Je Remercie

Frédéric Jean

Pierre Rouchon

Qui M'

Fernand Pelletier

Jean -Baptiste

Sandra Tamion

Arnaud Knippel

Khaled Lamia

Je voudrais remercier d'abord de tout mon coeur mon responsable de thèse Witold Respondek qui est à origine de ce travail. Il a dirigé mon mémoire de Master 2 avec gentillesse et ensuite accepté de me prendre en thèse. C'est un honneur pour moi de travailler avec lui et je ne peux qu'admirer son talent. Il a dirigé ma thèse avec beaucoup de patience et il a consacré beaucoup de temps à mon travail en étant toujours très disponible inconditionnellement à toutes les heures du jour et de la nuit, ce qui m'a énormément encouragé. Je le remercie d'avoir lu et corrigé très sérieusement beaucoup de versions préliminaires de mes travaux et ses redactions exceptionnelles ont donné plus de clarté aux résultats de ce mémoire. Je le remercie aussi, ainsi que son épouse Eva, de m'avoir beaucoup aidé durant ma présence à Rouen.

Introduction

Le contrôle des systèmes non linéaires constitue un domaine très actif de recherche en automatique et mathématique. Un système de contrôle non linéaire est un système d'équations non linéaires, décrivant l'évolution temporelle des variables du système sous l'action d'un nombre fini de variables indépendantes appelées contrôles de ce système. Un système de contrôle non linéaire peut être écrit généralement sous la forme :

Ξ : ẋ = f (x, u), (0.0.1) avec x ∈ X, l'état du système dans une variété differentielle et u est le contrôle à valeur dans U, une variété de dimension m appelée l'espace du contôle. Dans cette thèse, tous les systèmes que nous étudions sont des systèmes affines, i.e., des systèmes qui admettent la forme suivante Σ aff : ẋ = f (x) + m i=1 g i (x)u i = f (x) + g(x)u, ou des systèmes linéaires par rapport aux contrôles (f ≡ 0) de la forme suivante Σ lin : ẋ = m i=1 g i (x)u i = g(x)u.

Equivalence des systèmes par bouclage

Soit Σaff un autre système de contrôle défini par Σaff : ẋ = f(x) + m i=1 gi (x)ũ i = f (x) + g(x)ũ, où x ∈ X et ũ ∈ Ũ . Alors on dit que Σ aff et Σaff sont équivalents par bouclage s'il existe un difféomorphisme ϕ : X → X et un bouclage u = α(x) + β(x)ũ, où β(x) est une matrice m × m inversible et ũ = (ũ 1 , . . . , ũm ) ⊤ , tels que f = ϕ * (f + gα) et g = ϕ * (gβ).

Pour les systèmes Σ lin la relation entre f et f est évidemment absente. La distribution associée au système Σ aff , notée par D, est la distribution engendrée par les champs de vecteurs f, g 1 , . . . , g m , i.e., D = span{f, g 1 , . . . , g m } ⊂ T X.

Il est important de remarquer que pour les systèmes Σ lin , l'équivalence locale par bouclage coïncide avec l'équivalence des distributions D engendrée par les champs de vecteurs g 1 , . . . , g m . La linéarisation par bouclage est un outil très important dans la recherche des systèmes non linéaires dans le fait qu'on peut ainsi appliquer des propriétés d'un système linéaire à un système non linéaire. Le problème de la linéarisation par bouclage d'un système avec un contrôle a été résolu par Brockett [START_REF] Brockett | Feedback invariants for nonlinear systems[END_REF]. Jakubczyk et Respondek [START_REF] Jakubczyk | On linearization of control systems[END_REF], Hunt et Su [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] ont donné des conditions nécessaires et suffisantes de la linéarisation par bouclage pour un système affine avec multi-contrôles.

Système de contact canonique

Le but de cette thèse est d'étudier la géométrie et la structure des systèmes de contrôle non linéaires, en particulier les systèmes équivalents aux systèmes de contact canonique qui est une classe de systèmes non holonomes. Un système non holonome est un système soumis aux contraintes non holonomes qui ne sont pas intégrables. On rencontre ce type de contraintes par exemple dans les phénomènes de roulement sans glissement apparaissant pour les systèmes du type du robot mobile.

Considérons J n (R k , R m ), l'espace de n-jets des applications de classe C ∞ de R k dans R m avec les coordonnées canoniques données par (x 1 , . . . , x k , y 1 , . . . , y m , p σ j , 1 ≤ j ≤ m, 1 ≤ |σ| ≤ n), où x i , pour 1 ≤ i ≤ k, représentent les variables indépendantes et y j , pour 1 ≤ j ≤ m, représentent les variables dépendantes; σ = (σ 1 , . . . , σ k ) est un vecteur de nombres entiers non-négatifs tel que |σ| = σ 1 + • • • + σ k ≤ n et p σ j , pour 1 ≤ j ≤ m, correspondent aux derivées partielles ∂ |σ| y j ∂x σ . Notons p 0 j = y j , pour 1 ≤ j ≤ m. Chaque application lisse ϕ = (ϕ 1 , . . . , ϕ m ) de R k à R m définit une sous-variété dans J n (R k , R m ) par

p σ j =
∂ |σ| ϕ j ∂x σ (x 1 , . . . , x k ), pour 1 ≤ j ≤ m, 0 ≤ |σ| ≤ n et on l'appelle le n-graphe de ϕ. Tous les n-graphes sont sous-variétés intégrales, de dimension k, d'une distribution appelée la distribution de Cartan sur J n (R k , R m ), notée par CC n (R k , R m ), qui est annulée par les formes différentielles suivantes:

dp σ j - k i=1 p σ+1 i j dx i = 0, 0 ≤ j ≤ m, 1 ≤ |σ| ≤ n -1,
où σ + 1 i = (σ 1 , . . . , σ i + 1, . . . , σ k ). Il est clair que la distribution de Cartan est de rang constant. Nous appelons un système linéaire par rapport aux contrôles un système de contact canonique sur J n (R k , R m ) telle que sa distribution associée est engendrée par tous les champs de vecteurs de la distribution de Cartan CC n (R k , R m ).

Les applications des systèmes de contact canonique ou des distributions de Cartan sont de ce fait extrêmement nombreuses dans des domaines très différents tels que la mécanique, l'automatique, la thermodynamique, et l'électromagnétisme, etc. Par exemple, quand k = m = 1, le système de contact canonique sur J n (R 1 , R 1 ) peut s'écrire sous la forme du système de contrôle suivant ẋ0

0 = u 0 ẋ0 1 = x 1 1 u 0 ẋ1 1 = x 2 1 u 0 . . . ẋn-1 1 = x n 1 u 0 ẋn 1 = u 1 .
On obtient un système sous la forme chaînée -un système non holonome connu qui est extrêmement important en automatique et a été étudié par de nombreux scientifiques tels que Murray et Sastry [START_REF] Murray | Nonholonomic motion planning: Steering using sinusoids[END_REF], Samson [START_REF] Samon | Control of chained system: application to path-following and timevarying point stabilization of mobile robots[END_REF], Sørdalen [START_REF] Sørdalen | Conversion of the kinematics of a car with n trailers into a chained form[END_REF], etc. La distribution associée au système de contact canonique sur J n (R 1 , R 1 ) (la distribution de Cartan CC n (R 1 , R 1 )), est appelé aussi la forme normale de Goursat.

Nous étudions les systèmes de contact canoniques dans des cas différents afin de comprendre les différentes notions d'équivalence (par bouclage statique, par bouclage dynamique, par bouclage orbital, etc.) correspondant aux objets géometriques attachés aux systèmes de contrôles tels que les distributions, les distributions affines, les distributions caractéristiques, etc.

Platitude

La notion de la platitude différentielle a été introduite dans la théorie du contrôle par Fliess, Lévine, Martin et Rouchon [START_REF] Fliess | Sur les systèmes non linéaires différentiellement plats[END_REF], [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF] (voir aussi [START_REF] Aranda-Bricaire | Infinitesimal Brunovsky form for nonlinear systems with applications to Dynamic Linearization[END_REF], [START_REF] Isidori | A sufficient condition for full linearization via dynamic state feedback[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF], [START_REF] Pereira Da Silva | Relative Flatness and Flatness of Implicit Systems[END_REF], [START_REF] Van Nieuwstadt | Differential Flatness and Absolute Equivalence of Nonlinear Control Systems[END_REF]). Un système plat est un système pour lequel il est possible de trouver un système linéaire équivalent par bouclage endogène.

Considérons le système Ξ, défini par (0.0.1), nous lui associons l-prolongation Ξ l donnée par Ξ l : ẋ = f (x, u 0 ) u0 = u 1 . . . ul = u l+1 , pour un certain entier l, qui peut être considéré comme un système de contrôle sur X × U × R ml . Ses variables d'état sont (x, u 0 , u 1 , . . . , u l ) et ses m contrôles sont les m composantes de u l+1 . Notons ūl = (u 0 , u 1 , . . . , u l ).

Definition 0.0.1 Le système Ξ, défini par (0.0.1), est plat en (x 0 , ūl 0 ) ∈ X × U × R ml , où l est un nombre entier non négatif, s'il existe un voisinage O l de (x 0 , ūl 0 ) et m fonctions lisses, définies dans O l , h i = h i (x, u, u, . . . , u (l) ), 1 ≤ i ≤ m, et s'il existe un entier s, des fonctions lisses γ i , pour 1 ≤ i ≤ n, et des fonctions lisses δ i , pour 1 ≤ i ≤ m, tels que

x i = γ i (h, ḣ, . . . , h (s) ), 1 ≤ i ≤ n, u i = δ i (h, ḣ, . . . , h (s) ), 1 ≤ i ≤ m,
où h = (h 1 , . . . , h m ) ⊤ le long de chaque trajectoire x(t) définie par un contrôle u(t) tels que (x(t), u(t), u(t), . . . , u (l) (t)) ∈ O l . Le vecteur h = (h 1 , . . . , h m ) est appelé sortie plate de ce système. Si h i = h i (x, u 0 , u 1 , . . . , u r ), r ≤ l, le système est appelé (x, u, . . . , u r )-plat et, en particulier, x-plat si h i = h i (x). D'après la définition, le comportement du système plat est complètement caractérisé par la sortie plate et ses dérivées successives sous forme de relations algébriques. Ainsi le problème de la planification de trajectoire pour un système plat peut être résolu facilement grâce à la propriété de platitude. Il est connu qu'un système plat est toujours localement contrôlable. Notons qu'il est facile de montrer que le système de contact sur J n (R 1 , R 1 ), i.e., le système chaîné, est plat avec une sortie plate h = (h 1 , h 2 ) = (x 0 0 , x 0 1 ) en tous les points tels que u 0 = 0.

Rappelons les définitions suivantes :

Le drapeau des systèmes dérivés d'une distribution D est défini par la suite de distributions

D (0) = D et D (i+1) = D (i) + [D (i) , D (i) ], pour i ≥ 0.
Le drapeau de Lie est défini par une autre suite de distributions

D 0 = D et D i+1 = D i + [D 0 , D i ], pour i ≥ 0.
En parallèle, on peut aussi définir le système dérivé pour un système de Pfaff I = span {ω 1 , . . . , ω s }, engendré par s formes différentielles de degré 1 qui sont indépendantes partout. Son système dérivé est défini par

I (1) = span{ω ∈ I : dω ∧ ω 1 ∧ • • • ∧ ω s = 0}.
Le drapeau des systèmes dérivés de I est la suite de systèmes de Pfaff

I (0) ⊃ I (1) ⊃ • • • ⊃ I (i) ⊃ • • • ,
qui satisfait les relations I (0) = I et I (i+1) = (I (i) ) (1) . Si toutes les distributions du système dérivé sont de rang constant et D = I ⊥ , alors on a D (i) = (I (i) ) ⊥ , for i ≥ 0.

Nous allons maintenant présenter chapitre par chapitre les résultats contenus dans cette thèse.

Chapitre 1. Ce chapitre est dédié à la caractérisation complète de toutes les sorties plates pour un système avec deux contrôles qui est équivalent au système chaîné (le système de contact canonique sur J n (R 1 , R 1 )). Ce problème vient d'un phénomène que nous avons observé pour le système de la voiture (robot mobile avec une remorque)

Σ car :      ẋ ẏ θ0 θ1      =      cos(θ 1 -θ 0 ) cos θ 0 cos(θ 1 -θ 0 ) sin θ 0 sin(θ 1 -θ 0 ) 0      u 1 +      0 0 0 1      u 2 ,
où (x, y) sont les coordonnées caractésiennes du milieu de l'axe des roues arrières, et notons par θ 0 et θ 1 , respectivement, les orientations de l'axe principal de la voiture et les roues avant par rapport à x-axe. Rappelons que la platitude du système du robot mobile avec remorques roulant sans glissement a été montrée par Fliess, Lévine, Martin et Rouchon [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF] (voir aussi [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF]) et donc le système Σ car est x-plat. En effet, en chaque point de l'espace de configuration R 2 × S 1 × S 1 , ce système peut être transformé au système chaîné. En conséquence, autour de chaque point dans R 2 × S 1 × S 1 , il existe une x-sortie plate. Par contre, nous observons que chaque x-sortie plate possède un lieu singulier où il faut la remplacer par une autre x-sortie plate. Plus précisément, pour le système Σ car , le couple (h 1 , h 2 ) = (x, y) est une x-sortie plate pour les points où {q = (x, y, θ 1 , θ 2 ) | θ 1θ 0 = ± π 2 } et donc θ 1θ 0 = ± π 2 détermine le lieu singulier de la x-sortie plate (x, y). Cependant, l'équation θ 1θ 0 = ± π 2 ne définit pas les points de singularité pour le système Σ car . En effet, en ces points où θ 1θ 0 = ± π 2 , ce système est encore x-plat et la x-sortie plate peut être donnée par le couple ( h1 , h2 ) = (θ 0 , x sin θ 0y cos θ 0 ). Notons que cette deuxième x-sortie plate possède aussi un lieu singulier défini par {q = (x, y, θ 1 , θ 2 )| θ 1θ 0 = 0 , π}.

Toutes ces dernières analyses mènent naturellement aux questions suivantes : Combien de x-sorties plates possède le système? Peut-on les caractériser complètement?

Plus généralement, peut-on donner une caractérisation de toutes les x-sorties plates pour un système avec deux contrôles qui est équivalent au système chaîné (système de contact canonique sur J n (R 1 , R 1 ))?

Le problème de la platitude du système avec deux contrôles a été étudié et résolu par Martin et Rouchon [START_REF] Martin | Feedback linearization and driftless systems[END_REF]. Ils ont montré qu'un système avec deux contrôles est plat si et seulement si sa distribution associée D satisfait, dans un ouvert dense de M, la condition rang

D (i) = i + 2, 0 ≤ i ≤ n. (0.0.2)
Une structure de Goursat est une distribution de rang deux qui vérifie la condition (0.0.2) en tout point x ∈ M . Il est connu (voir von Weber [START_REF] Weber | Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen[END_REF], Cartan [START_REF] Cartan | Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes[END_REF], Goursat [START_REF] Goursat | Leçons sur le problème de Pfaff[END_REF]) que sur un ouvert dense de M, la condition (0.0.2) implique que D peut être transformée à la forme normal de Goursat (autrement dit, le système associé est localement équivalent au système chaîné). Kumpera, Ruiz et Giaro [START_REF] Giaro | Sur la lecture correcte d'un resultat d' Élie Cartan[END_REF] ont découvert l'existence des points singuliers pour le problème de transformer une distribution de rang deux à la forme normale de Goursat. Murray [START_REF] Murray | Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems[END_REF] a montré qu'un système avec deux contrôles est équivalent au système chaîné au point x 0 , autrement dit, la distribution associée est équivalente à la forme de Goursat, si et seulement si sa distribution associée satisfait la condition (0.0.2) et aussi la condition de régularité:

dim

D (i) (x 0 ) = dim D i (x 0 ), 0 ≤ i ≤ n. (0.0.3)
Alors, une question naturelle se pose : est-ce qu'un système dont la distribution associée D est une structure de Goursat est plat en les points où la condition de régularité (0.0.3) n'est pas satisfaite?

Theorem 0.0.2 Considérons un système Σ avec deux contrôles,

Σ : ẋ = f 1 (x)u 1 + f 2 (x)u 2 ,
où x ∈ M, une variété de dimension n + 2, pour n ≥ 1. Supposons que la distribution associée D = span {f 1 , f 2 } est une structure de Goursat, i.e., rang D (i) = i + 2, pour 0 ≤ i ≤ n. Alors les conditions suivantes sont équivalentes au point x 0 ∈ M (i) Σ est x-plat en (x 0 , ūl 0 ) ∈ M × R 2(l+1) , pour certain l ≥ 0; (ii) Σ est x-plat en (x 0 , u 0 ) ∈ M × R 2 ; (iii) dim D (i) (x 0 ) = dim D i (x 0 ), pour 0 ≤ i ≤ n et u 0 ∈ U sing (x 0 ); (iv) Σ est localement, autour du point x 0 , équivalent au système chaîné et u 0 ∈ U sing (x 0 ).

Remarque. Théorème 0.0.2 montre qu'un système tel que la distribution associée est une structure de Goursat est x-plat seulement aux points où la condition de régularité est satisfaite, i.e., dim D (i) (x 0 ) = dim D i (x 0 ), pour 0 ≤ i ≤ n et u 0 ∈ U sing (x 0 ) (voir chapitre 3 pour la définition de U sing ). Donc notre résultat donne une réponse négative pour la question posée par Martin et Rouchon [START_REF] Martin | Feedback linearization and driftless systems[END_REF] (voir aussi [START_REF] Martin | Feedback linearization and driftless systems[END_REF]) (concernant la xplatitude). Toute structure de Goursat peut être transformé à la forme normale de Kumpera-Ruiz ( [START_REF] Kumpera | Sur l'équivalence locale des systèmes de Pfaff en drapeau[END_REF], [START_REF] Mormul | Goursat flags: classification of codimension-one singularities[END_REF], [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF]). Alors notre résultat montre aussi que aucune forme normale de Kumpera-Ruiz singulière n'est x-plate.

Théorème 0.0.2 montre que les seules structures de Goursat x-plates sont celles qui satisfaitent la condition de régularitée (le système associé est équivalent au système chaîné). Alors nous allons caractériser toutes les x-sorties plates pour ce type de système avec deux contrôles.

Theorem 0.0.3 (Caractérisation des sorties plates)(Première version) Considérons un système Σ avec 2 contrôles, défini sur M, une variété de dimension n + 2, tel que sa distribution associée D satisfait rang D (i) = rang D i = i + 2, pour 0 ≤ i ≤ n. Fixons un point x 0 ∈ M et supposons que g 1 est un champ de vecteurs arbitraire dans D tel que g 1 (x 0 ) ∈ C n-1 (x 0 ), où C n-1 est la distribution caractéristique de D (n-1) , et ϕ 1 , ϕ 2 sont deux fonctions lisses définies dans un voisinage ouvert M du point x 0 . Alors (ϕ 1 , ϕ 2 ) est une x-sortie plate du système Σ en (x 0 , u 0 ), où u 0 ∈ U sing (x 0 ), si et seulement si les conditions suivantes sont satisfaites : (i) dϕ 1 (x 0 ) ∧ dϕ 2 (x 0 ) = 0, i.e., dϕ 1 et dϕ 2 sont indépendantes en x 0 ; (ii) (L g 1 ϕ 1 (x 0 ), L g 1 ϕ 2 (x 0 )) = (0, 0);

(iii) L c ϕ 1 ≡ L c ϕ 2 ≡ L c ( L g 1 ϕ 2 L g 1 ϕ 1
) ≡ 0, ∀ c ∈ C n-1 , où les fonctions ϕ 1 et ϕ 2 sont permutées telles que L g 1 ϕ 1 = 0. C'est toujours possible grâce à la condition (ii).

De plus, si (ϕ 1 , ϕ 2 ) satisfait (i) partout dans M et est une x-sortie plate en (x, u) pour chaque point x ∈ M et un certain u = u(x), où M est un ouvert dense dans M, alors Sing(ϕ 1 , ϕ 2 ) = {x ∈ M : (L g ϕ 1 (x), L g ϕ 2 (x)) = (0, 0)}.

Le Théorème 0.0.3 nous permet de vérifier si une paire de fonctions donnée (ϕ 1 , ϕ 2 ) est une x-sortie plate pour un système linéaire par rapport aux contrôles avec deux contrôles. Le théorème suivant va répondre à la question : y-a-t-il beaucoup de paires (ϕ 1 , ϕ 2 ) qui satisfont les conditions du Théorème 0.0.3? Theorem 0.0.4 Considérons un système Σ avec 2 contrôles, défini sur M, une variété de dimension n + 2, tel que sa distribution associée D satisfait rang D (i) = rang D i = i + 2, pour 0 ≤ i ≤ n. Fixons un point x 0 ∈ M et soit g 1 un champ de vecteurs arbitraire dans D tel que g 1 (x 0 ) / ∈ C n-1 (x 0 ). Alors, etant donné une fonction lisse ϕ 1 telle que L c ϕ 1 = 0, ∀ c ∈ C n-1 , et L g 1 ϕ 1 (x 0 ) = 0, il existe toujours une fonction ϕ 2 telle que (ϕ 1 , ϕ 2 ) forme une x-sortie plate de Σ en (x 0 , u 0 ), où u 0 ∈ U sing (x 0 ). De plus, si les deux paires (ϕ 1 , ϕ 2 ) et (ϕ 1 , φ2 ) sont des x-sorties plates de Σ, alors on a span {dϕ 1 (x), dϕ 2 (x)} = span {dϕ 1 (x), d φ2 (x)}, pour tous les points x dans un voisinage du point x 0 .

L'importance du Théorème 0.0.3 est qu'il nous permet non seulement de vérifier si une paire de fonctions donnée forme une x-sortie plate, mais aussi, avec Théorème 0.0.4, d'exprimer explicitement un système d'équations aux dérivées partielles du 1 er ordre dans le but de calculer toutes les x-sorties plates pour un système Σ avec deux contrôles (voir Section 1.3.2 pour les détails). En fin, nous appliquons nos résultats au système du disque roulant et du robot mobile avec remorques afin de montre comment calculer toutes les x-sorties plates en résolvant des équations aux dérivées partielles du 1 er ordre.

Chapitre 2. Ce chapitre est consacré à la modélisation, à la caractérisation, aux singularités et à la propriété de la platitude du système à n-barres dans l'espace R m+1 , qui généralise le système du robot mobile avec n remorques dans R 2 . Le système du type robot mobile avec contraintes de roulement sans glissement, comme un cas typique de système mécanique non holonome, est très important en automatique et a été étudié par de nombreux mathématiciens et automaticiens depuis les dernières vingtaine d'années (voir les livres [START_REF] Laumond | Robot Motion Planning and Control[END_REF] et [START_REF] Li | Nonholonomic Motion Plannging[END_REF]; les articles [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], [START_REF] Jean | The car with n trailers: Characterisation of the singular configurations[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], [START_REF] Laumond | Controllability of a multibody robot[END_REF], [START_REF] Murray | Nonholonomic motion planning: Steering using sinusoids[END_REF], [START_REF] Pasillas-Lépine | Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus[END_REF], [START_REF] Sørdalen | Conversion of the kinematics of a car with n trailers into a chained form[END_REF]). En 1991, Laumond [START_REF] Laumond | Controllability of a multibody robot[END_REF] a présenté un modèle cinématique du robot mobile avec remorques et a montré la controllabilité pour ce modèle. En 1994, Tilbury, Murray et Sastry [START_REF] Tilbury | Trajectroy generation for the n-trailer problem using Goursat normal form[END_REF] ont donné une caractérisation complète du système du robot mobile avec n remorques : autour des points réguliers, ce système peut être transformé en un système chaîné, autrement dit, il est localement équivalent au système de contact sur J n (R 1 , R 1 ). Les points réguliers sont caractérisés par la condition dim D (i) = dim D i , pour 0 ≤ i ≤ n, où D désigne la distribution associée à ce système. Respondek et Pasillas-Lépine [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF] ont montré qu'en dehors des points réguliers, le système du robot mobile avec n remorques peut être transformé à la forme normale de Kumpera et Ruiz. Une caractérisation complète du lieu singulier du système du robot mobile avec n-remorques a été donnée par Jean [START_REF] Jean | The car with n trailers: Characterisation of the singular configurations[END_REF].

Notre objectif dans ce chapitre est de généraliser le système du robot mobile avec remorques dans l'espace R m+1 (le système à n-barres), pour m ≥ 1, et de donner une caractérisation de ce système. Récemment, ce système, sa géométrie et, notamment, son lieu singulier ont été étudiés par Slayaman et Pelletier ( [START_REF] Slayaman | Bras articulé et distributions multi-drapeaux[END_REF], [START_REF] Slayaman | Articulated arm and special multi-flags (priprint)[END_REF]). Notre résultat est basé sur une nouvelle modélisation cinématique efficace pour ce système. Considérons le système à n-barres dans R m+1 tel que:

(1) La fin de la barre précédente coïncide avec la source de la barre suivante;

(2) La longueur de chaque barre est de 1;

(3) Chaque barre bouge sans glisser latéralement et la vitesse instantanée du point P i est parallèle au vecteur ----→ P i P i+1 , pour 0 ≤ i ≤ n -1.

x 1

x 2

x m+1 P 0 Supposons que les coordonnées des points P i , pour 0 ≤ i ≤ n, sont données par P i = (x 1 i , x 2 i , . . . , x m+1 i

). Alors l'espace de configuration de ce système peut être décrit complètement par les (n + 1)(m + 1) coordonnées (x 1 0 , . . . , x m+1 0 , x 1 1 , . . . , x m+1 1 , . . . , x 1 n , . . . , x m+1 n ) dans X = R (n+1)(m+1) . Les hypothèses | ----→ P i P i+1 | = 1, pour 0 ≤ i ≤ n -1, nous donnent les contraintes holonomes suivantes Ψ(x) = 0, où Ψ = {Ψ 1 , . . . , Ψ n } : X = R (n+1)(m+1) → R n est définie par

           Ψ 1 (x) = (x 1 1 -x 1 0 ) 2 + (x 2 1 -x 2 0 ) 2 + • • • + (x m+1 1 -x m+1 0 ) 2 -1 Ψ 2 (x) = (x 1 2 -x 1 1 ) 2 + (x 2 2 -x 2 1 ) 2 + • • • + (x m+1 2 -x m+1 1 ) 2 -1 . . . Ψ n (x) = (x 1 n -x 1 n-1 ) 2 + (x 2 n -x 2 n-1 ) 2 + • • • + (x m+1 n -x m+1 n-1 ) 2 -1.
(0.0.4) D'après ces n contraintes holonomes, le vrai espace de configuration du système à n-barres devient la sous-variété

Q = R m+1 × (S m ) n ⊂ X définie par Q = {x ∈ X : Ψ(x) = 0}.
En effet, le modèl Γ du système à n-barres Γ : q = m i=0 f i (q)u i , q ∈ R m+1 × (S m ) n , peut être défini par le système non holonome sans dérive Σ dans l'espace X = R (n+1)(m+1) ∆ : ẋ = n+m+1 i=1 g i (x)v i , x ∈ X, (0.0.5) où g 1 , . . . , g n+m+1 sont donnés par

g 1 = (x 1 1 -x 1 0 ) ∂ ∂x 1 0 + (x 2 1 -x 2 0 ) ∂ ∂x 2 0 + • • • + (x m+1 1 -x m+1 0 ) ∂ ∂x m+1 0 g 2 = (x 1 2 -x 1 1 ) ∂ ∂x 1 1 + (x 2 2 -x 2 1 ) ∂ ∂x 2 1 + • • • + (x m+1 2 -x m+1 1 ) ∂ ∂x m+1 1 . . . g n = (x 1 n -x 1 n-1 ) ∂ ∂x 1 n-1 + (x 2 n -x 2 n-1 ) ∂ ∂x 2 n-1 + • • • + (x m+1 n -x m+1 n-1 ) ∂ ∂x m+1 n-1 g n+i = ∂ ∂x i n , 1 ≤ i ≤ m + 1,
sous les contraintes holonomes Ψ(x) = (Ψ 1 (x), . . . , Ψ n (x)) = 0 définies par (0.0.4) (voir Section 2.2.2 pour plus de détails). Notons que nous n'avons pas donné l'expression explicite des champs de vecteurs f i , 0 ≤ i ≤ m, ni celle de la distribution associée D Γ . Toutes ses propriétés seront formulées et analysées en utilisant la distribution associée E = span {g 1 , . . . , g n+m+1 } du système ∆ et les contraintes holonomes Ψ(x) = 0. Observons aussi que notre modèle n'utilise pas de variables d'angle et ce caractère nous permet de pouvoir étudier les objets géométriques attachés au système à n-barres tels que la distribution, la distribution caractéristique, le système de Pfaff, etc. Quand m = 1, le système à n-barres retrouve une modélisation du célèbre système du robot mobile avec n -1 remorques.

Un système de contrôle Σ : ẋ = m i=0 u i f i (x), défini sur R (n+1)m+1 , est dit sous la forme m-chaînée s'il est representé par

ẋ0 0 = u 0 ẋ0 1 = x 1 1 u 0 • • • ẋ0 m = x 1 m u 0 • • • • • • • • • ẋn-1 1 = x n 1 u 0 • • • ẋn-1 m = x n m u 0 ẋn 1 = u 1 • • • ẋn m = u m .
Evidemment, la forme m-chaînée coïncide justement avec le système de contact canonique sur J n (R 1) est de rang constant nm + 1 et contient une sous-distribution involutive L dont le corang dans D (n-1) est constant et égal à 1, (iii) D(p) n'est pas contenue dans L(p). Parmi les trois conditions, la condition (iii) distingue les points singuliers et réguliers.

(i) D (n) = T M, (ii) D (n-
La caractérisation du système à n-barres dans R m+1 peut être exprimée par le théorème suivant :

Theorem 0.0.5 Le système à n-barres Γ dans R m+1 , pour m ≥ 1, est localement équivalent au système m-chaîné en chaque point x ∈ X = R (n+1)(m+1) qui satisfait Ψ(x) = 0 (c'est à dire x correspond à un point q ∈ Q) et (R1) m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, pour 1 ≤ i ≤ n -1, si m ≥ 2, (R2) m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, pour 2 ≤ i ≤ n -1, si m = 1.
Les conditions Ψ(x) = 0 impliquent que x est un point dans Q et les conditions (R1) et (R2) décrivent les points réguliers du système Γ dans Q. La condition m+1 j=1 (x j ix j i-1 )(x j i+1x j i ) = 0 est appelée la condition de régularité pour le système à n-barres. Notons θ i , pour 1 ≤ i ≤ n -1, l'angle orienté entre le vecteur ----→ P i-1 P i et ----→ P i P i+1 . Alors les conditions de régularité sont équivalentes à θ i = ± π 2 . Notre théorème montre que l'angle θ 1 = ± π 2 (i.e., la première barre est perpendiculaire à la deuxième bar) est une singularité pour le cas m ≥ 2 mais pas pour le cas m = 1. Il est intéressant de remarquer que la propriété de la contrôlabilité peut être obtenue comme un sous-résultat du Théorèm 0.0.5 :

le système à n-barres Γ est globalement contrôlable dans Q = R m+1 × (S m ) n .
La propriété de la platitude du système à n-barres dans R m+1 est aussi analysée dans ce chapitre. Dans le cas m = 1, la platitude du système du robot mobile avec remorques a été résolue par Fliess, Lévine, Martin et Rouchon [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF] (voir aussi [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF]). Dans le cas m ≥ 2, nous donnons le résultat suivant : Theorem 0.0.6 (Platitude du système à n-barre) Pour le système à n-barres Γ dans R m+1 , m ≥ 2, nous avons

(i) Γ est x-plat au point (q 0 , u 0 ) ∈ Q × R m+1 qui satisfait (1) Ψ(x) = 0 et m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, où q 0 ∈ Q est identifié avec un point x ∈ R (n+1)(m+1) satisfaisant Ψ(x) = 0; (2) u 0 est le contrôle tel que la vitesse Ṗ0 du point P 0 ne s'annule pas (et donc les vitesses Ṗi , 0 ≤ i ≤ n -1 sont non-nulles). (ii) P 0 = (x 1 0 , x 2 0 , . . . , x m+1

0

) est une x-sortie plate minimale de Γ en tout (q 0 , u 0 ) comme ci-dessus. (iii) Si h = (h 0 , . . . , h m ) est une x-sortie plate minimale en (q 0 , u 0 ), alors localement autour de q 0 on a span{dh 0 , . . . , dh m } = span{dx 1 0 , dx 2 0 , . . . , dx m+1 0 }.

Remarque. Il est important de remarquer la différence pour les x-sorties plates entre deux cas différents: dans le cas m ≥ 2, les coordonnées du point P 0 = (x 1 0 , x 2 0 , . . . , x m+1 0 ) forment la seule x-sortie plate minimale de Γ. Contrairement, au cas m = 1, la x-sortie plate minimale n'est pas unique. Chapitre 3. Le but de ce chapitre est de trouver des conditions nécessaires et suffisantes vérifiables pour qu'une distribution soit équivalente à la distribution de Cartan CC n (R 2 , R m ). En géométrie différentielle, la caractérisation intrinsèque de la distribution de Cartan CC n (R k , R m ) est encore un problème ouvert même si des solutions ont été obtenues dans certains cas particuliers. Le cas n = 1 et m = 1 a été résolu par Darboux [START_REF] Darboux | Sur le problème de Pfaff[END_REF] dans son théorème célèbre en généralisant les résulats de Pfaff [START_REF] Pfaff | Methodus generalis, aequationes differentiarum partialum, nec non aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles, completi integrandi[END_REF] et Frobenius [START_REF] Frobenius | Über das Pfaffsche problem[END_REF]; le cas n = 2, m = 1 et k = 1 par Engel [START_REF] Engel | Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen[END_REF]; le cas n > 2, m = 1 et k = 1 par E.von Weber, Cartan et Goursat (en point régulier), et par Libermann [START_REF] Libermann | Sur le problème d'équivalence des systèmes de Pfaff non complètement intégrables[END_REF], Kumpera et Ruiz [START_REF] Kumpera | Sur l'équivalence locale des systèmes de Pfaff en drapeau[END_REF], et Murray [START_REF] Murray | Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems[END_REF](en point quelconque); le cas n = 1 par Bryant [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF]; le cas général par Yamaguchi [START_REF] Yamaguchi | Contact geometry of higher order[END_REF]; le cas k = 1 par Gardner et Shadwick [START_REF] Gardner | The GS algorithm for exact linearization to Brunovsky normal form[END_REF], Murray [START_REF] Murray | Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems[END_REF], Tilbury et Sastry [START_REF] Tilbury | The multi-steering n-trailer system: A case study of Goursat normal forms and prolongations[END_REF], Aranda-Bricaire et Pomet [START_REF] Aranda-Bricaire | Some explicit conditions for a control system to be feedback equivalent to extended Goursat normal form[END_REF], Mormul [START_REF] Mormul | Multi-dimensional Cartan prolongation and special k-flags, in Geometric Singularity Theory[END_REF], Respondek et Pasillas-Lépine [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF].

Avant de présenter nos résultats, nous rappelons les concepts du rang d'Engel et du rang de Cartan. Le rang d'Engel, en point p, du système de Pfaff I = span {ω 1 , . . . , ω s } est l'entier le plus large ρ tel qu'il existe une 1-forme différentielle α dans I qui satisfait la condition suivante

((dα) ρ ∧ ω 1 ∧ • • • ∧ ω s )(p) = 0.
Le rang de Cartan de I est l'entier le plus petit k tel qu'il existe π 1 , . . . , π k ∈ Λ 1 (M) tels que

π 1 ∧ • • • ∧ π k ∧ ω 1 ∧ • • • ∧ ω m = 0, et dω ∧ π 1 ∧ • • • ∧ π k ≡ 0 mod I, ∀ ω ∈ I.
Theorem 0.0.7 Soit m ≥ 3 et D une distribution de rang m(n + 1) + 2 définie sur une variété M de dimension m 2 (n + 1)(n + 2) + 2. Alors D est localement, autour du point q ∈ M, équivalente à la distribution de Cartan CC n (R 2 , R m ) si et seulement si les conditions suivantes sont satisfaites : De plus, la sous-distribution involutive L de D de corang k, si elle existe, est unique et calculable.

(i) Chacune de ses distributions dérivées D (i) , pour 0 ≤ i ≤ n, est de rang constant m 2 (2n + 2 -i)(i + 1) + 2. (ii) Chacune de ses distributions dérivées D (i) , pour 0 ≤ i ≤ n -2, contient la distribution caractéristique C i+1 de D (i+1) dont le corang dans D (i) est de deux. De plus, la distribution caractéristique de D est réduite à {0}. (iii) Le rang d'Engel de (D (n-1) ) ⊥ est constant et égal à deux. (iv) dim (L + D (i) )(q) = dim L(q) + 2, pour 0 ≤ i ≤ n -2,
Afin de démontrer Theorème 0.0.7, nous proposons aussi une forme normale de Bryant étendue qui généralise la caractérisation de la distribution de Cartan CC 1 (R k , R m ). Chapitre 4. Cette dernière partie est consacrée à la linéarisation d'un système avec multi-contrôles par le bouclage orbital. La linéarisation par bouclage est un outil très important dans l'étude des systèmes non linéaires dans le fait qu'on peut ainsi appliquer les propriétés d'un système linéaire à un système non linéaire. Le problème de la linéarisation par bouclage d'un système avec un seul contrôle a été résolu par Brockett [START_REF] Brockett | Feedback invariants for nonlinear systems[END_REF]. Ensuite Jakubczyk et Respondek [START_REF] Jakubczyk | On linearization of control systems[END_REF] ainsi que Hunt et Su [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] ont donné des conditions nécessaires et suffisantes de la linéarisation par bouclage pour un système affine avec multi-contrôles. Pour les systèmes qui ne peuvent pas être linéarisés par bouclage, il est possible qu'ils soient linéarisés par bouclage avec un changement de temps qui s'appelle linéarisation par bouclage orbital. Ce problème a été proposé et étudié premièrement par Sampei et Furuta [START_REF] Sampei | On time scaling for nonlinear systems: applications to linearization[END_REF]. Malheureusement leurs conditions necessitent la resolution d'un système d'EDP non linéaire afin d'obtenir la transformation du changement de temps et donc elles ne peuvent pas être verifiées pour le système original. En 1998, Respondek [START_REF] Respondek | Orbital Feedback Linearization of Single-input Nonliear Control Systems[END_REF] a présenté des contitions nécessaires et suffisantes pour la linéarisation par bouclage orbital d'un système à un seul contrôle et ses conditions sont vérifiables pour le système original. Après son travail, Guay [START_REF] Guay | An algorithm for orbital feedback linearization of single-input control affine systems[END_REF] a donné en 2000 des conditions équivalentes avec une approche basée sur les formes différentielles et les a développées en 2001 à la linéarisation par bouclage orbital d'un système à multi-contrôles [START_REF] Guay | Orbital feedback linearization of multi-input control affine systems[END_REF]. Pour vérifier les conditions de Guay, on a besoin de chercher des générateurs convenables du système de Pfaff associés au système original.

