency[101].Additionally
thereareseveralparametersofthehumanwalkingmovementwhichcanbemeasured
andusedtoexplainandreconstructthewalkingmovementanditsproperties.Muscle
activationpatternsareusedtoevaluatemusclespeed,propulsiveenergyandthedevel-
opmentofforwarddynamicalmodels[198,134,194].Spinalcordinjuriesarestudiedto
obtainmoreknowledgeabouttheinﬂuenceofspinalneuroncircuitsonwalking[57,110].
Asitisverydiﬃculttoextractinformationaboutthesecomplexandinternalneuronal
structures,simpleranimalsarestudiedanddescribedmathematically[73].Theseﬁnd-
ingsareassignedtohumanlocomotioncharacteristicsandadapted[10,151].Another,
evenmorecomplex,componentwhichconstitutesanintegralpartofthewalkingtaskis
theaspectofperceptionandcentralprocessingandthecontroltasksofthebrain.The
rangeofexperimentswhichinvestigatetheinﬂuenceofperceptiononpostureorbalance-
maintenanceiswide.Oftentheseexperimentsaredoneinstance[65,15,170],orduring
walking[77,193].Fromtheseexperimentsmathematicalfunctionsofperceptionposture
relationsarederivedandapproximatedtodescribepossibleinterrelations[79,6,89].

Outline1.3

Themodelinthisworkisstructuredbottomup,startingwiththelowestconstraintsand
extendingthemodelfurtherforenhancedcapabilities.Themostbasicrequirementfora

troIn1ionduct

standingandsteppingmodelisthemechanics.Thesimplestandmosteﬀectivemechanics
aretheballisticgaitmodelsdrivenonlybygravityandtheseformthebasisforthemodel
developedhere.Thisiselaboratedinchapter2wherethesagittal(section2.2)andfrontal
(section2.3)planemodelsarepresented.Furthermovementgenerationcanbedivided
intosubsystems.Suchsubsystemswerefoundexperimentally,asitwasascertainedthat
movementsofcertainanimalscouldstillbeachievedifthoseanimalsweredecerebrated,
meaningthatthebrainnolongerplayedanyactiveroleinthelocomotiontasks.Inline
withtheseﬁndingsthemodelisdividedintoahigh-levelandalow-levelcomponent.
Thelow-levelmodelisextendedinchapter3withasimpleactuation.Accordingtobio-
logicalﬁndingsitcangenerateautonomousbasicrhythmicmovementssuchasstepping.
Inthisworktheoscillator-drivenmovements,suchasthosefoundinsimplelifeforms
aswellasinmorecomplexlifeformssuchasthecat(section3.2),areusedtogener-
atesteppingpatterns.Theactuationitselfiscorrelatedtotheantagonisticactuationof
musclesinordertostayclosetobiologypresentedin3.3and3.4.Thecomparisonto
humansteppingpatternsshowsmanysimilaritiesbutofcoursealsodiﬀerences.Forthe
desiredlateralcomponentforsteppingmovementsseveralsteppingpatternsarepossible
andshouldbegenerated.Thisincludessteppinginplacewithdiﬀerentpatternsand
lateralswayaswellasotherpossiblemovementsinthisplaneassteppingup/down(e.g.
aladder)andsteppingtotheside.Sodiﬀerentandrobuststeppingpatternscanbe
producedbyrelativelysimplemechanismswhichisshowninsection3.6,3.7and3.8.
Byanalyzingthesesteppingpatternsandﬁndingoutthelimits,suchasnointegration
ofanysensoryinformationandthereforenoinﬂuencebytheenvironment,thislow-level
steppingmodelisextendedfurther.Theextensionisaposturecontrolmodelwhich
integratesthesensorymodalitieswithacentralprocessingprocedure.Thismodelispre-
sentedinchapter4.Stanceandgaitarenotlocallyoptimizedtasksbutareoptimized
bystatisticalmeanstoensuregoodbalanceandrobustlocomotion.Tokeepthemodel
simple,awell-knownengineeringapproach,theKalmanﬁlter(section4.3),waschosen.
Thismodelintegratesmechanicsandsensorycues(section4.2)toachieveastablebut
sensitiveposturecontrol.Experimentalstudiesinsection4.4evaluateposturecontrol
characteristicsthatcanbeseenagaininthemodel.Thesimulationresultsaredetailed
insection4.5.
Finallyinchapter5,thisposturecontrol,whichmeanskeepingbalancewhilesensingthe
environment,isappliedtothemoving(stepping)model,whichtillthenhadnoabilityto
sensetheenvironmentorglobalbodypositions.Theaimofthisintegrationistocontrol
thegeneralwholebodyuprightpositioninthefrontalplane.Againasimpleapproach
ischosentoshowhowacompletemodelofabiologicallymotivatedstructurecanbe
realized.Sucharealizationneverreplicatesrealityexactlyorcompletelybutisamodelofsome
aspectsofrealitythatareknown.However,itcanbeusedtoexplainresultsfoundex-
perimentallyandtosucceedinunderstandingcomplexmovementresponsesinstanceand
stepping.Theworkpresentedissummedupinchapter6.Furtherpossibilitiesforthe
useofthismedio-lateralsteppingmodelandsomepossibleextensionsforfutureresearch
areoutlinedinsection6.1.

2PassiveMechanicalModels

aspOneect,posfrosiblemthemamecnnerhaofnicexas.Wamininglkingwdoalkingesofistocourseindeterminevolveitmecﬁrsthanismsfromsitsuchmostasmobviousclesus,
neuronalactivationandcomplexligamentconstructions,buttheabilityoftheactual
mechanicstoenablewalkingcanbestbeexaminedifthisaspectisinvestigatedseparately.
Itisafactthate.g.thebodymasses,theleglengthordiﬀerentlengthsoflegsinﬂuence
thethewmecalkinghanicstaskastprofoheyaundlyre.onlyForthisinﬂuencereasdonbygthereravitahavetionabelenfomarcens,yathepproaso-ccaheslletdobaobservllistice
ers.alkwThesearethemostsimplewalkingmodelsbuttheyneverthelessrevealaclosesimilarity
toenergeticnaturalcohstumanduewtoathelking.factTthaypicatlacctharuationisacteristicsexclusivoftelyhesegramovitadelstionaareltheandretheductbaionllistoicf
swingmovementtheyhaveduringwalking.Toshowtheabilitiesofmovementproduced
byballisticmodels,afrontal-planemodelisdevelopedandintroducedwhichfollowsthe
styleofanestablishedsagittal-planemodel.
InInsethectiofonllo2.2wing.2saectio2Dnmo2.1deltheforwiderballisticangeowfalkstaingte-of-willthe-arbetpreballisticsented.modelsThisismopresendelusested.
[38,111]asastartingpointastheseintegratemuchofthestate-of-the-artknowledge
forabotuthedevballisticelopedsagfroittanltawl-plaalkingnemomodel,dels.toTshihosw2Dthatmodeltheisintrousedducinedthisactthesisuationasmeachareferencenisms
canbeappliedtoasagittal-planemodelwithoutfurtherspeciﬁcations.Theballistic
modelsareimplementedinMATLAB.Theyareevaluatedwithrespecttotheirabilityto
providestablewalkingsolutionsforwalkingdownaslopeandotherparameterinﬂuences.
Thesteppingresultingmovmoemenvts,ementwhicshwillisbenewlyshowndevinelopedsectioninthis2.4.thesis,Thebawillllisticbeshomowdenlinfor2.3fro.nAstal
therearenostablesolutionsforthemedio-lateralmovementaccordingto[94]andasthere
isnoslopescenariocomparabletothatforthesagittalwalker,noperiodicmovements
willbeproducedandshown.Steppingmovementswillbepresentedlaterinsection3.4
ofchapter3onactuatedmodels.

2.1StateoftheArtofPassiveMechanicalModels

AﬁrstballisticwalkerwasevaluatedbyMochonandMcMahoninthe80’s[115,114,124].
Theyobservedtheswingphaseofastepandfromtheirballisticmodelwereabletoderive
characteristicdatawhichcorrelatedwithexperimentaldata,suchasswingphaselength

2PassiveMechanicalModels

tosteplengthrelation.Theyalsofoundcorrelationsbetweenthesteplengthandstep
frequencyandtheirinterdependencewithkneeandhipﬂexion.Later,intheearly90’s,
McGeerinitiatedanewseriesofballisticwalkers[111,112].Fromarimlesswheelhe
derivedtheballisticwalkerthatwalksdownashallowslope.Thiswasthemostsimple
formofawalkerwithtwostiﬀlegswhicharethemasterforthemodelspresentedinthe
following.AnothermodelcreatedbyMcGeerwasamodelwithknees,andMcGeerwas
alsotheﬁrsttodevelopa3Dballisticwalker[113].Coleman[19]andGarcia[38,39]
alsotooktheMcGeermodelastheirtemplate.Colemandevotedparticularlyattention
toinvestigatingthemovementoftherimlesswheelandfromthisresearchderivedhis
ballisticwalker[18].Garciafocusedparticularlyontheinﬂuenceofmodelparameters
onstabilizationandwalkingcharacteristics[37].Goswamiconducteddetailedresearch
intothestabilityofballisticwalkingmodels,asstabilityisnotaself-evidentproperty
ofthesemodels.3Dwalkersthatcanrundownaslopewithoutfallinghavealsobeen
developed.QuiteearlyonMcGeercametotheconclusionthatlateralstabilizationis
neededforstable3Dwalking.Thismeansthatthe3Dwalkersareeithernotcompletely
3Dornotcompletelypassive.Tobenotcompletely3Dmeansthattheyareboundto
certainmechanicalconstraintsinordertoachievelateralstabilization;inparticularthey
featureparallellegsinthefrontalplaneandnohipmovementrelativetothestanceleg
inthefrontalorsagittalplane.Allthisisexplainedindetailin[19](page135/136).
Anotherpossiblesolutionfora3DballisticwalkingmodelisgivenbyKuo[94].This
assumesthatthelateralstabilizationcannotbeachievedbyapassivemodelitself,but
hastobeachievedbyadditionalmedio-lateralactivationofthemodelasaformofactive
stabilization.Themodelwhichwasusedfortheactivestabilizationstillretainsmuchof
themodel’spassivity[92].
Theuseofballisticwalkingmodelsorpassivedynamicmodels,astheyarealternatively
called,isinterestingastheseprovideanopportunitytousewalkingmodelswhichcombine
relativelylowcomplexitywithnaturalbehavior.Asmentionedin[94,54,181]aballistic
modelcanbetakenasthebasisforanalyzingtocomprehendbipedalwalkingandthe
roleofadditionalactuationforenhancingwalkingabilities.
Asmentionedinchapter1,walkinghasimportantcharacteristicswhichhavebeentaken
intoaccount:(a)stabilityofwalkingmovements,(b)theenergyconsumptionofthesys-
temand(c)theappearanceofawalkingmovement.Togetanideaoftheinﬂuence,
walkingmechanicshaveonthesethreefeatures,ballisticmodelsareusedtodetermine
characteristics.Thestability(a)ofamovement,whichisthemostindispensablefactor
forsurvival,canbeachievedbytheseballisticmodelsasshownin[49,48,50,37,18,38].
However,theparametrizationofthemodelhasabiginﬂuenceonthesystemandexternal
disturbancescannotbehandledwell.Furthermorethestabilityrangeisnarrow.
Theenergyconsumption(b)ofballisticsystemsisideal,asonlytheenergyofgravitation
isaddedtothesystemtomakeitwalkandnoadditionalactivationisneeded.How-
ever,thegroundcontactmodelalsodeterminestheenergycharacteristics,e.g.therigid
groundcontactmodelpresentedinsection2.2.3canloseenergyduringcontactasitis
notpossibletopreservecontactimpulsesforcompletelynon-elasticcontacts.AsKuo[93]
andresearchersbeforehim[111,39]foundout,theadditionalenergythatisfedintothe
systemisanimpulsivetorqueaddedjustbeforeheelstrike.Thisisdiscussedinmore

2.2SteppingModelintheSagittalPlane

Figure2.1:AdoublependulumwithCartesianandPolarcoordinates,twosticksoflength
lwithpointmassesmandMandanglesΘandΦ.

detailtogetherwithenergyconservationofthesysteminsections2.3.2and2.3.3.
Thethirdpoint,theappearanceofthemovement(c),isnotsoeasilydetermined,but
relationssuchassteplengthtostepfrequencyarefoundtobeadequateandwalkingve-
locitycanbemodiﬁed[29].Itisalsomentionedinliteraturethattheballisticgait,which
isapendulumstyleofwalking,resemblesthenaturalappearanceofwalking[37,132].

2.2SteppingModelintheSagittalPlane

Ballisticwalkingisapendulumstyleofwalking.Duringtheswingphase,withonly
onesupportleg,astraight-leggedballisticwalkerisadoublependulum.Therefore,the
generalequationofapendulum2.2isusedandpresentedhere.Thiswillbeusedlater
insections2.2.2and2.3forthepassivedynamicequationswhichrepresenttheballistic
mechanicsofthesagittal-planeandthefrontal-planewalkers.

2.2.1LagrangianPrincipleofaPendulum

ThecoordinatesoftheuseddoublependulumarexandyintheCartesiancoordinate
systemandl,Θ,Φinapolarcoordinatesystem.Thisisseeninﬁgure2.1.TheEuler-
Lagrangediﬀerentialequationsarederivedforthesystem.TheLagrangianLis:
L=(T1+T2)−(L1+L2)whereTisthekineticenergyforthetwopendulumpartsand
Listhepotentialenergyforbothparts.TheEuler-Lagrangiandiﬀerentialequationwith

)(2.1

2PassiveMechanicalModels
astatevectorq=ΦΘisasfollows:
L∂L∂ddt∂q˙−∂q=0(2.1)
whichresultsinthependulumequation:
M(q)∗q¨+N(q,q˙)+G(q)=0(2.2)
whereM(q)isthemassmatrixorinertialmatrix,N(q,q˙)isthematrixofcentrifugal
theandpaCossivriolisemecterms,hanicsG.(q)Theistheformmatofrixthisofegqravuatioitatnisionalindepforcesendenandtqofhodescwribmaesnythesegmenstatetosf
mothedelpinendulumliterhaatures,ththeedetmatricesailedandderivsattateionvcanectorbearefoundaindapted[4,to122it.].Asthisisacommon

InversePendulumMechanicsforStance

Theinversependulumwhichisusedtorepresentastandingbodyorthestancelegmove-
menmasstMdurinaregwgivalkeninginxisandbrieﬂyycoderivordinaedintestheorbfolylothewing.angleTheΦcoandordinathetespofendulumthepelengtndulumhl.
TheLagrangianforthependulumisderivedwith:
L=21M∗v2−M∗g∗∗cosΦ(2.3)
wherevisthevelocityofthepointmassM.

v2=2∗Φ˙2
TheLagrangianisnowgivenby:
L=1M∗l2∗Φ˙2−M∗g∗l∗cosΦ
2andtheEuler-LagrangianequationofmotionwithsubstitutedLandaftersimpliﬁcation
to:leadsddt∂∂Φ˙L−∂∂ΦL=M∗l2∗Φ¨−M∗g∗l∗sinΦ=0(2.4)
2Nwhic(q,hq˙)desc=0ribandesthemamotrixtionG(qo)ft=heMin∗vg∗ertedl∗psin(Φ)endulum.thepWithendulummatrixequaM(qtion)=2.2Mis∗l,obtamatinedrix
intheformof:
M(q)∗q¨+N(q,q˙)+G(q)=0

2.2SteppingModelintheSagittalPlane

Figure2.2:Inversependulumwithasinglemassrepresentingthebodymass.

2.2.2Mechanicsofthe2DSteppingModelintheSagittalPlane

Themostsimplewalkingmodelisaballisticmodelconsistingoftwoknee-lesslegswith
pointmassesandathirdpointmass,thehip,whichjoinsthetwolegs[111,49,38].
Sothismodelhastheformofadoublependulum.Theballisticmodelisdrivenbythe
gravitationalforcebywalkingdownaslope.Theenergygainedbythechangeofpotential
energyintodownwardmovementisdissipatedatthegroundcontactattheendofeach
step.Theappearanceofthegaitisdeterminedbythemechanicalparameterssuchas
themassesandleglength,theslopegradientandtheinitialvaluestostartthemodel.
Theseinitialvalueshavetobechosencarefullysothattheenergygainedanddissipated
compensateeachother,leadingtoastablewalkingcycle.Thepassivedynamicsofthe
mechanicsarederivedfromthedoublependulumequations.Thedetailedequationsare
takenfrom[38],wherethelegmassesaresituatedattheendofthelegswhereaswith[49]
themassesaresituatedinthemiddleofthelegs.Thegeneralbasicequationforpassive
dynamics2.2isrewritten:

M(q)∗q¨+N(q,q˙)+G(q)=0
Asthegradientoftheslopeinﬂuencesthegravitationalforces,thevalueψgivestheslope
gradient.ThedetailedmatricesM(q),N(q,q˙),G(q)accordingtothemechanicalmodel
are:]38[of


M(q)=M+2∗m∗(1−cosφ)−m∗(1−cosφ)
m∗(1−cosφ)−m

(2.5)

2PassiveMechanicalModels

Figure2.3:oneThestep;2-Dtakballisticenfromw[alk38]er,(heaalsoviercalledline=theswing”simplestleg,ligwhalkteringlinemo=del”,stancesholegwing).

−m∗sinφ∗(φ˙2−2∗φ˙∗θ˙)
N(q,q˙)=2
m∗θ˙∗sinφ

.6)(2

andN(q,q˙)=−m∗sinφ∗(φ˙2−2∗φ˙∗θ˙)(2.6)
m∗θ˙2∗sinφ
andm∗gM∗g
G(q)=l∗(sin(θ−φ−ψ)−sin(θ−ψ))−l∗sin(θ−ψ)(2.7)
ml∗g∗sin(θ−φ−ψ)
Thisequationnowgivesthebodymechanicsofaballisticwalkerwhichisshowninﬁgure
2.3and2.1withlegmassesmandbodymassMandleglengthl.
Thewalkingcyclecanbedividedintotwophases:(a)thesinglesupportphaseorswing
phaseand(b)thedoublesupportorstancephase.Intheswingphase(a),theswingleg
swingsforwardaccordingtogravitation.Thiscanalsobeseeninﬁgure2.6.Thestepis
ﬁnishedandthetransitionbetweenthetwophases(a)and(b)takesplaceatthemoment
whentheswinglegjustgetsgroundcontact.Thisisthemomentofheelstrike,which
terminatestheswingphase(a)andstartsthedoublesupportphase(b).Theswingphase
iscompletelydeterminedbyequation2.2.Thedoublesupportphasestartswhentheheel
strikeoftheswinglegoccursandendswhentheformerstancelegtoesoﬀthegroundand
initiatesthenewswingphase.Thistransitionfrom(a)to(b)to(a)againisintegratedin
thegroundcontactmodel.Thegroundcontactmodelusedforthesagittal-planeballistic
walkerisdescribedinsection2.2.3.

2.2.3GroundContactModel

rigAsidinbomandiesywthatalkisinginstamondelsta,neotheusg[38round,54,con49,ta93ct,is111mo].deledasaninelasticimpactoftwo

2.3SteppingModelintheFrontalPlane

Ifoeqccurs.uationThis2.2groisundsimulatconetdactforocthecursifswingthephabosedy(a),hasitaiscerobservtainedangulawhetherrposgitroundion.Fcoornttachet
sagittal-planeballisticwalkerthisoccursif:
φ(t)−2∗θ(t)=0(2.8)
Whenthisangularconditionisattained,theheeljuststrikesthegroundandthetransition
fromconstellatswingionphasewhen(a)atostraigdohublet-leggesuppdworatlker,phaseina(b)matakesthematicaplace.lmeThereaning,istoalsoucahesnotherthe
grjustoundswingswithpastboththestalegs.nceThisleg.isFortheacstraaseight-whenleggφed(t)wa=lkerθ(t)this=case0iswheresimplythehaswingndledlegas
swingandnotasgroundcontact.Asthegroundcontactisinstantaneousitiscalculated
asadiscretetransitionoftheform:
q(t+τ)=H∗q(t−τ)(2.9)
whereqisthevectorofsystemstates,Histhetransitionmatrixofthegroundcontact,
qτis=a(Θv,eryΦ,Θ˙,small˙Φ).amoSounthetofstatimeteofandtheqissystemthevjustectorofaftersttateheaheengleslastrikndeadengularpendsveoloncittheies
istradensitionﬁnedmatraccorixdingHtoand[38the,18sta],tewithofthethesystconservemjusatiotnbofeforetorsiothenalheelmomenstrike.tandThethemamotrixdelH
ofaninelasticimpactoftworigidbodies:
−1cos(20∗θ)00
0002H=01+m/M∗sin(2∗θ)200(2.10)
−0−(1+cos(2∗θ˙))00
theAfterstancetheandcollisiosn,wingthelegstancesimplylegintbercecomeshangeathesinsweqinguatiolegnand2.10viceandvtheersa.initialThevaelonglescitiesof
forthenextsteparecalculatedanew.Asthegroundcontactismodeledasacomplete
nonelasticimpactoftworigidbodies,thesystemlosesenergywiththegroundcontact
grasavitthereatioisnalnoenergconservyaddedationboyfthemomenslopetum.totheIftsyhisstelosm,stofheenergysystemiswacomplksdownensatedthebsylotphee
atasteadypace.

2.3SteppingModelintheFrontalPlane

Intheprevioussectiontheforwardmovementofsteppingwasdescribed.Inthissection
themovementtothesideorthelateralmovementwillbeconsideredexplicitly.Walkingis
acomplex3-Dtaskwheretheinteractionbetweentheforwardmovementandthelateral
movementisnotjetknown.Toanalyzetheeﬀectofthelateralstabilityofwalking,one
possibilityisthatthelateralandsagittalstabilizingeﬀectsareonlyslightlyinterconnected

2PassiveMechanicalModels

andwhichbalawasnceshotaskwnbareyofKuosp[ec92ia,l94int].erIenstthiswhichresearclead,htogtheethermedio-latwitherKaluow’salkingassumptiomovn,emetontsa
steppingmodelinthefrontalplane.Inthefollowingthis2-Dfrontalmodelisdescribed
andderived,whichmovesonlyinamedio-lateraldirectionwithnoforwardmovement.
Thistypeofmovementincludessuchmovementssuchassteppinginplaceorsteppingto
ide.stheInthissectionthemechanicalprinciplesofthepresentedmodelaredescribedandthe
formulasusedarederived.Thebasicmechanicsareachainofrigidlinks,whicharetwo
legsjoinedbytwojointsatthehipasisshowninﬁgure2.4.Thismechanicalsystemhas1
or3DOF.Ifthebodyisstandingwithoutliftingaleg,theso-calleddoublesupportphase,
thetheangsystemleα.haIfsoonenlyleogneleavDOF:esttheheglaround,teralangtheleofso-calledswayinsinglethesuppfrontortalplaphanese,dethescribedsystemby
has3DOF:thelateralswayintheankleofthestanceleg(angleα),theupanddown
moﬁgurveemen2.5t.ofTothesimplifyhip(athengleβmec)ahandnicst,hethelateralmassesmovareemenalltpofotinthemassesswingplegos(itangleionedγin),tsheee
centerofeachlink,seealsoﬁgure2.4.

Figure2.4:The2-Dwalkingmodelformedio-lateralmovementsinthefrontalplane,
consistingoftwolegsandahipwithpointmasses.

Themechanicalsystemofthefrontalplaneisapendulumofthreesegmentswhereasthe
sagittalplanesystemisadoublependulum.Thismeansthatthegeneralequationforthe
singlesupportphaseofthefrontal-planemechanicsisapendulumequationasdescribed

2.3SteppingModelintheFrontalPlane

inequation2.2.ToderivetheformulasofthismechanicalconstellationtheLagrangian
formalismisused.Thefollowingsymbolsareused:
g:gravitationalforcewith9.8[N/s2]
m:massofalegwith11[kg]
M:bodymassrepresentedashipmass49[kg]
α,γ:anglesoftheleftandrightlegseeﬁgure2.5
β:angleofthehipseeﬁgure2.5
l,h:lengthofbodysegmentslegsandhipseeﬁgure2.4
q=(α,β,γ)Twhichisasbeforethestatevectorofthesystem.
Thecentralformulaexpressingtheequationofmotionforjointsandsegmentsdueto
gravitationandmechanicalconstraintsistakenfromsection2.2.1,equation2.2.The
systemisconservativewithtime-invariantconstraints.Theequationusedherehasthe
statevectorq=(α,β,γ)T,thisequationtakestheformofpendulumequation:
M(q)∗q¨+N(q,q˙)+G(q)+ucorrective=0
thewithansystemextethatnsionwillucorbreectivegenerawhictedhbisythethecoactrrectivuatioentoandrque.thereforItiseaistoractqueivelyaappliedpplietod
tothemechanicalsystem.Here,thetorqueisproducedbymuscularforcesthatare
Inthedeterminedfollobwingyatheneuraequalostioncillafortortsheystfroemntawl-hicplahnewillmecbehadenicsscribisedderivineddetawhicilinhressectionultsin3.3a.
pandendulum2.5isstusyleedasgenedescription,ralizedgivcoeninordinathetesyequastemtion.abTheove.LagTherangsysiantemformaseenlisminisﬁguresused2.4to
derivetheequation.Shortlydescribedtheprocedureisasfollows:determinationofthe
pCartotenetsiaialnceneorgyordinaofteseacbhysepgolarmenct,oordinacomputatesotionfeaofchthesegLagment,rangiancalculatLionandofﬁnathellydekineticrivationand
oftheEuler-Lagrangediﬀerentialequation,seealsoequation2.2.Thesestepsofderivation
aredetailedinthefollowing.

2.3.1Mechanicsofthe2DSteppingModelintheFrontalPlane
rdinatesoCo

HeretheCartesiancoordinatesofeachsegmentaredetermined.TheCartesiancoordi-
natesaredescribedbypolarcoordinates,seealsoﬁgure2.5.Coordinatesr=(xyz)Tof
thecenterofmassofbodysegmenta:
lsinαlα˙cosα
ra=lcosαr˙a=−l˙αsinα
00

2PassiveMechanicalModels

Figure2.5:The2-Dwalkingmodelinthefrontalplanewiththeanglesα,β,γasangular
jointpositionsforthethreejoints.

Coordinatesrofthecenterofmassofbodysegmentb:
2lsinα+hcosβ2lα˙cosα−hβ˙sinβ
rb=2lcosα+hsinβr˙b=−2lα˙sinα+hβ˙cosβ
00Coordinatesrofthecenterofmassofbodysegmentc:
2lsinα+2hcosβ+lsinγ2lα˙cosα−2hβ˙sinβ+lγ˙cosγ
rc=2lcosα+2hsinβ−lcosγr˙c=−2lα˙sinα+2hβ˙cosβ+lγ˙sinγ
00

Euler-Lagrangediﬀerentialequation

TheLagrangianListhediﬀerencebetweenkineticenergyTandpotentialenergyV:
L=T−V.AndtheEuler-Lagrangeequationisthesumofderivativesofit.TheEuler-
Lagrangeequation2.1canalsobetransformedintothederivativesoftheindividual

2.3SteppingModelintheFrontalPlane

energies,is:hwhicd∂∂q˙L−∂∂qL=
dt=d∂Ta−∂Ta+∂Va+d∂Tb−∂Tb+∂Vb+d∂Tc−∂Tc+∂Vc
dt∂q˙∂q∂qdt∂q˙∂q∂qdt∂q˙∂q∂q
.11)(2

DerivationoftheLagrangianEquation

ThekineticenergyTandthepotentialenergyVarecalculatedforeachsegment.The
kineticenergyTofbodysegmentais:
Ta=21mr˙a2=21m[(lα˙cosα)2+(−lα˙sinα)2]=21ml2α˙2
ThepotentialenergyVofbodysegmentais:
Va=mglcosα

ThekineticenergyTofbodysegmentbis:

Tb=1Mr˙b2
2=1M[(2lα˙cosα−hβ˙sinβ)2+(−2lα˙sinα+hβ˙cosβ)2]
2=1M[4l2α˙2cos2α+h2β˙2sin2β−4lhα˙β˙cosαsinβ+4l2α˙2sin2α+
2+h2β˙2cos2β−4lhα˙β˙sinαcosβ]
=1M[4l2α˙2+h2β˙2−4lhα˙β˙sin(α+β)]
2

ThepotentialenergyVofbodysegmentbis:
Vb=Mg(2lcosα+hsinβ)

ThekineticenergyTofbodysegmentcis:

2PassiveMechanicalModels

Tc=12mr˙c2
=1m[(2lα˙cosα−2hβ˙sinβ+lγ˙cosγ)2+(−2lα˙sinα+2hβ˙cosβ+lγ˙sinγ)2]
21=2m[4l2α˙2cos2α+4h2β˙2sin2β+l2γ˙2cos2γ−8lhα˙β˙cosαsinβ
+4l2α˙γ˙cosαcosγ−4hlβ˙γ˙sinβcosγ+4l2α˙2sin2α+4h2β˙2cos2β
+l2˙γ2sin2γ−8lhα˙β˙sinαcosβ−4l2α˙γ˙sinαsinγ+4hlβ˙γ˙cosβsinγ]
=1m[4l2α˙2+4h2β˙2+l2γ˙2−8lhα˙β˙sin(α+β)+4l2α˙γ˙cos(α+γ)−4hlβ˙γ˙sin(β−γ)]
2ThepotentialenergyVofbodysegmentcis:
Vc=mg(2lcosα+2hsinβ−lcosγ)

CalculationoftheDerivatives

NowthederivativeofLafterqandq˙willbedetermined.Thisisachievednotbyadding
alltheindividualpotentialandkineticenergiesandthenderivatingthetotal,butasin
equation2.11byderivingtheindividualenergiesandaddingthemafterwards.
Derivativesofkineticandpotentialenergyafterqandq˙ofthebodysegmenta:
ml2α˙ml2α¨
∂∂Tq˙a=0dtd∂∂Tq˙a=0
00−mglsinα
∂∂Tqa=0∂∂Vqa=0
0Derivativeofkineticandpotentialenergyafterqandq˙ofthebodysegmentb:
4Ml2α˙−2Mlhβ˙sin(α+β)
∂∂Tq˙b=Mh2β˙−2Mlhα˙sin(α+β)
04Ml2α¨−2Mlhβ¨sin(α+β)−2Mlhβ˙(α˙+β˙)cos(α+β)
dtd∂∂Tq˙b=Mh2β¨−2Mlhα¨sin(α+β)−2Mlhα˙(α˙+β˙)cos(α+β)
0

2.3SteppingModelintheFrontalPlane

−2Mlhα˙β˙cos(α+β)−2Mglsinα
∂∂Tqb=−2Mlhα˙β˙cos(α+β)∂∂Vqb=Mghcosβ
00Derivativeofkineticandpotentialenergyafterqandq˙ofthebodysegmentc:
4ml2α˙−4mlhβ˙sin(α+β)+2ml2γ˙cos(α+γ)
∂∂Tq˙c=4mh2β˙−4mlhα˙sin(α+β)−2mhlγ˙sin(β−γ)
ml2γ˙+2ml2α˙cos(α+γ)−2mhlβ˙sin(β−γ)
4ml2α¨−4mlhβ¨sin(α+β)−4mlhβ˙(α˙+β˙)cos(α+β)
dtd∂∂Tq˙c=4mh22β¨−4ml2hα¨sin(α+β)−4ml2hα˙(α˙+β˙)cos(α+β)
ml¨γ+2mlα¨cos(α+γ)−2mlα˙(α˙+γ˙)sin(α+γ)
+2ml2γ¨cos(α+γ)−2ml2˙γ(α˙+γ˙)sin(α+γ)
−2mhlγ¨sin(β−γ)−2mhlγ˙(β˙−γ˙)cos(β−γ)
−2mhlβ¨sin(β−γ)−2mhlβ˙(β˙−γ˙)cos(β−γ)
−4mlhα˙β˙cos(α+β)−2ml2α˙γ˙sin(α+γ)−2mglsinα
∂qTc=−4mlhα˙β˙cos(α+β)−2mhlβ˙γ˙cos(β−γ)∂qVc=2mghcosβ
−2l2α˙γ˙sin(α+γ)+2mhlβ˙γ˙cos(β−γ)mglsinγ
Theenergiesofallbodysegmentshavenowbeenderivedafterqandq˙,sothatequation
2.11cannowbecalculated.Afterthesimpliﬁcationofthisformulathesystemcanbe
writtenintheformofequation2.2whichwas:
M(q)∗q¨+N(q,q˙)+G(q)=0
InthefollowingthevaluescalculatedforthematricesM,NandGaregiven:

ml2+4Ml2+4ml2
M(q)=−2Mlhsin(α+β)−4mlhsin(α+β)
2ml2cos(α+γ)

−2Mlhsin(α+β)−4mlhsin(α+β)2ml2cos(α+γ)
Mh2+4mh2−2mlhsin(β−γ)(2.12)
−2mhlsin(β−γ)ml2

2PassiveMechanicalModels

−2Mlhα˙β˙cos(α+β)−2Mlhβ˙(α˙+β˙)cos(α+β)−4mlhβ˙(α˙+β˙)cos(α+β)−
N(q,q˙)=−2Mlhα˙β˙cos(α+2β)−2Mlhα˙(α˙+β˙)cos(α+β)−4mlhα˙(α˙+β˙)cos(α+β)−
−2mlα˙(α˙+γ˙)sin(α+γ)−2mhlβ˙(β˙−γ˙)cos(β−γ)−

−2ml2γ˙(α˙+γ˙)sin(α+γ)−4mlhα˙β˙cos(α+β)−2ml2α˙γ˙sin(α+γ)
−2mlhγ˙(β˙−γ˙)cos(β−γ)−4mlhα˙β˙cos(α+β)−2mlhβ˙γ˙cos(β−γ)(2.13)
−2ml2α˙γ˙sin(α+γ)+2mhlβ˙γ˙cos(β−γ)
−mglsinα−2Mglsinα−2mglsinα
G(q)=Mghcosβ+2mghcosβ(2.14)
γsinlgmEquatunderiongra2.2vitaistiontheandﬁnalformecmulahanicalexpreconstrassinginthts.eequaThetionsysotfemmotioisnofconservjoinativtsaendwithsegmetime-nts
invariantconstraints.Thisisonlythediﬀerentialequationforthesinglesupportphase.
Whentheswingleghitstheground,groundcontactoccurs.Thisequationdetermines
wtheasmecmenhationicnedsoafbogrveoundintheconptact,endulumpreseneqteduatioinntheisannextactivseactiotionno2.3ft.2.hepaThessuivcorerectisystemvewhicandh
thiswillbeintroducedandexplainedinchapter3section3.3.

Gr22.3.Contactound

Steppingis,asmentionedbefore,amovementconsistingoftwophases:theswingphase
andthesupportphaseofaleg.Thismeans,ifwalkingischaracterizedbyasequential
rhythmicmovement,itisasequenceofonelegsupportingandonelegswinging,followed
bInyathisshoresrtearcphasehthewheredoublebothsupplegsorthavphaesegroisundconstracontactinedandtobareeverythereforshoret,ssuppoortthatlegthes.
transitionfromswingtosupportlegismodeledinﬁnitesimallyshort.Thestatetransition
canbeseeninﬁgure2.6foronecompletestepcycle.
Theimpactisacontactbetweentworigidbodies.Ittakesplacewithoutanyslipping
andreboundingoftheleg.Thisimpacthappenswhentheswinglegtouchestheground.
Thewhenconthetinuousdiscretepartofimpactthebmoetwveenementhet,tgherounswdingandphasethe,isswingfollowlegedisbytrathensstaferrteedtrfromansitionthe
swingwithoutlegknetoestherequiressupportanlegexpliandcitlyvicevdeﬁnedersa.groThisundisacohntacybridttonogetnlinearsimilarsystem.movemeAnmotsdeasl
withknees.Intheliteraturevariouspossibilitiesforgroundcontactaredescribedsuch
astheincase[111of]ormecin[hanics54],wmohevreingtheintheimpactfrontalmomenplatne,isdethisﬁnedisbsimilayther.aThengulagrproundositioconn.tactIn
isdeﬁnedbytheangularpositionjustwhentheswinglegiscrossingthezeroground

2.3SteppingModelintheFrontalPlane

Figure2.6:Statetransitiondiagramforonecompletestepcycle:thestatechangebetween
doublesupportphase,wherebothlegshavegroundcontact,andsinglesupport
orswingphase,whereonlyoneleghasgroundcontact.

line(surface).Itispossiblefortheswinglegtodipbeneaththegroundsurfaceandto
getgroundcontactlateratthemomentwhentheswinglegcrossesthezerogroundline
again.Thisisimportantbecausenormalsteppinginplacewherethehipdropswhilethe
kneebends.Thegroundcontactisavoidedtillthehiprisesagainandthekneestretches.
Withoutkneesthiscantheoreticallybeachievedbydippingthelegintothegroundand
comingupagaintohaveanimpactforanewstep.Thisisthecaseforlevelgroundwhen:

l∗cos(α)−h∗sin(β)−l∗cos(γ)=0(2.15)
wherelistheleglength,histhehipwidthandtheanglesareasdeﬁnedinﬁgure2.5.If
groundcontactoccurs,thepositionandangleofthelegdonotchange,onlytheirangular
velocitychangesinstantaneously.Therearemanygroundcontactmodelsinliterature.
Herethemodelof[54]describedearlierin[68]isused.Thenewangularvelocityis
determinedbythefollowingequation:

M(q)∗q¨+N(q,q˙)∗q˙+G(q)=ucorrective+δFext(2.16)
withM,N,Gmatrices(mass,centrifugalforces,gravitation)ofthemechanicalsystemas
contactdeterminedpoinetarlierduringingsectioroundnscon2.2t.2act.andThe2.3constrandFainexttsisare,thethatexternaltheseforceexternalappliedforcestoarthee
(1)instantaneous,(2)impulsionalbutthat(3)thepositionremainscontinuousand(4)
ucorrective(thejointtorqueappliedbytheactuators)isnotimpulsional.Thisleadstothe

2PassiveMechanicalModels

factthattheintegrationofequation2.16resultsin:
M(q)∗(q˙+−q˙−)=Fext=E(q)T∗FT(2.17)
FNwithq˙+angularvelocityrightafterthecontactandq˙−beforecontactwheretheposition
staysthesameq+=q−.
+tFext=δFext(τ)dτaretheexternalappliedforcesconsistingofFTandFNwhichare
−tthetangentandnormalforces.Asthediﬀerenceofpositionq+−q−=0andthe
torquecorrectivedoesnotchangeintheinﬁnitesimallengthoftimeofcontact,thetermsof
N,G,torquecorrectivebecomezerowithintegration.ThematrixE(q)=∂∂qJisdetermined
whereJistheﬁnalpositionoftheswingleginCartesiancoordinates:
E(q)=∂J=2∗l∗cos(α)−2∗h∗sin(β)2∗l∗cos(γ)(2.18)
∂q−2∗l∗sin(α)2∗h∗cos(β)2∗l∗sin(γ)
Theconditionthattheswinglegdoesnotslipandtheimpactiscompletelyinelasticleads
tion:equatoE(q)∗q˙+=0(2.19)
Withequations2.17and2.19thereare5equationstosolvefor5unknownsq+,FT,FN
whichare:
q˙+M(q)−E(q)T−1M(q)∗q˙−
FT=E0∗0(2.20)
FNTheinvertibleoftheﬁrstmatrixontherighthandsideisdeﬁnedbecausethematrices
E(q)andM(q)havefullrankandarenonsingular.Thismeansthatequations:
q˙+=1∗M(q)∗M(q)∗q˙−
()detdeterminesthenewvelocityandequation:
FT=1∗−E(q)T∗M(q)∗q˙−
F()detNdeterminesthegroundcontactforces;heredet()isthedeterminantoftheﬁrstmatrixof
equation2.20tocalculatetheinverseofthisnonsingularmatrix.Inotherwordsthese
equationsleadtothetransformationoftheangularvelocitiesjustbeforetheground
contactq˙−totheangularvelocitiesjustafterthegroundcontactq˙+byequation:
q˙+=T∗q˙−(2.21)
whereTisthetransformationmatrixforthestatetransitionbetweensteps.

2.3SteppingModelintheFrontalPlane

2.3.3ExtendedGroundContactModelforActuatedMechanics

Inthenextchapterthesystemconsistsofboththemechanicsforsteppingandtheactor
component,theoscillatornetworkwhichappliesacorrectivetorquetojointsduringthe
swingphase.Iftheswinglegtouchesthegroundagain,groundcontacttakesplace.The
positionduringgroundcontactdoesnotvarybecausethereisnoimpulsivecorrective
torqueappliedtothesystem.Thestabilityofthewholesystemdependsontheenergykept
inthesystemfromsteptostep.Iftheenergydecreaseswitheachstep,thestepmovement
decreasesaswell.Thisdecreasecanleadtoinstabilitiesofthesteppingmovementswhich
shouldbecorrectedbytheactuation.Theactuationcannotcompensatethisenergyloss
inallcasesbecausetheoscillatorstateisnotnecessarilysynchronizedwiththepointof
timewhentheswinglegtouchestheground.Thisleadstothefactthatwhentheleg
touchesthegroundthestateoftheoscillatornetworkcanbediﬀerentforeachstep.This
synchronizationproblemleadstoavariationofhowmuchcorrectivetorquewasalready
appliedtothesystem.Anappropriatemeasureforthesynchronizationandtheoscillator
stateistheenergystateofthesystem.Overseveralstepstheenergyshouldbeaboutthe
sameatthebeginningofeachstepinodertoenablesteadyperiodicstepping.Toensure
thisenergyconstancyoverseveralstepsanenergycontrolcanbeintegrated,whichwill
beexplainedinthefollowing.
Duringtheswingphasethesystemiscontinuousandthecorrectivetorqueisgenerated
andappliedtothemechanicsinacontinuousway.Wherethereisinstantaneousground
contactoccursitisnotclearwhathappenstotheactivationleveloftheindividualneu-
rons.Theankleactuationoftheformersupportlegisnolongerusedaftertheground
contactwhenitbecomestheswingleg,becausetheankleoftheswinglegdoesnotreceive
anycorrectivetorques.Inthisresearchtheneuronalactivationstaysconstantduringthe
groundcontact.Iftheoverlaidfrequenciesofthemechanicaldynamicsandtheoscillator
actuationisnotsynchronousthisleadstoashiftbetweenthetwosystemswhichﬁnally
leadstoinstablemovementsolutions.Toenlargetherangeofstabilitythelackofsyn-
chronizationcanbecompensatedadditionally.Thisleadstoanextensionoftheground
contactmodel.Anenergytransitionruleisintroducedtoensurethattheenergyatthe
beginningofthestepisthesameasitwasatthebeginningofthelaststepbyapush-oﬀ.
Thesystemenergywhichislostoraddedtothesystemasaresultoflackingsynchro-
nizationbetweentheneuronactivationandgroundcontactmodeliscompensatedbythe
energytransitionrulewiththepush-oﬀ.Thiscanbeimaginedasarecoveryofsystem
energyEduringimpact,whichresultsincorrectivetorquesappliedduringpush-oﬀ.This
leadstovelocitiesofthepush-oﬀlegbeingadaptedimmediatelyaftertheimpact.The
equationforthisis:
E(stepi(1))−Epot(stepi+1(1))=Ekin(stepi+1(α˙(1),β˙(1),γ˙(1))(2.22)
whereEisthecompletepotentialandkineticamountofenergyofthesystem.Epotand
Ekinarethepotentialandkineticenergiesofthesystemrespectively.Thereareseveral
succeedingstepswithi=1...numberofsteps.Stepi(1)standsfortheﬁrststate
valuesofstepiwherevertherearestatevaluesforeachtimestep.Theinitialvalues

2PassiveMechanicalModels

++++++forthetheequanewtionst2.2ep2ifor+1thes(aetngulahreervvaeloluescitiesbyα˙theP,oβ˙wellanddogleγ˙g)trustareregioncalculatalgoedbrithmy[solv145ing],
whichsolvesanonlinearminimizationproblem.Theinitialvaluesforsolvingtheequation
+++arinitiaelatherefongularretheveoloutputscitiesα˙of++eq,β˙uatio++nand2.2γ˙0q++..TThishisenegivresgythetranenswitvionaluesruleqis,anwhichalternaaretivthee
toaboutsyncstahrbilitonizingyinthe3.5twandosysenlartemgessmecthereforhanicsetheandpactossibleuatiownoraskingprraopngeosedofinthethesteppingsection
del.mo

onmulatiSi2.4Results

forThenabaturallislticwawalklkingersmastepnydomorwneafactsloporsebygrinﬂuenceavitaationdnalconfotrorcesl.theItscanystem.beobNevjectederthelessthat
withthisballisticwalkeraquitenatural-lookingtypeofwalkcanbegeneratedwhich
isStaextbilityremeislycruciaenergyl,-foreﬃcienifthetandsyststemablefallssolutdown,ionsforneitherwanatlkinguralmovappemeneatsrancecannoberfound.energy
balancecanbeoptimized.Inhumansurvivingstrategiesstabilityofwalkinghasalways
bfooeed.nanTheimpstabilitortantyoffactoraperioasthedicamobilitvyementot,walksuchasstablywisalkingesse,nistialnoftoreahsunilytingderivanded.Ancollectingextra
section3.5isrequiredthereforewhichdeﬁnesthetermstabilityandthemathematical
prooAccordingf.Thetostabilittheyliterisvatureeriﬁed[49,for48,the50,fully126a,ct111uated,11fr2,on92ta],l-pthelanesagitmodelintal-planesectiomondel3.5.3of.
theballisticwalkerachievesstablesolutionsforcertainparameterconstellations.Inthe
followingthesimulatedperiodicmovementsproducedwiththemodelintroducedin2.2
areanalyzedandthecharacteristicsoftheseballisticperiodicmovementsaredetailed.

