Zero-momentpointtrajectorymodelingofabipedwalkingrobotusinganadaptiveneuro-fuzzysystemD.Kim,S.-J.SeoandG.-T.ParkAbstract:Abipedalarchitectureishighlysuitableforarobotbuilttoworkinhumanenvironmentssincesucharobotwillfindavoidingobstaclesarelativelyeasytask.However,thecomplexdynamicsinvolvedinthewalkingmechanismmakethecontrolofsucharobotachallengingtask.Thezero-momentpoint(ZMP)trajectoryintherobotsfootisasignicantcriterionfortherobotsstabilityduringwalking.IftheZMPcouldbemeasuredon-linethenitbecomespossibletocreatestablewalkingconditionsfortherobotandherealsostablycontroltherobotbyusingthemeasuredZMP,values.ZMPdataismeasuredinreal-timesituationsusingabipedwalkingrobotandthisZMPdataisthenmodelledusinganadaptiveneuro-fuzzysystem(ANFS).Naturalwalkingmotionsonatlevelsurfacesandupanddowna10slopearemeasured.ThemodellingperformanceoftheANFSisoptimizedbychangingthemembershipfunctionsandtheconsequentpartofthefuzzyrules.TheexcellentperformancedemonstratedbytheANFSmeansthatitcannotonlybeusedtomodelrobotmovementsbutalsotocontrolactualrobots.1IntroductionThebipedalstructureisoneofthemostversatilesetupsforawalkingrobot.Abiped,robothasalmostthesamemovementmechanismsasahumananditabletooperateinenvironmentscontainingstairs,obstaclesetc.However,thedynamicsinvolvedarehighlynonlinear,complexandunstable.Thus,itisdifculttogenerateahuman-likewalkingmotion.Therealisationofhuman-likewalkingrobotsisanareaofconsiderableactivity14.Incontrasttoindustrialrobotmanipulators,theinteractionbetweenawalkingrobotandthegroundiscomplex.Theconceptofazero-momentpoint(ZMP)2hasbeenshowntobeusefulinthecontrolofthisinteraction.ThetrajectoryoftheZMPbeneaththerobotfootduringawalkisaftertakentobeanindicationofthestabilityofthewalk16.UsingtheZMPwecansynthesisethewalkingpatternsofbipedrobotsanddemonstrateawalkingmotionwithactualrobots.Thus,theZMPcriteriondictatesthedynamicstabilityofabipedrobot.TheZMPrepresentsthepointatwhichthegroundreactionforceistakentooccur.ThelocationoftheZMPcanbecalculatedusingamodeloftherobot.However,itispossiblethattherecanbealargeerrorbetweentheactualZMPvalueandthecalculatedvalue,duetodeviationsinthephysicalparametersbetweenthemathematicalmodelandtherealmachine.Thus,theactualZMPshouldbemeasuredespeciallyifitistobeusedinatoparametersacontrolmethodforstablewalking.InthisworkactualZMPdatatakenthroughoutthewholewalkingcycleareobtainedfromapracticalbipedwalingrobot.Therobotwillbetestedbothonaatoorandalsoon10slopes.Anadaptiveneuro-fuzzysystem(ANFS)willbeusedtomodeltheZMPtrajectorydatatherebyallowingitsusetocontrolacomplexrealbipedwalkingrobot.2Bipedwalkingrobot2.1DesignofthebipedwalkingrobotWehavedesignedandimplementedthebipedwalkingrobotshowninFig.1.Therobothas19joints.ThekeydimensionsoftherobotarealsoshowninFig.1.Theheightandthetotalweightareabout380mmand1700gincludingbatteries,respectively.Theweightoftherobotisminimisedbyusingaluminiuminitsconstruction.EachjointisdrivenbyaRCservomotorthatconsistsofaDCmotor,gearsandasimplecontroller.EachoftheRCservomotorsismountedinalinkedstructure.Thisstructureensuresthattherobotisstable(i.e.willnotfalldowneasily)andgivestherobotahuman-likeappearance.AblockdiagramofourrobotsystemisshowninFig.2.Outrobotisabletowalkatarateofonestep(48mm)every1.4sonaatoororanshallowslopes.ThespecicationsoftherobotarelistedinTable1.ThewalkingmotionsoftherobotareshowninFigs.36.