Considérons le système suivant, pour m ≥ 2, Σ :

dx dt = f (x) + m i=1 g i (x)u i , f (x 0 ) = 0, (0.0.6) où x ∈ X = R (n+1)m+1 .
Dans cette thèse, le système Σ est dit linéarisable par bouclage orbital s'il est localement équivalent au système linéaire

Λ t : ẋ0 = 1 ẋ = Āx + m i=1 bi ūi .
Il est clair que le système Λ t n'est jamais controllable mais il peut être accessible (voir [START_REF] Isidori | Nonlinear Control Systems[END_REF], [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF] pour la définition) si et seulement si le sous-système ( Ā, B) est controllable, où B = (b 1 , . . . , b m ). Par la suite nous appelerons les indices de contrôlabilité et la forme canonique de Brunovsky de Λ t comme étant celles du sous-système ( Ā, B).

Définissons les distributions suivantes

G = span {g 1 , . . . , g m }, G j f = span {f, g i , ad f g i , . . . , ad j-1 f g i , 1 ≤ i ≤ m}, pour 1 ≤ j ≤ n + 1. Supposons que dim G n+1 f (x) = (n + 1)m + 1, alors nous avons dim G n f (x) = nm + 1. En conséquence, il existe m formes différentielles ω 1 , . . . , ω m ∈ Λ(X) définies uniquement par < ω j , g > = 0, ∀ g ∈ G n f < ω j , ad n f g i > = δ j i , pour 1 ≤ i, j ≤ m.
Grâce aux m formes differentielles, les fonctions suivantes peuvent être introduites :

T k,l i,j = < ω k , [ad n-1 f g i , ad l f g j ] > = ω k ([ad n-1 f g i , ad l f g j ]).
Le théorème suivant décrit des conditions nécessaires et suffisantes du problème de la linéarisation par bouclage orbital pour le système (0.0.6). Toutes ses conditions peuvent être vérifiées directement sur le système original.

Theorem 0.0.9 Le système non linéaire Σ, m ≥ 2, défini par (0.0.6), est localement équivalent par bouclage orbital au système linéaire Λ t dont tous les indices de contrôlabilité égalent n+1 si et seulement s'il satisfait, autour du point p, les conditions suivantes :

(OPL1) dim G n+1 f (x) = (n + 1)m + 1 (OPL2) [G j f , G j f ] ⊂ G j+1 f , 1 ≤ j ≤ n, (OPL3) [G, G 2 f ] ⊂ G 2 f , (OPL4) Les fonctions T k,l
i,j satisfont les conditions suivantes:

(i) T k,l i,j = 0, pour 1 ≤ k = i ≤ m, 1 ≤ j ≤ m, si l < n -1 1 ≤ i = k = j ≤ m si l = n -1 (ii) T i,l i,j = T k,l k,j , pour 1 ≤ l ≤ n -1, 1 ≤ i, j, k ≤ m t.q j = i, k si l = n -1, (iii) Quand m = 2, la distribution définie par B = span {g i , ad l f g i + b l i f, i = 1, 2 et 1 ≤ l ≤ n -1} doit être involutive, où b l i = T 1,l 1,i = T 2,l 2,i , pour 1 ≤ i ≤ 2, 1 ≤ l ≤ n -2 b n-1 1 = T 1,n-l 1,2 , b n-1 2 = T 2,n-l 2,1 .
Il est important de remarquer que la fonction γ qui décrit le changement de l'échelle de temps, par dt dτ = γ(x(t)), peut être construite directement à partir de la distribution B que l'on a définie dans ce théorème (voir les détails dans Chapitre 4).

Notre résultat implique aussi une relation intéressante entre la linéarisation par bouclage orbital du système (0.0.6) et la caractérisation de la distribution de Cartan CC n (R 1 , R m ). Il est connu que les systèmes de type Λ t , dont tous les indices de contrôlabilité sont constants et égaux à n + 1, peuvent être transformés par un changement de coordonnées et un bouclage à la t-etendue forme normale de Brunovsky (t-augmented Brunovsky canonical form)

Λ Br t : ẋ0 0 = 1 ẋ0 1 = x 1 1 • • • ẋ0 m = x 1 m . . . . . . . . . ẋn-1 1 = x n 1 • • • ẋn-1 m = x n m ẋn 1 = u 1 • • • ẋn m = u m .
A chaque système de contrôle défini par (0.0.6), il correspond à une distribution affine

A = f + G qui définit en chaque point x ∈ X un sous-espace affine A(x) = f (x) + G(x) de l'espace tangent T x X.
A = f + G ⊂ D telle que corang (G ⊂ D) = 1 et f (x) ∈ G(x). Si D satisfait les conditions suivantes (D1) D est localement équivalente à la distribution de Cartan CC n (R, R m ); (D2) C(D (1) ) = G,
(C1) D Σ est localement équivalente à la distribution de Cartan CC n (R, R m ); (C2) C(D (1) Σ ) = G.
Grâce aux deux Propositions, nous présentons une version complète pour Théorème 0.0.9 dans laquelle nous donnons des conditions équivalentes aux conditions (OPL1) -(OPL4) (voir Chapitre 4 pour les détails).

Chapter 1

Flat outputs of driftless two-input control systems

Introduction

The notion of flatness has been introduced by Fliess, Lévine, Martin and Rouchon in [START_REF] Fliess | Sur les systèmes non linéaires différentiellement plats[END_REF], [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF] in order to describe the class of control systems, whose set of trajectories can be parameterized by a finite number of functions and their time-derivatives. More formally, a system with m controls is flat if we can find m functions (of the state and control variables and their time-derivatives), called flat outputs, such that the evolution in time of the state and control can be expressed in terms of flat outputs and their time derivatives (see Section 1.2 for a precise definition and references).

As an introductory example, consider the nonholonomic car or, equivalently, a unicycle-like robot towing a trailer (see, e.g., [START_REF] Laumond | Controllability of a multibody robot[END_REF]). Denote by (x, y) ∈ R 2 the position of the mid-point of the rear wheels, and by θ 0 and θ 1 , respectively, the angles between the rear and front wheels and the x-axis. The controls u 1 and u 2 allow to move (forward and backward) the car and to turn. The car is subject to two nonholonomic constraints: the wheels are not allowed to slide. This leads to the following model given by a driftless, i.e., control-linear, system on R 2 × S 1 × S 1 :

Σ car :      ẋ ẏ θ0 θ1      =      cos(θ 1 -θ 0 ) cos θ 0 cos(θ 1 -θ 0 ) sin θ 0 sin(θ 1 -θ 0 ) 0      u 1 +      0 0 0 1      u 2 .
It is well known, as proved by Fliess, Lévine, Martin and Rouchon [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF] (see also [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF]) that the nonholonomic car is flat and that the position of the mid-point of the rear x y

θ 1 θ 0 Figure 1.1: nonholonomic car
wheels is a flat output. Indeed, the following coordinates change

x 1 = x x 2 = y x 3 = tan θ 0 x 4 = tan(θ 1 -θ 0 ) sec 3 θ 0 for θ 0 ∈ -π 2 , π 2 and θ 1 -θ 0 ∈ -π 2 , π 2
, followed by the invertible feedback

v 1 = u 1 cos(θ 1 -θ 0 ) cos θ 0 v 2 = u 1 sec 3 θ 0 sin(θ 1 -θ 0 ) (3 tan(θ 1 -θ 0 ) tan θ 0 -sec 2 (θ 1 -θ 0 )) + u 2 sec 3 θ 0 sec 2 (θ 1 -θ 0 ),
brings the system Σ car into the chained form:

         ẋ1 = v 1 ẋ2 = x 3 v 1 ẋ3 = x 4 v 1 ẋ4 = v 2 .
It is easy to see that the pair of functions h = (h 1 , h 2 ) = (x 1 , x 2 ) = (x, y) are flat outputs for the chained form. Indeed, we have

x 1 = h 1 , x 2 = h 2 , x 3 = ḣ2 ḣ1 , x 4 = 1 ḣ1 d dt ( ḣ2 ḣ1 ), v 1 = ḣ1 and v 2 = 1 ḣ1 ( ẋ4
). The applied transformation, consisting of a change of coordinates and feedback, is invertible which proves that, indeed, h = (x, y) are flat outputs of the nonholonomic car.

The presented procedure, to express the state and control in terms of h and its time-derivatives, exhibits three singularities: two in the state space (at θ 0 = ± π 2 and

θ 1 -θ 0 = ± π 2 )
and one in the control space (when u 1 (t) = v 1 (t) = 0). Let us analyze those singularities.

First, the singularity at u 1 (t) = 0 (at least at u 1 (t) ≡ 0) seems to be intrinsic: we cannot see how the angles θ 0 (t) and θ 1 (t) evolve if the observed point (x(t), y(t)) does not move.

Secondly, θ 0 = ± π 2 is not a singularity of the flat output h = (x, y). Indeed, around a point such that θ 0 = ± π 2 , we can choose the following coordinates change:

x 1 = y x 2 = x x 3 = cot θ 0 x 4 = -tan(θ 1 -θ 0 ) csc 3 θ 0 , for θ 0 ∈ (0, π) including θ 0 = -π 2 (or θ 0 ∈ (-π, 0), including θ 0 = -π 2 ) and θ 1 -θ 0 ∈ (-π 2 , π 2
), followed by a suitable feedback transformation, which brings the system Σ car into the chained form around θ 0 = ± π 2 . Therefore, the nonholonomic car is flat, with (x, y) as a flat output, at any q = (x, y, θ 0 , θ 1 ) where θ 1θ 0 = ± π 2 . Actually, the singularities at θ 0 = ± π 2 are related with the domain of inversion: when calculating θ 0 in terms of x, y, ẋ = u 1 cos(θ 1θ 0 ) cos θ 0 and ẏ = u 1 cos(θ 1θ 0 ) sin θ 0 , we have to invert either tan θ 0 = ẏ ẋ or cot θ 0 = ẋ ẏ .

Thirdly, and most interestingly, the nature of the singularity at θ 1θ 0 = ± π 2 is completely different. It is an intrinsic singularity of the flat output h = (x, y), see Section 1.3.1 for details, nevertheless the nonholonomic car is flat at θ 1θ 0 = ± π 2 if we choose another flat output! To see this, define two functions h1 (x, y, θ 0 ) = x sin θ 0y cos θ 0 h2 (x, y, θ 0 ) = x cos θ 0 + y sin θ 0 and consider the change of coordinates

y 1 = θ 0 y 2 = h1 (x, y, θ 0 ) y 3 = h2 (x, y, θ 0 ) y 4 = cot(θ 1 -θ 0 ) -h1 (x, y, θ 0 )
followed by the invertible feedback

v 1 = u 1 sin(θ 1 -θ 0 ) v 2 = u 1 (csc 2 (θ 1 -θ 0 ) -h2 (x, y, θ 0 )) -u 2 csc 2 (θ 1 -θ 0 ), which also brings Σ car into chained form, but this time around θ 1 -θ 0 = ± π 2 ,          ẏ1 = v 1 ẏ2 = y 3 v 1 ẏ3 = y 4 v 1 ẏ4 = v 2 .
Thus ( h1 , h2 ) = (θ 0 , x sin θ 0y cos θ 0 ) is another flat output of the nonholonomic car, valid around θ 1θ 0 = ± π 2 , but singular at θ 1θ 0 = 0, ±π (notice that the same singularity v 1 (t) = u 1 (t) = 0, as previously, occurs in the control space).

A series of natural questions arises: are there other flat outputs of the nonholonomic car and, if so, how many and how to describe them? More generally, how to characterize all flat outputs of any 2-input driftless control system and how to describe their singular loci and singular controls? The aim of this chapter is to give complete answers to those questions. This chapter is organized as follows. In Section 1.2, we define the crucial notion of flatness and recall a description of flat driftless 2-input systems. In section 1.3, we give our main results. We characterize all flat outputs of driftless 2-input systems and give a way of parameterizing them: it turns out that all flat outputs can be parameterized by an arbitrary function of intrinsically defined three variables. We also construct a system of 1 st order PDE's whose solutions are flat outputs of a given system. Still in Section 1.3, we illustrate our results by describing all flat outputs of some examples: nonholonomic car (1-trailer system) which we have just discussed and then the n-trailer system. We prove our results in Section 1.4.

Flatness of driftless two-input control systems

Throughout this chapter, the word smooth will always mean C ∞ -smooth. Consider a smooth nonlinear control system Ξ : ẋ = f (x, u), where x ∈ X, an n-dimensional manifold and u ∈ U, an m-dimensional manifold. Given any integer l, we associate to Ξ its l-prolongation Ξ l given by

Ξ l : ẋ = f (x, u 0 ) u0 = u 1 . . . ul = u l+1 ,
which can be considered as a control system on X l = X × U × R ml , whose state variables are (x, u 0 , u 1 , . . . , u l ) and whose m controls are the m components of u l+1 . Denote ūl = (u 0 , u 1 , . . . , u l ).

Definition 1.2.1 The system Ξ is called flat at a point (x 0 , ūl 0 ) ∈ X l = X × U × R ml , for some l ≥ 0, if there exist a neighborhood O l of (x 0 , ūl 0 ) and m smooth functions

h i = h i (x, u 0 , u 1 , . . . , u l ), 1 ≤ i ≤ m,
called flat outputs, defined in O l , having the following property: there exist an integer s and smooth functions γ i , 1 ≤ i ≤ n, and δ i , 1 ≤ i ≤ m, such that we have

x i = γ i (h, ḣ, . . . , h (s) ) u i = δ i (h, ḣ, . . . , h (s) ),
where h = (h 1 , . . . , h m ) ⊤ , along any trajectory x(t) given by a control u(t) that satisfy (x(t), u(t), u(t), . . . , u (l) (t)) ∈ O l .

The compositions γ i (h, ḣ, . . . , h (s) ) and δ i (h, ḣ, . . . , h (s) ) are, a priori, defined in an open set s+l) . The above definition requires that π(O s+l ) ⊃ O l , where π(x, ūs+l ) = (x, ūl ), and that for all such (x, ūs+l ), the compositions yield, respectively, x i and u i .

O s+l ⊂ X s+l = X × U × R m(
If h i = h i (x, u 0 , u 1 , . . . , u r ), r ≤ l, we will say that the system is (x, u, . . . , u r )-flat and, in particular, x-flat if h i = h i (x). In the case h i = h i (x, u 0 , u 1 , . . . , u r ), we will assume that they are defined on O r ⊂ X r = X × U × R mr , where π -1 (O r ) ⊃ O l and π stands for the projection π(x, u 0 , . . . , u r , . . . , u l ) = (x, u 0 , . . . , u r ).

The notion of flatness has been introduced in control theory by Fliess, Lévine, Martin and Rouchon [START_REF] Fliess | Sur les systèmes non linéaires différentiellement plats[END_REF], [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF] (see also [START_REF] Aranda-Bricaire | Infinitesimal Brunovsky form for nonlinear systems with applications to Dynamic Linearization[END_REF], [START_REF] Isidori | A sufficient condition for full linearization via dynamic state feedback[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF], [START_REF] Pereira Da Silva | Relative Flatness and Flatness of Implicit Systems[END_REF], [START_REF] Van Nieuwstadt | Differential Flatness and Absolute Equivalence of Nonlinear Control Systems[END_REF]), and has attracted a lot of attention because of its extensive applications in constructive controllability and trajectory tracking, compare [START_REF] Martin | Flat systems[END_REF] and references therein. A similar notion (of underdetermined systems of differential equations that are integrable without integration) has already been studied by Cartan [START_REF] Cartan | Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes[END_REF] and Hilbert [START_REF] Hilbert | Über den Begriff der Klasse von Differentialgleichungen[END_REF].

In this chapter, we deal only with two-input driftless (equivalently, control-linear) systems of the form

Σ : ẋ = f 1 (x)u 1 + f 2 (x)u 2 ,
on an (n + 2)-dimensional manifold M, where f 1 and f 2 are C ∞ -smooth vector fields independent everywhere on M and u = (u 1 , u 2 ) ⊤ ∈ R 2 . To this system, we associate the distribution D spanned by the vector fields f 1 , f 2 , which will be denoted by D = span {f 1 , f 2 }. Consider another 2-input driftless system

Σ : ẋ = f1 (x)ũ 1 + f2 (x)ũ 2 ,
where f1 and f2 are C ∞ -smooth vector fields on M . Form the matrices f (x) = (f 1 (x), f 2 (x)) and f(x) = ( f1 (x), f2 (x)). The systems Σ and Σ are feedback equivalent if there exist an invertible 2 × 2-matrix β, whose entries β ij , 1 ≤ i, j ≤ 2, are C ∞ -smooth functions on M, and a diffeomorphism ϕ :

M → M such that Dϕ(x) • f (x) • β(x) = f (ϕ(x)).
It is easily seen that Σ and Σ are locally feedback equivalent if and only if the associated distributions D = span {f 1 , f 2 } and D = span { f1 , f2 } are locally equivalent via ϕ, i.e.,

Dϕ • D(x) = D(ϕ(x)).
The derived flag of a distribution D is the sequence of modules of vector fields D (0) ⊂ D (1) ⊂ • • • defined inductively by

D (0) = D and D (i+1) = D (i) + [D (i) , D (i) ], for i ≥ 0.
The Lie flag of D is the sequence of modules of vector fields

D 0 ⊂ D 1 ⊂ • • • defined inductively by D 0 = D and D i+1 = D i + [D 0 , D i ], for i ≥ 0.
In general, the derived and Lie flags are different though for any point x, the inclusion D i (x) ⊂ D (i) (x) holds, for i ≥ 0.

A characteristic vector field of a distribution D is a vector field f that belongs to D and satisfies [f, D] ⊂ D. The characteristic distribution of D, which will be denoted by C, is the subdistribution spanned by all its characteristic vector fields. It follows directly from the Jacobi identity that the characteristic distribution is always involutive but, in general, it need not be of constant rank.

The problem of flatness of driftless 2-input systems has been studied and solved by Martin and Rouchon [START_REF] Martin | Feedback linearization and driftless systems[END_REF] (see also [START_REF] Martin | Feedback linearization and driftless systems[END_REF] and a related work of Cartan [START_REF] Cartan | Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes[END_REF]). Their important result proves that a system is flat if and only if its associated distribution D satisfies, on an open and dense subset M ′ of M, the conditions rank

D (i) = i + 2, 0 ≤ i ≤ n.
(1.2.1)

A distribution D is called a Goursat structure (also a "système en drapeau" in [START_REF] Kumpera | Sur l'équivalence locale des systèmes de Pfaff en drapeau[END_REF] and a Goursat flag in [START_REF] Mormul | Goursat flags: classification of codimension-one singularities[END_REF]) if it satisfies the conditions (1.2.1) at any point x ∈ M. It is known since the work of von Weber [START_REF] Weber | Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen[END_REF], Cartan [START_REF] Cartan | Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes[END_REF] and Goursat [START_REF] Goursat | Leçons sur le problème de Pfaff[END_REF] that the conditions (1.2.1) imply that on an open and dense subset M ′′ of M, the distribution D can be brought to the Goursat normal form, or equivalently, the corresponding control system is feedback equivalent to the chained form:

Σ chain :                  ż1 = v 1 ż2 = z 3 v 1 ż3 = z 4 v 1 . . . żn+1 = z n+2 v 1 żn+2 = v 2 . (1.2.2)
It is easy to see that Σ chain is x-flat with x-flat outputs chosen as h = (h 1 , h 2 ) = (z 1 , z 2 ) and provided that the control v 1 = 0 (compare Introduction, where we brought the nonholonomic car to the chained form for dim M = 4). Giaro, Kumpera and Ruiz [START_REF] Giaro | Sur la lecture correcte d'un resultat d' Élie Cartan[END_REF] were the first to observe the existence of singular points in the problem of transforming a distribution of rank two into the Goursat normal form. Murray [START_REF] Murray | Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems[END_REF] proved that the feedback equivalence of Σ to the chained form Σ chain (or, in other words, equivalence of the associated distribution to the Goursat normal form), around an arbitrary point x 0 requires, in addition to (1.2.1), the regularity condition (see Theorem 1.2.2 below) dim Let D be any distribution of rank two such that rank D (1) = 3 and rank D (2) = 4. Then there exists a distribution C 1 ⊂ D of corank one which is characteristic for D (1) , i.e., [C 1 , D (1) ] ⊂ D (1) . Indeed, the above rank assumptions imply that (after permuting f 1 and f 2 , if necessary) there exists a smooth function α such that (1) .

D (i) (x 0 ) = dim D i (x 0 ), 0 ≤ i ≤ n. ( 1
[f 2 , [f 1 , f 2 ]] = α[f 1 , [f 1 , f 2 ]] mod D
It follows that [f 2 -αf 1 , [f 1 , f 2 ]] = 0 mod D (1) and hence C 1 = span {f 2 -αf 1 }. Let U sing (x) be the 1-dimensional subspace of R 2 such that for any feedback control (u 1 (x), u 2 (x)) ⊤ = u(x) ∈ U sing (x), we have f 1 (x)u 1 (x) + f 2 (x)u 2 (x) ∈ C 1 (x) (clearly, U sing (x)
is spanned by (α(x), -1) ⊤ ). Any control u(t) ∈ U sing (x(t)) will be called singular and the trajectories of the system governed by a singular control remain tangent to the characteristic subdistribution C 1 . We have just given the definition of U sing (x) for dim M ≥ 4 (since we have used rank D (2) = 4). If dim M = 3, we define U sing (x) = 0 ∈ R 2 . Note that if l = 0, we will denote a fixed control value by u 0 (instead of more complicated u 0 0 ).

Theorem 1.2.2 Consider a 2-input driftless control system

Σ : ẋ = f 1 (x)u 1 + f 2 (x)u 2 , where x ∈ M, an (n + 2)-dimensional manifold, n ≥ 1. Assume that the distri- bution D = span {f 1 , f 2 } associated to Σ is a Goursat structure, that is, satisfies rank D (i) = i + 2, for 0 ≤ i ≤ n
, everywhere on M. Then the following conditions are equivalent:

(i) Σ is x-flat at (x 0 , ūl 0 ) ∈ M × R 2(l+1) , for a certain l ≥ 0; (ii) Σ is x-flat at (x 0 , u 0 ) ∈ M × R 2 ; (iii) dim D (i) (x 0 ) = dim D i (x 0 ) for 0 ≤ i ≤ n and u 0 ∈ U sing (x 0 );
(iv) Σ is locally, around x 0 , feedback equivalent to the chained form Σ chain and u 0 ∈ U sing (x 0 ).

We assume that D satisfies rank D (i) = i + 2, for 0 ≤ i ≤ n, so the characteristic distribution C 1 and the set of singular controls U sing are well defined. The above theorem implies that a driftless 2-input system is never flat at (x 0 , u 0 ) such that u 0 ∈ U sing (x 0 ). Therefore any x-flat outputs (ϕ 1 , ϕ 2 ) become singular in the control space (at u 0 ∈ U sing ) but they may also exhibit singularities in the state space M. To formalize this, assume that a pair of functions (ϕ 1 , ϕ 2 ) defined in an open set M ⊂ M are x-flat outputs at a point (x 0 , u 0 ) ∈ M × R 2 , that is, there exists a neighborhood

O 0 ⊂ M × R 2 , satisfying O 0 ⊂ π -1 (M)
, where π(x, u) = x, in which the conditions of Definition 1.2.1 hold. If a pair of functions (ϕ 1 , ϕ 2 ), defined in M ⊂ M, is an x-flat output at (x, u) for any x ∈ M and certain u = u(x), where M is open and dense in M, then by the singular locus of (ϕ 1 , ϕ 2 ), denoted by Sing(ϕ 1 , ϕ 2 ), we will mean the set of points x ∈ M such that (ϕ 1 , ϕ 2 ) are not x-flat outputs at (x, u) for any u ∈ R 2 .

The interest of the above theorem is two-fold. First, together with its proof, it will allow us to characterize all x-flat outputs of driftless 2-input systems (see Section 1.3). Secondly, it shows that a Goursat structure is x-flat at points x 0 satisfying dim D (i) (x 0 ) = dim D i (x 0 ), for 0 ≤ i ≤ n, only, that is, at regular points of D. Martin and Rouchon asked in [START_REF] Martin | Feedback linearization and driftless systems[END_REF] (see also [START_REF] Martin | Feedback linearization and driftless systems[END_REF]) whether a Goursat structure D is flat (dynamically linearizable) at points that do not satisfy dim D (i) (x 0 ) = dim D i (x 0 ). So our result gives a negative answer to their question (for x-flatness). Any Goursat structure can be brought to a generalization of the Goursat normal form, called Kumpera-Ruiz normal form (see [START_REF] Kumpera | Sur l'équivalence locale des systèmes de Pfaff en drapeau[END_REF], [START_REF] Mormul | Goursat flags: classification of codimension-one singularities[END_REF], [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF]). It follows that none of Kumpera-Ruiz normal forms is x-flat (except for the regular Kumpera-Ruiz normal form, that is, Goursat normal form). In particular, the system

             ẋ1 = x 5 u 1 ẋ2 = x 3 x 5 u 1 ẋ3 = x 4 x 5 u 1 ẋ4 = u 1 ẋ5 = u 2
which is historically the first discovered Kumpera-Ruiz normal form [START_REF] Giaro | Sur la lecture correcte d'un resultat d' Élie Cartan[END_REF], is not x-flat at any point of its singular locus {x ∈ R 5 : x = 0}. This answers negatively another question of [START_REF] Martin | Feedback linearization and driftless systems[END_REF].

It is known (see [START_REF] Jean | The car with n trailers: Characterisation of the singular configurations[END_REF], [START_REF] Laumond | Controllability of a multibody robot[END_REF], [START_REF] Mormul | Goursat flags: classification of codimension-one singularities[END_REF], [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF]) that the model of n-trailer system is a Goursat structure at any configuration point but is equivalent to the chained form out of the singular locus only, that is, if all angles θ i+1θ i between two consecutive wheels are not ± π 2 (except for θ 1θ 0 which is the most far from the top of the train). Therefore our theorem implies that at any singular configuration θ i+1θ i = ± π 2 , i ≥ 1, the n-trailer system is not x-flat.

The proof of Theorem 1.2.2 is given in Section 1.4.2 and is based on normal forms at singular points (introduced in [START_REF] Kumpera | Sur l'équivalence locale des systèmes de Pfaff en drapeau[END_REF], [START_REF] Mormul | Goursat flags: classification of codimension-one singularities[END_REF], and [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF] and called in the latter the Kumpera-Ruiz normal forms) and on the following result which is of independent interest (as is also proved in Section 1.4.2). It turns out that flat outputs and the chained form Σ chain are compatible: in fact, for any given pair of flat outputs (ϕ 1 , ϕ 2 ) of a system feedback equivalent to Σ chain , we can bring the system to the chained form Σ chain for which ϕ 1 and ϕ 2 serve as the two top variables. Proposition 1.2.3 Consider a driftless 2-input smooth control system Σ, defined on a manifold M of dimension n+ 2, whose associated distribution D satisfies rank D (i) = rank D i = i + 2, for 0 ≤ i ≤ n. Given any pair (ϕ 1 , ϕ 2 ) of flat outputs at (x 0 , u 0 ) ∈ M × R 2 , there exists a feedback transformation (Ψ, β) around x 0 bringing the system Σ into the chained form Σ chain , given by (1.2.2), such that ϕ 1 = z 1 and ϕ 2 = z 2 .

Characterization of flat outputs 1.3.1 Main Theorems

Recall a useful result due to Cartan [START_REF] Cartan | Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes[END_REF] whose proof can be found in, e.g., [START_REF] Martin | Feedback linearization and driftless systems[END_REF] and [START_REF] Pasillas-Lépine | Contact systems and corank one involutive subdistributions[END_REF]. i+1) and has constant corank one in D (i) . Theorem 1.2.2 implies that the only Goursat structures that are x-flat are those equivalent to the chained form (equivalently, whose associated distribution D is equivalent to the Goursat normal form). For this reason, we will consider in two theorems below such distributions only. Moreover, any distribution equivalent to the Goursat normal form obviously satisfies the assumptions of Lemma 1.3.1 and defines the involutive distribution C n-1 that is characteristic distribution for D (n-1) and of corank one in D (n-2) . Theorem 1.3.2 (Characterization of flat outputs, first version) Consider a driftless 2-input smooth control system Σ defined on a manifold M of dimension n + 2 whose associated distribution D satisfies rank D (i) = rank D i = i + 2, for 0 ≤ i ≤ n. Fix x 0 ∈ M and let g be an arbitrary vector field in D such that g(x 0 ) / ∈ C n-1 (x 0 ) and ϕ 1 , ϕ 2 be two smooth functions defined in a neighborhood M of x 0 . Then (ϕ 1 , ϕ 2 ) is an x-flat output of Σ at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ), if and only if the following conditions hold:

Lemma 1.3.1 (E. Cartan) Consider a rank two distribution D defined on a manifold M of dimension n + 2, for n ≥ 2. If D satisfies rank D (i) = i + 2, for 0 ≤ i ≤ n, everywhere on M, then each distribution D (i) , for 0 ≤ i ≤ n -2, contains a unique involutive subdistribution C i+1 that is characteristic for D (
(i) dϕ 1 (x 0 ) ∧ dϕ 2 (x 0 ) = 0, i.e., dϕ 1 and dϕ 2 are independent at x 0 ;

(ii) L c ϕ 1 ≡ L c ϕ 2 ≡ L c ( L g ϕ 2 L g ϕ 1 ) ≡ 0, for any c ∈ C n-1
, where the functions ϕ 1 and ϕ 2 are ordered such that L g ϕ 1 (x 0 ) = 0 which is always possible due to item (iii) below ; (iii) (L g ϕ 1 (x 0 ), L g ϕ 2 (x 0 )) = (0, 0); Moreover, if a pair of functions (ϕ 1 , ϕ 2 ) satisfies (i) everywhere in M and forms an x-flat output at (x, u) for any x ∈ M and certain u = u(x), where M is open and dense in M, then

Sing(ϕ 1 , ϕ 2 ) = {x ∈ M : (L g ϕ 1 (x), L g ϕ 2 (x)) = (0, 0)}.
(i) = rank D i = i + 2, for 0 ≤ i ≤ n.
Fix x 0 ∈ M and let ϕ 1 , ϕ 2 be two smooth functions defined in a neighborhood M of x 0 . Then (ϕ 1 , ϕ 2 ) is an x-flat output of Σ at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ), if and only if the following conditions hold:

(i) ′ dϕ 1 (x 0 ) ∧ dϕ 2 (x 0 ) = 0, i.e., dϕ 1 and dϕ 2 are independent at x 0 ; (ii) ′ L = (span {dϕ 1 , dϕ 2 }) ⊥ ⊂ D n-1 in M; (iii) ′ D(x 0 ) is not contained in L(x 0 ).
Moreover, if a pair of functions (ϕ 1 , ϕ 2 ) satisfies (i) ′ everywhere in M and forms an x-flat output at (x, u) for any x ∈ M and certain u = u(x), where M is open and dense in M, then

Sing(ϕ 1 , ϕ 2 ) = {x ∈ M : D(x) ⊂ L(x)}.
Remark 1. Notice that Theorem 1.3.3 is valid for any n ≥ 1 (i.e., dim M ≥ 3; if dim M = 3, then (iii) ′ is satisfied automatically) while Theorem 1.3.2 is true for n ≥ 2 only (i.e., dim M ≥ 4). In fact, in Theorem 1.3.2 we use the characteristic distribution C n-1 of D (n-1) but if dim M = 3, such a distribution does not exist and therefore Theorem 1.3.2 can not applied in that case. Remark 2. The two items (iii) and (iii) ′ describing the singular locus of an x-flat output (ϕ 1 , ϕ 2 ) are equivalent (which will be shown in the proofs of the two theorems) under the condition rank

D (i) = rank D i = i + 2, for 0 ≤ i ≤ n, i.e., {x ∈ M : (L g ϕ 1 (x), L g ϕ 2 (x)) = (0, 0)} = {x ∈ M : D(x) ⊂ L(x)}.
Remark 3. The conditions of both theorems are verifiable, i.e., given a pair of functions (ϕ 1 , ϕ 2 ) in a neighborhood of a point x 0 , we can easily verify whether (ϕ 1 , ϕ 2 ) forms an x-flat output of a control system under considerations and verification involves derivations and algebraic operations only (without solving PDE's or bringing the system to a normal form). Moreover, the theorems allow us to find the singular locus of a given flat output (ϕ 1 , ϕ 2 ).

A natural question to ask is if there is a lot of pairs (ϕ 1 , ϕ 2 ) which satisfy the conditions of Theorem 1.3.2 or 1.3.3? In other words, is there a lot of pairs (ϕ 1 , ϕ 2 ) which are x-flat outputs for a 2-input driftless control system? This question has an elegant answer given by the following theorem. Recall that C n-1 denotes the characteristic distribution of D (n-1) . Theorem 1.3.4 (Uniqueness of x-flat outputs) Consider a driftless 2-input smooth control system Σ whose associated distribution D satisfies rank D (i) = rank D i = i + 2, for 0 ≤ i ≤ n, locally around a point x 0 ∈ M, an (n + 2) dimensional manifold. Let g be an arbitrary vector field in D such that g(x 0 ) / ∈ C n-1 (x 0 ). Then for a given arbitrary smooth function ϕ 1 such that L c ϕ 1 = 0, for any c ∈ C n-1 , and L g ϕ 1 (x 0 ) = 0, there always exists a function ϕ 2 such that (ϕ 1 , ϕ 2 ) is an x-flat output of Σ at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ). Moreover, if for a given function ϕ 1 as above, the pairs (ϕ 1 , ϕ 2 ) and (ϕ 1 , φ2 ) are both x-flat outputs of Σ at (x 0 , u 0 ), then

span {dϕ 1 , dϕ 2 }(x) = span {dϕ 1 , d φ2 }(x),
for any x in a neighborhood of x 0 .

Remark. Observe that x-flat outputs (h 1 , . . . , h m ) and ( h1 , . . . , hm ) of a system with m controls such that span {dh 1 , . . . , dh m } = span {d h1 , . . . , d hm } can be considered as statically equivalent. Indeed, in that case there exist smooth functions H i and Hi of m variables such that h i = H i ( h1 , . . . , hm ) and hi = Hi (h 1 , . . . , h m ). It thus follows from Theorem 1.3.4 that for a given arbitrary ϕ 1 (satisfying the assumptions of the theorem), the choice of ϕ 2 is unique in the sense that all functions ϕ 2 giving x-flat outputs (ϕ 1 , ϕ 2 ) yield, actually, statically equivalent x-flat outputs.

Finding x-flat outputs

The importance of Theorem 1.3.2 is that it not only allows to check whether a given pair of functions forms an x-flat output but also, together with Theorem 1.3.4, to express explicitly a system of 1 st order PDE's to be solved in order to calculate all x-flat outputs for a given 2-input driftless system.

To this aim, choose n -1 vector fields c 1 , . . . , c n-1 spanning the characteristic distribution C n-1 of D (n-1) . Recall that C n-1 can be easily calculated as (see, e.g., [START_REF] Bryant | Exterior Differential Systems[END_REF])

C n-1 = {f ∈ D (n-1) : f dω ∈ (D (n-1) ) ⊥ },
where ω is any non-zero differential 1-form annihilating D (n-1) . Fix any vector field g of D, independent at x 0 with the characteristic distribution C n-1 of D (1) . According to Theorem 1.3.2 and Theorem 1.3.4, in order to find ϕ 1 , we have to solve the following system of 1 st order PDE's

L c i ϕ 1 = 0, 1 ≤ i ≤ n -1, L g ϕ 1 (x 0 ) = 0.
The above system possesses solutions (since C n-1 is involutive) and the space of solutions is that of functions of three variables (since corank C n-1 ⊂ T M) = 3). Now we will establish a system of equations for ϕ 2 . According to Theorem 1.3.2, it is given by

L c i ϕ 2 = 0, 1 ≤ i ≤ n -1 L c i L g ϕ 2 L g ϕ 1 = 0, 1 ≤ i ≤ n -1.
The last n -1 equations are equivalent to

(L g ϕ 1 )L c i L g ϕ 2 -(L g ϕ 2 )L c i L g ϕ 1 = 0, 1 ≤ i ≤ n -1.
Applying

L [c i ,g] ψ = L c i L g ψ -L g L c i ψ and taking into account that L c i ϕ 1 = L c i ϕ 2 = 0, we get (L g ϕ 1 )L [c i ,g] ϕ 2 -(L [c i ,g] ϕ 1 )L g ϕ 2 = 0,
which we rewrite as

L v i ϕ 2 = 0, 1 ≤ i ≤ n -1,
where the vector fields v 1 , . . . , v n-1 are given by

v i = (L g ϕ 1 )[c i , g] -(L [c i ,g] ϕ 1 )g, 1 ≤ i ≤ n -1.
We want to emphasize that the vector fields v i are easily calculable in terms of the vector fields g, c 1 , . . . , c n-1 and the chosen solution ϕ 1 . So finally, we have to solve the system L c i ϕ 2 = L v i ϕ 2 = 0 which, surprisingly, consists of 2(n -1) 1 st order PDE's on an (n + 2)-dimensional manifold. We will show below that this system reduces, actually, to n equations.

Reducing equations for ϕ 2

By Cartan's Lemma 1.3.1 we have

C 1 ⊂ C 2 ⊂ • • • ⊂ C n-1 and corank(C i ⊂ D ( i-1) ) = 1, for 1 ≤ i ≤ n -1.
Thus we can always choose vector fields c 1 , . . . , c n-1 such that

C n-1 = span {c 1 , • • • , c n-1 } and c n-1 (x 0 ) ∈ C n-2 (x 0 ), i.e., C n-2 = span {c 1 , • • • , c n-2 }.
Fix a vector field g ∈ D such that g(x 0 ) ∈ C n-1 (x 0 ) and clearly we have

D (n-2) = span {c 1 , • • • , c n-2 , c n-1 , g}.
We claim that the system

L c i ϕ 2 = 0, 1 ≤ i ≤ n -1 L v i ϕ 2 = 0, 1 ≤ i ≤ n -1, (1.3.1)
where

v i = (L g ϕ 1 )[c i , g]-(L [c i , g] ϕ 1
)g, can be reduced to a system of n equations. Since C n-2 is the characteristic distribution of D (n-2) , we get

[c i , g] = β i g mod C n-1 , 1 ≤ i ≤ n -2 (1.3.2)
where β i , for 1 ≤ i ≤ n -2, are smooth functions defined in a neighborhood of x 0 .