Thecharacteristicsofadownhillballisticwalkerareastancelegactingasapivotand
aswinglegswinginglikeapendulum.Thiscontinuousmovementisinterruptedwhen
theswinglegstrikestheground.Thenthediscretegroundmodeltransformsthemodel
statebeforethestrikeintothemodelstateafterthestrike,andthenextstepstartswith
theformerswinglegasnewstancelegandviceversa.Thenwhentheswinglegstrikes
thegroundagain,thisstepisﬁnishedandsoon.Theresultisarhythmicmovement
whichisrepeatedstepbystepperiodically.Groundcontactoccursattheprecisemoment
whenthesystemhasaspecialangularconstellation.Inasystemwithnoknees,the
swinglegalsonaturallyscuﬀsthegroundduringthestepse.g.whenthetwolegsarein
equalpositions,whichmeansthatθandφarebothzero.Thesesituationsarenottaken
intoconsiderationforgroundcontact.Thereareseveralpossibilitiesforavoidingsuch
situationssuchasshorteningtheswingleg[48,29]orcausingtheswinglegtomoveto
theside(laterally)[111].

2.4SimulationResults

Figure2.7:wSimalkulaer,tiontakingof8theastepsnglesdo(lewnft)aavnderyangulashallorwveloslopecitieswith(rig0ht).009ofa[rad]stablegradienballistict.

Figure2.8:Phaseplotforthesamesimulationof8stepsdownaveryshallowslopewith
0.009[rad]gradient.

2.4.1BallisticPeriodicMovementsintheSagittalPlane

Usingequations2.5,2.6,2.7fromsection2.2.2andgroundcontactequations2.8and2.9
fromsection2.2.3thefollowingresultshavebeencalculatedandareshowninﬁgures2.7
to2.14.InFigure2.7theangleandangularvelocityfor8stepsareshown.InFigure
2.8therelatedphaseplotofthesystemisshown.Itcanbeseenthatthephaseplot
isacyclicsolutionofaone-periodicsystemasalltheperiodicsolutionsareidentical

2PassiveMechanicalModels

repetitionsandeachsteplooksthesameasthelast.Astablesolutionsuchasthiscanbe
foundexperimentallyor,asGarcia[38]mentions,byamulti-variableNewtonRaphson
orgradientsearchmethodtoﬁndalocalminimalsolutionforanunconstrainednonlinear
function(MATLABfunctionfminunc).Thisdoeshowevermeanthattheinitialvalues
forthesearchhavetobealreadyclosetothesolutionasthisisalocalminimasearch.

SlopenglAe

tioOnenalimpfoorrcestantaﬀectparaingtmeterheofballisticthemowalkdeler.isAstheslopstudiedeangin[le38,,whic48]htheslopdetermineseanglethegrainﬂuencvitaes-
notquencyonly.Intheﬁgstauresbilit2.9yofandthe2.1s0ysteresultsmandcanthebepseeneriofordicitay,chbutangedalsoslothepestepangletolength0.01and[rfre-ad]
andtheresultingvariationsinsteplengthandstepfrequency.Abiggerangleproduces
wlongalkeringbstepsecoamesndaunstaslowerbleifsteptheslopfrequencyeang.leAsiscanenlabergedseeninfurtherﬁgureto02.1.0115and[rﬁgad].ureIn2.1ﬁg2uresthe
2.11and2.12thisﬁnallyleadstotheballisticwalkerfalling.

Variationofinitialvalues

Fortheslopeanglevariationitisshownthattheballisticwalkerquicklybecomesunstable.
Otherimportantparameterswhichareinterestingtovaryaretheinitialvaluesforthe
model,whicharetheanglesandangularvelocities.Asthemodelisasystemwhichisin
generalunstableandonlyhasverysmallparameterrangestoproducestablesolutions,itis
clearthattheinitialconditionsofsuchasystemhavealargeinﬂuenceonitsstability.The
resultsinthissubsectionareproducedwithaslopevalueofslopeangle=0.009[rad].
Figures2.13and2.14showhowachangeintheinitialvelocityofthestancefootθ˙about
0.004[rad]altersthemovementoftheballisticwalkerwiththethirdstep.Theresultis
fallingatthethirdstep.Theoriginalvalueofθ˙was0.199[rad],whichisseeninﬁgures
.2.8and2.7

2.4SimulationResults

Figure2.9:andChangangingulartheveloslopceitieasngleforto80.0steps1[rofad]walkgraingdienbutt,isleadsstilltosstligable.htlylargerangles

Figure2.10:Phaseplotofaballisticwalkerwithchangedslopeangleto0.01[rad]gradient.
Thisleadstoslightlylargeranglesandangularvelocitiesfor8stepsofwalking
stable.stillisbut

2PassiveMechanicalModels

Figure2.11:Simulationoftheanglesandangularvelocitiesofaballisticwalker,taking
5steps.Changingtheslopeto0.015[rad]gradientleadstoaninstability
whichresultsinafallatthe5thstep.

Figure2.12:Phaseplotofaballisticwalker.Thechangedslopewith0.015[rad]gradient
leadstoaninstabilitywhichresultsinalargeincreaseintheanglesand
angularvelocitiesatthe5thstep.

2.4SimulationResults

Figure2.13:Simulationoftheanglesandangularvelocitiesofaballisticwalkerwitha
variedinitialconditionforangularvelocityθ˙=0.195.Thisleadstounstable
walkingandafallatstep3.

Figure2.14:Phaseplotofaballisticwalkerwithvariedinitialconditionofinitialangular
velocityθ˙=0.195.

2PassiveMechanicalModels

2.5Conclusion

Themechanicalmodelpresentedinthischapterwasdesignedonballisticprinciples.In
generaltheaimwastodevelopthesimplestmodelwhichisabletoachievethedesired
mechanicalabilities.Theballisticmechanicsforthesagittalplanecanproducestepping
movementsonaslope.Toshowhowthedevelopedstructurescouldbeappliedtothe
sagittal-planemechanicsanestablishedmodelwasusedforreference.Thestabilityrange
ofthosesteppingmovementsisnarrowanddependsonthemechanics,theinitialval-
uesandtheslopegradient.Theinﬂuenceofinitialconditionsandslopegradientwere
simulatedtodemonstratetheirrelevanceforstability.Challengeswhichtheactuationpre-
sentedinthefollowingchapterwillhavetomeetaretoimprovestability,toexpandthe
possibleparameterrange,toenablediﬀerentsteppingpatterns,andtovarythestepping
movementstrategies.Anothertaskwillbefortheactuationtoattenuatetheinﬂuenceof
disturbancesappliedtothesystem.
Themechanicsintroducedforthefrontalplanedonotoﬀerasuitabledrivingmechanism
suchasaslope.Inadditiontothis,aswasearliermentionedbyKuo[94],themedio-
lateralwalkingmovementneedsanadditionalactuationtoachievestabilityandismainly
independentofthesagittal-planewalkingmovements.Thisleadstotheseparateevalu-
ationoffrontal-planesteppingmovements.Therequisiteactuationandtheabilitiesit
providesforthisnewmodelwillbedetailedinthenextchapter3.

3ActuationofPassiveMechanical
deMosl

Inthepreviouschaptertwoballisticmodelswerepresented,oneforthesagittalplaneand
oneforthefrontalplane.Thesagittalplanemodelachievedstablewalkingonaslope.
Withthefrontal-planemodelnostablewalkingwaspossible.Thesagittalwalkerrequired
ashallowslopetopowerit,butstablesolutionsofwalkingmovementswerefoundtobe
verysensitivetochangesintheslopegradientandtoinitialconditions.

thaActuattitionnooflongerapassivneedseamoslodelpetoenablespropoduceweringwaalkingndscotenps.trolFoftheurthermomodere,l,actuawhichtionmeaisnsa
ptheirossiblerobwusatynetosswstaithbilizeresptheecttdynoinitialamicsofconditiomecns,hanicsslop[87e,gra85,die92n,ts,41,12externa3]landdisturtoenhancebances
etc.,tonameonlyafewinﬂuences.Fortheresearchdescribedinthisthesisanoscillator
networkwasselectedforactuationbecauseitrepresentsrealbiologicalneuronalstructures
iswhicinhtroproducedducerhincorpoythmicratingmovantaemengotsnistlikiceastpplicatioepping.nAofnewthefraconttauatiol-plannetothesteppingjointsmodeandl
amecfeedbachanismskmecareahanismppliedwhicinhidenistbaicasledonfashionthetomtheuscularsagitreceptortal-planefeedbacmodelk.toTheproveselectthated
theyarealsopracticableforsteppingmovementsinthesagittalplane.Bothmodelswill
beevaluated,withrespecttotheirstabilityandtoparameterinﬂuencesonthestepping
movement.Inaddition,possiblesteppingpatternsandstrategiesarestudiedforthe
frontal-planesteppingmodelintroduced,togetherwiththeirresemblancetorealhuman
ts.emenvmostepping

Aselectionofstate-of-theartactuationmethodsforwalkingmodelsispresentedinthe
nextsection3.1.Theneuronalstructurewhichistakenasthebasisforthechosen
actuation,isdetailedfurtherinsection3.3.Theactuationiscoupledwiththemechanics
inpresensectiotednin3.4c.Thapterhecon2astrollaantabilitgoynisotficthemusclesystemforcesbyaappliedctuationtoandthethejointgas;inthisinisstadesbilitcribyaedre
mentionedabove.Thetermwalkingstabilityisdeﬁnedinsection3.5andtheusedproof
forstablesteppingpatternsisintroduced.Theactuationisappliedthemechanicsofboth
artheeshosagwnittalinasndectiofronnst3.7al-planeand3.8mo.del,andthesimulationresultsforsteppingmovements

3ActuationofPassiveMechanicalModels

3.1StateoftheArtofActuationMechanismsfor
WalkingModels

folloTherewaarepresevcaerallculatedpostrsibleajectoactuarytioornatypﬁxedes,forcontrinstaollancwe,[11,actuat102,ion17b].yjoTinhistttoyperquesisvthatery
dampcommoernlyelemenusedtsintoroboticinﬂuences.Anothethermovmethoemendt.ofIna[ctuat195]ionthisiswtheasmorealizeddelingfoofrsapringsagittaandl
themodelmobdelynotfollotowingfalltwforowsard,impleandrules:second,ﬁrst,thatthatthethesswingwinglegleisgnotmovpeslacedquicktolyoefarnoinughfronfort,
sowiththatthethelegnexlengtthstepandcasntepstilllengtbehbeingstabilized.variedIn[to29]obtanotainhetherapdesiredproachwisalkingdemoresnstult.ratedA
furtherpossiblemechanismistostoreenergyintheankletogiveandreleasethisenergy
inadditacoionntroofanlledymakindnneofrtoactuathetionsystemmeansviathaatthepush-oﬀpassivitduringyofthethewstaalknceingphasesystem[23].isTlost,he
hbutumaconntbrollaeingsbilitarey,ofstacoursebilityaacndtivroe,butbustnestheslearevelgaofained.ctivaAstionfoundduringby[w19a4],lkingtheismmusuchcleslessof
thanindicatesduringthataotherctuatiomovnemedurnts;ingwespecalkingiallyisthereducedswingtolegamminimusclesumaretoklittleeeptheactivenergyated.Thiscost
low,butneverthelessprovidesadditionalstabilityandrobustness.
Forthepresentresearch,actuationbyneuralnetworkswaschosen.Thisactuationbases
onthephysiologicalstructuresfoundinthecentralnervoussystem(CNS).Theseare
networksofneuronsthatgeneratesocalled”centralpatterns”.Neuronalcellstructures
thatcanproduceoscillationswithoutsensoryinputarecalledcentralpatterngenerators
CPGsaccordingto[30].TheseCPGsactivateneuronalstructuresandﬁnallythemuscles,
whereﬂying[15they3]orprowaducelkingrh[yt156hmic].Inmosvectioemennt3.2s.theThesbaesicmostvemenructurestsofcansucbehe.g.neurosnalwimmingoscill[ato51rs],
bareeentheexplainedsubjectinofmorevariodetausil.reseTaherches,actuatsucionhoasf[leg40,mo17v2,emen100,ts15b2y,12CPG9,183].structuresIn[17ha2s]
bipTheedalforwloardcomomovtionemendrivtencanbybeneuralvariedboscillatymoorsdeisluseparadtometers.achiev[e15sta2]ablendwa[40]lkingbuildpattertheirns.
neuronastructurellytodrivaendaptbiploedacomolwtionalkingtoenmovidelsronmenonbiotallogchaicalnges.parHoadigmsweverandtheyprobovidethcanoancendaptivtratee
onwalkinginthesagittalplane.Geneticalgorithmscanbeusedasin[70],tooptimize
theparametersfortheneuronalnetwork.

3.2ExamplesofOscillator-DrivenMovementsinBiology

Thesstructures,etupandusedfinuncttheioningfolloofwing,theareMadetsuokrivaedhere.oscillatorHence,netwaorkshorttoovrepreserviewentothefnebiolourgicalonal
protheirtotdetypesailedinamodenimalsofopiseratgiven.ioninTheretheaCNSrenoofhexpumanserimen[53ts].abButoutthereoscillatoarerexpsterimenructurestsaandnd

3.2ExamplesofOscillator-DrivenMovementsinBiology

Figure3.1:Mechanicsofthelampreyconsistofrigidsegmentsinalinefrom[55].

resultsabouttheneuronalstructuresofoscillatorsandtheirfunctionalityinanimals.In
thefollowingthetwoexamples’lamprey’and’cat’arepresentedtogiveanideaaboutthe
functionalityofsuchneuronalstructureswhichworklikemutuallyinhibitingoscillators.
Theanimals’movementisarhythmiconeandthiscanbedirectlycorrelatedwiththe
ucture.strlneurona

reypLam13.2.

Thelampreyisaneel-likeﬁsh,whichhasveryancientandthereforesimpleandlarge
neuronalstructures.Inthelampreyitwasﬁrstfoundandproventhatneuraloscillators
canproducearhythmicpattern,andthatthispatternproducestheswimmingmovement
ofthisﬁsh[51].TheseneuronalcellswerefoundintheCNSoftheﬁshanditwasproven
thatnobrainwasnecessarytoinstigatetherhythmicmovement,onlyaninitialimpulse
totheneuronaloscillators.Thisimpulsepromptstheneuronstoautonomouslyproducea
rhythmicactivationpatternforswimming.Additionally,thismotorpatternisreproduced
intheisolatedbrainstemcord[53].Theseneuronalstructuresareinterconnectedand
mutuallyinhibitorysothattheyareabletoproduceoscillations.Suchaneuronnetwork
isconnectedtothemusclesofthelampreyﬁsh,sothatthemotoneuronsofthemuscles
areactivated.Theactivationpatternalwaysleadstoacontractionofthemusclesonone
sideoftheﬁsh,whilethemusclesontheoppositesidejustrelax.
Thiscanbeseeninﬁgure3.1wheretheindividualjointmechanicsareinterconnected
withtheantagonisticmusclepairs.Inﬁgure3.2.1thecorrespondingneuronalstructures
consistingoffourneuronsconnectedtothemusclesareshownaswellastheresulting
movementofthelamprey.Inhibitoryneurons(I),excitatoryneurons(E)andinterneurons
(L)areassociatedwithanoscillatorwhichisconnectedtothemusclesviamotoneurons
(MN).Whilethemotoneuronofonesideisexcitedthemotoneuronontheothersideis
inhibited.Ononesidethemusclesarecontracted,whileontheothersidethemusclesare
notcontractedandthereforecanbestretched.Astheindividualoscillatorsareconsecutive
fromheadtotailoftheﬁsh,oneoscillatorsubsequentlytriggersthenextoscillatorwitha
shortlatencyinbetweenandsoon.Thisleadstoaphaseshiftbetweenthesegments.So
themovementofsequentialcontractionandrelaxationresultsinaserpentinemovement
swimming.for

3ActuationofPassiveMechanicalModels

Figure3.2:Neuronalinterconnectionofonesegment(oscillator)ofthelampreywiththe
musclesviathemotoneurons.

Cat23.2.

Therearesomeinterestingexperimentswithcatsthatgiveaninsightintohowlocomotion
functionsinthecat.Ithasbeenproventhatrhythmicalmotorpatternsaccordingtoa
CPGintheCNSproducerhythmicmovements[53,51,161].In[45]itisshownthatin
Sucdecehrebcatsratedarecafurttsheactivrmoreationablepatotternsdowcanastlkingillbmeovefoundmentthastonainducetreadwamill.lkingThismovemenﬁndingts.
hasproventhatthebasicmotorpatternisgeneratedbytheCNSwithoutanyhigh-level
sensoryinputsuchasvision,senseofequilibriumoroverallproprioception,butthat
rhyadditthmicionalmovhigh-levementelsgeneratensoryorisinputnotaninﬂuencesorganthisbutmosettingreo.rThelessaconceprincipleptofanoffuncautotnomoioning.us

Theseautonomouspatternsarea”substrateoflocomotion”[147].Accordingto[110]
themuscularﬂexionsandextensionsduringlocomotioninmammalianareproducedbya
CPGstructure.TheneuronaloutputsoftheCPG,intheformofneuronspikingactivity,
canbedeterminedduringlocomotionbutthestructureoftheCPGanditsneuronal
interactionsarenotknown[53].
ItisalsoclearthatthisCPGhasitslimits.Nevertheless,thecatisagoodexampleof
howrhythmicmovements,especiallywalking,areproducedinlowerlevelcontrolcircuits
thanthebrain,eveninhighervertebrates.TheCPGisonlyapossiblerepresentation
ofthesecontrolcircuits,asnocompletestructurebutthefunctioningofthissystemhas
beenproveninthecat[73].

ItisnotknownwhataCPGstructurelookslikeinhumanbeings,butasvertebrates
ashighlydevelopedascatshavesuchneuronalfunctionalities,thisisanindicationthat
evolutionismorelikelytoadvancethisestablishedstructurethantodiscardit.Therefore

3.3NeuralOscillatorModel

Figure3.3:Activationpatterngeneratedbymutuallyinhibitingneuronswithantagonistic
activationofﬂexorandextensormuscles.

thisideaofaCPGstructureisadoptedtocreateabipedsteppingmodelforrhythmic
ts.emenvmoleg

3.3NeuralOscillatorModel

Theactuationofthepassivemechanicalsystemisrealizedbyneuronaloscillatorswhich
generateantagonisticjointtorquesresemblingtomuscleactivation.Thefunctioningof
musclesisonlybasedoncontractionofthemusclesanditsﬁbers.Thisrequiresan
mantagusclesonisticatminimcompum:ositioannoftextensorhemusandcles.aFﬂexoror.exaIfonemplemausscleimpleconhingtractsetjoinhetntherequiresothertwiso
extendedandviceversa.Thisprincipleofﬂexorandextensoristransferredtotheneural
sensactivoatrs.ion.FigEaurechm3.3uscleshoiswsacontpairrolledofbﬂeyxmotorandoneuronsextensoandrthmeuscfleseedbacwhickofhamreusculaactivratedstretcbyh
amutuallyinhibitingpairofneuroncompounds.Eachneuroncompoundconsistsof
excitatory,inhibitoryandmotor-neurons.Theexcitatoryneuronofonepairactivates
themotoneuron,e.g.oftheﬂexormuscle,andtheinhibitoryneuroninhibitstheother
pair,e.g.oftheextensor.Thetwomotoneuronactivations,andconsequentlythemuscles,
arrigehtactivinﬁgatedurean3.3ta.Heregonisticathellynaturalthereforespikingandshoratewofalternaanactivtingeactivitneuronyiswhicshohiswn.seenonthe

3ActuationofPassiveMechanicalModels

Figure3.4:Schematicrepresentationofneuroninterconnectionswiththemuscularsystem
whichcontractsthejointmuscles.

Asimplemodelofthisfunctionalityhasonemotoneuronforoneﬂexorandoneforone
extensor.Thispairofﬂexorextensormotoneuronsisrepresentedbyamutuallyinhibit-
ingoscillatorconsistingoftwoneurons,eachintegratingtheinhibitoryandexcitatory
component.Thisisvisualizedinﬁgure3.4.
ThisprincipleofmutuallyinhibitingneuralnetworkswasintroducedbyMatsuoka[107,
108]theso-calledMatsuokaoscillator.Withdiﬀerentcombinationsofoscillatorsanetwork
isbuiltwhichgeneratesanactivationpattern.Ifthispatterniscoupledwithmuscle
activationthenamovementpatternofthemusculo-skeletalsystemcanbegenerated.This
integrationofoscillatornetworkandmuscleactivationwhichleadstoactuatedmovements
ofthemechanicalsystemwillbedescribedinsection3.4.

3.3.1TheMatsuokaOscillator

Theneuronmodelusedinthepresentstudyisacontinuous-timeneuronmodelasde-
scribedin[131].Themodelrepresentstheﬁringrateofaneuronbymeansofacontinuous
variableoftimewhichcorrespondstotheactivationofthemuscles.TheMatsuokaoscil-
latormodelintegratesmutualinhibition,excitationandexternalinpute.g.fromhigher
controllevelssuchassensorsandthebrain.Thisintegrationofnaturalneuronproperties
inatime-continuousrelativelysimplemathematicalmodelrepresentsanadvantageover
otherneuronssuchastheHodgkin-Huxleymodel[3].FurthermoretheMatsuokaoscil-
latorappliesthesepropertiestointerconnectedneurongroups,whichisusefulfordirect
ion.actuatonistictaganMutualinhibitionisrealizedbyweightingthesynapticconjunctions,whicharepositiveif
theyareexcitatoryandnegativeiftheyareinhibitory.Thepresentneuronmodelincludes
anadaptionovertime.Thismeansthatconstantexcitationdoesnotleadtoaconstant
outputbytheneurons,butdecreasesovertime.Theextendedequationforaninhibiting

3.3NeuralOscillatorModel

Figure3.5:Adaptiveneuroncharacteristicsvisibleforastepresponse,takenfrom[107].

neuronmodelasproposedin[107,108]is:
k1n˙i=∗(−ni−aij∗fj−b∗yi+ci∗si+exti)
Ta=1j1y˙i=∗(fi−yi)
Tbwithfi=max(0,ni−Δ)

.1)(3

where:i:thenumberoftheactualneuron
j:aselectednumberofthe1...kneuronsinthenetwork
nT,T::thethememtimecobranenstpantsotenoftialtheofosthecillatorneuroni(internalstateoftheneuron)
bafΔ::thetheﬁringthresholdratevofaluetheunderoutputwhicofhneurothenjneurondoesnotﬁre
saij::wtheeightimpulseofrainhibitoteoryfansynaptexternaiclinputconnectionsignatolaneuronjinthenetwork
c:weightofthesynapticconjunctions
yb::theadaptaadationptatioornrfatigateueforvariablesteady-stateﬁring
exti:orexternasensors,linputwhsichafromlsocahighnerlevdirectlyelssuchinﬂuenceasthethebrainneuronactivity
Table3.1:Parametersoftheneuronmodel.

Thismodeltakestheadaptationofaneuronintoaccount.Iftheneuronreceivesastep
input,theoutputﬁringrateinitiallyincreasesbutthendecreasestoalowerlevel,which
istheadaptationlevel.Thisisshowninﬁgure3.5.

3ActuationofPassiveMechanicalModels

Figure3.6:Asimple2-neuronoscillatornetworkanditsrhythmicactivityfrom[107]

Forexample,twoneuronsareconnected(j=1,2)andoneisﬁring,whichmeansahighn1
value,theadaptationrateleadstoadecreaseinthisvalueandthereforetheinﬂuenceof
theotherneuronincreasesasa12∗f2rises.Thereforethesecondneuroninhibitstheﬁrst
andafterawhilethesecondisﬁringatn2>0.Thisisseeninﬁgure3.6.Inthefollowing
thevalueΔissetatzero.Thisdeﬁnitionmeansnolimitationofgeneralitytothesystem.

3.3.2ConstraintsforOscillation
Therearesomemathematicallydeﬁnedconstraintstoguaranteeanoscillationmodefor
theneuronsofaneuronalnetworkwithaspecialparameterconﬁguration.Astable
rhythmicsolutioncanbeachievedfortheoscillatornetworkinequation3.1ifthefollowing
twoconstraintsaccordingto[107]arefulﬁlled:
sa1+ijb<sjifori,j=1....n(3.2)
wherenisthenumberofneuronsinthenetworkand
√aij∗aji>1+Tb(3.3)
TaTheoscillatornetworkspresentedinthefollowingareallparametrizedtomeetthese
criteriaforachievingastableoscillationsolution.

3.3.3BasicNetworkTypes
Inthefollowingthepropertiesofsomediﬀerentbasicoscillatornetworksaredescribedas
theyareusedformovementpatterngeneration:typeA)the2-neuronnetwork,whichis
anoscillatorandtypeB)the4-neuronnetworkswithdiﬀerentinterconnectionsbetween
thetwooscillators.TheﬁrstbasicnetworkA)isthe2-neuronnetworkseeninﬁgure3.6.
Eachneuronsuppressesandstimulatestheactivityoftheotherneuron.Thisoscillator
hasthecharacteristicthatonlyoneneuronﬁresatatime.Theoscillatorcorrelatesto
movementssuchassimplerhythmse.g,ﬂuttering,chewing,movingonelegwhich,as
mentionedabove,isasimpleantagonisticmovementbyﬂexorandextensor.
ThetypeB)networkwithtwooscillatorscanhavediﬀerentinterconnectionsbetweenthe
neuronsortheoscillators.Inthisstudythreekindsofinterconnectionsandtherelated

3.3NeuralOscillatorModel

Figure3.7:Three4-neuronnetworkswithdiﬀerentinterconnectionsproducediﬀerentac-
patterns.natiotiv

activationpatternaccordingto[85,107]areshown.Ifeachjointrequiresanantagonistic
oscillator(=pairofneurons),asymmetricnetwork,therearethreepossiblebasic4-neuron
networks,whichareshowninﬁgure3.7.

Intheﬁrstexampleinﬁgure3.7theinterconnectionsareanti-clockwisewithinhibitory
synapsesandcrosswisebetweenthetwocorners.Thisleadstoasequenceofactivation
fromneuron1,3,2,4.Inthesecondexampleinﬁgure3.7theinterconnectionsare
betweentheoscillatorpaironeachsideandbetweenthecongenerousneuronsoneach
side.Theresultingactivationisoscillationfortheneuronsoneachside1,2and3,4with
3identicalto2and4to1.Inthethirdexampleinﬁgure3.7theinterconnectionsareas
inthesecondexamplebutalsoincludeadditionalcrosswiseinterconnectionasintheﬁrst
lineplot.Thisleadstoathirdactivationpattern.Itisasequenceofshorteractivations
asintheﬁrstnetworkandwithloweramplitude.Thebasicactivationpatternsdescribed
willbeusedintheoscillatornetworksforthesteppingmodelpresentedinthenextsection.

3ActuationofPassiveMechanicalModels

Figure3.8:Interconnectionoftheoscillatorwiththemechanics.Actuationoftheankle
jointofthestanceleg.

3.3.4NeuronalOscillatorNetworksforWalking

First,oneoscillatorpatternforthesagittal-planemechanicsisshown.Asthisisasimple
modelofadoublependulum,onlyonenetworktoactuateandstabilizethismodelis
presented.Second,forthemainsubjectofthisresearch,thefrontal-planemodel,there
aremorevariationsofinterconnectionswhicharesuitable,so4diﬀerentnetworksare
duced.otrinAnoscillating2-neuronmodeliscalledanoscillator.Alloscillatornetworkswhichare
usedinthisworkalwaysconsistofseveral2-neuronnetworks.Ajointisalwaysantago-
nisticallyactuatedwhichmeansthatajointisactuatedbyone2-neuronoscillatorwhich
oscillatesandreactsthereforeantagonisticallyonthejoint.The2-neuronoscillatorsare
interconnectedtonetworkstointerrelatetheindividualjointmovements.

NetworksfortheSagittal-PlaneModel

Themechanicsusedforthepassivewalkingonaslopeshowninﬁgure2.3arethestructure
usedtointerconnectwiththeoscillatornetwork.Asthesagittalplaneisnotthemain
topicofthisthesisbutispresentedtoroundoﬀthemodelingofwalkingmovements,only

3.3NeuralOscillatorModel

onepossibilityforanactuationnetworkforthemechanicsisgiven.Inthemodelseenin
ﬁgure3.8onlytheanklejointisactuatedinthiscasebyoneoscillator.Theswinglegis
stillunactuatedsothereisnooscillatorforthehipjointmovement.Thisisrelatedtothe
factthattheswinglegreactsmainlyballisticduringtheswingphase[114].
Thissimplenetworkrelatesalsotothefactthatthemainmetaboliccostisthestep-
to-steptransitionaccordingto[28,95].Theactuationoftheanklejointenablesthe
systemtocompensatetheenergylostduringthestep-to-steptransitioninthepush-oﬀ
phaseandafterwards.Allothermovementsresultfromgravitationalforces.Thisis
moreoveroneofthesimplestactuationspossible.Neverthelessitshowsastronginﬂuence
onthemovementwithouteliminatingtheballisticprinciplecompletely.Theresultsshow
thatmovementvariabilityandanincreaseinrobustnessareachievedbythisactuation.
Anotherpossibleactuationwouldhavebeenactuationofthehipjoint,whichrelatesto
thehipstrategytobalanceforwardwalkingaccordingto[193].Anotherpossibilitywould
havebeentheactuationwitha4-neuronnetworktypeofanykindwhichactivatesankle
andhipincombination.Theresultspresentedinsection3.7areallgeneratedwiththe
anklenetworkshowninﬁgure3.8.
Asthenetworksareusedforallmodelsinstanceandwalkingorwalkinginplace,the
activationandtheinﬂuenceoftheankleneuronsdependontheotheroscillatorsbutalso
onthegroundreaction.Ifthereisnogroundreaction,therecanbenoforcetransmission
tothegroundandthereforetheankleoscillatordoesnothaveanyinﬂuenceonthesystem.
Thismeansthattheactivationfortheneuronsofthelegwithnogroundcontacthasno
eﬀectandtheyarethereforenotshownintheﬁgures.Inthesagittal-planemodelthis
alwaysreferstoneurons5and6andinthefrontal-planemodelthesearealwaysneurons
7and8(seeﬁgure3.9).

NetworksfortheFrontal-PlaneModel

Fplaneourmodiﬀerendel.tTheconﬁg4neturatworkionssaofreavisualineuronzedalinﬁgoscillatouresr3.1net0wtoork3.1ar3eandusedarewithdenottheedfrbyon(ata);l-
inactivitadditioyanrethesimvisualizedulatedintheneuronalsubﬁguractivesatdenoiontedpatterbyns(b)for.theStand-alostand-alonenemeanscaseofwithoutneuranonaly
inrecouplinglationortointertheactionlaterpwithositiothenmecforhanicscouplingalthougwithhthethelomeccatiohanics.noftheTheneurobasisnsisforshothesewn
diﬀerentneuronalnetworksisalwaystheconstellationofjointsandoscillatorsshownin
ﬁgure3.9.Inthisﬁguretherelationofneurontojointisvisualized.Foreachjoint
twoneuronsareappliedasanoscillator,whichoperatesinanantagonisticwaylikethe
extensorsandﬂexorsofthemuscularsystem.Alljointsarehingejoints,whichmeans
thatoneﬂexorextensorpairissuﬃcienttorealizethefullspectrumofjointmovement.
ThediﬀerentneuronconstellationsandtheiractivationpatternsaredenotedP1...P4.
Figure3.10(a)showstheﬁrstneuronalnetworkP1withrespecttotherelatedmechanics.
whicHerehtheisshohipwnisininﬁgureterconnected3.7.likTheetheanklessecondareco4-neuronnecntednettoworkthehipoutlinedwithinthessectioname3.3typ.3e,

3ActuationofPassiveMechanicalModels

Figure3.9:Frontal-planemechanicslinkedwiththeactuatingneuronsateachjoint.

ofnetwork,whichleadstoasynchronizationofactivationofthehipandankle.The
activationisinantagonisticrhythm.Atanygiventimeoneneuronofanoscillatoris
activeandtheotherisinactiveandalltheneuronsaresynchronized.Thisisshownfor
nthe1isneuractivonale,activneuronsationn2paandtternn3ofareneuroinhibitnalednetawndorktPheref1oinreﬁgnotureactiv3.1e,0(b).butWhenneuronnenur4onis
activeatthesametime.Theanklejointneuronsarecoupled,whichmeansthatneuron
n6isneurons,activeexceptwhenthatneurtheonan4nkleisaisctaivctaivted.atedTsheligahctivtlyaeartionlierisbutsynccohrontinnousuesfortillhiptheandactivankatedle
hipneuronsbecomeinactiveagain.

ThesecondoscillatornetworkP2isalsostructuredinlinewiththesecondexample
networktypeoutlinedinsection3.3.3,whichisshowninﬁgure3.7.Thehipandthe
anklesareconnectedasintheﬁrstnetworktypebutwiththehipconnectionchanged
crosswise.Thiscanbeseeninﬁgure3.11(a).Theresultingactivationpatternisthesame
andissynchronizedforthewholenetworkinactiveandinactiveneurons,butthereisa
changeintheneuronswhichareactivatedatthesametime.Thehipactivationisstill
pairwise,butnowactivewithneuronsn1andn3atthesametimeandankleneuronn5
activatedjustbeforethehipreacts.ThisisthesamecharacteristicasinpatternP1.
Thismeansforlatermovementsanearlier,longerbutlowerankleactivation.Thiscan
beseeninﬁgure3.11(b).

(a)Neuronalnetworkpattern1P1

3.3NeuralOscillatorModel

(b)NeuronalnetworkP1activitypattern.

Figure3.10:NeuronaloscillatorpatternP1

(a)Neuronalnetworkpattern2P2.

(b)NeuronalnetworkP2activitypattern.

Figure3.11:NeuronaloscillatorpatternP2

3ActuationofPassiveMechanicalModels

(a)Neuronalnetworkpattern3P3.

(b)NeuronalnetworkP3activitypattern.

Figure3.12:NeuronaloscillatorpatternP3

inThesectithirdonne3.3t.3w,orkwhictypheisP3shoiswninderivedﬁgurferom3.7.theThethird4-neuroexanmplenetwnetorkwsorakretyapedditiooutlinednally
crosswiseinterconnectedwhichcanbeseeninﬁgure3.12(a).Theresultingoscillator
activactivatationion.pattThisernissigniﬁesahipthatactivneuronsation,n1whicandhnis2oarefactdoubliveeduringtimebothduratioanctivasetphaheseankplusle
inactivephaseoftheankleneuronsn5andn6,whicharemutuallyactivatedrespectively.
Thisisseeninﬁgure3.12(b).

(a)Neuronalnetworkpattern4P4.

(b)NeuronalnetworkP4activitypattern.

Figure3.13:NeuronaloscillatorpatternP4
theTheﬁrstfourthbasicandlast4-neuronnetwnetorkwtorkypetypP4eforoutthelinedfroinntaselwctioalkning3.3.3meinchaﬁgnicsureis3.7.derivedHerefrtomhe

3.4ActivationofMechanicswithOscillators

inideatercoisnnecthattioneachissidedirectlyofthehipinﬂuencedisinbytheterconnectedanklecwitohnnecbtothion.ankleFiguresides,3.1so3(a)thatshothewshipall
areconnectionsynchros.nizeThedbutresultingtheankleactivatactivionatiopanistternphaseisacroshiftedsswiseasshoactivwneinhipﬁgurwheree3.1both3(b).sides

3.4ActivationofMechanicswithOscillators

Inoscillatthisorsectionetwnorktheiscopropuplingosed.ofThethefopallossivwingesdynamicubsectionsof3.4the.1meandcha3.4nical.2inmotrodelducewith(I)tthehe
generationoftorquesappliedtothemechanicsand(II)feedbackfromthemechanicsto
thetheocosmcillabinator.tion(I)offorceactivgeationerans.tionTheis(aII)fesimpleedbawckeighisteadcomadditivbinatioemecnsighanismnalofjowhicinthpaddsositionup
andsystemveloiscity.presenWithted(I)whicahndp(II),erformstherhmecythmichanicsmovandementhets.Inoscillatﬁgorurenet3.1w4ork,anoavcoerviewmpletofe
thesystemwiththeinterconnectionsbetweenmechanicsandneuraloscillatorsisshown.

Figure3.14:Systemoverviewoftheactuationofthepassivemechanicsbyneuronalos-
cillatorswithproprioceptivefeedback.

3.4.1JointTorqueGeneration

Theactivitygeneratedbytheoscillatornetworkistransformedtoatorquewhichisap-
pliedtothemechanics.Informula2.3thetorqueapplieddirectlytothejointsinaddition
tothegravitationalforcesistheucorrective.Insection3.3theantagonisticstructureofthe
muscularsystemwasdescribed.Putsimply,muscularinnervationincludesmotoneurons
whichcontractthecorrespondingmuscleﬁbersiftheyareactivated.Themusclecannot
dothecontrarymovement,thestretching.Thisstretchinghastobeperformedbyits
antagonist,whichinitsturniscontractedbymotoneuronactivation.Thecontraction
strengthdependsonthemotoneuronactivationlevelandthereforeonthequantityof
muscleﬁberswhichareactivated.Theactivationoftheoscillatorneuronsispropor-
tional.Theactivationoftheoscillatornetworkinducesthemotoneuronactivationand
thisgeneratesthemusclecontractionwhichinturnappliesatorquetotheattachedjoint.
ThediﬀerentactivationpatternsPileadtoadiﬀerentmusclecontractionandtherefore
toadiﬀerenttorquebeingappliedtothejoints.Ifthereare4oscillatorpairsasinﬁgure

3ActuationofPassiveMechanicalModels

3.9theforcegenerationforthefourjointsucorrective=uk,wherek=1...4numberof
joints,is:8
uk=wjk∗nj(3.4)
=1jwherewistheweightwithwhichtheoscillatorneuronnjinﬂuencesthecorresponding
.ktjoinIntgenerahetedcasejoinoftthetosagrquesittaalndmodeneuronsltheistorlessquegwithenerkatio=1n,is2.idenThejotical.intOtorqnlyuetheofntuhemberankleof
isankgele,nerawhictedhiswithtgenerawotedneuronsbytheaswseeigenhintedﬁgursumeof3.8the.twThisoisneuronsthetn1orqueandu1n2foratthethestaancnklee
joint.Thetorqueappliedtotheankleis:u1=w11∗n1+w21∗n2.Thehipjointisnot
actuated.Thereforetheappliedjointtorqueu2=0.
Nowthejointtorquesaredeterminedinthesamewayforthefrontal-planemodel.A
torqueisappliedtoeachjointk.Thistorqueiscomputedaccordingtoequation3.4.The
computedtorquesdependdirectlyontheneuronactivationlevelnandtheweighting.
Inthepresentedmodelitisalwaysthedirectlyconnectedneuronsjwhichcombinetheir
activationlevelstogenerateonejointtorque.Likeapairofﬂexorandextensormuscles,
thetorqueappliedtoajointisgeneratedbythecombinedsumofappliedtorques.For
beexampleseeninforﬁgurjoinetk3.9.=T1forherefortheefrmoonstatl-ofpthelanewemoighdel,tsarethesezeraroeasthethereneuroisnsnonin1andnterconnectio2ascann
betweene.g.,thefootneuronsandthehipactuation.Theweightvalueswhicharenot
aszerothaisndisthethereforjoinetactivwhicehahareswno12,grw22ound,w33,conw43tact,w51but,w61is.theThesjowingintkleg=a4nkle.isnotTheactuattorqeued
u4=0.Theothertorquesu1...u3areasfollows:
u1=w51∗n5+w61∗n6
u=w∗n+w∗n
u32=w1233∗n13+w2243∗n24
Table3.2:Thecorrectivetorquesappliedtothethreejoints.

Therearediﬀerentstrategiesinhumanbeingsdeterminingwhichmusclegroupsareac-
tivatedduringstanceandwalking.Thejointtorquescanthereforevaryintheweighting
whichdependsonthestrategyused.Ifthehipisactuatedmorethantheankle,the
weightsforthehipjointsarelargerthantheweightsfortheanklejoints.Insection3.8.2
thevariationoftheweightsaccordingtodiﬀerentstrategiesisinvestigatedinmoredetail
andtheresultingmovementpatternsareshown.

