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The rolling problem: overview and challenges YacineChitour,MauricioGodoyMolinaandPetriKokkonen 3 1 0 2 n a J 7 2 AbstractInthepresentpaperwegiveahistoricalaccount–rangingfromclassical tomodernresults–oftheproblemofrollingtwoRiemannianmanifoldsoneonthe ] other, with the restrictions that they cannot instantaneously slip or spin one with C respect to the other. On the way we show how this problemhas profited fromthe O developmentofintrinsicRiemanniangeometry,fromgeometriccontroltheoryand . h sub-Riemanniangeometry.Wealsomentionhowotherareas–suchasroboticsand t interpolationtheory–haveemployedtherollingmodel. a m [ 1 Introduction 1 v 0 Differential geometry has been inextricably related to classical mechanics, since 7 3 its very conceptionin the 18th century.As a matter of fact, back in the days, this 6 areaofresearchwasreferredtoasrationalmechanics.Thebasicideaofthispoint . of view is reasonablysimple: to a given mechanicalsystem M, one can associate 1 0 a differentiable manifold M in such a way that each possible state of the system 3 correspondstoauniquepointinM.Inthisway,eachpossiblevelocityvectorofM 1 atagivenconfigurationisrepresentedasatangentvectortoMatthecorresponding : v point.Theclassicaldictionarygoesasfollows: i X r YacineChitour a L2S, Universite´ Paris-Sud XI, CNRS and Supe´lec, Gif-sur-Yvette, 91192, France, e-mail: [email protected] MauricioGodoyMolina L2S, Universite´ Paris-Sud XI, CNRS and Supe´lec, Gif-sur-Yvette, 91192, France, e-mail: [email protected] PetriKokkonen L2S, Universite´ Paris-Sud XI, CNRS and Supe´lec, Gif-sur-Yvette, 91192, France and Uni- versity of Eastern Finland, Department of Applied Physics, 70211, Kuopio, Finland, e-mail: [email protected] 1 2 YacineChitour,MauricioGodoyMolinaandPetriKokkonen 1. Physicaldata(suchasmasses,lengths,etc.)ofelementsinMinduceaRieman- nianmetricinMrepresentingthekineticenergy. 2. Linearrestrictionsimposedonthepositionsof M(orthatcanbeintegratedto such)translatetosubmanifoldsofM. Inthelate19thcentury,physicistsnotedtherewereplentyofmechanicalsystems notconsideredbytheabovedictionary.Thesesystemswerenamednon-holonomic, opposedtoholonomicsystemswhicharedefinedinthesecondpointofthedictio- nary above. A mechanical system M is non-holonomicif its dynamics has linear restrictionsthatcannotbeintegratedtoconstraintsoftheposition.Forvariousex- amplesandabriefhistoricalbibliography,werefertheinterestedreadertothesur- vey[8].A well-knownearlyexampleofthesesystemsisthesphererollingonthe planewithoutslidingorspinning,studied(withsomevariants)byS.A.Chaplygin in the seminal works [16, 17]. Our aim in this paper is to give a general look at some of the most important breakthroughsin mathematics that gave us some un- derstandingofthegeneralizedversionofthissystemconsistingontwoRiemannian manifolds M and Mˆ of the same dimension rolling one against the other, not al- lowing instantaneous spins or slips. Nowadays these systems are often studied in connection to sub-Riemannian and Riemannian geometry [43, 48] and geometric controltheory[3]. The structure of the paper is the following. In Section 2 we recall two major playersin the studyof the mechanicalsystem describedaboveand earlydifferen- tial geometry:S. A. Chaplyginand E´. Cartan. Chaplyginstudies forthe first time theproblemfromamechanicalpointofviewandfindsfirstintegralsofmotionin differentsituations.Cartan’sdevelopmentandhiscelebrated“fivevariables”paper were notevidentlyconnectedto the rollingmodelat the time of theirpublication, see [10], nevertheless