The equation (1.3.2) implies that (recall that L c ϕ 1 = 0, for any c ∈ C n-1 )

L [c i , g] ϕ 1 = L β i g ϕ 1 = β i L g ϕ 1 . (1.3.3) Therefore, for 1 ≤ i ≤ n -2, v i = (L g ϕ 1 )[c i , g] -(L [c i ,g] ϕ 1 )g = (L g ϕ 1 )β i g -β i (L g ϕ 1 )g mod C n-1 = 0 mod C n-1 .
It follows that the equations

L c i ϕ 2 = 0, for c i ∈ C n-1 , imply L v i ϕ 2 = 0, 1 ≤ i ≤ n -2,
and therefore the system (1.3.1) is equivalent to the following system of n equations

L c i ϕ 2 = 0, 1 ≤ i ≤ n -1 L v n-1 ϕ 2 = 0, (1.3.4)
where

v n-1 = (L g ϕ 1 )[c n-1 , g] -(L [c n-1 ,g] ϕ 1
)g. Notice that ϕ 1 solves the system (1.3.4).

We are looking for a solution ϕ 2 of (1.3.4), independent with ϕ 1 , and by Frobenious theorem the system (1.3.4) possesses two independent solutions if and only if the distribution

L = span {c 1 , . . . , c n-1 , v n-1 } = C n-1 ⊕ span {v n-1 } is involutive.
Below we will show that L is, indeed, involutive and to this end it is sufficient to show that [c i , v n-1 ] ∈ L, for any 1 ≤ i ≤ n -1. Since D (n-2) = span {c 1 , . . . , c n-1 , g}, we have

D (n-1) = D (n-2) + [D (n-2) , D (n-2) ] = span {c 1 , . . . , c n-1 , g, [c n-1 , g]}. The fact that C n-1 = span {c 1 , . . . , c n-1 } is the characteristic distribution of D (n-1) implies that [c i , [c n-1 , g]] = ξ i g + τ i [c n-1 , g] mod C n-1 , 1 ≤ i ≤ n -1, (1.3.5)
where ξ i , τ i are smooth functions defined in a neighborhood of x 0 , and thus

L [c i ,[c n-1 ,g]] ϕ 1 = ξ i L g ϕ 1 + τ i L [c n-1 ,g] ϕ 1 , 1 ≤ i ≤ n -1. (1.3.6)
Observing that

L [c i ,g] ϕ 1 = L c i L g ϕ 1 -L g L c i ϕ 1 = L c i L g ϕ 1 L [c i ,[c n-1 , g]] ϕ 1 = L c i L [c n-1 ,g] ϕ 1 -L [c n-1 ,g] L c i ϕ 1 = L c i L [c n-1 ,g] ϕ 1 (1.3.7)
and applying the relations (1.3.2)-(1.3.6), we have, for

1 ≤ i ≤ n -2, [c i , v n-1 ] = [c i , (L g ϕ 1 )[c n-1 , g]] -[c i , (L [c n-1 ,g] ϕ 1 )g] = (L g ϕ 1 )[c i , [c n-1 , g]] + (L c i L g ϕ 1 )[c n-1 , g] -(L [c n-1 ,g] ϕ 1 )[c i , g] -(L c i L [c n-1 ,g] ϕ 1 )g = L g ϕ 1 (ξ i g + τ i [c n-1 , g]) + β i (L g ϕ 1 )[c n-1 , g] -(L [c n-1 ,g] ϕ 1 )β i g -ξ i (L g ϕ 1 )g + τ i (L [c n-1 , g] ϕ 1 )g mod C n-1 = (τ i + β i )v n-1 mod C n-1 which implies that [c i , v n-1 ] ∈ L , for 1 ≤ i ≤ n -2. Now consider the case i = n -1, that is, the Lie bracket [c n-1 , v n-1 ]. Applying (1.3.5) and (1.3.6) for i = n -1, we get [c n-1 , v n-1 ] = [c n-1 , (L g ϕ 1 )[c n-1 , g]] -[c n-1 , (L [c n-1 ,g] ϕ 1 )g] = (L g ϕ 1 )[c n-1 , [c n-1 , g]] + (L c n-1 L g ϕ 1 )[c n-1 , g] -(L [c n-1 ,g] ϕ 1 )[c n-1 , g] -(L c n-1 L [c n-1 ,g] ϕ 1 )g = (L g ϕ 1 )[c n-1 , [c n-1 , g]] -(L c n-1 L [c n-1 ,g] ϕ 1 )g = ξ n-1 (L g ϕ 1 )g + τ n-1 (L g ϕ 1 )[c n-1 , g] -ξ n-1 (L g ϕ 1 )g + τ n-1 (L [c n-1 ,g] )g mod C n-1 = τ n-1 v n-1 mod C n-1 , which implies that [c n-1 , v n-1 ] ∈ L.
In conclusion, the distribution L is involutive and hence the system (1.3.4) is solvable. Together with the analysis of Section 1.3.2, we get the following theorem.

Theorem 1.3.5 Assume that a control system Σ is x-flat at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ), that is, the associated distribution D is, locally at x 0 , equivalent to the Goursat normal form on an

(n + 2)-dimensional manifold M. Let C n-1 = span {c 1 , . . . , c n-1 } be the characteristic distribution of D (n-1) such that c n-1 (x 0 ) ∈ C n-2 (x 0 ) and g any vector field in D such that g(x 0 ) ∈ C n-1 (x 0 ). Then (i) For any smooth function ϕ 1 such that (Flat 1) L c i ϕ 1 = 0, 1 ≤ i ≤ n -1, L g ϕ 1 (x 0 ) = 0, the distribution L = span {c 1 , . . . , c n-1 , v} is involutive, where v = (L g ϕ 1 )[c n-1 , g] -(L [c n-1 ,g] ϕ 1 )g. (ii) A pair of functions (ϕ 1 , ϕ 2 ) forms an x-flat output of Σ at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ),
if and only if after a permutation (if necessary) ϕ 1 satisfies (Flat 1), dϕ 1 (x 0 ) ∧ dϕ 2 (x 0 ) = 0, and ϕ 2 satisfies

(Flat 2) L c i ϕ 2 = 0, 1 ≤ i ≤ n -1, L v ϕ 2 = 0.
Remark. In (ii) only one implication may need permuting ϕ 1 and ϕ 2 . Indeed, if (ϕ 1 , ϕ 2 ) satisfies (Flat 1) and (Flat 2), then it is an x-flat output (and no permutation is needed). If (ϕ 1 , ϕ 2 ) is an x-flat output, then at least one ϕ i , 1 ≤ i ≤ 2, satisfies L g ϕ i (x 0 ) = 0 and we choose ϕ 1 such that L g ϕ 1 (x 0 ) = 0.

Example 1.3.6 To illustrate the above-presented procedure of finding flat outputs, we will consider the case of 2-input system

ẋ = f 1 (x)u 1 + f 2 (x)u 2
on a 4-dimensional manifold M. Assume that the system is x-flat, that is, the associated distributions D = span {f 1 , f 2 } satisfies the conditions of Theorem 1.3.5. Choose a vector field c ∈ C 1 characteristic for D (1) and g ∈ D such that g(x 0 ) ∧ c(x 0 ) = 0.

According to above procedure we take as ϕ 1 an arbitrary solution of

L c ϕ 1 = 0 L g ϕ 1 (x 0 ) = 0
and, in order to find ϕ 2 , we have to solve

L c ϕ 2 = 0 L v ϕ 2 = 0 where v = (L g ϕ 1 )[c, g] -(L [c, g] ϕ 1
)g. Notice that the above system of three 1 st order PDE's contains a fourth one; indeed we have

L v ϕ 1 = (L g ϕ 1 )L [c,g] ϕ 1 -(L [c,g] ϕ 1 )L g ϕ 1 = 0. The system L c ϕ i = L v ϕ i = 0, 1 ≤ i ≤ 2, (1.3.8)
admits two independent functions ϕ 1 and ϕ 2 as solutions if and only if the distribution span {c, v} is integrable. A direct calculation shows that this is the case. All becomes clear: the involutive distribution span {c, v} is just the distribution L of Theorem 1.3.3 while ϕ 1 and ϕ 2 satisfying (1.3.8) are x-flat outputs since their differentials span L ⊥ . We also see that L is not unique: different choices of ϕ 1 lead to different vector fields v which, in turn, give different distributions L = span {c, v}, although all of them are involutive and thus define (via span {dϕ 1 , dϕ 2 } = L ⊥ ) non equivalent flat outputs. This is in a perfect accordance with Theorem 1.3. This leads to the following model given by a driftless system on

Q = R 2 × S 1 × S 1 : Σ disk :      ẋ ẏ θ φ      =      R cos φ R sin φ 1 0      u 1 +      0 0 0 1      u 2 = f 1 u 1 + f 2 u 2 .
A direct computation shows that rank

D (i) = rank D i = i + 2, for 0 ≤ i ≤ 2,
U sing = {u = (u 1 , u 2 ) ⊤ : u 2 = 0}.
Thus the singular control corresponds to rolling the disk along a straight line. Now let us calculate all its x-flat outputs by using the procedure given in the Section 1.3.2. We choose

c = f 1 = R cos φ ∂ ∂x + R sin φ ∂ ∂y + ∂ ∂θ ,
and take g = f 2 = ∂ ∂φ . Then as a first flat output we can take any function ϕ 1 satisfying the following system of equations

L c ϕ 1 = R cos φ ∂ϕ 1 ∂x + R sin φ ∂ϕ 1 ∂y + ∂ϕ 1 ∂θ ≡ 0 L g ϕ 1 (q 0 ) = 0.
Solving this system of equations, we get that ϕ 1 is any function of the form

ϕ 1 = ϕ 1 (φ, x -Rθ cos φ, y -Rθ sin φ)
satisfying ∂ϕ 1 ∂φ (q 0 ) = 0. Choose one such ϕ 1 and then ϕ 2 is any function independent with ϕ 1 that satisfies L c ϕ 2 = L v ϕ 2 = 0, where the vector field v is given by

v = (L g ϕ 1 )[c, g] -(L [c,g] ϕ 1 )g.
To illustrate this, choose the function ϕ 1 = x -Rθ cos φ around a point q 0 such that L g ϕ 1 (q 0 ) = Rθ sin φ = 0 and then

v = R 2 θ sin 2 φ ∂ ∂x -R 2 θ sin φ cos φ ∂ ∂y -R sin φ ∂ ∂φ .
Solving the system of equations L c ϕ 2 = L v ϕ 2 = 0, we get

ϕ 2 = ϕ 2 (x -Rθ cos φ, y -Rθ sin φ)
satisfying (dϕ 1 ∧dϕ 2 )(q 0 ) = 0. All such functions satisfy span {dϕ 1 , dϕ 2 } = span {dϕ 1 , d φ2 } and we can take, for instance, ϕ 2 = y -Rθ sin φ. Moreover, the singular locus of the x-flat output (x -Rθ cos φ, y -Rθ sin φ) is given by

Sing(ϕ 1 , ϕ 2 ) = {q ∈ Q : (L g ϕ 1 (q), L g ϕ 2 (q)) = (0, 0)} = {q ∈ Q : θ = 0}.
To see that sin φ = 0 is, indeed, not a singularity, we just permute ϕ 1 and ϕ 2 .

To consider another possibility, we choose ϕ 1 = φ and then we have

v = R sin φ ∂ ∂x -R cos φ ∂ ∂y .
Solving the system of equations L c ϕ 2 = L v ϕ 2 = 0, we get

ϕ 2 = ϕ 2 (φ, Rθ -x cos φ -y sin φ),
satisfying (dϕ 1 ∧dϕ 2 )(q 0 ) = 0. We can take, for instance, ϕ 2 = Rθ-x cos φ-y sin φ and a simple calculation shows that there does not exist singular point of the x-flat output (φ, Rθ-x cos φ-y sin φ) in the state space Q. In other words, (φ, Rθ-x cos φ-y sin φ) is an x-flat output at any point (q, u)

∈ Q × R 2 provided that u ∈ U sing (q).
For various choices of functions, our result allows to eliminate them as candidates for x-flat outputs. For example, we can conclude that if

(ϕ 1 , ϕ 2 ) is an x-flat output, then L c ϕ i ≡ 0, for i = 1, 2, where c = R cos φ ∂ ∂x + R sin φ ∂ ∂y + ∂ ∂θ is a characteristic
vector field of D (1) . It follows that independently of the choice of ϕ 2 , neither (x, ϕ 2 ), nor (y, ϕ 2 ), nor (θ, ϕ 2 ) can serve as an x-flat output.

A complete description of x-flat outputs for the nonholonomic car system

Come back to the example of the nonholonomic car Σ car that we analyzed in Section 1.1.

Σ car :      ẋ ẏ θ0 θ1      =      cos(θ 1 -θ 0 ) cos θ 0 cos(θ 1 -θ 0 ) sin θ 0 sin(θ 1 -θ 0 ) 0      u 1 +      0 0 0 1      u 2 ,
We choose as a characteristic vector field c = ∂ ∂θ 1 and take

g = cos(θ 1 -θ 0 ) cos θ 0 ∂ ∂x + cos(θ 1 -θ 0 ) sin θ 0 ∂ ∂y + sin(θ 1 -θ 0 ) ∂ ∂θ 0 .
As a first x-flat output we can take any function ϕ 1 satisfying L c ϕ 1 = ∂ϕ 1 ∂θ 1 ≡ 0 and

L g ϕ 1 (q) = 0, that is any function ϕ 1 = ϕ 1 (x, y, θ 0 ) such that L g ϕ 1 (q) = 0. Let us choose one such ϕ 1 then ϕ 2 satisfies L c ϕ 2 = L v ϕ 2 = 0, where the vector field v is given by v = (L g ϕ 1 )[c, g] -(L [c,g] ϕ 1 )g = - ∂ϕ 1 ∂θ 0 cos θ 0 ∂ ∂x - ∂ϕ 1 ∂θ 0 sin θ 0 ∂ ∂y + ( ∂ϕ 1 ∂x cos θ 0 + ∂ϕ 1 ∂y sin θ 0 ) ∂ ∂θ 0 .
Therefore ϕ 2 can be taken as any functions ϕ 2 (x, y, θ 0 ) satisfying L v ϕ 2 = 0 and (dϕ 1 ∧ dϕ 2 )(q) = 0. Given ϕ 1 as above, the space of solutions for ϕ 2 is thus parameterized by one function of two variables but any two solutions ϕ 2 and φ2 give statically equivalent flat outputs, that is span {dϕ 1 , dϕ 2 } = span {dϕ 1 , d φ2 }. On the other hand, different choices of ϕ 1 will lead to nonequivalent pairs (ϕ 1 , ϕ 2 ) of x-flat outputs.

To illustrate this, take

ϕ 1 = x, then v = cos θ 0 ∂ ∂θ 0 and L c ϕ 2 = L v ϕ 2 = 0
imply that ϕ 2 is any function of the form ϕ 2 (x, y) satisfying ∂ϕ 2 ∂y (q) = 0 (because of (dϕ 1 ∧ dϕ 2 )(q) = 0). All such functions satisfy span {dx, dϕ 2 } = span {dx, d φ2 } and we can take, for instance, ϕ 2 = y. This gives the well-known x-flat output (x, y).

To see another choice, take

ϕ 1 = θ 0 , then v = -cos θ 0 ∂ ∂x -sin θ 0 ∂ ∂y and the general solution of L c ϕ 2 = L v ϕ 2 = 0 is ϕ 2 = ϕ 2 (θ 0 ,
x sin θ 0y cos θ 0 ), which gives as an x-flat output (θ 0 , x sin θ 0y cos θ 0 ). Notice that the singular loci of the two choices of x-flat outputs are different. In fact, Sing(x, y) = {θ 1θ 0 = ± π 2 } and Sing(θ 0 , x sin θ 0y cos θ 0 ) = {θ 1θ 0 = 0, ±π}.

Now take ϕ 1 = x + θ 0 around cos θ 0 = 0, then v = -cos θ 0 ∂ ∂x -sin θ 0 ∂ ∂y + cos θ 0 ∂ ∂θ 0 . Thus the general solution of L c ϕ 2 = L v ϕ 2 = 0 is ϕ 2 = ϕ 2 (x + θ 0 , y - ln | cos θ 0 |)). We can take, for instance, ϕ 2 = y -ln | cos θ 0 | which gives a third x-flat output (x + θ 0 , y -ln | cos θ 0 |) of Σ car
and its singular locus is defined by

Sing(x + θ 0 , y -ln | cos θ 0 |) = {cos θ 0 = 0} ∪ {cos(θ 1 -θ 0 ) cos θ 0 + sin(θ 1 -θ 0 ) = 0}.

A complete description of x-flat outputs for the nonholonomic n-trailer system

Consider the kinematic model for a unicycle-like mobile robot towing n trailers such that the towing hook of each trailer is located at the center of its unique axle (with the assumption that the distances between any two consecutive trailers are equal). The n-trailer system is subject to nonholonomic constraints: it is assumed that the wheels of each individual trailer are aligned with the body and are not allowed to slip [START_REF] Laumond | Controllability of a multibody robot[END_REF] . This model and its control properties have attracted a lot of attention (see the books [START_REF] Laumond | Robot Motion Planning and Control[END_REF] and [START_REF] Li | Nonholonomic Motion Plannging[END_REF]; and the papers [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], [START_REF] Jean | The car with n trailers: Characterisation of the singular configurations[END_REF], [START_REF] Murray | Nonholonomic motion planning: Steering using sinusoids[END_REF], [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF], [START_REF] Sørdalen | Conversion of the kinematics of a car with n trailers into a chained form[END_REF]). We use here the following description introduced in [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF].

Consider the n-trailer system Σ n tr defined on R 2 × (S 1 ) n+1 , for n ≥ 0,

Σ n tr : q = f 1 (q)u 1 + f 2 (q)u 2 , q ∈ R 2 × (S 1 ) n+1 ,
where the vector fields f 1 and f 2 are given by

f 1 = π 0 cos θ 0 ∂ ∂x + π 0 sin θ 0 ∂ ∂y + n-1 i=0 π i+1 sin(θ i+1 -θ i ) ∂ ∂θ i f 2 = ∂ ∂θ n with π i = n j=i+1 cos(θ j -θ j-1
) and π n = 1. The configuration of this system is described by q = (x, y, θ 0 , . . . , θ n ) ∈ R 2 × (S 1 ) n+1 , where (x, y) denotes the position of the last trailer while θ 0 , . . . , θ n represent the angles between each trailer's axle and the x-axis. According to Theorem 1.2.2, the n-trailer system is locally x-flat at any (q 0 , u 0 ) such that dim D (i) (q 0 ) = dim D i (q 0 ), 0 ≤ i ≤ n (equivalently D is equivalent to the Goursat normal form at q 0 ) and u 0 ∈ U sing (q 0 ). The former condition yields cos(θ i,0θ i-1,0 ) = 0 and the latter means that u 0 = (u 10 , u 20 ) satisfies u 10 = 0. In other words, the n-trailer is x-flat along a trajectory q(t) besides those time instances t 0 at which the angle between two consecutive trailers becomes ± π 2 (besides the angle between the last trailer and one before the last which can be any) or those instance at which the velocity q(t 0 ) becomes tangent to ∂ ∂θ n , that is, the whole n-trailer movement stops.

Let D be the distribution associated to the n-trailer system Σ n tr , that is D = span {f 1 , f 2 }. It is easy to check that rank D (i) = i + 2, for 0 ≤ i ≤ n + 1 and that, see e.g., Lemma 4.10 in [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF], the characteristic distribution C i of D (i) , for 1 ≤ i ≤ n, is given by

C i = span {c 1 , . . . , c i } = span { ∂ ∂θ n , . . . , ∂ ∂θ n-i+1
}.

Taking

g = f 1 = π 0 cos θ 0 ∂ ∂x + π 0 sin θ 0 ∂ ∂y + n-1 i=0 π i+1 sin(θ i+1 -θ i ) ∂ ∂θ i , we have clearly D (i) = span {c 1 , . . . , c i+1 , g}, for 1 ≤ i ≤ n -1, around any point q 0 such that cos(θ i,0 -θ i-1,0 ) = 0.
According to the analysis performed in Section 1.3.2, and summarized in Theorem 1.3.5, as a first flat output around a given point q 0 ∈ R 2 × (S 1 ) n+1 , we can take any function ϕ 1 satisfying L c i ϕ 1 = ∂ϕ 1 ∂θ i ≡ 0, for 1 ≤ i ≤ n, and L g ϕ 1 (q 0 ) = 0, that is any function ϕ 1 = ϕ 1 (x, y, θ 0 ) such that L g ϕ 1 (q 0 ) = 0. Let us choose one such ϕ 1 and then ϕ 2 has to satisfy

L c i ϕ 2 = 0, 1 ≤ i ≤ n L vn ϕ 2 = 0, (1.3.9)
where

v n = (L g ϕ 1 )[c n , g] -(L [cn,g] ϕ 1
)g (notice that the dimension of the state space is n + 3). The conditions L c i ϕ 2 = 0, 1 ≤ i ≤ n, imply that ϕ 2 = ϕ 2 (x, y, θ 0 ) and now we consider the equation L vn ϕ 2 = 0. We have

g = ηg + n-1 i=1 π i+1 sin(θ i+1 -θ i )c n-i+1
,

where η = n-1 i=1 cos(θ i+1 -θ i ) and g = cos(θ 1 -θ 0 ) cos θ 0 ∂ ∂x + cos(θ 1 -θ 0 ) sin θ 0 ∂ ∂y + sin(θ 1 -θ 0 ) ∂ ∂θ 0 .
Recall that around q 0 under consideration θ i+1,0θ i,0 = ± π 2 . So η = 0 and by a direct calculation we check that L vn ϕ 2 = 0 if and only if L ṽn ϕ 2 = 0, where

ṽn = (L g ϕ 1 )[c n , g] -(L [cn,g] ϕ 1 )g = - ∂ϕ 1 ∂θ 0 cos θ 0 ∂ ∂x - ∂ϕ 1 ∂θ 0 sin θ 0 ∂ ∂y + ∂ϕ 1 ∂x cos θ 0 + ∂ϕ 1 ∂y sin θ 0 ∂ ∂θ 0 .
Given any ϕ 1 = ϕ 1 (x, y, θ 0 ) such that L g ϕ 1 (q 0 ) = 0, the space of solution of L ṽn ϕ 2 = 0 is clearly parameterized by one function of two variables. Moreover any two solutions ϕ 2 and φ2 give equivalent x-flat outputs span {dϕ 1 , dϕ 2 } = span {dϕ 1 , d φ2 } = L ⊥ , where L = span {c 1 , . . . , c n , v} = span {c 1 , . . . , c n , ṽ} (as we have already discussed in Section 1.3.3). This equivalence is very easy to proved and so we omit it here. Therefore ϕ 2 can be taken as any functions ϕ 2 = ϕ 2 (x, y, θ 0 ) satisfying L ṽn ϕ 2 = 0 and (dϕ 1 ∧ dϕ 2 )(x 0 ) = 0. The space of solutions is thus defined on {q : η 2 (q) = 0}, i.e., {q :

θ i+1 -θ i = ± π 2 }.
Take

ϕ 1 = x, then ṽn = cos θ 0 ∂ ∂θ 0 and L c i ϕ 2 = L ṽn ϕ 2 = 0, for 1 ≤ i ≤ n,
imply that ϕ 2 is any function of the form ϕ 2 (x, y) satisfying ∂ϕ 2 ∂y (q 0 ) = 0 (because of (dϕ 1 ∧ dϕ 2 )(q 0 ) = 0). All such functions satisfy span {dx, dϕ 2 } = span {dx, d φ2 } and we can take, for instance, ϕ 2 = y.

To see the other choice, take

ϕ 1 = θ 0 , then ṽn = -cos θ 0 ∂ ∂x -sin θ 0 ∂ ∂y and the general solution of L c i ϕ 2 = L ṽn ϕ 2 = 0 is ϕ 2 = ϕ 2 (θ 0 , x sin θ 0 -y cos θ 0 ).
It is interesting to notice that the family of all x-flat outputs of the n-trailer system coincides with the family of all x-flat outputs of the nonholonomic car, more precisely of the car defined as the tail consisting of the last and one-before-the-last trailers (those indexed , respectively, by i = 0 and i = 1) controlled by towing (forward and backward) and rotating the one-before-the-last trailer. Indeed, the family of x-flat outputs coincide because they are given by any ϕ 1 = ϕ 1 (x, y, θ 0 ) and any ϕ 2 (x, y, θ 0 ) satisfying L ṽn ϕ 2 = 0 (for the n-trailer) and L v 1 ϕ 2 = 0 (for the car) but, clearly, these two equations are equivalent (compare Section 1.3.4).

Proof of main theorems 1.4.1 Useful results

In this section we give a series of results that we will use in the subsequent sections when proving our theorems. We start with a weaker version of Proposition 1.2.3 proving that the statement of the latter holds on open and dense subset.

Lemma 1.4.1 Consider a driftless 2-input smooth control system Σ defined on a man- ifold M of dimension n + 2. Let ϕ 1 , ϕ 2 be two functions defined on M. If (ϕ 1 , ϕ 2
) is an x-flat output at (x 0 , u 0 ), then there exists an open neighborhood M of x 0 and an open and dense subset M such that around any point q ∈ M there exist coordinates (z 1 , . . . , z n+2 ) in which Σ is locally feedback equivalent to the chained form Σ chain , given by (1.2.2) such that ϕ 1 = z 1 and ϕ 2 = z 2 .

Proof:

Let (ϕ 1 , ϕ 2 ) be an x-flat output of Σ at (x 0 , u 0 ). There exists an open neighborhood M of x 0 such that (ϕ 1 , ϕ 2 ) is an x-flat output at (x, u) for any x ∈ M and u = u(x). It is known (see, e.g., [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF]) that the differentials of flat output are independent at x 0 and thus we put x 1 = ϕ 1 , x 2 = ϕ 2 , and complete them to a coordinate system ξ = (x 1 , x 2 , . . . , x n+2 ). Consider the (2

× 2)-matrix D = (D ij ) given by D ij = L g j ϕ i , 1 ≤ i, j ≤ 2.
It is immediate to see that rank D(q) ≤ 1, for any q ∈ M. Indeed, if the rank were two then by a suitable invertible feedback u = β(x)v we would get

ϕ 1 = x 1 , ẋ1 = v 1 ϕ 2 = x 2 , ẋ2 = v 2
which contradicts the flatness assumption because ϕ

(j) i = v (j-1) i
, for 1 ≤ i ≤ 2, and any j ≥ 1, and thus the coordinates x 3 , . . . , x n+2 cannot be represented as functions of ϕ (j) i , j ≥ 0. Therefore on an open and dense subset M ′ of M, rank D(q) = 1, for q ∈ M ′ . After applying around any q ∈ M a suitable invertible feedback u = β(x)v, the system becomes ẋ

= g 1 (x)v 1 + g 2 (x)v 2 ,
where the flat outputs and their derivatives are

ϕ 1 = x 1 , ẋ1 = v 1 ϕ 2 = x 2 , ẋ2 = ψ(x)v 1
with ψ(x) being a smooth function. Consider the vector fields

g 1 = ∂ ∂x 1 + ψ ∂ ∂x 2 + n+2 i=3 g 1i (x) ∂ ∂x i g 2 = n+2 i=3 g 2i (x) ∂ ∂x i .
We claim that for any q in an open and dense subset M ′′ of M ′ there exists ρ such that

L g 2 L µ g 1 ψ ≡ 0, for 0 ≤ µ ≤ ρ -2, L g 2 L ρ-1
g 1 ψ(q) = 0, and, moreover, that ρ = n for any q ∈ M ′′ . In other words, ρ = n is the relative degree of the single-input system ẋ = f + vg, where f = g 1 , g = g 2 and v = v 2 .

To prove our claim, first, observe that ρ exists on an open and dense subset M ′′ of M ′ . If not, on an open set in M ′ , we would have L g 2 L µ g 1 ψ ≡ 0 for any µ ≥ 0, which contradicts the flatness of Σ since v 2 could not be expressed in terms of ϕ

(j) i , i = 1, 2, j ≥ 0.
Thus ρ exists and is locally constant on an open and dense subset M ′′′ of M ′′ , with a priori different constant values on different connected components of M ′′′ . We claim that on each connected component the constant value of ρ is n. On one hand, we have ρ ≤ n since L g 2 ϕ 1 = L g 2 ϕ 2 = 0. On the other hand, if ρ < n, then we put ṽ2 = (L g 2 L ρ-1 g 1 ψ)v 2 + (L ρ g 1 ψ)v 1 which is an invertible feedback, because of the definition of ρ. Now knowing ϕ 1 and ϕ 2 , we can obtain

v 1 = φ1 , ψ = φ2 φ1 , by differentiation ρ -1 functions L ρ g 1 ψ = ψ (µ) , 1 ≤ µ ≤ ρ -1
, and finally the control ṽ2 . This gives 2 controls and ρ + 2 < n + 2 functions, so one function among x 3 , . . . , x n+2 is missing. This contradicts the flatness assumption. We thus have proved that on an open and dense subset M ′′′ of M ′ , the relative degree ρ is well defined and equals n.

Fix an arbitrary q ∈ M ′′′ , we can assume that ψ(q) = g 1i (q) = 0, for 3

≤ i ≤ n+2 (if not, we replace x i by x i -k i x 1 , for 2 ≤ i ≤ n+2,
where k 2 = ψ(q) and k i = g 1i (q), for 3 ≤ i ≤ n+2). We claim that the differentials of the functions x 1 , x 2 , ψ, L f ψ, . . . , L n-1 f ψ are independent at q ∈ M ′′′ . To this end, we will use the following result (see, e.g., Isidori [START_REF] Isidori | Nonlinear Control Systems[END_REF]): if two vector fields f and g and a function φ satisfy

L g φ = L g L f φ = • • • = L g L k-2 f φ = 0, L g L k-1 f φ = λ,
where λ is function, then for any 1

≤ j ≤ k -1, L ad j f g φ = L ad j f g (L f φ) = • • • = L ad j f g (L k-j-2 f φ) = 0, L ad j f g (L k-j-1 f φ) = -λ.
We apply this result to f = g 1 , g = g 2 , φ = ψ and k = n so, in particular, λ(q) = 0. Since, L g 2 ϕ 1 = 0 and L g 1 ϕ 1 = 1, it follows that

L ad g 1 g 2 ϕ 1 = L [g 1 ,g 2 ] ϕ 1 = L g 1 L g 2 ϕ 1 -L g 2 L g 1 ϕ 1 = 0
and by induction we prove easily that L ad j g 1 g 2 ϕ 1 = 0. Moreover,

ψ = L g 1 ϕ 2 implies that L adg 1 g 2 ϕ 2 = • • • = L ad n-1 g 1 g 2 ϕ 2 = 0 L ad n g 1 g 2 ϕ 2 (q) = (-1) n L g 2 L n g 1 ϕ 2 (q) = (-1) n L g 2 L n-1 g 1 ψ(q) = 0
Putting L ad n g 1 g 2 ϕ 2 = λ and evaluating the differential forms dϕ i , dL j g 1 ψ, 0 ≤ j ≤ n -1 on the vector fields g 2 , ad g 1 g 2 , . . . , ad n g 1 g 2 , g 1 we get (notice that

L g 1 ϕ 2 = ψ)          dϕ 1 dϕ 2 dψ dL g 1 ψ . . . dL n-1 g 1 ψ          g 2 , ad g 1 g 2 , . . . , ad n g 1 g 2 , g 1 =        0 0 • • • 0 1 0 0 • • • λ ψ • • • • • • • • • * * 0 (-1) n-1 λ • • • • • • • • • (-1) n λ * • • • • • • *       
The determinant of the above matrix is nonzero since λ(q) = 0, and therefore the functions ϕ 1 , ϕ 2 , ψ, L g 1 ψ, . . . , L n-1 g 1 ψ are independent. It follows that

z 1 = ϕ 1 z 2 = ϕ 2 z 3 = ψ z 4 = L g 1 ψ . . . z n+2 = L n-1 g 1 ψ
is a valid local change of coordinates in a neighborhood of any q ∈ M = M ′′′ , in which the system after applying feedback ṽ1

= v 1 , ṽ2 = (L n g 1 ψ)v 1 + (L g 2 L n-1 g 1 ψ)v 2 (invertible since (L g 2 L n-1 g 1 ψ)(q) = 0) takes the chained form Σ chain (1.2.
2). Although the above lemma was proved at generic points only, it implies the following result in a whole neighborhood of the point x 0 under consideration. Recall that for any Goursat structure D we denote by C n-1 the characteristic distribution of

D (n-1) (see Lemma 1.3.1). Corollary 1.4.2 Consider a Goursat structure D on M of dimension n + 2, that is, rank D (i) = i + 2, for 0 ≤ i ≤ n, hold everywhere on M. If the associated control system Σ is x-flat at (x 0 , u 0 ) ∈ M × R 2 ,
u 0 ∈ U sing , then for any x-flat output (ϕ 1 , ϕ 2 ) at (x 0 , u 0 ), there exists an open neighborhood M of x 0 in which we have L c ϕ i = 0, for i = 1, 2 and any c ∈ C n-1 .

Proof: Let (ϕ 1 , ϕ 2 ) be an x-flat output of Σ at (x 0 , u 0 ). By Lemma 1.4.1, there exists an open neighborhood M of x 0 and an open and dense subset M of M such that around any q ∈ M, the system Σ is feedback equivalent to the chained form Σ chain , given by (1.2.2), with ϕ 1 = z 1 and ϕ 2 = z 2 . We have C n-1 = span { ∂ ∂z 4 , . . . , ∂ ∂z n+2 } and hence L c ϕ i = 0, for i = 1, 2 and any c ∈ C n-1 on M and hence on M (since M is dense and the functions ϕ i as well as the distribution C n-1 are well defined on the whole M).

Proof of Theorem 1.2.2

We will show the implications (iii) ⇒ (iv) ⇒ (ii) ⇒ (i) ⇒ (iii).

(iii) ⇒ (iv): It is a well known result (proved by Murray in [START_REF] Murray | Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems[END_REF]) that dim (iv) ⇒ (ii): It is obvious for a system the chained form Σ chain given by (1.2.2), that ϕ 1 = z 1 and ϕ 2 = z 2 yield flatness for v 1 = 0 and the latter means that we can take in (ii) any u 0 ∈ U sing (x 0 ). (ii) ⇒ (i): Obvious.

D (i) (x 0 ) = dim D i (x 0 ), 0 ≤ i ≤ n,
(i) ⇒ (iii): This is the only difficult implication. Its proof will be based on Corollary 1.4.2 and on the result that assures that any Goursat structure can be brought to the following polynomial normal form, called Kumpera-Ruiz normal, as proved by Pasillas-Lépine and Respondek [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF] (see also Mormul [42]):

Theorem 1.4.3 (Kumpera-Ruiz normal form) Assume that n ≥ 2. Any Goursat structure defined on a manifold M of dimension n+2 is locally equivalent, at any point x 0 in M, to a distribution spanned in a small neighborhood of zero by a pair of vector fields that has the following form:

f 1 = m i=0 i-1 j=0 x j k j k i -1 j=1 (x i j + c i j ) ∂ ∂x i j+1 + ∂ ∂x i+1 1 , f 2 = ∂ ∂x 0 1 (1.4.1)
where the coordinates x i j , for 0 ≤ i ≤ m + 1 and 1 ≤ j ≤ k i , are centered at x 0 (that means that x(x 0 ) = 0); the integer m is such that 0 ≤ m ≤ n -2; and k i , for

0 ≤ i ≤ m -1, satisfy k 0 ≥ 1, . . . , k m-1 ≥ 1, k m ≥ 3, k m+1 = 1 and m+1 i=0 k i = n + 2. The constants c i j , for 1 ≤ j ≤ k i -1, are real constants.
Remark. In the above normal form, the integer m gives the number of singularities of the Kumpera-Ruiz normal form. When m = 0, the Kumpera-Ruiz normal form coincides with the Goursat normal form (since in this case all constant c i j can be eliminated).

In order to prove (i) ⇒ (iii), assume that Σ is x-flat at (x 0 , ūl 0 ), for a certain l ≥ 0, and that there exists an integer 2 ≤ i ≤ n such that dim D (i) (x 0 ) = dim D i (x 0 ). Since D is a Goursat structure and dim D (i) (x 0 ) = dim D i (x 0 ), for certain 2 ≤ i ≤ n, by Theorem 1.4.3, there exists a new coordinate system (x 0 1 , . . . , x 0 k 0 , . . . , x m 1 , . . . , x m km , x m+1 1 ) in which D takes, in a small neighborhood of zero, the Kumpera-Ruiz normal form with m ≥ 1 and m i=0 k i = n + 1 , i.e., D = span {f 1 , f 2 } where f 1 and f 2 are given by (1.4.1). A direct calculation shows that

D (j) = span { ∂ ∂x 0 1 , . . . , ∂ ∂x 0 j+1 , f 1 }, 1 ≤ j ≤ k 0 -1,
and

C j = span { ∂ ∂x 0 1 , . . . , ∂ ∂x 0 j }, 1 ≤ j ≤ k 0 -1. Observing that ∂ ∂x 0 k 0 , f 1 = m i=1 i-1 j=1 x j k j k i -1 j=1 (x i j + c i j ) ∂ ∂x i j+1 + ∂ ∂x i+1 1 and ∂ ∂x 1 1 = f 1 -x 0 k 0 ∂ ∂x 0 k 0 , f 1 - k 0 -1 j=1 (x 0 j + c 0 j ) ∂ ∂x 0 i+1 , we then get D (k 0 ) = D (k 0 -1) + [D (k 0 -1) , D (k 0 -1) ] = span ∂ ∂x 0 1 , . . . , ∂ ∂x 0 k 0 , ∂ ∂x 1 1 , m i=1 i-1 j=1 x j k j k i -1 j=1 (x i j + c i j ) ∂ ∂x i j+1 + ∂ ∂x i+1 1 ,
and

C k 0 = span { ∂ ∂x 0 1 , . . . , ∂ ∂x 0 k 0 }.
In the same way, we obtain that

C k i +j = C k i ⊕ span ∂ ∂x i+1 1 , . . . , ∂ ∂x i+1 j , for 0 ≤ i ≤ m, 1 ≤ j ≤ k i and k m + j ≤ n -1. Therefore, finally, the characteristic distribution C n-1 of D (n-1) is given by C n-1 = span ∂ ∂x 0 1 , . . . , ∂ ∂x 0 k 0 , . . . , ∂ ∂x m 1 , . . . , ∂ ∂x m km-2 = (span {dx m km-1 , dx m km , dx m+1 1 }) ⊥ .
For simplicity, we denote the coordinates x m km-1 , x m km , x m+1 1 by y 1 , y 2 , y 3 , respectively, and denote the remaining n -1 coordinates x i j by x 1 , . . . , x n-1 , that is,

(x, y) = (x 1 , . . . , x n-1 , y 1 , y 2 , y 3 ) = (x 0 1 , . . . , x 0 k 0 , . . . , x m 1 , . . . , x m-1 km , x m km , x m+1 1
).