3.4.2MuscleFeedbackAppliedtotheOscillator
Feoscillatedbackors.isIttheisdeinforpendenmatiotnonwhohicwhisthetraoscillatnsmittedornetwfromorkthereactsmectohanicalchangessystemofthetotme-he

3.4ActivationofMechanicswithOscillators
chanics.Thefeedbackiscombinedwithjointactuationwhichleadstoafeedbacksignal
correspondingtotheindividualjointmovements.Insection3.3theoscillatoractivation
representstheneuronactivationwhichdirectlyinducesmusclecontractioninlinewith
anantagonisticﬂexorandextensorpair.
First,Accordingthetmo[uscle178,le15ng9th]manduscularvelocitfeyedbacfeedbackiskofrepresenthemtedusclebytwospindlediﬀerenaﬀetrents,feedbacwhickhlooisps.a
proprioceptivefeedback.Thisfeedbackdirectlyinﬂuencesthemotoneurons.Thesecond
feedbackistheforcefeedbackfromthetendonorgansandinﬂuencestheinterneurons.
bacTheskeatndwostamechabilizationismns[9ar].ebesThesideeotproherprioreﬂexesceptivethefemaedbainckmelochaopsnismsallowforamstableuscularpostfeed-ure
controlifthefrequenciesarenottoohigh[178].Theadvantageisthatthiskindof
proprioceptivefeedbackdoesnotcostextraenergy,unlikeothermechanismssuchasco-
contractionofmusclesinordertoincreasetheintrinsicmusclevisco-elasticity[178].
Thecorresppresenondstmodirectlydeltorepresenthetsangtheularjoinpropriotpcoeptsition.ivemTheusclespindlespindlevelolengtcityhfeedbacfeedbackkiswhicalsoh
usedandthisisproportionaltotheangularjointvelocity.Theforcefeedbackisomit-
tedinordertosimplifythemodel;itisnotneededbecauseitimprovestheimpedance
characteristicofthemusclesystem[178],whichisnotacriticalfactorfortheproposed
del.momIntuscleheismocodenltracpresentedittedsaantasimplegonistisstructurstretceished.usedThisbasstedretconhingtheorfolloextensionwingisprinciple:sensedIfbya
proprioceptivesensorswhicharerepresentedasinhibitorysynapsesoftheneuronsinthe
neuronoscillatoritself.Proprioceptiveinformation,givenbythemusclesandthejoints,
canstablebeosusedcillafotingrmoexternavlement.sensoryThisinformasensorytioninformatoadapttionisthetheoscillafeedbactorknetsigwonalrktogivaenchietovtehea
oscillatorsasexternalexcitatoryinputsiintroducedinequation3.1.Asthisfeedback
dependsonproprioceptiveinformation,thefeedbacksignalischosentobeacombination
ofjointpositionandangularvelocityofthejointsasproposedin[52,172].Thefeedback
signalsiforeachoscillatorneuroni=1...8,asisshowninﬁgure3.9,isaweightedsum
offorathepairanofgularanptagositioonisticnsandoscillatovelornecities.uronsTheiandfolloi+wing1:equationsarethefeedbacksignal
si=±fd∗x±fdv∗x˙
si+1=fd∗xfdv∗x˙(3.5)
wherefdandfdvaretheweightinggainsoftheangularjointpositionandangularjoint
velocity.Theangleisoneofthethreeanglesofthemechanicalsystemα,β,γaccording
tothethenumbeinﬂuencrsofeoosftcillahetoroscsillarelatedtortoneuronseachonjointtheandmusangleclesfoandrtthehejofronintt.Seeal-planeﬁguremodel.3.9Sofor
neuronsn1andn2areinﬂuencedbythediﬀerencebetweenanglesαandβ,neuronsn3
andn4areinﬂuencedbythediﬀerencebetweenanglesβandγ,neuronsn5andn6by
angleαandneuronsn7andn8byangleγ,butonlyiftherelevantleghascontactwith
und.grotheForthesagittal-planemodelneuronsn1andn2areinﬂuencedbyangleΘasshownin
45

3ActuationofPassiveMechanicalModels

.3.8eﬁgurThefeedbackcompletestheloopofactivatedmechanicsaccordingtoﬁgure3.14.Accord-
ingtothetransitionchartofﬁgure2.6insection2.3.2thewholeprocessforonestepping
phaseisasfollows:themechanicaldynamicsreacttothegravitationalforcesplusthe
jointtorquesaddedadditionally.Thesejointtorquesareproducedbytheoscillatornet-
work,whichrepresentstheactivationlevelofthemuscles.Theactivationoftheoscillator
networkisinﬂuencedbytheproprioceptivemusclefeedback,whichrepresentsthemuscle
lengthandcontractionvelocity.Asteppingphaseisterminatedifgroundcontactoccurs.
Inthemodelproposedherethisgroundcontactismodeledasaninstantaneousevent.So
thenextsteppingphaseisthenextswingphaseoftheotherleg.Inthefollowingsection
3.5thesesteppingmovementsarecommentedinmoredetailwithespecialreferenceto
thestabilityofsuchsteppingmovements.

itStabil3.5y

Stabilityisaveryimportantcharacteristicofwalkingandsteppingingeneral.Asmen-
tionedinsection2.4thewalkingstabilitywasandiscrucialforhumansurvival.Besides
otherimportantcharacteristicsofwalkingthisisthemostimportantcharacteristic.In
thefollowing,thetermstabilityisdetailedfurtherinamathematicalwayandappliedto
steppingmovements.Furtheranumericalproofofstabilityisshownwhichisappliedto
walkingmovementsinthiswork.

3.5.1Poincare´Sections

Thesystemdiﬀisaerentialnonlinearequatdyionsnamicformecsystemhanicsinconandtinouosuscillatortime.sareWitheacnonlinear.hstepSogrotundhewcoanlkingtact
opropccursertieswhicofhthimaskhesythebridssyyssttememcadisconnnottinbeuous.deterSominethedstabybiliteigenyvandaluestheorfeaforesimpletheaJattracobianctor
matrixasforcontinuousnonlineardiﬀerentialequations.Walkingisaperiodicmovement
withsubsequentcontinuousperiodswhicharetheswingphases.Thismovementhasto
bepresentedasaperiodicsolution.Aperiodicsolutionissearchedwhichisstable.This
meansthatallmovementtrajectoriesstayintheneighborhoodofonecyclicmovement
trajectory,theperiodicorbit,iftheystartedinthecloseneighborhoodoftheperiodic
orbit.Itisanattractingorbitifallsolutionsconvergetothisorbitfortime→∞.The
palsoeriodiccalledasolutiolimitnisacycle.syTomptoticadeterminellystabletheifstitabilitisystaofblesucahndaanperioatdictractingorbittorbithe.PThisoincarise´
mapcanbeapplied[91,5].
ThePoincare´Mapconsidersanautonomousornon-autonomoussystemoftheform:
x˙=f(x)or˙x=f(x,t)withx(t0)=x0(3.6)

ThePoincare´MapPisdeﬁnedby:

ybilitSta3.5

xk+1=P(xk)(3.7)
wherePisthemappingfromsolutionxkofthesystemontosolutionxk+1wherekand
k+1denotethetime.Ifatime-periodicsystemisdividedintonsectionsofperiodTthe
systemcanbewrittenas:
x(T)=f(x,T)withx(t=t0+n∗T)=xk+n
wherethedurationofonesectionisthetimeTbetweentwopointsintimekandk+1.
Thesesectionsoftheorbitofthemotionfromtimektok+1canbedescribedbythe
Poincare´sections.Foraperiodicsolutionwhichstartsatxkattimekandreturnstothe
samestateinspacexk+1attimek+1thisisaﬁxedpointofthethePoincare´MapPwith
xk=xk+1.Thisimpliesthatthesteppingsystemreturnsafteracertaintimetoasolution
whereitcanbemappedtoanearliersolution.APoincare´sectionisaplanebecauseit
hasonedimensionlessthantheoriginalphasespaceandintersectswiththeorbits.For
steppingthismeansthate.g.atdoublesupportphasewhichisoneinstantintime,the
systemstateofthelastdoublesupportphaseorthebeforelastcanbemappedtothe
actual.Thesystemisdiscretelydividedintocontinuous-timeparts,thesinglesteps.
ForaperiodicorbitthePoincare´sectionsintersectwiththeorbitintheﬁxedpointx∗
foreachperiod.IngeneraltheintersectionpointscanbemappedbythePoincare´map
ontoeachother.ThePoincare´mapturnsacontinuousdynamicalsystemintoadiscrete
one.ThereforethePoincare´mapreducesthesearchforasta∗bleperio∗dicsolutio∗nto
thelinearizationof∗thePoincare´maparoundtheﬁxedpointxwithx=P(x).An
equilibriumpointxisLyapunovstableincasethat:
x(t0)−x∗<δ=⇒x(t)−x∗<∀t≥t0(3.8)
Theequilibriumpointx∗issaidtobelocallyasymptoticallystableifx∗isstableand,fur-
thermoreislocallyattractiveandthereexistsaδ(t0)whichmeansthatallsolutionsstart-
ingnearx∗tendtowardsx∗ast→∞.Mathematicallyspeakingthereexistsδ(t0,)>0
:thatsox(t0)−x∗<δ=⇒tlim→∞x(t)=x∗(3.9)
ThiscanbetransferedtothePoincaremapaccordingto[91]equation3.8and3.9are:
x0−x∗<δ=⇒P◦n(x0)−x∗<∀n≥0(3.10)
andx0−x∗<δ=⇒nlim→∞P◦n(x0)=x∗(3.11)
where◦nmeanstherepeatedapplicationofP,i.e.P◦n(x)=P◦P◦...◦P(x).
timesnThelinearizationofthePoincare´mapinthestablesolutionandﬁxedpointx∗isusedto

3ActuationofPassiveMechanicalModels

provethestabilityoftheﬁxedpointwithequation:
P∂P(x0)−x∗≈∂x0(x∗)(x0−x∗)(3.12)
Iftheeigenvaluesλiofthislinearizedmapareintherangeforalli:|λi|<1,theperiodic
solutionsarestable.Ifthereisatleastonewith|λi|>1,thentheperiodicsolutionis
unstable.

3.5.2StabilityProofAppliedtoPeriodicWalking

Inmovtheemenfollots.Twingherethearestasevbiliteraylwatheoryysofisproappliedvingwtoalkaingstabilistatbilityyprinoofliteraforptureerio[12dic6,w18a,lking37,
48,67].Theyallhandlewalkingasaperiodicmovementwithperiodicrecurringphases
whichcanbeanalyzedascontinuoussystemsovertime.Toanalyzethestabilityproperties
adiscretemappingmethodasaPoincare´mapisapplied.Thepresentedstabilityproof
isnumericandwasusedforallidentiﬁedstablesteppingmovementspresentedinthis
work.Asdescribedcitedby[48]from[69]agaitisstableifstartingfromasteadyclosed
Thisphasetrmeansajectory,thatanythephaﬁnitesedisturplanebofancteheleadsmovtoementanotherattrneactsarbythetrajetractorjectoryofiesinsimilarashapcertaine.
artheeatorobitalonestastablebilitytraofjectoarylimit(orbitw),alkingwhichcycleis,calledthelimitwholehcycleybroidfstheystseysmteism.Tmappoanaedblyzey
determiningthePoincare´mapandanalyzingwhethertheﬁxedpointisstableornot.If
theﬁxedpointx∗isstablethenthecompletestepcycleisalsotakentobestable.The
procedureforaperiodicgaitisasfollows:Themapofthehybridcycleisonestepwhich
starphasetsanwithdethendsainitiagainlcowithnditionthefornewtakiniteoialﬀovfthealuesswingrightlegafterrigththeafterdoublethesuppdoubleortsuppphaseort.
Ftooreqthisuatioalson3.7seeasﬁgx(urestep2.6k.+1)So=tPhe(x(stepsystemk))dividedwherexintoisthestepsstatecanvbeectorwrittenoftheaccorsystem,ding
x=[α,β,γ,α˙,β˙,γ˙]T.Ifthetrajectoryisperiodicitisvalidtosayxk=xk+1.
Soitfollowsaccordingtoequation3.7x∗=P(x∗).Tostatethattheﬁxedpointisreally
stablethefollowingderivationismadeaccordingto[48].ThenonlinearPoincare´mapping
functionPislinearizedbywritingitasTaylorserieswhichis:
P(x∗+δx)≈P(x∗)+∂P∗δx(3.13)
x∂P∂whereapplied∂xtoiseacthehgrstateadienofttheofcPyclicaccordingsolutiontoofthex∗.stTheatesgraanddienδxtisofaPsmacanllbpeercalculaturbattedionbδyx:i
∂P=Λ∗Γ−1(3.14)
x∂

ybilitSta3.5

withδx00...0
01δx0...0
Γ=......2......andΛ=(x1p−x∗)(x2p−x∗)...(xip−x∗)
0......δxi
whereiisthenumberofsystemstatesandxip=P(x∗+δxi)arethesolutionsforeach
disturbedstatepi.Thedistancebetweentheﬁxedpointsolutionx∗ande1)achperturbed
statesolutionxiattheendofaperiodisalsocalledmonodromymatrix.
Nowtheeigenvaluesofmatrixgradient∂∂xParedetermined.Iftheeigenvaluesarewithin
theunitcircletheconﬁgurationisstable.
doThisesnostatbilitpushyalstheomeasystemnsothauttoiffathestepbaissinpofertatturbedractioinnsometheswaystemyandwillthisbepattraerturbactedtionto
thestablesolution.Sointhisstepcycleandthefollowingcyclesthesystemwillreturn
tothesamegaitpatternortoasimilarstablepattern.

3.5.3FindingConﬁgurationsforStablePeriodicMovements
Asthewholesystemhas22stateswhichdependonabout40parametersitisnoeasy
tasktoﬁndastableconﬁguration.Thelastsubsections3.5.1and3.5.2describedhow
thetermstableorstabilityisdeﬁnedbymathematicalmeansandwhatthismeansfor
steppingandwalking.Ingeneralitcanbesaidthatagaitisstableifitisstablewith
respecttothecorrespondingﬁxedpointofthePoincare´Map.Thismeansthatasolution
ofthesystemexists:F(xk)=xk+n,wherekisthetimewherethesystemperiodically
returnstoe.g.thegroundcontactofastepandnisthen-periodicityofthesystem.
Butasteppingsystemcanalsobenon-fallingifitisnotstableasthiscouldbeachaotic
solutionwhichisstillanattractor.Suchasolutionisnotstableinamathematicalsense
buttherearealsonofallsduringstepping.
First,ashortoverviewwillbegivenofthewholeproceduretoﬁndaconﬁgurationfor
thesteppingsystemwhichappearstobestableandwhichthencanbeproventobe
stableornotstable.Theprocedureisasfollows:Atthebeginningadesiredfrequencyor
ω0=lwheregisthegravityandlisthelengthofthependulum.Inthemechanical
theapprogximateresonancefrequencyofthependulummechanicalsystemisdetermined
systemthisresonancefrequencyistakenfortheeigenfrequencyoftheswingleg.
Thefrequencyoftheisolatedoscillatorsystemisdirectlydeterminedbythetimeparam-
etersandthecouplingoftheoscillators.Thissystemcanbeadaptedtotherequired
frequencycharacteristicsbeforeitiscombinedwiththemechanicalsystem.
Next,thecompletesystemisobservedforchangingparameters,andtheparametersare
1)AddarouitindonallythepalesrioothdiecForundbitamewhnicthalismΦ˙atr=ixofDfa(x¯sys)∗temΦcanwherebeΦdeitsethrmineepderioasdlinicesaroluizationtionofoftthheesystesystemm
f(x,t).

3ActuationofPassiveMechanicalModels

generallyadaptedtoensureadesiredorcorrectperformanceofthesystem.Thisob-
servationisperformedexperimentallyinordertoestablishthefullrangeofobviously
possiblesolutions.Theperformanceofthesystemisobservedaccordingtothefollowing
criteria.Istheactivationseriesfortheneuronsintheintendedorder?Dothelevelsof
activationexistforallneurons,inotherwordsaretheactivationlevelsallpositive?And
aretheactivationlevelsofthesamedimension,inotherwordstheactivationlevelofone
oscillatorisnota100timeshigherthanthatoftheneighboringoscillator.Thisleadsto
alloscillatorshavingacomparableinﬂuenceontheactuationofthemechanics.Lastbut
notleastthesignsarecheckedtoensurethattheﬁrstdeterminationofthebodyangles
takestherightdirectionandtherightorder.
Twomethodshavebeenappliedtoﬁndaninitialﬁxedpoint:(1)theNewton-Raphson
methodand(2)thesecantmethod.TheNewton-Raphsonmethodonlyworksiftheinitial
guessiscloseenoughtothelaterﬁxedpoint,otherwisethemethoddoesnotconverge.
Theresultofthesemethodsisaninitialstatevectorx0,theﬁxedpoint,whichgivesa
goodapproximationofastablesolutionforaperiodicgait.ThismeansthatF(x0)≈x0.
Thenthesystemisthenprocessedwithx0astheinitialguessandreﬁnedtoastable
solutionbyiterativeapproximationtothelimitcyclesolutionofthesystem.Themethod
tocheckforstabilitypresentedinsection3.5.2isusedbyotherstudiesofgaitinthe
sagittalplanesuchas[48,49,38]whichcorrespondstothelocalstabilityoflimitcycles.
Insection3.8below,thesimulatedsteppingmovementswiththeproposedactuatedmod-
elsarepresented.Ifamovementisdenotedstablethisalwaysreferstothestability
proofintroducedhere.Amongstotherthingsthesubsequentsimulationresultsregard
theinﬂuenceofparametervariationsorexternalperturbationsonthestabilityofperiodic
ts.emenvmostepping

3.6SimulationofSteppingMovementsandVisualization

tegrThesatioimnsulatioarensdoanerebyalltheMimplemenATLABtedinsolvMAeroTLAB.de45Fwoitrhthethefollosetting:wingvarresults,iableallstepthesizein-,
absolutetoleranceAbsTol=1e−5andtherelativetoleranceRelTol=1e−7,the
otherintegrationvaluesaresettodefaultMATLABvalues.
TheHeretplothespresindividuaenteldplototvtisypuaeslizearethebrieﬂyresultscommenaretofedthetogsaivemeatbypetteerforovallerview.sectionsImpborteloanwt.
parametersofthesystemarethepositionsofthehinges,whicharerepresentedbyΦand
θforthesagittalmodelandwiththethreeanglesα,βandγforthefrontalmodel.
Thesepositionsareequaltotheanglesshowninﬁgures2.3and2.4.Thephaseplotisa
movisualizadelandtionagaoftinshetseα˙,pβ˙ositanionsdaγ˙gainforstthetherfroenlatatedlamonguladel.rTvehislocitploiestΦ˙is,θ˙aforcommonthesafogittarml
ofvisualizingthestabilityoflimitcycles.Ifasystemisstable,theshowncycletakes
thecycleforcanmbofeoseneenaslineconandvergnoetncmewitultiplehthislines.limitThecycleattratractiojectonryof.Inasolutioadditionntothethevelolimitcity

3.7SimulatedSteppingMovementsintheSagittalPlane

discontinuitiescanbeseenasdiscretevelocitychangeswhengroundcontactoccurs.
Afurtherplotistheplotagainsttimeoftheactivationlevelsoftheoscillatornetwork.As
theswingleghasnogroundcontact,theswinglegankleisnotactiveandthisactivation
levelisomittedintheplot.Forthesagittalmodelthereisonlythetwoneuronsatthe
ankleofthestancelegwhichareshown.Andforthefrontalmodelonlythesixneurons,
twoforthestanceankle,twoforthestancehiphingeandtwofortheswingleghip
hingeareshown.Finally,themovementplotdrawsastickﬁgurefordiﬀerenttimesteps
inthesameplottovisualizethemovement.Herethediscretizationofthestickﬁgure
movementisperformedwithamuchlowersamplingratethantheoriginalintegration
timestepsusedbytheMATLABodesolvers.Thisisbecausesinglelineshavetobe
seentoimaginethemovements,sothisplothasnodeﬁnitetimebaselinebutshowsa
motionsequence.Thewholesystemconsistsofthefourcomponents:mechanics,oscillator
network,torquegenerationandfeedbackfromthemechanicstotheoscillators.Inthe
followingsomemovementresultsforthissystemwillbeshown.Themovementofthe
mechanicalsystem,anglesandorangularvelocities,theoscillationpatternoftheneuron
networkactivationandthereactionofthesystemtoparameterchangessuchasfrequency
anddisturbancesappliedtothemechanicalsystemlikefootsliding.Inthefollowingall
simulationresultspresentedarecalculatedforthesagittalandfrontalmechanics,each
withaconstantsettingformechanicalparametersandmostoftheoscillatorparameters.
Anyvariationintheparametersisalwaysindicatedseparatelyforeachresult.

3.7SimulatedSteppingMovementsintheSagittalPlane

Inliteraturethereareseveralexamplesofactuatedwalkingmodelsinthesagittalplane
asmentionedatthebeginningofchapter3.Modelswithanactuatedhipjointaree.g.
asproposedin[127,158,157].Bipedwalkingmodelswithactuatedanklesareproposed
bye.g.[12,93].Inroboticsnormallyalljointsareactuatede.g.aspresentedin[174,11].
Thestabilityofunactuatedandactuatedsagittalplanemodelswasprovenformany
models.Stabilityanalysisexamplesofpassivemodelscanbefoundamongstothersin
[111,112,113,38,18,49,48]andforactuatedpassivemodelsin[67,126,174].Thereare
thereforevariouspossibleactuationswithstablesolutionsandthisopensupawiderange
ofactuationpossibilitiesforthesimpleballisticmodelofawalkingpendulumpresented
here.Oneofthosepossibilitiesisshownwithsomeparametervariations.Thestability
analysisisalwayscarriedoutwiththemethodandlimitcyclesolutionspresentedin
.3.5iontsec

3.7.1WalkingMovements

moConvtinemenuoustonanklevleelagrctuatiooundnofwithoutthethepassivgrewadienalkterofinaslothepe.sagitThetalpattplaenernleaofdsthetoamowvaemenlkingt
istransitioninﬂuencedthabtyotheccursactivinatiothendoupatbleternsuppoftheortphaankleseareoscillatofar.ctorsThethatgalsoroundconinﬂuencetacttandhe

3ActuationofPassiveMechanicalModels

systemsigniﬁcantly.Thedirectionofgroundreactionforcesinﬂuencestheinitialvelocity
valuesforthenextstepandthereforethegaitpattern.Theinitialvaluesdetermine
whetheramovementsolutionisattractedtoastablesolutionandwheretheﬁxedpoint
ofthissolutionis.Stabilityisalsoaﬀectedbytheparametersoffeedbackvaluesfdand
fdvaswellastheexternalinputssiorextitotheoscillator;theseparametershavea
principalinﬂuenceonthewalkingpattern.Inthefollowingageneralsetofparameters
isusedforeachsimulationequivalently.Theparametersandtheirvaluescanbeseenin
table3.3andtable3.4.
unitaluevameterparM70[kg]
m0[kg]
]m[1lg9.8[kg∗m/sec2]
Table3.3:Parametersofthemechanics.
Otherparametersarevariedandthosevariationsinﬂuencethewalkingpattern.The
parametersoftable3.3areasusedinequation2.2.2forthemechanics.Theequationis
malized.norTheconstantparametersfortheoscillatornetworkarethesameforallneurons.The
namesoftheparameterscorrelatewithequation3.1oftheMatsuokaoscillator:
aluevameterpara12,a211.5
2.5b8s][sec1TTab2[sec]
1.5fd-1.5fdvTable3.4:Parametersoftheneuronaloscillator.
Awalkingmovementgeneratedwiththeabovevaluesisasimpleforwardwalkingmove-
mentonlevelgroundwhichisshowninﬁgure3.15.Thesubplots(a),(b),(c)and(d)show
theangularpositionsandangularvelocitiesofthetwoanglesθ,φ,theneuronactivation,
thephaseplotandthemovementofthemechanicsofﬁgure2.3.Theresultwhichcan
beseenisthesuperimposingoftheneuronoscillationswiththemechanicaloscillations
andthereforeanewoscillation.Onlythegroundcontactproducesdiscontinuitiesinthe
velocities.Theanglesaresymmetric.Thetrajectoryisattractedtothecyclicsolution
afterafewsteps.Thisisseeninﬁgure3.15(c)wherethesingletrajectoriesconvergetoa
stablelimitcycleaftertheinitialtransienttimewherethelinesareseparateanddistinct.
Thediscreteeventofgroundcontactisvisualizedbyabreakinthelines,whichwouldbe
averticalconnectionlineifrepresentedbyasolidline.Intheplottedmovementinﬁgure
3.15(d)the1-periodicitycanbeseen.Everystepresemblesthepreviousone.

(a)Angularposition

lotphaseP(c)

3.7SimulatedSteppingMovementsintheSagittalPlane

(b)Oscillatoractivation

(d)Movement

Figure3.15:1-periodicstablesolutionforwalkingmovementinthesagittalplaneonlevel
ound.gr

3ActuationofPassiveMechanicalModels

(a)Angularposition

lotpPhase(c)

(b)Oscillatoractivation

(d)Movement

Figure3.16:2-periodicwalkingmovementwithamodulatedgroundcontactdirection
r.ectov

patAnotternsherpareossiblerepsoeatedlutioneveryisa2-secondperiodicstep.Tsolutiohenadisconsshotinwnuousinﬁgurestatetr3.16.ansitionHerecathenpbeerioseedicn
inthevelocitiesandtheanglesarenolongersymmetric.Thissolutiondiﬀersfromthe
forﬁrstthesolutioinitialnvwithelocitrespyofectthetothestanceglegroundisconexatctlyactcinovernditiotedn.forTevheerysecdirectionondofstep.thevThisectoris
likdueetaotheclubfootactuamotiovnemenitist.forWitcedhineactohthestepsathemestancedirectiolen,gstwhicartshisinliktheeanotherinitialtdirectionensionbutof
theankleforeachstep.

(a)Angularposition

lotphaseP(c)

3.7SimulatedSteppingMovementsintheSagittalPlane

(b)Oscillatoractivation

(d)Movement

Figure3.17:VStaariabletion1-poferiodicneuronwpalkingaramemotveremens=t1with0resultsparameterinafdvhigher=3.s5t,eTpbf=re1q.5.uency.

3.7.2VariationofParameters

Anotherpossiblestable1-periodicwalkingsolutionistakenandtheparametervariation
examined.Thevariationoftheexternalinputsoftheoscillatorsystemrepresentsa
variationofexternalinﬂuencessuchassensoryinputorotherhigh-levelcommands.This
variationinﬂuencesthestepfrequency.Inﬁgures3.17and3.18twoexampleswiths-
valuesof10and2respectivelyareshown.Figure3.19visualizesthevariationinstep
frequencyresultingfromvariationsinparameters.

3ActuationofPassiveMechanicalModels

(a)Angularposition

lotpPhase(c)

(b)Oscillatoractivation

(d)Movement

Figure3.18:V1-pariaeriotiondicofwalkingneuronmopavemenrametertswith=2rparaesultsmeterinfadvlo=wer3.5s,tTepb=1.frequency5..Stable

Figure3.19:Dependencyofstepfrequencyonvariationsinparametersfrom1to10.

3.8SimulatedSteppingMovementsintheFrontalPlane

Inandthepfolloossibilitieswingaofselethectiopropnofosedsimaulatctuatedionresmoultsdelisforpresenstepptedingtomodemovemennsttsrateinthetheafronbilitiestal
plane.Thecompletemodelusedforsimulation,consistsofthemechanicspresentedin
secusingtion2.3propriocandeptiviseamctuateduscularbyfetheedbaocsckaillastorexplainednetworksinPseictiopron3.4ducing.2andjoint3.4t.1orq.uesThisandis
visualizedinﬁgure3.14.Withtheinstantaneousgroundcontactofsection2.3.2this
bec(swingomesphaase)hybridwhichissystemshownwithintheﬁgurestat2.6es.doTheublegrsuppoundorcotntactphasewillandbemosingledeledsuppwithortphaslighset
variationsasthegroundcontactfore.g.steppinginplaceorsteppinguphastobevaried
accordingtothegroundlevel,groundreactionforcesanddirectionofinitialvelocities
pataccorternsdingaretothepresenmovtedementotshowdirection.thevariaSobbiliteloywininmosevctioemennt3.8typ.1esthreeofthediﬀemorendetl.movNext,emenint
section3.8.2theinﬂuenceofthevariationofsingleparametersisshownwhichimplythe
pareossibilitappliedyofexternalextendingperturthisbatiomonsdeltowiththeexternalsystem.conThetroleﬀaectndofinput.suchIpnserturectiobation3.8ns.5onttherehe
movementanditsstabilityaretested.Insection3.8.6themovementpattern’stepping
inrealplascube’isjects.compaThisredshotowsareathatlitmoisvpemeossiblentpatttosimernulaterecordedmovinemeanntexppaterternsimentalwhichsetuparevwithery
similartorealsteppingmovementswiththeproposedmodel.Finally,insection3.10the
possibilitiesandrestrictionsoftheproposedmodelarediscussedforfurtherextensionsof
del.motheTheproposedmodelhasseveralparameterswhichareinthefollowingonceexplainedin
thewhicrhelaaretedusedequatotions.simulaHeretethethefovlloalueswingforresults.theparaTheremetarerizatiothenofmecthehamonicaldelsparaaremetgiveresn,
bodymassM,legmassesm,leglengthl,hipwidthhandgravitationalforceg.The
valuesoftheseparametersareforallsimulationsidentical.Fortheusedvaluesseetable
3.5Theconstantparametersfortheoscillatornetworkarethesameforallneurons.The

unitaluevameterparmM1419[[kkgg]]
l0.5[m]
h0.1[m]
g9.8[kg∗m/sec2]
Table3.5:Parametersofthemechanics.

nameoftheparameterscorrelateswithequation3.1oftheMatsuokaoscillator.The
parametervaluesareshownintable3.6:

3ActuationofPassiveMechanicalModels

aparameterv2.5alue
3.5b2c2d4sTable3.6:Parametersoftheneuronaloscillators.

3.8.1SimulationofDiﬀerentMovementPatterns

Theplane.propTheosedmovmoemendelistsadevreelopofaedtogeneralgenerasortet,rhastheythmicparsteppingametermosettingvemenoftstheintmohedelfronwtalas
notespeciallyadaptedtoexperimentalhumandata,becausethismodelshalldemonstrate
generalpotentialsofthisrelativesimplemusculoskeletalmodel.Sotheﬁrstresultarethe
tyfollopeswingoffoursteppinggeneralmovtypemenestofpatternspatternswarehichpresencanbted:eachievedwiththismodel.Inthe
•Steppinginplacewithdroppingofhip
•Steppinginplacewithliftingofhip
idesaStepping••Steppingupase.g.onaladder
areTheseageneralrefourmovsetemenppingtsinmovtheemenfrontstalwhicplahne.canbTheedonesteppingbyevineryhplaceumancanbebeingdoneandinwthicwho
diﬀerentways,themorenaturalwayisbydroppingthehipbutitisalsopossibletolift
thehipatthebeginningofstepandthenletitdropagaintogetgroundcontact.Inﬁgure
3.20theposition,activationandphaseplotcanbeseenforcase(1)steppinginplacewith
droppinghip.Herethehipdropssothelegdipsintothegroundandcomesupagaintill
itseeisninlevelthewithphasetheplogrottheundsurfsteppingacewhicmovhisementdetecisated1-paesrtiohedicglimrounditcocyclentact.movAsemencant.bSoe
alltrajectoriesareidenticalandthegaitissymmetricandstable.
Anotherexamplecanbeseeninﬁgure3.21wherethephaseplotisnotalimitcyclebut
nothemovsymmetricementonelookseither.stableThisfors8teppinsteps.gInmovthisemencasetscanonbpeeraion-dicpserioodiclutiongisait.Orfounditisandan
systeminstablewouldconﬁgcollauratpsioneorthaittisisreallycloseatocahaostaticbleattsoractivlutionebutsolutioafn.terIfsevtheeralsystemmoreisstepsunstatblehe
butcanthoughdo8steps,whichisaccordingtothewholebodymechanicsstillastable
physicalsteppingsolution,thiskindofinstabilitycanbehandledbyahigh-levelcontrol
thatwillbeproposedinchapter5.
Athirdshownpossiblesteppinginplacemovementpatternisseeninﬁgure3.22.Thisis
aslowly’driftingdrifts’moinvoneementdirectiowhichn.isInsurelythepunstaositionble.plotThisaslowdriftingincreasecanboresedecenraseasetheofthephaseangplotles

(a)Angularpositions

lotpPhase(c)

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Oscillatornetworkactivations

(d)Movement

Figure3.20:Simulationofstablemovementsteppinginplacewithdroppinghip:simu-
latedwiththeparametersTa,Tb=0.05,fd=0.5,fdv=1,initialstatevector
init1andankleandhipstrategyF12.

3ActuationofPassiveMechanicalModels

(a)Angularpositions

lotphaseP(c)

(b)Oscillatornetworkactivations

(d)Movement

Figure3.21:Solutionofanothersteppinginplacemovementwithdroppinghip.The
simsolutioulatnediswithnotprtheoveparntoametersbeTstablea,Tbbut=0.05there,fdis=no0.5,obfdvvious=1f.a2,llriskinitialeithestatr:e
vectorinit1andankleandhipstrategyF12.

(a)Angularpositions

lotpPhase(c)

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Oscillatornetworkactivations

(d)Movement

Figure3.22:Steppinginplacewithdroppinghipwithadriftmovementinonedirectionas
thelimping:stepsmoimvemenulatedtwisiththeunsymmetricalparametersbetwTea,enTbr=igh0t.05and,fdleft=leg0..4,fThisdv=is1.lik1,e
initialstatevectorinit1andankleandhipstrategyF10.

denotesthisdrift.Withthisconstellation,severalstepscanbegenerated,whicharestill
astablewholebodyconstellationbutafterseveralstepsthesystembecomesinstable.
Theresultingmovementlookslikeanasymmetriclimping.Asmentionedabovethisdrift
andinstabilitmoveymencant.beInprevtheelonw-tedlevwithelmaushigculoh-slevkeleletalcontmoroldelthatthesepnsesositiontheowfhothelebowholedypbodyositionis
included.notTheoscillatthreeornetexaworkmplesbutabowithveofdiﬀerensteppingtfeedinbacplacekoremoxveternmentsoscillatoarergeneinputratedext.withThethestrasategyme
forhiporankleactivationwasforallthreecasesauniformhipandanklestrategy.
Thenextexampleofmovementissteppinginplacewithliftinghip.Steppinginplace
withliftingthehipstartsnotwiththeusualdropofthehipbutincontraryaliftingup
ofthethemechiphanics.againsThetgrstavitartingation.hipvThiseloccitieshanghaevinetomobveementhetconnetraredsyanotherdirectioninitialtoliftvatheluehipfor

3ActuationofPassiveMechanicalModels

(a)Angularpositions

(c)lotphaseP

(b)Oscillatornetworkactivations

(d)Movement

Figure3.23:Stablesymmetricsteppinginplacemovementwithliftinghip:simulated
withtheparametersTa,Tb=0.1,fd=1.2,fdv=1.69,initialstatevector
init1andhipextensorstrategyF14.

isandnotnotsotonatdropural,itwhicthereforehhatheppensactivaatccoionrdingleveltoofgrathevitahiptionaljointsforhacevs.etoAsbethismucmohvemenhighert
thaninthesteppinginplacewithdroppinghip.Thishigheractivationlevel,whichcan
behigherclearlyhipselevenelinenaﬁgurblesethe3.23,systemisabtoout2react.5agtimesainstthemagniﬁedgravitarelatedtionaltoforcﬁgureseand3.20.liftTthehe
hip.Therefore,theenergylevelatthebeginningofastephastobehigherthanitisfor
droppinghip.Theamplitudeofthehipmovementishigherandtheamplitudeoftheleg
movementisverylow.Ascanbeseeninphaseplot3.23,(c)thepresentedsolutionis
alsoastablesolutionwhereallperiodictrajectoriesareattractedtothelimitcycle.The
ofgaitthiisssyhipmmelifttupric,stsoe,ppingtheinlimitplacecycleisfothartαtheandhipγvareloecitcongryisuent.highlyAnodisconthertinchauousinracteristicthe
groundcontact.Thismeansifthelegtouchesgroundwithavelocityv,thelegleaves
groundwithavelocityintheotherdirection−v.Thiscanbebestseeninthephaseplot
3.23(c)wherethegreenphasespacelineishighlydiscontinuousforthehip.

3.8SimulatedSteppingMovementsintheFrontalPlane

Thethirdsteppingmodeissteppingtotheside.Thisintentiontosteptothesidehasto
beclearatthebeginningsotheinitialvaluescanbeadaptedtosteppingtothesideand
thegroundcontactmodelisadapted.Themodelforthegroundcontactwaspresentedin
section2.3.2,heretheinitialvelocityvaluesforthenextsteparecalculatedbyequation
2.20andthoughbyequation2.21.Inthecaseofsteppingasidetheforceappliedtothe
mechanicswhengroundcontactoccurshasanotherdirectionthaninthecaseofstepping
incasewithoutmovingthebodyinthefrontalplane.Thismovementtothesideislike
asteeringcommandofanuppercontrolcenterwhichsays’gototheleftorright’.It
isanintentionalsignal.Thisleadstoagroundreactionforcewhichisnotinamoreor
lessverticaldirectionbutthetangentialcomponentofthisforceishigher.Sotheinitial
movementofthenewstanceleghasadirectionalsohorizontalsothatthewholebody
movestotheside.Thisisrealizedbyajointtorqueofthenewandoldstanceankleα
andγ,whichhaveaninitialvaluethathasthesamedirectionandkeepsthishorizontal
movementvectorforeachstep.Sothegeneralmovementdirectionisdeterminedbythe
initialvalueandthehorizontalvectordirectionofthegroundcontactinitialmovement.
Thisisdeterminedintheequationfortheenergypreservationdetailedin2.3.3.The
equationforthisisaccordingtoequation2.22:
E(stepi(1))−Epot(stepi+1(1))=Ekin(stepi+1(−α˙(1),β˙(1),−γ˙(1))
wherethevelocitiesofαandγhavethesamedirectionbutoppositetothedirectionof
thelaststep.Thisconﬁgurationofthegroundcontactleadstoasteppingtotheside
movementsshowninﬁgure3.24.Thismovementisalsosimulatedwithdroppinghipsteps
alikeﬁgure3.20.
Anotherpossiblewalkingpatterntothesideiswithsmallshortstepsthataremore
controlledbytheanklesthanthehip.Astheanklesarenotsuchastrongactuatoras
thehipthestepsaresmaller.Thissteppingmovementtothesidewithsmallerstepsand
anklestrategyF11isseeninﬁgure3.25.
Theforthandlaststeppingtypeisthesteppingup.Thisisasteppinginplacebuthere
thehipisliftedase.g.forsteppingupaladder.Furthermore,thegroundcontactis
diﬀerentcomparedtosteppinginplace.Ifthewholebodystepsupwards,theground
contacthastobeadaptedtoastructure,e.g.likealadderwhichcanbesteppedup.
Forthismovementthegroundcontactwasmodiﬁedasfollows:Foreachsteptheground
contactconditionismovedbyadeltaupwards.So,fortheﬁrststepthegroundlevelis
zeroforthenextitisdeltahigherandsoon.Theconditionaccordingtoequation2.15is
adaptedbytheadditionaltermdeltaΔ,whichmeansahighergroundlevelthenbefore.
Theequationforthisis:

l∗cos(α)−h∗sin(β)−l∗cos(γ)=Δ(3.15)
Thefourtypesofsteppingmovementscanbevariedmainlybytheinitialvaluesandthe
groundcontactwhichdeﬁnesthenewinitialconditionforthenextstepandthestrategyof
actuationasankleorhipstrategy.Foralimitcyclestabilityofthosesteppingmovements,
thefeedbackparametersareessential.Sointhenextsection3.8.2someparametersare

3ActuationofPassiveMechanicalModels

(a)Angularpositions

lotphaseP(c)

(b)Oscillatornetworkactivations

(d)Movement

Figure3.24:Steppingtothesidewithdroppinghip,simulatedwiththeparametersTa=
0.1,Tb=0.3,fd=0.6,fdv=3.5,initialstatevectorinit8andhipstrategy
.10F

(a)Angularpositions

lotpPhase(c)

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Oscillatornetworkactivations

(d)Movement

Figure3.25:Anothersteppingtothesidemovementwithdroppinghiphasmorehip
0.3actua,Tbtio=n0.w1,hicfdh=res0.ult6,sfindv=sma4.5,llerinitialsteps:statesimvulateectodrwitinhit8theandparaanklemetersstraTategy=
.11F

3ActuationofPassiveMechanicalModels

(a)Angularpositions

phaseP(c)lot

(b)Oscillatornetworkactivations

(d)Movement

Figure3.26:Steppingupwardswithrisinghipwhichislikegoingupaladderandisrelated
tothegroundlevelcondition:simulatedwiththeparametersTa=0.3,Tb=
0.1,fd=0.85,fdv=3.8,initialstatevectorinit8andhipstrategyF10.

3.8SimulatedSteppingMovementsintheFrontalPlane

Figure3.27:Inﬂuenceofoscillatorinputsonstepfrequencywithdurationofonethird
ofthestepcycle(lowercurve)ordurationzero(uppercurve)ofthedouble
e.sphaortsupp

variedsystematicallytoanalyzetheinﬂuenceofthoseparametersonthesystem.

3.8.2InﬂuenceofParameterChangesonMovementPatterns

ExternalOscillatorInput

Averyinterestingparameteroftheoscillatornetworkistheinputswhichcanbea
controlinputfromhigherlevelsasthebrainandsensors.Inthefollowingthevariationof
parametersanditsinﬂuenceonthesteppingmovementisshown.Inﬁgure3.27arising
ofsforthesamemodelparametrizationresultsinavaryingofstepfrequency.Thereis
adirectlinearinterconnectionbetweentheoscillatorinputsandthestepfrequency.
Anadditionalfactorwhichinﬂuencesthestepfrequencyisthegroundcontact.The
groundcontactpresentedinthisworkisaninstantaneouscontactsothelengthoftimeof
thecontactiszero.Ifthiscontactdurationwouldvarybetweenzeroandaboutonethird
oftimeofthestepcycle,whichisanormalvalueforslowwalking,thestepfrequency
isinﬂuenced.Itisalinearindirectproportionwhichdescribesanincreasingduration
ofgroundcontactwhichleadstoadecreaseinstepfrequency.Thisisshowninﬁgure

3ActuationofPassiveMechanicalModels

3.27withthesecondline,whichistheresultofvaryingtheparameterswithalonger
groundcontactdurationofaboutonethirdoftheswingtimeinsteadofzero.Thisshall
onlydemonstratethattheabsolutevalueofstepfrequencydependslargelyontheground
contacttimebutthefrequencychangebyparametervariationisvalidindependentofthe
groundcontacttime.

ActuationStrategies

ofThethejointoscillatotorquergactiveneratioationnlevelstransferarevfuncaried.tionThethatvaproriatioducesnisthemotivtoraquestedbyforexeapcehrjoimeinnttallyout
foundhipandanklestrategiesinhumans[66,59,60].Theanklestrategymeansthat
theanklemusclesareactivatedprimarilyandthehipstrategythatthehipmusclesare
usedprimarily.Sodiﬀerentmusclegroupsareprevalentlyusedforthesamemovement
depsimilarendenrestulton.theWhasttratstraegytegyaispersusedonbyusesa.personThismedepansendsthatondiﬀediﬀerenrenttfastractors,tegiese.g.trleadainingtoa
toandproavge.eInthethetwoexpstraerimentegiestsoandf[66their,59,mix60]ture.themTrauscleiningactiveﬀeatioctsnhawvasealsomeasuredbeenfboyundEMGto
studiedinﬂuenceforthestpanceosturandepconotstrolurestrconategtroly[a60nd].notIn[fo66r,59],stepping,thehipbutaitndcaannklebestraassumetegiesdarthate
similarneuronalpatternchangesoccurinsteppingandthattherearediﬀerentstrategies
howtoperformaspecialmovement.Forthesagittalwalkingmovementtherearesome
foranahiplysisoandfwanklealkingastctuatrategioniesinduring[194,g13ait.3],Aswhicinhshostancewtandhatsagittheretalaregaitdiﬀerenthosetstrstrategategiesies
arefoundforthefrontal-planemovementsimilarstrategiescanbeexpected.
Forthisreason,inthefollowing8diﬀerentactuationstrategiesareevaluatedtostudy
theTheeqinﬂuencuatioenoftocactuahangetionthewaeighpplietcdhatngesorquesandforsttherategiehingessonwtheasinlotrow-levduceledwithsteppingequamodetionl.
3.4.Theproducedtorquesforthesinglehingesareshownintable3.4.1.Theweighting
valueswofthisequationandtablearevariedaccordingtodiﬀerentstrategiesofhipand
leankn.uatioactF10F11F12F13F14F15F16F17
w6030306052013030
w221260303060305016030
w3320303060520530
ww432020903030306060510420051407300
51wstr61ategy20h90aa30&ha60&h20a&h&ext&ns5a0&h&ext150a&h&ns60a&ext
Table3.7:Theweightingfactorsforthejointtorquegenerationaccordingtoequation3.4
fordiﬀerentstrategies.

