It Takes Two Neurons to Ride a Bicycle

# Fermat's Library | It Takes Two Neurons To Ride a Bicycle annotated/explained version. Source: [https://fermatslibrary.com/s/it-takes-two-neurons-to-ride-a-bicycle](https://fermatslibrary.com/s/it-takes-two-neurons-to-ride-a-bicycle)  ItTakesTwoNeuronsToRideaBicycle MatthewCook ∗ Abstract Pastattemptstogetcomputerstoridebicycleshaverequiredaninor\- dinateamountoflearningtime\(1700practiceridesforareinforcement learningapproach\[1\],whilestillfailingtobeabletorideinastraight line\),orhaverequiredanalgebraicanalysisoftheexactequationsof motionforthespecificbicycletobecontrolled\[2,3\]\.Mysteriously,hu\- mansdonotneedtodoeitherofthesewhenlearningtorideabicycle\. Herewepresentatwo\-neuronnetwork 1 thatcanrideabicycleinade\- sireddirection\(forexample,towardsadesiredgoaloralongadesired path\),whichmaybechosenorchangedatruntime\. Justaswhenapersonridesabicycle,thenetworkisveryaccuratefor longrangegoals,butintheshortrunstabilityissuesdominatethebehav\- ior\.Thishappensnotbyexplicitdesign,butarisesasanaturalconse\- quenceofhowthenetworkcontrolsthebicycle\. 1Introduction Thetaskofridingabicyclepresentsaninterestingchallenge,whetherforhumanorfor computer\.Wedonothavegreatinsightastohowwerideabicycle,andwedonothave muchusefuladviceforsomeonewhoislearning\. Infact,inthecourseofthisproject,Ihadthechancetoridea“virtualbicycle”onthe computer,andIwassurprisedtofindhowcounterintuitiveitis\.Ihadthoughtthat,knowing perfectlywellhowtorideabicycleinreallife,itwouldbenoprobleminsimulation\. However,inreallifetheremustbeadditionalinertialcuesthatIsenseorleaningactions thatImakewhicharemissingfromthesimulation,sinceIhadtolearn,asiffromscratch, whatcuestoattendtoandhowtoreacttothem\.Ieventhoughtatfirstthattheremustbea buginthesimulator,sincetoturnrightIfoundIhadtopushthehandlebarstotheleft\.Of course,ifyoustoptothinkaboutit,thatisexactlycorrect\.Toturnright,thebicyclehasto leantotheright,andtheonlywaytomakethathappen 2 istoshiftthepointofcontactwith thegroundtotheleft,whichrequiresaninitialpushtotheleft\.Butthen,oncethebicycleis leaningtotheright,itwillitselfpushthehandlebarstotherightduetohowitisconstructed forstability, 3 withaforceevengreaterthanyourinitialpushtotheleft,somaintaininga ∗ CaliforniaInstituteofTechnology,MailStop136\-93,Pasadena,CA91125cook@paradise\.caltech\.edu 1 Actually,thetitleofthispaperisunproven\.Wehavenotruledoutthepossibilitythatasingle neuroncouldrideabicycle\. 2 Thisisignoringthetorqueeffectduetothespinningofthefrontwheel,butifyoutakethatinto account,ittoohasexactlythesameeffectastheeffectdescribedabove\(pushingtotheleftmakes youleantotheright\)\. 3 Seefootnote7onpage6\.  constantgentleleftwardpushdoesindeedcausethebicycletoturntotheright\.Similarly, tocomeoutoftherightwardturn\(oreventomaintainit\),youneedtopushthehandlebars gentlytotheright\. Inthispaperweoutlinethevariousportionsofourproject,whichhasledustofinda surprisinglycompetenttwo\-neuronnetwork\.Differentportionsofthisworkarelikelyto beofinteresttodifferentpeople\.Thereadershouldfeelfreetoskipoversectionsthatare notofinterest—thepaperhasbeenorganizedsothatskippingaheadshouldnotresultina lossofunderstandingwhenreadinglatersections\. 2Methodology:OverviewoftheSimulatorSystem 2\.1ThePhysics Inordertoallowustoexperimentwithdifferentbicyclecontrollers,wefirsthavetosetup avirtualbicycleforthemtocontrol\.Theequationsofmotionforabicyclearesomewhat complex\[2\],soitseemsnomorecomplicatedandmuchmoreusefultojustwriteageneral robotsimulator,whichcanreadadescriptionofanarbitraryrobot\(rigidbodieslinked byhinge\-likeconnections\),andsimulatehowthatrobotwillmovegiventheforcesbeing appliedtoit\.Thisentailscalculatingthemomentsofinertiaforeachrigidbody,simulating themotionofasinglerigidbodygivenforcesactingonit\[5\],andsolvingasystemof equationsateachstepforhowthehinge\-likeconnectionscanapplyforcestothepartsof therobotsothatthealignmentandco\-locationrequirementsofthehingesaremet\. 4 2\.2TheBicycleRobot Oncewehavesuchageneralpurposephysicssimulator,thenwecanturntosettingupa robot,inthiscaseabicycle\.Abicycleiscomposedoffourrigidbodies:thetwowheels, theframe,andthefrontfork\(thesteeringcolumn\)\.Eachadjacentpairofpartsisconnected withajointthatallowsrotationalongadefinedaxis,andthewheelsareconnectedtothe groundbyrequiringthattheirlowestpointmusthavezeroheightandnohorizontalmotion \(nosliding\)\. Figure1:Thevirtualbicycle\. Beyondspecifyingtheconstructionandconnectionsthatformthebicycle,weneedtode\- cidewhatsensoryinputshouldbeavailabletothecontroller,andhowthecontroller’s outputsshouldbeconvertedintoforcesonthebicycle\(inroboticsterms,whatthesensors 4 Eventhesupportofthewheelbythegroundcountsasahinge\-likeconnectionforthispurpose\. andactuatorsshouldbeforthisnon\-holonomicunder\-actuatedsystem\)\.Forourbicycle, weallowalltheeasilyperceivablequantitiestobeavailabletoaninterestedcontroller: Position,heading,speed,angleofthehandlebars\(anditsrateofchange\),andtheamount thebicycleisleaning\(anditsrateofchange\)\.Foractuators,weallowatorqueontheback wheelandatorqueonthehandlebars\.Humansalsomakegooduseofleaningtooneside ortheotherwhentheyride,butwewillnothavesuchacontrolontheriderlessbicycle\. Also,wedonotallowthecontrollertoknowthespecificsofthebicycle,suchasitsexact proportionsorthemassesofitsparts\. 2\.3TheController Oncewehavesetuptherobotbicycle,wecanturntothetaskofinterest:Designinga controllerforthebicycle\.Wewantthecontrollertosolvethesameproblemthatahuman solveswhenridingthebicycle\.Thehumanknowswheretheyare,whichwaytheyare going,howfasttheyaregoing,howthebicycleisleaning,andsoon,butasweknowfrom experiencethehumandoesnotneedtoknowthespecificsoftheconstructionofthebicycle\. Herewearefinallyfacedwithaproblemthatwedonot,apriori,knowhowtosolve\.So westareattheceilingforawhile,andwheneverwearestruckwithsomeinspiration,we quicklywriteacontrollerbasedonit\. Therearethreemainstylesofcontroller\(prescient,human,andtwo\-neuron\)thathaveled ustointerestingresultsorobservations,andwewilldiscusstheminthenextthreesections\. Noneofthemmadesignificantuseofthespeed—theyallmanagedtocontrolthebicycle usingjustthehandlebars\.Wewillnotdiscussherethosecontrollerswhichdidnothingbut crashthebicycleateveryopportunity\. 3ThePrescientController:ALookatReinforcementLearning Oneinterestingideaforacontroller,giventhattheentiresystemisbeingsimulated,isto letthecontrollercheatbygivingitaccesstothesimulator\.Thiscouldnotbedonewitha controllerforabicycleintherealworld,soitisnotofinterestforapplications,butwecan certainlytryitinthesimulatedworldtoseewhathappens\. Inparticular,wecantrythefollowingalgorithmforthecontroller:Ateachstep,first simulateandcomparethreeactions\.Theactionsonlydifferinhowthehandlebarsare pushedatthefirstinstant:pushedleft,pushedright,ornottouched\.Theremainderofeach ofthethreeactionsistodonothinguntilthebicyclecrashes\.Thesethreeactionscanthen becomparedonthebasisofwhichonecausesthebicycletoremainuprightforthelongest time,whichoneresultsinthemostprogresstotheright,orwhateverothercriterionone decidestooptimize\.Aftersimulatingtheresultsofthethreeactions,thecontrollerdecides whattodoatthisinstantbasedonthoseresults\.