-Figures3and4areshowfrontandsideviewsoftherobot,respectivelywhentherobotisonaatsurface.Figure5isasnapshotoftherobotwalkingdownaslopewhereasFig.6isasnapshotoftherobotwalkingupaslope.ThelocationsofthejointsduringmotionareshowninFig.7.ThemeasuredZMPtrajectoryisobtainedfromten-degree-of-freedom(DOF)dataasshowninFig.7.TwodegreesoffreedomareassignedtothehipsandanklesandoneDOFtoeachknee.Usingthesejointangles,acyclicwalkingpatternhasbeenrealised.Ourrobotisabletowalkcontinuouslywithoutfallingdown.Thejointanglesinthefour-stepmotionofourrobotaresummarisedintheAppendix.2.2ZMPmeasurementsystemTheZMPtrajectoryinarobotfootisasignicantcriterionforthestabilityofthewalk.Inmanystudies,ZMPcoordinatesarecomputedusingamodeloftherobotandinformationfromtheencodersonthejoints.However,weemployedamoredirectapproachwhichistousedatameasuredusingsensorsmountedontherobotsfeet.Thedistributionofthegroundisreactionforcebeneaththerobotsfootiscomplicated.However,atanypointPonthesoleofthefoottothereactioncanberepresentedbyaforceNandmomentM,asshowninFig.8.TheZMPissimplythecentreofthepressureofthefootontheground,andthemomentappliedbythegroundaboutthispointiszero.Inotherwords,thepointPonthegroundisthepointatwhichthenetmomentoftheinertialandgravityforceshasnocomponentalongtheaxesparalleltotheground1,7.Figure9illustratestheusedsensorsandtheirplacementonthesoleoftherobotsfoot.ThetypeofforcesensorusedinourexperimentsisaFlexiForceA201sensor8.Theyareattachedtothefourcornersoftheplatethatconstitutesthesoleofthefoot.SensorsignalsaredigitisedbyanADCboard,withasamplingtimeof10ms.Measurementsarecarriedoutinrealtime.Thefootpressureisobtainedbysummingtheforcesignals.UsingthesensordataitiseasytocalculatetheactualZMPvalues.TheZMPsinthelocalfootcoordinateframearecomputedusing(1).Whereeachfiistheforceatasensorriisthesensorpositionwhichisavector.ThesearedenedinFig.10.Inthegure,Oistheoriginofthefootcoordinateframewhichislocatedatthelower-left-handcornertheleftfoot.ExperimentalresultsareshowninFigs.1116.Figures11,13and15showthex-coordinateandy-coordinateoftheactualZMPpositionsforthefour-stepmotionoftherobotwalkingonaatoorandalsodownandupaslopeof10,respectively.Figures12,14and16showntheZMPtrajectoryoftheone-stepmotionoftherobotusingtheactualZMPpositionsshowninFigs.11,13and15.Asshowninthetrajectories,theZMPsexistinarectangulardomainshownbyasolidline.Thus,thepositionsoftheZMPsarewithintherobotsfootandhencetherobotisstable.3ZMPtrajectorymodellingInmanyscienticproblemsanessentialsteptowardstheirsolutionistoaccomplishthemodellingofthesystemunderinvestigation.Theimportantroleofmodellingistoestablishempiricalrelationshipsbetweenobservedvariables.Thecomplexdynamicsinvolvedinmakingarobotwalkmakethecontroloftherobotcontrolachallengingtask.However,ifthehighlynonlinearandcomplexdynamicscanbecloselyproducedthenthismodellingcanbeusedinthecontroloftherobot.Inaddition,modelling,canevenbeusedinrobustintelligentcontroltominimisedisturbancesandnoise.3.1ANFSFuzzymodellingtechniqueshavebecomeanactiveresearchareainrecentyearsbecauseoftheirsuccessfulapplicationtocomplex,ill-denedanduncertainsystemsinwhichconventionalmathematicalmodelsfailtogivesatisfactoryresults