Theabbreviationsarea:anklestrategy,h:hipstrategy,ext:theextensorsgainishigher

3.8SimulatedSteppingMovementsintheFrontalPlane

thantheﬂexorgainandns:verylowswinglegactivation.
The8actuationweightingpatternsarehipstrategyF10,anklestrategyF11andF17
andtherestiscombinedhipandanklestrategy.Thecombinedstrategyisdividedinto
equalactivationwithhigherF13andloweractivationlevelF12.Forthecombinedstrat-
egythereexistsalsoalowactivationbutnearlynoswinglegactivationF14andhigh
activationwithnoswinglegactivationF16wheretheextensorsareenforced.Finally,
F15isacombinationwithlessswinglegactivationandhigherankleactivationandmore
hipextensoractivation.Thehipjointk=2hasalwaystwoequalvaluesforextensor
andﬂexororahigherextensorvalue.Thishighervaluewislikeaninitialtensionofthe
jointaccordingtoexpectedloadslikethegravitationalforces.Thenamedstrategiesare
listedinthetable3.4.1withtheweightingfactorswaccordingtoequation3.4.Weights
wwhicharenotshownareallzero.
Anexampleofatorquetransferfunctionvariationforasteppinginplacemovementis
showninﬁgures3.28and3.29.
Inﬁgure3.28(g-i)thesteppinginplacemovementwithahipandanklestrategy,which
wasalreadypresentedinﬁgure3.20,isshown.Thesameconﬁgurationofallparameters
wastakenforallshownplotsin3.28and3.29.Theonlyvariationisthetorquegeneration
functionwhicharecombinationsofhipandanklestrategiesrepresentedbyF10...F17.
Naturally,theshownconﬁgurationsarenotallstablebecauseforthisalsootherparame-
tershavetobeadaptede.g.thefeedbackgainsortheoscillatortimeandgainparameters.
ThesameactuationstrategybutwithhighergainsastherearethepairslowgainF12
andhighergainF13orF14andF16.Thishigherorlowergainneedsanadaptionof
parametersasthefeedback.Alltorquegenerationfunctionsarecomparedtothereference
functionF12.Thelinesofplot3.28and3.29,witheachthreeplots,showthefollowing:
F10:Thehipjointismainlyactuatedwhichleadstoanasymmetricmovementthathas
ahigherhipamplitudebutthenearlyunactuatedankleandswinglegjointshow
smalleramplitudes.Theinitialtiltedanklepositionandtheanklepositionbythe
ﬁrsthiptransientisnotcorrectedbutiskeptduringthewholemovement.
F11:Theanklejointismainlyactuated.Thisdeterminesasymmetricmovementbe-
causetheinitialanklepositioncanbeadaptedtothehipmovement.Theamplitude
oftheanklemovementissmallerastheanklesaremoreactuated.Thediﬀerence
tothecombinedhip,ankleandswinglegactuationofF12isnotverystrong.
F13:ThesameactuationasinF12withahighergainisusedsothemovementis
notstablewiththesameparametersetting.Butitcanbeseenthatthegeneral
amplitudeofthehipandanklemovementismuchsmaller.
F14:Thisisacombinedhipandanklestrategywithmoreactuationoftheextensorsand
nearlynoswinglegactuation.Thismovementislikewisenotstablebutthestance
andswinglegamplitudeismorecenteredwhichmeansitismorelikeanatural
pendulumswingingaroundthezeropoint.

3ActuationofPassiveMechanicalModels

F15:Actuateshipandanklewithemphasisontheextensorbutonahigherleveland
theswinglegisalsoactuatedabit.Thisleadstoanasymmetricsteppingmovement
withalargerhipmovementonthesidetowhichthebodyistilted.AsinF10the
initialanklepositioncannotbestraightenedbecausetheankleextensoractuation
istooweakcomparedtotheﬂexoractuation.
F16:Heretheankleandhipareactuatedbutwithamuchlargergainandtheswingleg
isnearlynotactuated.Thisleadstoasymmetricsteppingmovement.Thehigher
gainleadstosmalleramplitudesandthereforealsoahigherstepfrequencysothe
swingleghasnotmuchtimetoswingfree.
F17:Moreankleactivationthanhipbutswinglegisactuatedthesameasthehip.This
leadstoaresultinbetweenF11andF12.

3.8.3DiﬀerentFeedbackGains

Asmentionedabove,thefeedbackgainfactorsforpositionandvelocityfeedbackofequa-
tion3.5canbevaried.Thisvariationmainlyinﬂuencesthestabilityoftheresulting
movement.Soifthefeedbackgainisonlyslightlychanged,theattractivebasinofthepe-
riodicsteppingsolutionisnotleft,sothetrajectoriesareattractedtothelimitcycleafter
severalperiods.Thiscanbeseeninasolutionwhichneedssomestepsforthetransient
eﬀectafterthetrajectoryisattractedtothelimitcycle.
Thischangeofthefeedbackgaincanbeinterpretedasifthereoccursachangetothe
system,likealoadaddedtothesystemorthegeometryofthesystemischangedby
somethingorsimplythestrategyofreactionischangedbytrainingorbetterbeneﬁts.
Thesechangesresultinanadaptionofthesystemtothenewconstellation.Another
importantpointisthatformostofthestablemovementsalittlevariationofthefeedback
doesneitherresultinagreatchangeofthemovementnorinaninstantinstability.This
meansthatthesystemactsrobusttosmallchangesinthefeedback.Thiscanbeseenin
ﬁgure3.31wherethephaseplotsareshownandinﬁgure3.30wheretheangularpattern
ofthemovementsisvisualized.Itwasvariedthefeedbackgainfdvandfd.Thefeedback
0ga.1ininftdvhewashovarizonriedtalbyseriesstepsofof0.plots.1inThethevinertervticalalssoferiesvariaofplotiontsaare:ndffodr=fd[0b.y3,a0.4,0stepsiz.5,0e.6]of
andfdv=[0.6...1.2].Thiswasalwaysdoneforthesamesystemconstellationwiththe
parameters:F12,P1,s=4Ta,Tb=0.05.
Inthephaseplotsitcanbeseenthatthereareseveralconnectedstablesolutionsforthe
variationofthefeedbackgains.Themovementpatternchangesslightlyinappearancebut
notprofoundlyincharacteristics.So,avariedfeedbackinthesamebasinofattraction
doesnotinﬂuencethemovementfundamentaluntilitleavesthebasinofattractionofa
limitcycleandisthereforeinstable.

(a)AngularpositionsF10

(d)AngularpositionsF11

(g)AngularpositionsF12

(j)AngularpositionsF13

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)OscillatoractivityF10

(e)OscillatoractivityF11

(h)OscillatoractivityF12

(k)OscillatoractivityF13

(c)PhaseplotF10

(f)PhaseplotF11

(i)PhaseplotF12

(l)PhaseplotF13

Figure3.28:straInﬂuenctegieseFof10.join..tFto13.rqueHerveatheriatsiontrategieaccordingsaretoaccordidiﬀengrentottorqtableue3.8tra.2:nsfehip,r
ankle,ankleandhipwithlowactuation,andankleandhipwithhigherac-
els.levtiontua71

3ActuationofPassiveMechanicalModels

(a)AngularpositionsF14

(d)AngularpositionsF15

(g)AngularpositionsF16

(j)AngularpositionsF17

(b)OscillatoractivityF14

(e)OscillatoractivityF15

(h)OscillatoractivityF16

(k)OscillatoractivityF17

(c)PhaseplotF14

(f)PhaseplotF15

(i)PhaseplotF16

(l)PhaseplotF17

Figure3.29:Inﬂuenceofjointtorquevariationaccordingtodiﬀerenttorquetransfer
strategiesF14...F17.Herethestrategiesareaccordingtotable3.8.2:mix-
tureofankleandhipstrategywithdiﬀerentswinglegactuationandhip
72extensoractivationandthelastexampleisanklestrategywithhigherexten-
sorthanﬂexorlevels.

eFigur

0:3.3

3.8

lateduSim

Stepping

Movements

the

talronF

Plane

feePlotsdbacofktofhepositionsteppingfdmofromveme0.3nttoa0.6nglesandforvevloaricitatioyfndvoffromthe0pr.6tooprio1.2.ceptive

nActuatio

eFigur

1:3.3

ivssaPe

hanicalMec

delsMo

Phaseplotforthesamemovementsforvariationofproprioceptivefeedback
ofpositionfdfrom0.3to0.6andvelocityfdvfrom0.6to1.2.Stabilityis
achievedforlimitcyclesolutions.

3.8SimulatedSteppingMovementsintheFrontalPlane

3.8.4DiﬀerentOscillatorPatterns

Insection3.3,fourdiﬀerentoscillatornetworktypesareproposed.Diﬀerentnetworkshave
diﬀerentpropertiesbutthosediﬀerencesdonotautomaticallyleadtodiﬀerentmovement
patternsbecauseamovementisacomplexcombinationofalltheparametrizedinﬂuences.
InthefollowingaselectionofmovementsproducedbydiﬀerentoscillatornetworkP1...P4
areexemplaryexplainedandanalyzedtoshowcharacteristics.

Intheﬁgures3.32,3.33,3.34and3.35foursteppingsolutionsareshownforthefour
diﬀerentoscillationpatterns.

Itphasecanbbeetwseeneenthahiptacandrosswlegiseangneuroularnmoinvteraement.ctionThispaistternbasecauseP2tleadshetoneuroanmoreactivasymmetrtionoicf
theresulttwooftlegshisisisthatsymmetricthehipsyncahromplitudenousisandlargtheer.hipneuronactivationisshifted.Another

init0init1init2init3init4init5init6init7init8
α0.0257000000.025700
β0.0002000000.000200
γ0.0238000000.023800
α˙0.03540.02380.10.10.238-0.10.3540.10.1
β˙-0.01450.0354000.035400.01450.010.01
γ˙-0.0602-0.0145000.145-0.10.1020.10.1
n11.78691.78691.98-1.991.7869-1.98-1.7869-22
n2-6.0811-6.0811-1.981.99-6.0811-1.986.08112-2
f10.75250.75250.70.70.75250.7-75250.70.7
f20.3790.3790.70.70.3790.70.3790.70.7
n3-12.89042.89042-22.89042-12.89042-2
n40.6977-0.6977-22-0.6977-20.6077-22
f30.12720.12720.70.70.12720.7-0.12720.70.7
f45.16815.16810.70.75.16810.75.16810.70.7
n5-0.4533-0.4533-2-2-0.4533-24.5332-2
n60.18930.18932-20.18932-1.8932-2
f50.66590.66590.70.70.66590.70.66590.70.7
f60.01050.01050.70.70.01050.7-0.01050.70.7
n74.6201-4.6201-2-2-4.6201-24.62012-2
n8-15.862815.86282215.86282-15.86282-2
f72.25922.25920.70.72.25920.72.25920.70.7
f80.17330.17330.70.70.17330.70.17330.70.7
Table3.8:Diﬀerentinitialvaluesforthesystem.

3ActuationofPassiveMechanicalModels

(a)Angularposition

lotphaseP(c)

(b)Oscillatornetworkactivation

(d)Movement

Figure3.32:SimulationofpatternP1withtheparametersF11,s=4,fd=
0.5andfdv=1resultsinastablesteppinginplacemovementwithdrop-
pinghipandlargerstanceandswinglegamplitude.

(a)Angularposition

lotpPhase(c)

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Oscillatornetworkactivation

(d)Movement

Figure3.33:0.Sim48ulaandtionfofdvpa=1.ttern2reP2sultswithinathestableparameterssteppingFin11,placesmo=v4,ementfdwit=h
droppinghip.Itshowsahigheractivationofthehipwithlargeramplitude
anddiﬀerentvelocitiesduringthemovement.

3ActuationofPassiveMechanicalModels

(a)Angularposition

phaseP(c)lot

(b)Oscillatornetworkactivation

(d)Movement

Figure3.34:SimulationofpatternP3withtheparametersF11,s=4,fd=
0.48andfdv=1.2resultsinastablesteppinginplacemovementwhichis
moredynamic.Thehigherthedegreeofinteractionbetweentheneuronsthe
morethejointmovementisinﬂuenced.Thehipmovementdependstherefore
moreonthestanceandswinglegmovement.

(a)Angularposition

lotpPhase(c)

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Oscillatornetworkactivation

(d)Movement

Figure3.35:0.Sim5ulaandtionfofdv=pa1.5tternPresults4inwithathestableparsteametersppinginF11,placesmo=veme4,ntfdwith=
thereactiohignhesttimesdynaandmic.theHvereloeevcities,enmorespeecianeuronllyofinthelegs,terconnectioaremnsuchleadthigher.ofaster

3ActuationofPassiveMechanicalModels

3.8.5StabilityofMovementswithExternalPerturbations

Stabilityisthemostimportantcharacteristicofthepresentedsteppingmovements.Fur-
theritisinterestinghowrobustthemovementsareregardingtoexternalperturbations,
whichareappliedtothesystem.Innaturetheseperturbationscouldbeeventslikeun-
evengroundoccurringsuddenlyandunexpectedly,aslippingofthelegorablowfrom
anexternalsource,whichdisturbsthebodymovementdirectly.Suchexternaldistur-
bancestothesystemarecommonandofteninnaturalwalking.Thereisnopreactionto
thesedisturbancesastheyarenotforeseenbutonlyareaction.Itwillbeshownthatan
oscillator-drivenmechanicsisgenerallyabletoregainstabilityafteraperturbationifitis
notsostrongtopushthesystemoutoftheattractiveregion.Thisabilityleadstoamore
robustsystemforawiderangeofmovements.Inthefollowingthreesimpleperturbations
areshown.Eachangleα,βandγisperturbedoncebyasuddendiscretechange.
Inﬁgure3.36theperturbationofstanceangleαisshown.Thestancefoote.g.slipsaway
tothesideoppositetotheinducedmovement.Thisperturbationleadstoadisturbance
ofthesystemwhichiscompensatedaftershorttime.Thenexttwostepshavetostabilize
thesystemagain.Thecompensationworkoftheﬁrststepissmallerthanthatofthe
second.Afterwardsthesteppingmovementisstableagain.
Inﬁgure3.37thestancelegisdisturbedatthesameinstantbutthedirectionofdistur-
banceisopposite.Herethedisturbancecanbeimaginedase.g.againaslippingofthe
stancelegbutnowtowardstheswingleg.Thisdisturbanceevenleadstolessperturbance
ofthesteppingsystembecauseitisinthesamedirectionasthenaturalmovementwould
havebeen.Thestableconﬁgurationisregainedveryquickly.Aninstantslippingofthe
stancelegindirectionofthemovementandintheoppositedirectionhavebeenapplied
withtheresult,ifthestancelegslipsthedirectionoftheforeseenmovementthereisnot
muchreactionofthesystem.Thesteplengthandamplitudeisenlargedabitbutthe
successivestepsareagainasnormal.Bycontrasttheperturbationagainstthemovement
directionleadstoafarbiggerdisturbanceofthesystembutnotintherelatedstepbut
inthefollowing.Thestepisshorterbecausethemusclefeedbackreactsonthesudden
angularchangeandthesuccessivethreestepsareneededtocompensatethisdisturbance
becausethehipmovementtakesagreatpartofthecompensationpart.
Inﬁgure3.38theswinglegisdisturbedatthebeginningoftheswingphase.This
happensforexampleiftheswingleggetsstuckorcaughtbyanobstaclejustafterthe
pushoﬀphase.Thisdisturbanceleadstoarealdisturbanceofthesystem.Theground
contactaheadisreachedwithcompletelydiﬀerentangularvalues.Thefollowingstepis
outofbalanceandthereforethesecondfollowingstephastocompensateandregulate
thesteppingmovementwithabighipandstancelegcountermovement.Thereafterthe
systemhasfounditsstableconﬁgurationagainandthesteppingmovementissymmetric
gain.amuniforandInﬁgure3.39theswinglegisdisturbedattheendingoftheswingphaselikeifan
obstaclejustpreventsthenormaldoublesupportphaseheavily.Thisdisturbanceleads
torelativelysmalldisturbancesofthesystem.Thegroundcontactaheadisretardedonly

3.8SimulatedSteppingMovementsintheFrontalPlane

aconlitttleactabit.ndtheThewhonextlesangtepleshowsconstellatlittleionisinﬂuenceonlywvhiceryhcoslighmestlyfromomdiﬁed.themoAftediﬁedrwgardsroundthe
stepcycleisbacktotheoriginal.
Inﬁgure3.40thehipisdisturbedwhichisadiscretechangeoftheangleβ.Thisdis-
turbanceisthemostsevereonebecausethehipmassisthebiggestandititnotso
wpell-ensabationlancedmovovemenerttheofstathencewholeglebtoobdyethequicsyklystemregstabilizedainsagstaain.bilityButandaftereturronensaftbigercoonem-
moresteptothenormalsteppingmovement.
Thisdemonstrationofthreediﬀerentsystemperturbationshallonlydemonstratethat
thesystemisrobustagainstperturbations.Thereisalwaysacompensationmovement
mowherevementhetregaamplitudeinedafterandcompdurationensatiodepnemondsvoemenntthesistypageainofpstableerturbaandtioon.ftThehesamesteppingtype
asthemovementwasbeforetheperturbation.

(a)Angularposition

lotpPhase(c)

(b)Oscillatornetworkactivation

Figure3.36:Perturbationofangleαofthestancelegwhichisaslippingtotheside.
Parametersusedforsimulationare:patternP1,torquegenerationstrategy
F12,feedbackfdv=1andfd=0.5.

3ActuationofPassiveMechanicalModels

(a)Angularposition

P(c)lotphase

(b)Oscillatornetworkactivation

Figure3.37:Perturbationofangleαofthestanceleg,whichisasliptowardstheotherleg.
Parametersusedforsimulationare:patternP1,torquegenerationstrategy
F12,feedbackfdv=1andfd=0.5.

(a)Angularposition

lotpPhase(c)

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Oscillatornetworkactivation

Figure3.38:Perturbationofangleγoftheswingleg,whichislikecausedbygettingstuck.
Parametersusedforsimulationare:patternP1,torquegenerationstrategy
F12,feedbackfdv=1andfd=0.5.

3ActuationofPassiveMechanicalModels

(a)Angularposition

haseP(c)lotp

(b)Oscillatornetworkactivation

Figure3.39:Perturbationofangleγoftheswinglegattheveryendoftheswingphase.
Parametersusedforsimulationare:patternP1,torquegenerationstrategy
F12,feedbackfdv=1andfd=0.5.

(a)Angularposition

lotpPhase(c)

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Oscillatornetworkactivation

Figure3.40:Perturbationofangleβofthehip,whichislikeadirectpushtothehip.
Parametersusedforsimulation:patternP1,torquegenerationstrategyF12,
feedbackfdv=1andfd=0.5.

3ActuationofPassiveMechanicalModels

imageoeVid(a)

(b)Extractedpossiblemarkers

Figure3.41:Thevideoimageisﬁlteredandthresholdedtoreceiveimageregionsforpos-
ers.rkmasible

3.8.6ComparisonofSimulationDatawithRealSteppingData

Lastbutnotleastthisbiomechanicalmodelwasmadetorepresentcharacteristicsof
realsteppingmovements.Therefore,themodeldataarecomparedtorealdatawhich
aregainedfromexperimentallyraiseddata.Theexperimentalsetuptogetthedatais
shortlyexplainedandtheresultingmovementdataarecomparedqualitativelywiththe
.dataionulatsimIntheexperimentmovementdatawascollectedviavideotrackingofmarkersputonthe
joints.Asthisisforaroughcomparisonofrealandsimulateddata,themarkerswere
putontheclotheswherethejointsandtheinterestinghingesarewellvisible.Thiscan
beseeninﬁgure3.41(a).Thefeetarejustmarkedaroundtheankletoseethelifting
movementofthelowerleg,nottogetanklemovements.Then,themarkersareputto
theknees,thehipsandadditionallytwomarkersareputabovethemiddleofthepelvis
togettheupperbodymovement.Thesteppingpersonisrecordedwithacamerawith
aframerateof25fpsandaresolutionof720x576.Afterwardsviaimageprocessingin
MATLABtheseeightmarkersareextractedandidentiﬁedandputtogethertoastick
ﬁgurewhichrepresentsthesteppingsubject.Forthisseeﬁgure3.42(b).
TheimageprocessinginMATLABconsistsofthefollowingprocessingsteps:Theimage
seeninﬁgure3.41(a)isdecomposedintoitsthreecolorchannelsR,GandB.Afterwards,
theGreenChannelistransformedintothe’hiv’colorspace.Thisimageissimplythresh-
oldedtogetthepossibleregionsfortheyellowmarkerswhichisseeninﬁgure3.41(b).

3.8SimulatedSteppingMovementsintheFrontalPlane

(a)Determinationofboundaryandcenters

(b)Selectionofmarkersforstickﬁgure

Figure3.42:Theextractedregionsareclassiﬁedandobservedoverseveralimagestomatch
theregionstomarkersandafterwardsconnectthemasastickﬁgure.

Thesingleregionsareclassiﬁedaccordingtotheirsize,boundaryandcentroid.Inthe
followingonlythesearchedmarkersareextractedﬁnallyasseeninﬁgure3.42(a).The
foundcentroidsareobservedoverseveralsubsequentimagesandmatchedtothem.In
thefollowingthecentroidsarematchedtothepositiononthebodyandafterwardsthey
areconnectedtoastickﬁgurewhichispresentedinﬁgure3.42(b).Fromthisstickﬁgure
theanglesα,βandγarereconstructed.Theanglesaredeﬁnedinthesamewayasinthe
mechanicssectioninﬁgure2.4.Thehipandfeetmarkeraretakentocalculatetheangles.
Thekneemarkersdonotchangetheangleslargelyasthekneeismoreorlessinline
withthehipandfeetwithaslightdeviationaccordingtothejointpositioningasknock
kneesorbowlegsofthesubjects.Theangularvelocitiescanalsobereconstructedwith
theinformationoftheframerate.Thereceivedanglesarecomparedincharacteristicand
phasetothesimulateddata.Thetwosteppingmodessteppinginplacewithdropping
hipandsteppingasidehavebeencompared.

3ActuationofPassiveMechanicalModels

(a)Angularpositions

(b)Phaseplot

Figure3.43:Videotracking:angularpositionandphaseplotofexperimentaldataof
place.instepping

(a)Angularpositions

(b)Phaseplot

Figure3.44:Simulation:angularpositionandphaseplotofsimulateddataofsteppingin
place.

Steppinginplaceisamoreorlesssymmetricmovementwherethelegsaremovedal-
ternatinginapendulummovement.Themovementofthelegsisthoughdiametrical
timeshifted.Onelegmovestowardstheotherandbackagainduringonestepwhile
themovementofthetwolegsisopposite.Thismeansthattheplotofthestanceand
swinglegangleissymmetricifthesameifthetimeshiftisomitted.Thismovementis
representedbythesimulationveryappropriatelywhichisseeninﬁgure3.43and3.44.
Thecharacteristicsofthelegandhipmovementisthesameandthephasesarealsoa
goodmatchwherethehipdropsbeforetheswinglegswingsbackagain.Thestanceleg
oscillatesinthesamesequenceasthehipdropsdown.Andthetwolegsdoapendulum
movement.Theamplitudesoftheangleshaveagoodmatchingrelationonlythevelocity
insimulationishighertheninreality.Thisisbecausethereisnoslowdownbyground
contactandnoenergystorageine.g.muscleﬁbersorjointsduringtheswingphase.This
meansthataverysuitablesolutionofthemodelwasfoundtorepresentsteppinginplace.

(a)Angularpositions

3.8SimulatedSteppingMovementsintheFrontalPlane

(b)Phaseplot

Figure3.45:Videotracking:angularpositionandphaseplotofexperimentaldataof
.sidethetostepping

(a)Angularpositions

(b)Phaseplot

Figure3.46:Simulation:angularpositionandphaseplotofsimulateddataofsteppingto
ide.sthe

3ActuationofPassiveMechanicalModels

Characterizingsteppingtothesideisalegmovementofthestancelegintheopposite
directionoftheswingleg(whichmeansthesameangularpositionbecausetheanglesare
Theopposlegitemocovunementedtassshahovewnsoinmeﬁglauretency2.4to).Teachishomothervemenandttheleadship.toThethesimsidewulaaytedmodavetamenhat.s
hipthishaslegamomvuchemenhigt,herbutathemplitudemovemerelantedtstaroetsheyncleghronoamplitusudeswithoutthananinylatrealitency.y.ThisAndrefetrhes
tomovtheemenfacttslikthatethereelasticareﬁbenorstlatenciesructuresomoradeledgroundnorarconetactthereawithnyastructdeterminedureswhicdurahdelation.y
Theshiftsobnlyetwsteenlegructuressandinhipthearemothedeltywpehicofhosccanillaprotorducnetewbigorkasdiﬀerenctheesconnectioinlanstenciesinﬂuenceand
thetimingrelationsofactivation.Andtheweightingaccordingtodiﬀerentactuation
strategiesisalsoanimportantfactorinﬂuencingthelatenciesandshifts.Thesetwo
sidewinﬂuencingaysfasteppingctorsawerlargeernotpartspofeciallytheadahipsptedhiftforisthedonesidewdurinaysgmothevemendoublet.Insupptheortnaphaturalse
whichisnotreproducedbythemodel’sinstantaneousgroundcontact.Thisleadsto
simulationdatawhichisdiﬀerenttothevideo-trackingdataalthoughbothshowsideways
ﬁgurstepping.e3.46.Theexperimentaldatacanbeseeninﬁgure3.45andthesimulateddatain

oncussiDis3.9

Theproposedmodelconsistsofthefrontalorsagittalmechanicsactuatedbyanoscil-
latornetworkofMatsuokaoscillatorsandantagonisticjointtorquegenerators,plusa
muscularfeedbackmechanismbasedonpositionandvelocityinformation.Thismodel
alreadyprovidesonalowlevelmanyofthetypicalsteppingmovementsfoundinbiology.
”Low-level”meanswithouttheuseofanyhighercontrolsuchasthebrainandhigh-level
sensors.IncontrasttothemodelsofGengetal.[41,40,144],whereonlythesagittal
planeismodeled,thepresentedresearchmodelsthefrontalplaneandanalyzesitinde-
tail.Gengmodelsabiologicallymotivatedrobot,whichmeansthattheneuronsdonot
incorporateadaptationeﬀectsbuthavedirectreﬂexivecouplingtothepositionwiththe
aimofachievingfastreactiontimesforeachstep.Anotherexampleofneuronallydriven
mechanicsisgivenbyRighettiandIjspeert[152].HereHopfoscillatorsareusedwhich
arecoupledinchainswithamasteroscillatortoguaranteephaseshiftsbetweentheos-
cillators.However,thestabilityofthelateralmotioniscontrolledbythesensoryoutput
ofthegyrosrepresentingavestibularsensor,andnotbyanautonomousCPGpattern
asinthepresentedmodel.AlsoinMiyakoshietal.[123]lateralstabilizationisachieved
byahigh-levelPD-controlscheme.Noneoftheseothermodels,inliterature,evaluated
thepossibilitiesoftheneuro-mechanicalmodelforperforminglateralsteppingmovements
andtheabilitiesofthemodelwhichdependontheparametersoftheoscillatorsystem.
Thereissofarnootherworkknowntotheauthorwhichstudieslateralsteppingpatterns
onthebasisoftheneuronalactuation;thesearethefoursimulatedpatterns:steppingin
placewithdroppingandliftinghip,steppingsidewaysandsteppingup.Forthepurpose
ofvalidationtheactuationconceptwasalsotestedsuccessfullywiththesagittalmodel.

Discussion3.9

Kuo[94]arguesthatstabilizationofthefrontalplaneandofthesagittalplanemovements
arelargelyindependent.Thereare,however,nobiologicalstudiesaboutthemechanismof
theseparationortheinteraction.Thiscombinationwouldthereforebeasuitablesubject
forfutureresearch.
Threecharacteristicsofsteppingmovements:stability,frequency(orvelocity)andground
contactareanalyzed.Asstabilityistheprimarycharacteristicinvestigatedwhenanalyz-
ingsteppingpatternsandstrategies,itisoftenstudiedinliterature[127,128,49,48,38,
39].StabilitywasprovenusingthenumericmethodproposedbyGoswami[48].There
wereshownstablemovementsandunstableoneswhichcandoseveralstepsbeforefalling.
Thethreemainsteppingmovementswhicharepossiblewiththefrontalmechanics,step-
pinginplace(withliftinganddroppinghip),steppingtothesideandsteppingup,were
realizedasautonomoussteppingmovementsoftheoscillator-drivenmechanics.Thethree
steppingtypeswererealizedwithdiﬀerentinitialconditionsordiﬀerentgroundcontacts.
Thusmeansthatingeneralthereisnodiﬀerencebetweensteppingwithdroppinghipor
steppingwithliftinghip.Inadditiontheknownstrategiesofhip,ankleormixedactua-
tionweretestedaccordingtoHoraketal.[59,60].Horakpostulatesthatthesechanging
strategiesoccurasaresultoflearningandexperience.Thismeansthattheautomated
rhythmicmovementsarenotﬁxedbutcanbeadaptedinlinewithlearningeﬀects.This
learninghasnotbeenimplementedinthepresentedresearch,butthestrategychanges
whichhavebeentestedherecouldalsobeimplementedaslearnedstrategies,becausethe
strategyinﬂuencesperformancefactorssuchaseﬃciencyorappearanceofthestepping
movement.ThemodeldevelopedbyGengetal.[40]showsthatlearningalgorithmscan
beusedtoadaptactuationtotheenvironment.
Oneimportantmodelparameteristheexternalinputtotheneuronss.Thisinputwas
proventochangethestepfrequencytogetherwiththedurationofthegroundcontact.By
isolatingparametersitispossibletotunethestepfrequencywithoutgreatlyinﬂuencing
othersteppingparameters.AsManoonpong[105]mentions,thespeedvariation,which
mainlycorrelateswiththestepfrequency,couldnotbeeasilyadaptedinearlierneuro-
mechanicalmodelsasin[171].
InHoraketal.[66]andMu¨lleretal.[133],thehipandanklestrategiesusedinposture
controlandsagittalwalkingarestudied.Therearenostudiesknowntotheauthorwhere
medio-lateralstrategiesareanalyzed.Withthemodelpresentedhereitwasshownthat
diﬀerentjointactuationstrategiesinﬂuencetheappearance,thephaseandthecoordina-
tionofthesteppingmovements.Theseresultscanbeagoodstartingpointforexperimen-
talstudiesofmedio-lateralhipandanklestrategiesbecausemedio-lateralstabilizationis
evenmoredependentonactivestabilizationthansagittalstabilization[6,94].
Anaturalsystemofwalkingisoftenexposedtodisturbinginﬂuences,someofwhichcan
beforeseenbutmanyofwhichoccurunexpectedlyandsuddenly.Suchperturbationsof
thesystemweresurveyedwhetherandhowthesystemreacts.Thethreeangleswere
disturbedindiﬀerentdirectionsorindiﬀerentpointsintimeofthestepcycle.The
systemcompensatesfortheperturbationwithinafewstepsandreturnstoastablestep
cycle.Thisrobustbehaviorisabigadvantageofusingdistributedactuationfeedback.

3ActuationofPassiveMechanicalModels

Distributedwalkingsystemsasin[171,105]werealsofoundtodisplayrobustbehavior.
Lastbutnotleast,thesimulatedmovementswerecomparedwithrealexperimentaldata
receivedfromavideotrackinganalysis.Thiscomparisonwascarriedoutinordertocheck
forqualitativesimilaritiesanddiﬀerences.Formedio-lateralsteppingmovementsthere
arenoexperimentalwalkingpatternsknowntotheauthor.Itcanbeconcludedthat
therearesimulationsolutionswhichresembletheexperimentalmovementsquitewellas
regardsphase,periodicity,amplitudeandappearance,buttherearealsomanysolutions
foundwhicharequitediﬀerent.Humansteppingisaveryindividualmovementwhich
adaptstochangesandinﬂuencesofthemovementapparatusandshowsawiderangeof
variablemovementswhichareneverrepeatedexactlythesameway.

3.10Conclusion

Inationthisbcyhaantpteragoanisticcombinajointtiontorqofuetheapplicapassivtionemecandhanicsmuscle-oftcypehapterfeed2bacwithkwaasneuralpresenatcetd.u-
Thismodelisalsocalledthelow-levelmodelbecauseitdoesnotincludehigh-levelpos-
turecontrolfunctionsofthebrain.Simulationofthelow-levelmodelwasperformedto
evaluatethecharacteristicsofneuralactuationandtorevealtheproperties,potentialand
shortcomingsofthemodel.
Theactuationconceptwasappliedeﬀectivelybothtothefrontal-planemechanicsand
tothesagittal-planemechanics.Thisshowsthattheconceptisgeneralandcanbeap-
pliedtodiﬀerentmechanicsandrhythmicmovementpatterns.Acombinationofthose
tingwotheplanesismedio-laterapossiblelbutsteppingisleftmovforemenfuturtseofrethesearfcrh.ontaThisl-planethesismocodel.ncenIntratesgeneraonlsittwudy-as
demonstratedthatthestabilityrangeandthesteppingvariabilitywereincreasedbythe
concept.ntioactuaSimulationofthefrontalplaneshowedthatthreetypicalsteppingmovementssuchas
asteppingladder)incanbplaceepro(drduced.oppingInandadditionliftingthehip),strsteppingategieswtohichthearesideuseadndforsteppingactuationupwe(e.g.re
testedaccordingtothehip,ankleandmixedstrategiesfoundexperimentally(inthelit-
erature).Thesestrategiescanbeappliedsuccessfullyandresultindiﬀerentstepping
mostravtegiesements.andConsequensteppingtlypa,ittternsispousesd.sibleThistocaninﬂuencebeacthehievloewdo-levnelamohigdelherbleyvvelabyryingteitherhe
adaptingoroverrulingautomaticmovementstosuitconditionswhichhavebeenexperi-
encedandlearned,suchaseﬃciencyandstability,ortotakeconsciousdecisionssuchas
theselectionofsteppingupbecausethereisanobstacleorsteppingtothesidebecause
thistematisanicallyinstrinucﬂuetnceion.movAnalyemensistofctheharaparacteristics:metersasceFirstlyr,tainedfeedbathrckeegpaainsdirrametersectlywhichinﬂuencesys-
stability.Secondly,initialconditions,aswellasthegroundcontact,whichtriggersanew
initialconditionforthenextstep,inﬂuencethedirectionofmovement.Thirdly,parame-
btereinsistegrateddirectlyinapropohigh-levrtionaleltomothedeltosteppinginﬂuencefrequencyprecis.elyThesethosearethreeparproametersperties.whichcThisouldis

3.10Conclusion

lefttofutureresearchbecausethiscorrelatesstronglywiththeintentionofwalking,which
wasnotsubjectoftheresearchperformedinthisthesis.
toItwtheasrealprovhenumathantitissteppingposmosiblevtoemensimtswulahictehhavsteppingebeenmovanalyemenzetsd.whichareverysimilar

Inconclusionthepresentedmodelaccountsforseveralcharacteristicsofhumanstepping
movements,thoughtherearealsosomelimitations.Onelargelimitationthathasalready
beenmentionedisthe”low-level”aspect.Thismeansthattherearenohigh-levelfunctions
integratedinthemodelpresentedsofar.Thereforethewholebodypositionremains
unknownandenvironmentalinﬂuencesarenottakenintoaccount.Ifinappropriateinitial
conditionsarechosenorthefeedbackgainsarenotadaptedcompletely,thisleadsto
unstablemovements.Aftersomestepsthesteppingmodelwillfalloverormighteven
tumblewiththeﬁrststep.Inthenextchapter4ahigh-levelposturecontrolmodelis
developedtostabilizethewholebodypositionandtointegratesensoryperceptioninto
themodel.Thismodelalsobasesonbiologicallyprinciplesandstructuresbutatthesame
timeaimstokeepthehigh-levelposturecontrolrelativelysimple.

4High-LevelPostureControl

Sensorsmeasuretherelationbetweentheenvironmentandthebodyortherelationship
betweenindividualbodyparts.High-levelposturecontrolisacontrolwhichbasesonthe
andinformathetiorelanprotionshipvidedbetbyweenthestehem.sensorsThisandmeaonnskthanotwledgethereisofatheknobowledgdy,tehewhicenhvhasironmenbeetn
gainedbyexperienceandmemorizedandthisisabletorelatethesensoryinformationto
acontext.Thebrainhastocorrelatetheinternalbodystateswiththeexternalworldand
inprotegducesrationthetaskrighistnareacmetdionofhigh-levtheelinhereternalbsysecausetemtsucohinfterunctnalionsandasexsensternaloryeveninformats.Thistion
andprocessingspinalcoorrdalevknoelswaledgndearbaseearethereforelocafotedundhiginherthethanbrain.theThemecohanbicjectivalelevoefls,thisreﬂexhigh-levlevelsel
informationprocessingisposturecontrolofthebodyforstandingandstepping.
Pthatosturheigh-leconvtelrolspinecthisiﬁcacontionstextforalwtheaysmovemeamennstaconretrollingdeﬁned,thate.g.thethebodydirectiondoesornotspfalleedoorf
movement.Inthecaseofstanding,thetaskistomaintainstablestanceandthetarget
positioindividuanisl”legastandingngle,butuprighinfot”.rmatForiononsteppingmoindividuavemenlptosstitheionsreisisnogadirectheredtcboyntrallolsofensoreachs
toandaisingenerategrlbatedodytopositiostabilizensucthehwasholethisuprare,ighte.g.bodythepositiozeron.momenThettecpoinhnicatl(ZequivMP),alentthes
centerofpressure(COP)orthecenterofmass(COM).Thosevaluesrepresentthewhole
bodypositionwhicharecontrolledinordertoachieveuprightsteppingandstanding
ts.emenvmomenThetsinomeasuremenrdertotscoordinaregisteredtebthemytheandtosensouserythemsystemsforarehigh-leveltransmittbodyedmoinvtobemenodytcomorrec-ve-
tions.Thesensorysystemswhichareaddressedinthefollowingarethevisualsense,the
vsoryestibulasensre.Thesense,theprmeasuremenoprioctseptiveregisteredsensebyandtheseforstafourte-osensf-tohery-arsytstemomsdels,are:theretinalsomaimagtosen-e,
headaccelerations,relationsbetweenbodypartssuchasjointsandmusclesandﬁnally
othersensormeasurementssuchastemperature,hapticfeedbacketc.
Thesensoryinformationusedinthefollowingisderivedfromthefourfollowingsensitive
systems:•thevestibularsense,whichsensesbodyaccelerationsinall3translationsand3
rotationsandwhichislocatedintheheadintheinnerear.
•themeasuresvisualthesense,envwhicironmenhtasenselsmovligehtmenprtsoinducingrelatitheontoretinathelheadimageandandeyemovsubsequenementtlys.

Itissituatedinthehead.

•theproprioceptivesense,whichsensesthebodyitselfanditsrelationse.g.injoints
andinthemuscularfeedbackasalreadymentionedinsection3.4.2.Itisdistributed
overthewholebodyandisnotcentralizedinoneorgan.

•thetempsomaerature,tosensorypainandsense,vibwhicratiohnandincludesprallopriocskineptiosensnoasrsanddescribedmeasuresin[18e.g.5].pressure,

Therearemanyexamplesandpossibilitiesforintegratingsensorymodalitiesintoahigh-
inlevtegelpratioosturencofunctionsntrol.forAsthethecroeismplexnohproumaofntphattosturheerecoisnatrolbiolosystem,gicalathesenalogymoofdelssensorare
expideasfoerimenrsotsslutionstepbywithstepatogreaterexplainorclesserharadegcteristicsreeofpofosspibilitosturey;contheytrol.needPtoosturbeevoerriﬁedstanceby
controlarethereforethesubjectofintensivestudy,includingexperimentsandclinical
ﬁndingsaswellasmodelexplanations[88,120,80,97,180].Themodelpresentedinthis
bwoordyksctoates,nsistsenofvirothenmenstensorandytmeahesurrelatioemenntbetpawrt,eenthwhicehm,prandotvidesransmittsinformathistiontoabtheouthigth-he
levsenselopryrocesinforsingmatiocompnonandent,usestheittobrain.deterTheminethigh-lehewvaelytprohecessingsystemcoreampctsonentotkineeptegtheratesbotdyhe
inbalance.Forthishigh-levelmodelnewexperimentsoneyemovementandartiﬁcial
vtoevestibulaaluatrestimthepoulatiostnurearecontevarolmoluateddeltogproethervided.withIntegraﬁndingstionfofrsomensortheyliterameasurementureintorderand
processedposturecontrolcommandsleadstoastancemodelwhichmaintainsbalanceand
isabletodemonstratetheinﬂuenceofsensoryinputsonposture.Byextendingthevisual
expsensoryerimencuetalbyresultsanaarelternativreproesduced.ensorynonlinearity,accordingtostate-of-the-artmodels,

Insection4.1someideasformodelsandrelatedexperimentalstudiesarepresentedto
provideinsightintothecomplexityandabstractionofsensorintegrationposturemodels.
Itisalsoderivedwhyastatisticalestimatorischosenasthesensorintegrationmodel.

Section4.2presentsthesensorymodelswhichareusedinthisworkandwhichvalidate
experimentalresults.Thesearethetransferfunctionsusedtorelaytheenvironmentand
bodyinformationperceivedbythesensorsforinternalprocessing.Insection4.3themodel
fortheinternalestimationisderived.AKalmanﬁlterestimationmodelisintroducedand
extendedwithanonlinearsensorypart.Thehigh-levelposturecontrolmodelisapplied
andtestedusinganegativefeedbackofoptimalcontrolaspresentedinsubsection4.3.4
andalsothemechanicsofaninversependulum.Thisapplicationisdetailedinsection
4.3.2.Thesimulationresultsofthismodelareestablishedaccordingtoexperimental
ﬁndingsregardingpostureresponsetovisualandvestibularstimulations.Theﬁndingsare
deliveredfromtheliteratureandfromtheauthor’sownexperiments.Theseexperimental
studiesandresultsaredetailedinsections4.4.2and4.4.3.Themodelisimplemented
andsimulatedinMATLAB.Simulationresultsarecomparedtoexperimentalresultsin
section4.5.Finally,insection4.7thesimulatedresultsarediscussed.