\(Eachdifferentcriterionisthusthebasis foradifferentcontroller\.\) Thesesimulationsweretriedwithandwithoutrandommildforces\(“wind”\)beingapplied tothebicycle\.Theoriginalmotivationforthiswassothatthecontrollerwouldnotbeable torelyonanabsolutelyperfectpredictionofthefuture\.Itmightalsohelpthecontrollerto haveamore“continuous”behavior,sinceoverthecourseofseveralconsecutiveinstants,it wouldbegettingaroughestimationoftheprobabilitydistributionforsuccessofeachofits actions,leadingtothecontrollertakingasimilarlydistributedaction\.However,suchwind turnedoutinfacttohavenosignificanteffectontheresults\. Inthelanguageofreinforcementlearning,suchacontrollerisexactlywhatyouwould getafteronestepofpolicyiteration,ifyoustartwiththenullpolicyofnevertouching thehandlebars,andallowyourselfthreeactionsateachstep\(pushleft,pushright,orno  Figure2:Instabilityofanunsteeredbicycle\.Thisshows800runsofabicyclebeingpushedtothe right\.Foreachrun,thepathofthefrontwheelonthegroundisshownuntilthebicyclehasfallen over\.Theunstableoscillatorynatureisduetothesubcriticalspeedofthebicycle,whichlosesfurther speedwitheachoscillation\. push\)\.Thenifthecontrollerlearnsthevaluefunctionforthispolicy\(whichinpractice wouldrequirelotsofexperiencewithnottouchingthehandlebars,butwhichwesimulate bygivingthecontrolleraccesstothesimulator\),itcanthenactgreedilywithrespecttothat valuefunction\.Thisamountstoonestepofpolicyiteration,andatleastforthegoalofnot fallingover,anoptimalpolicyisindeedobtainedafterasingleiteration\(i\.e\.,itsuccessfully doesn’tfalldown\)\.However,itdoesnotdothisinaconventionalway,saybyridingina straightline,butrathermanagestomaintainstabilityatnear\-zerospeedbydoingstunts withthefrontwheel,forexamplebyspinningthehandlebarsincircles\(thehandlebars andfrontwheeldonotbumpintotheframeforourbicycle,andtherearenocablestoget twisted,sowhynot?\)\.Amovieofthisbizarrebehaviorcanbeseenat: http://www\.paradise\.caltech\.edu/∼cook/Warehouse/RecursiveBike\.avi Despitemanyattemptsatformulatingasensiblevaluefunction,wefounditdifficultto getsensiblebehavioroutofthebicycle\.Byrewardinguprightness,thebicyclewouldstop ridingnormallyandstartdoingstuntsasdescribedabove\.Ifwetriedtodiscouragethisby rewardingspeed,thebicyclewouldswoopfromsidetoside,whereeachswoopresultsina temporaryincreaseinspeed\.Ifwetriedtodiscouragethisbyrewardinggoinginastraight line,thebicyclewoulddothisverynicely,butofcourseitwouldfalloverrightaway,as avoidingthefallwouldhaverequireddeviatingfromthestraightline\.Ofcourse,onecould tryweightedcombinationsoftheseorotherideas,butthenthequestionstartstobenot howlongitwilltakethecontrollertolearntoridethebicycle,buthowlongitwilltakeus tolearnhowtoprogramthecontrollertogetittoridenormally\.Ashasbeenpointedout bypeoplewhohaveworkedwithreinforcementlearning,itcanbeaverytrickybusiness tryingtopickagoodvaluefunction\. Evenifwehadhadmarveloussuccesswiththismethod,itwouldnotbeimmediately