In (x, y)-coordinates, the control system Σ associated to the Kumpera-Ruiz normal form (1.4.1), reads as

Σ KR :                        ẋ1 = u 1 ẋ2 = γ 2 (x, y)u 2 . . . ẋn-1 = γ n-1 (x, y)u 2 ẏ1 = x r β 1 (x, y)u 2 ẏ2 = x r β 2 (x, y)u 2 ẏ3 = x r β 3 (x, y)u 2
where β 1 , β 2 , β 3 and γ j , for 2 ≤ j ≤ n -1, are smooth functions defined in a neighborhood of 0 ∈ R n+2 , 1 ≤ r ≤ n -1 is an integer, and the characteristic distribution C n-1 is given by

C n-1 = span ∂ ∂x 1 , . . . , ∂ ∂x n-1 = (span {dy 1 , dy 2 , dy 3 }) ⊥ .
By our assumption, Σ KR is x-flat at (0, ūl

0 ) ∈ R n+2 × R 2(l+1) and let (ϕ 1 , ϕ 2 ) be an x-flat output defined in a neighborhood O of 0 ∈ R n+2 . Being a Goursat structure, Σ KR satisfies dim D (i) (z) = dim D i (z), 0 ≤ i ≤ n,
∂ϕ 1 ∂x i = ∂ϕ 2 ∂x i = 0, 1 ≤ i ≤ n -1, holds in O ′ and since O ′ is dense in O, also in O. It follows that ϕ i = ϕ i (y 1 , y 2 , y 3 ) for i = 1, 2.
Moreover, the fact that (ϕ 1 , ϕ 2 ) is an x-flat output at (0, ūl 0 ) implies that ϕ 1 , ϕ 2 , φ1 , φ2 must be independent at (0, u 0 ) ∈ R n+2 × R 2 , where u 0 = (u 10 , u 20 ), (see, e.g., [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF] and [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF]). Calculating the derivatives φi , for i = 1, 2, we get φi =

3 j=1 x r β j u 2 ∂ϕ i ∂y j = F i (x, y 1 , y 2 , y 3 , u 2 ), i = 1, 2. Then d φi = dF i = n-1 j=1 ∂F i ∂x j • dx j + 3 j=1 ∂F i ∂y j • dy j + ∂F i ∂u 2 • du 2 , i = 1, 2
and at (0, u 0 ),

d φi (0, u 0 ) = dF i (0, u 0 ) = ∂F i ∂x r (0, u 0 ) • dx r , i = 1, 2, which implies that (dϕ 1 ∧ dϕ 2 ∧ d φ1 ∧ d φ2 )(0, u 20 ) = 0,
independently of the value of u 20 , which gives a contradiction. Therefore if a system associated to a Goursat structure is x-flat at (x, ūl 0 ), for some ūl 0 , then we have dim Proof: Sufficiency: Take any 2-input system whose associated distribution satisfies dim D (i) (x) = dim D i (x) = i + 2 everywhere in a neighborhood of x 0 and choose two functions fulfilling (i) -(iii). We can bring Σ to the chained form Σ chain , given by (1.2.2), in coordinates (z 1 , . . . , z n+2 ), transforming x 0 into 0 ∈ R n+2 , and its associated distribution is given by D = span {g 1 , g 2 }, where

D (i) (x 0 ) = dim D i (x 0 ), 0 ≤ i ≤ n.
g 1 = ∂ ∂z 1 + z 3 ∂ ∂z 2 + • • • + z n+2 ∂ ∂z n+1 g 2 = ∂ ∂z n+2 .
The characteristic distribution C n-1 is given by

C n-1 = span { ∂ ∂z 4 , . . . , ∂ ∂z n+2 },
and the condition

L c ϕ i = 0, for any c ∈ C n-1 , of item (ii) implies that ϕ i = ϕ i (z 1 , z 2 , z 3 ), for i = 1, 2. Item (iii) implies that there exists ϕ i (say ϕ 1 , if not we permute) such that L g 1 ϕ 1 (0) = ∂ϕ 1 ∂z 1 (0) = 0. Due to (i) we can complete ϕ 1 and ϕ 2 by a function ϕ 3 (z 1 , z 2 , z 3 ) such that (dϕ 1 ∧ dϕ 2 ∧ dϕ 3 )(0) = 0 to introduce new coordinates zi = ϕ i (z 1 , z 2 , z 3 ), 1 ≤ i ≤ 3, followed by zi = z i , 4 ≤ i ≤ n. We have żi = ψ i (z 1 , z 2 , z 3 , z 4 )u 1 , 1 ≤ i ≤ 3,
where

ψ i = L g 1 ϕ i . Since ψ 1 (0) = L g 1 ϕ 1 (0) = 0, we apply invertible feedback ũ1 = ψ 1 u 1 to get Σ :                ż1 = ũ1 ż2 = ψ2 ũ1 ż3 = ψ3 ũ1 żi = zi+1 ũ1 ψ 1 , for 4 ≤ i ≤ n + 1 żn+2 = u 2 . Notice that the characteristic distribution C n-1 = span { ∂ ∂ z4 , . . . , ∂ ∂ zn+2 } (since zi = ψ i (z 1 , z 2 , z 3 ), for 1 ≤ i ≤ 3). It thus follows from L c L g 1 ϕ 2 L g 1 ϕ 1 = 0, c ∈ C n-1 , that ψ2 = ψ 2 (z 1 , z 2 , z 3 , z 4 ) ψ 1 (z 1 , z 2 , z 3 , z 4 )
is actually a function of z1 , z2 , z3 only. Moreover, the condition rank

D (n) = rank D n = n + 2 implies that ∂ ψ3 ∂ z4 (0) = 0 and a direct calculation shows that L g2 L µ g1 ψ2 ≡ 0 for 0 ≤ µ ≤ n -2 and L g2 L n-1 g1 ψ2 (0) = 0, where g1 = ∂ ∂ z1 + ψ2 ∂ ∂ z2 + ψ3 ∂ ∂ z3 + z5 ψ 1 ∂ ∂ z4 + • • • + zn+2 ψ 1 ∂ ∂ zn+1 g2 = ∂ ∂ zn+2 .
Therefore the function ψ2 satisfies the condition of ψ from the proof of Lemma 1.4.1, and following that proof we can thus bring the system to the chained form (1.2.2), with (ϕ 1 , ϕ 2 ) = (z 1 , z2 ) which proves that (ϕ 1 , ϕ 2 ) is indeed an x-flat output at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ).

Necessity: Assume that Σ is x-flat at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ), and let (ϕ 1 , ϕ 2 ) be its x-flat output defined in a neighborhood M of x 0 . It is well known (see [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF]) that dϕ 1 (x 0 ) ∧ dϕ 2 (x 0 ) = 0. By Lemma 1.4.1, we can bring Σ, around any point q ∈ M (open and dense in M), into the chained form Σ chain , given by (1.2.2), with

ϕ 1 = z 1 , ϕ 2 = z 2 , and x 0 is transformed into z 0 = 0 ∈ R n+2 . We have C n-1 = span { ∂ ∂z 4 , . . . , ∂ ∂z n+2 }.
By a direct calculation we get that L g ϕ 1 (0) = 0 for any g ∈ D such that g(0) ∈ C n-1 (0), and that

L c ϕ 1 ≡ L c ϕ 2 ≡ L c ( L g ϕ 2 L g ϕ 1 ) ≡ 0, for any c ∈ C n-1
and g as above, which gives the item (ii) on M. Now observe that the flat outputs ϕ 1 , ϕ 2 are well defined in M and so is the characteristic distribution C n-1 (since the distribution D associated to Σ satisfies rank

D (i) = i + 2, 0 ≤ i ≤ n everywhere in M). It follows by continuity that L c ϕ 1 ≡ L c ϕ 2 ≡ (L g ϕ 1 )(L c L g ϕ 2 ) -(L g ϕ 2 )(L c L g ϕ 1 ) ≡ 0 holds also on M thus implying (ii) on M.
It remains to prove (iii). Bring Σ, locally around x 0 ∈ M, into the chained form Σ chain , given by (1.2.2) (which is always possible by the assumption of theorem). Then item (ii), which we have just proved on M, implies that

ϕ i = ϕ i (z 1 , z 2 , z 3 ), for 1 ≤ i ≤ 2 and g = ∂ ∂z 1 + z 3 ∂ ∂z 2 + z 4 ∂ ∂z 3 mod C n-1 . If for i = 1 or 2, we have L g ϕ i (0) = 0, then (iii) holds. So assume that L g ϕ 1 (0) = L g ϕ 2 (0) = 0 implying ∂ϕ 1 ∂z 1 (0) = ∂ϕ 2 ∂z 1 (0) = 0. We have φ1 = ( ∂ϕ 1 ∂z 1 + z 3 ∂ϕ 1 ∂z 2 + z 4 ∂ϕ 1 ∂z 3 )u 1 = (a + bz 4 )u 1 φ2 = ( ∂ϕ 2 ∂z 1 + z 3 ∂ϕ 2 ∂z 2 + z 4 ∂ϕ 2 ∂z 3 )u 1 = (c + dz 4 )u 1
and thus φ1 (0, u 0 ) = φ2 (0, u 0 ) = 0, where u 0 stands for the value of the nominal control. Defin M = {z ∈ M : (L g ϕ 1 (z) = L g ϕ 2 (z) = (0, 0)}. On M we, clearly, have

z i = γi (ϕ 1 , ϕ 2 , φ1 , φ2 ), 1 ≤ i ≤ 3.
We claim that around 0 there must exist smooth functions γ i , for 1

≤ i ≤ 3, such that z i = γ i (ϕ 1 , ϕ 2 , φ1 , φ2
) that coincide with γi on M. Indeed, from the definition of flatness it follows that z i = γ i ( φs 1 , φs 2 ), for some s ≥ 0. But if s > 1, then there exists nontrivial relations γ i = γi on M contradicting differential independence of ϕ 1 and ϕ 2 (see [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF]). We have

ϕ 1 = ϕ 1 (z 1 , z 2 , z 3 ), ϕ 2 = ϕ 2 (z 1 , z 2 , z 3 ) and since ∂ϕ 1 ∂z 1 (0) = ∂ϕ 2 ∂z 1 (0) = 0, it follows that z 1 = γ 1 (ϕ 1 , ϕ 2 , φ1 , φ2 )
, where γ 1 depends explicitly on φi , but the composition γ 1 (ϕ 1 , ϕ 2 , φ1 , φ2 )(z 1 , z 2 , z 3 , z 4 , u 1 ) depends actually on z 1 , z 2 , z 3 only. Notice that φ1 and φ2 are linear with respect to u 1 and affine with respect to z 4 which implies that (adbc) ≡ 0 (in order that it does not depend on z 4 ), and that γ 1 is a function of a + bz 4 c + dz [START_REF] Arnold | Geometrical Methods in the Theory of Ordinary Differential Equations[END_REF] (in order that it does not depend on u 1 ). However neither a + bz 4 c + dz 4 nor c + dz 4 a + bz 4 is smooth at 0. Therefore L g ϕ i (0) = 0 for at least one 1 ≤ i ≤ 2 and the item (iii) holds as well. The above analysis also shows that at any point x ∈ M that satisfies L g ϕ 1 (x) = L g ϕ 2 (x) = 0, the pair (ϕ 1 , ϕ 2 ) fails to be an x-flat output. In other words,

Sing(ϕ 1 , ϕ 2 ) = {x ∈ M : (L g ϕ 1 (x), L g ϕ 2 (x)) = (0, 0)}.
Now we will show how Proposition 1.2.3 follows from Theorem 1.3.2. Proof: (of Proposition 1.2.3) Let (ϕ 1 , ϕ 2 ) be a pair of x-flat outputs at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ). Then (ϕ 1 , ϕ 2 ) satisfy the items (i) -(iii) of Theorem 1.3.2 and we can follow the procedure described in the sufficiency part of that theorem in order to bring Σ into the chained form Σ chain such that ϕ 1 = z 1 and ϕ 2 = z 2 .

Proof of Theorem 1.3.3

Proof: Sufficiency: We will prove separately the cases dimM = 3 and dimM ≥ 4.

Case (I): dim M = 3

Let ϕ 1 , ϕ 2 be any functions satisfying (i) ′ and (ii) ′ . Introduce coordinates x 1 = ϕ 1 , x 2 = ϕ 2 and complete them to a coordinate system (x 1 , x 2 , x 3 ), centered at 0 ∈ R 3 . Choose two vector fields g1 , g 2 such that D = span {g 1 , g 2 }, L = span {g 2 }. We have g1 (0) ∈ L(0) and thus there exits a function ϕ i , for i = 1, 2, such that L g1 ϕ i (x 0 ) = 0, say L g1 ϕ 1 (x 0 ) = 0. Define

g 1 = 1 L g1 ϕ 1 g. The associated control system ẋ = u 1 g 1 (x) + u 2 g 2 (x) is ẋ1 = u 1 ẋ2 = ψ(x)u 1 ẋ3 = η(x)u 2
where ψ and η are smooth functions. We have η(0) = 0 and we can suppose that ψ(0) = 0 (if not, replace x 2 by x 2ψ(0)x 1 ). The condition dim D (1) (0) = 3 implies that g 1 , g 2 and [g 1 , g 2 ] are independent at 0 ∈ R 3 and hence ∂ψ ∂x 3 (0) = 0. Replacing x 3 by ψ and applying feedback to normalize ψ, we get

ẋ1 = u 1 ẋ2 = x 3 u 1 ẋ3 = u 2 for which (ϕ 1 , ϕ 2 ) = (x 1 , x 2
) is an x-flat output at (0, u 0 ), where u 0 = (u 10 , u 20 ) ⊤ such that u 10 = 0. To see that u 10 = 0 is not a singular control, introduce the new

coordinate x2 = x 2 -x 1 x 3 to get ẋ1 = u 1 ẋ2 = -x 1 u 2 ẋ3 = u 2
for which ( φ1 , φ2 ) = (x 2 , x 3 ) is an x-flat output at (0, u 0 ) such that u 20 = 0. If follows that the singular control is U sing = {(0, 0)} only.

Case (II): dim M ≥ 4 Take any 2-input system whose associated distribution satisfies dim

D (i) (x) = dim D i (x) = i + 2
everywhere in a neighborhood of x 0 and choose two functions fulfilling (i) ′ -(iii) ′ . We can bring Σ to the chained form (1.2.2) in coordinates (z 1 , . . . , z n+2 ), transforming x 0 into 0 ∈ R n+2 , and its associated distribution is D = span {g 1 , g 2 }, where

g 1 = ∂ ∂z 1 + z 3 ∂ ∂z 2 + • • • + z n+2 ∂ ∂z n+1 g 2 = ∂ ∂z n+2 .
The characteristic distribution C n-1 is given by

C n-1 = span { ∂ ∂z 4 , . . . , ∂ ∂z n+2 }.
Item (ii) ′ implies that C n-1 ⊂ L = (span {dϕ 1 , dϕ 2 }) ⊥ . Indeed, if there was a vector field f ∈ C n-1 such that f / ∈ L, then D (n-1) = L + span{f } and hence

D (n) = D (n-1) + [D (n-1) , D (n-1) ] = L + span{f } + [f, L] = L + span{f } = D (n-1)
which contradicts the condition rank

D (n) = n + 2 . Therefore C n-1 ⊂ L holds indeed.
Consequently we have L c ϕ i ≡ 0, for i = 1, 2 and any c ∈ C n-1 , which implies that

ϕ i = ϕ i (z 1 , z 2 , z 3 ) for i = 1, 2. Moreover observing that g 2 ∈ C n-1 ⊂ L, by item (iii) ′ we conclude, g 1 / ∈ L, which implies that there exist ϕ i , 1 ≤ i ≤ 2, such that L g 1 ϕ i (0) = 0, says i = 1.
In other words, we have

L g 1 ϕ 1 (0) = ∂ϕ 1 ∂z 1 (0) = 0. By (i) ′ we can complete ϕ 1 and ϕ 2 by a function ϕ 3 (z 1 , z 2 , z 3 ) such that (dϕ 1 ∧ dϕ 2 ∧ dϕ 3 )(0) = 0 to introduce new coordinates zi = ϕ i (z 1 , z 2 , z 3 ), 1 ≤ i ≤ 3, followed by zi = z i , 4 ≤ i ≤ n.
The remaining part of the proof follows the same line as that of Theorem 1.3.2.

Necessity: Assume that Σ is x-flat at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ), and (ϕ 1 , ϕ 2 ) is an x-flat output at (x 0 , u 0 ), defined in a neighborhood M of x 0 . It is well known (see [START_REF] Fliess | A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF], [START_REF] Pomet | A differential geometric setting for dynamic equivalence and dynamic linearization[END_REF]) that dϕ 1 (x 0 ) ∧ dϕ 2 (x 0 ) = 0. Now we will prove the item (ii) ′ . Clearly, Lemma 1.4.1 applies and thus there exists an open and dense subset M ⊂ M with the properties claimed by the lemma. Around any x ∈ M, there exists a local coordinate system (z 1 , . . . , z n+2 ) such that z 1 = ϕ 1 , z 2 = ϕ 2 in which Σ takes the chained form Σ chain (1.2.2) and x is transformed into 0 ∈ R n+2 . Then by a simple computation we get

D (n-1) = span { ∂ ∂z n+2 , ∂ ∂z n+1 , . . . , ∂ ∂z 3 , ∂ ∂z 1 + z 3 ∂ ∂x 2 }.
Hence 1) . Now consider a point x ∈ M \ M. The distributions L = (span {dϕ 1 , dϕ 2 }) ⊥ and D (n-1) are of constant rank and they are well defined at any point of M. Assume that L(x) = (span {dϕ 1 (x), dϕ 2 (x)}) ⊥ ⊂ D (n-1) (x), then, because of constant ranks of L and D (n-1) , the inclusion does not hold at any x in an open neighborhood Õ of x. Clearly, M ∩ Õ = ∅ which gives a contradiction. In conclusion, we have L(x) = span {dϕ 1 (x), dϕ 2 (x)} ⊥ ⊂ D (n-1) (x) for any point x in M. Observe that D(x 0 ) ⊂ L(x 0 ) holds if and only if there exists a vector field g 1 ∈ D such that

L = (span {dϕ 1 , dϕ 2 }) ⊥ = (span {dz 1 , dz 2 }) ⊥ = span { ∂ ∂z n+2 , ∂ ∂z n+1 , . . . , ∂ ∂z 3 } ⊂ D (n-
(L g 1 ϕ 1 (x 0 ), L g 1 ϕ 2 (x 0 )) = (0, 0),
which is just the item (iii) of Theorem 1.3.2. This shows the equivalence of the two singular loci defined in Theorem 1.3.2 and Theorem 1.3.3 and proves, due to Theorem 1.3.2, the necessity of (iii) ′ .

Proof of Theorem 1.3.4

Proof: The results of Section 1.3.2 and Section 1.3.3 show that for a given arbitrary smooth function ϕ 1 such that L c ϕ 1 = 0, for any c ∈ C n-1 , and L g ϕ 1 (x 0 ) = 0, there always exists a function ϕ 2 , independent with ϕ 1 , such that (ϕ 1 , ϕ 2 ) is an x-flat output of Σ at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ). By Proposition 1.2.3, we can introduce new coordinates by z 1 = ϕ 1 , z 2 = ϕ 2 and complete them to a coordinate system (z 1 , . . . , z n+2 ) in which our original system Σ takes, via a feedback transformation, the chained form

Σ chain :                  ż1 = v 1 ż2 = z 3 • v 1 ż3 = z 4 • v 1 . . . żn+1 = z n+2 • v 1 żn+2 = v 2
Suppose that there exists another function φ2 such that (ϕ

1 , φ2 ) = (z 1 , φ2 ) is an x-flat output of Σ at (x 0 , u 0 ), u 0 ∈ U sing (x 0 ). Clearly, (ϕ 1 , φ2 ) is also an x-flat output of Σ chain at (0, v 0 ), v 0 ∈ U sing (0). Take g = ∂ ∂z 1 + z 3 ∂ ∂z 2 + • • • + z n+2 ∂ ∂z n+1
and let C n-1 be the characteristic distribution of D (n-1) , where D is the associated distribution of Σ chain . By the result of Section 1.3.2 and Section 1.3.3, the function φ2 must satisfy the equations

L c φ2 = 0, ∀ c ∈ C n-1 , L v φ2 = 0. where v = (L g ϕ 1 )[c n-1 , g] -(L [c n-1 ,g] ϕ 1 )g and c n-1 = ∂ ∂z 4
. Solving the above system of equations, we get φ2 = { φ2 (z 1 , z 2 ) :

∂ φ2 ∂z 2 (0) = 0}. Thus span {dϕ 1 , dϕ 2 }(z) = span {dz 1 , dz 2 }(z) = span {dz 1 , ∂ φ2 ∂z 1 (z)dz 1 + ∂ φ2 ∂z 2 (z)dz 2 }(z) = span {dϕ 1 , d φ2 }(z),
for any z in a neighborhood of 0. Correspondingly, for the original system Σ, the above equality is true for any point in a neighborhood of x 0 .

Let F lat(ϕ 1 ) be the codistribution spanned by the differentials of all x-flat outputs determined by function ϕ 1 , i.e., F lat(ϕ 1 ) = span {dϕ 1 , dϕ 2 }, where ϕ 2 is any function such that (ϕ 1 , ϕ 2 ) is an x-flat output. Clearly, F lat(ϕ 1 ) is well defined because of Theorem 1.3.4.

Corollary 1.4.4 If F lat(ϕ 1 ) = span {dϕ 1 , dϕ 2 }, then F lat(ϕ 1 ) = F lat(ϕ 2 )
This corollary is easy to prove and here we omit its proof.

Chapter 2

The geometry and flatness property of the n-bar system

Introduction

The well known n-trailer system was proposed by [START_REF] Laumond | Controllability of a multibody robot[END_REF] to model a unicycle-like robot towing trailers. This nonholonomic model has attracted a lot of attention and has been a source of inspiration to study its various properties: controllability ( [START_REF] Laumond | Controllability of a multibody robot[END_REF]), structure ( [START_REF] Jean | The car with n trailers: Characterisation of the singular configurations[END_REF], [START_REF] Mormul | Goursat flags: classification of codimension-one singularities[END_REF], [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF], [START_REF] Sørdalen | Conversion of the kinematics of a car with n trailers into a chained form[END_REF]), flatness ( [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], [START_REF] Jakubczyk | Invariants of dynamic feedback and free systems[END_REF]), motion planning and tracking ( [START_REF] Laumond | Robot Motion Planning and Control[END_REF], [START_REF] Murray | Nonholonomic motion planning: Steering using sinusoids[END_REF], [START_REF] Pasillas-Lépine | Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus[END_REF]), optimal control ( [START_REF] Laumond | Robot Motion Planning and Control[END_REF]), etc. In this chapter we propose its generalization, which we call the n-bar system, consisting of a "train" of n rigid bars subject to nonholonomic constraints (see a detailed description in Section 2.2.2 below). We study the geometry of the model of the n-bar system and prove that around any regular configuration (that is, none of the angles between two consecutive bars is ± π 2 ), the control system associated to the n-bar system is feedback equivalent to the m-chained form. This implies that the n-bar system is flat around any regular configuration and we show that the cartesian position of the source point of the last (from the top) bar is a flat output. We show also that all other minimal flat outputs are equivalent to that one. This is in contrast with the n-trailer system for which the position of the last trailer is also a flat output but there is a whole family of not equivalent flat outputs (parameterized by one function of three variables, see Chapter 1). As a by-product of our consideration we deduce the global controllability of the n-bar system since it is accessible at any (regular or not) configurations. We send the reader to [START_REF] Slayaman | Bras articulé et distributions multi-drapeaux[END_REF] and [START_REF] Slayaman | Articulated arm and special multi-flags (priprint)[END_REF] for another, although similar, model for the n-bar system (called there an articulated arm) and for a detailed analysis of singular configurations. This chapter is organized as follows. We define our model of the n-bar system in Section 2.2. We provide geometric notions and recall a characterization of Cartan distributions CC n (R, R m ) (m-chained forms) and given our first main result, equivalence of the n-bar system in R m+1 to the m-chained form, in Section 2.3 and its proofs in Section 2.4. The flatness property of the n-bar system is analyzed in Section 2.5. In Section 2.6, we give a proof of technical Lemma 2.4.1 which is used in Section 2.4.

n-bar system in R m+1

In this section, a model of the n-bar system will be proposed with which we will work throughout this chapter. In order to do that, we analyze first the model of a rigid bar moving in R 3 and then we will extend it to the general case of n-bar system moving in R m+1 .

Model of a rigid bar moving in R 3

Consider a rigid bar moving in R 3 which is described by a vector --→ P 0 P 1 , where

P i = (x 1 i , x 2 i , x 3
i ) for i = 0, 1. It it assumed that the endpoint P 1 of the bar can rotate freely when the source point P 0 remains fixed and that the bar moves in such a way that the direction of the instantaneous velocity of P 0 is parallel to the direction of the bar --→ P 0 P 1 . For simplicity, the length of the bar is assumed to be one. No matter how the bar moves, its position can always be determined uniquely by the source point P 0 and the endpoint P 1 . Therefore the configuration of the bar system can be described completely by x = (x 1 0 , x 2 0 , x 3 0 , x 1 1 , x 2 1 , x 3 1 ) ∈ R 6 . At the same time, the assumption of | --→ P 0 P 1 | = 1 implies that these six variables must satisfy the following holonomic constraint

ψ(x) = (x 1 1 -x 1 0 ) 2 + (x 2 1 -x 2 0 ) 2 + (x 3 1 -x 3 0 ) 2 -1 = 0 (2.2.1)
which is the equation of the unit sphere S 2 in R 3 centered at P 0 . The holonomic constraint (2.2.1) reduces the dimension of the configuration space of the bar system to five. In fact, if we consider only the rotation of the endpoint P 1 with respect to the source point P 0 , the former belongs always to the sphere S 2 with center P 0 and radius 1. Hence the configuration of the bar can be described completely by the position of the point P 0 ∈ R 3 together with the position of the point P 1 in a unite sphere S 2 . In other words, the true configuration space is the cartesian product Q = R 3 × S 2 , which is a submanifold of R 6 given by

Q = {x ∈ R 6 : ψ(x) = 0}.
Notice that the equation (2.2.1) implies that there exists an index j, for 1 ≤ j ≤ 3, such that x j 1x j 0 = 0, say j = 1. Then the assumption that the direction of the instantaneous velocity ( ẋ1 0 , ẋ2 0 , ẋ3 0 ) of P 0 is always parallel to --→ P 0 P 1 introduces two nonholonomic constraints:

(x 2 1 -x 2 0 ) ẋ1 0 -(x 1 1 -x 1 0 ) ẋ2 0 = 0 (x 3 1 -x 3 0 ) ẋ1 0 -(x 1 1 -x 1 0 ) ẋ3 0 = 0 which geometrically means that the velocity ( ẋ1 0 , ẋ2 0 , ẋ3 0 , ẋ1 1 , ẋ2 1 , ẋ3 1 )
⊤ is always annihilated by the two following differential 1-forms:

(x 2 1 -x 2 0 )dx 1 0 -(x 1 1 -x 1 0 )dx 2 0 (x 3 1 -x 3 0 )dx 1 0 -(x 1 1 -x 1 0 )dx 3 0 .
The distribution E on R 6 , annihilated by the above two differential forms, is given by

E = span {g 1 , • • • , g 4 },
where

g 1 = (x 1 1 -x 1 0 ) ∂ ∂x 1 0 + (x 2 1 -x 2 0 ) ∂ ∂x 2 0 + (x 3 1 -x 3 0 ) ∂ ∂x 3 0 g 2 = ∂ ∂x 1 1 , g 3 = ∂ ∂x 2 1 , g 4 = ∂ ∂x 3 1 ,
and it defines the control-linear (driftless, in other words) system on R 6

∆ bar : ẋ = 4 i=1 g i (x)v i , x ∈ R 6 ,
where u 1 , u 2 , u 3 , u 4 are four arbitrary functions of time can be interpreted as controls (and will do so throughout). In order to obtain a kinematic model of the rigid bar we have to constrain the system ∆ bar to its true configuration space Q. The crucial observation that the intersection T Q ∩ E defines a distribution D of constant rank equal to 3 on Q and thus gives rise to the kinematic model of the rigid bar moving in R 3 , Γ bar : q(t) ∈ D(q(t)), q ∈ R 3 × S 2 .

Locally, we can choose three independent vector fields f 1 , f 2 , f 3 on Q such that D = span {f 1 , f 2 , f 3 } which yields a local representation of the bar system

Γ bar : q = f 1 (q)u 1 + f 2 (q)u 2 + f 3 (q)u 3 , q ∈ R 3 × S 2 .
To summarize, the model of the rigid bar system Γ bar is defined by the system ∆ bar , together with the holonomic constraint

ψ(x) = (x 1 1 -x 1 0 ) 2 + (x 2 1 -x 2 0 ) 2 + (x 3 1 -x 3 0 ) 2 -1 = 0.
Notice that we do not express the distribution D explicitly, however, all its properties can be deduced by means of the distribution E and the holonomic constraint (2.2.1).

The configuration space R 3 × S 2 of the rigid bar system Γ bar can also be interpreted in a natural way through the classical rigid body theory. To analyze the motion of a rigid body, we choose an inertial reference frame (O, x, y, z) and a body reference frame (O ′ , x ′ , y ′ , z ′ ) that is fixed to move with the body (thus the point O ′ coincides with the source point P 0 of the rigid bar system). Then the position of the body is specified by the vector r = --→ O ′ O ∈ R 3 , along with the orientation of the orthogonal frame {x ′ , y ′ , z ′ } relative to {x, y, z}, which is determined by a (3 × 3) orthogonal matrix R belonging to the group of special orthogonal matrices

SO(3, R) = {R ∈ Gl(3, R) | RR ⊤ = I, det R = 1}.
Now consider the rigid bar moving in R 3 and let the origin O ′ be the source point P 0 . Notice that the bar --→ P 0 P 1 can be looked at as such a thin rigid body that its rotations are reduced to the rotations of the unit vector

--→ O ′ z ′ of the z ′ -axis, which is --→ P 0 P 1 = --→ O ′ z ′ .
The configuration space is thus the quotient SO(3)/SO(2) ∼ = S 2 with SO(2) corresponding to the rotations preserving the plane x ′ O ′ y ′ . We find again that the configuration of the bar system is described by the position of the point O ′ = P 0 ∈ R 3 together with the position of a point of a unit sphere S 2 , that is, the configuration space is R 3 × S 2 .

Model of the n-bar system moving in R m+1

In this section we will consider the n-bar system moving in R m+1 , as shown on Figure 2.2, and derive a kinematic model for it. It is assumed that all n components of the n-bar system are attached in such a way that P i is the source point of the (i+1)-th bar and simultaneously the endpoint of the i-th bar and that the instantaneous velocity of the point P i is parallel to the vector ----→ P i P i+1 , for 0 ≤ i ≤ n -1. Furthermore, each rigid bar is assumed to have length one. The coordinates of P i in R m+1 are given by

P i = (x 1 i , x 2 i , • • • , x m+1 i ), 0 ≤ i ≤ n.
x 1

x 2

x m+1 P 0

P 1 P 2 P n-2 P n-1 P n Figure 2.2: the n-bar system in R m+1
Clearly, the configuration of the n-bar system can be described completely by the (n + 1)(m + 1) coordinates

(x 1 0 , • • • , x m+1 0 , x 1 1 , • • • , x m+1 1 , • • • , x 1 n , • • • , x m+1 n ) ∈ X = R (n+1)(m+1) .
Due to the assumption | ----→

P i P i+1 | = 1, for 0 ≤ i ≤ n-1, we have the following holonomic constraints Ψ(x) = 0, where Ψ = {Ψ 1 , • • • , Ψ n } : X = R (n+1)(m+1) → R n is given by            Ψ 1 (x) = (x 1 1 -x 1 0 ) 2 + (x 2 1 -x 2 0 ) 2 + • • • + (x m+1 1 -x m+1 0 ) 2 -1 Ψ 2 (x) = (x 1 2 -x 1 1 ) 2 + (x 2 2 -x 2 1 ) 2 + • • • + (x m+1 2 -x m+1 1 ) 2 -1 . . . Ψ n (x) = (x 1 n -x 1 n-1 ) 2 + (x 2 n -x 2 n-1 ) 2 + • • • + (x m+1 n -x m+1 n-1 ) 2 -1.
(2.2.2)

Under these n holonomic constraints, the true configuration space of the n-bar system becomes the regular embedded submanifold

Q = R m+1 × (S m ) n ⊂ X defined by Q = {x ∈ X : Ψ(x) = 0}.
Moreover, the equation Ψ(x) = 0 implies that for any 1 ≤ i ≤ n, there always exists

1 ≤ σ(i) ≤ m + 1, such that x σ(i) i -x σ(i)
i-1 = 0. Now the assumption that the instantaneous velocity of the point P i is parallel to the vector ----→ P i P i+1 , for 0 ≤ i ≤ n -1, imposes the following nonholonomic constraints on the n-bar system: the velocity of the system along any trajectory is annihilated by the following differential 1-forms

Ω j i = (x j i -x j i-1 )dx σ(i) i-1 -(x σ(i) i -x σ(i) i-1 )dx j i-1 , for 1 ≤ i ≤ n, 1 ≤ j ≤ m + 1 and j = σ(i).
The distribution E annihilated by all forms Ω j i is given by

E = i,j
ker Ω j i = span {g 1 , . . . , g n+m+1 },

where

g 1 = (x 1 1 -x 1 0 ) ∂ ∂x 1 0 + • • • + (x m+1 1 -x m+1 0 ) ∂ ∂x m+1 0 g 2 = (x 1 2 -x 1 1 ) ∂ ∂x 1 1 + • • • + (x m+1 2 -x m+1 1 ) ∂ ∂x m+1 1 . . . g n = (x 1 n -x 1 n-1 ) ∂ ∂x 1 n-1 + • • • + (x m+1 n -x m+1 n-1 ) ∂ ∂x m+1 n-1 g n+i = ∂ ∂x i n , 1 ≤ i ≤ m + 1, (2.2.3)
which defines the control-linear system on X = R (n+1)(m+1)

∆ : ẋ = n+m+1 i=1 g i (x)v i , x ∈ X. (2.2.4)
To obtain a kinematic model of the n-bar system we have to constrain the system ∆ to the regular submanifold Q ⊂ X. Consider the embedding Φ : Q → X such that Φ(q) = q, for any q ∈ Q. Let J be the codistribution spanned by all differential forms Ω j i , i.e., J = span

{Ω j i , 1 ≤ i ≤ n, 1 ≤ j ≤ m + 1, j = σ(i)}.
(2.2.5)

Clearly J = E ⊥ and the pull back Φ * maps J into a codistribution I on Q,

I = Φ * J = span {ω j i , 1 ≤ i ≤ n, 1 ≤ j ≤ m + 1, j = σ(i)}, (2.2.6)
where ω j i = Φ * Ω j i . Define a distribution D by D = I ⊥ . Notice that D is just the intersection T Q ∩ E and is of constant rank equal to m + 1 (see Section 2.4.2 for details) and thus gives rise to a control-linear system

Γ : q = m i=0 f i (q)u i , q ∈ R m+1 × (S m ) n , (2.2.7)
where locally D = span {f 0 , . . . , f m }, which describes completely the n-bar system moving in R m+1 . To summarize, the model Γ describing the n-bar moving in R m+1 , for m ≥ 1, is defined by the control-linear system △, given by (2.2.3)-(2.2.4), together with the following holonomic constraint

Ψ(x) = (Ψ 1 (x), . . . , Ψ n (x)) = 0,
where Ψ i , 1 ≤ i ≤ n, are defined by (2.2.2). Notice that, like in the case of R 3 described in Section 2.2.1, we give explicit expression neither for the distribution D nor for the vector fields f i , for 0 ≤ i ≤ m. All their properties will be formulated and analyzed in terms of the distribution E and the holonomic constraint Ψ(x) = 0.

Another, although similar model for the n-bar system (called articulated arm) has been very recently introduced and studied in [START_REF] Slayaman | Bras articulé et distributions multi-drapeaux[END_REF], and [START_REF] Slayaman | Articulated arm and special multi-flags (priprint)[END_REF]; in the former a detailed analysis of the singular locus (see also section 2.3.2) has been performed.

The presented model of the n-bar system in R m+1 is a natural generalization of the well known n-trailer system on R 2 . The latter is a model for a unicycle-like mobile robot towing n-trailers such that the tow hook of each trailer is located at the center of its unique axle (with the assumption that the distances between any two consecutive trailers are equal). The n-trailer system is subject to nonholonomic constraints: it is assumed that the wheels of each individual trailer are aligned with the body and are not allowed to slip [START_REF] Laumond | Controllability of a multibody robot[END_REF]. This model and its control properties have attracted a lot of attention (see the books [START_REF] Laumond | Robot Motion Planning and Control[END_REF] and [START_REF] Li | Nonholonomic Motion Plannging[END_REF]; and the papers [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], [START_REF] Jean | The car with n trailers: Characterisation of the singular configurations[END_REF], [START_REF] Murray | Nonholonomic motion planning: Steering using sinusoids[END_REF], [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF], [START_REF] Sørdalen | Conversion of the kinematics of a car with n trailers into a chained form[END_REF]).

Clearly the nonholonomic constraint that the wheel cannot slip on the plane R 2 can be equivalently rephrased that the instantaneous velocity of the middle point of the i-th trailer axle, say point P i , is parallel to the vector ----→ P i P i+1 joining the two consecutive axles. These are exactly our nonholonomic constraints imposed for the n-bar system, the only difference being to allow the vectors ----→ P i P i+1 to move in R m+1 and not on the plane R 2 .

Equivalence of the

D (0) = D and D (i+1) = D (i) + [D (i) , D (i) ], for i ≥ 0.
The Lie flag of D is the sequence of modules of vector fields

D 0 ⊂ D 1 ⊂ • • • defined inductively by D 0 = D and D i+1 = D i + [D 0 , D i ], for i ≥ 0.
In general, the derived and Lie flags are different; though for any point p in the underlying manifold the inclusion D i (p) ⊂ D (i) (p) holds, for i ≥ 0.

Two distributions D and D defined on two manifolds M and M, respectively, are equivalent if there exists a smooth diffeomorphism ϕ between M and M such that (ϕ * D)(p) = D(p), for each point p in M.