4High-LevelPostureControl

4.1StateoftheArtofSensorimotorPostureModels

Theideasformodelspresentedinthefollowingallinvolvetheuseofsensorymodels.
Thesesensorymodelsdiﬀer,butthekindofinformationtheyprovideissimilarandoften
ofthesamecharacteristic.Nospecialexplanationisgivenofanyofthesensorymodels
usedinthestate-of-the-artposturecontrolmodelspresented;mentionisonlymadeofthe
typeofsensoryinformationintegratedinthemodel.
Verystraightforwardstancecontrolisprovidedbymodelswhichintegratethesensory
signalsinaPID-controlmodel.IntheresearchpublishedbyPeterkaetal.[138,139,140]
abiologicallymotivatedmodelisdevelopedwhichweightstheinformationprovidedby
eachsensorindividuallyandintegratestheinformationbysummation.Thissensoryin-
formationistime-delayedandthenusedforPIDcontrolofthebodypositionbycorrective
jointtorqueinput.Thiscontrolactsparalleltothepassivemuscledynamicswhichare
stabilizedwithpositiveforcefeedbackinordertoinvestigatetheirqualitativeinﬂuence
onposturecontrol.ThepurposeofthismodelisexplainedinPeterka[140](p.6)”Our
relativelysimplemodelsallowedustoapplysystemsidentiﬁcationmethodsinordertoes-
timatetherelativecontributions(sensoryweights)ofvarioussensoryorientationcuesin
diﬀerentenvironmentalconditions”.[61]alsoproposesaPIDmodelwithsensoryweight-
ingtoexplaindiﬀerencesinstandingwitheyesopenandeyesclosed.Otherexamples
ofreweightedmultisensoryinputscombinedwithPIDcontrolaregivenbyMergneret
al.[119,120,118,109].Theyusethemodeltointroduceanonlinearrelationbetween
sensoryinputsandpositioncontrol.Thenonlinearityofthesensorysystemismodeled
bythresholdswhichleadtononlinearreactionsinthepositionresponse.Thesensory
cuesforvisualperceptionaremorecloselyinvestigatedin[120],whereasthesensorycues
forproprioceptionhavebeenstudiedinBeckeretal.[7,116],alwaysincombination
withvestibularsensorcues.ThestudiesconductedbyBeckerintroduceanotheraspect
withregardtosensorycombination.Theaveragedweightingforsensorycuefusionis
opposedbyacognitive”eigenmodel”ofvestibularperception.Thiscognitivemodelis
usedtoexplainthediscrepanciesingain(ratioofachievedtodesiredrotationposition)
betweenhighandlowstimulationvelocityandduration.Itwasfoundthatlongerand
slowerstimulusofpassiverotationortreadmillsteppingonarotatingplatformleadto
anoverestimationofthesubject’sownrotationposition;thisstandsincontrasttosingle
reweightinganddecreasinggaintheoryoutlinedabove.
Stancestabilizationcanalsobeachievedusingfuzzycontrol[74,106],whichisnotasclear-
cutanddirectasclassiccontrolmethodsandisthereforealsocalled”soft-computing”.
In[74]fuzzycontrolisappliedtointernalcontrolwhichonlyrelatestointernalvalues
suchasproprioceptivesensation.In[106]fuzzycontrolisappliedtoamoregeneraltask
thangaitandisthereforeextendedbyalearningability.Butitisalsoreducedtosimple
measurementssuchaslengthandanglefromtheproprioceptivesense.
Anothergroupofpositioncontrolmodelsforhumanstanceandgaitarethestatistical
estimationmodels.Theestimationofsystemstatesisusedasfeedbacktostabilizethe
system.Estimationrequiresmodelknowledgeofthesystemandthesensors.Thisis

4.1StateoftheArtofSensorimotorPostureModels

representedhereasKalmanestimationorinamoregeneralwayasBayesestimation.The
Bayesapproachforheadpositionestimationwithseveralsensorsisdetailedin[47,99].
Forboth,thestatisticalideaisthatthecontrolofuprightstanceisabsolutelyessentialfor
humansurvival.Frequentfallswouldhavebeenaserioushandicapforescape,defenseor
othersimpledailytasks.Itisnotimportantthattheindividualmovementorcontroltask
isoptimizedabsolutelybutthatthestatisticalprocedureisoptimizedandrobustover
allcontroltasks.Thissolutionisthereforeaninterestingapproachforbiologicalmodels.
Wolpert[98]statesforcognitionmodelsthat:”...BayesianDecisionTheory.Thistheory
deﬁnesoptimalbehaviourinaworldcharacterizedbyuncertainty,andprovidesacoherent
wayofdescribingsensorimotorprocesses.”(p.319)

andKalmanobservaﬁlterstionwhicﬁltershbaseappliedontothetecBayhnicalesiantheosystemsry[a16re2].commoThenlinearmodelsKaforlmaneﬁstimaltertionhas
beeKalmannprovﬁlteredtoapprgiveoachthewasopttherefimaloerestimasetionlectedforforthiswhitethesisGaussianinorderprotocessesdev[elop190].ahigTheh-
levelcontrolmodelincorporatingmultisensoryprocessingandmodelknowledge.Another
notreasonreprisothaducetthewithtocompleteosimplesporectrumstaticofafuncpproationacheslity.theForsensthisoryreainfosonarmationstatisticafusionlmododeesl
basedonposturecontrolseemedanappropriateapproachhere.

ExamplesofKalmanestimationmodelsforbiologicalstancecontrolwithmultiplesensors
aree.g.thosedevelopdebyvanderKooijetal.[179,180],Jekaetal.[136,14]andKuo
[96,6,97].In[179]theKalmanﬁlterisnotlinearbutextendedbyanonlinearnoise
covariancedescription.Thisnonlinearityleadstoadecreasinggainofswayresponse
withincreasingsensoryinputs.In[96,6,97]thesensorsarestate-dependentsensory
modelswhichareintegratedinthelinearposturecontrolmodelwithanestimationfor
thesagittalplanestance.Theoptimalfeedbackcontrolstrategyisvariedwithdiﬀerent
controlobjectivesandtimedelaystoestablishthediﬀerencesfordiﬀerentcontrolstrate-
gieswhicharecomparedwithexperimentalﬁndings.Theexperimentallyfoundhipand
anklestrategiesarereproducedbyslightlychangingoneparameterwhichdeﬁnestheratio
ofCOMcontrolobjectivestoangularposition.In[97]thesensorymodelsaredeveloped
further.Thisthesisexamineswetherthelossofasensorymodalitysuchasvisionorthe
vestibularsensestimulatesincreasedposturalresponse.Thechangeinsensorymodalities
withincreasingageisalsomodeledandexplainedbydecreasingsignal-to-noise-ratios,
whichleadtolessinformationbeinggatheredbysensoryinput.In[14]thediﬀerence
betweenpresenceandlackofmodel-knowledgeoftheenvironmentarecompared.The
conclusionisthatanunmodeledenvironmentgivessimulationresultswhicharecloserto
experimentaldata.
Anotherquestionrelatestotheforminwhichtheinformationfromthesensorsispro-
cessed,especiallytheinformationfromvisualsensors.Thisisinvestigatedinthecontrol
modeldescribedbyOieetal.[80].Velocityseemstobethemosteﬀectiveinformation
type.Freemanetal.[34]alsofoundthevelocityinformationtobethemainprocessing
cueinvisualmotionperception.

4High-LevelPostureControl

4.2SensoryModels

Thesensorsnamedbeforecanbedescribedbydiﬀerentsensoryandcognitionmodels
whicharefurtherintegratedintothehigh-levelposturecontrolconcept.
Therearethreemaintypesofsensorswhichinﬂuencewalkingmovementsandwhichare
detailedfurther.First,thereisthevestibularsenselocatedintheinnerear.Thevestibu-
larsenseconsistsoftheotolithswhichsensethetranslationalaccelerationsofthehead
andthesemicircularcanalswhichsensetherotationalaccelerationsofthehead.Sec-
ond,thereisthevisualsense,theeyes.Herearetinalimageisproducedandprocessed.
Andthird,thereisthecomplexofsomatosensoricandproprioceptivesensorswhichcan-
notbenamednorlocatedasthiscomplexisnotcentralizedinoneorganandtheexact
mechanismsandinteractionsarenotcompletelyknown.Inthisthesisthesomatosensoric
perceptionisbasedonthedeﬁnitionof[150,185],perceptionofinﬂuencesoftheskine.g.
tactileinputs,temperatureinputsandnotonthebodysurfacebutdeeperstructureslike
theproprioceptivesensorycues.Theproprioceptionisthesensingofone’sownproperties.
Thismeanssensingoftherelativepositionoftwobodysegmentsortheappliedtorques
toajointwhicharerelatedtothemuscleandjointsensors.Inchapter3themuscular
stretchreceptorshavealreadybeenexplainedinsection3.4.2.Thissensoryfeedback
whichisdirectlycoupledtothelow-levelmuscularactivationloopbelongstothelocal
proprioceptivesensors.Thedirectjointandmusclefeedbackusedinthelow-levelmodel
hasalsoahigh-levelcomponentastheproprioceptioniscertainlyaninputtohigh-level
decisionsandintentions.Andtherearealsootherpartsoftheproprioceptiveperception
whicharemorehigh-levelandthereforeintegratedwithothersensorycues.Thismeans
thattherearedeterminedcommandsforawholebodypositionandbalancecontrol.
Inthisthesisonlytheproprioceptionisconsideredaccordingto[150].Neithertheso-
matosensorycuesnore.g.theauditivesensorycuesaremodeledandintegratedinthe
modelasitisconcernednottobethemostimportantinformationforposturecontrol.

4.2.1VestibularSense

Toexplainthecomponentsofthevestibularorganinasimplemanneritcanbesaid:the
vestibularorganconsistsofasymmetricpairoftwoorganspositionedsymmetricallyinthe
head.Theexactpositioncanbeseeninﬁgure(4.1).Eachvestibularorganconsistsofthe
threesemicircularcanalsandtheotoliths,maculaeandsacculae.Thesemicircularcanals
servetomeasuretherotationalaccelerationsandtheotolithstodetectthetranslatoricand
gravitationalaccelerations.Inastandingpositionthemovementsofaninversependulum
arerelativelysmallwhichleadstotheapproximationofthevestibularsensation,which
isinthefollowingonlyrotationalaccelerations.Theotolithsaresubjectofresearchin
literatureine.g.[31,76,75].Forthepresentedmodelonlythesemicircularcanalsare
deled.moThevestibularsemicircularcanalsareawellanalyzedsenseandtherearemanyexamples
inliteraturetonameonlyafew[44,166,81].

delsMoSensory4.2

Thesemicircularcanalsaccordingto[62,192,13]areorderedinanearlyorthogonal
positiontoeachotherseeﬁgure(4.1).Acanalisatubeﬁlledwithﬂuid(endolymphe).
Amembrane(cupula)whichisspannedacrossthetubecrosssectionismovedbyhead
acceleration.Finally,thismovementcausesintheaﬀerentnervesanactionpotential
whichencodestheheadvelocityandacceleration.Thesepotentialsaretransmittedto
thebrainstembyfrequencymodulationandrelativetimingoftheactionpotentials.

Figure4.1:Thevestibularorgan[13].

Thetransferfunctionforthesemicircularcanalsasfoundin[192]andin[13]isderived
fromaﬂuid-ﬁlledtorsionalpendulum.Theangleofcupulardeﬂectionxsccisdetermined
bythediﬀerenceofangularheadmovementxheadwhichisinaninversependulummodel
inidentheticalstoemicirculathearngulacanarl.systemThesumstateofxtorqsubtrauescwtehicdhbyisthethemoheadvemenacceletofrationendolymphemultiplieXde
bythemomentofinertiaisidenticaltothesumoftheresultingviscousandtheelastic
torques.Theviscoustorquesaredeﬁnedaccordingtotheﬂuidmechanicsofalaminar
toﬂowrqueinisathindeﬁnedtubewhicaccordinghretosultsapinatendulumorquemoprodelpowrtiohernalethetothetorqueﬂuidisvelopropcitoy.rtionaTheltelaosttheic

4High-LevelPostureControl

angulardeﬂectionofthemembranexscc.Sotheresultingmodelisgiveninequation4.1
x¨scc(t)=x¨(t)−K1∗x˙scc(t)−K2∗xscc(t)(4.1)
whereK1andK2aretheproportionalgainsfortheviscousandelastictorquesincombi-
nationwiththeinertia.Amoredetailedexplanationofthisequationisgivenin[192,13].
ThisformulaistransformedtoLaplacianformandthegainsK1andK2aretransformed
tothetimeconstantsTscc1andTscc2.WhereTscc1∗Tscc2=K12andTscc1+Tscc2=KK21.The
equationisasfollows:
yscc(s)Tscc1∗Tscc2∗s
x˙(s)=(Tscc1∗s+1)∗(Tscc2∗s+1)(4.2)
Equation4.2standsforthetransferfunctionofthesemicircularcanal,whereaccordingto
theothersensorymodelsx˙isthesystemorheadvelocityandysccisthesensedmembrane
(cupula)deﬂectionresultingfromheadmovement.Tsccaretwotimeconstantswhich
describethetorsionalpendulummodel.Againthissystemcanbegiveninstatespace
formandtheequationthereforeis:
x˙scc(t)=Ascc∗xscc(t)+Bscc∗uscc(t)(4.3)
yscc(t)=Cscc∗xscc(t)+Dscc∗uscc(t)
wherexsccisthestateofthevestibularorganandtheinputusccistheheadvelocityx˙.
ThematricesAscc,Bscc,Cscc,Dsccareconstantmatrices.Theexternalappliedvestibular
stimulationwhichisaddedviatheskintothenervoussignalisthereforenotincludedin
theBscctermbutinthefollowingtransferfunction.Thetransferfunctionfortheaﬀerent
nervesisapproximatedwithaﬁlterwithequationwhichisgivenaccordingtoGoldberg
]:44[al.etyvest(s)yvest(s)ta∗(tr∗s+1)
yscc(s)+GVS(s)=uvest(s)=gnata∗s+1(4.4)
wheretaisthetimeconstantforadaptationandtrforthehighfrequencybehaviorand
gnaisagainfactorofthevestibularnervetransferfunction.
˙xvest=Avest∗xvest(t)+Bvest∗uvest(t)(4.5)
yvest=Cvest∗xvest(t)+Dvest∗uvest(t)
wherexvestisthestateofthevestibularnerveandthenervesignaluvestconsistsofthe
naturalimpulseofthesemicircularcanalsandthegalvanicstimulationwhichisdirectly
appliedonthenerveoverelectrodespinnedtotheskin.

ceptionrioProp24.2.

Theandproftenoprionotceptioexactlynisatraceasenseble.notInloliteracatedtureinonethereoragarenmanbutydistrstudiesibutedforoprveropriothebceptivodye

010

delsMoSensory4.2

inﬂuencetothebodysystemforbalance,orientationandpositioncontrol[77,64,119,
121,117,82,199].In[180,97]theproprioceptivecueonhigh-levelisrepresentedbya
bandpasstransferfunction.Thisbandpass-ﬁltermodelaccordingtoKuo[97]isusedhere.
Theproprioceptivetransferfunctioncouplesthecompletebodypositione.g.angular
position,COM,COPorZMPpositionwiththesensedpositionandthesensoryoutputs.
HeretheequationinLaplaceandstatespaceformis:
yprop(s)=(Tsp∗s+1)(4.6)
x(s)(Tsp∗α∗s+1)
whichiswritteninstatespaceformas:
xprop(t)=Aprop∗xprop(t)+Bprop∗uprop(t)(4.7)
yprop(t)=Cprop∗xprop(t)+Dprop∗uprop(t)
withthestateofthebodyxandthestateoftheproprioceptivesensorysystemxprop.
ypropisthemeasurementoutputoftheproprioceptivecue,Tspisatimeconstant,αagain
constant.upropisaninputtotheproprioceptivesystemwhichisthebodystatexhere
theangularpositionofthebodycenterofmass,Aprop,Bprop,CpropandDpropareconstant
s.ricematAnotherproprioceptivesystemwhichneedstobeexplained,isthesensorysystemof
theeyemovements.Asthevisualsysteminformationisacombinationofretinalimage
information,proprioceptionandvestibularcoupling,thisisbestseenandstudiedincom-
pensatoryeyemovementstostabilizethegazee.g.onatarget.Thesearenon-guided
movementswhicharenotvoluntarybutreﬂexmovementsaccordingtoainternalcoupling
ofeyemotorcontrolwiththevestibularsysteme.g.inthevestibulooccularreﬂexVOR
[184].Smootheyepursuitwithsloweyemovementsareofinterestinthiswork.Sothe
proprioceptivesenseforsuchmovementsismodeled.Heretheeyemovementtransfer
functionismodeledasanestimationoftheeyevelocitywhichisgivenbytheeﬀerence
copyofthemotorcommand.
Anideaof[155]ofaeyepursuittransferfunctionisatime-delayedlowpassﬁlteredve-
locitysignalwithagain.Theearlierideaofavisualmodelof[154]wasalsotakenby
[97]torepresentthevisualsensorywithanintegratedmodelforgeneralvisualvelocity
perception.Inthisworktheinteractionbetweentheretinalimageandtheeyemovement
isstudiedwhichleadstotwoseparatemodelsforrepresentationofhumanvisualand
pursuitmovementperceptionwhichareintegratedinthesystemwhichrepresentsthe
visualmotionperception.
Forasmoothpursuiteyemovementtheeyeconsciouslyfollowsatarget.Theeyevelocity
dependsontheheadmovementandonthetargetvelocityu˙t.Ifthegazeisalwaysﬁxed
tothetargetitisassumed:

withconst=0.75...0.95.

yeye=(u˙t−x˙head)∗const

.8)(4

110

4High-LevelPostureControl

Theconstfactoralreadyconsidersacertaindegreeofinaccuracyandtimedelay.As
Niemeieretal.[135]mentions,thesereasonsfortheunderestimationofthesignalyeye
maybebecauseofthecalculationofthissignalwhichbasesonthecomparisonofan
eﬀerencecopywiththeproprioceptivesignal.Andofcoursethereisalwaysatimedelay
alsointhevisualprocessingcuewhichismainlythereasonforadelayedandtherefore
inaccuratepursuitmovement[137].Iftheeyemovementiscorrelatedtothesensedimage
velocitythereisanadditionaltransferfunctionwhichrepresentsthisvelocitysensation.
ThisisstudiedinRobinsonetal.[155]whereasimplelowpasscharacteristicwithdelay
isthesimplestinternalmodeloftheeyevelocitysignal.Equation4.8isexpandedbythe
lowpasscharacteristicswhichgivesthetransferfunctionusedinthefollowingwork:
1yeey(u˙t−x˙head)=const∗Tv∗s+1(4.9)
withTvisthetimeconstantofthelowpassﬁlter.Thisequationapproximateseyepursuit
movementswhicharebelowathresholdvelocityvalue.Ifthevelocityoracceleration
becomestohighasmoothpursuitisnolongerguaranteed.Theeyescanmovewithvery
quick,socalledsaccadic,eyemovements.Inthiscasethismodelwillnolongerrepresent
theeyemovements.Intheexperimentsitwasguaranteedthatonlysmootheyepursuit
movementsaretested.
Thestatespaceequationforthetransferfunctionaboveisgivenwith:
x˙eye(t)=Aeye∗xeye(t)+Beye∗ueye(t)
yeye(t)=Ceye∗x˙eye(t)(4.10)

SenseVisual34.2.

Thevisualsystem,theeyes,senselight,whichmeansthatimagesareprojectedtothe
retinawhichareprocessedforfurtherproperties.Iftheimagemovesontheretinathis
iscalledretinalsliporretinalvelocity.Therearemanyphenomenonsofvisualsensation.
Inofthethefosurrllowingoundingsonlyintherelasensingtiontoofthevelobcoitdyyoisranmoveimpmenorttanwilltbinforematioconsidered.nandMoveinﬂuencesments
opticstandingﬂow.andAssteppingfoundbyta[sks.46,47The]itmovalsoemencantbcanebeextraextrctedactedasaofcothembinatioretinalnofsliptoherteyhee
movementsignalandtheretinalsignalwhichtogetherrepresenttheinternalmeasurement
ofvelocity.Thedetectionofmotioncanbeachievedbytheretinalslipoftheprojected
eﬀeimagree.nceIfcoptheyaeyndescopursuemparedanreaobﬀjecerenttlythetosigthenalretinatolconimatrogelthemotioeynetogmotioettnheistakdiﬀerenceena.s
aThestatiodiﬀerencenaryretinarepreslenimagtsethewhicsenshisingproofducedmotion.byInexppursuingeraimenmotsinving[46ta]itrgetisisshopewnrcteivehatd
asmovingtarget.Butamovingretinalimagewhichisproducedbyeyemovementsis
perceivedasstationary[46].Whichleadstotheconclusionthatsubjectscanclearly
separatethebackgroundmovementfromtheobjectmovement.
Theclassicalmodeltorepresentandinterprethumanmotionperceptionduringcombined

210

MoSensory4.2dels

eyemovementyeyeandretinalimagemovementyretisaccordingtoHolstandMittelstaedt
[188,187]alinearcombinationofthosetwovelocities.ThiswasextendedbyFreemanand
Banks[35]bygainfactorstoexplaincertainphenomenonsofhoweyemovementinﬂuence
theperceivedvelocities.Thevelocitymeasurementisthediﬀerenceoftheampliﬁedretinal
yretandeyevelocityyeye.Thisleadstotheequationwhichinterpretsvisualperceived
velocityofmotionyvisasaweighteddiﬀerence:
yvis=gr∗yret−ge∗yeye(4.11)
withgainfactorsgrandgewithggre<1.Thismodeldoesnotreﬂectthemotionsensation
forhighervelocities.Forhighervelocitiestheperceptionbecomesclearlynonlinearand
thereisaneﬀectofsaturation[177,165].Thenonlinearrelationofperceivedvelocity
dependentoneyemovementandretinalvelocityisdeﬁnedbyTuranoandMassof[177]
whichisquasilinearnearthezerovelocityandasymptotictoamaximumvalueRm/2for
higherpositiveornegativevelocities.Herethemappingofrealeyeandretinalvelocities
totheinternalestimatedvelocitiesisnonlinearandsaturating.Theformuladerivedin
[177]torepresentthismappingisasfollows:
yvis=fr(x˙ret)−fe(x˙eye)(4.12)
1111
yvis=Rm∗1+exp(−gr∗x˙ret)−2−Rm∗(1+exp(−ge∗x˙eye−gi∗x˙ret)−2)
wherethegaingidescribestheinﬂuenceoftheretinalvelocityontheeyemovement.
Goltzetal.[47]showinexperimentsthattheperceptionofvelocitycannotberepresented
byasummationofretinalandeyevelocitybutthatthereisamultiplicativeterm.This
indicatesthatthereisadependenceofspatialstructureoftheretinalimageandtheeye
velocitysignal.Intheexperimentsof[47]thevelocityperceptionofanobjectinspaceis
representedbyanonlinearcombinationbetweenthedIretinalimageandtheeyevelocityfor
retinalimagepartswheretheilluminationgradientsdxareunidirectional.Theformula
is:dI+dIdI
yvis=dx∗dt+dx∗yeye(4.13)
whereIistheretinalimageintensitywhichhasaspatialderivationaccordingtothe
locationontheretinadIdxwhichrepresentsthedirectionofthemovementandatime
gradientdtdIwithwhichthisimagemovesalongtheretina.Theyeyeisthevelocityofthe
eyemovement.Ifthebrainknowsthespatialandthetimegradientitcancomputethe
imagevelocitydtdx=dxdI−1∗dtdIbuttheinverseofvectordxdIisnotuniquelydeﬁned.The
least-square-ﬁttedsolutionistakenwhichisaMoore-Penrosepseudoinverselabeledby
+.()TheretinalprocessingisdescribedinYangetal.[197]asanapproximatedlowpassﬁlter.
Yangsaysthatthetransferfunctionfromlightenteringthehumaneyewhichisafterwards
samplingthecontinuousspatialvariationbyseveralcelltypesanditsresamplingcanbe
describedbyalowpassﬁltering.Theprimaryprocessingofretinalinformationyretis

310

4High-LevelPostureControl

thereforedescribedbythefollowingtransferfunction:
1yetruret=1+tr∗s(4.14)
whereuretisthespatialvariationoflightontheretina.Hereonlythevelocitycomponent
isofinterest.Thereforeuretisthevelocityofthevisualstimuliwhichisprocessedon
theretina.Inthenextparagraphitisexplainedthatthisstimuliisacombinationofeye
vareelocityrepresenyeyetedandbyubexternaandlwhicinputshhatosthealreaeydyeasbeene.g.premosenvtedemenintsothefcthelassenicavlironmenequattionwhic4.11h
torepresentvelocityperception.Thetimeconstantoftheretinalﬁlteringistr.
Instatespacenotationthistransferfunctionis:
x˙ret(t)=Aret∗xret(t)+Bret∗uret(t)(4.15)
yret(t)=Cret∗xret(t)+Dret∗uret(t)

.15)(4

Thedeviatchaionofracteristicsthevisouaflpvisualerceptionvelocitylikpelihooerceptiodisnpropfoundorintiona[l167to]aarelogthatarithmicthevstaelondarcityd
function.Thepriorprobabilitydecreaseswithavelocitypowerlawwhichbringsabout
saturationeﬀectsforhighvelocities.
Thisleadstothefollowingmodel,representingthesensingofvisualmotion,whichwas
developedfortheposturalmodelinthisthesis.Themeasuredvisualmotionistheveloc-
ityyviswhichisingeneralasumofanestimationoftheretinalvelocityyretandtheeye
velocityyeye.Theproportionalitytotherealvelocityisalogarithmicfunctionwhichis
likderivelihoedodfromhasthelogaWebrithmicer-Feccharahnerlawcteristics.Thisandcotherrelatesfunctiotontheof[ﬁnding177]ofhas[16a7,19similar7]cthahatrac-the
teristic.TheWeber-Fechnerlawisexplainedbelowinthenextsubsectioninmoredetail.
Themeasuredmotionbythevisualsensorysystemiscalculatedasfollows:
yvis=cvis(yret,yeye)=(4.16)
=(sign(yret)∗ln(|yret|/xr0+1)−(sign(yeye)∗ln(|yeye|/xe0+1)
wheretherepresentationofeyeandretinalmovementiseachcorrelatedtoathresh-
boldecaxruse0,xte0he.yaThoseretoothrsemasholl.ldsInstand[167f]orforthemoretinavlememonvtsemewhicnththiscanvjausluetnotwasbegivenmeasuredwith
0.3[rithmicdeg/secwhic].hForleadslartogeravsatelouratiocitiesntheeﬀecpt.erceivTheedfavsteloercittheydomoesvenotmentrasistheelesselinearlyrisbutthelogainga-
oftheperceivedmovementwhichalsomeansthattheinﬂuenceofaperceivedmovement
doesincreaseﬁrstnearlylinearlyandthenlessanlesserwithincreasingvelocity.Theﬁnal
modelrepresentingperceivedvisualvelocityyvisisthereforedeterminedbythediﬀerence
oftheeyevelocityandtheretinalimagevelocityyretwhichisinﬂuencedbytheexternal
visualscenewhichwillberepresentedasavisualbackgroundwithdeﬁnedvelocityub.
Themeasurementscaleofthemodelisalogarithmicone.

410

Weber-FechnerLaw

delsMoSensory4.2

FeThechnerWebtoer-FdescecribhnerelathewrewlaastionintrboetwduceeendbtheyErobnsjtectiveHeinricphyhsicaWleberandandsubGjectivustaevperTheoceivedord
stimuli.ThelawconsistsoftheWeberlawwhichsaysthatthejustnoticablechangeof
astimulusinrelationtotheactualstimulusisconstant:
dxp=k∗dxs
xsthewherecdxhangepisoftthehejusstimtulus.noticableThismeansdiﬀerencesthatptheerceivrelaed,tionxsisofthethestimactualulusstcimhangulus,eandrelateddxstois
theactualstimulusvalueatthatinstant,multipliedbyaconstantfactork,isproportional
tothejustnoticablediﬀerence.Fechnerfoundthatthisrelationislogarithmic.The
Ferelatchnerionlafactowrkextendsisindepthiselandenwtbofyintheteagratctualion,stimunderulusxthethisassumptioleadstonthethatWtebheer-coFecnsthneanrt
s:wlaxxp=k∗lnxs+c=k∗lnxs−k∗lnxs0=k∗lnxss0(4.17)
wherecistheintegrationconstantwhichalsocanberepresentedbyalogarithmicxs0
multipliedbyaconstantfactor,xs0istheminimalthresholdofperception.
TheWeber-Fechnerlawisappliedtomanysensesasthehapticsense,thetastesense
orsensvaerytioncowhicmmonlyhisproforpothertiosenalnsingtotohefloglightarithmintensitofythe.Tsthimisulus.descrInibesthisthetheintsisenstheityWofebter-he
Fechnerlawisappliedtorepresentvisualvelocityperception.Thesensedvisualmovement
yvTheishassensoraloghasartheithmiclogcaritharahmiccteristicpropertywhicthathformeanslowterhattsensorhevaluemeasurementhetratnsisferlogafunctrithmic.ionis
nearlylinearbutforhighersensorvaluesthetransferfunctionbecomesclearlynonlinear.
Forveryhighvelocitiesthisisnotapplicablebutthevelocityrangeappliedinthisthesis
referstonormaleverydayvisualmotionsofmovingobjectswhichdonotexceedthese
nges.ra

Simoncellietal.[167]proposeanoptimalobserverconstructtorepresentthevisualspeed
perception.Theprecisenoisecharacteristicsareunknown.AlsoRaoetal.[148]found
goodbehaviorofoptimalestimationapproachesforrepresentingthevisualperception.
Theinformasttioatisticanlandtheobservermoisdelnaormedprioarskano’bestwledgeguess’whicofhisthewcitedorldbyfor[167the]afromctual[186].sensoryIn
thisthesisthevisualsensorycueastheothersensorycuesareintegratedinanoptimal
estimatortheKalmanﬁlterwhichisproposedinthenextsection4.3.1.TheKalmanﬁlter
iscuesalinearwhicheisstimatodetailedrandinthashetobKalmaeextnendeddescriptiotonrepresinseentctiosucnh4.3no.3.nlinearvisualperception

105

4High-LevelPostureControl

4.3EstimationforPostureControl

Thestatisticalestimation,theKalmanﬁlter,isdetailedinthefollowingandapplied
totheposturecontrol.Thisdemonstratestheintegrationofsensoryinformationinto
posturecontrol.Thesensorintegrationisshownforlinearvisualsensorymodelsand
pnonlineaosturerconwhictrolhisleadderivtoedansextuppeonsiortingnofandthecompleKalmanmentﬁlter.ingtAhenegaproptiveertiesofeedbacfthekloopKalmanfor
stateestimation.Thesumofallthisistheposturecontrolmodel,developedtosimulate
sensorimotordependenciesforthetaskofkeepingthebodyinbalance.

4.3.1TheKalman-FilterTheory

TosumupthefunctioningoftheKalmanﬁltershortlyitcanbesaid:theKalmanﬁlter
estimatesthenextsystemstatesbyusingaprioriknowledge.Theestimatedstatedepends
ontheconditionalprobabilitydensityfunction.Thisaprioriknowledgeiscompletedby
usingthemeasurementsofthesystemstatesasinnovativeinformation.Additionally
theinnovationiscomparedbysubtractiontotheaprioriexpectedmeasurements.The
resultingdiﬀerenceiscalledresidual.TheresidualisweightedbytheKalmangainand
addedtotheaprioriknowledgeaboutthesystemtoreceivetheﬁnalestimate.Theneeded
aprioriinformationisthesystemandsensorydynamics,thenoisestatisticsofmodeland
measurements,theinitialvaluesforsystemstatesanderrorstatistics.AllKalmanFilter
equationscanbeappliedtocontinuoustime-variantsignals(KalmanBucyFilter)or
alsotodiscretesignals.InthefollowingthediscreteKalmanﬁlterisintroduced.The
quantizationisdoneinstepsk.Soitisdetermined:t(τ)=tkandthenextquantization
stepist(τ+δ)=tk+1withδisthequantizationstepsize.Thecontinuous-timematrices
havetobediscretizedwithanappropriatemethode.g.anEulermethod.Thebiological
systemismoreacontinuoustimesystembuttheimplementationisdiscrete.
TheKalmanestimationﬁlterbasesonalinearmodelrepresentationofthesystem:
xk+1=A∗xk+B∗uk+W∗wk(4.18)

xAk::vinectoterarofctionbsystemetweenstatestheoflastthestatmoedelxkatandtimetheknextstepxk+1
uk:vectorofexternalinputstothesystem
B:enﬁlterviromatnmenrixttoontherepresensystemttheinteractionofexternalinﬂuencese.g.fromthe
wWk::vNoiseectorogainfstomactrixhasticwhicnohiseﬁlterwhicshtheiswhitenoiseGaeﬀecustssianonnotheisesystwithemastameatesnofzero
Table4.1:Kalmanvectorsandmatricesforthestatemodel

Theoutputofthesysteminformofmeasurementsisrepresentedbyalinearrelation.It

610

4.3EstimationforPostureControl

consistsofthemeasurementsandtheexternalinputsandnoiseattimestepkby:

yk=C∗xk+D∗uk+vk

.19)(4

yCk::vﬁlterectormatofrixmeawsurhicehmenrelattsesattimsystemekstateswithmeasurementoutput
uk:vectorofexternalinputstothemeasurementsystem
D:ﬁltermatrixwhichrepresentstheinteractionofexternalinﬂuenceswiththe
tsmeasuremenvk:vectorofstochasticnoisewhichiswhiteGaussiannoisewithamedianofzero
Table4.2:Kalmanvectorsandmatricesforthesensorymeasurements

Thestatisticalerrorsourcesasnoisearemodeledandtheerrorbetweenestimationand
realmeasuredvaluesisupdatedwithatimevaryinggain,theKalmangain,toreceive
thenextestimation.Thecharacteristicsofthenoisemodelsarealwayswhite,Gaussian
andzeromean.Thecovariancesofthenoiseisdescribedwith:
E{wk}=E{vk}=0(4.20)

E{wkwjT}=QE{vkvjT}=R(4.21)
E{vkwkT}=0(4.22)
withEistheexpectationandE{x}istheexpectedvalueofx,jisaskaindexoftime.
QandRarethenoisecovariancematricesofthesystemandthemeasurementnoise.In
equation4.20thezero-meanofthenoiseprocessispresented.Equation4.21determines
thecovariancesofnoiseandinequation4.22itisshownthatthenoiseprocessesofsystem
andmeasurementsareuncorrelated.TheerrorcovariancematricesQandRhavetobe
determinedforthemodeltodeterminethereliabilityofe.g.ameasurement.Thebigger
asingleRvalueis,thelessasinglemeasurementisweightedfortheinnovationofnew
estimation,becausethereliabilityislowandviceversa.TheKalmanﬁlterminimizes
theexpectederrorbetweenestimationandrealstatebyminimizingtheerrorcovariance
matrixP.TominimizethiserrorcovarianceP,theRiccatiequationisused.Thediscrete
is:tionequaiRiccatPk=A∗Pk−1∗AT+W∗Q∗WT−Pk−1∗CT∗(C∗Pk−1∗CT+R)−1(4.23)
ThiscalculationoftheerrorcovariancematrixPisrealizedintwosteps,theprediction
(equation4.24)andthecorrection(equation4.25)asitwascalledby[190]whichisone
aprioriestimationandafterwardtheaposterioriupdate.
Pk−=A∗Pk−1∗AT+W∗Q∗WT(4.24)

107

4High-LevelPostureControl

Figure4.2:moThedeltwoandphasestheupofdaatingKaandlmancorrﬁlter:ectionthebyprethedictioinputnofaccoractuadingltosensorytheintdaerta.nal

and

Pk+=(I−Kk∗C)∗Pk−

.25)(4

ThisminimizationistransferredtotheoptimalKalmangainwhichampliﬁesthemea-
surementresidualtoupdatetheestimation.ThisoptimalKalmangainKiscalculated
bythemultiplicationoftheerrorcovariancewiththemeasurementandthemeasurement
noisecovariancematrix.TheKalmangainminimizestheaposteriorierrorcovariancePk+
with:Pk+=E{(xk−xˆk)(xk−xˆk)T)}(4.26)
SotheKalmangainhasthefollowingformula:
Kk=Pk−∗CT∗(C∗Pk−∗CT+R)−1(4.27)
Theestimationofthenextsystemstatexˆistherefore:

xˆk=xˆk−1+Kk∗(yk−C∗xˆk−1)withxˆk−1=A∗xk−1+B∗uk−1

810

.27)(4

.28)(4

4.3EstimationforPostureControl

TheschematicoftheKalmanequationscanbeseeninﬁgure4.3.1.TheKalmanﬁlter
predictsasystemwithanunderlyingstochasticprocesswhichiscaseforsystemswhere
onlynoisemeasurementsoftherealstatesofthesystemareavailable.Thediscrete
Kalmanﬁlterisarecursivestatisticestimationmethodwhichminimizesanoptimization
criteriontheerrorcovariancematrixP.Theaprioriestimationxˆdependonallapriori
knownmeasurements.Therecursiononlytakestheprecedingestimationvaluewhich
isalreadyameanvalueoverallpastestimationvalues.ThisisaMarkovprocessas
onlythelastvalueisdirectlyusedforthecalculations.TheKalmangainisrelatedto
thecovariancesofthemeasurementmodelC,theexpectederrorcovariancesPandthe
measurementnoisecovariancesR.Thiscanbeseeninequation4.27.Ifthesystem
andmeasurementmodelsdonotmodelthereality,themeasuredstatesdivergefrom
therealstates.Thisisinterpretedasmeasurementnoise.Theestimationprocedure
onlyslowlyconvergesinthiscasebecausethesystemnoisecovariancematrixQissmall.
IfQisincreasedtheconvergenceisfasterbutthesystemismoresensitivetosystem
noise.So,theestimationqualitybecomeslessandsystemerrorsarelesslikelytobe
detected.Ifthemeasurementnoisecovariancesaremodeledthisrepresentsthereliability
ofthemeasurements,ifRgetssmallerthemeasurementsaretakentobemorereliable.
ThereforetheKalmangainweightsthemeasurementsmore.

4.3.2ApplicationoftheKalmanFiltertotheStanceModel

Thepresentedsensorymodelscannowbeintegratedinaposturecontrolmodel.The
posturewhichshallbecontrolledistheuprightstance.Themechanicsofstancecan
berepresentedbyaninvertedpendulum.Thisrepresentationofastandingpositionis
oftenusedinliteraturease.g.in[86,138,120,136,179].Inthisthesisthefrontal-plane
mechanicsareanalyzedindetail.Theinversependulummechanicscanbeusedforboth
stancemodelsinsagittalandinfrontalplane.Thecharacteristicsofthependulummodel
willalwaysdependonthespecialstancepositionconstraintsforthefeet.Sothereisof
courseadiﬀerenceifstanceisanalyzedwithfeetsidebysideinanarrowpositionorina
widepositionorifthestancepositionisevenatandemfootposition(onefootisplaced
infrontoftheotherstandingononeline).Inthefollowingthestancepositionisavery
narrowpositionfeetsidebysideandthelateralstanceswayisanalyzed.Sotheinverse
pendulummodelisidenticaltothemechanicsofthefrontal-planemodelofsection2.3
ifthephaseisdoublesupportandthefeetpositionisidenticalwhichmeansonepoint.
Forthesteppingmovementsthedoublesupportphasewasalwaysmodeledasadiscrete
eventwhichoccursinstantaneouslybetweentwoswingphases.Nowthedoublesupport
phaselastforthewholestance.Theequationfortheinvertedpendulummechanicsis
explainedinsection2.2.1inequation2.2.
Themovementofthemechanicsismeasuredbythesensors.Theproprioceptionsen-
sorsmeasuretheswayangleΦ,thevestibularsensormeasuresthechangeoftheangular
velocityΦ˙andfortheeyemovement,thevisualsensormeasuresthevelocitydiﬀerence
betweenthevisualworldandtheselfmotion.Thisinformationmeasuredbythesensors
isnowintegratedbythebrainusingastatisticalestimationusingmodelknowledgeac-

910

4High-LevelPostureControl

Figure4.3:Anoverviewoftheposturecontrolsystemwithmechanics,sensors,statistical
estimationandintegrationandthefeedbackcontroller.

cordingtoexperiencesandtheseactualmeasurements.Theintegrationresultsinmotor
temcommaofnds,mecherehanics,mearepresensurtedemeasnts,ankinletetgratorques,ionandthatstacommabilizendtgehenerstaationdingnisbshoodywn.inThisﬁgursys-e
.4.3

Thepartoftheintegrationisacomplextaskofthebrainandonlylittleisknownabout
theinternalprocesses.Forreasonsalreadynamedintheintroductionofthischapter
andbecauseitiswellknowninliteraturetheintegrationinthisthesisisimplemented
asanestimationﬁlterwithmodelknowledge.TheestimationisrealizedbyaKalman
ﬁlter.Thereforethemodelknowledgeofthesystemwhichisrepresentedbyaninverse
pendulumandthesensorswhicharedescribedandmodeledasin4.2accordingtotheir
physicalandcognitiveproperties.Thismodelknowledgeisusedtogetherwiththesensory
measurementsinaKalmanﬁltertoestimatethesystemandsensorystatesapriori.This
estimationislikeanexpectationandaprioriknowledgeofnotyetreceivedandprocessed
systeminformationonbaseofstatisticalknowledge.Thecontinuoussensorymodels
aretransformedinastatespacerepresentationasrequiredfortheKalmanﬁlterand
discretizedbytheForwardEulerMethodwiththefollowingequation:

011

A=I+δ∗FandB=δ∗Ξ

).29(4

4.3EstimationforPostureControl

whereIistheidentitymatrixandδisthetimestepbetweent(τ)=tkandt(τ+δ)=tk+1.
ThestatespacetransformationwassimulatedinMATLABwiththetf2ssfunction.The
sensorymeasurementsarenotonlyinﬂuencedbythesystemstatesandtheexternalin-
ﬂuencesbutbythelastinternalstateastheyhaveintegratingorderivatingproperties.
ThisleadstotherepresentationoftheKalmansystemstateswhichincludestheme-
chanicalstatesaswellasthesensorystates.ThestatevectoroftheKalmanﬁlterto
estimatethesystemisx=(x,˙x,xprop,xscc,x˙scc,xvest,xeye,xvis)whichis
xthemechanicalangularstateofthebody,x˙itsvelocity,xpropthestateofthepropri-
oceptivesensor,xscc,x˙scc,xvestthevestibularstatestosenseangularaccelerations,
xeyetheproprioceptivestateoftheeyemovementandxvisthevisualsensorystateof
theretinalvelocity.Themeasuredvaluesbythefoursensorymodelsarethevector
y=(yprop,yscc,yeye,yvis)whichistheproprioceptivelymeasuredposition,theac-
celerationoftherollmovementsensedbythevestibularsemicircularcanals,thesensed
eyevelocityandthemeasuredretinalvelocitywhichgivesaretinalsignalandisprocessed
forvisualvelocitymeasurement.Asexternalinputstothesystemfromtheenvironment
thevectorisu=(uc,ut,ub,uvest)whicharethecorrectivetorqueappliedtothe
jointsofthemechanics,thetargetvelocityofthetargetwhichisﬁxatedbytheeyes,the
backgroundvelocitywhichisvisuallysensedandthegalvanicstimuliwhichisappliedto
thevestibularnerve.

