Robotics: Science and Systems 2006 Philadelphia, …

1 Designmethodologiesforcentralpatterngene rators:anapplicationtocrawlinghumanoidsL udovicRighettiandAuke JanIjspeertBiologicallyInspiredRoboticsG roupSchoolofComputerandCommunicationScie ncesEcolePolytechniqueF ed eraledeLausanne(EPFL)- Systemsofcouplednonlinearoscillatorsinsp iredfromanimalcentralpattern generators (CPGs)are increasinglyusedforthecontroloflocomotio ninrobots, ,suchsystemspresentinterestingcharacteri sticslike limitcyclebehavior( ),synchronization, are nowgoodmethodologiesfordesigningsystemst hatexhibitspecificgaits, ,howevertechniquestomodulatetheshapeofth erhythmicsignalsina controlledwayare ,wepresenta methodforshapingthesignalsofanoscillator ysystemaccordingtoseveralcriteriathatare relevantforlocomotioncontrol(butwhichcou ldalsobeusefulforotherapplications).Thes ecriteriaincludebeingabletoadjusttherela tive durationsofascendinganddescendingphasesi nacycle, , applythemethodtothedesignofa ,wedesignthesystemtoproducestabletrot-li ke designedsuchthatthespeedoflocomotioncanb eadjustedbyvaryingthedurationofthestance phasewhilekeepingthedurationoftheswingph aseconstant,like INTRODUCTIONT hisworkispartoftheRobotCubproject,a 5-yearEuro-peanprojectwhosepurposeis tobuilda 54-degreesoffreedomhumanoidrobotwiththec ognitive abilitiesofa child[1].

2 Theprojecthastwo maingoals:first,tocreateanopenandfreely- availablehumanoidplatformforresearchinem bodiedcognition,andsecond,tostudycogniti ve toa child,therobot(calledtheiCub) designmethodologyforthecrawlingcontrolle r, basedontheCentralPatternGenerator(CPG) ( ,respiration)[2],[3].AlthoughCPGsarecont rolledbysimpledescendingpathsfromhigherp artsofthebrain,they areabletogeneratethesignalsthatcontrolth ecomplex [4],[5].ModelsofCPGsforroboticsapplicati onshave provensuccessful,especiallyforlocomotion controlwheretheyareusedtogeneratejointtr ajectories[6] [8].Theiradvantageisthatitiseasytomodula tethetrajectoriesforlocomotionandtheyhav e , veryfewdesignmethodologiesarecurrentlyav ailabletoconstructthem[9],[10].Inparticu lar, ,todesignthecontrollerwestudythecrawling behaviorofinfantsinordertoextractimporta ntprinciplesforourcontroller. Thenwepresenta CPGfora specifictask, designedouroscillatorfromtheobservationt hatthegaitpatternofanimalsandhumanscanbe separatedintotwodistinctphasesforeachlim b.

3 Isa well-knownfactthatwhenquadrupedschangeth eirspeedoflocomotion,they mightchangetheirgaitandthedurationofthes tancephase,butthedurationoftheswingphase tendstoremainthesame[11].However, , animportantfeature,sinceduringtheswingph ase,onelimbisoff theground, ,wepresenta couplingschemebasedontheanalysisofthecra wlingpatternofrealinfantstoreproducea a trot-like , itappearsthatthereexistsa correlationbetweenthemovementofa limbduringitsstancephaseandtheswingphase oftheoppositelimb. We reproducethisinfluenceinthecouplingschem ewepresentandweusethetheoryofsymmetricdy namicalsystems[12] [14] physicssimulationoftheiCub, ,wefirstreviewdataoncrawlingininfants(Se ctionII).Wethenpresentthedesignapproachb ehindourmodelofcoupledoscillators(Sectio nIII).Thedesignisdoneincrementallywithfi rsttheconstructionofa nonlinearoscillatorwithtwo controlledtimescales,thentheadditionofin ter-limbinfluencesbetweenoscillatorsofop positelimbs,andfinallytheadditionofinter -limbcouplingsbetweenthecompletefour-osc illatorsystemforimplementingthetrot-like rigidarticulatedbodysimulationoftheiCub, andcomparedtotheoriginalinfantcrawlingga its(SectionIV).

4 Thepaperconcludeswitha shortdiscussion(SectionV).II. CRAWLINGININFANTSV eryfewstudiesaboutcrawlinginbabieshave [15],[16]butthey allfocusonthecognitive developmentofinfantsthroughlocomotionand nonehave [15].Indeed,thisgaitisthemostwidespreado neamonginfantsandforafirststudyofcrawlin git ,withthephaserelationofa [15].Inordertostudycrawlingininfants, , witha morenaturalcoordinateframeforthecontrolo fa Angle (rad)(a) Angle (rad)(b) Angle (rad)(c) Angle (rad)(d) reconstructionofa crawlingsequencefromtherecordingsofa crawlingbaby. We plotthejointangles(inradian)ofthe4 , , , showsa thatstandardcrawlingisa trot-like meansthatthediagonallimbs( )areinphaseandhalfa , thistrot-like gaitis , ,thestancephaseisreallylongcomparedtothe swingphase,it representsabout70%ofa studybetweenthekinematicsofcrawlingbabie sandmonkeyscanbefoundin[17] (asdefinedinFigure1),wenoticethatduringt hestancephasethejointslowsdownorevensome timesstopsduringtheswingphaseoftheopposi telimb.