we present them from our point of view. In Section 3, we briefly present Nomizu’s breakthrough introduction of the dynamics of rolling in higherdimensions,throughembeddedsubmanifoldsofEuclideanspaceanditsrela- tiontoCartan’sdevelopment.InSection4wepresenthowtheproblemwasbrought backtolifewhencontroltheoryseesindifferentialgeometryausefultooltotreat thecontrollabilityissueoftherollingmodelintwodimensionsandsomegeometric consequences of optimality conditions. Section 5 surveys how the higher dimen- sionalrollingsystem was re-discoveredand howit appearsnaturallyin geometric interpolation.Finally in Section 6 we present the latest results that have been ob- tainedconcerningthecontrollabilityofthesystemanditssymmetries.Weconclude withabriefdiscussiononsomegeneralizationsandopenproblems. 2 The early years:Mechanics andthe new differential geometry Thefirsttimethe problemofa ballrollingonthe planewasconsideredasworthy ofstudy wasin the seminalpapersof S. A. Chaplygin[16, 17], one ofthe fathers of non-holonomicmechanics.The resultswere consideredsurprisinglydifficultat thetime,andfor[16]ChaplyginwontheGoldMedalofRussianAcademyofSci- Therollingproblem:overviewandchallenges 3 ences.Themainresultsheobtainedwerefirstintegralsofmotionforthesystemin several geometric situations. Even these seemingly elementary problems contains unexpected difficulties and bottlenecks when trying to obtain closed formulae for thedynamics.Asstatedin[16],afterobservingthatthedifferentialequationofthe dynamicscanbeintegratedinquadratures.Essentiallyatthesametime,E´.Cartan was developing his coordinate-freedifferentialgeometry. With this new language hewasabletoproposeandstudymanyproblems,mostoftenrelatedtothesearch ofinvariantsofgeometricsystems.Inthissurvey,wewillonlyfocusintwoofhis manyideas:thesearchforinvariantsandsymmetriesforcontrolsystemswithtwo controlsandfivedegreesoffreedom,andthedefinitionofaffineRiemannianholon- omythroughthedevelopmentofacurve.Bothofthisideaswillappearseveralother timesinthissurvey. 2.1 Chaplygin’s ball In the year 1897 the work [16] written by S. A. Chaplygin was published. This papers is one of a series of research articles in which Chaplygin analyzed non- holonomicsystems.Alsoofparticularrelevancetothissurveyisanotherpaper[17]. Inparticularhewasinterestedinstudyingfirstintegralsandequationsofmotionfor differentsystemsofrollingballs. Toillustratehisresults,Chaplyginwasabletofindanintegralofmotionforthe systemofahomogeneoussmallballofmassm andahomogeneoussphereofmass 1 m ,inwhichtheballrollswithoutslippinginsidethesphere. We willthinkofthe 2 dynamicsoccurringinEuclidean3-space.LetObethecenterofthesphere,letGbe thecenterofthemovingballandAthepointofcontactbetweenthetwo.Introducing thequantitiesa=dist(O,G)andb=dist(O,A),thenonehastheintegralsofmotion: (cid:229) 2 m y dxi x dyi +M b 1 b da a db =const. i i i (cid:18) dt − dt (cid:19) (cid:18)a− (cid:19)(cid:18) dt − dt (cid:19) i=1 Where A=(a ,b ,g ) with respect to a fixed frame OX Y Z , and the points G= ′ ′ ′ (x ,y ,z )andO=(x ,y ,z )withrespecttoamovingframeAXYZ,withaxesat 1 1 1 2 2 2 alltimesparalleltothoseinOXY Z.AdditionallyM=m +m denotesthemass ′ ′ ′ 1 2 ofthesystem. Theequationsofmotionarecomplicatedanditserveslittlepurposetowritethem downhere.Nevertheless,thereisaninterestinghistoricalremarkatthispoint.After arrivingataverycomplicateddifferentialequationtodescribethedynamicsofthe system,Chaplyginobservesitcanbewrittenintheform dv +vF (z )+Y (z )=0, dz 4 YacineChitour,MauricioGodoyMolinaandPetriKokkonen forsomeappropriatefunctions F andY afteraseriesofchangesofvariables.He thenventurestosay [...]and,therefore,canbeintegratedinquadratures.Wewillnotwriteoutthesequadratures sincetheyarerathercumbersome. Asfarasweknow,theintegrationofdifferentialequationsconnectedtotheproblem ofrollingballsisstillanareaofactiveresearch,seeforexample[13]. 