9.InthislightweintendtouseasystemtomodeltheZMPtrajectory.Thefuzzyinferencesystemisapopularcomputingframeworkthatisbasedontheconceptsoffuzzysettheory,fuzzyif-thenrules,andfuzzyreasoning.WewillusetheSugenofuzzymodelinwhichsinceeachrulehasacrispoutput,theoveralloutputisobtainedviaaweightedaverage,thusavoidingthetime-consumingprocessofdefuzzication.Whenweconsiderfuzzyrulesinthefuzzymodel,theconsequentpartcanbeexpressedbyeitheraconstantoralinearpolynomial.ThedifferentformsofpolynomialsthatcanbeusedinthefuzzysystemaresummarisedinTable2.Themodellingperformancedependsonthetypeofconsequentpolynomialusedinthemodelling.Moreover,wecanexploitvariousformsofmembershipfunctions(MFs),suchastriangularandGaussian,forthefuzzysetinthepremisepartofthefuzzyrules.Theseareanotherfactorthatcontributestotheexibilityoftheproposedapproach.ThetypesofthepolynomialareasfollowsAblockdiagramofthemodellingsystemisshowninFig.17.Theproposedmethodisrstusedtomodelandthencontrolapracticalbipedwalkingrobot.Toobtainthefuzzyrulesforthefuzzymodellingsystemwemustnotesthatthenonlinearsystemtobeidentiedisabipedwalkingrobotwithteninputvariablesandeachinputvariableshastwofuzzysets,respectively.Forthefuzzymodel,theif-thenrulesareasfollows:whereAi，Bi，,Jiinthepremisepartoftheruleshavelinguisticvalues(suchassmallorbig)associatedwiththeinputvariable,x1,x2,x10；respectively.Fj(x1,x2,x10)；istheconstant,orrst-orderconsequentpolynomialfunctionforthejthrule.AsdepictedinFig.18,twotypesofMFswereexamined.OneisthetriangularandtheotherisGaussian.Figure19isanadaptiveneuro-fuzzyinferencesystem10architecturethatisequivalenttotheten-inputfuzzymodelconsideredhere,inwhicheachinputisassumedtohaveoneofthetwoMFsshowninFig.18.NodeslabelledPgivetheproductofalltheincomingsignalsandtheselabelledNcalculatetheratioofacertainrulesringstrengthtothesumofalltherulesringstrengths.ParametervariationinANFISisoccuredusingeitheragradientdescentalgorithmorarecursiveleast-squaresestimationalgorithmtoadjustboththepremiseandconsequentparametersiteratively.However,wedonotusethecomplexhybridlearningalgorithmbutinsteadusethegeneralleast-squaresestimationalgorithmandonlydeterminethecoefcientsintheconsequentpolynomialfunction.3.2SimulationresultsApproximatelymodelswereconstructedusingtheANFS.Thenaccuracywasquantiedintermsoftheremean-squarederror(MSE),values.TheANFSwasappliedtomodeltheZMPtrajectoryofabipedwalkingrobotusingdatameasuredfromoutrobot.TheperformanceoftheANFSwasoptimisedbywaryingtheMFandconsequenttypeinthefuzzyrule.ThemeasuredZMPtrajectorydatafromourrobot(showninFigs.3241AintheAppendix)areusedastheprocessparameters.WhentriangularandGaussianMFsareusedinthepremisepartandaconstantintheconsequentpartthen,thecorrespondingMSEvaluesarelistedinTable3.WehaveplattedourresultsinFigs.2025.ThegeneratedZMPpositionsfromtheANFSareshowninFigs.20,22and24foraatleveloor,walkingdowna10slopeandwalkingupa10slope,respectively.InFigs.21,23and25,wecanseethecorrespondingZMPtrajectorieswhicharegeneratedfromtheANFS.Forsimplicity,theprocessparameterofbothkneescanbeignored.Asaresult,wecanreducethedimensionofthefuzzyrulesandtherebylowerthecomputationalburden.InthiscasethesimulationconditionsoftheANFSanditscorrespondingMSEvaluesaregiveninTable4.FromtheFiguresandTablesthatpresentthesimu