5 Itisasiftheswingphaseofa limbwasinhibitingthemovementoftheopposit elimb. Thisobservationis alsosupportedbythedatashownin[17], ,theexactcontrolofthisjointis lessimportant( ).Theelbowjointsarefoldingduringtheswing phase,toallowthearmtoreachafurtherregion infrontofthebabybutdonotmove nottostudyindetailthecrawlingsequenceoft hebaby, buttoextractthefeaturesthatseemimportant inordertoreproducethesamegaitina toemphasizefromtheseobservationsandfromt hestudyof[17]arefirstthatthecrawlinggait isa trot-likegaitintermsofphaserelationsbetw eenthelimbsbutwithastancephasethatis ,thereisa correlationbetweentheswingphaseofa limbandthearrestofmovementofthehip(orsho ulder)jointoftheoppositelimb. Third, MODELI nthissectionweconstructa constructa CPGmodel,wedefinea , weknowthatduringlocomotionatvariousspeed s, (andthechangeofgait)influencethespeedofl ocomotion[3],[11].Thus,wewouldlike ,wewouldlike ourCPGtogenerateatrot-like gait, , theCPGmusthavepropertiesthatmakesitsuita bleforthecontrolofa thereforewanttheCPGtoshow limitcyclebehaviorandtobestableagainstpe rturbations, alsowanttobeabletosmoothlymodulatethegen eratedtrajectoryinfrequencyandinamplitud etohave a thefollowingproperties Smoothmodulationofthegeneratedtrajectory infre-quency andamplitude Independentcontrolofthedurationoftheswin gandstancephases(theascendinganddescendi ngphases) Trot-like gaitwitha stancephasemuchlongerthantheswingphase Inhibitionofthemovementofthehipandshould erjointsduringtheswingphaseoftheopposite limbs firstpresenta modelofa ,theequationofmotionofthejointanglecanbe expressedas_x=y(1)_y= Kx(2)Thefrequency wanta durationofthestancephasedifferentfromthe durationoftheswingphase, constantsaccordingtothephase, canthuswritea generalspringconstantasK=kstance+ (kswing kstance)1eby+ 1(3)

6 Wheretheexponentialfunctionworksasa stepfunctionwhichselectseitherkswingorks tanceaccordingtothesignofthevelocityofmo vementy, a periodicorbitsaroundtheunstablecenter0an dthusthesystemis canpointtheflowtowardoneperiodicorbitbyc onstrainingthetotalenergyofthesystem,sin ceit definesthemaximumvaluexcantake ina (Kx2+y2)(4)whichisthesumofthepotentialan dkineticenergiesofthesystem(wetake themassequaltoone).Aty= 0wehaveE=12Kx2, whichgivesxmax= q2EK. We canchooseatotalenergysuchthatxmaxisbound edtoa certainvalue,E= 2K2andxmax= .Inordertoconstrainttheamplitudeofoscill ations,weadda thenrewritethewholesystemas_x=y(5)_y= y( 2K (Kx2+y2)) Kx(6)where isa constantcontrollingthespeedofconvergence oftheenergyofthesystem12(Kx2+y2)tothewan tedtotalenergy12 (Kx2+y2), thendifferentiationwithrespecttotimegive s_E=12_Kx2+ y2(K 2 E)(7) 11X0 020 20 TimeY(a)kstance=kswing 202X0 020 40 TimeY(b)kstance= , in3(a)weplottheoscillationswhenkstance=k swing= 4:(2 )2, in3(b)weplotkstance=13kswing=4:(2 )2.

7 Ineachplotweshowtheoscillationsxandtheve locityy. Attimet= 1:5, weperturbthesystembysettingxandytoa randomvalue, ,Kcanbeapproximatedasa switchingfunction,whosevalueequalseither kswingwheny <0orkstancewheny >0. Thus,fory6= 0, weseethattheflowisalwaysdirectedtowardE= K 2. Wheny= 0,Kchangesitsvaluefroma alwaysdirectedtowardK +y2=K 2. Itiscomposedoftwo half-ellipsesthatsharethesamesemi-minora xis (sothey areconnected)andwithfociaty= pkswing 1andy= pkstance , withtheboundedenergy, weassurethattheoscillatorisstableandthat wecancontroltheamplitudeoftheoscillation swhichareequalto . , (e byi+ 1)(e kyj+ 1)+kswingebyi+ 1(8)whereidenotestheoscillatorthatisinhi bitedandjtheoppositeoscillator,kcontrols thespeedofslowdownoftheoscillator. Withthiscouplingscheme,whenonelimbstarts itsswingphase, thismomentandtheenergydampingterm . x1(t) =x2(t) =x3(t) =x4(t)UnstablefI;(12)(34)gx1(t) =x2(t) =x3(t+T2) =x4(t+T2)UnstablefI;(13)(24)gx1(t) =x2(t+T2) =x3(t) =x4(t+T2) ;(14)(23)gx1(t) =x2(t+T2) =x3(t+T2) =x4(t) ,wederivedthepossiblepatternofsynchroniz ationaccordingtothepossiblesubgroupsofsp atialsymmetry.