An alternative description of the above definitions can also be given using the dual language of differential forms. A codistribution I of rank s on a smooth manifold M (or a Pfaffian system) is a map that assigns smoothly to each point p in M a linear subspace I(p) ⊂ T * p M of dimension s. Such a field of cotangent s-planes is spanned locally by s pointwise linearly independent smooth differential 1-forms ω 1 , . . . , ω s on M, which will be denoted by I = span {ω 1 , . . . , ω s }. In the case of constant rank we can (locally) identity codistribution and Pfaffian system which we will do throughout this chapter. Two codistributions (Pfaffian systems) I and Ĩ defined on two manifolds M and M , respectively, are equivalent if there exists a smooth diffeomorphism ϕ between M and M such that

I(p) = (ϕ * Ĩ)(p) for each point p in M.
For a codistribution I, its derived flag I (0) ⊃ I (1) ⊃ • • • can be defined by

I (0) = I and I (i+1) = {ω ∈ I (i) : dω ≡ 0 mod I (i) }, for i ≥ 0,
provided that each element I (i) of this sequence has constant rank. In this case, it is immediate to see that the derived flag of the distribution D = I ⊥ coincides with the sequence of distributions that annihilate the elements of the derived flag of I, that is

D (i) = (I (i) ) ⊥ , for i ≥ 0.
Consider J n (R, R m ), the space of n-jets of smooth maps from R into R m and denote its canonical coordinates by x 0 0 , x 0 1 , . . . , x 0 m , x 1 1 , . . . , x 1 m , . . . , x n 1 , . . . , x n m , where x 0 0 represents the independent variable and x 0 i for 1 ≤ i ≤ m, represent the dependent variables, and x j i , for 1 ≤ i ≤ m and 1 ≤ j ≤ n, correspond to the ordinary derivatives

d j x 0 i d(x 0 0 ) j . Any smooth map ϕ from R into R m defines a submanifold in J n (R, R m ) ∼ = R (n+1)m+1 by the relation x j i = ϕ (j)
i (x 0 0 ), for 1 ≤ i ≤ m and 0 ≤ j ≤ n, where ϕ (j) i (x 0 0 ) denotes the j-th derivative with respect to x 0 0 of the i-th component ϕ i of ϕ. This submanifold is called the n-graph of ϕ.

The Cartan distribution on J n (R, R m ), denote by CC n (R, R m ), is the completely nonholonomic distribution spanned by the following family of vector fields

∂ ∂x 0 0 + n-1 j=0 m i=1 x j+1 i ∂ ∂x j i , ∂ ∂x n 1 , . . . , ∂ ∂x n m
or, equivalently, annihilated by the following family of differential forms:

dx j i -x j+1 i dx 0 0 , 0 ≤ j ≤ n -1, 1 ≤ i ≤ m.
It turns out that all n -graphs are integral submanifolds of dimension 1, that is integral curves, of the Cartan distribution CC n (R, R m ).

The problem of characterizing distributions that are locally equivalent to the Cartan distribution CC n (R, R m ) has been studied and solved in the following way by Pasillas-Lépine and Respondek [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF] (see also [START_REF] Mormul | Goursat flags: classification of codimension-one singularities[END_REF], [START_REF] Pasillas-Lépine | Contact systems and corank one involutive subdistributions[END_REF], [START_REF] Yamaguchi | Contact geometry of higher order[END_REF]). Recall that the Engel rank, at a point p, of a codistribution I = span {ω 1 , • • • , ω s } is the largest integer ρ such that there exists a 1-form α in I for which ((dα

) ρ ∧ ω 1 ∧ • • • ∧ ω s )(p) = 0. Theorem 2.3.1 A distribution D of rank m + 1, with m ≥ 2,
on a manifold M of dimension (n + 1)m + 1 is equivalent, in a small enough neighborhood of a point p in M, to the Cartan distribution CC n (R, R m ) if and only if the following conditions hold:

(i) D (n) (p) = T p M; (ii) D (n-1
) is of constant rank nm+1 and contains an involutive subdistribution L n-1

that has constant corank one in

D (n-1) ; (iii) D(p) is not contained in L n-1 (p).
Moreover, if m ≥ 3, L n-1 exists if and only if the Engel rank of (D (n-1) ) ⊥ equals one and rank C(D (n-1) ) = (n -1)m and is given as

L n-1 = F 1 + • • • + F m ,
where

F i = {f ∈ D (n-1) : f dω i ∈ (D (n-1) ) ⊥ } and ω i 's are any differential 1-forms such that I (n-1) = (D (n-1) ) ⊥ = span{ω 1 , . . . , ω m }.
Remarks 1. The involutive subdistribution L n-1 , whose existence is claimed by (ii), is unique (if it exists) and it will be called the canonical involutive subdistribution in D (n-1) . The uniqueness and involutivity of L n-1 follow from a result of Bryant [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF] and have been shown in [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF]. Remarks 2. Item (i) and (ii) describe the essential geometric property of distributions equivalent to the Cartan distribution CC n (R, R m ) while the condition (iii) distinguishes regular points p at which D(p) ⊂ L n-1 (p) form singular points, where this last condition is violated.

The case m = 1 is excluded from Theorem 2.3.1 because if an involutive subdistribution of corank one L n-1 ⊂ D (n-1) exists it cannot be unique and therefore there is no a canonical one. However, a "non-canonical" version of Theorem 2.3.1 holds for m = 1 as well, as proved in [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF]: a rank-two distribution is equivalent to the Cartan distribution CC n (R, R), called also a chained normal form or Goursat normal form if and only if there exists a distribution L n-1 satisfying the conditions (i), (ii) and (iii) of Theorem 2.

Σ : ẋ = m i=0 f i (x)u i = f (x)u, x ∈ M, and
Σ : ẋ = m i=0 fi (x)ũ i = f (x)ũ, x ∈ M ,
where u = (u 0 , . . . , u m ) ⊤ ∈ R m+1 , ũ = (ũ 0 , . . . , ũm ) ⊤ ∈ R m+1 and the rows f = (f 0 , . . . , f m ) and f = ( f0 , . . . , fm ) are formed by C ∞ -smooth vector fields f i and fi , 0 ≤ i ≤ m, on M and M , respectively. We say that Σ and Σ are feedback equivalent if there exists a diffeomorphism ϕ : M → M , x = ϕ(x), and a feedback transformation u = β(x)ũ where β(x) is an invertible C ∞ -smooth (m + 1) × (m + 1)-matrix such that

Dϕ(x) • f (x)β(x) = f (ϕ(x)).
Definition 2.3.2 An (m + 1)-input driftless control system Σ : ẋ = m i=0 u i f i (x), defined on R (n+1)m+1 , is said to be in the m-chained form if it is represented by

ẋ0 0 = u 0 ẋ0 1 = x 1 1 u 0 • • • ẋ0 m = x 1 m u 0 • • • • • • • • • ẋn-1 1 = x n 1 u 0 • • • ẋn-1 m = x n m u 0 ẋn 1 = u 1 • • • ẋn m = u m .
A system in the m-chained form is also called the canonical contact system on J n (R, R m ). In fact, the vector fields f 0 , . . . , f m of the m-chained form coincide with those generating the Cartan distribution CC n (R, R m ) given in Section 2.3.1.

To any control-linear system Σ, we associate the distribution spanned by all its vector fields D Σ = span {f 0 , . . . , f m }. The (local) feedback equivalence of Σ and Σ coincides with the (local) equivalence of the associated distributions D Σ and D Σ.

Therefore the statement that a control system Σ is locally feedback equivalent to the m-chained form (equivalently, to the canonical contact system on J n (R, R m )) will always mean that the associated distribution D Σ is locally equivalent to the Cartan distribution CC n (R, R m ).

In this section we will formulate our first main result. See [START_REF] Slayaman | Bras articulé et distributions multi-drapeaux[END_REF] and [START_REF] Slayaman | Articulated arm and special multi-flags (priprint)[END_REF] for another approach to the problem of equivalence of the n-bar system (called there the articulated arm system) to the Cartan distribution (and, more generally, to the multi-flag system).

Theorem 2.3.3 The n-bar system Γ moving in R m+1 , for m ≥ 1, defined by (2.2.7), is locally feedback equivalent to the m-chained form at any point x ∈ X = R (n+1)(m+1) satisfying Ψ(x) = 0 (that is, at x corresponding to a point q ∈ Q) such that

(R1) m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, for 1 ≤ i ≤ n -1 , if m ≥ 2; (R2) m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, for 2 ≤ i ≤ n -1 , if m = 1.
Moreover, at any point q ∈ Q (equivalently, at any point x ∈ X = R (n+1)(m+1) satisfying Ψ(x) = 0), the n-bar system satisfies the condition (i) and (ii) of Theorem 2.3.1.

Remark 1. Let D Γ be the distribution associated to the n-bar system Γ. Define the regular locus Reg(Γ) of Γ as Reg(Γ) = Reg(D Γ ). Then Theorem 2.3.3 implies that the regular locus of Γ is given by

Reg(Γ) m≥2 = {x ∈ X : Ψ(x) = 0, m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, 1 ≤ i ≤ n -1} Reg(Γ) m=1 = {x ∈ X : Ψ(x) = 0, m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, 2 ≤ i ≤ n -1}.
The condition Ψ(x) = 0 implies that x is a point of Q while the conditions (R1) and (R2) identify the regular points of Γ in Q. It is obvious that Reg(Γ) is open and dense in the configuration space Q for both m ≥ 2 and m = 1. Remark 2. The regularity condition m+1 j=1 (x j ix j i-1 )(x j i+1x j i ) = 0 has a clear interpretation for the n-bar system. Let θ i , for 1 ≤ i ≤ n -1, denote the angles of the (i + 1)-th bar with respect to the i-th bar, i.e., the angles between the vectors ----→ P i-1 P i and ----→ P i P i+1 on the plane spanned by them. Then clearly the regularity conditions mean that θ i are different from ± π 2 , in other words, the i-th bar is not perpendicular to the (i + 1)-th one. Using the angles θ i , the regular locus can be rewritten as

Reg(Γ) m≥2 = {q ∈ Q : θ i = ± π 2 , 1 ≤ i ≤ n -1} Reg(Γ) m=1 = {q ∈ Q : θ i = ± π 2 , 2 ≤ i ≤ n -1}.
It is interesting to observe the difference between the planar (m = 1) and all other cases (m ≥ 2). Namely, the angle ± π 2 between the last two bars (the two most far from the controlled one) is a singularity for m ≥ 2 but is not for the planar case. The latter implies, in particular, that the 2-bar system in R 2 is transformable into the chained form even if the bars are perpendicular. This is not true any longer if we consider the 2-bar system in the space R m+1 , m ≥ 2 (in R 3 , for instance). Of course, the 2-bar system in R 2 is just the 1-trailer system (a unicycle-like mobile robot towing one trailer or, equivalently, a nonholonomic car) and it is well known that the system can be brought into the chained form even if the axles are perpendicular. In other words, the rank 2 distributions on 4-dimensional manifolds with the growth vector (2, 3, 4) have no singularities, a result that goes back to [START_REF] Engel | Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen[END_REF]. Remark 3. The property of controllability of the n-bar system can also be obtained from Theorem 2.3.3. Indeed, we have the following corollary.

Corollary 2.3.4 The n-bar system Γ is globally controllable on

Q = R m+1 × (S m ) n .

Proof of Theorem 2.3.3

It is well known that (see, e.g., [START_REF] Jean | The car with n trailers: Characterisation of the singular configurations[END_REF], [START_REF] Murray | Nonholonomic motion planning: Steering using sinusoids[END_REF], [START_REF] Pasillas-Lépine | On the geometry of Goursat structures[END_REF], [START_REF] Sørdalen | Conversion of the kinematics of a car with n trailers into a chained form[END_REF] and Chapter 1 of this thesis) that the n-trailer system (that is, the n-bar system on R m+1 for m = 1) is locally feedback equivalent to the chained form (that is, its associated distribution is equivalent to the Cartan distribution CC n (R, R)) if and only if the angles between the consecutive trailers are not ±π/2, except for the angle between the last two ones (equivalently, except for the angle between --→ P 0 P 1 and --→ P 1 P 2 ). Therefore, we will prove only the case m ≥ 2 here. The proof of Theorem 2.3.3, for m ≥ 2, will use Theorem 2.3.1. First, the conditions (i) and (ii) of Theorem 2.3.1 will be verified for the n-bar system Γ. Secondly, we will prove that the regular locus Reg(Γ) m≥2 is just the set of the points which satisfy the condition (R1) of Theorem 2.3.3. For simplicity, we denote the associated distribution D Γ by D and clearly we have D ⊥ = I where I is defined by (2.2.6).

Notations

For convenience, we express J , the codistribution given by (2.2.5), in the following form

J = span {Ω j i , 1 ≤ i ≤ n, 1 ≤ j ≤ m + 1, j = σ(i)} = span {Ω 1 , Ω 2 , . . . , Ω n },
where

Ω i = (Ω 1 i , . . . , Ω σ(i)-1 i , Ω σ(i)+1 i , . . . , Ω m+1 i ), for 1 ≤ i ≤ n.
Similarly, the codistribution I, given by (1.6), is expressed as

I = span {ω j i , 1 ≤ i ≤ n, 1 ≤ j ≤ m + 1, j = σ(i)} = span {ω 1 , ω 2 , . . . , ω n },
where ω i = (ω 1 i , . . . , ω

σ(i)-1 i , ω σ(i)+1 i , . . . , ω m+1 i ), for 1 ≤ i ≤ n.
The following symbols will be used in this chapter: [START_REF] Altafini | Some properties of the general n-Trailer[END_REF]

dX i = (dx 1 i , . . . , dx m+1 i ) (2) ∧dX i = dx 1 i ∧ • • • ∧ dx m+1 i (3) ∧d Xσ(i) i = dx 1 i ∧ • • • ∧ dx σ(i)-1 i ∧ dx σ(i)+1 i • • • ∧ dx m+1 i (4) ∧Ω i = Ω 1 i ∧ • • • ∧ Ω σ(i)-1 i ∧ Ω σ(i)+1 i ∧ • • • ∧ Ω m+1 i (5) ∧ω i = ω 1 i ∧ • • • ∧ ω σ(i)-1 i ∧ ω σ(i)+1 i ∧ • • • ∧ ω m+1 i .

Condition (i):

D (n) = T Q Let Λ k (Q) denote the space of smooth differential k-forms on Q and put Λ(Q) = ⊕Λ k (Q). An ideal of Λ(Q) is a subspace V of Λ k (Q) such that if α ∈ V , then α ∧ β ∈ V for any β ∈ Λ(Q).
Lemma 2.4.1 Consider the codistribution I defined by (2.2.6) and let (Φ * dX i ) be the ideal in Λ(Q) generated by Φ * dx 1 i , . . . , Φ * dx m+1 i , then we have:

(a) dω j i ∧ ω i ∧ ω i+1 = 0, for 1 ≤ i ≤ n -1, 1 ≤ j ≤ m + 1, and j = σ(i); (b) ∧ω i = (-1) m (x σ(i) i -x σ(i-1) i ) m-2 (∧Φ * d Xσ(i) i-1 ) mod (Φ * dX i ), for 1 ≤ i ≤ n -1; (c) dω j i ∧ ω i ∧ • • • ∧ ω 1 = 0, for 1 ≤ i ≤ n, 1 ≤ j ≤ m + 1 and j = σ(i) .
The proof of Lemma 2.4.1 is quite tedious and is given in Section 2.6. Below we will show how Lemma 2.4.1 implies the condition D (n) = T Q.

If i = n, the condition (c) shows that dω j n ∧ ω 1 ∧ • • • ∧ ω n = 0, for 1 ≤ j ≤ m + 1 and j = σ(n), which imply that all differential forms ω 1 , ω 2 , . . . , ω n are independent. Recall that each ω i consists of m differential 1-forms. Therefore I is of constant rank nm and thus

rank D = dim Q -rank I = (n + 1)m + 1 -nm = m + 1.
Observe that

dω j n = Φ * dΩ j n = Φ * dx j n ∧ Φ * dx σ(n) n-1 -2Φ * dx j n-1 ∧ Φ * dx σ(n) n-1 -Φ * dx σ(n) n ∧ Φ * dx j n-1 .
It is obvious that the differential forms dω 1 n , . . . , dω

σ(n)-1 n , dω σ(n)+1 n
, . . . , dω m+1 n are linearly independent on Q. Consequently, all the differential formes

dω j n ∧ ω n ∧ • • • ∧ ω 1 ,
for 1 ≤ j ≤ m + 1 and j = σ(n), are also linearly independent on Q. From condition (a), we get easily that

dω j i ∧ ω 1 ∧ • • • ∧ ω n = 0, 1 ≤ i ≤ n -1, 1 ≤ j ≤ m + 1
, and j = σ(i). (2.4.1) Recall that the derived codistribution I (1) is given by

I (1) = {ω ∈ I : dω ≡ 0 mod I}.
For any ω ∈ I, we have

ω = n i=1 m+1 j=1,j =σ(i) α i j • ω j i ,
where

α i j ∈ C ∞ (Q). Applying (2.4.1), we deduce dω ∧ ω 1 ∧ • • • ∧ ω n = d   n i=1 m+1 j=1,j =σ(i) α i j • ω j i   ∧ ω 1 ∧ • • • ∧ ω n = m+1 j=1, j =σ(n) α n j • dω j n ∧ ω 1 ∧ • • • ∧ ω n .
The differential forms dω j n ∧ ω n ∧ • • • ∧ ω 1 , for 1 ≤ j ≤ m + 1 and j = σ(n), are linearly independent and thus dω ∧ ω 1 ∧ • • • ∧ ω n = 0 implies that all coefficients α n j must vanish, i.e., α n j = 0, for 1 ≤ j ≤ m + 1, and j = σ(n). This shows that, indeed, any ω ∈ I (1) can be always expressed by a linear combination of the differential forms ω 1 , ω 2 , . . . , ω n-1 . In other words, we have

I (1) = span {ω 1 , ω 2 , . . . , ω n-1 }.
In the same way, it can be shown that I (2) = span {ω 1 , ω 2 , . . . , ω n-2 } . . .

I (n-1) = span {ω 1 } I (n) = 0.
The expressions of the derived flag

I (0) ⊃ • • • ⊃ I (n)
show immediately that rank I (i) = (ni)m, and hence rank D (i) = (i + 1)m + 1. In particular, rank

D (n) = (n + 1)m + 1 which implies that D (n) = T Q. 2.4.3 Condition (ii) : D (n-1) contains a corank one involutive subdistribution L n-1 ⊂ D (n-1)
Define a distribution K on X = R (n+1)(m+1) by

K = span { ∂ ∂x 1 1 , . . . , ∂ ∂x m+1 1 , . . . , ∂ ∂x 1 n , . . . , ∂ ∂x m+1 n } ⊂ T X.
Let L be the distribution on Q defined by L ⊥ = Φ * (K ⊥ ) , i.e.,

K -→ (K) ⊥ Φ * -→ L ⊥ -→ L Lemma 2.4.2
The distribution L is involutive and contained in D (n-1) .

Proof: Since K is involutive and Φ * (K ⊥ ) = L ⊥ , the involutivity of L is evident. Let J n-1 be the codistribution defined on X = R (n+1)(m+1) by

J n-1 = span {Ω j 1 , 1 ≤ j ≤ m + 1, j = σ(1)},
where

Ω j 1 = (x j 1 -x j 0 )dx σ(1) 0 -(x σ(1) 1 -x σ(1) 0 )dx j 0 . Since K ⊥ = span {dx 1 0 , . . . , dx m+1 0 }, it is obvious that J n-1 ⊂ K ⊥ .
Recall that we showed in Section 2.4.2

I (n-1) = span {ω j 1 , 1 ≤ j ≤ m + 1, j = σ(1)} = Φ * J n-1 .
Then clearly we have Φ * (J n-1 ) ⊂ Φ * (K ⊥ ) and therefore 1) .

I (n-1) ⊂ L ⊥ which implies that L ⊂ D (n-1) . Lemma 2.4.3 L is of constant corank one in D (n-
Proof: Observe that

L ⊥ = Φ * (K ⊥ ) = span {Φ * dx 1 0 , . . . , Φ * dx m+1 0 }. Since Φ * dx 1 0 ∧ • • • ∧ Φ * dx m+1 0 = 0, we have rank L ⊥ = m + 1 and thus rank L = dim Q -rank L ⊥ = (n + 1)m + 1 -(m + 1) = nm.
In Subsection 2.4.2, we proved rank D (n-1) = nm + 1 which yields corank (L ⊂ D (n-1) ) = 1.

Define L n-1 = L, then the condition (ii) of Theorem 2.3.1 holds.

2.4.4

The regular locus of the n-bar system in R m+1

Proposition 2.4.4 The regular locus of the n-bar system Γ in R m+1 , for m ≥ 2, is given by

Reg(Γ) m≥2 = {x ∈ X : Ψ(x) = 0, m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, 1 ≤ i ≤ n -1}.
For simplicity of exposition, we assume throughout this section that σ(i) = m+1, for 1 ≤ i ≤ n, i.e.,

x m+1 i -x m+1 i-1 = 0, 1 ≤ i ≤ n.
We can always achieve this by a suitable choice of coordinates. Under this assumption, the codistribution

J = E ⊥ on R (n+1)(m+1) is J = span {Ω j i , 1 ≤ i ≤ n, 1 ≤ j ≤ m}, where Ω j i = (x m+1 i -x m+1 i-1 ) dx j i-1 -(x j i -x j i-1
) dx m+1 i-1 . In Section 2.4.3, the subdistribution L n-1 satisfying the conditions (i) and (ii) of Theorem 2.3.1 was constructed. Since m ≥ 2, L n-1 is unique (see [START_REF] Pasillas-Lépine | On the geometry of control systems equivalent to canonical contact systems: regular points, singular points and flatness[END_REF]). Therefore the singular locus of the system Γ is defined by

Sing(Γ) = {q ∈ Q : D(q) ⊂ L n-1 (q)}. From the relation (L n-1 ) ⊥ = Φ * K ⊥ (defining L n-1 = L, see Section 2.4.3) and I = D ⊥ = Φ * E ⊥ = Φ * J (see Section 2.2.
2), we conclude that the singular locus of the n-bar system Γ can be expressed as

Sing(Γ) = {q ∈ Q : D(q) ⊂ L n-1 (q)} = {q ∈ Q : (L n-1 ) ⊥ (q) ⊂ I(q)} = {q ∈ Q : Φ * K ⊥ (q) ⊂ Φ * J (q)}.
Notice that Φ * K ⊥ = span {Φ * dx 1 0 , . . . , Φ * dx m+1 0 }, and

Φ * J = span {Φ * Ω j i , 1 ≤ i ≤ n, 1 ≤ j ≤ m},
where Φ * Ω j i = (x m+1 i -x m+1 i-1 )Φ * dx j i-1 -(x j i -x j i-1 )Φ * dx m+1 i-1 . Since L n-1 and I are both of constant rank, we have immediately that the relation Φ * K ⊥ (q) ⊂ Φ * J (q) holds if and only if

Φ * dx 1 0 ∧ • • • ∧ Φ * dx m+1 0 ∧ Φ * Ω 1 ∧ • • • ∧ Φ * Ω n = 0. (2.4.2)
By a straightforward calculation, it is easy to see that (2.4.2) is equivalent to

Φ * dx 1 0 ∧ • • • ∧ Φ * dx m+1 0 ∧ Φ * Ω 2 ∧ • • • ∧ Φ * Ω n = 0. Denoting S = Φ * dx 1 0 ∧ • • • ∧ Φ * dx m+1 0 ∧ Φ * Ω 2 ∧ • • • ∧ Φ * Ω n , we thus have Sing(Γ) = {q ∈ Q : S(q) = 0}.
Lemma 2.4.5 Define the differential form Θ k by

Θ k = Φ * dx 1 k ∧ • • • ∧ Φ * dx m+1 k ∧ Φ * Ω k+2 ,
then we have

Θ k = (-1) m (x m+1 k+2 -x m+1 k+1 ) m-1 • λ k+1 • (∧Φ * d Xm+1 k ) ∧ Φ * dx 1 k+1 ∧ • • • ∧ Φ * dx m+1 k+1 , for 0 ≤ k ≤ n -2, where ∧Φ * d Xm+1 k = Φ * dx 1 k ∧ • • • ∧ Φ * dx m k and λ k+1 = (x m+1 k+2 -x m+1 k+1 ) + x 1 k+1 -x 1 k x m+1 k+1 -x m+1 k (x 1 k+2 -x 1 k+1 ) + • • • + x m k+1 -x m k x m+1 k+1 -x m+1 k (x m k+2 -x m k+1 ).
Proof: Differentiating the relation

Ψ k+1 = (x 1 k+1 -x 1 k ) 2 + (x 2 k+1 -x 2 k ) 2 + • • • + (x m+1 k+1 -x m+1 k ) 2 -1 = 0 we obtain dx m+1 k = dx m+1 k+1 + x 1 k+1 -x 1 k x m+1 k+1 -x m+1 k (dx 1 k+1 -dx 1 k ) + • • • + x m k+1 -x m k x m+1 k+1 -x m+1 k (dx m k+1 -dx m k ).
Therefore, on the configuration space Q, we have

Φ * dx m+1 k = Φ * dx m+1 k+1 + x 1 k+1 -x 1 k x m+1 k+1 -x m+1 k (Φ * dx 1 k+1 -Φ * dx 1 k )+• • •+ x m k+1 -x m k x m+1 k+1 -x m+1 k (Φ * dx m k+1 -Φ * dx m k ).
Recall that Ω j k+2 = (x m+1 k+2x m+1 k+1 )dx j k+1 -(x j k+2x j k+1 )dx m+1 k+1 , for 1 ≤ j ≤ m, and substituting the latter into Θ k , we get

Θ k = Φ * dx 1 k ∧ • • • ∧ Φ * dx m+1 k ∧ Φ * Ω k+2 = Φ * dx 1 k ∧ • • • ∧ Φ * dx m k ∧ (Φ * dx m+1 k+1 + x 1 k+1 -x 1 k x m+1 k+1 -x m+1 k Φ * dx 1 k+1 + • • • + x m k+1 -x m k x m+1 k+1 -x m+1 k Φ * dx m k+1 ) ∧ Φ * Ω 1 k+2 ∧ • • • ∧ Φ * Ω m k+2 = (x m+1 k+2 -x m+1 k+1 ) m • (∧d Xm+1 k ) ∧ Φ * dx m+1 k+1 ∧ Φ * dx 1 k+1 ∧ • • • ∧ Φ * dx m k+1 -(x m+1 k+2 -x m+1 k+1 ) m-1 (x 1 k+2 -x 1 k+1 ) x 1 k+1 -x 1 k x m+1 k+1 -x m+1 k • (∧d Xm+1 k ) ∧Φ * dx 1 k+1 ∧ Φ * dx m+1 k+1 ∧ Φ * dx 2 k+1 ∧ • • • ∧ Φ * dx m k+1 -(x m+1 k+2 -x m+1 k+1 ) m-1 (x 2 k+2 -x 2 k+1 ) x 2 k+1 -x 2 k x m+1 k+1 -x m+1 k • (∧d Xm+1 k ) ∧Φ * dx 2 k+1 ∧ Φ * dx 1 k+1 ∧ Φ * dx m+1 k+1 ∧ Φ * dx 3 k+1 ∧ • • • Φ * dx m-1 k+1 ∧ Φ * dx m k+1 -• • • -(x m+1 k+2 -x m+1 k+1 ) m-1 (x m k+2 -x m k+1 ) x m k+1 -x m k x m+1 k+1 -x m+1 k • (∧d Xm+1 k ) ∧Φ * dx m k+1 ∧ Φ * dx 1 k+1 ∧ • • • ∧ Φ * dx m-1 k+1 Φ * dx m+1 k+1 = (-1) m (x m+1 k+2 -x m+1 k+1 ) m-1 • λ k+1 • (∧Φ * d Xm+1 k ) ∧ Φ * dx 1 k+1 ∧ • • • ∧ Φ * dx m+1 k+1
where

λ k+1 = (x m+1 k+2 -x m+1 k+1 ) + x 1 k+1 -x 1 k x m+1 k+1 -x m+1 k (x 1 k+2 -x 1 k+1 ) + • • • + x m k+1 -x m k x m+1 k+1 -x m+1 k (x m k+2 -x m k+1 ).
Proof of Proposition 2.4.4: Applying lemma 2.4.5, we obtain

S = Φ * dx 1 0 ∧ • • • ∧ Φ * dx m+1 0 ∧ Φ * Ω 2 ∧ • • • ∧ Φ * Ω n = Θ 0 ∧ Φ * Ω 3 ∧ • • • ∧ Φ * Ω n = (-1) m (x m+1 2 -x m+1 1 ) m-1 • λ 1 • (∧Φ * d Xm+1 0 ) ∧ Φ * dx 1 1 ∧ • • • ∧ Φ * dx m 1 ∧ Φ * dx m+1 1 ∧Φ * Ω 3 ∧ • • • ∧ Φ * Ω n = (-1) m (x m+1 2 -x m+1 1 ) m-1 • λ 1 • (∧Φ * d Xm+1 0 ) ∧ Θ 1 ∧ Φ * Ω 4 ∧ • • • ∧ Φ * Ω n = (-1) 2m 2 i=1 (x m+1 i+1 -x m+1 i ) m-1 • (λ 1 • λ 2 ) • (∧Φ * d Xm+1 0 ) ∧ Φ * d Xm+1 1 ∧Φ * dx 1 2 ∧ • • • ∧ Φ * dx m 2 ∧ Φ * dx m+1 2 ∧ Φ * Ω 4 ∧ • • • ∧ Φ * Ω n = (-1) 2m 2 i=1 (x m+1 i+1 -x m+1 i ) m-1 • (λ 1 • λ 2 ) • (∧Φ * d Xm+1 0 ) ∧ Φ * d Xm+1 1 ∧Θ 2 ∧ Φ * Ω 5 ∧ • • • ∧ Φ * Ω n = • • • = (-1) (n-1)m n-1 i=1 (x m+1 i+1 -x m+1 i ) m-1 • n-1 k=1 λ k • (∧Φ * d Xm+1 0 ) ∧ • • • ∧ Φ * d Xm+1 n-2 ∧ Φ * dx 1 n-1 ∧ • • • Φ * dx m+1 n-1 .
Notice that the differential forms Φ * d Xm+1 ) m-1 = 0, so it can be concluded that S(q) = 0 if and only if there exists 1 ≤ k ≤ n -1 such that λ k = 0. The latter holds if and only if

m+1 j=1 (x j k -x j k-1 )(x j k+1 -x j k ) = 0 which, in fact, is equivalent to the orthogonality ----→ P k-1 P k ⊥ ----→ P k P k+1 . Therefore Sing(Γ) = {q ∈ Q : S(q) = 0} are equivalent to Sing(Γ) = {x ∈ X : Ψ(x) = 0, ∃ 1 ≤ i ≤ n -1 s.t. m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0}, and
Reg(Γ) = {x ∈ X : Ψ(x) = 0, m+1 j=1 (x j i -x j i-1 )(x j i+1 -x j i ) = 0, 1 ≤ i ≤ n -1}.
2.5 Flatness of the n-bar system in R m+1

Consider a smooth nonlinear control system

Ξ : ẋ = f (x, u)
where x ∈ X, an n-dimensional manifold and u ∈ U, an m-dimensional manifold.

Given any integer l, we associate to Ξ its l-prolongation Ξ l given by

Ξ l : ẋ = f (x, u 0 ) u0 = u 1 . . . ul = u l+1
which can be considered as a control system on X l = X × U × R ml , whose state variables are (x, u 0 , u 1 , . . . , u l ) and whose m controls are the m components of u l+1 . Denote ūl = (u 0 , u 1 , . . . , u l ).

Definition 2.5.1 The system Ξ is called flat at a point (x 0 , ūl 0 ) ∈ X l = X × U × R ml , for some l ≥ 0, if there exist a neighborhood O l of (x 0 , ūl 0 ) and m smooth functions

h i = h i (x, u 0 , u 1 , . . . , u l ), 1 ≤ i ≤ m,
called flat outputs, defined in O l , having the following property: there exist an integer s and smooth functions γ i , 1 ≤ i ≤ n, and δ i , 1 ≤ i ≤ m, such that we have

x i = γ i (h, ḣ, . . . , h (s) ), 1 ≤ i ≤ n u i = δ i (h, ḣ, . . . , h (s) ), 1 ≤ i ≤ m,
where h = (h 1 , . . . , h m ) ⊤ , along any trajectory x(t) given by a control u(t) that satisfy (x(t), u(t), u(t), . . . , u (l) (t)) ∈ O l .

The compositions γ i (h, ḣ, . . . , h (s) ) and δ i (h, ḣ, . . . , h (s) ) are, a priori, defined in an open set O s+l ⊂ X s+l = X ×U ×R m(s+l) . The above definition requires that π(O s+l ) ⊃ O l , where π(x, ūs+l ) = (x, ūl ), and that for all such (x, ūs+l ), the compositions yield, respectively, x i and u i .

If h i = h i (x, u 0 , u 1 , . . . , u r ), r ≤ l, we will say that the system is (x, u, . . . , u r )-flat and, in particular, x-flat if h i = h i (x). In the case h i = h i (x, u 0 , u 1 , . . . , u r ), we will assume that they are defined on O r ⊂ X r = X × U × R mr , where π -1 (O r ) ⊃ O l and π stands for the projection π(x, u 0 , . . . , u r , . . . , u l ) = (x, u 0 , . . . , u r ).

Let h 1 , . . . , h m be flat outputs of a system Ξ. It has been observed in [START_REF] Respondek | Symmetries and minimal flat outputs of nonlinear control systems[END_REF] that there exist integers k 1 , . . . , k m such that span {dx 1 , . . . , dx n , du 1 , . . . , du m } ⊂ span {dh

(j) i , 1 ≤ i ≤ m, 0 ≤ j ≤ k i },
and if at the same time span {dx 1 , . . . , dx n , du 1 , . . . , du m } ⊂ span {dh

(j) i , 1 ≤ i ≤ m, 0 ≤ j ≤ µ i }, then k i ≤ µ i , for 1 ≤ i ≤ m.
The m-tuple (k 1 , . . . , k m ) will be called the differential m-weight of h = (h 1 , . . . , h m ) and k = m i=1 k i will be called the differential weight of h. Definition 2.5.2 Flat outputs of Ξ at (x 0 , ūl 0 ) are called minimal if their differential weight is the lowest among all flat outputs of Ξ at (x 0 , ūl 0 ).

Let U sing (x) be the m-dimensional subspace of R m+1 such that for any feedback control (u 0 (x), . . . , u m (x)

) ⊤ = u(x) ∈ U sing (x) we have m i=0 f i (x)u i (x) ∈ C 1 (x)
, where C 1 ⊂ D = span {f 0 , . . . , f m } is the characteristic subdistribution of D (1) . Any control u(t) ∈ U sing (x(t)) will be called singular and the trajectories of the system governed by a singular control remain tangent to the characteristic distribution C 1 .

The following theorem, given in [START_REF] Respondek | Symmetries and minimal flat outputs of nonlinear control systems[END_REF], characterizes the minimal flat outputs for systems that are feedback equivalent to the m-chained form (i.e., the canonical contact system on J n (R, R m )), with m ≥ 2.

Theorem 2.5.3 Consider the driftless control-linear system

Σ : ẋ = m i=0 f i (x)u i ,
defined on a manifold X and let D = span {f 0 , . . . , f m } be the distribution associated to Σ. If Σ is locally feedback equivalent, at x 0 ∈ X, to the m-chained form, with m ≥ 2, then the following conditions are equivalent:

(i) (L n-1 ) ⊥ = span {dh 0 , . . . , dh m } around x 0 , where L n-1 denotes the subdistribution that is involutive and of corank one in D (n-1) ; (ii) h 0 , . . . , h m are minimal flat outputs of Σ at (x 0 , u 0 ), where u 0 ∈ U sing (x 0 ).

Recall that the n-bar system, as defined in Section 2.2.2, evolves on its configuration manifold Q = R m+1 × (S m ) n , whose points q ∈ Q correspond to those x ∈ R (n+1)(m+1) that satisfy Ψ(x) = (Ψ 1 (x), . . . , Ψ n (x)) = 0, where Ψ i , for 1 ≤ i ≤ n, are given by (2.2.2). The following theorem describes the flatness property of the n-bar system Γ. Theorem 2.5.4 (Flatness of the n-bar system) For the n-bar system Γ moving in R m+1 , where m ≥ 2, we have

(i) Γ is x-flat at any (q 0 , u 0 ) ∈ Q × R m+1 satisfying
(1) Ψ(x) = 0 and m+1 j=1 (x j ix j i-1 )(x j i+1x j i ) = 0, where q 0 ∈ Q is identified with a point x ∈ R (n+1)(m+1) satisfying Ψ(x) = 0;

(2) u 0 is such that the instantaneous velocity Ṗ0 of the point P 0 is nonzero (and thus the instantaneous velocities Ṗi , 0 ≤ i ≤ n -1, are nonzero). (ii) The coordinates of the source point of the last bar P 0 = (x 1 0 , x 2 0 , . . . , x m+1 0 ) are minimal x-flat outputs of Γ at any (q 0 , u 0 ) as above. (iii) If h 0 , . . . , h m are any minimal x-flat outputs at (q 0 , u 0 ), then locally around q 0 we have span{dh 0 , . . . , dh m } = span{dx 1 0 , dx 2 0 , . . . , dx m+1 0 }.

The item (iii) is in contrast with the planar case m = 1, where minimal flat outputs are not unique and their totality is actually parameterized by an arbitrary function of three variables (See a detailed analysis in Chapter 1). Another difference between the planar case (m = 1) and higher-dimensional case (m ≥ 2) is that in the latter all angles ± π 2 between two consecutive bars form singularities while in the former the angle ± π 2 between the last and one before the last bars (trailers) --→ P 0 P 1 and --→ P 1 P 2 is not a singularity.