2.2 Cartan’s“five variables”paper AranklvectordistributionDonann-dimensionalmanifoldMor(l,n)-distribution (where l <n) is, by definition, an l-dimensional subbundle of the tangent bundle TM,i.e.,asmoothassignmentq D definedonMwhereD isanl-dimensional q q 7→ | | subspaceofthetangentspaceT M.TwovectordistributionsD andD aresaidtobe q 1 2 equivalent,ifthereexistsadiffeomorphismF :M MsuchthatF D =D 1 q 2 F(q) foreveryq M.Localequivalenceoftwodistribu→tionsisdefineda∗nalo|gously.| ∈ Cartan’sequivalenceproblemconsistsinconstructinginvariantsofdistributions withrespecttotheequivalencerelationdefinedabove.AseminalcontributionbyE´. Cartan in [14] was the introductionof the “reduction-prolongation”procedurefor buildinginvariantsandthecharacterizationfor (2,5)-distributionsvia afunctional invariant (Cartan’s tensor) which vanishes precisely when the distribution is flat, thatis, whenitislocallyequivalentto the(unique)gradednilpotentLiealgebra h ofstep3withgrowthvector(2,3,5). Inthesamepaper,Cartanalsoprovedthatinthissystemthereishiddenareal- izationofthe14-dimensionalexceptionalLiealgebrag .Toexplainwheredoesit 2 appear,letusrecallthataninfinitesimalsymmetryofan (l,n) distributionD isa − vectorfieldX VF(M)suchthat[X,D] D.Nowconsiderthe(unique)connected ∈ ⊆ and simply connected nilpotent Lie group H with Lie algebra h. The two dimen- sionalsubspaceofhthatLiegeneratesit,canbeseenasa(2,5) distributiononH. − Ingeneral,a(2,5) distributionthatisbracketgeneratingisnowadaysknownasa − Cartandistribution.Inthissetting,thefollowingtheoremtakesplace. Theorem1 (Cartan1910). The Lie algebra of symmetries of the flat Cartan distribution is precisely g , andthis situation is maximal,that is, for generalCartan 2 distributionsthedimensionoftheLiealgebraofsymmetriesis 14. ≤ Moreover, Cartan gave a geometric description of the flat G -structure as the 2 differentialsystem that describes space curves of constant torsion 2 or 1/2 in the standardunit3-sphere(seeSection53inParagraphXIin[14].) The connection between this studies by Cartan and the rolling problem comes from the fact that the flat situation described above occurs in the problem of two 2-dimensional spheres rolling one against the other without slipping or spinning, assumingthattheratiooftheirradiiis1: 3,see[12]forsomehistoricalnotesanda thoroughattemptofanexplanationforthisratio.Infact,whenevertheratiooftheir Therollingproblem:overviewandchallenges 5 radiiisdifferentfrom1: 3,theLiealgebraofsymmetriesbecomesso(3) so(3), × thus droppingits dimension to 6. A complete answer to this strange phenomenon aswellasa geometricreasonforCartan’stensorwasfinallygivenin tworemark- able papers [52, 53] (cf. also [4]), where a geometric method for construction of functionalinvariantsofgenericgermsof(2,n)-distributionforarbitraryn 5isde- ≥ veloped.Ithasbeenrecentlyobservedin[5]thattheLiealgebraofsymmetriesofa systemofrollingsurfacescanbeg inthecaseofnon-constantGaussiancurvature. 