Fortheinversependulummodeltheuprightstanceiscontrolledbythecorrectivetorque
appliedtotheanklejointwhichcanbeseeninﬁgure2.2withuc=τa.Theeyesand
thevestibularorganaresituatedintheheadandarethereforemodeledatthetopofthe
inversependulum.Theangleandangularvelocityofthependulumarethereforethesame
asmeasuredbythesensors.Asstimulationstothesystemdiﬀerentsignalsareapplied.
Thevisualﬁxationtargetisamovingpointwhichisﬁxatedwiththeeyessoapursuit
movementistheresult.Theequationforthisrelationwasgivenin4.9.Wheretheinput
utisthevelocityofthepursuedtargetandtheoutputistheeyevelocityyeye.Theretinal
imageisthecombinationofthediﬀerentvisibleobjectmovements.Inthisworkthere
isusedastationaryormovingbackgroundandapursuedtargetwhichwasalsomoved
orstationary.Themodelforsensoryinformationprocessingforvisionwasderivedin
section4.2.Theinputisthebackgroundmovementubandtheeyemovementyeyewhich
arecombinedtotheoutputyvis.Inthelinearcasethevisualsensorycueisdetermined
byequation4.11inthenonlinearcasebyequation4.16.Finally,thevestibularsensecan
bestimulatedartiﬁciallybyanexternallyappliedstimulustothevestibularsystemuvest.
Thisexternallyappliedstimulusissummeduptothevestibularsignaloftherotational
acceleration.Thiswasexplainedinequation4.3and4.5.
ThesystemmodelandsensorymodelsareintegratedintheKalmanﬁlterinthefollowing.
Allthestatesareintegratedinthestatevectorx,thesysteminputsarethevectoruand
themeasurementsofasinglesensorysystemarerepresentedbyvectory.Thegeneral
systemequationfortheKalmanﬁlteristherefore:
xk+1=A∗xk+B∗uk(4.30)

111

4High-LevelPostureControl

andthemeasurementsystemis:
yk=C∗xk+D∗uk(4.31)
Thesinglematricesofequation4.30and4.31areasfollows.ThestatematrixAofall
is:sestat

Amech00000
BpropAprop0000
B0A000
A=Bvest∗sccDscc0Bvest∗sccCvestAvest00(4.32)
−Beye000Aeye0
00000Avis
Thematrixtoapplytheexternalinputstothemechanicalsystemandthesensorsis
mechmatrixBwith:B000
0000
B=000Bvest(4.33)
0Beye00
0−BvisBvis0
ForthemeasurementmatrixCthemeasurementtermsofallsensorsareintegratedin
onematrixwhichis:
DpropCprop000
eeyeeyC=−DDvest00Cv0estC000(4.34)
0000Cvis
TheexternalinputtothemeasurementsystemismappedbymatrixD:
0000
eeyD=00D000Dv0est(4.35)
0−DvisDvis0

.35)(4

Thisinputssttoatethespacesystemomdelandoftitshesystemmeasuremenandts.itsseWitnsohrsthisdescribsystemesthemodelinﬂuencethefolloofthewingresexternalults
aresimulatedandcomparedtoexperimentallyfoundcausalities.Anoverviewofthe
systemandKalmanﬁltercombinationcanbeseeninﬁgure4.4.Thisﬁgureshowsthat
thesensorysystemmeasuresthesystemstateswhichistheinputfortheKalmanﬁlter
innovation,theresidual.TheoutputoftheKalmanﬁlteristheestimationofthesystem
statesxˆ.

112

eFigur

:4.4

4.3

EstimationforPostureControl

AschematicoverviewoftheKalmanﬁltercomponentsincombinationwith
systemandsensors.Thediscreteestimationofthestatexwithmeasurement
correctionbythesensoryoutput.

311

4High-LevelPostureControl

4.3.3ExtendedKalmanforNonlinearSensoryModels

Asherewloasgaalreadyrithmicallymen.tThisionedleaindsstoectiothenno4.2thenlinearvisualmeasuremensensorytcuefunctioismon4.1deled6forthenonlinearvisualyl,
perceptionofvelocity.Inthelinearcasethestatespacerepresentationofthevisual
cueisdescribedbythelowpassretinalprocessingofequation4.14or4.15andthelinear
summationoftheeyemovementwiththeenvironmentalmovementgivenbyequation4.11
senswhicohryleacdsuestothetheKamalmantricesﬁlterAvis,isBvis,CextendedvisandinDthevismentiomeasuremennedatbovematr.Fix.ortheThenonlinegeneralar
isprothecedurelinearforizaantionaextendedroundKathelmanapriorﬁlteriisstatheteestimalinearization.tionItaroisunddoneawboryckingapoinlculatingt.tThishe
Jacobianattheaprioristateestimation.ForanonlinearmeasurementfunctionCthis
is:∂∂xc11∂∂xc21...∂∂xcn1
x∂∂C=......(4.36)
xˆ∂∂cx1m∂∂cx2m...∂∂xcnmˆx
C∂ThiscalculatelinearizedtheusualmeasuremenKalmantequatresultsionsinfortheKalmaJaconbianerrorcomatrixva∂riance,xwhicgainhcanandbeestimausedtionto
asdescribedabove.Allfunctionsarefurthermodeledlinearlyexceptthevisualvelocity
measurement.Thereforethevelocityperceptionislinearizedaroundtheestimatedve-
locityvalue.Incaseofthevisuallogarithmicfunctioncvis(yret,yeye)theJacobianmatrix
Cvisiscalculatedasfollows:
∂xxˆxˆ
∂cvis=∂∂xcveyise∂∂xcrviets=Csigeyne(∗xˆxˆeyeyee)∗+Cxeey0eCsigrnet(∗xˆxˆrretet)+∗Cxrr0et=c1visc2vis(4.37)
ThisequationleadstoamodiﬁedmeasurementmatrixCofthesensorymodelwith:
DpropCprop000
C=−DDveyeste00Cv0estC0eye00(4.38)
000c1visc2vis
Inmenthetmafollotrixwingwhereathesnothenlinearlinearposptouresturecontconroltrmooldelmodiselgenerausestedthemawithtrixthisgsivenensorinequameasure-tion
.44.3

eNoisofInﬂuence

TheKalmanestimationmodeldeﬁnesnoisecovariancesσqandσrandtheerrorcovariance
matricesQandRwhicharedeﬁnedinequation4.21.Theydeﬁnethereliabilityofthe
systemandmeasurements.Especiallythemeasurementcovariancescanbeadaptedto
showdiﬀerentsensoryweightings.Oieetal.arguein[136]thatiftwoormoresensors
measurethesamesystemstatethisredundancyleadstoaweightingofallofthesesensory

411

4.3EstimationforPostureControl

outmeanputs.valueAwweighouldtingbecolessuldbnoisyeathameanntvhealueosinglefallsensorymeasuremenoutputs.tsoftheThecsameovariastancete.sThisare
arereducedgivenarelatedhigtherothenreliabilitumbyerofandmeasosurtheements.measuremenThistsbringsareawbeoiguthtedthatmortheemeainsturheemomendetsl
moanddeltoinﬂuencebestitresemmoreble.hisKiemelexp[13erimen6]taalsollyfounddeﬁnesthetransfervaluesfunctiooftnshe.Acovacoriansncesequenceinohisf
thothisseofresemthepblanceositisionthatmeathesuremecovnats.riancesofthevelocitymeasurementsissmallerthan

Fortheproposedmodelthemeasurementnoisecovariancematrixischosenasfollows.
ThegeneralmeasurementcovarianceσRisweighteddiﬀerentlytothesinglemeasurement.
Thesumofoverallweightsisalwaysone:
σ/n000
R0cσR/nc00
R=00wc∗σR/nc0(4.39)
000σR/(nc∗wc)
withformanction=ndelivumberederofbyavtheailaretbleinalsensorysystemcueswhicandhwcancisbethetheasamebstractioasntheofeyeamounmotvofemenin-t
orconﬂicting.So,ifthereismuchretinalinformationavailablethiscausesadecreaseof
tiothenerroprorccoess.varOniancetheandcontrathereforryifethetheeyveisuamolvseemennsotryandcuetishemoreretinalrelevimaantgeforhavetheoppestimaosing-
informationtheuseofthevisualsensorycueislessreliablesotheerrorcovariancein-
creasesandfortheestimationthiscuehaslessinﬂuence.Thisisespeciallyimportantif
thesensorycuesofeyemovementandretinalimagehavetobeintegratedtogetagood
estimationfortheself-motion.

4.3.4OptimalLinearQuadraticRegulator

Thefeedbackoftheestimatedvaluestothecorrectivetorqueappliedtothemechanics
isrealizedbyafeedbackloop.ThisfeedbackisasimplePD-feedbackcontrollerwhich
basesonanoptimalitycriterion.
Thesystemofapendulumcanbeapproximatedbyalinearsystemmodelforsmallangles.
TheKalmanestimatorestimatesallthesystemstatesofthesystem.Thisisthebasisfor
anoptimalcontroller.Theoptimalcontrollerisafeedbackcontrollerwhichfeedsback
allthesystemstateswhichareoptimallyweighted.Thisoptimalityisdeﬁnedduetoan
optimalitycriterionorperformancecriterionwhichcanbe(1)theminimizationofenergy,
(2)regulationofthesystemoutputwithminimizationofthedistancetoadesiredoutput
value,(3)time-basedminimizationoftransitionsorsomemoregeneralcriteriaas(4)
theLagrangiancriterionorthe(5)Mayerschescriterionaccordingto[36].Forthelinear
quadraticregulatorLQRthiscriterionJminimizesthequadraticperformancecriterion

511

4High-LevelPostureControl

whichiscalculatedwiththefollowingequation:
TJ=1∗x(T)T∗S∗x(T)+1[x(t)T∗Q(t)∗x(t)+u(t)T∗R(t)∗u(t)]dt(4.40)
22t0withx=x−xewhichisthedeviationofthestancestatexfromthedesiredstatein
positionandvelocity.Fortheinversependulumstancemodelthesystemstatex=(Φ).u
isthecorrectivetorqueappliedtotheinversependulumwhichshallalsobeminimized.In
thepresentmodelaccordingto[36]theenergydeﬁnedbytheinputtothesystemandthe
systemstatedeviationaccordingtoanintendedpositionareoptimized.Theweighting
matrixQisasymmetricpositivesemideﬁnitematrixandthematrixRissymmetric
positivedeﬁnite.Sfortheweightingoftheterminalpenaltycostisapositivesemideﬁnite
matrix.Theﬁnalstatewhichshallbereachedisnormallyneverachievedbecauseof
inaccuracyofthesensors,themodeletc..Thereforeintheperformancecriteriontheﬁnal
stateisweightedbyStoreceiveaminimaldeviation.Wherethematricescanbe,butdo
nothavetobe,time-dependent.TominimizeJtheRicattiequationisused.Thisusageof
theRicattiequationresemblestheoptimizationintheKalmanFilter.Optimalfeedback
controlisoftencombinedwiththeKalmanﬁlter[86,96,179,162,36].AndtheKalman
ﬁlterprovidesallstatesofthesysteminanestimationwhichinanaturalsystemare
normallynotcompletelymeasuredbysensorsandthereforenotavailableforanoptimal
control.Afeedbackofthefullstatevectorofthesystemisrequiredforoptimalcontrol.
TheRicattiequationisgivenbyequation4.23whichistheterm:
U˙=U∗B+R−1∗BT∗U−U∗A−A∗U−Q
whichissolvedforU.Buttosolvetheequationtheinitialconditionhastobedeﬁned
U(t0)=S[36].Theresultingfeedbackgainisgivenbytheterm:
u=−K∗xwithK=R−1∗BT∗U(4.41)
x˙Thefeedbackoftheoptimallinearregulatoristheweightedangularpositionvectorof
systemstatesxinanegativefeedbackloop.ThegainKisoptimizedaccordingtothe
performancecriterionwhichisminimizedbytheRicattiequation.Thechoiceofthe
matrixR=B∗IastheapplicationofthetorquetothesystemisdeﬁnedbyBand
thefurtherapplicationisduetolinearfactorsrepresentedbytheidentitymatrixI[96].
ThematrixQdeﬁnestheregulationperformanceandRthecontroleﬀort.Thematrix
QisweightedrelativelytoRbyaweightingfactor.ThesinglematrixelementsofQare
reducedtotheelementoftheoriginasproposedinKuo[96,97].Thecostfunctionswhich
arederivedarethreefunctions.Theﬁrstistheminimizationoftheangularpositionsfrom
thezeropositiontheverticalposition.Thisleadsto:
c1=(x∗xT)2Q1=0I00(4.42)

611

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

whereIistheidentitymatrixandxangularstatesofthesystem.Second,theangular
velocityshouldbeminimized:
c2=(x˙∗x˙T)2Q2=00(4.43)
I0wherex˙aretheangularvelocitystatesofthesystem.Andthirdthepositionofthecenter
ofmassshouldbeinahorizontalpositionwhichiswithinthesupportareasaplusasmall
adjacentextraareaδawhichstandsforthenon-criticalswayarea.Thissupportareais
fortheinvertedpendulumjust−δa<sa<δa.Forthefrontal-planemodelthisisdeﬁned
astheareabetweenthestancelegandtheswinghipjointinthedirectionofthex-axis
plustheadjacentarea.Thisleadstothecostfunction:

2c3=(xcom∗x)2Q3=xcom0(4.44)
00

wherexTcomisavectoroftheinﬂuenceontheCOMofthesinglemechanicalposition
states.Thisdependsonthemechanicalmodel.
Thiscontrolproblemisappliedtotheinversependulumaswellaslatertothemechanical
frontal-planemodelwiththreelinks.Theinversependulummodelasdeﬁnedin2.2and
ﬁgure2.2hasthesameCOMpositionasthepositionofthesinglemassofthependulum.
So,adeviationfromtheangularpositionandtheCOMisthesame.Thismeansthat
anadditionalCOMpenaltycostfunctionislikedoublingtheangulardeviationpenalty
function.ThisleadstothefollowingmatrixQ:
Q=wQR∗(µ1∗Q1+µ2∗Q2+µ3∗Q3)(4.45)
withwQRisapositiveweightingfactorofthematrixQinrelationtomatrixRandthe
factorsµaretheweightingfactorsofthesinglecostfunctionstoeachother.Asdeﬁned
inKuo[96]thesumoftheseweightingfactorsshallbeµ1+µ2+µ3=1toguaranteenot
toinﬂuencetheintermatrixweightingwQRwiththisfactor.Fortheinversependulum
03/2modelthematrixisdeﬁnedas:Q=wRQ∗01/3becauseallthreecostfunctions
havebeenweightedequally.

4.4ExperimentallyFoundInﬂuencesofSensoryCueson
olControstureP

Psensosturoryeconinformatrolmotion.delsThewhicthruenesusestheandKaaccuralmancyﬁltofertrhisepresmecenhatnisinmtegracantiononlymecbehavnisalidatmseodf
bandypstanceostureandrespponseerceptionexpexperimenterimenswhictshwithconcernsubjects.visual,Thersomaeisatosensoricwiderangeand/oforpverceptionestibular

711

elHigh-Lev

eFigur

811

:4.5

ostureP

The

oltrCon

lmanaK

ltronco

orestimat

the

system

via

binatiocomn

coerrectiv

with

input

rollertcon

eth

kfeedbac

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

informationprocessing.Thoseperceptioncueshaveofcourseadelayintimebetweenthe
realsensoryimpressionandtheuseoftheinformationforamotorcommand.Thisdelay
isgiveninliteraturewithvaryingvaluese.g.by[180]withabout100msande.g.by[142]
withupto200ms.Inthefollowingespeciallythevisualcueincludingproprioceptiveeye
movementinformationinsection4.4.2andthevestibularprocessingcueareinsection
4.4.3areregarded.Theinﬂuenceofthestimulationofthosetwocuesandthecombination
isevaluatedinclinicalandotherexperimentalstudies.

PresentationsandPlots14.4.

Interestingpropertiesofthemodelarepresentedbyfourdiﬀerentplottypes:

(a)Theplotsposturwhicheshorespwonseswayofrespoindividuanse,esltstimimatedulasttionsatesareandsanalyensorzeydobyutputsignaloverovertime.time

(b)Thefrequencyresponseispresentedbyamplitudeoralsocalledgainandphaseof
thetransferfunctionH.Thetransferfunctionisdeﬁnedasfollows:
H(jw)=So(jw)(4.46)
Si(jw)
whereSoistheoutputsignalwhichisheretheangularsignalofthesway.And
thisangularsignalisFFT-transformedintothefrequencyrange,soitresultsin
So(jw).ThesameappliesfortheinputsignalSiwhichisherethestimulusapplied
tothesystemgivingSi(jw).ForthepresentedresultstheMATLABfunction’ﬀt’
wasused.Theamplitude(gain)andphaseofthetransferfunctionH(jw)was
calculatedwiththeMATLABfunctions’abs’and’phase’.Whichusethefollowing
n:tiocomputagain(H)=abs(H)=|H|=sqrt(real(H(jw))2+imag(H(jw))2)(4.47)
phase(H)=angle(H)=arctanrimageal((HH))(4.48)
(c)Thestimulusoverresponseamplitudeplotshowstheamplituderelationbetweenthe
stimulusandthepostureresponse.Especiallynonlineareﬀectslikethesaturation
withincreasingstimulusamplitudecanbeseeninthisvisualization.Thesway
amplitudeisdirectlycorrelatedtotheswayangleofthebodyorpendulumandthe
increasingfactoristhestimulusamplitude.

(d)Fortheswayresponse,theRootMeanSquareRMSoftheswayangleisusedto
detectstatisticalrelevantdiﬀerencesbetweensinglestimulationmodes.Foreach
trialanRMSisdeterminedandthesingleconditionscanbecomparedbythe

911

4High-LevelPostureControl

quantitativevalueoftheconditions’RMS.ForRootMeanSquaremeasurementthe
followingformulaisused:

2RMSx=k=1...n(xk)
nwherexisastatevaryingovertimewiththetimestepsk=1...n.

4.4.2InﬂuenceofVisualPerceptionwithEyeMovementson
PolControsture

tArtheofState

.49)(4

espTheeciallyvisualinpercerelatioptniontoistheanimpsurrounortandings.tfaActorswofellstthecabilizatioombinatnandionoofrienvistatualionretinaofplinfoosturr-e
ofmatheadionawithndeyeyeeinandheadheadpmoositiovenmenontsstainﬂuenceisncesthestudied.swaItywrespasonse.foundInt[hat72]heathedrotainﬂuencetion
andeyeorientationinﬂuencetheswayofthebodywhichisalignedinthesamedirection.
Thegazedirectionhadasigniﬁcantinﬂuenceontheswaydirection.Andgazeandhead
andirectioninﬂuenceareofsigganiﬁczeandirectlytioncorinrelatedthewhicneutrahlwahesaadlspofoositioundnwinha[t77].wasButexpla[72]inediddnowithttﬁndhe
’neutralormostnatural’conﬁguration.Probably,thiscorrelateswiththeresultfoundin
[77head]oforielessntatioinﬂuencneinﬂuenceofisoandlatedginﬂueazenceofmanipulathetioncomasbinedincoheadntraandsttoeyeaorienhightlyations.signiﬁcant
Anothercharacteristiccanbetheswayamplitudeandfrequencyresponseofstanceex-
periments.Diﬀerentamplitudesandfrequenciesofvisualstimulationshaveaninﬂuence
onthestancesway[170,149,136,139].In[149,136]thevisualstimulationisaﬁeldof
trianglesmovingwithasinusoidaloscillationplusatransversalvelocitycomponent.The
aincreasenonlineaofrtrmaanslatnner.ionalWithveloinccitreyacsinghasngestimtheulusveCOMlocitswyaty,heaCOMnalyzedswinaythresepofronnsetaldecrplane,easeind
atthestimulusfrequencyoftheoscillation.Thisrelationindicatesthatthehigherthe
velocitythelessistheweightingofthesensoryinput.Anotherinterestingﬁndingof
[149,43]isthatevenavisualinputwithhighvelocityorlargemovementamplitudepro-
videsinformationwhichcanbeusedforthestancestabilization.Theresultsshowthat
thethanclosstimedueyelationconditiosituantionshaswtheithfashighesttmoswvingayaorhigmplitudehaandmplitudevariavisbilitualy.stimItisulatioevenns.higher
Adetailedanalysisoftheinﬂuenceofstimulusfrequencyandamplitudeonanterior-
posteriorswayfrequencyresponseisfoundinPeterkaetal.[139,141].Thetwostimuli
supportsurfaceandvisualsurroundingareappliedtonormalsubjects.Thevisualand
proprioceptivestimulusattainsimilarcharacteristics.So,withincreasingstimulusam-
plitudetheswayresponsesaturates.Forfrequencyvariationofvisualstimulus,theFast
FourierincreasesDFTslightratlynsandformedthanstimdecruluseasesresprapidonsely.wasPhaseevalisuatedleadingfor(>gain0)aandtthephase.beginningGainﬁrstand

012

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

forhigherfrequencieslagging(<0).Aperfectresponseofthesystemtothestimulus
wouldbeasway-responsewithgain1andphase0asisalsomentionedby[139].
Experimentswithvisualstimuluswhichinduceseyepursuitandcombinethereforeeye
movementwithvisualretinalperceptionhavebeenmadebyGlasaueretal.[43].Here
thesinusoidalmovingvisualtargetisfollowedwiththeeyesbyasmoothpursuit.The
eyepursuitmovementmatchedthetargetmovementveryclosely.Theadditionallyadded
backgroundwhichproducesretinalimagegivestheindicationthattheeyemovement
doesinﬂuencethemedio-lateralbodyswaysigniﬁcantly.Thisleadstoassumptionsthat
thereisaswellretinalimageinformationwhichprovidesareferenceoftherelationbody
toenvironmentasadditionaltheeyemovementwhichisoftencoupledtoeverydayslife
taskswhichalsoindicatesareferencebetweenbodyandenvironment.Notonlytheretinal
motionbutalsotheeyemovementinﬂuencestheposturalswayaswasfoundbyIvanenko
etal.[72].Theswayresponsealwaysdivergesindirectionofgaze.Thisinﬂuencewas
testedincombinationwithheadtrunkrotation.Theeyemovementinﬂuenceonposture
happensalsoifnorelationbetweenbodyandenvironmentisgivenlikein[78]withahead-
ﬁxedvisualpursuitstimulus.Vestibularneuritispatientscouldbestabilizedbyﬁxating
aheadﬁxedstimulus[78].Withmovingeyesduringpursuitmovements,thepostural
swayincreasesagainevenmorethanswayindarknesswhichisshowninGlasaueretal.
[43]withhealthysubjects.Theinteractionofretinalandeyemovementinformationis
notclear.Asubjectisabletodistinguishself-motionandenvironmentalmotion[27].
Though,howmuchoftheusedeyemovementinformationandretinalﬂowinformationis
usedandhowthisinformationisintegratedarestillsubjecttoactualresearch.Relations
betweenbackgroundmovementandamovingtargetwhichisﬁxatedarestudiedin[43].
Theﬁxationofaspaceﬁxedtargetresultsinthelowestposturalswayanswerwhere
theeyepursuitwithnospacereferenceormovingspacereferenceshowedthehighest
swayresponse.Incaseswheretheeyemovement(ﬁx/pursuit)wascontradictorytothe
referencemovement(moving/ﬁx)theswayresponsewasinbetweenthetwocasesnamed
beforeandsimilartotheconditiondarknesswherethereisneitherretinalinputnoreye
movement.Thiscouldsupportthetheorythatevenwithcontradictoryinformationthere
isstillinformationthatcanbeusedtostabilizestanceorthatthevisualinformationis
notusedsuchasinthecasedarkness.Thesingleconditionsofpursuitandbackground
informationareshowninﬁgure4.6from[43].
AcontraryﬁndingofStoﬀregenetal.[169,168]isthattheeyepursuitfrequencyis
notcoupledtotheswayfrequencyandthatswayvariabilitywasreducedwhensubjects
pursuitatarget.ThefrequencyrangeofStoﬀregenis(0.5,0.8and1.1[Hz])foran
amplitudeof11[deg].ThisrangeaswasalreadymentionedbyStoﬀregen[168]couldbe
toosmalltoreceiverepresentativeresults.

ImplementedOwnExperiments

Togetfrequencyacompaadditioringnalandexperimenextendedtsknohavwebledgeeneofpmadeosturefortswhisaythesis.correlationTherefortoeey,ethemovexpemeneri-t

112

4High-LevelPostureControl

Figure4.6:NineconditionsE1-E3andF1-F3andH1-H3witheyepursuitwithorwithout
backgroundandadditionalheadrotationareshown.Thegraphicsaretaken
[from].43

mentsofGlasaueretal.[43]havebeenextendedtoﬁndoutiffrequencyandamplitude
variationofeyepursuitmovementshaveaninﬂuenceonthepostureresponserelatedto
themovementvariation.Especiallytheeyemovementsarestudiedinmoredetailforlower
frequenciesanddiﬀerentamplitudes.LowerfrequenciesincomparisontothoseStoﬀre-
genevaluatedaremotivatedbecausesmoothpursuitmovementscanonlybeguaranteed
forlowfrequencies.In[1]thehorizontaleyepursuitforfrequenciesintherangeof0.07
...0.42[Hz]withanamplitudeof22.5[deg]havebeenstudied.Forthehighestfrequen-
cieswithnormalsubjectsthepursuitwasinterruptedbysaccadicmovements.Therefore
thefrequenciesandamplitudeshavebeenchosenlowerforthepursuitexperimentsmade
.orkwthisforTheexperimentsetupisasfollows:Healthysubjectsstandingincompletedarknessona
Kistlerposturographieplatform(Model9286AA)andpursuingasinusoidalmovinglight
pointonatranslucentscreen.Thedistanceofthescreentothesubjectwas0.7m.The
postureresponsewasmeasuredbymeasuringtheCOPviatheKistlerplatformandthe
headpositionviaanoptical3Dtrackingsystem(fromIntersenseModelIS-600)forahead
ﬁxedmarker.Swayistheswayinmedio-lateraldirection.Thetaskwasalwaystoﬁxate
withtheeyesthestationaryormovingtargetpointwhichisequivalenttothecondition
“E1”inGlasaueret.all[43].Allsubjectsgotthesameinstructionshowtoachievethe
standingpositionandtopursuitthevisualtarget.Therearethreeseriesofexperiments.
First,ashortserieswith6subjectsand5conditionsforevaluationofthedesign.Second,

212

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

Figure4.7:MeanRMSoftheCOPforall5conditionswith95%conﬁdenceintervalshown
bars.theyb

adiﬀerenlongetrserstanceieswcoithndition13subandjects8asubndje10cts.Theconditiocons.nditioTnshird,areathelwayssametheascsonecotrondlcobutnditionwith
Thestandingotherinconditiocompletensaredacorknessmabinationdnsﬁxatofaionofmplitudeatarangetdlightfrequencpoinytvariawhichtionisofnotthemotaving.rget
themovemenﬁxatiot.npTheointprogandrestsionhereafterofonea2tria5lsewcoas5ndssecodurandstionofofdarknessthestimafterwulusardsaccording5secotondstohef
isradiﬀerentndomizedconditforions;allsubwithjaects.5secoThendconditpauseionsbetofwetheenﬁreacsthexpcondition.erimentsTheeriesareconditioasnfolloorderws:

conditionc1c2c3c4c5
darkﬁxationa1f1a2f1a2f2
Table4.3:Conditionsc1...c5oftheexperimentswithcontrolconditiondarknessand
diﬀerentamplitudeandfrequencyconditions

witha1=2.5[deg]anda2=12[deg]angleofthemaximummedio-lateralvisualtargetpoint
amplitudeandthestimulusfrequencieswithf1=0.33[Hz]andf2=0.0833[Hz].Results
oftherootmeansquareRMSofthecenterofpressureCOPcanbeseenforthe5
conditionsinﬁgure4.7withthe95%conﬁdenceintervalshownasbarsaroundthemean
RMSvalue.Calculationofthe95%conﬁdenceintervalovernsamplesxi:

n1xi±icdf(1−q,µ,σ)∗std√(x)withstd(x)=1(xi−x¯)2(4.50)
ni=12nn−1

312

4High-LevelPostureControl

whereicdfistheinversecumulativedistributionfunctionwhichsigniﬁesthe1−2q-quantile
ofthenormaldistributionwithmeanµandvarianceσ.
CalculationoftheMeanSquaredErrorofasamplemeanovernsamplesxi:
nMSE(X)=E{X−µ)2}=σ2/nwithX=1xi(4.51)
n=1iTheeyemovementstimulatedswayresponsesc3,c4,c5haveahigherRMSthanthe
ﬁxationcondition(F(3,15)=4.884,p=0.0145).Thiswasalsofoundinotherstudies
[pro43,14duced6].aOneCOPresultswayofresthepostimnseulawhictionhcdidonditionotnsshowwasapthaeatkanatathemplitudestimoulusfa1=2.5frequency[deg].
Thiscouldbeduetothefactthatthissmallamplitudewasveryclosetothesignalto
noiseratiosonoresponseisvisible.Thiscanbeseeninﬁgure4.8.Forahigheramplitude
andbothfrequenciesacorrelationinthefrequencycouldbefound.

Figure4.8:FrequencyswayresponseoftheCOPandthefrequencytransferedbythe
ulus.stim

tioStanrtingofthefroeymethispursuitlittlestimstudyulusthewasfolloevawingluated.expTheerimenstecondforexamplitperimenudeatndconsisfreqtsueofncy10vcoarian--
ingditiotansble:withwithdiﬀetherentstimulusfrequencyamplitudesandaa1mplitude=6[degv]ariaandtionsa2=12whic[dehga]reasnglehoofwntheinthemaximfolloumw-

412

conditionc1c2c3c4c5c6c7c8c9c10
darkﬁxa1f1a1f2a1f3a1f4a2f1a2f2a2f3a2f4
Table4.4:10conditionsofthesecondexperiments.

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

tudAmpli(a)e

haseP)(b

Figure4.9:Transferfunctionamplitudeandphaseforfourdiﬀerentstimulusfrequencies
amplitudes.owtand

medio-lateralvisualtargetpointandthestimulusfrequencieswithf1=121=0.0833[Hz],
f2=91=0.111[Hz],f3=61=0.1667[Hz],f4=31=0.33[Hz].Inthisexperiment13
normalsubjectsparticipated.Inﬁgure4.9thetransferfunctionofposturalresponseto
andvisualphasestim<uliHin(jthew)isdefrequencyﬁnedrasangeinissshoectiown.n4.4The.1.traItnsisfershofunwncttheionpwithostitsuralgrainesp|oH(nsjewo)|f
thediﬀerenCOtPandamplitudestheheada1,ap2ositio.Thenvataluesfoursignifydiﬀerenthetmestimantulatioransfernfrequenciesfunctionofv1er...allf4subforjectwtso
andthebarsshowthe95%conﬁdenceinterval.

Theamplitudeofthetransferfunctionlookssimilarandfrequencyappearstohavea
decreasingeﬀect.Forthephaseplotforthetwohigherfrequencieswithbiggeramplitude
a2thephaseisdecreasingwhichsigniﬁesalagoftheresponse.Ifthefrequencytransposed
signameansloftheeaachtmplituderialandofcothenditionfunctioisnevaH(jluatedw),aatsigtheaniﬁcanccotrdingdiﬀerestimnceulusbetweenfrequencythewhicsingleh
conditionsc3...c10canbefound.Forthegainfunctiontheoveralleﬀectissigniﬁcant
with(F(7,84)=2.917,p=0.0088).Forthephasefunctionofthefrequencyresponse

125

4High-LevelPostureControl

Figure4.10:Themedianvaluesof|So(jwstim)|with95%conﬁdenceintervalbars.

itissigniﬁcantwith(F(7,84)=4.513,p=0.0003).Ifthesameevaluationisdonefor
theconditionsc7...c10whichhavethehigheramplitudea2.Theoveralleﬀectforthe
gainfunctionissigniﬁcantwith(F(3,36)=3.804,p=0.0182),andthephasefunction
with(F(3,36)=4.829,p=0.0063).Further,witha2-factorrepeatedmeasurement
ANOVA,inthegainfunctionthefrequencyfactorisalsosigniﬁcantwith(F(3,36)=
3.80,p=0.018)withoutinteraction.Inthephasefunctionthefrequencyissigniﬁcant
with(F(3,36)=4.096,p=0.0134).Theamplitudefactorshowsnosigniﬁcanteﬀectin
thegainfunction.Inthephasefunctiontheamplitudeeﬀectissigniﬁcantwith(F(1,12)=
14.56,p=0.0025).Thepost-hocScheﬀeTestshowsasigniﬁcantdiﬀerenceforthephase
functionbetweenthetwoamplitudelevelsandbetweenfrequencylevel1and3.Thefour
levelsf1:(mean±MSE=0.0024±0.0571∗10−5),f2:(mean±MSE=0.0024±0.527∗
10−5),f3:(mean±MSE=0.0013±0.102∗10−5),f4:(mean±MSE=0.0016±0.320∗10−5)
.Forthisseeﬁgure4.10.

InGlasaueretal.[43]theRMSofthelateralCOPswayvalueisevaluated.So,inthefol-
lowingtheRMSvaluesareshownforthe10conditionswith95%conﬁdenceintervalbars.
TheRMSvaluesareevaluatedbyrepeatedmeasurementvarianceanalysis.Theoverall
eﬀectissigniﬁcantwith(F(9,108)=3.390,p=0.0011).Furtherthedarkconditionissig-
niﬁcantlydiﬀerentfromthevisualstimulationcondition(F(8,108)=3.405,p<0.0017).

Thenextﬁgure4.11showstheRMSoverallconditions.Thereisnosigniﬁcanteﬀect
forthefrequencyoramplitudefactorfoundbutanunexpecteddiﬀerenceisvisible.The
RMSswayresponseisnotlowestfortheﬁxationconditionbutallstimulationconditions
havealowerswayresponse.Thedarknessconditionhasthehighestswayresponse.The
diﬀerencebetweentheconditionsc1(mean±MSE=0.0199±0.00193∗10−5)andc2-c10

612

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

Figure4.11:MeanRMSswayresponseforallconditionsc1toc10with95%conﬁdence
interval.Thereisadecreasingtendencywithhigherfrequencies.

(mean±MSE=0.016±0.717∗10−5)isareductionofsway.Forfrequencyvariation
forbothamplitudesnosigniﬁcantresponsevariationcouldbefound.Aswellforthetwo
diﬀerentamplitudesthereisnodiﬀerentswayresponsefoundoverallstimulusfrequencies.
Inplot4.12itcanbeseenthatthetendencyforbothamplitudesoverallfrequenciesis
decreasingandthatthehigheramplitudedecreasesevenmore,butnotsigniﬁcantly.To
seethetwoamplitudesincomparisontheplot4.12showsthetwoamplitudesoverall
frequenciesbesideeachother.ThehigheramplitudeleadstoasmallerRMSsway.Till
nowitwasassumedthatthetwofactorsamplitudeandfrequencyhaveanindependent
inﬂuenceontheswayresponse.Ifnoweverythingisrecalculatedasactualvisualtarget
velocityitcanbeseenthatanincreasingvelocityleadstoadecreasingRMSresponse.
Themaximumvelocitiesaredeterminedinthezerotargetpositionforallconditionswhich
leadsto6diﬀerentvelocities.Theconditionsc5andc7resultinthesamevelocityandthe
conditionsc6andc9.Inﬁgure4.13theRMSswayresponsetothe6velocitiesisshown.
Thestanceconditionmayinﬂuencetheswayresponsesigniﬁcantly.Therefore,athird
experimentwiththesame10conditions,8normalsubjectsandadiﬀerentstancecondition
wasevaluatedtoseeeventualdiﬀerencesaccordingtothestancecondition.Nowstanceis
anormalnarrowstancewiththefeetclosesidebysideonarubberfoamof20cmthickness.
Therestoftheexperimentalsettingisidentical.Asbeforethegaindoesnotvarymuch
withfrequencyandtheamplitudehasaslightlydecreasingeﬀect.Inthephaseplotitcan
beseenthatthephaselaghasbecomebiggerbecausethedampingeﬀectoftherubber
foamisseen.Butthephasedecreasesasbeforewiththefrequency.Nooveralleﬀectis

712

4High-LevelPostureControl

Figure4.12:MeanRMSswayforbothamplitudesa1anda2,variedwiththefourfre-
quenciesf1...f4.Forthehigheramplitudeadecreasingeﬀectwithrising
visible.isfrequency

812

Figure4.13:MeanRMSswayresponseforthe6velocitiesofthepursuittarget.

4.4

ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

etudampli(a)

hasep)(b

Figure4.14:Tandrtansfeworfuncamplittionudesamplita1andudea2and.Thephaseaformplitudefourshodiﬀerenwstadecstimreaulussingeﬀfrequenciesectfor
thegain.Thefrequencyhasadecreasingeﬀectonthephaseandfora1on
912.gainthe

4High-LevelPostureControl

Figure4.15:Themedianvaluesof|So(jwstim)|with95%conﬁdenceintervalbars.

foundforthefrequencygainandphasefunctionaswellasfortheRMSfunction.This
isthecasebecausetheswayresponseismuchlessinthismorestablestancecondition.
Theoveralleﬀectforthefrequencygainiswith(F(7,49)=2.09,p=0.0617)justnot
signiﬁcant.Thisisthesameforthe2factorvarianceanalysiswhichhassimilarclose
resultstosigniﬁcanceforbothfactorsbutnointeraction.Inﬁgure4.15itcanbeseen
thatthevariancewithinaconditionismuchhigherandsothediﬀerencesarenotsoclear.
Inﬁgure4.17thereisnoeﬀectoftheamplitudeseen.Though,withinoneamplitude
especiallyfora2atendencyofdecreasingRMSswaywithincreasingfrequencycanbeseen.
Thisstanceconditionisgenerallymorestable,butwiththefoamrubberunderground
thevarianceofthemeasurementsincreaseswhichisseeninallplotscomparedtothe
tandemstanceconditionofexperimentE2.Theeﬀectisthattherearenosigniﬁcant
diﬀerencesbetweentheconditionsbutthecharacteristicsareverysimilartotheresults
seeninexperimentE2.Thismakesclearthatthestanceconditionisaveryimportant
conditionwhichcanreduceorenforceeﬀectsandwhichcanbeverynoisy.
Concludingitcanbesaid,thatnosigniﬁcantinﬂuenceoffrequencyoramplitudevalue
couldbeprovenontheRMSvaluebutonthefrequencyswayresponseasigniﬁcant
inﬂuenceofthefrequencyvalueisdetermined.Thetendencyfoundisdecreasingsway
responsewithrisingfrequencyinallconditions.Thestanceconditionchangehasan
inﬂuenceontheabsolutevaluesbutnotonthetendencyofdecrease.Thedarkness
conditionissigniﬁcantlydiﬀerentofallvisualstimulatedconditions,theswayresponse
RMSissigniﬁcantlyhigher.

013

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

Figure4.16:MeanRMSswayresponseforallconditionswithmeanRMSvaluesand95%
l.atervinconﬁdence

Figure4.17:MeanRMSswayresponseforthetwodiﬀerentamplitudesa1anda2in
comparison.Adecreasingswayresponseeﬀectisregisteredasthefrequency
oftheamplitudea2rises.

113

4High-LevelPostureControl

4.4.3InﬂuenceofVestibularPerceptiononStanceControl

BasicsandStateoftheArt

Vlarestibulasensor.rperStimceptioulantionisofherethisthepsensorerceptiocannbofeaacngulahievredaccbyelerae.g.,tioncahangcteingsofontthehevsuppestibu-ort
surface,whichalsoresultinproprioceptiveimpressions,orbygalvanicvestibularstim-
ulatmastoionidGunderVS.TeahechGofVStheissanubarject’stiﬁcialeaers.lectroAnelestimctrulicsatiotimnuluswhichofisaboutdirec0tly.25a-1pplied[mA]toisa
nervusuallyelyingused.beloThiswthestimsurfaulusceinisducdepesolaanrized.electricalThisstimdepouluslarizaontiontheskinleadstandoasthereforensatetionalhe
theinputvtoestibulathesrorensorganyprobutcesstheingdepsyostelarizamwhictionhofwtashenoatﬀerorigentvinatedestibulabyarrealnervesens[44o].ryTinputhisode-f
pﬁringolararizatiotenleadsmeanstoasthatensatapionositivwehicahnoisdeartiﬁcialincreasesbecatheusaeﬀethisrentmovﬁringemenrtate.pThiserceptioincnreacouldsed
notbereproducedwithnaturalphysicalstimulation.Thesingleperceptionvectorsof
eacwhichhissemictheircpularerceivcanaedlmoofvtheemenvtwaestibulasarnalyorgzedaninanddetatheilbyresultingFitzpatricsumketoftal.hes[e32v].ectorThes
vectorialsensationwhichisproducedbyabilateralbipolarGVSstimulusisshownin
.84.1eﬁgur

Figure4.18:Theelectricalstimulationproducesdiﬀerentmovementperceptionvectors
inthediﬀerentvestibularorganparts.Thevectorscombinetoproducethe
movementﬁnallyperceived.

vTheectorproandducaedrotaimpresstionaiolnmorevsememenblestinarothellinghormoizonvtaemelntplane.intheThefronmaintalplamovneemewithnt,itsthemainroll

213

4.4ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

inthefrontalplanewillbethestimulationfactorforthischapter.Thedecomposition
ofthisvectorisdetailedin[189].Thepolaritywhichmeansthattherightsideisthe
anodeandtheleftthecathodeorviceversainﬂuencesthedirectionofsensoryinput.
Thisdirectionindicatesiftherollmovementistotherightsideortheleft.Thestimulus
inducesarollmovementtothecathodalside.Themechanismofgalvanicstimulationand
itseﬀectontheorganismisexplainedindetailinFitzpatricketal.[33,189,32].
InBentetal.[8]theGVSstimulusappliedinstanceinducesabodyswayresponse
whichwasarollofthebodysegmentshead,trunkandpelvis.Theunexpectedﬁnding
wasthattheconditioneyesclosedoreyesopendidnotinﬂuencetheswayresponsevery
muchwhichisopposedtoformerﬁndingsofDayandBonato[25]whichfoundareducing
inﬂuenceofvisioninputontheswayresponse.
ThevestibularresponsetoGVSwithdiﬀerentamplitudeswasstudiedin[65].Anincreas-
ingamplitudeshowedanincreasingswayresponseoftheCOP.Additionally,somatosen-
sorylosssubjectshavebeentestedbesidenormalsubjectsandtheyshowedanevenlarger
swayresponsetothestimuli.Theincreasefoundwasinbothcasesalinearrelationwith
theslopedependingonthegradeofsomatosensoryloss.AfurtherstudyofHlavacka
[64,63]showedthattheCOPswayresponsetoastimulushasalargedelayofabout
onesecond.Thecombinationofproprioceptivefeedbackfromthelegsandthevestibular
informationarelinearlycombined[64].
AccordingtoDayetal.[26,32]thecontinuoustimeswayresponseofhead,trunkand
pelvisisalateralsidewardsmovementstartingwithsomelatency120ms.Thenaftera
shorttimeofsidewardstiltingthepositionstaysconstantandafterthestimulusceased
thebodyreturnstoitsoriginalposition.