Proof:

The items (i), (ii) and (iii) are natural consequences of Theorem 2.3.3 and Theorem 2.5.3. Theorem 2.3.3 assures that for m ≥ 2, the n-bar system Γ is locally feedback equivalent to the m-chained form at any point q 0 that corresponds to x ∈ R (n+1)(m+1) satisfying Ψ(x) = 0 and m+1 j=1 (x j ix j i-1 )(x j i+1x j i ) = 0, for 1 ≤ i ≤ n -1. Moreover, it was proved in Section 2.4.3 that around q 0 , the subdistribution L n-1 , which is involutive and of corank one in D (n-1) , is given by

(L n-1 ) ⊥ = span {Φ * dx 1 0 , . . . , Φ * dx m+1 0 } = span {dΦ * x 1 0 , . . . , dΦ * x m+1 0 }.
Notice that on the configuration space Q, we have always that Φ * x j 0 = x j 0 , for 1 ≤ j ≤ m + 1. Thus according to Theorem 2.5.3, the coordinates of P 0 = (x 1 0 , x 2 0 , . . . , x m+1 0 ) are minimal x-flat outputs of Γ around q 0 which implies immediately that Γ is x-flat at (q 0 , u 0 ) for some control u 0 . Before we characterize the control u 0 , notice that Theorem 2.5.3 implies that for control systems that are feedback equivalent to the m-chained form, for n ≥ 2, m ≥ 2, the minimal flat outputs are equivalent in the sense that for any two families of minimal flat outputs (h 0 , . . . , h m ) and ( h0 , . . . , hm ), we have span {dh 0 , . . . , dh m } = span {d h0 , . . . , d hm }. In view of this and the item (ii) of Theorem 2.5.4, any minimal flat outputs (h 0 , . . . , h m ) of the n-bar system satisfy span {dh 0 , . . . , dh m } = span {dx 1 0 , dx 2 0 , . . . , dx m+1 0 } in R m+1 for n ≥ 2, m ≥ 2. This proves (iii). Now we are going to characterize the control u 0 . According to the definition of the flat output, the entire state and all the controls should be functions of the coordinates x 1 0 , x 2 0 , . . . , x m+1 0 and their derivatives. Recall the system ∆ given be (2.2.3) and (2.2.4) ∆ :

         ẋj 0 = v 1 (x j 1 -x j 0 ) . . . ẋj n-1 = v n (x j n -x j n-1 ) ẋj n = v n+j , 1 ≤ j ≤ m + 1,
and consider the system of equation for the

x j 0 -variables                    ẋ1 0 = v 1 (x 1 1 -x 1 0 ) ẋ2 0 = v 1 (x 2 1 -x 2 0 ) . . . ẋm+1 0 = v 1 (x m+1 1 -x m+1 0 ) Ψ 1 (x) = m+1 j=1 (x j 1 -x j 0 ) 2 -1 = 0.
(2.5.1)

A direct computation shows that

v 1 = ( ẋ1 0 ) 2 + • • • + ( ẋm+1 0 ) 2 1 2 = η 1 (P 0 , Ṗ0 ), (2.5.2)
for some function η 1 . Substituting (2.5.2) into (2.5.1), we get

x 1 1 = x 1 0 + ẋ1 0 v 1 = ϕ 1 1 (P 0 , Ṗ0 ) x 2 1 = x 2 0 + ẋ2 0 v 1 = ϕ 2 1 (P 0 , Ṗ0 ) . . . . . . x m+1 1 = x m+1 0 + ẋm+1 0 v 1 = ϕ m+1 1 (P 0 , Ṗ0 ),
for some functions

ϕ i 1 , for 1 ≤ i ≤ m + 1. Put ϕ 1 = (ϕ 1 1 , . . . , ϕ m+1 1
), we thus have

P 1 = (x 1 1 , . . . , x m+1 1 ) = (ϕ 1 1 (P 0 , Ṗ0 ), . . . , ϕ m+1 1 (P 0 , Ṗ0 )) = ϕ 1 (P 0 , Ṗ0 ).
In the same way, we obtain that, for 2 ≤ i ≤ n,

v i = η i (P i-1 , Ṗi-1 ) = ηi (P 0 , Ṗ0 , P (2)
0 , . . . , P

and

P i = ϕ i (P i-1 , Ṗi-1 ) = φi (P 0 , Ṗ0 , P (2)
0 , . . . , P (i) 0 ), for some functions ηi and φi . Finally, the controls v n+j , for 1 ≤ j ≤ m + 1, can be expressed by

v n+j = ẋj n 2.6 Proof of Lemma 2.4.1 (a). dω j i ∧ ω i ∧ ω i+1 = 0, for 1 ≤ i ≤ n -1, 1 ≤ j ≤ m + 1
, and j = σ(i).

Proof: For simplicity, we assume that j < σ(i). The case of j > σ(i) can be proved in the same way. Since Ω j i = (x j ix j i-1 )dx

σ(i) i-1 -(x σ(i) i -x σ(i)
i-1 )dx j i-1 , we have

dΩ j i = dx j i ∧ dx σ(i) i-1 -2dx j i-1 ∧ dx σ(i) i-1 -dx σ(i) i ∧ dx j i-1 .
Therefore

dΩ j i ∧ Ω i = dΩ j i ∧ Ω 1 i ∧ • • • ∧ Ω m+1 i = (-1) σ(i) (x σ(i) i -x σ(i) i-1 ) m (∧dX i-1 ) ∧ dx j i +(-1) σ(i)+1 (x j i -x j i-1 )(x σ(i) i -x σ(i) i-1 ) m-1 (∧dX i-1 ) ∧ dx σ(i) i = δ 1 + δ 2 ,
where

δ 1 = (-1) σ(i) (x σ(i) i -x σ(i) i-1 ) m (∧dX i-1 ) ∧ dx j i δ 2 = +(-1) σ(i)+1 (x j i -x j i-1 )(x σ(i) i -x σ(i) i-1 ) m-1 (∧dX i-1 ) ∧ dx σ(i) i .
Suppose that σ(i) ≤ σ(i + 1), a direct computation shows that

δ 1 ∧ Ω i+1 = (-1) σ(i)+m-1 (x σ(i) i -x σ(i) i-1 ) m (x j i+1 -x j i )(x σ(i+1) i+1 -x σ(i+1) i ) m-1 (∧dX i-1 ) ∧ x j i ∧ x 1 i ∧ • • • ∧ x j-1 i ∧ x σ(i+1) i ∧ x j+1 i ∧ • • • x σ(i+1)-1 i ∧ x σ(i+1)+1 i ∧ • • • ∧ x m+1 i = (-1) m+σ(i)+σ(i+1)-1 (x σ(i) i -x σ(i) i-1 ) m (x j i+1 -x j i )(x σ(i+1) i+1 -x σ(i+1) i ) m-1 (∧dX i-1 ) ∧ dX i , and
δ 2 ∧ Ω i+1 = (-1) σ(i)+m (x j i -x j i-1 )(x σ(i) i -x σ(i) i-1 ) m-1 (x j i+1 -x j i ) (x σ(i+1) i+1 -x σ(i+1) i ) m-1 (x σ(i) i+1 -x σ(i) i ) (∧dX i-1 ) ∧ dx σ(i) i ∧ dx 1 i ∧ • • • ∧ dx σ(i)-1 i ∧ dx σ(i+1) i ∧ dx σ(i)+1 i ∧ • • • ∧ dx σ(i+1)-1 i ∧ dx σ(i+1)+1 i ∧ • • • ∧ dx m+1 i = (-1) m+σ(i)+σ(i+1) (x j i -x j i-1 )(x σ(i) i -x σ(i) i-1 ) m-1 (x j i+1 -x j i ) (x σ(i+1) i+1 -x σ(i+1) i ) m-1 (x σ(i) i+1 -x σ(i) i )(∧dX i-1 ) ∧ dX i .
Differentiating the constraint

Ψ i = (x 1 i -x 1 i-1 ) 2 + (x 2 i -x 2 i-1 ) 2 + • • • + (x m+1 i -x m+1 i-1 ) 2 -1 = 0, we get (x 1 i -x 1 i-1 )dx 1 i -(x 1 i -x 1 i-1 )dx 1 i-1 +• • •+(x m+1 i -x m+1 i-1 )dx m+1 i -(x m+1 i -x m+1 i-1 )dx m+1 i-1 = 0,
which implies that Φ * dX i-1 and Φ * dX i are dependent on the configuration space Q. Then

dω j i ∧ ω i ∧ ω i+1 = Φ * (dΩ j i ∧ Ω i ∧ Ω i+1 ) = Φ * (δ 1 ∧ Ω i+1 + δ 2 ∧ Ω i+1 ) = (-1) σ(i)+m+σ(i+1) • [(x j i -x j i-1 )(x σ(i) i -x σ(i) i-1 ) m-1 (x j i+1 -x j i ) (x σ(i+1) i+1 -x σ(i+1) i ) m-1 (x σ(i) i+1 -x σ(i) i ) -(x σ(i) i -x σ(i) i-1 ) m (x j i+1 -x j i )(x σ(i+1) i+1 -x σ(i+1) i ) m-1 ] • (∧Φ * dX i-1 ) ∧ Φ * dX i = 0.
The case of σ(i) > σ(i + 1) can be treated in a similar way.

(b). ∧ ω i = (-1) m (x σ(i) i -x σ(i) i-1 ) m-2 (∧Φ * d Xσ(i) i-1 ) mod (Φ * dX i ), for 1 ≤ i ≤ n -1, where (Φ * dX i ) denotes the ideal generated by Φ * dx 1 i , . . . , Φ * dx m-1 i .
Proof: Consider the case i = 1 and denote σ(1) = k, then

Ω j 1 = (x j 1 -x j 0 )dx k 0 -(x k 1 -x k 0 )dx j 0 ,
for 1 ≤ j ≤ m + 1 and j = k. Differentiating the constraint

Ψ 1 = (x 1 1 -x 1 0 ) 2 + (x 2 1 -x 2 0 ) 2 + • • • + (x m+1 1 -x m+1 0 ) 2 -1 = 0, we obtain dx k 0 = - m+1 j=1,j =k x j 1 -x j 0 x k 1 -x k 0 dx j 0 mod (dX 1 ),
and hence

ω j 1 = Φ * (Ω j 1 ) = - (x j 1 -x j 0 ) 2 x k 1 -x k 0 + (x j 1 -x j 0 ) Φ * dx j 0 - m+1 l=1,l =k,l =j x l 1 -x l 0 x k 1 -x k 0 (x j 1 -x j 0 )Φ * dx 1 0 mod (Φ * dX 1 ).
Since Φ * dx 1 0 , . . . , Φ * dx k-1 0 , Φ * dx k+1 0 , . . . , Φ * dx m+1 0 are independent on Q, comparing the coefficients of Φ * dx j 0 it is easy to see that

∧ω 1 = ω 1 1 ∧ • • • ω k-1 1 ∧ ω k+1 1 ∧ • • • ∧ ω m+1 1 = (-1) m 1 (x k 1 -x k 0 ) m • |A| Φ * dx 1 0 ∧ • • • Φ * dx k-1 0 ∧ Φ * dx k+1 0 ∧ • • • ∧ Φ * dx m+1 0 ,
where |A| is given by, in fact, a special case of the following determinant J m ,

J m = a 2 1 + a 2 k a 1 a 2 • • • a 1 a k-1 a 1 a k+1 • • • a 1 a m+1 . . . . . . . . . . . . . . . . . . . . . a k-1 a 1 a k-1 a 2 • • • a 2 k-1 + a 2 k a k-1 a k+1 • • • a k-1 a m+1 a k+1 a 1 a k+1 a 2 • • • a k+1 a k-1 a 2 k+1 + a 2 k • • • a k+1 a m+1 . . . . . . . . . . . . . . . . . . . . . a m+1 a 1 a m+1 a 2 • • • a m+1 a k-1 a m+1 a k+1 • • • a 2 m+1 + a 2 k with a i = x i 1 -x i 0
, where a k = 0 and m+1 i=1 a 2 i = 1. By a straightforward calculation, we obtain that

J m = a 2m-1 k • (a 2 1 + • • • + a 2 m+1 ) = a 2(m-1) k ,
and thus 1) . Therefore, we have

|A| = (x k 1 -x k 0 ) 2(m-
∧ω 1 = (-1) m (x k 1 -x k 0 ) m-2 (∧Φ * d Xk 0 ) mod (Φ * dX 1 ) = (-1) m (x σ(1) 1 -x σ(1) 0 ) m-2 (∧Φ * d Xσ(1) 0 ) mod (Φ * dX 1 ).
Similarly, it can be proved, for 1 ≤ i ≤ n -1, that

∧ω i = (-1) m (x σ(i) i -x σ(i) i-1 ) m-2 (∧Φ * d Xσ(i) i-1 ) mod (Φ * dX i ). (c). dω j i ∧ ω i ∧ • • • ∧ ω 1 = 0, 1 ≤ j ≤ m + 1, j = σ(i) Proof:
We give here the proof for i = n and the other cases can be shown in the same way. For simplicity, we assume that j < σ(n). A direct computation shows that

dΩ j n ∧ Ω n = (-1) m+σ(n)-1 (x σ(n) n -x σ(n) n-1 ) m dx j n ∧ dX n-1 +(-1) m+σ(n) (x j n -x j n-1 )(x σ(n) n -x σ(n) n-1 ) m-1 dx σ(n) n ∧ dX n-1 .
Moreover, differentiating the constraint

Ψ n = (x 1 n -x 1 n-1 ) 2 + (x 2 n -x 2 n-1 ) 2 + • • • + (x m+1 n -x m+1 n-1 ) 2 -1 = 0 we get dx σ(n) n = - m+1 l=1,l =σ(n) x l n -x l n-1 x σ(n) n -x σ(n) n-1 dx l n mod (dX n-1 ). Therefore dω j n ∧ ω n = Φ * (dΩ j n ∧ Ω n ) = (-1) m+σ(n)-1 [(x σ(n) n -x σ(n) n-1 ) m Φ * dx j n + (x j n -x j n-1 ) 2 (x σ(n) n -x σ(n) n-1 ) m-2 Φ * dx j n + m+1 l=2,l =σ(n),l =j (x σ(n) n -x σ(n) n-1 ) m-2 (x j n -x j n-1 )(x l n -x l n-1 ) Φ * dx l n ] ∧ Φ * dX n-1 = (-1) m+σ(n)-1 η j • Φ * dx j n ∧ Φ * dX n-1 +(-1) m+σ(n)-1 m+1 l=2,l =σ(n),l =j η l • Φ * dx l n ∧ Φ * dX n-1 ,
where

η j = (x σ(n) n -x σ(n) n-1 ) m + (x j n -x j n-1 ) 2 (x σ(n) n -x σ(n) n-1 ) m-2 , η l = (x σ(n) n -x σ(n) n-1 ) m-2 (x j n -x j n-1 )(x l n -x l n-1 ). Since Φ * dx 1 n , . . . , Φ * dx σ(n)-1 n , Φ * dx σ(n)+1 n , . . . , Φ * dx m+1 n are independent on Q, all dif- ferential forms Φ * dx i n ∧ Φ * dX n-1 ∧ ω n-1 ∧ • • • ∧ ω 1 , for 1 ≤ i ≤ m + 1 and i = σ(n)
, are also independent. Therefore, it is enough to prove

η j • Φ * dx j n ∧ Φ * dX n-1 ∧ ω n-1 ∧ • • • ∧ ω 1 = 0, (2.6.1)
to conclude that

dω j n ∧ ω n ∧ • • • ∧ ω 1 = (-1) m+σ(n)-1 η j • Φ * dx j n ∧ Φ * dX n-1 ∧ ω n-1 ∧ • • • ∧ ω 1 +(-1) m+σ(n)-1 m+1 l=2,l =σ(n),j η l • Φ * dx l n ∧ Φ * dX n-1 ∧ ω n-1 ∧ • • • ∧ ω 1 = 0.
Notice that η j = (x

σ(n) n -x σ(n) n-1 ) m + (x j n -x j n-1 ) 2 (x σ(n) n -x σ(n) n-1 ) m-2 = 0 is obvious since x σ(n) n -x σ(n)
n-1 = 0. Hence in order to prove (2.6.1), it is sufficient to show that

Φ * dx j n ∧ Φ * dX n-1 ∧ ω n-1 ∧ • • • ∧ ω 1 = 0.
Differentiating the constraint

Ψ n-1 = (x 1 n-1 -x 1 n-2 ) 2 + (x 2 n-1 -x 2 n-2 ) 2 + • • • + (x m+1 n-1 -x m+1 n-2 ) 2 -1 = 0 implies the following relation dx σ(n-1) n-1 = dx σ(n-1) n-2 mod (dx j n-1 , dx j n-2 , 1 ≤ j ≤ m + 1, j = σ(n -1)).
Recall the condition (b) of Lemma 2.4.1,

∧ω i = (-1) m (x σ(i) i -x σ(i) i-1 ) m-2 (∧Φ * d Xσ(i) i-1 ) mod (Φ * dX i ),
and a straightforward calculation shows that

Φ * dx j n ∧ Φ * dX n-1 ∧ ω n-1
Chapter 3

Cartan distribution for surfaces 3.1 Introduction

Consider J n (R k , R m ) ∼ = R N
, for a suitable N, the space of n-jets of smooth maps from R k into R m and denote its canonical coordinates by

(x 1 , . . . , x k , y 1 , . . . , y m , p σ j , 1 ≤ j ≤ m, 1 ≤ |σ| ≤ n),
where x i , for 1 ≤ i ≤ k, represent independent variables and y j , for 1 ≤ j ≤ m, represent dependent variables; the vector of non-negative integers σ = (σ 1 , . . . , σ k ) is a multi-index such that |σ| = σ 1 + • • • + σ k ≤ n and p σ j , for 1 ≤ j ≤ m, correspond to the partial derivatives ∂ |σ| y j ∂x σ , where ∂x σ stands for

∂ σ 1 x 1 • • • ∂ σ k x k . Denote p 0 j = y j , for 1 ≤ j ≤ m. Any smooth map ϕ = (ϕ 1 , . . . , ϕ m ) from R k into R m defines a submanifold in J n (R k , R m ) by the relation p σ j = ∂ |σ| ϕ j ∂x σ (x 1 , . . . , x k ),
for 1 ≤ j ≤ m and 0 ≤ |σ| ≤ n. This submanifold is called the n-graph of ϕ. It turns out that all n-graphs are integral submanifolds, of dimension k, of a distribution called the Cartan distribution on J n (R k , R m ) and denoted by CC n (R k , R m ). The Pfaffian system that annihilates

CC n (R k , R m ), which is called the canonical contact Pfaffian system on J n (R k , R m ), is given in the canonical coordinates of J n (R k , R m ) by dp σ j - k i=1 p σ+1 i j dx i = 0, for 1 ≤ j ≤ m, and 0 ≤ |σ| ≤ n -1,
where σ + 1 i = (σ 1 , . . . , σ i + 1, . . . , σ k ).

A natural problem is to characterize those distributions which are (locally) equivalent to a Cartan distribution. This problem was posed by Pfaff [START_REF] Pfaff | Methodus generalis, aequationes differentiarum partialum, nec non aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles, completi integrandi[END_REF] in 1814 and is still open in its full generality although many important particular solutions have been obtained.

In the case of n = m = 1 and an arbitrary k, the solution has been obtained by Darboux [10] in his famous theorem generalizing earlier results of Pfaff [START_REF] Pfaff | Methodus generalis, aequationes differentiarum partialum, nec non aequationes differentiales vulgares, utrasque primi ordinis, inter quotcunque variabiles, completi integrandi[END_REF] and Frobenius [START_REF] Frobenius | Über das Pfaffsche problem[END_REF]. The case n = 2, m = 1 and k = 1 was solved by Engel in [START_REF] Engel | Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen[END_REF]. The case n ≥ 2, m = 1 and k = 1 was solved by E. von Weber [START_REF] Weber | Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen[END_REF], Cartan [START_REF] Cartan | Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes[END_REF] and Goursat [START_REF] Goursat | Leçons sur le problème de Pfaff[END_REF] (at generic points) and by Libermann [START_REF] Libermann | Sur le problème d'équivalence des systèmes de Pfaff non complètement intégrables[END_REF], Kumpera and Ruiz [START_REF] Kumpera | Sur l'équivalence locale des systèmes de Pfaff en drapeau[END_REF], and Murray [START_REF] Murray | Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems[END_REF] (at an arbitrary point). The case n = 1 with k and m arbitrary has bee studied by Gardner and solved by Bryant [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF], [START_REF] Bryant | Exterior Differential Systems[END_REF]. The case k = 1, with n and m arbitrary (that is, characterization of Cartan distributions for curves) has been studied by Gardner et Shadwick [START_REF] Gardner | The GS algorithm for exact linearization to Brunovsky normal form[END_REF], Murray [START_REF] Murray | Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems[END_REF], Tilbury et Sastry [START_REF] Tilbury | The multi-steering n-trailer system: A case study of Goursat normal forms and prolongations[END_REF], Aranda-Bricaire and Pomet [START_REF] Aranda-Bricaire | Some explicit conditions for a control system to be feedback equivalent to extended Goursat normal form[END_REF], Mormul [START_REF] Mormul | Multi-dimensional Cartan prolongation and special k-flags, in Geometric Singularity Theory[END_REF], Respondek and Pasillas-Lépine [START_REF] Pasillas-Lépine | Contact systems and corank one involutive subdistributions[END_REF], [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF]. The general case of arbitrary n,m, and k has been studied by Yamaguchi [79].

This chapter is devoted to the problem of when a given distribution is locally equivalent to the Cartan distribution in the case k = 2, that is, Cartan distributions for surfaces, and is organized as follows. In Section 3.2, we define the Cartan distribution for surfaces. In Section 3.3, we give our main results. In Section 3.4, we recall the Bryant theorem (describing the case n = 1, with k and m arbitrary) and propose its extension that will be useful in the proof of our main result. Finally, we prove our results in Section 3.5.

Cartan distributions for surfaces

A distribution D on a C ∞ -smooth manifold M is a map that assigns smoothly to each point q ∈ M a linear subspace D(q) ⊂ T q M. Given a distribution D, we will denote by Γ(D) the submodule of the module V ∞ (M) of C ∞ -smooth vector fields consisting of C ∞ -smooth sections of T M with values in D, i.e., Γ(D) = {f ∈ V ∞ (M) : f (q) ∈ D(q)}, for any q ∈ M. We will say that a distribution D is C ∞ -smooth if D(q) = Γ(D)(q) for any q ∈ M. D will be called nonsingular if dim D(q) is constant and in this case we will say that D is of rank k, where dim D(q) = k for any q ∈ M.

The derived flag of a distribution D is the sequence of modules of vector fields D (0) ⊂ D (1) ⊂ • • • defined inductively by

D (0) = Γ(D) and D (i+1) = Γ(D (i) ) + [Γ(D (i) ), Γ(D (i) )], for i ≥ 0.
The Lie flag is the sequence of modules of vector fields

D 0 ⊂ D 1 ⊂ • • • defined inductively by D 0 = Γ(D) and D i+1 = Γ(D i ) + [Γ(D 0 ), Γ(D i )], for i ≥ 0.
In general, the derived and Lie flags are different; though for any point q in the underlying manifold the inclusion D i (q) ⊂ D (i) (q) holds, for i ≥ 0.

A Pfaffian system on a smooth manifold M is a differential system ω 1 = • • • = ω m = 0, where ω i are differential 1-forms on M. We associate to it the codistribution I by putting I(q) = vect R {ω 1 (q), . . . , ω m (q)} ⊂ T * q M. Conversely, let I be a smooth codistribution that assigns smoothly to each point q in M a linear subspace I(q) ⊂ T * q M of dimension m. Such a field of cotangent m-planes is spanned locally by m pointwise linearly independent smooth differential 1-forms ω 1 , . . . , ω m on M, which will be denoted by I = span {ω 1 , . . . , ω m }. So in the case of constant rank we can (locally) identity codistributions and Pfaffian systems which we will do throughout this paper. For a Pfaffian system I, we can define its derived flag I (0) ⊃ I (1) ⊃ • • • by

I (0) = I and I (i+1) = {ω ∈ I (i) : dω ≡ 0 mod I (i) }, for i ≥ 0,
provided that each element I (i) of this sequence has constant rank. In this case, it is immediate to see that the derived flag of the distribution D = I ⊥ coincides with the sequence of distributions that annihilate the elements of the derived flag of I, that is

D (i) = (I (i) ) ⊥ , for i ≥ 0.
Remark: If D is a smooth nonsingular distribution, then with no confusion we can identify D and Γ(D) and will do it throughout this chapter. If, however, D is singular or non-smooth, then the distinction between D and Γ(D) is essential. Notice that if D is C ∞ -smooth, then the function q → dim D(q) is lower semi-continuous (in other words, it can drop at a point but not jump up) because if vector fields belonging to D are independent at a point so they are in a neighborhood. In the paper, however, we will have to deal with non-smooth distributions D for which q → dim D(q) jumps up at a point (and thus is not lower semi-continuous). This will be the case if I is a smooth codistribution of a non-constant rank and thus D = I ⊥ , defined as D(q) = (I(q)) ⊥ is non-smooth for which q → dimD(q) is actually upper semi-continuous. Indeed, dim D(q) = ndim I(q), where q → dim I(q) is lower semi-continuous because I was assumed to be smooth.

Given a distribution D of constant rank k on a n-dimensional manifold M, choose locally m = nk differential 1-forms ω 1 , . . . , ω m such that

I = D ⊥ = span {ω 1 , . . . , ω m }.
We define the characteristic distribution C(D) of D pointwise by C(D)(q) = {v ∈ D(q) : v dω(q) ∈ I(q), ∀ ω ∈ I}.

Notice that, in general, C(D) may not be a smooth distribution and dim C(D)(q) may jump up at a point. If C(D) is nonsingular (in other words, of constant rank), then it can be equivalently defined as the module of its sections:

Γ(C(D)) = {g ∈ Γ(D) : [g, f ] ∈ Γ(D), ∀ f ∈ Γ(D)}.
Denote by (x 1 , x 2 ) coordinates on R 2 and by (y 1 , . . . , y m ) coordinates on R m . According to Introduction, we denote by

(x 1 , x 2 , p σ j , 1 ≤ j ≤ m, 0 ≤ |σ| ≤ n), the canonical coordinates on J n (R 2 , R m )
, where σ = (σ 1 , σ 2 ) such that σ 1 , σ 2 ≥ 0 and |σ| = σ 1 + σ 2 ≤ n and (by definition) p 0 j = y j , for 1 ≤ j ≤ m. As we already explained in the Introduction, the surface {y j = ϕ j (x 1 , x 2 )}, 1 ≤ j ≤ m, defined by any smooth map ϕ from R 2 into R m , can be lifted to the surface

p σ j = ∂ |σ| ϕ j ∂x σ (x 1 , x 2 ) , in J n (R 2 , R m
), called the n-graph of the surface {y j = ϕ j (x 1 , x 2 )}. As we pointed out in the Introduction, we can endow J n (R 2 , R m ) with a (nonintegrable) distribution. Namely, a surface in J n (R 2 , R m ), that projects to R 2 , is the lift of a surface in R m if and only if it is an integral submanifold of the Cartan distribution

CC n (R 2 , R m ) on J n (R 2 , R m
), called also the Cartan distribution for surfaces, spanned by the following vector fields

∂ ∂x 1 + n-1 |σ|=0 m j=1 p σ+1 1 j ∂ ∂p σ j , ∂ ∂x 2 + n-1 |σ|=0 m j=1 p σ+1 2 j ∂ ∂p σ j , ∂ ∂p σ j , 1 ≤ j ≤ m, |σ| = n.
The annihilator of CC n (R 2 , R m ) is the Pfaffian system, called the canonical contact Pfaffian system for surfaces, spanned by

dp σ j -p σ+1 1 j dx 1 -p σ+1 2 j dx 2 , 1 ≤ j ≤ m, 0 ≤ |σ| ≤ n -1.
We have dim J n (R 2 , R m ) = m 2 (n + 1)(n + 2) + 2; the Cartan distribution CC n (R 2 , R m ) and its annihilator are, respectively, of rank m(n + 1) + 2 and m 2 n(n + 1). The aim of this chapter is to identify all distributions D of rank m(n + 1) + 2 on R N where N = m 2 (n + 1)(n + 2) + 2, that are locally equivalent to the Cartan distribution CC n (R 2 , R m ), that is, for which there exists a local diffeomorphism ϕ :

R N → J n (R 2 , R m ) such that ϕ * D = CC n (R 2 , R m ).

Characterization of Cartan distributions for surfaces

Main result

The crucial notion will be that of the Engel rank. Definition 3.3.1 The Engel rank, at a point q, of a Pfaffian system I = span {ω 1 , . . . , ω m }, denoted by rank E I(q), is the largest integer k such that there exists a 1-form α in I for which

((dα) k ∧ ω 1 ∧ • • • ∧ ω m )(q) = 0.
In Section 3.3.2, we will describe relations of the Engel rank with Cartan rank and, in particular, with the existence of involutive subdistributions of corank k in D = I ⊥ . Theorem 3.3.2 Assume that m ≥ 3. A distribution D of rank m(n + 1) + 2 on a manifold M of dimension m 2 (n + 1)(n + 2) + 2 is locally, around a point q of M, equivalent to the Cartan distribution for surfaces CC n (R 2 , R m ) if and only if the following conditions hold: 1) that exists due to (iii).

(i) Each element D (i) , for 0 ≤ i ≤ n, has constant rank equal m 2 (2n+2-i)(i+1)+2. (ii) Each element D (i) , for 0 ≤ i ≤ n -2, contains the characteristic distribution C i+1 of D (i+1) that is of constant corank two in D (i) . Moreover, the characteristic distribution of D is empty. (iii) The Engel rank of (D (n-1) ) ⊥ is constant equal two. (iv) dim (L n-1 + D (i) )(q) = dim L n-1 (q) + 2, for 0 ≤ i ≤ n -2, where L n-1 is an involutive subdistribution of corank two in D (n-
Remark 1: The condition (iii) is the essential condition identifying Cartan distribution for surfaces. We want to underline two of its features. First, it is easily verifiable in terms of a given distribution D (see the definition of the Engel rank). Secondly, it has an elegant geometric interpretation: it is equivalent to the existence of an involutive subdistribution, which we denote by L n-1 , of corank two in D (n-1) . We will discuss the geometric interpretation of the condition (iii) in Section 3.3.2 Remark 2: The condition (ii) implies that each D (i) , 0 ≤ i ≤ n -2, also contains an involutive subdistribution L i of corank two in D (i) . Checking the existence of such an involutive distribution does not require, however, calculating the Engel rank of D (i) because the condition (ii) assures that, indeed,

L i = C i+1 = C(D (i+1)
). In view of this, the particularity of L n-1 is seen: in fact, L n-1 is not a characteristic distribution of D (n) = T M and therefore its existence has to be verified by calculating the Engel rank of D (n-1) . Remark 3: The condition (iv) is a sort of regularity condition. If the conditions (i) -(iii) are satisfied everywhere on M, then (iv) holds in an open and dense subset M of M. In this sense, (i) -(iii) are structural conditions that guarantee equivalence to a Cartan distribution for surfaces at generic points and (iv) distinguishes regular points from singular ones. Given a distribution and a fixed point q ∈ M, the condition (iv) can be checked algebraically because the involutive subdistribution L n-1 is unique (if it exists) and can be calculated; we will discuss that in Section 3.3.2 below.

The proof of the above theorem will be given in Section 3.5. It is based on the Bryant normal form, which we will discuss in Section 3.4.

Involutive subdistributions of corank k

The condition (iii) of Theorem 3.3.2, saying that the Engel rank of D (n-1) equals two, is essential for identifying Cartan distributions for surfaces. In this section, we will relate that condition to that expressed via the Cartan rank and to other geometric properties, in particular, to the existence of an involutive subdistribution of corank two.

Definition 3.3.3 The Cartan rank of a Pfaffian system I = span {ω 1 , . . . , ω m } at q ∈ M, denoted by rank C I(q), is the smallest integer k for which there exist π 1 , . . . ,

π k ∈ Λ 1 (M) such that π 1 ∧ • • • ∧ π k ∧ ω 1 ∧ • • • ∧ ω m (q) = 0, and dω ∧ π 1 ∧ • • • ∧ π k ≡ 0 mod I, for any ω ∈ I.
The following theorem answers the question: when does a given distribution D of constant rank k + mk, for m ≥ 3, contain an involutive subdistribution L ⊂ D that has constant corank k in D? Moreover, if any of the equivalent conditions (i)-(iv) is satisfied, then the involutive subdistribution L of corank k in D is unique and can be calculated as

L = F1 + • • • Fm ,
where Fi = span {f ∈ D : f dω i ∈ D ⊥ } and ω i are any differential 1-forms such that D ⊥ = span {ω 1 , . . . , ω m }; in fact, it is enough to take in the above sum defining L only two terms corresponding to any 1 ≤ i = j ≤ m.

84

Notice that, in general, rank C I(q) = rank E I(q) even if both ranks are constant (see [START_REF] Bryant | Exterior Differential Systems[END_REF] for examples). The equality of both ranks is a consequences of the assumptions D (1) = T M and C(D) = 0 and was proved by Bryant [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF]. The proof of the above theorem is given in Section 3.5.3.

Extended Bryant normal form

The Bryant normal form is the canonical form of the Cartan distribution CC 1 (R k , R m ) (that is, for 1-jets). Our proof of Theorem 3.3.2 is based on an extension of the Bryant normal form. In this section we will recall the Bryant normal form (and for completeness we will provide its alternative proof), then we will give its extension and, finally, provide a dual version of that extension which we will use directly in the proof of our main result, Theorem 3.3.2.

Bryant normal form

The question of which Pfaffian systems are locally equivalent to the canonical contact Pfaffian system on J 1 (R k , R m ) (equivalently, the problem to characterize the Cartan distributions CC 1 (R k , R m )) has been studied by Gardner since 1972 and Bryant. In this subsection, we will recall the solution to this problem given by Bryant [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF] in 1979.

If for a constant rank Pfaffian system I, the characteristic distribution C(D) of the distribution D = I ⊥ is of constant rank, then its annihilator (C(D)) ⊥ is a Pfaffian system, called the retracting codistribution of I and denoted by R(I), in other words, R(I) = (C(D)) ⊥ . The following result (see, e.g., [START_REF] Bryant | Exterior Differential Systems[END_REF]) interprets the rank of R(I) in terms of the minimal number of coordinate variables needed to express I. Lemma 3.4.1 Let I be a Pfaffian system on M such that the rank of I and of its retracting codistribution R(I) are constant. If rank R(I) = s, then around any q ∈ M there are coordinates (x 1 , . . . , x r , y 1 , . . . , y s ), where r+s = n such that I is generated by 1-forms that are expressed via the coordinates y 1 , . . . , y s only. Moreover, s = rank R(I) is the minimal integer with that property.

This Lemma shows that given a Pfaffian system, the rank of its retracting system equals, indeed, the minimal number of variables needed to describe the system (see [START_REF] Bryant | Exterior Differential Systems[END_REF]).

The following characterization of the canonical contact Pfaffian system on J 1 (R k , R m ) was proved by Bryant in [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF] (see also [START_REF] Bryant | Exterior Differential Systems[END_REF]).

Theorem 3.4.2 (Bryant normal form) Assume that m = 2. A Pfaffian system I = span {ω 1 , . . . , ω m } of constant rank m, defined on a manifold of dimension m + k + mk and such that I (1) = 0, is locally equivalent to the canonical contact Pfaffian system on J 1 (R k , R m ), called also the Bryant normal form,

dp 0 1 -p 1 1 dx 1 -• • • -p k 1 dx k = 0 . . . dp 0 m -p 1 m dx 1 -• • • -p k m dx k = 0 (3.4.1)
if and only if its Engel rank is constant and equals k, and the rank of its retracting codistribution R(I) is constant and equals m + k + mk.

Above, the variables p i j , where 1 ≤ i ≤ k and 1 ≤ j ≤ m, stand for p 1 i j , where 1 i = (0, . . . , 1, . . . , 0), with the i-th component being equal to one.

For completeness, we will give an alternative proof of Theorem 3.4.2 (Bryant normal form for m ≥ 3) based on Theorem 3.3.4, and on Lemma 3.5.3.

Proof: Let D be the distribution defined by D = I ⊥ . Obviously, the condition that the rank of the retracting system R(I) equals m + k + mk, implies that C(D) = 0. Since m ≥ 3 and the Engel rank of D ⊥ equals k, it follows by Theorem 3.3.4 that D contains an involutive subdistribution L that has corank k in D. Therefore, there exists a coordinate system (x 1 , . . . , x k , p i j ), for 0 ≤ i ≤ k and 1 ≤ j ≤ m, such that

D = L ⊕ span {f 1 , . . . , f k }, where L = span { ∂ ∂p i j , 1 ≤ i ≤ k, 1 ≤ j ≤ m}, and
f i = ∂ ∂x i + a i1 ∂ ∂p 0 1 + • • • + a im ∂ ∂p 0 m , for 1 ≤ i ≤ k,
for some functions a ij . We can assume that all functions a ij satisfy a ij (q 0 ) = 0 replacing, if necessary, p 0 j by p 0 j -k i=1 a ij (q 0 )x i . The condition C(D) = 0 implies that D satisfies the assumptions of Lemma 3.5.3 (see the next section), with r = km and l = k. Therefore rank ∂a ∂p (q) = km, for any q in a neighborhood of q 0 , where a = (a 11 , . . . , a 1m , a 21 , . . . , a km ) ⊤ and

p = (p 1 1 , . . . , p 1 m , p 2 1 , . . . , p k m ) ⊤ . It follows that the map defined by xi = x i , 1 ≤ i ≤ k, p0 j = p 0 j , 1 ≤ i ≤ k, pi j = a ij , 1 ≤ i ≤ k, 1 ≤ j ≤ m, is a local diffeomorphism transforming q 0 into 0. We have ϕ * L = span { ∂ ∂ pi j , 1 ≤ i ≤ k, 1 ≤ j ≤ m} and ϕ * f i = ∂ ∂ xi + m j=1 pi j ∂ ∂ p0 j mod L.
It follows that (we skip the tildes)

ϕ * D = span { ∂ ∂ xi + m s=1 pi s ∂ ∂ p0 s , ∂ ∂ pi j , 1 ≤ i ≤ k, 1 ≤ j ≤ m} thus proving that ϕ * D is the Cartan distribution CC 1 (R k , R m ) expressed in z = (x, p)- coordinates.

Extended Bryant normal form

We give in this section, a generalization of Bryant's Theorem 3.4.2 and then provide a dual version of that generalization.

Theorem 3.4.3 (Extended Bryant normal form, first version) Assume that m = 2. Let I = span {ω 1 , . . . , ω m } be a Pfaffian system defined on a manifold M of dimension m + k + mk + l such that I (1) = 0. If I satisfies, in a neighborhood of a point q ∈ M, the following conditions:

(i) The Engel rank of I is constant and equals k;

(ii) The rank of the retracting codistribution R(I) is constant and equals m+k +mk, then there exist local coordinates (z, y) = (x, p, y) = (x i , p i j , y 1 , . . . , y l ), for 0 ≤ i ≤ k, 1 ≤ j ≤ m, around q such that I takes in (z, y) = (x, p, y)-coordinates the Bryant normal form (3.4.1).