2 2.3 Cartan’sdevelopment E´. Cartan in [15] defined a geometric operation, that he called development of a manifoldontoatangentspace,inordertodefineholonomyintermsof“Euclidean displacements”,i.e.,elementsofE(n).Inhisownwords: Quand on de´veloppe l’espace de Riemann sur l’espace euclidien tangent en A le long d’un cycle partant de Aet yrevenant, cet espace euclidien subit un de´placement et tous lesde´placementscorrespondant auxdiffe´rentscyclespossiblesformentungroupe,appele´ grouped’holonomie. Aninterpretationofthisquoteintermsofmanifoldsrollingfollowsnaturally.For a givenloopg : [0,t ] M onann dimensionalRiemannianmanifoldM, onecan rollMagainsttheEuc→lideanspaceRn obtaininganewcurvegˆ: [0,t ] Rn,called the developmentof g . By parallel transporting along g any orthonorm→alframe of T g(0)M, we obtain a rotation Rg O(n). The fact that gˆ is not necessarily a loop in|ducesatranslationTg correspon∈dingtothevectorgˆ(t ) gˆ(0).We concludethat wecanassociatetog anelement(Rg,Tg)oftheEuclidea−ngroupofmotionsE(n). ThesubgroupHolaff(M)ofE(n)consistingofallsuch(Rg,Tg)obtainedbyrolling alongall absolutelycontinuousloopsg is knownas the affine holonomygroupof MandtheorthogonalpartHol(M) O(n)ofitistheholonomygroupofM. ⊆ It is known that if M is complete and with irreducible Riemannian holonomy group,theaffineholonomygroupcontainsalltranslationsofT M,see[37,Corol- x | lary7.4,ChapterIV].Inotherwords,undertheirreducibilityhypothesis,therota- tionalpartoftheaffineholonomypermitstorecoverthetranslationalpart,andthis consistsofallthepossibletranslationsinT M. x | PerhapssomethingthatmighthavebeennotexpectedbyCartanisthatthiscon- cept of developmentwould play a fundamentalrole in the definition of Brownian motiononamanifold,andthesubsequentexplosionofinterestthatstochasticanal- ysis in Riemannian manifoldshas had in later decades, see [29]. For a long time, mathematicians have had the intuition that by rolling an n-dimensional manifold M alongagivencurvey(t)inRn withtheEuclideanstructure,onewouldobtaina curveinMwhichresemblestheoriginalcurvey(t),see[27].Themainoutstanding idea(asfarasweknowduetoMalliavin)wastouseCartan’sdevelopmentthrough theorthonormalframebundleandWiener’smeasure,see[50]. The idea of how to define Brownian trajectories on manifoldsis similar to the interpretation given above. Intuitively, one can draw a Brownian path B(t) in Rn, 6 YacineChitour,MauricioGodoyMolinaandPetriKokkonen and then one can consider the system of M rolling against Rn following the path B(t).TheprecisedefinitionusesalessregularversionofCartan’sdevelopmentand paralleltransport. This naive notion allows one to recover the Laplace-Beltrami operator D of M the manifold.It is often interpretedas if Brownianpaths are the “integralcurves” forD . Of coursethis assertionlacks ofmathematicalprecision, butit introduces M the idea that second order differential operators induce “diffusions” on the manifold.This pointof view hasbeen exploitedsignificantlyin the study ofstochastic differentialequationsonmanifolds,see[7]. 3 A “forgotten” breakthrough An important contribution to the understanding of the problem of rolling without slipsorspinscametolightinthepaper[45]byK.Nomizu.Hisaimwastogivea mechanicalinterpretationofcertaindifferentialgeometricinvariantsusingthissys- tem.HemainlyfocusesinsubmanifoldsofRN withtheusualEuclideanstructure, andsowillwealongthissection. He begins with a simple generalconsideration:as a motion occurring in a Eu- clideanspaceRN withoutdeformingobjects,arollingcanbeseenasacurveinthe EuclideangroupE(N),thatisafunction[0,t ] t f E(N)givenby t ∋ 7→ ∈ C c f = t t , (1) t (cid:18)0 1(cid:19) where f =Idistheidentitymatrixof(N+1) (N+1),C O(N)andc RN. 0 t t × ∈ ∈ Hecallssuchtypesofcurves1-parametricmotions. Fora given 1-parametricmotion f , he observedthat there is a naturaltime- t dependentvectorfield X associated{to }it. Foranarbitrarypoint