ImplementationofOwnExperiments

sis.ThisThecouldstimalsoulusbevdurateriﬁedionwinasthechosenstancetobande1ste[sec]ppingwhicexhpiserimabenouttsthemadetimeforthethisbothe-dy
swayresponseneedstoreachthemaximumtiltposition.Theresultingswayresponseis
thereforeanincreasingandafterwardsdecreasingtiltmovement.Thestimulusamplitude
wasconditio1n[mA].overThisalltissrials.hownThereinﬁgurhaveeb4.1een9.Thetestedda6tahealtshowhynsubareajectsmeanwithresp2tronseialsforforeaeacchh
condition.The6conditionsare,stance,walkinginplaceandjogginginplaceeachin
side.combinaThetionwithmeasuremenGVStofwiththebtheodyanoswdaalywaelectrosattadeoinedncebyonmeathesurleftingandthetoncerunkonmothevremenightt
bmaryakerstereowith0vision.30tra[Hz].Theckedgmaalvrkanicer.stiThismulusformeasuremensteppingtdelivconditers3Dionsdawatasaoflwatheystracstartkeedd
inthemomentwhentheheel-strikeoftherightfootoccurred.Thesubjectswhereall
blindfolded.Thecharacteristictrunkswayresponseovertimeofonesubjectisgivenin
ﬁgure4.20.Inallcasesaclearbodyswayinmedio-lateraldirectioncanbeseen.Italso
ofcanthebessystemeenthaatppeawhers.ntAnheboexplanadyrettionurnsfortothistheorcouldiginabelptheositfoionlloawing.slightWhenovtheerswingingbody
reactstotheGVSstimuluswithsomedelaythecorrectivemovementalsocontrolledby

313

elHigh-Lev

eFigur

413

9:4.1

ostureP

Meant,hrig

trConol

continuousbodyswayovertimeforanodalstimulation
shownforstance,steppingandjogginginplace.

deiss

left

nda

eFigur

0:4.2

4.4

ExperimentallyFoundInﬂuencesofSensoryCuesonPostureControl

Constance,tinstuouseppibongdyandswajyooggveringtinimeplaforceoneforaonenodalsinglestimsubulatjeciont.

side,

forwnsho

513

4High-LevelPostureControl

vtheenttheotherbsodyensoryfromcuesfalling.(especIfiathellystimproprioulusceisption)stoppedprothisducesacouncountertemorvmoemevntemenisttoadaptepre-d
agThisainwwithouldsoalsomedelaexplayinsothetherereducisaedsmaeﬀeclltovofGerswingVSofwiththeasystemdditionalinthevisualothersensorydirection.cues.
TheThebswoadyyswrespayonsegoeslaostsnfaobroutabtoutwoosneecondssecondwhicandhistthenhedodeublecreasoesfthewithstimtheulusceaseduraofttion.he
jogstimgingulusinforplaceabitisthmoatrethetshanwaaysecoamplitnd.udeTheisincreased.diﬀerenceInbettwheeencaseofstancesteandppingorsteppingjoggingor
theGVSbostimdyswulusayaicstsofenlargtheedbecsystemauseinofthetheswmomenayingtofofheel-thestrtrikunkethewithsystemeachisstepclearandlyifmorthee
alsoinstablehighearsinthancasforeofstastance.nceSo,withthisbotcanhfeet.explainThethehigdynamicheroGVSftheinducsystemedswaforyamplitsteppingudeis.
InthecontextofthepresentedGVSstimulusexperimentsdiﬀerentmomentsoftimefor
thestimulusstarthavebeentested.Andoneresultwas,thatitmakesadiﬀerencewhat
pointinthestepcycleforthestimulusstartischosen.E.g.ifforjoggingmovementsthe
stimuluswasstartedintheﬂightphasewhichmeansafterthefootleavesthegroundthe
stimulusresultshadalotofvariance.Thisisexplainedbythefactthatthepossibilities
ofactivebalancecontrolduringtheﬂightphasearenotboundtogroundreactionforces
andthereforeverydiﬀerenttothenormalstancecontroltorques.Betweensteppingand
jogginginplacetheGVSstimulusproducesverysimilarswayresponses.Thismightindi-
catethatthebalancecontrolformedio-lateralisnotdiﬀerentduringthosetwomovement
terns.patItuluscantobethesummecathoddauplside.thattheThelatsewraalyborespdysonsewayisresploweronseforisstancaccordingethantofortheGVSsteppingstim-or
jogginginplacebutequalincharacteristic.Thedurationofbodyswaytothe1second
stimdecreaseulusisswaabyomoutvoneemensectboefndoreforincreareturningseotofbothedyorswiginaaylapndositionaboutwthehichesamendstimewithaforlittthele
.erswingingvoInvisualthestimfollowuliingtosesimctulaiontethethemostancedelandppostureosturerespcononsestrolamonddelcomparwasetstimhemulatotedthebyrespGVSonsesand
.tallyerimenexpfound

4.5SimulatedSwayResponsesforVisualandVestibular
ationStimul

theThemoequatiodelnsconsistsdescribingof(1)thethesebonsodyrymodynadalitmicsiesof4.5an,in4.7ve,rse4.1p0andendulum4.11orequa4.1tio6n,(32.4),t(2)he
theprofeecessingdbackandcontpredictivrollerewithpartequaoftionthe4.4Ka1.lmanThisﬁltermodelwithwhiceqhuatioisnspresen4.30tedtoin4.3ﬁgur5eand4.3(4is)
calculatedwiththegeneralparametersgiveninsection4.5.1.Interestingpropertiesofthe
modelarepresentedbyfourdiﬀerentplottypes:(a)thesignalinthetimerange,(b)the
transferfunctioninthefrequencyrangeoralsocalledfrequencyresponses,(c)relational

613

4.5SimulatedSwayResponsesforVisualandVestibularStimulation

plotstimulusoverswayresponseamplitudeand(d)theRootMeanSquareplot.Those
plotshavealreadybeenusedandexplainedinthesectionofexperiments4.4.1.
Inthefollowingfourdiﬀerentsimulationsarepresented.First,thevestibularstimulation
withGVSincomparisontotheexperimentsof4.4.3.Second,thevisualprocessingof
retinalimagevelocityforthelinearandnonlinearvisualsystemmodel.Third,thesensor
integrationisshownbythecombinationofdiﬀerentstimuliforeyemovementandvisual
backgroundmovementaccordingtotheexperimentsbyGlasaueretal.[43].Fourth,the
eyemovementissimulatedfordiﬀerentfrequenciesandvelocitiesincomparisontothe
experimentsin4.4.2.Beforethesimulationresultsarepresented,theparametrizationof
themodelisaddressedinthenextsection.

4.5.1ParametersofthePostureControlModel

ThelinearposturKalmaenconesttrolimatomordelincousedmhebinatreionconsistswithothefftheeedbacinvkersecopntrolendulumrelatedmotodeltahendotpti-he
malitpresenytedcritberyiontheJwhicfrequencyhisrespdeﬁnedonseaineqmplitude,uation4.4phase2andanda4.4sw3.ayTherespomonsdeleovsimerulatimetionandis
thestimulusinﬂuenceofdiﬀerentstimuliontheswayresponse.Theresultspresented
herehavebeensimulatedinMATLAB.Thestate-spaceequationsusedareequations2.2,
ar4.4e1,in4.3dis2,crete4.33,fo4.3rm.4Theandsa4.35mplingoftheratemeocfhathenicsmeaschwellanicsasoandfthethewhoKalmalensysteestimatmisrion;ealizetheyd
ofwith0.1time[sec]stepswhicohf0.0leads01to[seca].tTimeheKdelaaylmanofe100stima[msection].proThiscessisdiﬀerencesampledinsawithmplingtimestepsrates
standsforthetimedelayproducedbysensoryandneuronalprocessingwhichisgivenwith
valuesabout100[msec][180].Theparameterswhichareﬁxandusedforsimulationare
theparametersfortheequationsgivenaboveandthesensortransferfunctionswhichare
givenintheaccordingsubsectionsof4.2.Furtherparametersarethoseofthemechanical
system,thependulum,andtheerrorcovariancesfortheestimation.
Ifindepthereendenisntotrfurialstherforeeaxchplanacondittion,iontheorgsivtenimresulus.ultIfsartheemomeandelisresultsreferrceadtolculatedaslinearfromor5
nonlinear,thisstandsforthelinearmodelingornonlinearmodelingofthevisualvelocity
sensorycue.Allothermodelsarethesameoverallsimulationresults.
Toshowthebehaviorofthelinearestimationthesinglesignalsproducedinthemodelare
ofsho0.wn2[Hfozr]oandneeaxnample.amplitudeTheofsystem2[degw].asThestimKaulalmantedwithﬁlteraestimatvisualionbacwkgasroundcalculatmoveedmenwithts
thethesnoiseensorsco.vaTheriancessensoryQcov=signa0.ls005withandRsystemcov=sta0.tes05.andThesyexternastelmstimstatesuliareinputaremeasuredshowbny
inﬁgure4.3.TheestimatedKalmanﬁltervaluesofthesystempositionandvelocityis
plottedinﬁgure4.21.
Intheleftplot(a)itcanbeseentherealsystemstatesandtheirestimations.The
positioestimatednveloestimatiocitynsigresemnalmorblesethedeviarealtionpcanositionbeveryseenacloselyndespwitheciaallylittlesmalldelanoisyy.Insignalthe

713

4High-LevelPostureControl

(a)Systemstateandestimation

(b)Sensoryoutput

Figure4.21:SystemstatesareestimatedbytheKalmanﬁlterandmeasuredbythesen-
sorymodalities.Thegraphsshowtherealandestimatedbodyposition,
velocityandthefoursensoryoutputs.Sensorsandsystemarenoisy.

partsinaccuraareciesmos.oInthedthebylefttheplotestimat(b)theionsebnsoecauseryofstatesthecaquannbetizatseenionforeﬀaectvisualandthestimmouladeletion.d
Herethestateofproprioceptionwhichstandsforpositionmeasurement,thevestibular
statewhichisproportionaltothevelocitymeasurement,theeyevelocitystatewhichis
acombinationofbodyandstimulusvelocityandﬁnallythevisualmeasurementstate
whichperformsnonlinearly,areshowntogetherwiththeiractualposition,velocityand
uli.stimlexterna

4.5.2VestibularStimulation

Inthefollowingtheresultsformodelsimulationsfordiﬀerentsensoryinputmodalities
areshown.First,thesystemresponseisshownforthecasethatallsystemsensorsare
availableandprovidecorrectinformation.Theresultsshownarealwaysaveragedresults.
For10trialsthesamesituationhasbeensimulatedandthencalculatedthemeanover
those10trials.Thisisdonetoreducetheseeninﬂuenceofnoiseintheswayresponses.

ThevestibularsensewasstimulatedbyaGVSof1secondduration.Theexperimental
resultshavebeengiveninsection4.4.3.Thesimulationwascalculatedforthecasethat
thereisnovisualinputtothesystemlikeintheconditioneyesclosed.TheGVSstimulus
Instarﬁgtsureatse4.2co2nditc7.anThebenoseenisecthaovtariathenceCOsMputangtotheularressystemponsaereσQreacts=0to.00the5sandtimσRulus=0.with05.
delay.Theresponseisaswaytoonesidecomingbacktothezeroposition.Thenthere
isanoverswingingofthesystemwhichresultsinaswaytotheothersidebeforegoing
bebacskaidtothatthethezeropdelaosyitedion.respCoonsemparisevdertoythesimilar.experTheimendelatallyyisfoasundfoundswayby[resp65]onseaboitutcanone

813

4.5SimulatedSwayResponsesforVisualandVestibularStimulation

Figure4.22:SimulatedswayresponseofCOMforagalvanicstimulationforonesecond.
Thegraphsshowthevestibularsensorysignalandthecorrectivetorquede-
termined(above)inrelationtotheresultingbodymovement(below).

second.Andthereisoftenalsoanoverswingtotheothersideasthestimulationside.
Thethenaduranothertionsefocorandonetocomesecondbackstimtoulustheiszeroabpoutositioonenfollosecondwedforbytheanoﬁrstversdevwingiatio.nThisandis
thesameasfoundinmyexperiments.

Alsoalongerstimulusissimulatedinﬁgure4.23.Heretheswayresponseisalsodelayed
andoflongerduration.Afteradelayofaboutonesecondtheswayincreasestillitstays
atalevelwithaslightdecrease.Thisfactcorrelateswiththeﬁndingsof[26,32].When
theGVSstimulusisstoppedthebodyreturnstothenormalbodypositionwithdelay
andaverysimilarrateasthedecreaseofbodysway.

Inﬁgure4.24thestimulusof7secondsisappliedwithdoubledstimulusamplitude.This
leadsalsotoahigherswayamplitudeandlesseﬀectsofnoisydisturbances.Theincrease
ofthebodyswayresponsetoincreasingstimulusamplitudewasalsofoundinHlavacka
[65].Thecharacteristicoftheswayresponseisthesameasforlowerstimulusamplitude
asshownbeforeinﬁgure4.23.

AninterestingﬁndinginthesimulationsforlongerGVSstimuliisthatwiththestopping
ofthestimulusaﬁrstsmallswayofthebodyindirectionofthestimulusisseenbefore
theswaygoesbacktothezeroposition.ThiswasaﬁndingduringmyGVSexperiments
thatthesuddenstopoftheGVSincreasestheswaybeforedecreasingit.Thiscouldbe
thecaseifthechangeofthestimulusalsoinnervatesaninformationwhichisprocessed
forfurtherposturecontrol.

913

4High-LevelPostureControl

Figure4.23:SimulatedswayresponseofCOMforalongergalvanicstimulationof7sec-
onds.determinedThegra(abphsoves)hoinwtherelavtionesttoibulartheresensorysultingsignaboldyandmothevemencorrectivtaselotorngqueas
thestimulustakes(below).

Figure4.24:Simwithulahightederswaaymplitude.responseTheofgraCOMphsforshoawtgalvhevanicestibularstimulatiosensorynofs7ignalsecoandsnd
mothevcoremenrectt(biveelotorw),quewhicdehtiserminedlargerb(abovecausee)inoftrelahetionhighertoathemplitude.resultingbody

014

4.5SimulatedSwayResponsesforVisualandVestibularStimulation

nGai(a)

)(bhaseP

Figure4.25:Linearmodel:Frequencyswayresponseofthebodytoasinusoidalstimulus
withdiﬀerentfrequenciesandamplitudes.Thisispresentedasgainand
phasefunctionofthetransferfunctionsinthefrequencydomain.

4.5.3RetinalStimulation

Second,anothersimulatedstimulationisvisualstimulationwhichstandsfortheretinal
perceivedmovement.Inthefollowingtwostimuliaredistinguished.First,eyepursuit
stimulusut,avisualtargetpointwhichisﬁxatedwiththeeyesandfollowedwhenmoving.
Andsecond,avisualbackgroundwhichproducesadiﬀerentretinalimagethanthetarget.
Theprocessedvisualinformationisalwaysthevisuallymeasuredmovementvelocity.
Thesimulationofthesystemisstimulatedbyavisualhorizontalbackgroundstimulus
whichisasinusoidalmovementofthebackground.Thisproducesaretinalmovingimage.
Thefrequencyofthestimuluswasvaried.Thevaluesare0.03,0.05,0.12,0.3,0.5,0.7,1[Hz].
Theamplitudewasalsovariedfroma1toa6.Themodelissimulatedwiththeamplitudes
1,3,5,8,12,18[deg].Inthefollowingthefrequencyresponsewithmarginandphaseis
shown.ThisplotwaschoseninresemblancetoPeterka[139]butwithahorizontalvisual
stimulusfortheswayresponseinthefrontalplane.Itcanbeseeninﬁgure4.25thelinear
modelresponseandthenon-linearmodelresponseinﬁgure4.26.Forbothcasesarising
frequencycausesarapiddecreaseofamplitudeofthetransferfunction.Inﬁgure4.25
thefrequencyresponseshowsadecreasinggainfunctionwithasuccessiveplateau.For
thenonlinearmodelthisisverysimilarandwithincreasingamplitudethegaindecreases
sligh.lytThephaseofthetransferfunctionshowsaﬁrstslowly,thenfasterdroppingfunction.This
signiﬁesthatthereisaphaselagoftheresponsetothestimulus,meetingtheexpectations
oftheory.Thephasestartsalreadyinthenegativebecausethemodelincludesadelay

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4High-LevelPostureControl

Gain(a)

P)(bhase

Figure4.26:Nonlinearmodel:Frequencyswayresponseofthebodytoasinusoidalstim-
ulusphasefwithunctiodiﬀerennoftthetfrequencieransfersandfunctionamplitudes.intheThisfrequencyispredosentedmain.asgaNoinwathend
gainisamplitudedependent.

of0.1secondswhichisalwayspresentedintheprediction.Thisleadstoadelayinthe
processingoftheposturecommandwhichthereforeproducesthecorrectivetorquealways
delayed.Anotherpossibilitywherethedelayisgeneratedisthesensoryprocessing.Then
theestimationcouldalsobemorepredictionandbringsaphaseleadforveryslowstimuli
becausetheexpectationofastimuluswouldalreadyproduceareactiontopreventthe
destabilizationbythisinput.Thishypothesiswasnottestedinthisthesisandisleftto
futureresearch.Inthenonlinearmodeltheloweramplitudesandfrequenciesleadtoan
increaseinphasebutastheswayresponsecontainsahighlevelofnoise,duetothegreat
ampliﬁcationofthenonlinearfunction,thisraiseisalsoduetonoise.
Verylowfrequenciescouldnotbereasonablysimulatedbecausethequantizationerror
liesby0.01Hz.Lowerfrequenciesthan0.03[Hz]arethereforenotsimulated.Second,
accordingtothepropertiesofalinearsystem,theCOMswayamplitudefordiﬀerent
stimuliamplitudesisalinearrelationwithdiﬀerentgradientsfordiﬀerentfrequencies
whichisseeninﬁgure4.27.
Ifthevisualvelocityprocessingismodelednonlinearlythisinﬂuencestheswayresponse
nonlinearly.Inﬁgure4.28thenonlinearsystemresponsecanbeseenfordiﬀerentampli-
tudes.Forlowamplitudestheresponseincreasesbutforhigheramplitudestheresponse
saturates.Thesaturationisduetothenonlinearlogarithmiccorrelationofthevisual
processingtothevisualstimuluswhichwasdescribedearlierin4.2.Thetestedstimuli
herehadanamplitudeof(1,3,5,8,12,18[deg])ofangularmaximum.Thefrequencyis
simulatedwith0.2[Hz].Asaturationeﬀectisalsodocumentedinliterature[139,136].

214

4.5SimulatedSwayResponsesforVisualandVestibularStimulation

Figure4.27:Linearmodel:AngularCOMswayresponseislinearovervaryingstimulus
amplitudeswithdiﬀerentgradientsfordiﬀerentfrequencies.

Figure4.28:Nonlinearmodel:AngularCOMswayresponseovervaryingstimulusampli-
tudesforallfrequencies.

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4High-LevelPostureControl

Figure4.29:tudesNonlineawitrhmolowdel:noisBeocodyvsarwayiancesrespQonsecov=to0.00visua05landstimRulicovw=ith0.00diﬀ5.erentampli-

Oneexampleofactualswayresponsepositionsisshowninﬁgure4.29.Thenonlinearityof
thevisualsensorycueleadstothedecreasingeﬀectofswayamplituderaiseforincreasing
stimulusamplitudesa1toa6.Asaturationofswayresponseisperformedwithincreasing
stimulusamplitude.Thevisualsensationofmovementdecreaseswithincreasingmove-
ment.Thisphenomenonisalsofoundinmanyintensity-basedvisualperceptiontasks
andinthisresearchadaptedasexplainedinsection4.2.3.

4.5.4EyeMovementStimulation

Third,theeyemovementissimulatedaccordingtotheowneyemovementexperiments
presentedin4.4.2.Theeyestimulusproducesasmoothpursuitmovementoftheeyewhich
followsamovingtarget.Thetargetmovementissinusoidal.Relatedtotheexperiments
theamplitudeandfrequencyofthetargetmovementisvaried.Thesimulatedposition
frequenciesaref1=0.08[Hz],f2=0.11[Hz],f3=0.17[Hz],f4=0.33[Hz]andtheposition
amplitudesarea1=6[deg]anda2=12[deg].Thenoisecovariancesaremodeledwith
Qcov=0.05andRcov=0.1.Thesimulationresultsarerepresentedintheamplitudeand
phaseoffrequencyresponseandtheRMSofCOMposition.TheCOMpositionisthe
angularpositionoftheinversependulum.Inﬁgure4.30thefrequencyresponseisseen.
Itcanbeseenthatthegaindecreaseswithincreasingfrequency.Thisrelatesalsoto
theﬁndingsoftheexperimentwhichindicateadecreaseofgain.Thoughthedecreasing
gaineﬀectinthemodelismuchclearerthanintheexperiment.Thegradientinthe

414

4.5SimulatedSwayResponsesforVisualandVestibularStimulation

nGai(a)

haseP)(b

Figure4.30:Frequencyswayresponseofthebodytoasinusoidalstimulusoftheeye
pursuitcreasesforwithdiﬀerenincreastingfrequenciesfrequency.andThepamplitostionudes.amplitudeThegdoainesandnotphamakseeande-y
.diﬀerence

simulationdependsontheeyemovementgainoftheprocessingwhichcouldbedamped
orintegratedwithfurtherinformationbeforeusedfortheposturecontrol.Hereinthe
modeltheeyemovementinformationisdirectlyusedforthesensorintegrationwhich
canbeareasonforamoredirectandenforcedinﬂuenceoftheeyemovementcompared
totheexperiment.Anotherreasoncanbethattheexperimentmeasuresareinﬂuenced
bymorethanonlytheeyepursuitandproducethereforeanoisydiminishedresponse.
Theamplitudevariationdoesmakelittlediﬀerenceinthesimulatedgainorphase.This
indicatesthattheproportionofstimulustoswayresponseislinearovertheamplitudes.If
thefrequencyisclosetothecutoﬀfrequencyofthelowpassﬁltertheamplitudemakesa
diﬀerencebecausethetransferfunctionisnolongerlinear.Thephasefunctionalsoshows
adecreasewhichmeansmorelaggingwithincreasingfrequencywhichisalsoexpected.
Thisphasedecreaseismorevariableinexperimentaldata.Butintheexperimentswith
higheramplitudea2asteadydecreasedownto-175[deg]showsthesameeﬀect.Forthe
lowamplitudetheswayresponseisnotsocloselycorrelatedandthereforethephaselag
ismuchmorevariable.Thephasedecreaseofthesimulationforamplitudea2isabout
thesameinsimulation.Aloweramplitudea1leadstoaslowerphasedecreasewhichis
duetothelowervelocityofthestimulus.
Inthenextﬁgure4.31thesimulatedmeanRMSvaluesoftheCOMcanbeseen.In
ﬁgure4.31itcanbeseenthattheRMSvaluedecreaseswithincreasingfrequencyand
withdecreasingamplitude.Thedecreasehasabendnearthecutoﬀfrequencyofthe
eyemovementlowpassﬁlter.Thislowpassﬁlterisasimpliﬁcationofreality,thisbend
wouldbeexpectednotsoclearandsharpinreality.Intheexperimentsalsoatendencyof
decreaseisseenespeciallyfortheconditionwithhigheramplitudea2.Thebestcorrelation
iswithhigheramplitudeandthenormalstanceconditiononfoamrubberwhichisalso

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4High-LevelPostureControl

Figure4.31:forBodytheswfouayrrespstimonseulus(frdiseqplaueyenciesdbayndthetwomeanaRMSmplitudesofofCOMtheeyeangulapurrsvuitaluesex-)
pamplerimenitudets.trTheiggersswaayrisedecreasesinthewswitahyrespincreasingonseforlofrequencywafrequenciesndan.increasein

closesttothesimulatedmodel.IngeneralthehigheramplitudeproducesahigherRMS
swayresponsebutthiseﬀectdecreaseswithrisingfrequency.Whytheresultsarelike
thiscanbestbeseenfor4examplesoftherealmeantimesignalsofbodyresponse
vtoelocitinputyissignadirectlylut.proForportthisionalseetothethefoswlloaywingamplitudeplots4.32but.Oneindirectlycanseepro,pthatortiotnalhetotargtheet
swayfrequencyaccordingtothelowpasscharacteristic.Inthemodelthedecreasewith
increasingamplitudecouldonlybeachievedbythenonlinearlogarithmicvisualsensory
cue.visualInpthiserceptioneyepurissuitthereforexpeerimennottvistheiblevisualforhighinputgaisinsosmallftheandtrantheseﬀferectfuncoftthision.nonlinear

TheexperimentofStoﬀregenetal.[169]useshigherfrequenciesf1=0.5,f2=
0.8,f3=1.1[Hz],thosefrequencyconditionshavealsobeentestedwiththemodel.
ThefrequencytransferfunctionresultsaresimilartotheresultsabovebuttheRMS
valuesincreasingdiﬀeror.Thisdecreasingisseenintendencﬁguryereco4.33gniza.Inble.theThisﬁgurehappitenscanbbeceauseseenthethatlowtheparsescishanor-
acteristicsappliedtotheeyemovementsensationreducethesehigherfrequenciestoa
ﬁxasimilartionrespconditonseiongahasinaandthigherhusvtheariaeﬀectbilityiswhicnothwdiﬀerenouldt.alsFourtcoherrresp,itondcantobetheseenﬁndingthatstheof
Stoﬀregenetal.[168].

614

4.5SimulatedSwayResponsesforVisualandVestibularStimulation

f1a1(a)

1fa2(c)

3fa1)(b

3fa2)(d

Figure4.32:Simulatedsignalsofangularswaypositionandvelocityinrelationtothe
visualinpututoftargetmovementforeyepursuit.

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4High-LevelPostureControl

Figure4.33:BodyswayresponsemedianRMSfortheexperimentsof[169]withhigher
frequenciesinrelationtotheﬁxationcondition.

4.5.5CombinedRetinalandEyeMovementStimulation

Fourth,thedependenceonthestimulusissimulatedaccordingtotheexperimentsof
Glasauer[43].Thevisualstimulusconsistsofasinusoidalmovingpoint,asmentioned
above,whichispursuedbytheeyes,andabackgroundwhichisoﬀoronandmoving
togetherwiththepointorstable.TheconditionsthereforeareassetdowninGlasauer
].43[al.et(a)Theswayresponseofeyesclosedconditionissimulated.Then(b),theswayresponse
toeyesopenandﬁxationofastableorsinusoidallymovingpointispresented.And(c),
theswayresponseofastableorsinusoidallymovingpointwithabackgroundisshown.
Thesimulationshavebeenrunwithlinearandnonlinearvisualandeyeproprioception
sensoryparts.ThesixvisualconditionsF1,F2,F3,E1,E2andE3(dark,ﬁxation,ﬁxed
targetwithmovingbackground,movingtargetwithoutbackground,movingtargetwith
ﬁxedbackground,movingtargetandbackground)explainedinﬁgure4.6of[43]have
beensimulatedwiththefollowingparameters:Qcov=0.0005;Rcov=0.005,stimulus
amplitude3[deg],stimulusfrequency0.2[Hz].Thenoiseandnoisecovarianceshavebeen
chosentobeverysmalltoavoidtheinﬂuenceofnoiseontheresponseandtoseethe
characteristicsmoreclearly.Fivetrialshavebeenaveragedtoshowtheresults.Inﬁgure
4.34thesixconditionscanbeseenwithmeanswayresponseandtheappliedstimulus.The
meanresponseiscalculatedfrom5trials.TheRMSwasalsocalculatedandthemedian
RMSofswayanglewith95%conﬁdenceisshowninﬁgure4.35.TheconditionsF2,F3
areclearlydiﬀerenttotheconditionsE.Thisisbecausetheinﬂuenceofvisualinﬂuenceis
noneinF1andconstantinF2whichmeanszerovelocity.Thebackgroundmovementof
F3isdiminishedbytheeyemovementwhichisﬁxandlinear.InconditionsE1toE3the

148

eFigur

4:4.3

4.5

SimulatedSwayResponsesforVisualandVestibularStimulation

F3Theco(red)nditionswithwitresultinghvarangyingularstimbouladyptionositioncombina(blue)tionsandE1velotocitE3yand(green).F1to

914

4High-LevelPostureControl

Figure4.35:MedianRMSswayvaluesforall6visualstimulationconditionsE3andF1
toF3withhigherrelativevisualgain.

Figure4.36:MedianRMSswayvaluesforall6visualstimulationconditionsE3andF1
toF3withlowerrelativevisualgain.

015

Discussion4.6

amountofvisualinformationdiﬀersbutinconditionE1andE3theinformationoftheeye
movementandthevisualbackgroundareequalandthereforetheeﬀectisreinforced.The
eﬀectofE3ishigherthaninE1becausetheamountofvisualinformationisinE1reduced
toonemovingpointbutinE3itisapointandthewholebackgroundmovingtogether.
TheconditionE5showsthemovingpointwithstablebackgroundheretheinformation
oftheeyemovementconﬂicttheretinalmeasuredmovementofthebackground.The
inﬂuenceofeyepursuitonposturecontrolisthereforenotsostrongifitismeasuredas
conﬂict,oreventhecuewiththerightinformation,herethebackground,isfavored.This
canbeseeninthediﬀerencebetweenthetwoplots4.35and4.36herethegeneralﬁxed
weightofthevisualvelocitymeasurementwasdecreased.Thereforethevisualinﬂuence
decreasesandtheeyemovementgetsmoreweight.Ifthevisualinformationgivescorrect
informationtostabilizethebodylikeinF2andE2thebodyswayincreases,elseitstays
ordecreases.Inotherwordsthereismoreinformationavailablewhichiscorrect,related
totheenvironmentandbodyrelationandwhichcanbeusedtostabilizethebodymore
eﬀectively.TheeﬀectthatF3stabilizesthebodybetterthanconditionE2couldbe
attributedtotheinﬂuenceofeyemovementontheposturecontrol.Iftheeyemovement
isveryimportantfortheposturalstabilizationthiswillleadtoastrongerinﬂuenceofthe
sensoryinputofeyemovementthanofthevisualmeasuredmovement,becauseinthis
casenoadditionalvisualorlearnedcontextinformationcouldbeusedtoenforceonecue.

cussiDis4.6on

Vestibular,visualandeyemovementstimulationsweretestedintheauthor’sownexper-
imentsandexaminedinliteratureandwereevaluatedinsimulationusingtheposture
controlmodelpresented.Thegeneralcharacteristicsofthestancemodelaredescribed
assensorintegrationintheKalmanestimationprocesswithadelayinprocessing,and
negativefeedbackcontrolwithoptimizationcriteria.
Galvanicstimulationleadstoverygoodswayresponsesimilaritiescomparingsimulation
withexperiments.Thetimingandcharacteristicsoftheswayresponsecorrelatehighly
withtheexperimentalﬁndingsandtheory.
Eyemovementsandvisualprocessingaremorecomplexsensorycues.Nevertheless,in
manyconditionsthesimulationresultsshowthesamecharacteristicsasthosefoundin
experimentsandintheliterature.Retinalvelocitystimulationleadstosimilarfrequency
responsefunctions.Onlythephaseisdiﬀerent,andthisisexplainedbythefactthat
theestimationprocessingincorporatesonlyonedelay,whichisaroughsimpliﬁcationof
.yrealitThesamephaseasthatfoundbye.g.,Peterka[139],cannotbereproduced.Thiscould
beduetothesimpledelaymodelingorduetothefactthatitisthefrontalswayresponse
whichisevaluatedratherthanthesagittalsway.Thisissuﬃcient,however,forthelater
balancecontrolfunctionofthesteppingmovementandthephasefunctionrepresents
realityqualitatively,butnotquantitatively.

115

4High-LevelPostureControl

Thesimulationofvariouseyemovementstimulationsﬁndsthattheswayresponsegain
decreasesasstimulusfrequencyincreases.Thisisthesameﬁndingasintheexperimen-
btaetlweenresults.eyeInmovtheementliteratandurepthereosturalareswtawy.oconGlasatraueryrhetypaotl.he[ses43,ab14o6ut]ptheostulacorrteselathationt
theycorrelateandStoﬀregenetal.[169]saysthatthisisnotthecase.Inthemodel
devexplainedelopedbyherethethelowdecreapasssingﬁltereﬀingectooffthetheeyeyeemomovvemeemennts.tsoTnhistheexplaswayinsresptheonseﬁndingcanbofe
[tha169t]astherewellisaaslacthekofresultscorrelaoftionthebexpetwereenimetnhetsepryeesenmotedvemehere;ntstheandposexplanaturaltionswaisynotbut
thethatpresentheretisstudylow-pabutssactorrelahighertion.frequenciesEyemoovf0emen.5ts0.8wanderet1.1ested[Hz]onusedtheinmodeStoﬀrleusgedenetin
al.[169]inrelationtoaﬁxedstimuluswithafrequencyofzero.Inthiscase,asthe
valuesfrequenciesdonotarediﬀeralreamdyuchhig;hinandparticulathebrodythereswaisynoresponseevidenciseoftherefswaoyrevresperyonselow,tethendinRgMtoS
wasdecreaseestablishedwithbincreaetwseeningtfrheequencfrequenciesy.ThisinistheaprobastudiesbleexpublishedplanatbyionofStoﬀregwhyennoetal.[diﬀerence169].

Thenthereisstillthequestionofwhethereyestimulationleadstoanincreaseordecrease
inbodyswayresponse.Aswasshowninthe3diﬀerentexperimentsbotharepossible,
[16whic9],hwhicmeanshﬁndthestudiesdecreasinginb[o43dy],swawhicyhhavetoascertainbeincrconsidered.easingbTohedyswexplanaay,atiosnwfoellrasthesine
contradictoryﬁndings,accordingtothemodelproposedhere,isthattheswayresponse
dependsonboththefrequencyandamplitude.Thehigherthefrequencyandthesmaller
theamplitude,thelowertheRMSoftheswayresponse.Butiftheamplitudeisvery
low,itisalsoveryclosetothesignal-to-noiseratio.Thiscanalsobeseeninﬁgure4.8
inconditionc3wheretheamplitudeistoosmall.Asisseeninthesimulationresultsin
ﬁgure4.33,wheretheconditionﬁxediscomparedtotheothers,and4.36,withconditions
F2andE1,themeanRMSvalueoftheswayforﬁxationandeyemovementisnotalways
equal.Themodelproposedheremakesitclearthatswayresponseasaresultofeye
movementconditionsdependsonthefrequencyandamplitudeofthestimulusandalso
thattheﬁxationvariesbecauseofthenoisevariances.Thisisonepossiblereasonforthe
contradictingresultsoftheexperimentsbuttherearealsootherfactorswhichexplainthis
eﬀect.Forexamplethetypeofvisualstimuluse.g.,size,contrast,information,andthe
inﬂuencefromothersourcessuchaslevelofattentiontoataskorthestancecondition.
Thiscannotbetestedwiththemodelpresented,becausethosefactorsarenotmodeled;
alargereyemovementstudywhichtakessuchfactorsintoaccountwouldbeoffurther
use.Asnotede.g.inPeterka[141],anadditionalﬁxationofthebodyitselfreducesthe
noiseintheswaydata,andthishelpstoisolatethebodyswayresponse.
Thecombinedeyemovementandretinalvelocityprocessingwasrepresentedbyanonlin-
earmodelrelatedtothelogarithmicperceptionlawofWeber-Fechner.Thisextensionof
themodelreproducednonlinearsaturationeﬀectswhichareexplainedbyothermodels,
suchas[180]wherestatenoisecovariancesareadapted,orin[136]wheresensoryweights
areadjustedtominimizethemeansquareofthecontrolorin[118]wherethresholdfunc-

215

nConclusio4.7

tionsarereweighted.Inthepresentedresearchthevisualvelocityprocessingismodeled
nonlinearlyaccordingtotheWeber-Fechnerlaw.Theeyemovementsensorycuemodel
islinearaccordingtotheamplitudebehavior.Howeverforstimuliwhichalreadyleadto
phighlyosturenoconnlineartrolberespcauseonsesof,thethesaeyeturamotionvemeneﬀects.tcueaThisutomaisanticallyexplahasnatioangrforeatertheinﬂuencediﬀerenceson
establishedforconditionsF3andE2of[43].Thesaturationeﬀectsinpostureresponse
ofvisualstimuliamplitudecanalsobereproducedbythisnonlinearity.Theevaluation
ofandthiscomhybiponatiothesisnsneewithdstootherbesensostudiedrycueins.furtherexperimentsforeyevisioncombinations

4.7Conclusion

feeUsingdbacthekcconhotsrol,enthesensors,modelhighpres-leveneltedprochereessingproforvidesstaatesolutionestimatiforonba(Kalmalancingn)inandtheLup-QR
rightstanceusinghigh-levelintegratedsensoryinformation.Inotherwordsthismodel
enablestheenvironmenttoinﬂuenceposturecontrolandinadditionaholisticpostureis
determinedforthestabilizationofthewholebody.
vestibulaAccordingrtoandthevisuasimlulastionensorysresultstimitulatcanionsbeatsaidthethatsamethemotimedelstatakbilizesiningtotheaccounptostureseveralfor
uprightstance.Thesesimulationswerecomparedwithrealexperimentsandexperimental
ﬁndingsintheliteraturetoshowthatthebehaviorisverysimilar.
Themodelisthereforeabletotakediﬀerentenvironmentalconditionsintoaccount.It
canadapttodiﬀerentsensationsituationssuchas:missingsensorycues(eyesclosed),
atadisturbsked(eyesensopursuitrycues(ginﬂuencalvesanicvisvioestn),oibularrstimstimuliulatiosituan),tionssensosucryhascuesabeingvisualbacinﬂuencedkgroundby
andeyemovementsconveyingthesameordiﬀerentinformationabouttheenvironment,
leadingtoconﬂictingmeasurements.Thesesituationscanhappeninreallifeandthey
areusedheretoevaluatetheposturecontrolsystem.Extendingvisualsensorycuebya
blogyaaritcommohmicnnopnlineariterceptionylawaccoun(Wtsebforer-Ftheechnestimr).ulationeﬀectsandexplainsthenonlinearity
Althoughthemodelpresentedherewastestedandevaluatedforbalanceinstance,the
maintainingofequilibriumgenerallyrepresentsthesametaskduringsteppingmovements.
Thesensoryinformationisneededforthepresentstudy,althoughthespecialinﬂuenceof
thesensorypresencuetresstimearculushtheinputsstimonulussteppingeﬀectopnosturbaelanceisnowtasevexaminaluaeteddextinraorderinttohiswguaork.ranIteen
thattheposturemodelexplainstypicalstimulatedposturalresponses;theseconstitutea
suitablebasisforfurtherperceptionexperimentsaswellasforextendedposturecontrol
tasks.Themodeldevelopedhereforposturecontrolisabletodeterminethewholebody
position,afeaturewhichthelow-levelneuro-mechanicalmodellacks.Inthefollowing
chapter,thesamemodelisappliedtothesteppingtask.

153

5LoIntegrationw-LevelMoofdHelisgh-Leveland

Thehigh-levelposturecontrolmodeluseshigh-levelsensoryinformationtostabilizethe
wholebody.Incontrasttothisthelow-levelmodelautonomouslygeneratesstepping
movementswithouttakingtheoverallpositionintoaccount.Theobjectiveoftheinte-
grationinthischapteristoimprovethestabilityofthelow-levelmodelbyintegrating
integhigh-levrateeldtosensacoryhieveeinformanlargedtionamondvingknorwangeledgbey.Inincotrpheorafollotingwing,sensorythosecues,twoandmothdeuslsaren-e
isaablingsupperperceptioositionnooffhigtheh-bolevdyelinandlorelatiow-nlevtoeltheactuaenvirtiononmenwhicht.depTheendsoconceptntheofovineralltegrabotiondy
balance.Onlyifthebalanceisconsideredtobeatriskorconsciouslyinﬂuenced,the
high-levelcontrolinﬂuencesthesteppingtask.
Inthefollowingtheintegrationofhigh-levelandlow-levelmodelsrepresentsapossibility
istoexaexpaminedndthebywloorkinokinggatrangtheeoffollothewingsteppingfourmoexemplarydel.Theprpoblemserforofmancetheoflothew-levineltegramodetionl:
1.isFirst,antheinitialvstabilitalueypdeproblem;endsonincothentrainitstialtovathisluesaashtheumanblimiteingcyccalenstapproartaximastationble
steppingmovementwithanynormalinitiallegposition.
2.Second,anasymmetricsteppingpatternisnotcorrectedinthelow-levelmodelbut
repeatsitselfwhichleadstoadriftinthemovement.
3.Third,unsuitablefeedbacktuningcanleadtoinstability.
4.Fourth,thesidewardmovementtendstodriftbecausethedirectionofmovementis
determined.notSection5.1presentsanumberofstate-of-the-artmodelswhichcombinesteppingmove-
thementsmodelwithabndodythestamostbilizatiocon.mmonHereaprinciplesspecialofmenrobtionotics,isamasderoboofticstheiscotherrelatﬁeldionbwithetwteehen
larsitiongestcovnatrrietolyofconceptlow-levuseledfoandrthehigh-inlevetegralcontiontrolisderivprinciples.ed.InInsseectioctionn5.35.2thetheintsupegraerptedo-
modelcomposedoftheindividualcomponentsisintroduced,adaptedtothestance-control
conceptandextendedtoincludethepossibilityofadditionalhipcontrol.Theresultsof
thenotsimleast,ulatioinnofsectionsthe5.5fourandcas5.6esthepropoconsedtrolaboveconceptisisdetaileddiscussinedsecatndion5.4summed.And,upolanstbasbutis
ofthesefourrepresentativestabilizationcases.