applicabletoamorerealisticsituation,duetoitsrelianceongettinganswersfromthe simulatorto“whatif?”questions,effectivelyhavingaccesstoanoracle\.However,it wouldhaveprovidedahintthatreinforcementlearningcouldbeusedeffectively\.Asit was,thehintwegotwasthatreinforcementlearningwouldhaveahardtimeofit\.Thisis alsothehintweweregettingfromRandløvandAlstrøm’snicepaper\[1\],wheretheytried variousreinforcementlearningmethodstogettheircontrollertolearntorideabicycle,but evenwiththebestmethodsittookthousandsofpracticeridestolearnnottofalldown, andthousandsmoretolearntoridetowardsagoal,andevenafterhaving“learned,”the  controllerappearedtohavearatherdrunkenbehavioratbest\. 4TheHumanController Onevenerablemethodoflearningistobetaughtbyanexpert,perhapsbyexample\.The obviousexpertinthiscasewouldbeaskilledhuman\.Toenablethis,wehadthecomputer presentareal\-timegraphicaldisplayofthebicycle,allowingahumantousethekeyboard tocontrolthepedalingandthepushingofthehandlebars\.Asformyself,thisledtothe experiencedescribedintheintroduction\(onpage1\),whereIdiscoveredthatcontrolling thebicycleisquitecounter\-intuitive\. 5 Thehumancontrollerswhotriedtolearntodrivethevirtualbicyclefoundsubjectivelythat itwasimportanttopayattentiontotheangleatwhichthebicyclewasleaning,andtocon\- centrateonmanipulatingthisangle\.Thisobservationbecamethebasisforthetwo\-neuron controllerdescribedbelow\.Noneofthehumansbecameproficientatgettingthebicycleto travelinaprecisedirection,askilldemonstratednicelybythetwo\-neuroncontroller\.One cannotruleoutthepossibilitythatthehumansmighthavegainedthisskillwithfurther practice\. Therearetwowaysinwhichwetriedtousethehumanexpertise\.Onewaywasbyrecord\- ingwhatwasgoingonduringoneoftheirrides\.Collectingthisdataandthenanalyzingit forpertinentcorrelationsyieldedsurprisinglylittleinthewayofusefulinformation\. 12345678910111213141516 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 12345678910111213141516 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1:t 2:x 3:y 4:ΘHheadingL 5:sHspeedL 6:ΓHanglewithverticalL 7:ΑHhandlebarangleL 8:s i HintendedspeedL 9:Τ h HtorqueonhandlebarsL 10:t  11:x  12:y  13:Θ  14:s  15:Γ  16:Α  Figure3:Thecovariancebetweenmanyofthequantitiesavailabletothecontroller\.Themain diagonalhasbeensettozerosoasnottobesovisuallydistracting\.Thereisgoodcorrelationbetween thecontroller’soutputτ h \(9\)andtheamountofturning ˙ θ\(13\),whichmightseemtosolvetheproblem immediately,butunfortunatelythecausalityinthisrelationshipgoesthewrongway:thecontroller needstoadjustτ h inresponsetoγtomaintainstability,whereγ\(6\)and ˙ θ\(13\)happentobestrongly correlated\.\(Thiscorrelationbetweenγ\(6\)and ˙ θ\(13\)canbeimmediatelyunderstoodbasedonthe simplestphysicsofcentrifugalforce,butwedonotwantourcontrollertoneedknowledgeofany physicallaws\.\)Thequiteusefulcausalrelationbetweenτ h \(9\)and˙γ\(15\)isvisible,butthereislittle heretoclueoneintoitsimportanceforsuccessfulcontrol\. Theotherwayinwhichwetriedtousehumanexpertisewasbyhavingthehumanssubjec\- tivelydescribehowtheyweretryingtocontrolthebicycle\.Sincetheyhadjustgonethrough theexperienceofhavingtolearnhowtodoit,theywerereasonablywellabletodescribe 5 Thisatleastexplainsinpartwhyittakesacertainamountofpracticeforahumantobecomea proficientbicycleriderintherealworld\.  