Robotics: Science and Systems 2006 Philadelphia, PA, USA ...

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).

2 Thesecriteriaincludebeingabletoadjustthe relative durationsofascendinganddescendingphasesi nacycle, , applythemethodtothedesignofa ,wedesignthesystemtoproducestabletrot-li ke designedsuchthatthespeedoflocomotioncanb eadjustedbyvaryingthedurationofthestance phasewhilekeepingthedurationoftheswingph aseconstant,like INTRODUCTIONT hisworkispartoftheRobotCubproject,a 5-yearEuro-peanprojectwhosepurposeis tobuilda 54-degreesoffreedomhumanoidrobotwiththec ognitive abilitiesofa child[1].Theprojecthastwo maingoals:first,tocreateanopenandfreely- availablehumanoidplatformforresearchinem bodiedcognition,andsecond,tostudycogniti ve toa child,therobot(calledtheiCub) designmethodologyforthecrawlingcontrolle r, basedontheCentralPatternGenerator(CPG) ( ,respiration)[2],[3].

3 AlthoughCPGsarecontrolledbysimpledescend ingpathsfromhigherpartsofthebrain,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.

4 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).

5 Wethenpresentthedesignapproachbehindourm odelofcoupledoscillators(SectionIII).The designisdoneincrementallywithfirstthecon structionofa nonlinearoscillatorwithtwo controlledtimescales,thentheadditionofin ter-limbinfluencesbetweenoscillatorsofop positelimbs,andfinallytheadditionofinter -limbcouplingsbetweenthecompletefour-osc illatorsystemforimplementingthetrot-like rigidarticulatedbodysimulationoftheiCub, andcomparedtotheoriginalinfantcrawlingga its(SectionIV).Thepaperconcludeswitha shortdiscussion(SectionV).II. CRAWLINGININFANTSV eryfewstudiesaboutcrawlinginbabieshave [15],[16]butthey allfocusonthecognitive developmentofinfantsthroughlocomotionand nonehave [15].

6 Indeed,thisgaitisthemostwidespreadoneamo nginfantsandforafirststudyofcrawlingit ,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.

7 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.

8 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)

9 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)wheretheexponentialfunctionworksasa stepfunctionwhichselectseitherkswingorks tanceaccordingtothesignofthevelocityofmo vementy, a periodicorbitsaroundtheunstablecenter0an dthusthesystemis canpointtheflowtowardoneperiodicorbitbyc onstrainingthetotalenergyofthesystem,sin ceit definesthemaximumvaluexcantake ina (Kx2+y2)(4)whichisthesumofthepotentialan dkineticenergiesofthesystem(wetake themassequaltoone).

10 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. Ineachplotweshowtheoscillationsxandtheve locityy.