8 Foreachsubgroup, , assurethatthisslowdownwillbefastenoughwh enx'0, wechangethedampingtermsoit hasa veryhighvaluewhenx'0anda dothis,wetransform intoa Gaussianfunctioncenteredaround0. i= (1 + e x2i)(9)where isthedampingconstant, controlsthechangeofthedampingaround0and iwecannowindependentlycontrolthegenerald ampingtermthatconstrainsthetotalenergyof thesystemandthedampingwhenx'0, oftheCPGI nadditiontothecouplingschemeforinhibitio n,wehave tointroducea couplingtomaintainthephaserelationsbetwe eneachlimb. We wanta halfa periodoutofphaserelationbetweenoppositel imbs( )andanin-phaserelationbetweendiagonallim bs( ).To designsucha network,weusethetheoryofsymmetriccoupled cellnetworks[12] [14].Bylookingonlyatthesymmetriesofa networkofcoupledoscillators, ,thesymmetriesofthenetworkinducethatthec orrespondingordinarydifferentialequation s(ODEs)describ-ingthenetworkhave thesamesymmetry. Inthiscasewecandistinguishtwo certainsetofODEswhicharethesymmetries suchthatforany solutionx(t)ofthesetofODEs x(t) =x(t).

9 Thespatio-temporalsymmetriesarethesymmet ries'whichpreserve theorbitofa solution,whichmeansthatifx(t)isasolution withorbitfx(t)g, then'x(t) x1 x2 x3 x4 Time(a) symmetry0102030 x1 x2 x3 x4 Time(b)fI;(12)(34)gsymmetry0102030 x1 x2 x3 x4 Time(c)fI;(13)(24)gsymmetry0102030 x1 x2 x3 x4 Time(d)fI;(14)(23) showthe4 possiblepatternsofsynchrony (a)and6(b)attimet= 10sweadda perturbationof0:01tox1, weseethatsucha (c),whichisthecrawlingpattern,attimet= 10weadd1:0tox1andattimet= 20wesetthestatevariablesofeachoscillator ata randomvaluebetween[ 2;2], it (d)is a pacegait,attimet= 10sweadda randomnoisebetween[ 0:2;0:2]oneachxi. Foralltheexperiment,wesetkswing=kstance= 2,c1 =c2 = 1:0, = 100, = 10, = 0:45,b=k= 100and = ,ifx(t)isa periodicsolution,then'x(t) ,if wenumberthelimbsasinFigure4,wewanttheper mutationofthediagonallimbs(13)(24)tobea spatialsymmetryand((12)(34);12)and((14)( 23);12)tobespatio-temporalsymmetrieswith halfa canconstructa ,weknowthatthecrawlinggaitis a periodicsolutionofany :H/KTheorem[14]Let bethesymmetrygroupofa H bea andonlyifH=KiscyclicandKisanisotropy , ,wehave =H= I; (13)(24);0 ; (12)(34);12 ; (14)(23);12 andK= I; (13)(24);0 We clearlyseethatH=K =Z2is cyclicandthusthetrot-likegaitexistsasa solutionofthesystemaslongaswechooseacoup lingschemesuchthatKisanisotropy subgroup(whichiseasy).

10 We justhave tochoosea couplingsuchthatthetrotgaitisstable,butw ealreadyknowthatit ,weaddstandardsubtractivecouplingbetween theseoscillatorsinordertoenforcethehalfa ,wealsoaddadditive make thedesiredphaseshiftsbetween2 oscillatorsstable[18].Figure4 (10)_yi= iyi(Ki( 2 x2i) y2i) Kixi c1yj+c2yk(11)Ki=kstance(e byi+ 1)(e kyj+ 1)+kswingebyi+ 1(12) i= (1 + e x2i)(13)wherei=1:::4denotestheithoscilla tor,jtheoppositeoscillatorandkthediagona loscillator, , , notethattheonlystablepatternofoscillatio nwitha smallregionofstabilitythatislimitedandfo ra randomnoisebetween[ 0:2;0:2], , a canalsosmoothlymodulatethefrequencyofthe patternbychangingindependentlythefrequen cy ofthe0102030 x1 x2 x3 x4 Time(a)Modulationofthefrequency0102030 x1 x2 x3 x4 Time(b) (a)wemodulatethefrequency 2, att= 10wesetkswing= 4kstance=4 2, whichcorrespondstoa doublingofthespeedoftheswingandatt= 20wesetkstance= 4kswing= 4 2.