whattheiralgorithmwas,andasitturnedouttheyhaddevelopedsimilartechniques,based oncarefullyadjustingtheangleatwhichthebicycleisleaning\. Basedonthereportsofthehumans,thetwo\-neuronnetworkcontrollerwasimplemented, anditworkedalmostimmediately,havingfewparameters,andnotbeingoverlysensitive toanyofthem\.Thismethodisthesubjectofthenextsection\. 5TheTwo\-NeuronNetwork Herewepresentatwo\-neuronnetworkwhichcanoperatethebicyclecompetentlyovera rangeofspeeds\.Theoutputofthefirstneuronisfedintothesecondneuron,whoseoutput isconnectedtoanactuatorwhichappliesthespecifiedamountoftorquetothehandlebars\. Asinputstothenetwork,weprovidethedesiredheadingθ d ,aswellasthecurrentheading θandthedegreetowhichthebicycleiscurrentlyleaningγ,alongwiththeirderivatives ˙ θ and˙γ\. 6 Duetothenatureoftheproblem,wewilluseanetworkthatiscontinuousbothintimeand invalues\.Aunit’soutputisdeterminedsimplybyathresholdingfunctionofaweighted sumoftheunit’sinputs\.Ifnecessary,thiscanbeinterpretedasamean\-firing\-ratemodel, butwewillnotexploreissuesofnetworkrealismhere\.Giventhatsuchasmallnetwork ofthistypesuffices,thereseemslittledoubtthatmorerealisticnetworkscouldsolvethe problemaswell\. Thetaskforthenetworkwillbetomakethebicycletravelinthedesireddirection\.This canthenbeusedbyhigher\-levelplanningsystemstomakethebicycleheadtowardsagoal, ortofollowapathbyheadingtowardsasequenceofwaypoints\. Inordertosetthebicycle’sheadingθasdesired,weneedtobeabletocontrol ˙ θ\.Weknow fromfigure3that ˙ θisstronglyrelatedtoγ,theamountthebicycleisleaning,sowecantry tocontrol ˙ θindirectlybysimplycontrollingγ\. 7 Tocontrolγ,weneedcontrolover˙γ\.Andindeed,ouractuator,whichcanexertadesired torqueonthehandlebars,happenstohavereasonablecontrolof˙γ\.Thereisnotadirect orfixedcorrespondence,butasageneralrule,duringstableriding,ahigherclockwise torqueonthehandlebarswillcausethebicycletostartleaningmoretotheleft\.Thus,by settingthetorqueaccordingtohowwewouldlikeγtochange,weshouldbeabletohaveγ convergetowardsitsdesiredvalue\.\(Notethatitdoesn’tmakeabigdifferenceiftheactual convergenceisjusttowardssomeapproximationofγ’sdesiredvalue—theexactdesired valueisnotcriticalforthismethodofcontroltosucceed\.\) Thefirstneuroninourcircuitwilloutputthedesiredγ,withthenonlinearitybeingapplied sothatthebicycledoesn’ttrytoleantoofarover\. 8 Thesecondneuroninourcircuitwill outputthedesiredtorquetobeappliedtothehandlebars\. Thefirstneuronwilltakeasinputsθandθ d \(whichonecanassumetobewithin±πofθ, 6 Actually,aswewillsee,thenetworkdoesnotevenneedtouse ˙ θ\. 7 Oneofthereasonsthatcontrollingγworksisduetotherealisticbicyclegeometry\.Realbicycles aredesignedtobestable,whichallowsaridertoridewithoutholdingthehandlebars,simplyby controllingtheamountthebicycleisleaning\.Wenotethatonetypicalimportantfactorinstabilityis thattheaxisofrotationofthefrontforkshouldpassbelowthehubofthefrontwheelbutaboveits pointofcontactwiththeground,afeaturewehaveduplicatedonthevirtualbicycle\. 8 Althoughmostofusdonothavedirectexperiencewiththis,bicyclescanbecomequiteunstable iftheyareinastateofextremeleaning\.Inreallife,usuallythewheelsskidoutfromunderusbefore thispointisreached\.Inthissimulator,skiddingdoesnotoccur,butsincethiscontrollerspecifically avoidsstatesofextremeleaning,ittherebyavoidstheproblementirely,regardlessofwhetherthe problemisthatofslippingorthatofbecomingunstable\.  