y RN we define t ∈ (X) := dfu(x) ,wherex= f 1(y).Usingequation(1),onecanseethat(X) = t y du t− t y (cid:12)u=t Sy+v,where(cid:12)S = dCtC 1 o(N)andv = Sc +dct RN.Thecorresponding t t (cid:12) t dt t− ∈ t − t t dt ∈ elementoftheLiealgebrae(N) ddftt ft−1=(cid:18)S0t v0t(cid:19) (2) iscalledtheinstantaneousmotion.Slipsandspinscannowbeencodedintermsof thevectorfieldX andtheinstantaneousmotion. t Definition1. Theinstantaneousmotion(2)iscalledaninstantaneous: standstillifS =0andv =0, t t • translationifS =0andv =0, t t • rotationifthereexistsapoi6nty RN suchthat(X) =0andS =0. • 0∈ t y0 t 6 Therollingproblem:overviewandchallenges 7 With this at hand, it is possible to define rolling without slipping (skidding in Nomizu’sterminology)norspinningbetweenMn,Mˆn֒ RN. → Definition2. Let f bea1-parametricmotionsuchthat f (M)istangenttoMˆ at t t apointy Mˆ.As{sum}ethat(X) =0andS =0.Themotion f isarollingiffor t ∈ t yt t 6 t anypairoftangentvectorsX,Y T N ∈ yt S(X),Y =0, (3) t h i andforanypairofnormalvectorsU,V T Mˆ ∈ y⊥t S(U),V =0. (4) t h i Anequivalentwayofstatingconditions(3)and(4)isthatS mapsT Mˆ toT Mˆ t yt y⊥t andalsomapsT Mˆ toT Mˆ. y⊥t yt This definition allowed Nomizu to find a very concrete realization of Cartan’s development.Forthecaseofsurfacesrollingontheplane,hisresultreads Theorem2 (Nomizu1978).Let x be a smoothcurve on a surface M which does t not go through a flat point of M. There exists a unique rolling f of M on the t tangent plane S at x such that y = f (x) is the locus of point{s o}f contact. The 0 t t t curvey isthedevelopmentofthecurvex intoS . t t As a consequence of this result, Nomizu noticed that there is a natural kinematic interpretationoftheLevi-CivitaconnectionforasurfaceM,comingfromtherolling formulation: a vector field U(t) along the curve x is parallel with respect to the t Levi-CivitaconnectionofMifandonlyifC(U(t))isaconstantvectorforallt. t As a matter of fact, he was able to extend this result to higher dimensionsand gave conditionsunderwhich rollingsexist in terms of the shapes of the submani- folds,thatis,intermsofbothintrinsicandextrinsicdata. For reasons unknown to us, this paper seems to have been forgotten over the years. Nomizu’s definition of higher dimensionalrolling is equivalentto Sharpe’s oneinSubsection5.1andmanyofhisobservationshavebeenrediscoveredin[48, AppendixB].Nevertheless,thereisnoreferencetothepaper[45]inSharpe’sbook. 4 Revival:The two dimensional caseand robotics The aim of this section is to put in context the study of the rolling model for the caseoftwodimensionalmanifolds,andhowtheyappearednaturallyinproblemsof sub-Riemanniangeometry,roboticsandgeometriccontroltheory. 8 YacineChitour,MauricioGodoyMolinaandPetriKokkonen 4.1 Rigidityofintegralcurves inCartan’sdistribution In the celebrated paper [11], R. Bryant and L. Hsu studied curves on a manifold Q ofdimensionn 3 tangentto a (2,n) distributionD. Theidea was to analyze thespaceW (p,q)≥ofdifferentiablecurve−sinQconnectingtwopointsp,q Qand D beingtangenttoD(calledD-curvesbythem).ThespaceW (p,q)isendow∈edwith D itsnaturalC1 topology.TheideathatD-curvescanbe“rigid”playsafundamental roleintheirpaper. Definition3. AD-curveg : [0,t ] QisrigidifthereisaC1-neighborhoodU ofg inW (g (0),g (t ))sothateveryg →U isareparametrizationofg .Wesaythatg is D 1 locallyrigidifeverypointofI=[0∈,t ]liesinasubintervalJ Isothatg restricted ⊂ toJ isrigid. Theirmainresultgoesasfollows. Theorem3 (Bryant & Hsu 1993).Let D be a non-integrablerank 2 distribution on a manifold Q of dimension (2+s) 3. Suppose further that the distribution ≥ D =[D,D] (which hasrank3) is nowhereintegrable.Then there alwaysexist D- 1 curvesthatarelocallyrigid. Theygiveamoreprecisedescriptionofsuchcurvesintermsofprojectionsofchar- acteristic curvesin a dense subsetof the annihilatorof D , but stating it precisely 1 wouldnotservethepurposesofthisexposition. Forus,themostrelevantpartoftheirworkistheirsectiononexamples,inpar- ticulartheirstudyofsystemsofCartantypeandofrollingsurfaces. Recallthatabracketgenerating(2,5) distributionissaidtobeofCartantype. − InotherwordsDisaCartandistributionifD hasrank3andD =[D ,D]hasrank 1 2 1 5.AsaconsequenceofTheorem3,theyobservethatthereisexactlya5-parameter familyoflocallyrigidD-curves.Infacttheybrieflydiscussaremarkablegeometric behavioroccurringinthissituation:ifMisconnected,thenanytwopointsofMcan bejoinedbyapiecewisesmoothD-curve,whosesmoothsegmentsarerigid. After all these observations, they devote themselves to the analysis of two oriented surfaces M and Mˆ endowed with Riemannian metrics rolling one over another without slipping or twisting. Let F and Fˆ be the oriented orthonormal frame bundles of M and Mˆ. Bryant and Hsu considered the “state space” manifold Q=(F Fˆ)/SO(2), where SO(2) acts diagonally on the Cartesian product. Anelementin×Qisatriple(x,xˆ;A),wherex M,xˆ Mˆ andA: T M TMˆ isan x xˆ orientedisometry.Theirformulationis asfol∈lows.C∈onsidera curveg→: [0,t ] Q givenbyg (t)=(x(t),xˆ(t);A(t)),thentheno-slipconditionreadsA(t)(x˙(t))=→x˙ˆ(t). Theno-twistconditionrequiressomemorecare.Let e ,f : [0,t ] TM beapar- 1 1 → allelorthonormalframealongthecurvex(t)andlet e (t)=A(t)(e (t)), f (t)=A(t)(f (t)), 2 1 2 1 betheorthonormalframealongxˆ(t)obtainedviaA.Therollinghasno-twistwhen- everthemovingframee ,f isalsoparallel(alongxˆ). 2 2 Therollingproblem:overviewandchallenges 9 An important insight for the problem was expressing the no-twist and no-slip conditionsintermsofa(2,5) distributionDonQ.Leta ,a ,a bethecanonical 1 2 21 1-formsofMonF andsimila−rlyb ,b ,b forMˆ,see[49].Recallthattheseforms 1 2 21 satisfytheso-calledstructureequations da =a a , db =b b , 1 21 2 1 21 2 ∧ ∧ da = a a , db = b b , 2 21 1 2 21 1 − ∧ − ∧ da =ka a , db =kˆb b , 21 1 2 21 1 2 ∧ ∧ where k and kˆ are the Gaussian curvatures of M and Mˆ respectively. With all of this,onecanconsiderthedistributionD˜ onF Fˆ definedbythePfaffianequations × a b =a b =a b =0. 1 1 2 2 21 21 − − − Thedistributiontheywerelookingforcorrespondstothe“push-down”imageofD˜ underthesubmersionF Fˆ Q.Asmoothcurveg : [0,t ] Qdescribesarolling withoutslippingortwist×ingi→fandonlyifg isaD-curve. → AremarkablefactisthatthedistributionDisofCartantypewheneverk kˆ =0, − 6 whichisanopensetinQ.Onthisset,thecorresponding5-parameterfamilyofrigid curvesdescribestherollingofMˆ againstMfollowinggeodesics. 4.2 Non-holonomyinrobotics Thetraditionalmodelingofamechanicalsystemconsidersconfigurations(orstates) of thismechanicalsystem as pointsq of a smoothfinite-dimensionalmanifoldM, and the correspondingvelocities q˙ T M are subject to locally independentcon- q ∈ straintsinthePfaffianform A(q)q˙=0, (5) whereA()isanm nmatrixofreal-valuedanalyticfunctions,wherem<n.Con- · × straintsaresaidtobeholonomiciftheirdifferentialformgivenby(5)isintegrable. Inthiscase,thereexistintegralsubmanifoldsofdimensionn mthatareinvariant. − Iftheconstraintsarenotholonomicatsomeq M,thentherewillexistanintegral 0 ∈ submanifoldcontainingq ofdimensionn m+kwith0<k m.Theintegerkis 0 − ≤ referredtoasdegreeofnon-holonomy.If k=m, theconstraints,andbyextension thesystem,aresaidtobemaximallynon-holonomic(see[44]). There is