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ZeromomentpointtrajectorymodelingofabipedwalkingrobotusinganadaptiveneurofuzzysystemD.Kim,S.J.SeoandG.T.ParkAbstractAbipedalarchitectureishighlysuitableforarobotbuilttoworkinhumanenvironmentssincesucharobotwillfindavoidingobstaclesarelativelyeasytask.However,thecomplexdynamicsinvolvedinthewalkingmechanismmakethecontrolofsucharobotachallengingtask.ThezeromomentpointZMPtrajectoryintherobotsfootisasignificantcriterionfortherobotsstabilityduringwalking.IftheZMPcouldbemeasuredonlinethenitbecomespossibletocreatestablewalkingconditionsfortherobotandherealsostablycontroltherobotbyusingthemeasuredZMP,values.ZMPdataismeasuredinrealtimesituationsusingabipedwalkingrobotandthisZMPdataisthenmodelledusinganadaptiveneurofuzzysystemANFS.Naturalwalkingmotionsonflatlevelsurfacesandupanddowna10°slopearemeasured.ThemodellingperformanceoftheANFSisoptimizedbychangingthemembershipfunctionsandtheconsequentpartofthefuzzyrules.TheexcellentperformancedemonstratedbytheANFSmeansthatitcannotonlybeusedtomodelrobotmovementsbutalsotocontrolactualrobots.1IntroductionThebipedalstructureisoneofthemostversatilesetupsforawalkingrobot.Abiped,robothasalmostthesamemovementmechanismsasahumananditabletooperateinenvironmentscontainingstairs,obstaclesetc.However,thedynamicsinvolvedarehighlynonlinear,complexandunstable.Thus,itisdifficulttogenerateahumanlikewalkingmotion.Therealisationofhumanlikewalkingrobotsisanareaofconsiderableactivity1–4.Incontrasttoindustrialrobotmanipulators,theinteractionbetweenawalkingrobotandthegroundiscomplex.TheconceptofazeromomentpointZMP2hasbeenshowntobeusefulinthecontrolofthisinteraction.ThetrajectoryoftheZMPbeneaththerobotfootduringawalkisaftertakentobeanindicationofthestabilityofthewalk1–6.UsingtheZMPwecansynthesisethewalkingpatternsofbipedrobotsanddemonstrateawalkingmotionwithactualrobots.Thus,theZMPcriteriondictatesthedynamicstabilityofabipedrobot.TheZMPrepresentsthepointatwhichthegroundreactionforceistakentooccur.ThelocationoftheZMPcanbecalculatedusingamodeloftherobot.However,itispossiblethattherecanbealargeerrorbetweentheactualZMPvalueandthecalculatedvalue,duetodeviationsinthephysicalparametersbetweenthemathematicalmodelandtherealmachine.Thus,theactualZMPshouldbemeasuredespeciallyifitistobeusedinatoparametersacontrolmethodforstablewalking.InthisworkactualZMPdatatakenthroughoutthewholewalkingcycleareobtainedfromapracticalbipedwalingrobot.Therobotwillbetestedbothonaflatfloorandalsoon10slopes.AnadaptiveneurofuzzysystemANFSwillbeusedtomodeltheZMPtrajectorydatatherebyallowingitsusetocontrolacomplexrealbipedwalkingrobot.2Bipedwalkingrobot2.1DesignofthebipedwalkingrobotWehavedesignedandimplementedthebipedwalkingrobotshowninFig.1.Therobothas19joints.ThekeydimensionsoftherobotarealsoshowninFig.1.Theheightandthetotalweightareabout380mmand1700gincludingbatteries,respectively.Theweightoftherobotisminimisedbyusingaluminiuminitsconstruction.EachjointisdrivenbyaRCservomotorthatconsistsofaDCmotor,gearsandasimplecontroller.EachoftheRCservomotorsismountedinalinkedstructure.Thisstructureensuresthattherobotisstablei.e.willnotfalldowneasilyandgivestherobotahumanlikeappearance.AblockdiagramofourrobotsystemisshowninFig.2.Outrobotisabletowalkatarateofonestep48mmevery1.4sonaflatfloororanshallowslopes.ThespecificationsoftherobotarelistedinTable1.ThewalkingmotionsoftherobotareshowninFigs.3–6.Figures3and4areshowfrontandsideviewsoftherobot,respectivelywhentherobotisonaflatsurface.Figure5isasnapshotoftherobotwalkingdownaslopewhereasFig.6isasnapshotoftherobotwalkingupaslope.ThelocationsofthejointsduringmotionareshowninFig.7.ThemeasuredZMPtrajectoryisobtainedfromtendegreeoffreedomDOFdataasshowninFig.7.TwodegreesoffreedomareassignedtothehipsandanklesandoneDOFtoeachknee.Usingthesejointangles,acyclicwalkingpatternhasbeenrealised.Ourrobotisabletowalkcontinuouslywithoutfallingdown.ThejointanglesinthefourstepmotionofourrobotaresummarisedintheAppendix.2.2ZMPmeasurementsystemTheZMPtrajectoryinarobotfootisasignificantcriterionforthestabilityofthewalk.Inmanystudies,ZMPcoordinatesarecomputedusingamodeloftherobotandinformationfromtheencodersonthejoints.However,weemployedamoredirectapproachwhichistousedatameasuredusingsensorsmountedontherobotsfeet.Thedistributionofthegroundisreactionforcebeneaththerobotsfootiscomplicated.However,atanypointPonthesoleofthefoottothereactioncanberepresentedbyaforceNandmomentM,asshowninFig.8.TheZMPissimplythecentreofthepressureofthefootontheground,andthemomentappliedbythegroundaboutthispointiszero.Inotherwords,thepointPonthegroundisthepointatwhichthenetmomentoftheinertialandgravityforceshasnocomponentalongtheaxesparalleltotheground1,7.Figure9illustratestheusedsensorsandtheirplacementonthesoleoftherobotsfoot.ThetypeofforcesensorusedinourexperimentsisaFlexiForceA201sensor8.Theyareattachedtothefourcornersoftheplatethatconstitutesthesoleofthefoot.SensorsignalsaredigitisedbyanADCboard,withasamplingtimeof10ms.Measurementsarecarriedoutinrealtime.Thefootpressureisobtainedbysummingtheforcesignals.UsingthesensordataitiseasytocalculatetheactualZMPvalues.TheZMPsinthelocalfootcoordinateframearecomputedusing1.Whereeachfiistheforceatasensorriisthesensorpositionwhichisavector.ThesearedefinedinFig.10.Inthefigure,Oistheoriginofthefootcoordinateframewhichislocatedatthelowerlefthandcornertheleftfoot.ExperimentalresultsareshowninFigs.11–16.Figures11,13and15showthexcoordinateandycoordinateoftheactualZMPpositionsforthefourstepmotionoftherobotwalkingonaflatfloorandalsodownandupaslopeof10,respectively.Figures12,14and16showntheZMPtrajectoryoftheonestepmotionoftherobotusingtheactualZMPpositionsshowninFigs.11,13and15.Asshowninthetrajectories,theZMPsexistinarectangulardomainshownbyasolidline.Thus,thepositionsoftheZMPsarewithintherobotsfootandhencetherobotisstable.3ZMPtrajectorymodellingInmanyscientificproblemsanessentialsteptowardstheirsolutionistoaccomplishthemodellingofthesystemunderinvestigation.Theimportantroleofmodellingistoestablishempiricalrelationshipsbetweenobservedvariables.Thecomplexdynamicsinvolvedinmakingarobotwalkmakethecontroloftherobotcontrolachallengingtask.However,ifthehighlynonlinearandcomplexdynamicscanbecloselyproducedthenthismodellingcanbeusedinthecontroloftherobot.Inaddition,modelling,canevenbeusedinrobustintelligentcontroltominimisedisturbancesandnoise.3.1ANFSFuzzymodellingtechniqueshavebecomeanactiveresearchareainrecentyearsbecauseoftheirsuccessfulapplicationtocomplex,illdefinedanduncertainsystemsinwhichconventionalmathematicalmodelsfailtogivesatisfactoryresults9.InthislightweintendtouseasystemtomodeltheZMPtrajectory.Thefuzzyinferencesystemisapopularcomputingframeworkthatisbasedontheconceptsoffuzzysettheory,fuzzyifthenrules,andfuzzyreasoning.WewillusetheSugenofuzzymodelinwhichsinceeachrulehasacrispoutput,theoveralloutputisobtainedviaaweightedaverage,thusavoidingthetimeconsumingprocessofdefuzzification.Whenweconsiderfuzzyrulesinthefuzzymodel,theconsequentpartcanbeexpressedbyeitheraconstantoralinearpolynomial.ThedifferentformsofpolynomialsthatcanbeusedinthefuzzysystemaresummarisedinTable2.Themodellingperformancedependsonthetypeofconsequentpolynomialusedinthemodelling.Moreover,wecanexploitvariousformsofmembershipfunctionsMFs,suchastriangularandGaussian,forthefuzzysetinthepremisepartofthefuzzyrules.Theseareanotherfactorthatcontributestotheflexibilityoftheproposedapproach.ThetypesofthepolynomialareasfollowsAblockdiagramofthemodellingsystemisshowninFig.17.Theproposedmethodisfirstusedtomodelandthencontrolapracticalbipedwalkingrobot.Toobtainthefuzzyrulesforthefuzzymodellingsystemwemustnotesthatthenonlinearsystemtobeidentifiedisabipedwalkingrobotwithteninputvariablesandeachinputvariableshastwofuzzysets,respectively.Forthefuzzymodel,theifthenrulesareasfollowswhereAi,Bi,,,,Jiinthepremisepartoftheruleshavelinguisticvaluessuchassmallorbigassociatedwiththeinputvariable,x1,x2,,x10respectively.Fjx1,x2,,x10istheconstant,orfirstorderconsequentpolynomialfunctionforthejthrule.AsdepictedinFig.18,twotypesofMFswereexamined.OneisthetriangularandtheotherisGaussian.Figure19isanadaptiveneurofuzzyinferencesystem10architecturethatisequivalenttotheteninputfuzzymodelconsideredhere,inwhicheachinputisassumedtohaveoneofthetwoMFsshowninFig.18.NodeslabelledPgivetheproductofalltheincomingsignalsandtheselabelledNcalculatetheratioofacertainrulesfiringstrengthtothesumofalltherulesfiringstrengths.ParametervariationinANFISisoccuredusingeitheragradientdescentalgorithmorarecursiveleastsquaresestimationalgorithmtoadjustboththepremiseandconsequentparametersiteratively.However,wedonotusethecomplexhybridlearningalgorithmbutinsteadusethegeneralleastsquaresestimationalgorithmandonlydeterminethecoefficientsintheconsequentpolynomialfunction.3.2SimulationresultsApproximatelymodelswereconstructedusingtheANFS.ThenaccuracywasquantifiedintermsoftheremeansquarederrorMSE,values.TheANFSwasappliedtomodeltheZMPtrajectoryofabipedwalkingrobotusingdatameasuredfromoutrobot.TheperformanceoftheANFSwasoptimisedbywaryingtheMFandconsequenttypeinthefuzzyrule.ThemeasuredZMPtrajectorydatafromourrobotshowninFigs.32–41AintheAppendixareusedastheprocessparameters.WhentriangularandGaussianMFsareusedinthepremisepartandaconstantintheconsequentpartthen,thecorrespondingMSEvaluesarelistedinTable3.WehaveplattedourresultsinFigs.20–25.ThegeneratedZMPpositionsfromtheANFSareshowninFigs.20,22and24foraflatlevelfloor,walkingdowna10slopeandwalkingupa10slope,respectively.InFigs.21,23and25,wecanseethecorrespondingZMPtrajectorieswhicharegeneratedfromtheANFS.Forsimplicity,theprocessparameterofbothkneescanbeignored.Asaresult,wecanreducethedimensionofthefuzzyrulesandtherebylowerthecomputationalburden.InthiscasethesimulationconditionsoftheANFSanditscorrespondingMSEvaluesaregiveninTable4.FromtheFiguresandTablesthatpresentthesimulationresults,wecanseethatthegeneratedZMPtrajectoryfromthefuzzysystemisverysimilartoactualZMPtrajectoryofmeasuredforourwalkingrobotshowninFigs.11–16.ThedemonstratedhighperformanceabilityoftheANFS,meansthatANFScanbeeffectivelyusedtomodelandcontrolapracticalbipedwalkingrobot.3.3ComparisonsWenowcomparetheperformanceofANFSwithnumericalmethodsincludingthreetypesofstatisticalregressionmodels.Foreachstatisticalregressionmodel,fourdifferentcasetypeswereconstructed.Theirgeneralformsinthecaseoftwoinputsaregivenaswheretheciaretheregressioncoefficients.ThecorrespondingMSEvaluesaregiveninTables5–7whichrevealsthattype2givesthebestresultsforthexandycoordinatesforalltheconsideredwalkingconditions.ThegeneratedZMPpositionsandtheircorrespondingtrajectonsgeneratedusingthetype2regressionmodelareshowninFigs.26–31.WecanconcludethattheANFSdemonstratedaconsiderablybetterZMPtrajectorythanthestatisticalregressionmodels.4ConclusionsTheANFSmodellingattheZMPtrajectoryofapracticalbipedwalkingrobothasbeenpresented.ThetrajectoryoftheZMPisanimportantcriterionforthebalanceofaIEEProc.ControlTheory
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