Proof: Recall that the characteristic distribution is always involutive. Since rank R(I) equals m + k + mk, there exists, by Lemma 3.4.1, a local coordinate system (z, y) = (z 1 , . . . , z m+k+mk , y 1 , . . . , y l ), centered at (0, 0) ∈ R m(k+1)+k × R l such that

ω i = m+k+mk j=1 ω i j (z)dz j , 1 ≤ i ≤ m.
Therefore ω 1 , . . . , ω m can be considered locally as differential 1-forms on R m+k+mk , the latter equipped with z-coordinates. Define the codistribution (Pfaffian system) Ĩ spanned by them locally around 0 ∈ R m+k+mk . Assume for a moment that Ĩ satisfies the conditions of Theorem 3. (1). For any α ∈ Ĩ, we have

α = m i=1 α i (z)ω i ∈ I,
which gives (since

I (1) = 0) dα ∧ ω 1 ∧ • • • ω m = 0,
thus implying that Ĩ(1) = 0.

(2). The condition rank E Ĩ = k holds if and only if

(1) (dα) k+1 ∧ ω 1 ∧ • • • ∧ ω m = 0, for any α ∈ Ĩ;
(2) There exists α ∈ Ĩ such that (dα

) k ∧ ω 1 ∧ • • • ∧ ω m = 0.
On one hand, since rank E I = k, for any Ω = m i=1 η i (z, y)ω i ∈ I, we have

(d Ω) k ∧ ω 1 ∧ • • • ω m = 0, which implies that for any α = m i=1 α i (z)ω i ∈ Ĩ, we have (dα) k+1 ∧ ω 1 ∧ • • • ω m = 0. On the other hand, the condition rank E I = k implies that there exists a 1-form Ω = m i=1 η i (z, y)ω i ∈ I such that (d Ω) k ∧ ω 1 ∧ • • • ω m (0, 0) = 0 (recall that the coordinates are centered at zero). Define a differential 1-form Ω ∈ Ĩ by Ω = m i=1 c i ω i
where the constant coefficients c i are given by c i = η(0, 0). We have d Ω = m i=1 c i dω i and thus

(d Ω) k ∧ ω 1 ∧ • • • ω m (0) = (d Ω) k ∧ ω 1 ∧ • • • ω m (0, 0) = 0 which implies that rank E Ĩ = k.
(3). If in a neighborhood of 0 ∈ R m(k+1)+k there exists a point q such that rank R( Ĩ)(q) = m + k + mks where s > 0, it follows that rank C( Ĩ⊥ )(q) = s > 0 implying that there exists 0 = ṽ ∈ Ĩ⊥ (q) such that ṽ dω(q) ∈ Ĩ(q) for any ω ∈ Ĩ. It follows that v = (ṽ, 0), where 0 stands for the y-components of v, satisfies v ∈ I ⊥ (q) and v ∈ C(I ⊥ )(q) at any q = (q, q) for any q corresponding to the y-components, in particular at q = (q, 0). This contradicts the assumption rank C(I ⊥ )(q) = 0 in a neighborhood of q 0 = 0. Therefore we have rank R( Ĩ) = m + k + mk.

In the dual language of vector fields and distributions, in particular, using D instead of I, as well as C(D) instead of R(I), and calculating vector fields annihilated by the differential forms of (3.4.1), we obtain the following result:

Theorem 3.4.4 (Extended Byrant normal form, dual version) Assume that m = 2. Let D be a distribution of constant rank k + mk + l defined on a manifold M of dimension m + k + mk + l such that D (1) = T M. If D satisfies,
in a neighborhood of a point q ∈ M, the following conditions:

(i) the Engel rank of D ⊥ is constant and equals k, (ii) the rank of the characteristic distribution C(D) of D is constant and equals l, then there exist local coordinates (z, y) = (x, p, y) = (x 1 , . . . , x k , p 0 1 , . . . , p k 1 , p 0 2 , . . . , p k 2 , . . . , p 0 m , . . . , p k m , y 1 , . . . , y l ) around q such that

D = D ⊕ C, where D is the Cartan distribution CC 1 (R k , R m ) expressed in z = (x, p)-coordinates, that is, D = span { ∂ ∂x i + m s=1 p i s ∂ ∂p 0 s , ∂ ∂p i j , 1 ≤ i ≤ k, 1 ≤ j ≤ m} and C = C(D) = span { ∂ ∂y } = span { ∂ ∂y 1 , . . . , ∂ ∂y l }
is the characteristic distribution of D.

Proof: Since F is involutive and has constant rank nr 2 , there exist, by Frobenius theorem, local coordinates (x 1 , . . . , x n-r 1 -r 2 , p 1 1 , . . . , p r 1 1 , p 1 2 , . . . , p r 2 2 ), in a neighborhood O of q ∈ M, such that F ⊥ = span {dp 1 2 , . . . , dp r 2 2 }. Moreover, the condition that E is involutive and has constant corank r 1 + r 2 implies that there exist r 1 + r 2 smooth functions ϕ 1 , . . . , ϕ r 1 +r 2 in O such that

E ⊥ = span {dϕ 1 , . . . , dϕ r 1 +r 2 },
The relation E ⊂ F yields immediately that F ⊥ ⊂ E ⊥ , i.e., span {dp 1 2 , . . . , dp r 2 2 } ⊂ span {dϕ 1 , . . . , dϕ r 1 +r 2 }.

which implies that there exist r 1 functions among the ϕ i 's (say ϕ 1 , . . . , ϕ r 1 , after a reordering) such that span {dϕ 1 , . . . , dϕ r 1 , dp 1 2 , . . . , dp r 2 2 } = span {dϕ 1 , . . . , dϕ r 1 +r 2 }.

Introduce new coordinates by replacing p i 1 by ϕ i , for 1 ≤ i ≤ r 1 , and in new coordinates we have clearly

F ⊥ = {dp 1 2 , . . . , dp r 2 2 } E ⊥ = {dp 1 1 , . . . , dp r 1 1 , dp 1 2 , . . . , dp r 2 2 }.
Recall that we have defined the characteristic distribution C(E) of a distribution E pointwise as C(E)(q) = {v ∈ E(q) : v dω(q) ∈ I(q), ∀ ω ∈ I}, where I = E ⊥ . Lemma 3.5.3 Let E be a distribution of rank r + l defined on a smooth manifold M of dimension r + s, for l ≤ s. Suppose that at any point q ∈ M, the characteristic distribution C(E) of E satisfies C(E)(q) = 0. Assume that in a local coordinate system (p 1 , . . . , p r , y 1 , . . . , y s ) in a neighborhood O(q 0 ) of a point q 0 ∈ M, the distribution E is spanned by

E = span { ∂ ∂p 1 , . . . , ∂ ∂p r , f 1 , . . . , f l },
where

f 1 = f 1 1 ∂ ∂y 1 + f 2 1 ∂ ∂y 2 + • • • + f s 1 ∂ ∂y s . . . f l = f 1 l ∂ ∂y 1 + f 2 l ∂ ∂y 2 + • • • + f s l ∂ ∂y s (3.5.1) for 1 ≤ i ≤ m and 1 ≤ j ≤ s. Then rank ( ∂f ∂p )(q) = r,
for any q ∈ O(q 0 ), where p = (p 1 , . . . , p r ) and the map f : O(q 0 ) → R ls is formed by

f = (f 1 1 , . . . , f s 1 , f 1 2 , . . . , f 1 l , . . . , f s l ) ⊤ .
Proof: Assume that at a point q ∈ O(q 0 ), we have rank ( ∂f ∂p )(q) = r ′ < r and, without loss of generality, that the first r ′ columns of ( ∂f ∂p )(q) are independent. For any r ′ + 1 ≤ i ≤ r, we have

∂f ∂p i (q) = r ′ j=1 c j i ∂f ∂p j (q),
for some constants c j i ∈ R. Define the vector fields

g i = ∂ ∂p i - r ′ j=1 c j i ∂ ∂p j , r ′ + 1 ≤ i ≤ r.
We have, [g i , f t ](q) = 0, for r ′ + 1 ≤ i ≤ r and any f t , 1 ≤ t ≤ l, which implies that for any ω j ∈ I = E ⊥ , we have (g i dω j )(q) = 0 and hence g i (q) ∈ C(E)(q), for r ′ +1 ≤ i ≤ r, thus contradicting the assumption C(E)(q) = 0. Therefore, rank ( ∂f ∂p )(q) = r for any q ∈ O(q 0 ).

Proof of Theorem 3.3.2

Proof: For simplicity, we will write p σ 1 ,σ 2 instead of p (σ 1 ,σ 2 ) and p 0 instead of p (0,0) . If D is equivalent to the Cartan distribution for surfaces CC n (R 2 , R m ), then by a direct calculation, we can easily check that the conditions (i) -(iv) are satisfied.

Consider now a distribution that satisfies, at a point q ∈ M, the conditions (i) -(iv). To start with, observe that the distribution D (n-1) (of rank m 2 (n + 1)(n + 2)m + 2, by (ii)) satisfies the conditions of Theorem (3.4.4) with k = 2 and l = m 2 (n + 1)(n + 2) -3m. Indeed, (iii) implies that the Engel rank of (D (n-1) ) ⊥ equals two and by (ii) its characteristic distribution C n-1 is of constant rank given by rank

C n-1 = rank D (n-2) -2 = m 2 (n + 1)(n + 2) -3m = c n-1 ,
(we used rank D (n-2) = m 2 (n + 1)(n + 2) -3m + 2 following from (i)). Therefore, by Theorem 3.4.4, D (n-1) is the direct sum of its characteristic distribution C n-1 and a distribution locally equivalent to the Cartan distribution CC 1 (R 2 , R m ). It follows that there exists a local coordinate system, in a neighborhood O(q) of q, denoted by (x 1 , x 2 , p σ j ), for 1 ≤ j ≤ m, 0 ≤ |σ| ≤ n, such that

D (n-1) = span ∂ ∂p σ j 2≤|σ|≤n , ∂ ∂p 1,0 j , ∂ ∂p 0,1 j , f n-1 , g n-1 , 1 ≤ j ≤ m ,
where the vector fields f n-1 and g n-1 are given by

f n-1 = ∂ ∂x 1 + m j=1 p 1,0 j ∂ ∂p 0 j g n-1 = ∂ ∂x 2 + m j=1 p 0,1 j ∂ ∂p 0 j .
Moreover, the characteristic distribution C n-1 is given by

C n-1 = span ∂ ∂p σ j , 1 ≤ j ≤ m, 2 ≤ |σ| ≤ n .
Observe that the coordinates y i , 1

≤ i ≤ l = c n-1 of Theorem 3.4.4 correspond to p σ j , for 1 ≤ j ≤ m, 2 ≤ |σ| ≤ n (notice that the number of p σ j 's is, indeed, c n-1 ). Clearly, the involutive subdistribution L n-1 of corank two in D n-1 is given by L n-1 = span ∂ ∂p σ j , 1 ≤ j ≤ m, 1 ≤ |σ| ≤ n .
For a notational convenience we put C n = L n-1 . We have

C 1 ⊂ • • • ⊂ C n-1 ⊂ C n = L n-1 ,
(see Lemma 3.5.1) and thus the coordinates p σ j can be chosen, due to a successive application of Lemma 3.5.2, such that we have, for 1 ≤ j ≤ m,

C i = span ∂ ∂p σ j , 1 ≤ j ≤ m, n -i + 1 ≤ |σ| ≤ n .
Now the proof follows from Lemma 3.5.4 given below. Notice that the hypothesis of Lemma 3.5.4 holds for k = 1. Therefore a successive applying the lemma (n -1)times (for k from 1 up to n -1) yields local coordinates (x 1 , x 2 , p σ j ), for 1

≤ j ≤ m, 0 ≤ |σ| ≤ n, in which D = D (0) = C 1 ⊕ span {f 0 , g 0 },
where

C 1 = span { ∂ ∂p σ j , 1 ≤ j ≤ m, |σ| = n}, and
f 0 = ∂ ∂x 1 + n-1 |σ|=0 m j=1 p σ+1 1 j ∂ ∂p σ j g 0 = ∂ ∂x 2 + n-1 |σ|=0 m j=1 p σ+1 2 j ∂ ∂p σ j ,
which shows that D is locally equivalent to the Cartan distribution for surfaces CC n (R 2 , R m ). Assume that for a certain 1 ≤ k ≤ n, there exist coordinates

(x 1 , x 2 , p σ j ), for 1 ≤ j ≤ m, 0 ≤ |σ| ≤ n, centered at 0 ∈ R N , where N = m 2 (n + 1)(n + 2) + 2, such that L n-1 = span ∂ ∂p σ j , 1 ≤ j ≤ m, 1 ≤ |σ| ≤ n C i = span ∂ ∂p σ j , 1 ≤ j ≤ m, n -i + 1 ≤ |σ| ≤ n for 1 ≤ i ≤ n -1 and D (n-k) = C n-k+1 ⊕ span {f n-k , g n-k } where f n-k = ∂ ∂x 1 + k-1 |σ|=0 m j=1 p σ+1 1 j ∂ ∂p σ j g n-k = ∂ ∂x 2 + k-1 |σ|=0 m j=1 p σ+1 2 j ∂ ∂p σ j .
(Recall that for a notational convenience we put, for

k = 1, C n-k+1 = C n = L n-1 ).
Then we can modify the coordinates in such a way that in new coordinates

(x 1 , x2 , pσ j ), for 1 ≤ j ≤ m, 0 ≤ |σ| ≤ n, centered at 0 ∈ R N , we have L n-1 = span ∂ ∂ pσ j , 1 ≤ j ≤ m, 1 ≤ |σ| ≤ n C i = span ∂ ∂ pσ j , 1 ≤ j ≤ m, n -i + 1 ≤ |σ| ≤ n for 1 ≤ i ≤ n -1 and D (n-k-1) = C n-k ⊕ span {f n-k-1 , g n-k-1 } where f n-k-1 = ∂ ∂ x1 + k |σ|=0 m j=1 pσ+1 1 j ∂ ∂ pσ j g n-k-1 = ∂ ∂ x2 + k |σ|=0 m j=1 pσ+1 2 j ∂ ∂ pσ j .
Proof: (of Lemma 3.5.4) To explain the idea of the proof, we will show the lemma for k = 1 and then, following similar arguments, for an arbitrary 1 ≤ k ≤ n.

Suppose that the assumptions of the lemma hold for k = 1. The condition (ii) of Theorem 3.3.2 implies that corank (C n-1 ⊂ D (n-2) ) = 2 and thus there exist two vector fields f n-2 and g n-2 such that locally

D (n-2) = C n-1 ⊕ span {f n-2 , g n-2 } = C n-2 ⊕ span ∂ ∂p σ j , 1 ≤ j ≤ m, |σ| = 2 ⊕ span {f n-2 , g n-2 }.
We have

D (n-1) = C n-1 ⊕ span ∂ ∂p σ j , 1 ≤ j ≤ m, |σ| = 1 ⊕ span {f n-1 , g n-1 }, (3.5.2)
then the relation 1) implies immediately that

f n-2 , g n-2 ∈ D (n-2) ⊂ D (n-
f n-2 = s 1 f n-1 + s 2 g n-1 + m j=1 (A 1 j ∂ ∂p 1,0 j + A 2 j ∂ ∂p 0,1 j ) mod C n-1 g n-2 = t 1 f n-1 + t 2 g n-1 + m j=1 (B 1 j ∂ ∂p 1,0 j + B 2 j ∂ ∂p 0,1 j ) mod C n-1 where s 1 , s 2 , t 1 , t 2 , A i j , B i j , for 1 ≤ i ≤ 2, 1 ≤ j ≤ m, are smooth functions defined in a neighborhood O of 0 ∈ R N .
The condition (iv) of Theorem 3.3.2, applied for i = n -2, implies that f n-2 and g n-2 are independent at 0 ∈ R N , modulo L n-1 . Hence (s 1 t 2s 2 t 1 )(0) = 0 and thus by a suitable choice of f n-2 and g n-2 (and by renaming A i j and B i j as well as taking a smaller O, if necessary), we can assume that

f n-2 = f n-1 + m j=1 (A 1 j ∂ ∂p 1,0 j + A 2 j ∂ ∂p 0,1 j ) g n-2 = g n-1 + m j=1 (B 1 j ∂ ∂p 1,0 j + B 2 j ∂ ∂p 0,1 j
).

We will now analyze the functions A i j and B i j , for 1 ≤ i ≤ 2 and 1 ≤ j ≤ m. First, we replace p 1,0 j by p 1,0

j -A 1 j (0)x 1 -B 1 j (0)x 2 and p 0,1 j by p 0,1 j -A 2 j (0)x 1 -B 2 j (0)x 2 in order to obtain A 1 j (0) = A 2 j (0) = B 1 j (0) = B 2 j (0) = 0. This change of coordinates preserves L n-1 and C i , for 1 ≤ i ≤ n -1. Secondly, the characteristic distribution of D (n-2) is C n-2 = span ∂ ∂p σ j , 1 ≤ j ≤ m, 3 ≤ |σ| ≤ n . It follows that, for any 1 ≤ j ≤ m and 3 ≤ |σ| ≤ n, ∂ ∂p σ j , f n-2 = ∂ ∂p σ j , g n-2 = 0
and hence

A i j = A i j (x 1 , x 2 , p σ ), B i j = B i j (x 1 , x 2 , p σ ), 0 ≤ |σ| ≤ 2, (3.5.3)
for i = 1, 2 and 1 ≤ j ≤ m, where p σ = (p σ 1 , . . . , p σ m ). Thirdly, a direct calculation gives 1) , it is easy to conclude that B 1 j = A 2 j , for 1 ≤ j ≤ m. Now consider the distribution

[f n-2 , g n-2 ] = (A 2 1 -B 1 1 ) ∂ ∂p 0 1 + (A 2 2 -B 1 2 ) ∂ ∂p 0 2 + • • • + (A 2 m -B 1 m ) ∂ ∂p 0 m mod L n-1 . Since [f n-2 , g n-2 ] ∈ D (n-
E n-2 = span ∂ ∂x 1 + m j=1 p 1,0 j ∂ ∂p 0 j + A 1 j ∂ ∂p 1,0 j + A 2 j ∂ ∂p 0,1 j , ∂ ∂x 2 + m j=1 p 0,1 j ∂ ∂p 0 j + B 1 j ∂ ∂p 1,0 j + B 2 j ∂ ∂p 0,1 j , ∂ ∂p σ j , 1 ≤ j ≤ m, |σ| = 2
around 0 ∈ R 6m+2 , equipped with coordinates (x 1 , x 2 , p σ j ), for 1 ≤ j ≤ m and 0 ≤ |σ| ≤ 2. It is well defined (because of (3.5.3)) and it satisfies the assumptions of Lemma 3.5.3 with r = 3m, l = 2 and s = 3m + 2. Indeed C(E n-2 ) = 0 because C(D (n-2) ) = C n-2 and, moreover, the coordinates (p i ) for 1 ≤ i ≤ r, correspond to (p σ j ), for 1 ≤ j ≤ m and |σ| = 2, and the coordinates y i , 1 ≤ i ≤ s, to (x 1 , x 2 , p σ j ), 1 ≤ j ≤ m and 0 ≤ |σ| ≤ 1. It follows by Lemma 3.5.3 that the map given by xi

= x i , i = 1, 2 pσ j = p σ j , 1 ≤ j ≤ m, |σ| = 2 p2,0 j = A 1 j , 1 ≤ j ≤ m p1,1 j = A 2 j = B 1 j , 1 ≤ j ≤ m p0,2 j = B 2 j , 1 ≤ j ≤ m is a local diffeomorphism, denoted by ϕ, in a neighborhood of 0 ∈ R m 2 (n+1)(n+2)+2
. It is easy to see that ϕ preserves the nested sequence of distributions

C 1 ⊂ • • • ⊂ C n-1 ⊂ C n = L n-1 , that is, ϕ * C i = span ∂ ∂ pσ j , 1 ≤ j ≤ m, n -i + 1 ≤ |σ| ≤ n ,
for 1 ≤ i ≤ n, and that the vector fields f n-2 and g n-2 become, respectively,

ϕ * f n-2 = ∂ ∂ x1 + m j=1 p1,0 j ∂ ∂ p0 j + p2,0 j ∂ ∂ p1,0 j + p1,1 j ∂ ∂ p0,1 j ϕ * g n-2 = ∂ ∂ x2 + m j=1 p0,1 j ∂ ∂ p0 j + p1,1 j ∂ ∂ p1,0 j + p0,2 j ∂ ∂ p0,1 j .
To prove the general case, which follows the same line, assume that in a coordinate system (x 1 , x 2 , p σ j ), 1 ≤ j ≤ m, 0 ≤ |σ| ≤ n, centered at 0 ∈ R N , we have

D (n-k) = C n-k+1 ⊕ span {f n-k , g n-k } and C i = span ∂ ∂p σ j , 1 ≤ j ≤ m, n -i + 1 ≤ |σ| ≤ n , for 1 ≤ i ≤ n (recall that we denoted L n-1 = C n ),
where

f n-k = ∂ ∂x 1 + k-1 |σ|=0 m j=1 p σ+1 1 j ∂ ∂p σ j g n-k = ∂ ∂x 2 + k-1 |σ|=0 m j=1 p σ+1 2 j ∂ ∂p σ j .
By the condition (ii), we have corank (C n-k ⊂ D (n-k-1) ) = 2 which implies that there exist two vector fields f n-k-1 , g n-k-1 such that

D (n-k-1) = C n-k ⊕ span {f n-k-1 , g n-k-1 } = C n-k-1 ⊕ span { ∂ ∂p σ |σ|=k+1 } ⊕ span {f n-k-1 , g n-k-1 }.
Since f n-k-1 , g n-k-1 ∈ D (n-k-1) ⊂ D (n-k) , they can be expressed as

f n-k-1 = s 1 f n-k + s 2 g n-k + m j=1 (A 1 j ∂ ∂p k,0 j + A 2 j ∂ ∂p k-1,1 j + • • • + A k+1 j ∂ ∂p 0,k j ) mod C n-k g n-k-1 = t 1 f n-k + t 2 g n-k + m j=1 (B 1 j ∂ ∂p k,0 j + B 2 j ∂ ∂p k-1,1 j + • • • + B k+1 j ∂ ∂p 0,k j ) mod C n-k ,
where s 1 , s 2 , t 1 , t 2 , A i j , B i j , for 1 ≤ i ≤ k + 1 and 1 ≤ j ≤ m, are smooth functions defined in a neighborhood O of 0 ∈ R N . The condition (iv), applied for i = nk -1, implies that f n-k-1 and g n-k-1 are independent at 0 ∈ R N , modulo L n-1 . Hence (s 1 t 2s 2 t 1 )(0) = 0 and thus by a suitable choice of f n-k-1 and g n-k-1 , we can assume that

f n-k-1 = f n-k + m j=1 (A 1 j ∂ ∂p k,0 j + A 2 j ∂ ∂p k-1,1 j + • • • + A k+1 j ∂ ∂p 0,k j ) g n-k-1 = g n-k + m j=1 (B 1 j ∂ ∂p k,0 j + B 2 j ∂ ∂p k-1,1 j + • • • + B k+1 j ∂ ∂p 0,k j
).

We will now analyze the functions A i j , B i j , for 1 ≤ i ≤ k +1 and 1 ≤ j ≤ m. First, for 0 ≤ i ≤ k and 1 ≤ j ≤ m, we will replace p i,k-i by p i,k-i -A i+1 j (0)x 1 -B i+1 j (0)x 2 in order to obtain A i+1 j (0) = B i+1 j (0) = 0. This change of coordinates preserves L n-1 and C i , for 1 ≤ i ≤ n -1. Secondly, the characteristic distribution of

D (n-k-1) is C n-k-1 = span ∂ ∂p σ j , 1 ≤ j ≤ m, k + 2 ≤ |σ| ≤ n . It follows that, for any 1 ≤ j ≤ m and k + 2 ≤ |σ| ≤ n, [ ∂ ∂p σ j , f n-k-1 ] = [ ∂ ∂p σ j , g n-k-1 ] = 0
and hence

A i j = A i j (x 1 , x 2 , p σ ), B i j = B i j (x 1 , x 2 , p σ ), 0 ≤ |σ| ≤ k + 1, (3.5.4) for 1 ≤ i ≤ k + 1, 1 ≤ j ≤ m
, where p σ = (p σ 1 , . . . , p σ m ). Thirdly, a direct calculation shows that

[f n-k-1 , g n-k-1 ] = m j=1 (A 2 j -B 1 j ) ∂ ∂p k-1,0 j + (A 3 j -B 2 j ) ∂ ∂p k-2,1 j + • • • +(A k+1 j -B k j ) ∂ ∂p 0,k-1 j mod C n-k+1 .
On the other hand, [f n-k-1 , g n-k-1 ] ∈ D n-k = C n-k+1 ⊕ span {f n-k-1 , g n-k-1 }, which implies that B i j = A i+1 j , for 1 ≤ i ≤ k and 1 ≤ j ≤ m. Now consider the distribution

E n-k-1 = span ∂ ∂x 1 + m j=1 k-1 |σ|=0 p σ+1 1 j ∂ ∂p σ j + A 1 j ∂ ∂p k,0 j + A 2 j ∂ ∂p k-1,1 j + • • • + A k+1 j ∂ ∂p 0,k j , ∂ ∂x 2 + m j=1 k-1 |σ|=0 p σ+1 2 j ∂ ∂p σ j + B 1 j ∂ ∂p k,0 j + B 2 j ∂ ∂p k-1,1 j + • • • + B k+1 j ∂ ∂p 0,k j , ∂ ∂p σ j , 1 ≤ j ≤ m, |σ| = k + 1 . around 0 ∈ R m 2 (k+2)(k+3)+2
, equipped with the coordinates (x 1 , x 2 , p σ j ), for 1 ≤ j ≤ m and 0 ≤ |σ| ≤ k + 1. The distribution E n-k-1 is well defined (because of (3.5.4)) and, moreover, it satisfies the assumption of Lemma 3.5.3 with r = (k + 2)m, l = 2 and s = m 2 (k + 1)(k + 2) + 2. Indeed, C(E n-k-1 ) = 0 because C(D (n-k-1) ) = C n-2 and, moreover, the coordinates p i , 1 ≤ i ≤ r, correspond to p σ j , 1 ≤ j ≤ m, |σ| = k + 1, (notice that ♯{σ : |σ| = k + 1} = k + 2), and the coordinates y i , 1 ≤ i ≤ s, to (x 1 , x 2 , p σ j ), 1 ≤ j ≤ m and 0 ≤ |σ| ≤ k. It follows by Lemma 3.5.3 that the map ϕ

given by xi = x i , i = 1, 2 pσ j = p σ j , 1 ≤ j ≤ m, |σ| = k + 1 pk+1,0 j = A 1 j , 1 ≤ j ≤ m pk,1 j = A 2 j = B 1 j , 1 ≤ j ≤ m . . . p1,k j = A k+1 j = B k j , 1 ≤ j ≤ m p0,k+1 j = B k+1 j , 1 ≤ j ≤ m, is a local diffeomorphism, denoted by ϕ, in a neighborhood of 0 ∈ R m 2 (n+1)(n+2)+2
. It is easy to see that the local diffeomorphism ϕ preserves the nested sequence of distributions

C 1 ⊂ • • • ⊂ C n-1 ⊂ C n = L n-1 ,
that is,

ϕ * C i = span ∂ ∂ pσ j , 1 ≤ j ≤ m, n -i + 1 ≤ |σ| ≤ n ,
for 1 ≤ i ≤ n, and that the vector fields f n-k-1 and g n-k-1 become, respectively,

ϕ * f n-k-1 = ∂ ∂ x1 + k |σ|=0 m j=1 pσ+1 1 j ∂ ∂ pσ j ϕ * g n-k-1 = ∂ ∂ x2 + k |σ|=0 m j=1 pσ+1 2 j ∂ ∂ pσ j .
3.5.3 Proof of Theorem 3.3.4

Proof: Let I be a codistribution defined by I = D ⊥ and locally spanned by differential 1-forms ω 1 , . . . , ω m , that is, I = span{ω 1 , . . . , ω m }.

(i) ⇒ (iv): Since rank C I = k, there exist π 1 , . . . , π k ∈ Λ 1 (M) such that The condition C(D) = 0 implies that the differential forms ω i , π j , η i j are independent everywhere. To see this, assume that there exists a point q ∈ M such that the differential forms ω i , π j , η i j , for 1 ≤ i ≤ m and 1 ≤ j ≤ k are dependent at q. Since dim M = m + k + mk, there exists a vector 0 = v ∈ T q M such that < η i j (q), v >=< π l (q), v >=< ω r (q), v >= 0. It follows that v ∈ I ⊥ (q) = D(q) and that dω i (v, w) = 0, for any w ∈ D(q), thus implying that 0 = v ∈ C(D)(q) and contradicting the assumption C(D)(q) = 0 for any q ∈ M. Now taking the exterior derivative of both sides of (3.5.6), we get k j=1 η i j ∧ dπ j = 0 mod I, π 1 , . . . , π k which implies for any 1 ≤ j ≤ k,

π 1 ∧ • • • ∧ π k ∧ ω 1 ∧ • • • ∧ ω m = 0,
dπ j ≡ 0 mod (I, π 1 , . . . , π k , η i 1 , . . . , η i k ), (3.5.7)
where (•) stands for the ideal in the exterior algebra. Notice that the above system is formed by m relations, obtained for 1 ≤ i ≤ m by the corresponding differential forms ω 1 , . . . , ω m . Let F be a distribution defined by

F = span {ω 1 , . . . , ω m , π 1 , . . . , π k } ⊥ = span {f ∈ D : π j (f ) = 0, 1 ≤ j ≤ k}.
Obviously, F satisfies corank (F ⊂ D) = k and now we will show that F is involutive.

Define the distributions F i , for 1 ≤ i ≤ m, as

F i = span {ω 1 , . . . , ω m , π 1 , . . . , π k , η i 1 , . . . , η i k } ⊥ = span {f ∈ D : π j (f ) = 0, η i j (f ) = 0, 1 ≤ j ≤ k}.
The relation (3.5.6) implies that for any 1 ≤ s ≤ m,

dω s = k j=1 η s j ∧ π j + m i=1 A s i ∧ ω i . (3.5.8)
On one hand, for any f, g ∈ F i and 1 ≤ s ≤ m,

dω s (f, g) = k j=1 η s j ∧ π j (f, g) + m i=1 A s i ∧ ω i (f, g) = 0,
on the other hand,

dω s (f, g) = L f ω s (g) -L g ω s (f ) -ω s ([f, g]) = -ω s ([f, g]).
Therefore ω s ([f, g]) = 0 for 1 ≤ s ≤ m and any f, g ∈ F i . Moreover, since dπ j ≡ 0 mod (I, π 1 , . . . , π k , η i 1 , . . . , η i k ), by an analogous argument as above, we obtain that dπ j (f, g) = -π j ([f, g]) = 0, for 1 ≤ j ≤ k. Therefore it can be concluded that for any f, g ∈ F i , we always have [f, g] ∈ F which implies that

[F i , F i ] ⊂ F , 1 ≤ i ≤ m.
We claim that F = m i=1 F i . The relation m i=1 F i ⊂ F is obvious. Recall that the differential forms ω 1 , . . . , ω m , π 1 , . . . , π k , η i j , for 1 ≤ i ≤ m and 1 ≤ j ≤ k, are independent everywhere so they generate Λ 1 (M) over C ∞ (M). Choose vector fields f 1 1 , . . . , f 1 k , . . . , f m 1 , . . . , f m k , such that f i j , for 1 ≤ i ≤ m and 1 ≤ j ≤ k, satisfy

η l s (f i j ) = 1 i = l, j = s 0 otherwise.
and, moreover,

F = span {f 1 1 , . . . , f 1 k , . . . , f m 1 , . . . , f m k }.
Then the subdistributions F i , for 1 ≤ i ≤ m, are given by

F i = span {f 1 1 , . . . , f 1 k , . . . , f i-1 1 , . . . , f i-1 k , f i+1 1 , . . . , f i+1 k , . . . , f m 1 , . . . , f m k }.
Clearly, for any f i j ∈ F , where 1 ≤ i ≤ m and 1 ≤ j ≤ k, we have f i j ∈ F s , for any s = i and 1 ≤ s ≤ m. This implies that F ⊂ m i=1 F i and thus we get

F = F 1 + • • • + F m .
Moreover, it is easy to see that it is enough to take in the above sum only two terms corresponding to any 1 ≤ i = j ≤ m.

In order to prove involutivity of F , consider any two vector fields f l j , f r s in F = m i=1 F i . Since m ≥ 3, there always exists F i such that i = l and i = r and thus If rank (L ∩ L) = mkk, then there exist vector fields h 1 , . . . , h mk-k , f 1 , . . . , f k , g 1 , . . . , g k such that L = span {h 1 , . . . , h mk-k , f 1 , . . . , f k } L = span {h 1 , . . . , h mk-k , g 1 , . . . , g k }.

f l j , f r s ∈ F i . Recall that [F i , F i ] ⊂ F and thus we get [f l j , f r s ] ∈ F , proving that [F , F ] ⊂ F , i.
On one hand, L ∪ L = span {h 1 , . . . , h mk-k , f 1 , . . . , f k , g 1 , . . . , g k } ⊂ D.

On the other hand, we have

rank (L ∪ L) = rank L + rank L -rank (L ∩ L) = k + mk = rank D.
This implies that D = span {h 1 , . . . , h mk-k , f 1 , . . . , f k , g 1 , . . . , g k } and, moreover, h i for 1 ≤ i ≤ mkk, are characteristic vector fields of D which contradicts the condition C(D) = 0.

Assume now that rank (L ∩ L) = mkr where 1 ≤ r < k. Then there exist vector fields h 1 , . . . , h mk-r , f 1 , . . . , f r , g 1 , . . . , g r such that L = span {h 1 , . . . , h mk-r , f 1 , . . . , f r } L = span {h 1 , . . . , h mk-r , g 1 , . . . , g r }.

The involutivity of L implies that, without loss of generality, we can choose the h i 's and f j 's that commute. Therefore there exist local coordinates (x, y) = (x 1 , . . . , x mk , y 1 , . . . , y m+k ), in a neighborhood of any point q ∈ M , such that for 1 ≤ i ≤ mkr and 1 ≤ j ≤ r,

h i = ∂ ∂x i , f j = ∂ ∂x mk-r+j , g j = ∂ ∂y j + k+m l=r+1 g l j (x, y) ∂ ∂y l mod L
(the form of g 1 , . . . , g r follows from their independence). Since L is involutive, it is easy to see that ∂g l j ∂x i ≡ 0, for 1 ≤ i ≤ mkr, 1 ≤ j ≤ r and r + 1 ≤ l ≤ k + m which implies that g l j = g l j (x mk-r+1 , . . . , x mk , y). Clearly, L ∪ L = span {h 1 , . . . , h mk-r , f 1 , . . . , f r , g 1 , . . . , g r } ⊂ D, and thus there exist kr vector fields g r+1 , . . . , g k such that D = span {h 1 , . . . , h mk-r , f 1 , . . . , f r , g 1 , . . . , g r , g r+1 , . . . A direct computation gives

∂G ∂x =                0 • • • 0 ∂g r+1 1 ∂x mk-r+1 • • • ∂g r+1 1 ∂x mk • • • • • • • • • • • • • • • • • • 0 • • • 0 ∂g k+m r ∂x mk-r+1 • • • ∂g k+m r ∂x mk ∂g k+1 r+1 ∂x 1 • • • ∂g k+1 r+1 ∂x mk-r ∂g k+1 r+1 ∂x mk-r+1 • • • ∂g k+1 r+1 ∂x mk • • • • • • • • • • • • • • • • • • ∂g k+m k ∂x 1 • • • ∂g k+m k ∂x mk-r ∂g k+m k ∂x mk-r+1 • • • ∂g k+m k ∂x mk                = 0 G 12 G 21 G 22 .
Since m ≥ 3, we have rank ∂G ∂x ≤ rank G 21 + rank G 12 ≤ m(kr) + r < mk, which contradicts rank ∂G ∂x (q) = mk. Therefore L and L coincide on M, which is open and dense, and hence (being of constant rank) coincide everywhere on M.

(iv) ⇒ (iii) is obvious.

(iii) ⇒ (i): Choose, locally, m differential 1-forms ω 1 , . . . , ω m and k differential 1-forms π 1 , . . . , π k such that

π 1 ∧ • • • ∧ π k ∧ ω 1 ∧ • • • ∧ ω m =
0, and

F ⊥ = span {π 1 , . . . , π k , ω 1 , . . . , ω m } D ⊥ = span {ω 1 , . . . , ω m } Since dim M = k + m + mk,
locally there exist mk differential 1-forms η 1 , . . . , η mk such that π 1 , . . . , π k , ω 1 , . . . , ω m , η 1 , . . . , η mk are independent everywhere. Then we can choose m + k + mk vector fields g 1 , . . . , g k , h 1 , . . . , h m , f 1 , . . . , f km , satisfying

π i (g j ) = 1 i = j 0 i = j , 1 ≤ i, j ≤ k, ω i (h j ) = 1 i = j 0 i = j , 1 ≤ i, j ≤ m, and
η i (f j ) = 1 i = j 0 i = j , 1 ≤ i, j ≤ mk,
such that D = span {g 1 , . . . , g k , f 1 , . . . , f km } and F = span {f 1 , . . . , f km }. For simplicity, we denote {Ω 1 , . . . , Ω k+m } = {π 1 , . . . , π k , ω 1 , . . . , ω m }.

Then, for any 1 ≤ i ≤ m, the 2-form dω i can be expressed as

dω i = k+m j=1 A i j ∧ Ω j + l<s, B i ls η l ∧ η s , (3.5.9)
for some differential 1-forms A i j and some functions B i ls . On one hand, for any vector fields f l , f s ∈ F , 1 ≤ l < s ≤ km, we have

dω i (f l , f s ) = B i ls .
On the other hand,

dω i (f l , f s ) = L f l ω i (f s ) -L fs ω i (f l ) -ω i ([f l , f s ]) = ω i ([f l , f s ]) = 0.
Therefore we get B i ls ≡ 0 and thus

dω i = k+m j=1 A i j ∧ Ω j = m j=1 α i j ∧ ω j + k s=1 β i s ∧ π s ,
for some differential 1-forms α i j and β i s , which gives immediately that

dω ∧ π 1 ∧ • • • ∧ π k ∧ ω 1 ∧ • • • ∧ ω m ≡ 0
for any ω ∈ I = D ⊥ . By the definition of the Cartan rank, it can be conclude that rank C D = k.

(ii) ⇔ (i) has been proved by Bryant in [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF].