a more convenientway for control theory to describe the constrained system. If G(q)denotesa matrix whosecolumnsforma basis forthe annihilating distributionofA(q),thenalladmissiblevelocitiesq˙ A(q) T Mcanbewritten ⊥ q ∈ ⊂ aslinearcombinationsofthecolumnsofG(q), n m (cid:229)− q˙=G(q)w= g(q)w, (6) i i i=1 10 YacineChitour,MauricioGodoyMolinaandPetriKokkonen wherewisavectorofquasivelocitiestakingvaluesinRn m.Whenquasivelocities − canbe assignedvaluesatwill in time,functionscan beregardedas controlinputs of the driftless, linear-in-the-control,nonlinear system defined by (6). A physical actuatorisassociatedtoeachcontrolinput,e.g.amotorforelectromechanicalsys- tems. The issue of non-holonomy of the original system, i.e. non-integrability of (5),canbeaddressedbystudyingthedistributionD spannedbythethevectorfields g’sandmorepreciselythecorrespondingLiealgebrageneratedbytheg’s.Ifthe i i system ismaximallynon-holonomic(orcompletelycontrollable),anytwo config- urations q and q of its n-dimensionalmanifold can be connected along the flows ′ ofn mvectorfields.Fromanutilitarianengineersviewpoint,the latterdefinition − mayberephrasedasann-dimensionalnon-holonomicsystemcanbesteeredatwill using less thanactuators.Thisformulationunderscoresthe appealingfact thatde- viceswithreducedhardwarecomplexitycanbeusedtoperformnontrivialtasks,if non-holonomyisintroducedonpurpose,andcleverlyexploited,inthedevicedesign (see[44]). Non-holonomyofrollingisparticularlyrelevanttoroboticmanipulation,oneof themaingoalsofwhichistomanipulateanobjectgraspedbyarobotend-effectorso astorelocateandre-orientitarbitrarily,theso-calleddexterityproperty.Dexterous robotichandsdevelopedsofaraccordingtoananthropomorphicparadigmemploy far too many joints and actuators (a minimum of nine) to be a viable industrial solution.Non-holonomyof rollingcan be used to alleviate this limitation.In fact, whilerollingbetweenthesurfacesofthemanipulatedobjectandthatoffingershas been previously regarded as a complication to be neglected, or compensated for, someworks(see,inparticular,[1,6,18,24,39,40]andthereferencestherein)tried toexploitrollingforachievingdexteritywithsimplermechanicalhardware. Introducingnon-holonomyonpurposeinthedesignofroboticmechanismscan be regarded as a means of lifting complexity from hardware to the software and control level of design. In fact, planning and controlling non-holonomic systems is in general a considerably more difficult task than for holonomic systems. The veryfactthattherearefewerdegrees-of-freedomavailablethanthereareconfigura- tionsimpliesthatstandardmotionplanningtechniquescannotbedirectlyadapted tonon-holonomicsystems.Fromthecontrolviewpoint,non-holonomicsystemsare intrinsicallynonlinearsystems,inthe sensethattheyare notexactlyfeedbacklin- earizable,nordoestheirlinearapproximationretainthefundamentalcharacteristics ofthesystem,suchascontrollability(see[44]). Thesystemofrollingbodiesconsideredheredifferssubstantiallyfromtheclass of chained form systems or differentially flat systems (see Rouchon [46]). Con- sider, for example, the case such of the plate-ball system (i.e. a ball rolling on a plane without slipping or spinning), which is a classical problem in rational mechanics,broughttotheattentionofthecontrolcommunitybyBrockettandDai[9]. Montana [42] derived a differential-geometric model of the rolling constraint between general bodies, and discussed applications to robotic manipulation. Li and Canny[38]showedthattheplate-ballsystemiscontrollable,andthatthesameholds fortworollingspheres,providedthattheirradiiaredifferent.