415

5.1StateoftheArtofIntegrationModels

Inroboticstherearemanyintegrationconceptsastheseareessentialforbuildingastable
walkingrobot.TherearesomeimpressiveexamplesofwalkingrobotssuchastheHonda’s
ASIMO[173],robotsJOHNNYanditssuccessorLOLAdevelopedbytheTechnische
Universit¨atM¨unchen[42]orthebiologicallymotivatedRunBot[41].Inroboticsystems
nodiﬀerentiationisnecessarilymadebetweenhighlevelandlowlevel,bute.g.theexact
position,velocityoracceleration,areusedinthesamewayasoverallinformationina
centralcontrolunitoracentrallymastereddistributedcontrolsystem.
Controlandgenerationofactuationisoftendividedintopatterngeneration,e.g.by
trajectorygenerationandoverallbalancecontrol.Therearevariousexamplesofthis
division,e.g.trajectoryplanningandcontrol[173,42,104]whichisusedinASIMOand
JOHNNYtocontrolthemovementsofalllinks.InRunBotsensoryperceptionisdirectly
integratedintheneuronalactortocontrolthelinkmovementdirectlyandlocally[105].
Theoverallbehavioristrainedbyahigh-levellearningalgorithmfortheneuronalsystem.
Onediﬀerencebetweentheoreticalmodelsandtheroboticsystemsnamedaboveisthe
groundcontact.Intheorythegroundcontactisoftenmodeledinstantaneously,asin
thisstudy,butinarealrobotthedoublesupportphasehasadeﬁnedduration.This
durationisveryimportantforstepcontrolandthesmoothnessofstepping[160]because
thisphaseincludestheslow-downofthelaststepandthespeed-upwiththenextstep.In
realsystemsthisphaseisnotasabruptaswithaninstantaneousmodelandisdependent
onseveralgroundcontactconstraintstocontrolstepparameters[103].
Alargegroupofrobotsarebasedontrajectorieswhichareeitherprecalculated[83,164]or
calculatedonlineonbasisofmodels[84,191].Thosetrajectorycalculationsdependonthe
modelandrequireaccuratestateinformatione.g.theinertiaofthelinks,accelerationof
bodypartsetc.InSobotkaetal.[164,196]theprecalculatedgaittrajectoriesaremodiﬁed
onlinebyaJacobicompensation.Withthismethodtrajectoriesareadaptedinthecase
ofunexpecteddeviationsfromtheprecalculatedsituations.Thisisachievedbyadding
totheprecalculatedvalueajointspacetransformationwhichcorrectsthemovementto
followonedirection.
TheoverallbalancecontrollersareagroupwhichKajita[84]termesZMP(ZeroMoment
Point)controllersbecausetheZMPisoftenusedtocontrolthebalance.Thismeans
tomaintainbalancetheZMPiscontrolledwhichmeansitiskeptintheallowedrange
forexamplethesupportfootareaoradesiredposition.Manysimilarbutdiﬀerent
explanationshavebeengivenfortheZMP.Onepracticalexplanationisthatpublished
byArakawaandFukuda[2],whostatethattheZMPisthepointonthegroundwhere
allmomentsgeneratedbyreactionforcesandreactiontorquesareinbalance,thepoint
atwhichtheirsumiszero.ThisZMPcontrolrequireseitheranexactknowledgeofthe
dynamicsofthebodymechanicsandtheirstates,oramodel-basedapproachincluding
prediction.Thelattergroupofsystemswhichuseroughknowledgeintheformofamodel
ofthebodydynamics,e.g.theZMP,aremainlydependentonfeedbackinformation.
Inthiscontextaninvertedpendulummodelisoftenusedtorepresentsimpliﬁedbody

515

5IntegrationofHigh-LevelandLow-LevelModels

].84[hanicsmecIfasystemisgenerallyunstableorhasaninternaldynamicaccordingtothetheory
ofnon-minimal-phasesystems,suchasaninvertedpendulumwithunstablepole-zero
compensation,astaticstatefeedbackisnotsuﬃcienttostabilizethesystem[71].As
invariantcontrolfeedbackisnotsuﬃcienttostabilizesuchzero-dynamicsystems[71]the
directrelationbetweencontroltorquesandZMPdynamicsinaloophastobebroken
bypredictionsorsubstitutions.Therearethereforseveralcontrolapproachesinrobotics
whichcombineZMPfeedbackcontrolwithanadditionalapproach.InrobotJOHNNY
[104]theZMPcontrolissubstitutedbydirectcontactforcecontrol,onceithaslefta
permittedrangefortheZMParea.AnotherexampleisfoundinSobotka[163],wherea
nominalZMPcontrolissubstitutedbyprecalculatedtrajectories.Thissubstitutionoccurs
whentheZMPleavestheallowedZMParea.Thebalancecontrolisthendeterminedby
invariantcontrolofdegree-oneZMPfeedbacklinearization.Amulti-controlapproachis
usedbyKimetal.[90],wherethenominaltrajectoryplanningiscomplementedbythree
additionalcontrollerstoachievebalancecontrol.OnecontrolleristheZMPcompensator,
whichcompensatestheinstabilityonthebaisofthepolesofzero-dynamicsystem.
Anotherclassofcontrolstrategieswhichareappliedtogeneratewalkingmovementsare
themodelswhichusepredictivecontrol,oftenincombinationwithoptimalcontrol,to
achievee.genergyeﬃcientgaitpatternsortoenforceperiodicityofthetrajectoryfora
periodicgait.AnexampleisgiveninMorimotoandAtkeson[130],whereoptimizationis
achievedbyapplyingamodiﬁedcriterionoflowtorquesandperiodicitywhichincludes
disturbances,andthereforeresultsinrobuststablewalking.In[176]thebodydynamics
aremodeledbylinearizedpendulumdynamicsandthemovementispredictedinorder
tocontroljointimpedance.Kajitaetal.[84]usesacombinationofmodel-basedZMP
controlwithpredictivemovementcontroltogeneratestablesteppingmovements.In
Wieber[191],too,amethodisproposedforgeneratingwalkingmovementsbyprediction
oftheZMPmovementandoptimizationofmaintainingtheCOM(CenterofMass)ata
t.heightnconstaThebiologistCruse[24]oncesaidthatacharacteristicbehaviorofthebiologicalsystem
istheautonomyofthemovementparts.Thismeansthatnotonlythebrainbutallparts
likee.g.muscles,neuronshavetheirownrulesandplanstofollowtoachieveaﬁnal
successivemovement.
Thereforeinthemodelofthisthesisthelow-levelcomponentforsteppingisleftau-
tonomousaspresentedinchapter3.Thereforethelow-levelsteppingcontrolitselfisnot
directlyinﬂuencedbutindirectlybecausethetwolevelsaresuperposed.Thesuperposi-
tionofhighandlowlevelactuationisdependentontheoverallbodybalance.Onlyifthe
balanceisconsideredatrisktheadditionalhigh-levelactuationisapplied.Further,the
modelofthebodydynamicsinthebrainissimpliﬁedverymuchtotheabstractinverted
pendulummodeloftheCOM.Thehigh-levelsensorintegrationpartofchapter4isused
topredictandcontroltheCOMtostayinanormalrange.Withnormalarangeiscon-
sideredtobestablewithhighprobabilityandthefallriskislow.Thecontrolisachieved
bysuperpositionoflow-levelactuationwithhigh-levelcorrectivetorquesifthenormal
rangeoftheCOMisleft.Theinteractionofthesetwopartsisshowninprincipletogive

615

5.2ControlStrategyfortheSteppingModel

andideaaboutthepossibleinﬂuencesanddependenciesandnottogiveafullanalysisof
thecontrolproblem.

5.2ControlStrategyfortheSteppingModel

corrTheectiofourn,(2)scenarioprevsendesctiornibeofdindriftingtheintmorovemenductiotsno(3f)thisimpropcerhapter,fee(1dba)cinikgatialinsaconditiond(4)ns
steppingtothesidearepresentedtoshowtheapplicationofthehigh-levelcontrolmodel
tothelow-levelsteppingmodel.Thisapplicationneedsaslightlyextendedcontrolconcept
thanthelinearquadraticfeedbackcontrollerofthestancecontrol.Thisisbecausenow
notwhichonlyhavethetostabencelegconsidered.butInalsothehipfolloandwingswitingislegnotmothevoemebntsjectivareetocpartonstrooflthethesysgeneraltem
isrhgyenethmicratedbsteppingythemolowv-levemeneltbutneuronathelglevlobalel.sIftoneabilityimagandinesbaalance.steppingAsbbefoodyreittheisobsteppingvious
thatthehigh-levelsensorsonlyperceivemovementsofthewholebodywhichinturn
leadstothebraingeneratedcorrectivecommands.Thiswholebodyisrepresentedbythe
centerofmass(COM).Biologicallythiscanbeinterpretedasthebalancepointofabody
accordingtogravity.Thisisselectedbecausethevestibularorproprioceptivesensors
prelaterceionivebtheetweenwholethebodywholeswabyodyacceleratmovemionentorpandositiontheamondvtheementvisuaoflssurroensepeundings.rceivestThishe
isasimpliﬁcationofthebody.Thebraindoesnothaveanaccuratemodelofallbody
partswhichleadstothefactthattheposturecontrolKalmanestimationbasesonan
inversependulummodelrepresentingtheCOMmovementaccordingtothestanceleg.
Thissimpliﬁcationcomesupforinaccuraciesandnotwellexperiencedmovementsand
aninadequaupandciesdoowfnthemovbraemenintiscomputsensedatiobnsy.actualAdditiosensnallyorythedatahipandmovCOMementpredicwhichtrionevsultsaluesin
tocontroltheverticalhipmovementinrelationtotheCOM.
tioThenalgeconerantrolliffeedbactheksystemcontrolworksconceptintheappliednormaltorathengelowandlevanelaisdditiodivnalidedinhigh-letwo:velnoconaddi-trol
ifthesystemleavesthisnormalrangeandisatarisktobecomeinstable.So,whenev-
erythingis’normal’thesteppingisanautomaticautonomousmovement.Butifstability
isisthreatappliedenedasorsuperpsomethingositionisofcothensaciousctuallyinmovtendedement.anIanthedditionafollolwingcorrectivtheecoconceptntroloftsignalhe
’normal’rangeandtheaccordingcontrolisintroduced.

5.2.1FeedbackLinearizationTheory

ASISOnegaortiveMIMOfeedbacsysktemislinearizatderivionedfoinrathetimefolloinvwing.ariantWithcontrothelsystemnonlinearforansysteminputiswrittoutputen

715

5IntegrationofHigh-LevelandLow-LevelModels

withinputvectoruandoutputvectoryas:
q˙=f(q)+g(q)∗u(5.1)
y=h(q)(5.2)
whereqisthevectorofsystemstatesandfisafunctionofthedynamicsofthesestates,g
isthemappingofthestatesandtheinputsonthestatesandhtheoutputfunction.Now
theoutputyisdiﬀerentiatedtilltheoutputisanyfunctionfoftheinput∂y/∂q=f(u).
is:tiontiadiﬀerenThey˙=dh(q)q˙=dh(q)f(q)+dh(q)g(q)∗u(5.3)
dqdqdqIfnowtheequationofthen-thdiﬀerentiationissetto∂yn/∂qn=f(u)=a(q)+b(q)∗u=
v(q)whereaandbarematricesandbiscalledtheinvertiblematrixforMIMOsystems.
Therefromthevalueofucanbedeterminedanalyticallyby:
u=b−1∗(−a+v)(5.4)
AcommontermtodescribethediﬀerentiationistheLiederivative.Thederivativein
equation5.3iscomputedusingthechainrule.TheLiederivativeofh(q)isdeﬁnedwith
respecttof(q)as:
Lfh(q)=dh(q)f(q)(5.5)
dqAndsimilarly,theLiederivativeofh(q)isdeﬁnedwithrespecttog(q)as:
Lgh(q)=dh(q)g(q)(5.6)
dqThisnotationleadstotheexpressionofy˙:
y˙=Lfh(q)+Lgh(q)∗u=v(q)(5.7)
ntimesdiﬀerentiationoftheoutputyleadsto:
y=h(q)=z1
y˙=Lfh(q)=z˙1=z2
y¨=Lf2h(q)=z˙2=z3(5.8)
...y(n)=Lfnh(q)+LgLfn−1h(q)∗u=z˙n=v(q)

815

)(5.7

.8)(5

5.2ControlStrategyfortheSteppingModel

Thecontrolinputucanbederivedasbeforeinequation5.4with:
1u=LgLfn−1h(q)(−Lfnh(q)+v)

Todesignvalineartermcanbeusede.g.atermoftheform:
v(q)=−k0∗y−k1∗y˙−k2∗y¨...
Withthisdesigntheobjectiveisthatthevaluesruntotheirdesiredvalues:y→yd.
Ifthereareinternaldynamics(zero-dynamics),sothatthesystemrequiresaperfect
pmodelossibilittoyacishievaneaarodditiobustnalcorontrol,bustnessthentermtheaodderderdoftothetheconcontrotlroltearmccordinghastotobe[58]raised.e.g.Aa
dynamicextensionwhichconsidersthezero-dynamics.

5.2.2AppliedFeedbackLinearizationforHipMovements

Asthemodeldoesnotcontainatrunkmodel,arepresentativemovementofthewhole
bodyisgivenbythecenterofmassCOM.Themodeledangularpositionsandvelocities
aretheCOMpositionandvelocitiesinmedio-lateraldirection.Additionally,thereisa
modeloftheverticalmovementoftheCOMasbiggerhipmovementsresultinavertical
movementwhichcannotbeachievedbyaninversependulummodel.Itisavertical
movementaccordingtogravitation.
TheverticalpartoftheCOMcomywhichisalsoperceivedbythesensorycuesisdeter-
y:bminedy=comy=f(α,β)=1/MG∗((2∗M∗l+3∗m∗l)∗cos(α)+(M∗h+2∗m∗h)∗sin(β)(5.9)
wheretheanglesandmassesareasintroducedinthemechanicssection2.3inﬁgure
2.4and2.5.ThederivativeofcomyaftertheverticalpositionandvelocityoftheCOM
isbuilttogetadirectrelationbetweenthecomyandtheexternalinputubtothehip
joint.Thisisthefeedbacklinearization,accordingtoequation5.3.Theup-and-down
COMmovementforbighipmovementscanbedescribedasamovementoriginatedby
gravitationandexternalinputub.Thehipmovementoriginatesfromgravityandapplied
jointtorqueswhichinteracttostabilizethehipwhichleadtothefollowingsimpliﬁed
equation:x˙01x0
x˙21=00∗x21+ub−g(5.10)
wherethevectorxaretheverticalhippositionandvelocity.They=comyisdiﬀerentiated
accordingtoy˙=com˙y=∂∂cox1my∗x˙1+∂∂cox2my∗x˙2twotimesuntiltheinputuispartofthe

915

5IntegrationofHigh-LevelandLow-LevelModels

equationaccordingtoequation5.8whichleadsto:
y=z1y∂z2==Mv∗cos(x1)∗x˙1=Mv∗cos(x1)∗x2=Lf1(5.11)
x∂z3=−Mv∗sin(x1)∗x22−M∗cos(x1)∗x˙2=Lf2+Lg(Lf1)∗u
withMv=M∗h+2∗m∗h.Byinsertionofequation5.10intoequation5.11thefollowing
btained:oisionequatLf2+Lg(Lf1)∗u=Mv∗sin(x1)∗x22−Mv∗cos(x1)∗g+M∗cos(x1)∗ub=v(5.12)
Nowv=k0∗(comy−comy0)=k0∗(y−yd)isalinearcontrollerwithydisthedesired
andnormalpositionoftheCOMcomy0.Thisleadstotheequationofcontrolinputub:
ub=v+tan(x1)∗x2+g=−k0∗(y−yd)+tan(x1)∗x˙2+g(5.13)
Mv∗cos(x1)2Mv∗cos(x1)1
Thiscalculatesthecorrectiveinputforthehipmovementwherex1isthehipposition
derivedfromtheangularhipmovementwhichisderivedasx1=h∗sin(β)(forthissee
ﬁgure2.5).ThisisproportionaltotheactualCOMpositionminusthecomy0valuewhich
dependsmainlyontheactualstancelegangle.x2istheverticalvelocityofthehipwhich
isapproximatedbythevelocityoftheverticalCOMmovementwhichisvalidifthehip
movementgetslarger.Thisisexactlythecasewhentheadditionalcontrolisused.

5.2.3AppliedPreviewControlandOptimizationCriteria
pendulummovementaccordingtoequation2.4withthestatevectorq=ΦcomΦ˙com.
First,thepreviewcontrolisusedfortheCOMposition.TheCOMismodeledasaninverse
IntheKalmanestimationtheCOMmovementispreviewedtobecontrolledbyanopti-
mizationcriterionlikeinsection4.3.4.TheCOMmovementisperceivedbythesensory
systemandthusintegratedintheposturecontrol.
TherealCOMofthefrontalplanemechanicalbodyiscalculatedasfollows:
Φcom=tan(comx)=Mc∗sin(α)+Mv∗cos(β)+m∗l∗sin(γ)(5.14)
comyMc∗cos(α)+Mv∗sin(β)−m∗l∗cos(γ)
withMc=3∗m∗l+2∗M∗landMv=2∗m∗h+M∗h.TheCOMposition
determinedinequation5.14isthesensedbodyCOMmotionbyproprioceptive,vestibular
andvisualsenses.Thisbodymotionisrepresentativeforonesinglestepfromdouble
supportphasetothenextdoublesupportphaseandisapproximatedbyaninverted
pendulummotion.ThemodeledCOMisrepresentedbytheinversependulumwithangle
andangularvelocitywiththefootoftheCOMpenduluminthestancefoot.Themodeled
COMmassisrepresentedbyonesinglemasswhichisdeterminedbythesumofallreal
bodymasseswhichisMG=2∗m+M.ThemodeledCOManglerelatesmainlytothe

016

5.3AppliedIntegrationModel

stancelegangleasthehipmassonlycontributestothesinusofthehipanglewhichis
asmallvalue.ThedesiredCOMangleisΦdandthedesiredangularCOMvelocityis
zero.Theprinciplesoftheoptimalcontrolaccordingtoquadraticminimizationcriteria
wasexplainedinsection4.3.4ofchapter4high-levelbalancecontrol.Thecriterionto
minimizetheangulardeviationofpositionandvelocitycanbedirectlyappliedtotheCOM
inversependulumpositionandvelocity.Theperformanceisspeciﬁedbytheoptimization
indexderivedfromequation4.40.TheﬁrstJaisforthependulumCOMmovementand
thesecondJbfortheverticalhipmovement:
∞Ja=qiT∗Qx∗qi+uaT∗R∗ua
k=i∞Jb=eiT∗Qe∗ei+ubT∗R∗ub0
k=iThetwocriteriatooptimizetheCOMmovementbyJaaretherefore:
c1,c2=q∗qTwithq=[Φcom,Φ˙com]withQx=wRQ∗00.50.05(5.15)
withwQRisapositiveweightingfactorofthematrixQxinrelationtomatrixR.AndRis
theunitymatrix.Thethirdcriterionc3tominimizeJbistheoptimizationofthevertical
motionoftheCOMvalue,becausethisisnotadequatelymodeledwiththependulum
equation.Theverticalhipmovementaccordingtothehipanglewhichish∗sin(β)is
notmodeled.Sothecomygoesupanddownduringastep.Ifsomethingunexpected
happensthecomyisdestabilizedmorethanduringanormalstepandthereforehasto
bestabilizedindependentlyofthependulumCOMmovement.Thereforethecomyis
feedbacklinearizedwiththeinputubtogettherelationbetweeninputandoutput.The
valueofthecontrolfactork0isdeterminedbytheoptimizationcriterion.Thevertical
COMpositioncomyisaddedasathirdcriterionwhichdependsonthevalueofthevertical
hippositionx1andbecomesminimalbythehipanglegoingtobezero:
ce=(x1)2Qe=10(5.16)
00ub0iscalculatedfortheboundaryconditioncomy0oftheverticalCOMmovement.If
equation5.13iscalculatedwiththisvalue,thisgivestheestimationofthefactork0for
thefeedbacklinearizedverticalCOMcontrolbyoncesolvingtheequationfork0.

5.3AppliedIntegrationModel

Thecontrolissplitupintotwoparts:thelow-levelcontrolwhichisalreadyrealizedbythe
muscularpositionvelocityfeedbackfunctionandanadditionalcorrectivecontrolwhich
isappliediftherangeofnormalmovementsisleft.Thisrangeofnormalmovements

116

5IntegrationofHigh-LevelandLow-LevelModels

.18)(5

isdeﬁnedaccordingtotheinversependulummodelforthestancelegandanormal
droppingandliftingofthehipwhichiscommonforsteppingmovements.Foreachof
themovementdirectionsarangewithupperandlowerboundsisdeﬁnedwhereinthe
movementis’normal’andthoughnotcontrolledadditionallywithcorrectivecontroluc.
uuThesuperpositionis:
a1u=uact+uc=u2+ub(5.17)
0u3Thesuperposedtorquesareaccordingtothenormalrangeasfollows:
uc(1)=0,ifΦcomlow<xˆ<Φcomhighor
ifΦ˙lcomow<xˆ<Φ˙higcomh
uc(1)=uaelse
u=uc(2)=0,ifcomylow<xesti(comy)<comyhigh(5.18)
uc(2)=ubelse
uc(3)=0
giveninsection4.3.4.HerexˆistheestimationoftheCOMstatesΦcomΦ˙com.ubis
withua=K∗xˆaccordingtotheminimizationofJausingmethodandequation4.41
calculatedwithequation5.13byusingk0calculatedwithequation5.16andxesti(comy)
istheestimationoftheverticalCOMdisplacementaccordingtotheestimationmodelof
theCOM.ThevaluesΦlcomow,Φhigcomh,Φ˙lcomow,Φ˙higcomhcomylowandcomyhigharetheupperand
lowerboundariesfortherangeoftheCOMmovementwhichisconsideredtobe’normal’
andthereforestable.IftheverticalCOMmovementislargerthanthenormalrangeit
iscertainlyduetohipmovement.So,thediﬀerencebetweenthedesiredverticalCOM
positionandtheactualpositionisrelatedtothesinusofthehipangleβ.Thedesired
valueofthetheabsoluteverticalbodyCOMdependsalsoonthestanceleg,becausethe
moreverticalthestancelegisthehigherisdesiredvalueyd=comy0.Thecomydepends
ontheinputtorqueofthehipubasdescribedinequation5.13.Thevaluecomyisnot
staticandchangesthecontrolconditiontherefore.Thevaluesofuc(1)arederiv0edfrom
thepredictivecontroloftheoptimizationcriteriaverysimilartotheposturecontrolin
stancebefore.Thevalueuc(2)isderivedfromthefeedbacklinearizedcontrolofequation
.35.1Thecompletesystemisshowninﬁgure5.1wherethehigh-level(blue)andlow-level
(orange)partsareintegrated.Thetorquegenerationisthesuperpositionoflow-level
oscillatororiginatedtorquesuactandhigh-leveltorquesuc=ua,ub,0whichare
addediftheCOMpositionleavesthenormalandasstableconsideredregion.Onlow-
levelthebodystatesareperceivedviathemuscularfeedbackfunctiononhigh-levelthey
areperceivedviathesensors.Thehigh-levelstatisticalestimationestimatestheCOM
positionandvelocityasanglesandangularvelocitiesΦ,Φ˙anditsestimatedvertical

216

5.3AppliedIntegrationModel

corrcompectivonenetˆconcomtryol0towhicrques.hisIncompatheredfollowitwinghtthehreerealexsensedamplesvaofluesupcomyerimptooseddetermcorrectivinethee
controlareshowntodemonstratetheeﬀectonsteppingstabilityandsidewardsstepping
t.emenvmo

Figure5.1:Integrationoflow-(orange)andhigh-level(blue)systems.Thesensorsare
dividedintolow-levelfeedbackfordirectmuscularfeedbackandhigh-level
feedbackforothersensorycuesprocessedinhigherlevels.Herethehighlevel
estimationisabstractedtotheCOMpositionandvelocityΦ,Φ˙,comy.The
high-levelcontrolisappliedbysuperpositiontothelow-leveljointtorques
.utac

316

5IntegrationofHigh-LevelandLow-LevelModels

5.4SimulationofLow-LevelSteppingMovementswith
High-LevelPostureControl

Thefourexampleswhicharestabilizedorimprovedwiththeintegrationofhigh-level
informationviasensorsandposturecontrolprocessingare:
(1)badinitialconditions,
(2)asymmetricsteppingpatternswhichleadtoadrift,
(4)improperfeedbackgainswhichleadtoadestabilizationand
(3)steppingtotheside.
Formoreconveniencetheoriginallow-levelsimulationsarealwaysrepeatedlyshownbelow
theimprovednewsimulationsinsmallersize.Thereferencedﬁguresincludebothﬁgures,
theoriginalﬁguresandthesmallrepeatedﬁgureswithtwodiﬀerentreferencenumbers.
Inﬁgure5.2case(1)withbadinitialconditionscanbeseenwithadditionalhigh-level
control.Itshowstheoriginallow-levelmovementinﬁgure5.3whichisrelatedtothe
caseseeninﬁgure3.29.Ifunsuitableinitialconditionsarechosen,whicharethough
stillrealisticangularstartingpositions,nocorrectionoftheinitialinadequacycanbe
achieved.Thishappensbecausetheautonomouslow-levelsteppingmodelonlycontinues
orreproducestheinitialconditionswiththeprovidedlocalfeedbackandnocorrectionof
theglobalposition.Inthisstudy,thehigh-levelposturecontrolisusedtocompensate
initiallyunsuitablevaluesbycorrectingthem.Thereafter,themovementispushedback
totheattractivebasinofthelow-levelsteppingmovementandismaintainsstablewithout
anyadditionalhigh-levelcontrol.Thevaluesareallinthenormalrangeagainandare
notevaluatedasarisktofall,duetounsafeCOMpositions.Theadditionalcontrolat
thebeginningdrivesthesystemintoastablemovement.
Inﬁgure5.4case(2)withanasymmetriclegmovementleadstoadrift,whichresultsin
increasinglegandhipangles.Thiswouldbelikealimpingwithaslowincreaseofthe
limpinglegangleduetotheasymmetry.Afterseveralmorestepsthiswouldleadtoa
falldown.Theoriginalﬁgureforthismovementcanbeseeninﬁgure5.5whichisrelated
toﬁgure3.22.TheresultingCOMmovementisadriftandespeciallyanincreaseofthe
lateralCOMmovement.Withtheadditionalcontrolthismovementcanbeadaptedtoa
steppingmovementwithsymmetricsteps;afteralargerﬁrstreactioninordertocorrect
theinitialasymmetricmovement.Thefollowingsymmetricstepsarestablebecausethey
areattractedtoastablelimitcyclemovementafterafewsteps.
Steppingisunstableifthefeedbackisnotmodeledcorrectlyasincase(3)orifthe
feedbackisnotcorrectbecauseofe.g.,alongerinjury.Figures3.30and3.31andthe
repeatedlyshownﬁgure5.7visualizetheoriginalmovementwithunsuitablefeedback(in
ﬁgure3.31itistheﬁrstplotinthesecondlinewithfdv=0.7,fd=0.3).Theinstability
occursbecausethefeedbackisnotappropriateforthemovementandactuationpattern.
Withasuperpositionofthehigh-levelcontrolastablesolutionofsteppingisachieved.In

416

5.4SimulationofLow-LevelSteppingMovementswithHigh-LevelPostureControl

(a)Angularpositions

P(c)lotphase

(b)OscillatoractivationF14

(d)Movementofthreestepswithinitial
overshooting

Figure5.2:Steppinginplacewithbadinitialconditionsisconvergedtoastablestepping
inplacemovementbypushingthesolutionbacktotheattractivebasinofthe
limitcyclewiththeadditionalhigh-levelCOMcontrol.

(a)Angularpositions(b)OscillatoractivationF14

phaseP(c)lot

Figure5.3:Theoriginalsteppinginplacemovementwithbadinitialconditionsisunsta-
ble.

516

5IntegrationofHigh-LevelandLow-LevelModels

(a)Angularpositions

lotphaseP(c)

(b)Oscillatoractivation

(d)Movementoftwostepsafterset-
tlingasstablesolution

Figure5.4:Steppingmovementwhichisunsymmetricalandthereforedriftsandmovesto
theside.WithadditionalCOMcontrolinitialsteppingtothesideisconverted
toperiodicandsymmetricsteppinginplace.

theﬁrststepwhenthenormalmovementrangeisleft,thesuperposedcontrolstabilizes
themovement.Thiscanbeseeninthefollowingﬁgure5.6.
Thestabilizationisveryquickbecausethemodelistooideal.Duringaninjurythe
actuationwouldbeweakenedandnotactingwithfullstrengthandthelocalmuscular
structuresandmechanicswouldbeslightlychanged.Thisisnotmodeled,andthecontrol
isalsooptimal.Thesuperposedcontrolleadsthereforetoaninstantadjustmentofthe
deﬁcits.Inﬁgure5.8thecase(4)isshown,asidewardssteppingmovement.Here,theoriginal
movementisshowninﬁgure5.9.Thesidewardsmovementisnotstablebecausethe
singleangleshaveadrift.Thelegstogethermovementisnotexactlytheoppositeofthe
legsapartmovementwhichleadstothesidewardsmovement.Withthehigh-levelcontrol
severalstepstothesidecanbeachievedwithoutafallortendencytoinstability.The

616

(a)Angularpositions

(b)Oscillatoractivations

Discussion5.5

lotphaseP(c)

Figure5.5:Originalsteppinginplacemovementwithadriftmovementinonedirection
asthestepmovementisunsymmetrical.

graphicsshow20stepstotheside.

oncussiDis5.5

Thehigh-levelposturecontroldeﬁnesacontrolfortheoverallbodypositionwhichis
supposedtobeasstableaspossible.Theintegrationbetweenhighandlowlevelwas
achievedbyasuperpositionprinciple.Ifthebodyleavesthenormalmovementrangeand
isatrisktofallorbecomeinstable,additionaltorquewasappliedtobringthesystem
backtonormalrange.Therearenogivenvaluesforthetrajectoriesbecausetheyare
determinedonlineinthelowlevelandtheyarenotpredeﬁnedbutvariableanddynamic.
Thisisdiﬀerenttotheclassicroboticsystemsase.g.in[143,173,29,11].Or,as
Lydoireetal.said“Bipedrobotcontroltechniquesareusuallybasedonthetrackingof
pre-computedreferencetrajectories.Therefore,toachieveautonomyinlocomotion,itis
necessarytostoreasetoftrajectorieshandlingallthepossiblesituationsandevents...”
[103](p.749).So,theadditionalcontrolcannotgenerateacontrolinputwhichisclose
totheplannedtrajectorycontrol[163]orsubstitutedbyadirectcontrol[104]butitis
additional.Itissuperposedwiththelow-levelbutdoesnotinﬂuenceitdirectly.Tosay
itinotherwordsthetwoactuationsaresuperposedlikeanemphasizingofsomethingor
tooverrulethelowercontrolbutnevertoreplaceit.
Thenecessityofexactstatedata(e.gpositions)andofcoursetheexchangeofalldata
betweencontrolandlocationofdatagenerationleadstoahighcomplexityinrobotics
whichwasmentionedbyKajitaetal.[84].Withtherelativelysimpleadditionalcon-
trolconceptproposedinthiswork,itispossiblethatthesystemdoesallthestepping
movementswhileenlargingitsstabilityrangewithlowcomplexityandlowdataexchange
rate.Thisisachievedbyusingasimplemodeltoapproximateandestimatethewhole
bodyCOMtodeterminethehigh-levelcontrol.Thereforetheamountofexchangeddata
isreduced,whicharetheCOMsensedpositionandvelocity.

716

5IntegrationofHigh-LevelandLow-LevelModels

(a)Angularpositions

lotphaseP(c)

(b)Oscillatoractivation

(d)Movement

Figure5.6:Steppinginplacewithslightlyunsuitablemuscularfeedbackgainswithfdv=
0.7andfd=0.3.Thesuperposedhigh-levelcontrolleadstoastablestepping
tern.pat

(a)Angularposition

(b)Oscillatoractivation

lotpPhase(c)

Figure5.7:Orbackiginaglainswsteppingithfdvin=pla0.c7eandmovfdemen=0t.3,withwhichslighletalydtounsinsuittableabilitmy.uscularfeed-

816

(a)Angularpositions

lotphaseP(c)

Discussion5.5

(b)Oscillatoractivation

(d)Movementoftheﬁrst11stepsto
deiseth

Figure5.8:Thegraphsshowsteppingtothesidewhichoriginallyhaddriftinglegangles.
Withsuperposedhigh-levelhipandCOMcontrolthemovementstayswithin
thenormalregion.

(a)Angularpositions

(b)Oscillatoractivation

lotphaseP(c)

Figure5.9:Originalsteppingtothesidemovementwithdriftinglegangles.

916

5IntegrationofHigh-LevelandLow-LevelModels

feeThedbasimck,ulatiounnusualaresultsctuatioshonwstthatrategiesinapprandopriateasymmetricconditiopnsositioasnsinitiaalrecosnditiouccessnsfully,aﬀecstatebi-d
lized.Whenthemovementinstabilityreachestooriskyandextremevaluesthelow-level
controlissupportbyhigh-levelcontrol.Inallsimulatedcasesthemovementcouldbe
mobrovughementbt.ackThistotheleadsstabletoaatprevtractiventeionofregionabnoofrmaltheplimitositions.cycleItortisoanotnoduermaltorﬁxangecoon-f
straintsase.g.in[103](initialfootpositionanddistance,relationofCOMtostance)but
withasimplesuperpositionprinciple.
Intoocoextrememparisopntoositio[17ns]noareprecisconsidered.epositioThenofproapolegsedorhigach-onlevtaelctcopnotrointlisstracalculategyistednootnlyto
guaranteeastablerangeforallpositionrangesandasitisexactlydeﬁnedandmodeled
intheroboticswithe.ganalyticsolutionsofinversekinematics[191]butitpushesthe
systembackintonormalrangeandthesystemgoesbacktoalow-levelstablemovement
ifthereexistsalow-levelstablemovement.

5.6Conclusion

Concludingitcanbesaid,thatmanyshortcomingsofthelow-levelactuatedstepping
modelascriticalinitialvalues,accuratefeedbackgains,changeofactuationstrategyand
steppingtothesidecanbeachievedwithasuperposedhigh-levelposturecontrol.Forall
simulationsitischaracteristicthattheadditionalcontrolismainlyappliedtostabilize
thestancelegwhichisthemostcriticalparameterofkeepingbalance.Alsothehip
movementsarehigh-levelcontrolledbutthisismainlytopreventanunnaturaldynamic
rangeofthehipandtosmooththemovements.Theoriginalunstablesteppingpatterns
areallstabilizedbybringingthesystembacktotheattractionbasinofthelimitcycle
n.solutioThelowleveloscillatorsarenotdirectlyinﬂuencedbythehigh-levelposturecontrol,only
thevariedpositionsoflegsandhipinﬂuencetheoscillatoractuationviathemuscular
feedback.Iftheoscillatoractuationisnotsupposedtobesolow-levelbutisinﬂuenced
bythehighlevelmoredirectly,likeanadditionalcontrolinput,thiswouldleadtoa
mechanismwhichrelatesthetwolevels.Especially,forinﬂuenceswhichchangethestep-
pingmovementforlongerlikeinjuriesortrainingadirectinteractionofbothmechanisms
constitutesaninterestingenhancement.
Thehigh-levelcontrolproposedinthisworkconsidersthestabilityofwholebodyposture
whichleadstoimprovedstabilityofsteppingpatterns.However,acontrolofthestepping
movementwhichisconsciouslyinﬂuencedastheincreaseofsteppingfrequency,thechange
ofsteppingstrategyorthedirectionofmovementisnotyetconsidered.Though,there
havebeenshowntheinterfaceparameterstoinﬂuencethesteppingpattern.Andthe
variationsofthoseinterestingparametershavebeenanalyzedtoshowthepossibilities
buttheimplementationofsuchparametervariationsonahigh-levelislefttofuture
h.researc

017

6SummaryandFinalConclusion

Inthisthesisanewmodelforfrontal-planesteppingmovementswasdevelopedinor-
dertoevaluatemedio-lateralmovementsduringgait,andinvestigateinﬂuencesexerted
bysystematicparameterchangesonthemodel,itsstabilityanditsmovementabilities.
Themodelingwascarriedoutonthebasisofbiologicalprinciplesandusingabottom-up
approach,whichmeansthatthestartingpointfortheintroducedmodelisassimpleas
possible,andthemodelisenhancedconsistentlythroughouttheworktoextenditsabili-
tiesandperformance.Thisapproachstandsincontrasttoconventionalroboticsolutions.
Theprincipalextensionstothemodelwerealsoappliedtoasagittal-planewalkingmodel
toshowthatthemodelpresentedisalsoapplicabletoothermovementplanes,andto
showthatthefoundationshavealreadybeenlaidfortheplannedintegrationofthetwo
planestoa3Dmodelinfuture.

Duetothebottom-upapproachthemodelwassplitintoalow-levelandahigh-level
modelcomponentinlinewithbiologicalprocesseswherelow-leveltasksarethemore
automatictasksandhigh-leveltasksareprimarilydirective.Forthelow-levelmodelbal-
listicmechanicaldynamicsareapplied.Thedisadvantagessuchasasmallstabilityrange,
dependencyoninitialvaluesandgravitationalinputhavebeenimprovedbyactuatingthe
mechanics.Thetransitionfromapassivetoanactivemodelwasachievedbycreatinga
neuronaloscillatorstructurewithmuscularfeedbackandjointtorquegenerationworking
onanantagonisticprinciple.Theparametersofthemodelwerevariedtoidentifychar-
acteristicparametersforspecialfunctions.Inthisway,parametersforvaryingactuation
strategy,stepfrequency,steppingpatternsandsteppingstabilityhavebeenidentiﬁed.
Themostcriticalfactorwiththelow-levelsteppingmodelwasfoundtobethestabilityof
thesteppingsolutionsi.e.ensuringthatnofalloccurs.Inthisworkvariouspossibilities
forachievingstablesteppinginplacewithdroppingorliftinghip,steppingaside,and
steppingupwardswereproposed.Themovementswerecomparedwithvideotracking
dataofrealsteppingmovementsandfoundtobeverysimilar,especiallyforthestepping
inplacemovement.Themovementswerealsotestedunderdisturbinginﬂuencessuchas
slipping,gettingstuckorsustaininganexternalpush;themodelisfoundtohaverobust
reactionsandtoreturntoastablesolutionifthedisturbanceisnottoostrong.Sta-
bilityandperformanceweremuchbetterthanwiththepassivemodelbuttherestayed
stillsomelimitationswhichresultfromlackingperceptionoftheoverallcontextofthe
steppingmovement.Duetotheprinciple’keepitsimple’theadditionofanothermodel
levelleadedtoafurtherextensionandimprovementofthelow-levelmodelandnotlike
inmanyotherresearchtheelaborationofthelow-levelmechanicsandactuation.This
wasrealizedinthehigh-levelmodel.

117

6SummaryandFinalConclusion

Thehigh-levelmodelwasdevelopedtorepresentasensor-drivenperceptionofthewhole
bodypositionandtoestablisharelationshipbetweentheenvironmentandthebodyto
accomplishposturecontroltasks.Thebasisforthismodelismodelknowledgeinthe
formofstatisticalestimationandsensormodelsderivedfrombiologicalexamples.The
combinationofbodymovementsandenvironmentalinﬂuencesperceivedbythesenses
withastatisticalestimation,basedonexperiences,inafeedbackcontrolloopwasproposed
asthehigh-levelposturecontrolmodel.Theextensionofthevisualcuebyanonlinearity
derivedfromtheWeber-Fechnerlawtakesthenonlinearswayresponseeﬀectsintoaccount.
Toevaluatetheperformanceofthesensor-drivenposturecontrolmodel,twoexperiments
withrealsubjectshavebeenperformed,oneforvestibularstimulationandanotherfor
visualpursuitstimulation.Theexperimentaldataforpostureresponseduringstimulation
werereproducedandveriﬁedbythehigh-levelposturemodelsimulation.

Toimprovetheperformanceandenhancetheabilitiesofthelow-levelmodel,thetwo
modelswereintegratedbyasuperpositioncontrolconcept.Thecontrolinﬂuencesmainly
thestanceleg,whichisthemostcriticalparameterformaintainingbalance,andthehip
movement.Thesuperpositionconceptdoesnotinﬂuencethelow-levelactuation,butthe
twolevelsaresuperimposed.Thismeansthatincasethestabilityisatriskanadditional
high-levelcontrolissuperimposed.Thissuperpositioniscomparablewithanoverrulingof
thelow-levelautonomousmovementgenerationbyahigh-levelcommand.Itwasshown
thatthisintegrationleadstoimprovedstabilityofthesteppingmovements.Stability
ofmovementsisnolongermainlydependentontheinitialvaluesandthisleadstoan
increasedrangeofstablesolutionsandthepossibilityofinﬂuencingposturebysensory
.cues

Inconclusionitcanbesaidthatthisrelativelysimplemodelofthefrontalplanecan
provideawidespectrumofmovements,producingstable,realistic,ﬂexibleandrobust
medio-lateralsteppingsolutions.Movementsarenotpredeﬁnedbutdeterminedonline
accordingtodynamicconstraintssuchasmechanics,externalinﬂuences,generalopti-
mizationcriteria(e.g.stayingupright),actingwithamovementstrategy(e.g.selecting
ankleorhipstrategy),orchoosingageneralmovementpattern(e.g.steppinginplaceor
side).thetoTheinﬂuenceofperceptionandhigh-levelcontrolonthetaskofposturecontrolrequires
anadditionalmodelforanadditionaltask.Thebasisofthehigh-levelmodelisnotas
obviousasthemechanicsandisevenmorecomplexthanthelow-levelneuronalstruc-
tures.Thismeansthatitcanonlybeevaluatedbye.g.conductingexperimentswith
realsubjects.Theintegrationbystatisticalestimationcombinedwithoptimalcontrol
reproducestheauthor’sownexperimentsandalsoexperimentalresultsfromthelitera-
ture.Nonlinearitiesintheswayresponsecanbereproducedbyextensiontononlinear
perceptionrules.Thisabstractionofhigh-levelprocessingtoastatisticalestimationisa
generalapproachwhichleavesfurtherscopefordevelopingenvironmentalormodelchar-
acteristicsandotherprobabilitydistributions.
Stabilityisakeyfactorforsteppingmovementswhichcanbeenlargedbybringingsolu-
tionstotheattractivebasinofastablesolutionintheformofhigh-levelposturecontrol
implementedassuperpositionoflow-levelcontrolandadditionalcorrectivecontrol.This

217

outloO6.1k

allbutismeansmuchthatsofterﬁndingbecausestableofthesolutiowidernsisstabilitnotaycritrangice.alinitialvalueproblemanylonger

okOutlo6.1

Althoughthemodelsdevelopedinthisthesisprovidestableandvariablesolutionsfor
steppingmovements,thereisstillscopeforfurtherenhancementsoftheabilitiesandper-
formanceofthemodels.Thisthesispresentsageneralmodelforstudyingtheinﬂuence
ofsensorycuesonmedio-lateralsteppingandthiscannowbeappliedtofurtherexper-
imentsofstimulationsinﬂuencinglateralsteppingstability.Thesuperpositionprinciple
showedthedesiredbehavior,butfurtherconclusionsaboutahigh-levelcontrolconcept
shouldbetestedandexpanded;specialsensoryinputsinparticularwouldbeasuitable
subjectforfutureresearch.Oneimportantfactoraﬀectingabilitiesandperformanceis
withoutadoubtthemechanics.Thegroundcontactandtheenergystorageinthejoints
duringthedoublesupportphaseduetoelasticpropertiesarekeyfactorsforgenerationof
eﬃcientwalkingmovements.Themechanicaldynamicsaselaboratedinchapter2havea
largeinﬂuenceonthemovement.Manyresearchgroupsstudyexclusivelythemechanics
andthespecialpropertiesconnectedwiththese,sotheextensionofmechanicswould
certainlyimprovecharacteristicslikenaturalappearance,energyeﬃciencyandinsome
casesstability.Anotherpossibilityforfurtherdevelopment,forwhichthefoundationhas
alreadybeenlaid,becausethetwomechanicalplanesaremodeledandactuatedbythe
sameprinciple,isthecombinationofthetwo2Dmodelstoa3Dmechanicalmodel.This
mechanicalcombinationwouldincreasethecomplexityofthemodelimmenselyandthe
overlyingbiologicalstrategiesforcombiningthetwomovementplanesarenotyetknown.
SomestartingpointshavebeenproposedbyKuo[6,94]forﬁndingtherelationbetween
theactuationoflateralandsagittalmovementsinordertostabilizethem.Experiments
intometaboliccostsarevaluableforgaininganinsightintotheamountofactuation,but
thetypeofactuationneedstobestudiedinfurtherexperimentsandmodels.
Theperceptionexperimentsshouldbeextendedtoincludefurthervisualandvestibular
stimulations,becausetheinﬂuenceofthese,especiallyonthesidewardmovementduring
stepping,canbestudiedinmoredetailonthebasisofthemodelwhichhasbeendeveloped.
Onediﬃcultywithsuchexperimentsisthehighswayvariabilitywhichoccursduring
steppingmovementsduetothestepdynamics.Thismeansthatanyadditionalbody
swayinthisconnectionisdiﬃculttoextractbutasinthecaseoftheGVSexperiments,
theadditionalswayisshortandbigenoughtomeasure.OthermeasuressuchasCOP
orraisedmuscularactivationlevelscouldalsobechosentomeasuremedio-lateralsway
response.Onegeneralapplicationwhichisveryappropriateforneuronallyactuated
mechanicalmodelsarelearningalgorithms.Theselearningalgorithmscanbeusedtoﬁnd
moreandbetterorﬁttersolutionsforsteppingmovements;thesecouldforexamplebe
anadaptionofneuronfrequencytofrequenciessuitableforsteppingdynamicsorneuron
activationlevels.Asthewholebodypositioninthepresentthesisiscontrolledonboth
alowlevelandahighlevel,thelearningcouldalsobeusedtoadaptthelow-levelmodel

317

6SummaryandFinalConclusion

tolonger-lastinghigh-leveldestabilizationcaseswhichare

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