althoughthisisnotessential\),andsimplycalculatethedesiredchangeinheadingθ d −θ, multiplybyaconstant,applyathreshold,andthenoutputtheresult,whichwewilldenote byγ d \. Thesecondneuronwilltakeasinputsγ d ,γ,˙γ,anditsownoutput,whichwewilldenote byφ\.Itwillcalculatetheamountbywhichγshouldbechanged,γ d −γ,andcomparethat toaconstanttimesthecurrentrate˙γatwhichγischanging\.Thedifferencebetweenthese, howmuch˙γshouldbeadjustedby,isthenscaledandsentasthecontroller’soutputτ h of howmuchtorquetoapplytothehandlebars\. γ d =σ\(c 1 θ d −c 1 θ\) τ h =c 2 γ d −c 2 γ−c 3 ˙γ Figure4:Theequationsforthetwo\-neuronnetwork\.σdenotesathresholdingfunction\.Thethree constantsc i needtobesetbytheimplementorinlightofthebicycle’sstabilitycharacteristics,but thenetwork’sbehaviorisnottoosensitivetotheirprecisevalues,soitisactuallyquiteeasytogeta workingnetwork\. 6Results Thetwo\-neuronnetworkcontrollerdoesremarkablywellatcontrollingthebicycle,aswe canseeinfigure5\. Figure5:Thepathtakenbythebicyclewhentoldtoaimatthesuccessivewaypointsshown\.Each timethebicyclegotwithinacertaindistanceofitscurrenttargetwaypoint,thetargetwouldchange tobecomethenextwaypointinthesequence\.Thisdistancecanbeseenasthedistancebetweenthe lastwaypointandtheendofthebicycle’spathshown\.Theirregularityinthewritingisduetothe author’smessyhandwritingwhentryingtowritewithamouse,andisnotthefaultofthebicycle\. Althoughthetwo\-neuronnetworkcontrollerworkswellforarangeofspeeds,onething thecontrollerdoesnotdoistotrytodampentheinstabilitiesthatcanarisewhenriding tooslowlyorintoosharpofaturn\.\(Thiswouldprobablyrequireathirdneuronthatis dedicatedtothistask\.\) 7FutureDirections:MoreAutomatedLearning Thefuturedirectionsofinteresttoushavetodowithusingthissystemtounderstandmore aboutlearning\.Thereareotherinterestingfuturedirections,suchasbuildingarealbicycle robot,whichweprobablywillnotpursue\.  Oneobviousthingtodowiththissystemistohaveitlearnandtuneitsparameterswith experience\.Ideally,ifthesystemisplacedonaslightlydifferentbicycle,itshouldquickly 9 learnhowtosuccessfullyoperatethenewbicycle\. Thisnetworkwasdesignedinanad\-hoc,iftraditional,way\.First,ahumantriedtocontrol thebicyclewiththesimulator\.Aftermanyattempts,thehumanfinallybecameasomewhat skilledoperatorofthebicycle,abletoavoidfallingdownbutstillnotabletoheadreliably towardsadesiredgoal\.Nonetheless,thehumanatthispointwasabletodescribethekey parameterswhichwerebeingattendedto,andbasedonthis,thetwo\-neuronnetworkwas designed\.Theskillofthisnetworkimmediatelyexceededthehuman’sskill\. However,wewouldliketotakethehumanoutofthisloop\.Wewouldlikethecomputer tobeabletofigureoutonitsownwhatsimplenetworkmightwork,usingaminimumof experience\(i\.e\.,aminimumnumberofcrashesbeforemasteryofthebicycle\),andusing nodetailedknowledgeofthephysicalsystem\.Ourattemptssofarhavebeenstatistically based,andhavenotbeenverysuccessful\.Wefeelthatweneedtohaveacausalmodel forwhatisobserved,wherethedirectionofcausalityispartofwhatisobserved\.Causal networks\(beliefpropagationnetworks\)mightbeagoodrepresentationforsuchknowledge, buttherealissueishowtohavesuchanetworkbeautomaticallydesignedforus\.Itisour opinionthatageneralsolutiontothisproblemwouldhavemanyapplications\. Acknowledgments IwouldliketothankShukiBruckandErikWinfreeformanyhelpfuldiscussions\.This workwassupportedinpartbythe“AlphaProject”thatisfundedbyagrantfromthe NationalHumanGenomeResearchInstitute\(GrantNo\.P50HG02370\)\. 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