In order to prove L = F1 + • • • + Fm , it is enough to show that Fi = F i , for 1 ≤ i ≤ m, where F i have been defined when proving the implication (i) ⇔ (iv). Recall the equations (3.5.8)

dω i = k j=1 η i j ∧ π j + k s=1 A i s ∧ ω s .
According to the definition of the distribution F i , for any f ∈ F i , we have

ω i (f ) = 0, π j (f ) = 0, η i j (f ) = 0 for 1 ≤ i ≤ m and 1 ≤ j ≤ k. Therefore f dω i = k j=1 η i j (f ) • π j -π j (f ) • η i j + m s=1 A i s (f ) • ω s -ω s (f ) • A i s 104 = m s=1 A i s (f ) • ω i ∈ I = D ⊥ ,
which implies that f ∈ Fi and thus F i ⊂ Fi . On the other hand, for any f ∈ Fi ⊂ D, we have clearly ω s (f ) = 0, for 1 ≤ s ≤ m and

f dω i = k j=1 η i j (f ) • π j -π j (f ) • η i j + m s=1 A i s (f ) • ω s -ω s (f ) • A i s = k j=1 η i j (f ) • π j -π j (f ) • η i j + m s=1 A i s (f ) • ω s .
The above expression belongs to D ⊥ (since f dω i ∈ D ⊥ ) and therefore η i j (f ) = 0 and π j (f ) = 0 because the differential forms η i j , π j , and ω s are independent. This implies that f ∈ F i and thus Fi ⊂ F i implying that Fi = F i .

105 Independently, Guay adapted an approach using exterior differential systems to derive necessary and sufficient conditions for the orbital feedback linearization of single-input systems [START_REF] Guay | An algorithm for orbital feedback linearization of single-input control affine systems[END_REF] and then of multi-input systems [START_REF] Guay | Orbital feedback linearization of multi-input control affine systems[END_REF]. Verifying Guay's conditions requires the search of suitable generators for the associated Pfaffian system. This chapter considers the orbital feedback linearization for specific multi-input affine control systems. Both necessary and sufficient conditions are established so that they can be verified in terms of the original system. In addition, the time re-scaling function can be constructed using those conditions.

The chapter is organized as follows. The definitions and notations are given in Section 4.2. The main result is presented in Section 4.3. In Section 4.4, a complete version of the main theorem is derived due to an analysis of the distributions associated to any control-affine system. Two illustrating examples are proposed in Section 4.5. All proofs are given in Section 4.6.

Orbital feedback equivalence

Consider a control-affine system, with m controls, of the following form

Σ : dx dt = f (x) + m i=1 g i (x)u i , (4.2.1)
where x ∈ X, a C ∞ -manifold of dimension d, and f and g i , for 1 ≤ i ≤ m, are C ∞ -vector fields on X. Throughout this chapter, smooth means C ∞ -smooth.

Define a new time scale τ such that

dt dτ = γ(x(t)),
where γ is a smooth R-valued function on X satisfying γ(•) = 0. Then in the new time scale τ , the system Σ can be rewritten as

dx dτ = dx dt • dt dτ = γ(x)f (x) + m i=1 γ(x)g i (x)u i = γ(x)f (x) + m i=1 g i (x)v i ,
where v i = γ(x)u i , 1 ≤ i ≤ m, are feedback modified controls. A time re-scaling allows, as we have just seen, to multiply the drift f by an arbitrary nonvanishing function while its action on control vector fields g i can be compensated by a suitable feedback transformation u i (t) → γ(x(t))u i (t) = v i (t).

Consider another control system of the same form evolving on a manifold X Σ :

dx dt = f(x) + m i=1 gi (x)ũ i .
The two systems Σ and Σ are said to be orbital feedback equivalent if there exists a diffeomorphism ϕ : X → X, an R m -valued smooth function α = (α 1 , . . . , α m ), a smooth matrix β = (β j i ), 1 ≤ i, j ≤ m, invertible everywhere, and a smooth function γ, satisfying γ(•) = 0, such that

f = ϕ * (γf + m i=1 α i g i ) (4.2.2)
and for 1

≤ i ≤ m, gi = ϕ * ( m j=1 β j i g j ), (4.2.3)
where ϕ * denotes the tangent map of ϕ, i.e.,

(ϕ * f )(x) = Dϕ(ϕ -1 (x)) • f (ϕ -1 (x)).
The system Σ defined by (4.2.1) is said to be orbital feedback linearizable if it is orbital feedback equivalent to a linear system of the form

Λ : ẋ = Ax + m i=1 b i ũi + c. (4.2.4)
There are two reasons to add the extra constant vector field c to the dynamics of the linear control system

ẋ = Ax + m i=1 b i u i .
The first is to consider it around a non-equilibrium point x 0 . Then in coordinates x = xx 0 , the system becomes

ẋ = Ax + m i=1 b i u i + c,
where c = Ax 0 (of course c can be incorporated into Ax if the system is controllable and is considered globally on R d , but this may become impossible if we work locally only). The second reason occurs when dealing with linear time-varying systems of the form

ẋ = A(t)x + m i=1 b i (t)u i .
Add time as a new variable x 0 = t, that is, ẋ0 = 1. If all controllability indices are time-invariant then, as it is well known ( [START_REF] Brunovský | A classification of linear controllable systems[END_REF], [START_REF] Kailath | Linear systems[END_REF], [START_REF] Morse | Structure of index-invariant systems[END_REF], [START_REF] Silverman | Controllability and observability in timevariable linear systems[END_REF]), we can bring the system, via a linear change of coordinates and feedback (both time-varying) to the form

Λ t : ẋ0 = 1 ẋ = Āx + m i=1 bi ūi , (4.2.5)
which in the augmented state space x = (x 0 , x) is of the form (4.2.4) with c = (1, 0, . . . , 0) ⊤ . We will call the system Λ t time-augmented linear system. Throughout this chapter, whenever we speak about orbital feedback linearization, we will always mean orbital feedback equivalence to the time-augmented linear control system Λ t . Of course, the system Λ t is never controllable but it can be accessible [START_REF] Isidori | Nonlinear Control Systems[END_REF], [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF] (the latter if and only if the linear subsystem ( Ā, B) is controllable where B is the matrix whose columns are ( b1 , . . . , bm )). We will speak about controllability indices and Brunovsky canonical form of Λ t meaning the respective objects of ( Ā, B).

Notice that we have defined time re-scaling using any non vanishing function γ. Of course in many problems, it would be more natural to use exclusively positive functions γ which preserve not only unparameterized system's trajectories but also the direction of the time arrow along the trajectories. It is actually easy to observe that if Σ is orbital feedback equivalent to a time-augmented linear system Λ t via a negative time re-scaling function, it is so via a positive time re-scaling one. To see it, assume that Σ is orbital feedback equivalent to a time-augmented form (4.2.5) via a feedback transformation and a time re-scaling function given by γ < 0. Then the same feedback transformation completed by the time re-scaling function defined by -γ > 0 brings Σ into ẋ0 = -1

ẋ = -Āx - m i=1
bi ūi , and it is enough to apply the linear isomorphism replacing x 0 by -x 0 . Notice that, in general, the pairs ( Ā, B) and (-Ā, -B) are not conjugated via a linear isomorphism but are feedback equivalent and can be brought to the same t-augmented Brunovsky canonical form Λ Br t (see the definition at the beginning of Section 4.4). More precisely, we have Proposition 4.2.1 The following are equivalent, locally around x 0 ∈ X:

(i) Σ is orbital feedback equivalent to a t-augmented linear system; (ii) Σ is orbital feedback equivalent, with γ < 0, to the t-augmented Brunovsky canonical form Λ Br t ; (iii) Σ is orbital feedback equivalent, with γ > 0, to the t-augmented Brunovsky canonical form Λ Br t ;

The aim of this chapter is to find checkable geometric conditions for the problem: when is the multi-input, i.e., m ≥ 2, system Σ :

dx dt = f (x) + m i=1 g i (x)u i , f (x 0 ) = 0 (4.2.6)
where x ∈ X, locally orbital feedback equivalent to a time-augmented linear system Λ t with equal controllability indices? The statement of the problem requires that f (x 0 ) = 0, where x 0 is a given point around which we work, and that dim X = (n + 1)m + 1, where all controllability indices (which coincide, by assumption) equal n + 1. We will assume those two conditions throughout.

Main result

Put ad 0 f g i = g i and inductively ad j+1 f g i = [f, ad j f g i ], for 1 ≤ i ≤ m and j ≥ 0. We define the following distributions

G = span {g 1 , . . . , g m }, G j f = span {f, g i , ad f g i , . . . , ad j-1 f g i , 1 ≤ i ≤ m}, for 1 ≤ j ≤ n + 1.
If we suppose dim G n+1 f (x) = (n + 1)m + 1 in a neighborhood of x 0 (which we will, indeed, assume below), then we have dim G j f (x) = jm + 1, for 0 ≤ j ≤ n. It follows that m differential 1-forms ω 1 , . . . , ω m ∈ Λ(X) are defined uniquely by

ω j (h) = 0, for any h ∈ G n f , ω j (ad n f g i ) = δ j i (4.3.1)
for 1 ≤ i, j ≤ m. Using those differential forms, we introduce the following R-valued functions:

T k,l i,j = ω k ([ad n-1 f g i , ad l f g j ]) (4.3.2)
for any 0 ≤ l ≤ n -1 and any 1 ≤ i, j, k ≤ m such that i = j in the case l = n -1. Of course the functions T k,n-1 i,i can also be defined (actually, they vanish), but we exclude them to simplify formulations of results. We have obviously

T k,n-1 i,j = -T k,n-1 j,i , (4.3.3)
for any 1 ≤ i, j, k ≤ m. Our main result on orbital feedback linearization is as follows:

Theorem 4.3.1 The multi-input control-affine system Σ, defined by (4.2.6) for m ≥ 2, is locally orbital feedback equivalent to a time-augmented linear system Λ t , with all controllability indices equal to n+1, if and only if it satisfies, in a neighborhood of x 0 ,

(OPL1) dim G n+1 f (x) = (n + 1)m + 1; (OPL2) [G j f , G j f ] ⊂ G j+1 f , for 1 ≤ j ≤ n; (OPL3) [G, G 2 f ] ⊂ G 2 f ; (OPL4) The functions T k,l
i,j satisfy the following conditions:

(i) T k,l i,j = 0, for 1 ≤ k = i ≤ m, 1 ≤ j ≤ m, if l < n -1 1 ≤ i = k = j ≤ m, if l = n -1, (ii) T i,l i,j = T k,l k,j , for 1 ≤ l ≤ n -1, 1 ≤ i, j, k ≤ m s.t j = i, k if l = n -1,
(iii) when m = 2, then additionally, the distribution defined by

B = span {g i , ad l f g i + b l i f, i = 1, 2 and 1 ≤ l ≤ n -1}
must be involutive, where

b l i = T 1,l 1,i = T 2,l 2,i , for 1 ≤ i ≤ 2, 1 ≤ l ≤ n -2 b n-1 1 = T 1,n-l 1,2 , b n-1 2 = T
(OFL3) ′ [G, G j f ] ⊂ G j f , 2 ≤ j ≤ n + 1.
But it can be proved that (OFL2) and (OFL3) imply (OFL3) ′ for any j ≥ 3.

Remark 2. We would like to emphasize two important features of Theorem 4.3.1. First, all its conditions are easily checkable in terms of the original system. Secondly, if the conditions (OFL1) -(OFL4) hold for Σ, the time re-scaling function γ can be constructed as follows:

(1). Put B = span {g i , ad l

f g i + b l i f : 1 ≤ i ≤ m, 1 ≤ l ≤ n -1},
l i = T k,l k,i , where 1 ≤ k ≤ m is any integer such that k = i when l = n -1 (actually, T k,l k,i
is the same function for any k because of (OFL4)(ii)). We will prove below (see Lemma 4.6.3 and Proposition 4.6.4) that for m ≥ 3, B is an involutive subdistribution of G n f of corank one and, for m = 2, we assume that in (OFL4)(iii). (2). Choose m + 1 smooth functions φ 0 , φ 1 , . . . , φ m such that span {dφ 0 , dφ 1 , . . . , dφ m } = B ⊥ , where the codistribution B ⊥ denotes the annihilator of B.

(3). The construction of B implies immediately that f (x 0 ) ∈ B(x 0 ). Thus there exists a function φ i , for a certain 0 ≤ i ≤ m, such that L f φ i (x 0 ) = 0 where L f φ i denotes the Lie derivative of φ i along the vector field f . (4). The time re-scaling function is given by γ = 1 L f φ i .

Note that the time re-scaling is not unique. Indeed, take any smooth function φ such that d φ ∈ B ⊥ and L f φ(x 0 ) = 0. Then γ = 1 L f φ defines a time re-scaling function that renders the system ẋ = f (x) + m i=1 g i (x)ũ i , with f = γf , feedback linearizable and, conversely, any γ achieving that goal is of the form γ = 1 L f φ , where φ is as above. Remark 3. Obviously, a control system that is feedback linearizable is always orbital feedback linearizable. To see how it is reflected in conditions (OFL1) -(OFL4), put G j = span {g i , ad f g i , . . . , ad j-1 f g i , 1 ≤ i ≤ m} and observe that Σ, defined on X of dimension (n + 1)m + 1, is locally feedback equivalent to a t-augmented linear system Λ t if and only if it satisfies the following conditions

(FL1) dim G n+1 (x) = (n + 1)m; (FL2) dim G n+2 (x) = (n + 1)m; (FL3) [G j , G j ] ⊂ G j , 1 ≤ j ≤ n.
Indeed, the condition (FL1) implies that Σ decomposes into 1-dimensional system ẋ0 = 1 followed by a (n + 1)m-dimensional subsystem that is feedback linearizable due to the standard linearizability conditions (FL2) and (FL3) (for the latter see [START_REF] Isidori | Nonlinear Control Systems[END_REF], [START_REF] Jakubczyk | On linearization of control systems[END_REF], [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF]). Now it is immediate to see that (FL3) implies (OFL2), (OFL3) and (OFL4) (with all T k,l i,j = 0), while (FL1) and (FL2) are related with (OFL1) via f (x 0 ) ∈ G n f (x 0 ), which reflects the time-augmented form of Λ t .

From distributions to control-affine systems and back

In the previous section we characterized control-affine systems that are orbital feedback equivalent to a time-augmented linear system Λ t , with all controllability indices equal. Any such system Λ t (with the common value of the controllability indices being n + 1) can be transformed, via a linear change of coordinates and linear feedback, to the t-augmented Brunovsky canonical form

Λ Br t : ẋ0 0 = 1 ẋ0 1 = x 1 1 • • • ẋ0 m = x 1 m . . . . . . ẋn-1 1 = x n 1 • • • ẋn-1 m = x n m ẋn 1 = u 1 • • • ẋn m = u m .
The drift and the control vector fields of the above form are, respectively,

f cc = ∂ ∂x 0 0 + n-1 j=0 m i=1 x j+1 i ∂ ∂x j i , g cc 1 = ∂ ∂x n 1 , • • • , g cc m = ∂ ∂x n m , (4.4.1)
(B1) ′ D (n) = T X; (B1) ′′ rank D (n-1) = nm + 1 and there exists an involutive subdistribution L n-1 of D (n-1) such that corank (L n-1 ⊂ D (n-1) ) = 1.

From distributions to control-affine systems

Recall that G = span {g 1 , . . . , g m } denotes the control distribution spanned by the control vector fields. Observe that the set of all admissible velocities of the controlaffine system Σ at x ∈ X is the affine subspace A(x) = f (x)+G(x) of the tangent space T x X. Therefore to any control affine system there corresponds an affine distribution A = f + G which assigns to any x ∈ X the affine subspace A(x) of T x X. Feedback equivalence can be rephrased, using the notion of affine distributions, in an invariant way that does not depend on particular control parameterizations of Σ. Indeed, two control-affine systems Σ and Σ are feedback equivalent if and only if there exists a diffeomorphism ϕ such that à = ϕ * A, provided that the control distributions G and G are of constant rank. Similarly, they are orbital feedback equivalent if there exists a diffeomorphism ϕ and a function γ(•) = 0 such that à = ϕ * (γA).

Here γA means that all vector fields of A are multiplied by γ, i.e., γA = γf + G.

Consider an arbitrary constant rank distribution D on X. To any affine distribution A ⊂ D, where A = f + G such that corank (G ⊂ D) = 1 and f (x) ∈ G(x), we attach the control system Σ A , given up to orbital feedback.

Proposition 4.4.2 Consider a constant rank distribution D in which we choose an affine distribution

A = f + G ⊂ D such that corank (G ⊂ D) = 1 and f (x) ∈ G(x). If the distribution D satisfies (D1) D is locally equivalent to the Cartan distribution CC n (R, R m ); (D2) C(D (1) ) = G,
then the associated control-affine system Σ A is locally orbital feedback equivalent to the t-augmented Brunovsky canonical form Λ Br t .

Remark. It seems that the assertion of that theorem may a priori depend on the choice of f but actually it does not. Indeed, if for one vector field f , the corresponding system Σ A is orbital feedback linearizable it is so for any f such that span {f

} + G = span { f } + G = D,
Σ ) = G Σ .

Proof: Necessity. Without loss of generality, we can assume that

Σ : ẋ = f (x) + m i=1 g i (x)u i
is locally orbital feedback equivalent to the t-augmented Brunovsky canonical form

ẋ = f cc (x) + m i=1 g cc i (x)ũ i ,
where the vector fields f cc and g cc i , for 1 ≤ i ≤ m, are given by (4.4.1). This proves that the distribution D Σ is locally equivalent to CC n (R, R m ).

Sufficiency. Since D Σ is locally equivalent to CC n (R, R m ) = D, there exists a local diffeomorphism ϕ : X → X such that ϕ * D Σ = D = span {f cc , g cc 1 , . . . , g cc m }.
In other words, there exists an everywhere invertible matrix-valued smooth function

M = (µ j i ), 0 ≤ i, j ≤ m, such that ϕ * (µ 0 0 f + m j=1 µ j 0 g j ) = f cc ϕ * (µ 0 i f + m j=1 µ j i g j ) = g cc i .
From the structure of CC n (R, R m ) = D = span {f cc , g cc 1 , . . . , g cc m }, it is easy to see that G = span {g cc 1 , . . . , g cc m } is the characteristic distribution of D(1) . Therefore the characteristic distribution of D

(1) Σ is C(D (1) Σ ) = span {µ 0 i f + m j=1 µ j i g j , 1 ≤ i ≤ m}.
The condition (C2) gives C(D

Σ ) = G Σ = span {g 1 , . . . , g m } which implies

µ 0 1 = • • • = µ 0 m = 0.
Define a smooth matrix β = (β j i ) by β j i = µ j i , for 1 ≤ i, j ≤ m. Clearly the function µ 0 0 cannot be zero at x 0 and β is invertible because the matrix M(x) is invertible everywhere.

are given in terms of the distribution D Σ associated to Σ. Equivalence of D Σ to the Cartan distribution CC n (R, R m ) can be tested using either the conditions (B1) -(B2) of Theorem 4.4.1 or the conditions (B1) ′ , (B1) ′′ and (B2) of Remark following it. The latter lead to the conditions (C1) ′ -(C4) ′ and the former to (C1) ′′ -(C3) ′′ .

Examples

In this section we will illustrate our results on orbital feedback linearizability by two examples.

Example 4.5.1 Consider a control system described on R 7 , equipped with the coordinates (x, y 1 , y 2 , z 1 , z 2 , w 1 , w 2 ) by Σ :

                     ẋ = 1 ẏ1 = z 1 + w 1 ẏ2 = z 2 + y 1 w 2 ż1 = w 1 ż2 = w 2 ẇ1 = u 1 ẇ2 = u 2
where (u 1 , u 2 ) ⊤ ∈ R 2 is the control. This system is not feedback linearizable because it does not satisfy the conditions (FL1) -(FL3) given in Section 4.3 (see the calculation below). We will show that it fulfils however the conditions of Theorem 4.3.1 almost everywhere and thus it is locally orbital feedback linearizable on an open and dense subset of R 7 . We have

f = ∂ ∂x + (z 1 + w 1 ) ∂ ∂y 1 + (z 2 + y 1 w 2 ) ∂ ∂y 2 + w 1 ∂ ∂z 1 + w 2 ∂ ∂z 2 , g 1 = ∂ ∂w 1 , g 2 = ∂ ∂w 2 .
A direct calculation gives

ad f g 1 = -∂ ∂y 1 -∂ ∂z 1 ad f g 2 = -y 1 ∂ ∂y 2 -∂ ∂z 2 ad 2 f g 1 = ∂ ∂y 1 + w 2 ∂ ∂y 2 ad 2 f g 2 = (1 -z 1 -w 1 ) ∂ ∂y 2 .
Then it is easily seen that the distribution G 2 = span {g 1 , g 2 , ad f g 1 , ad f g 2 } is not involutive and hence Σ is not feedback linearizable. On the other hand, the distributions

G = span {g 1 , g 2 }, G j f = span {f, g 1 , g 2 , . . . , ad j-1 f g 1 , ad j-1 f g 2 },
for 1 ≤ j ≤ 3, satisfy the conditions (OFL1) -(OFL3) of Theorem 4.3.1 around any point in R 7 such that 1z 1w 1 = 0. Now we examine the condition (OFL4) of Theorem 4.3.1. Define the differential forms ω 1 , ω 2 by

ω j (h) = 0, for any h ∈ G 2 f , ω j (ad 2 f g i ) = δ j i ,
for 1 ≤ i, j ≤ 2. Solving the above equations, we obtain

ω 1 = -z 1 dx + dy 1 -dz 1 ω 2 = 1 1 -z 1 -w 1 ((z 1 w 2 -z 2 )dx -w 2 dy 1 + dy 2 + w 2 dz 1 -y 1 dz 2 ). Taking into account that [ad f g 1 , ad f g 2 ] = ∂ ∂y 2 , we get T 1,1 1,2 = ω 1 ([ad f g 1 , ad f g 2 ]) = 0, T 2,1 1,2 = ω 2 ([ad f g 1 , ad f g 2 ]) = 1 1 -z 1 -w 1 .
Notice that for the system Σ, the conditions (i) and (ii) of (OFL4) are trivially satisfied and thus it remains to verify the condition (iii) of (OFL4). That is, the distribution

B = span {g 1 , g 2 , ad f g 1 + b 1 1 f, ad f g 2 + b 1 2 f }
should be involutive, where the functions b 1 1 and b 1 2 are given, respectively, by

b 1 1 = T 2,1 2,1 = -T 2,1 1,2 = 1 z 1 + w 1 -1 b 1 2 = T 1,1 1,2 = 0. Substituting b 1 1 and b 1 2 into B, we get B = span {g 1 , g 2 , ad f g 1 + b 1 1 f, ad f g 2 + b 1 2 f } = span {g 1 , g 2 , f + (z 1 + w 1 -1)ad f g 1 , ad f g 2 } = span {g 1 , g 2 , h 1 , h 2 },
where

h 1 = f + (z 1 + w 1 -1)ad f g 1 = ∂ ∂x + ∂ ∂y 1 + (z 2 + w 2 y 1 ) ∂ ∂y 2 + (1 -z 1 ) ∂ ∂z 1 + w 2 ∂ ∂z 2 h 2 = ad f g 2 = -y 1 ∂ ∂y 2 -∂ ∂z 2 .
Computing the Lie brackets

[g 1 , g 2 ] = 0, [g 1 , h 1 ] = 0, [g 1 , h 2 ] = 0, [g 2 , h 1 ] = -h 2 , [g 2 , h 2 ] = 0, [h 1 , h 2 ] = 0,
we conclude that the distribution B is indeed involutive and therefore the system Σ is orbital feedback linearizable. The time re-scaling function γ is constructible. To this end, choose three functions φ 1 = xy 1 , φ 2 = y 1 z 2y 2 , φ 3 = e x (z 1 -1) so that they satisfy B ⊥ = span {dφ 1 , dφ 2 , dφ 3 }.

Since L f φ 1 = (1z 1w 1 ) = 0 out of the set {z 1 + w 1 = 1}, a time re-scaling function γ can be taken as

γ = 1 1 -z 1 -w 1 .
By a direct calculation we can verify that the system Σ with respect to the new time scale dt = γdτ , i.e., the system Σ 1 , which is Σ with f replaced by γf , satisfies the feedback linearizability conditions (FL1) -(FL3) (see Remark 3 following Theorem 4.3.1).

Notice that we can also choose Consider a rigid bar moving in R 3 such that the instantaneous velocity of the bar is parallel to its direction. Let (x, y, z) ∈ R 3 denotes the coordinate position of the source point of the bar, ϕ denotes the angle between the bar axis and the plane XOY and θ denotes the angle between the x-axis and the projection of the bar on the XOY -plane.

γ 2 = 1 L f φ 2 , in {1 -z 1 -w 1 = 0} ∩ {z 2 = 0} or γ 3 = 1 L f φ 3 , in {1 -z 1 -w 1 = 0}
We assume that we control both angular velocities and that the bar moves forward only with a velocity of norm one.

Σ bar :

             ẋ = cos ϕ cos θ ẏ = cos ϕ sin θ ż = sin ϕ θ = u 1 φ = u 2
where θ ∈ S 1 and -π 2 < ϕ < π 2 . Note that this model is not defined at the point ϕ = ± π 2 . Therefore the state is (x, y, z, ϕ, θ) ∈ X = R 3 × (-π 2 , π 2 ) × S 1 and the drift and the control vector fields are, respectively,

f = cos ϕ cos θ ∂ ∂x + cos ϕ sin θ ∂ ∂y + sin ϕ ∂ ∂z , g 1 = ∂ ∂θ , g 2 = ∂ ∂ϕ .
A direct computation gives

ad f g 1 = cos ϕ sin θ ∂ ∂x -cos ϕ cos θ ∂ ∂y ad f g 2 = sin ϕ cos θ ∂ ∂x + sin ϕ sin θ ∂ ∂y -cos ϕ ∂ ∂z .
It is easy to find that the distribution G 2 = span {g 1 , g 2 , ad f g 1 , ad f g 2 } is not involutive and thus Σ bar is not feedback linearizable because it does not satisfy the conditions (FL1) -(FL3) given in Section 4.3. However it is locally orbital feedback linearizable.

Clearly, the distributions

G = span {g 1 , g 2 }, G j f = span {f, g 1 , g 2 , .
. . , ad j-1 f g 1 , ad j-1 f g 2 }, for 1 ≤ j ≤ 2, satisfy the conditions (OFL1) -(OFL3) of Theorem 4.3.1 around any point in X. Moreover the condition (OPL4) is trivially satisfied because all the functions T k,l i,j vanish identically. Indeed, recall that m = 2 and n = 1 so l = 0 since l ≤ n -1. Then we have,

T k,l i,j = ω k ([ad n-1 f g i , ad n-1 f g j ]) = ω k ([g i , g j ]) = 0 for 1 ≤ i, j, k ≤ 2, since [g 1 , g 2 ] = [ ∂ ∂θ , ∂ ∂ϕ ] ≡ 0.
Consequently, the involutive distribution B coincides with the distribution G = span {g 1 , g 2 }. In order to find a suitable time re-scaling function, we choose φ 1 = x, φ 2 = y, φ 3 = z such that

B ⊥ = span {dφ 1 , dφ 2 , dφ 3 }. If θ = ± π 2 , then L f φ 1 = L f x = cos ϕ cos θ > 0.
Therefore a time re-scaling function is given by γ = 1 cos ϕ cos θ .

Multiplying the right hand side of Σ bar by 1 cos ϕ cos θ and putting v = tan θ, w = tan ϕ cos θ , we obtain finally the linear system by a suitable feedback

             ẋ = 1 ẏ = v ż = w v = ũ1 ẇ = ũ2 .
In the case θ = ± π 2 , the orbital feedback linearization can be realized in a similar way with the time re-scaling function given by γ = 1 cos ϕ sin θ , since L f φ 2 = L f y = cos ϕ sin θ = 0. As we have explained in Remark 2 following Theorem 4.3.1, the choice of the time re-scaling function γ is not unique. In fact, we can choose any ϕ such that dϕ ∈ B ⊥ , that is any ϕ = ϕ(x, y, z), such that L f ϕ(x 0 ) = 0 and re-scale the time via γ = 1 L f ϕ .

[G, G j f ] ⊂ G j f , 3 ≤ j ≤ n + 1.
Proof: We proceed by induction. If j = 3 we have

G 3 f = span {f, g i , ad f g i , ad 2 f g i , 1 ≤ i ≤ m}.
Given an arbitrary g ∈ G, in order to show that [g,

G 3 f ] ∈ G 3 f , it is enough to prove [g, ad 2 f g i ] ∈ G 3 f , 1 ≤ i ≤ m.
Computing that Lie bracket, we have, by the Jacobi identity,

[g, ad 2 f g i ] = [g, [f, ad f g i ]] = [[g, f ], ad f g i ] + [f, [g, ad f g i ]] = -[ad f g, ad f g i ] + [f, [g, ad f g i ]]. The condition [G, G 2 f ] ⊂ G 2 f implies that [g, ad f g i ] ∈ G 2 f
. Moreover, since the vector fields ad f g, ad f g i and f belong to G 2 f , it follows from the condition (OFL2) that

[ad f g, ad f g i ] ∈ G 3 f , [f, [g, ad f g i ]] ∈ G 3 f ,
and thus [g, ad 2 f g i ] ∈ G 3 f . Assume now that the statement is true for j

= k, i.e., [G, G k f ] ⊂ G k f . Consider G k+1 f = G k f ⊕ span {ad k f g i , 1 ≤ i ≤ m} 123 = span {f, g i , ad f g i , . . . , ad k-1 f g i , ad k f g i , 1 ≤ i ≤ m}.
For any g ∈ G, by the Jacobi identity we have, for 1

≤ i ≤ m, [g, ad k f g i ] = [g, [f, ad k-1 f g i ]] = [[g, f ], ad k-1 f g i ] + [f, [g, ad k-1 f g i ]] = -[ad f g, ad k-1 f g i ] + [f, [g, ad k-1 f g i ]]. The induction assumption [G, G k f ] ⊂ G k f implies that [g, ad k-1 f g i ] ∈ G k f , for 1 ≤ i ≤ m. Moreover, since ad f g, ad k-1 f g i and f belong to G k f , the condition (OFL2) implies that [ad f g, ad k-1 f g i ] ∈ G k+1 f , [f, [g, ad k-1 f g i ]] ∈ G k+1 f . Thus [g, ad k f g i ] ∈ G k+1 f , for 1 ≤ i ≤ m, which yields [G, G k+1 f ] ⊂ G k+1 f
and the lemma follows by an induction argument.

Recall that in Section 4.3 we have defined, under (OFL1), the differential forms ω 1 , . . . , ω m and with their help, the functions T k,l i,j were also well defined for any 0 ≤ l ≤ n -1 and any 1 ≤ i, j, k ≤ m such that i = j in the case l = n -1.

n f such that corank (B ⊂ G n f ) = 1 and [B, B] ⊂ G n f , then B = G ⊕ H,
where G = span {g 1 , . . . , g m } and H = span {ad j f g i + b j i f, 1 ≤ i ≤ m, 1 ≤ j ≤ n -1} for some suitable C ∞ -functions b j i defined on X.

Proof: To begin with, we will show that G = span {g 1 , . . . , g m } ⊂ B locally around x 0 . Assume that there exist g ∈ G such that g(x 0 ) ∈ B(x 0 ), since B is of constant rank, we have g(x) ∈ B(x), for any x in a neighborhood O of (x 0 ). Then the the condition corank (B ⊂ G n f ) = 1 implies that G n f = B ⊕ span {g} in O. Moreover, Lemma 4.6.1 shows that g is a characteristic vector field of G n f and therefore, locally around x 0 ,

[G n f , G n f ] = [B, B] + [B, g] ⊂ G n f ,
which implies that G n f is involutive. This gives a contradiction and thus G ⊂ B. Since rank B = rank G n f -1 = nm , there exists a rank (n -1)m distribution

H = span {h 1 i , . . . , h n-1 i , 1 ≤ i ≤ m} ⊂ B such that B = G ⊕ H.
, h0 ) = (ad f g i , . . . , ad n-1 f g i , f ) B = (ad f g i , . . . , ad n-1 f g i , f ) * Id .
A straightforward calculation shows that . By the condition (OFL1), the vectors ad f g 1 (x 0 ), ad 2 f g 1 (x 0 ) and ad n f g i (x 0 ) are linearly independent. Then the fact ad n f g i ∈ G n f implies that h(x 0 ) ∈ G n f (x 0 ) which contradicts the relation (4.6.1). Therefore rank B 1 = (n -1)m and Lemma 4.6.2 holds.

hj i = ad j f g i + bj i • ad f g 1 , (i, j) = (1 ,
The following version of a result of Bryant [START_REF] Bryant | Some aspects of the local and global theory of Pfaffian systems[END_REF] was given by Pasillas-Lépine and Respondek in [START_REF] Pasillas-Lépine | Contact systems and corank one involutive subdistributions[END_REF], [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF] in order to study the Cartan distribution CC n (R, R m ). i,j , defined by (4.3.2), satisfy the following conditions: (OFL4) (i) T k,l i,j = 0, for

1 ≤ k = i ≤ m, 1 ≤ j ≤ m, if l < n -1 1 ≤ i = k = j ≤ m, if l = n -1 ,
(OFL4) (ii) T i,l i,j = T k,l k,j , for 1 ≤ l ≤ n -1, 1 ≤ i, j, k ≤ m s.t i = j, k = j if l = n -1

Remark. The condition (OFL1) assumes that the distributions G n+1 with h j i = ad j f g i + b j i f where b j i are C ∞ -functions defined on X. Compute the following Lie brackets, for any 1 ≤ i, j ≤ m and 1 ≤ l ≤ n -1,

[h n-1 i , h l j ] = [ad n-1 f g i + b n-1 i f, ad l f g j + b l j f ] = [ad n-1 f g i , ad l f g j ] -b l j • ad n f g i + (L ad n-1 f g i b l j ) • f +b n-1 i • ad l+1 f g j -(L ad l f g j b n-1 i ) • f + (b n-1 i L f b l j -b l j L b n-1 i )f = [ad n-1 f g i , ad l f g j ] -b l j • ad n f g i + b n-1 i • ad l+1 f g j + η • f
ω k ([h n-1 i , h l j ]) = ω k ([ad n-1 f g i , ad l f g j ]) -b l j • ω k (ad n f g i ) + b n-1 i • ω k (ad l+1 f g j ) + η • ω k (f ) = T k,l i,j -b l j • ω k (ad n f g i ) + b n-1 i
• ω k (ad l+1 f g j ).

It follows that T k,l i,j = b l j • ω k (ad n f g i )b n-1 i

• ω k (ad l+1 f g j ).

(4.6.3)

We will consider separately the case l < n -1 and l = n -1.

If l < n -1, then obviously ad l+1 f g j ∈ G n f and so ω k (ad l+1 f g j ) = 0, for any 1 ≤ k ≤ m. If k = i, by the definition of ω k , we have ω k (ad n f g i ) = 0. Thus the equation ( 4 and the result follows by (4.6.5).

The relations (4.6.4) and (4.6.6) show that we always have T i,l i,j = T k,l k,j , for any 1 ≤ l ≤ n -1 and any 1 ≤ i, j, k ≤ m such that j = i and j = k in the case l = n -1.

(ii) =⇒ (i): Assume that a system Σ satisfies (OFL1) -(OFL3) and that the functions T k,l i,j , 1 ≤ i, j, k ≤ m, 1 ≤ l ≤ n -1, satisfy the conditions (OFL4)(i) -(OFL4)(ii). For any 1 ≤ i ≤ m and 1 ≤ l ≤ n -1, define the functions b l i = T k,l k,i , (4.6.7)

where we take any 1 ≤ k ≤ m if l < n -1 and any k = i if l = n -1 (actually (OFL4)(ii) implies that the functions T k,l k,i , do not depend on k). Define a distribution B by

B = span {g i , ad l f g i + b l i f, 1 ≤ i ≤ m, 1 ≤ l ≤ n -1}.
We claim that B satisfies item (i) of Lemma 4.6.4. The relations B ⊂ G n f and corank(B ⊂ G n f ) = 1 are obvious. Therefore it remains to prove that [B, B] ⊂ G n f . Denote h l i = ad l f g i + b l i f . Lemma 4.6.1 says that G is contained in the characteristic distribution of G n f and thus any g ∈ G ⊂ B and any h l j , 1 ≤ j ≤ m, 1 ≤ l ≤ n -1, satisfy [g, h l j ] ∈ G n f . So we need only to prove that for any vector fields h r i and h l j , for 1 ≤ i, j ≤ m, 1 ≤ l, r ≤ n -1, we have [h r i , h l j ] ∈ G n f , (4. 6.8) which is equivalent to ω k ([h r i , h l j ]) = 0, 1 ≤ k ≤ m. Notice that in the case r < n -1 and l < n -1, the relation (4.6.8) is obvious since

h r i ∈ G n-1 f , h l j ∈ G n-1 f
and then the condition (OFL2) yields directly

[h r i , h l j ] ∈ [G n-1 f , G n-1 f ] ⊂ G n f .
Now consider the case r = n -1. Following the same calculation as in (4.6.2), we get

ω k ([h r i , h l j ]) = ω k ([h n-1 i , h l j ]) = ω k ([ad n-1 f g i , ad l f g j ]) -b l j • ω k (ad n f g i ) + b n-1 i • ω k (ad l+1 f g j ) + η • ω k (f ) = T k,l i,j -b l j • ω k (ad n f g i ) + b n-1 i
• ω k (ad l+1 f g j ).

We will consider separately the case l < n -1 and l = n -1. If l < n -1, notice that ad l+1 f g j ∈ G n f and thus we have ω k (ad l+1 f g j ) = 0, for any 1 ≤ k ≤ m. If k = i, the definition of ω k gives that ω k (ad n f g i ) = 0 and moreover, by the condition T k,l i,j = 0, we get ω k ([h n-1 i , h l j ]) = 0.

If k = i, the definition of ω k gives that ω k (ad n f g i ) = ω k (ad n f g k ) = 1. Then the relation (4.6.7) implies that ω k ([h n-1 i , h l j ]) = T k,l i,jb l j = 0. Now assume that l = n -1. If k = i, j, from the definition of ω k and the assumption (OFL4)(ii), we find ω k (ad n f g i ) = 0, ω k (ad n f g j ) = 0 and T k,n-1 i,j = 0 which imply

ω k ([h n-1 i , h n-1 j ]) = T k,n-1 i,j -b n-1 j • ω k (ad n f g i ) + b n-1 i
• ω k (ad n f g j ) = 0.

If k = i and i = j, we have ω k (ad n f g i ) = ω k (ad n f g k ) = 1 and ω k (ad n f g j ) = 0. Taking into account the equality T k,n-1 k,j = b n-1 j , we get

ω k ([h n-1 i , h n-1 j ]) = ω k ([h n-1 k , h n-1 j ]) = T k,n-1 k,j -b n-1 j = 0.
The case k = j and i = j follows from the previous one by permuting i and j. Therefore the relation (4.6.8) holds for r = n -1, any 1 ≤ i, j ≤ m and any 1 ≤ l < n -1. Finally the case l = n -1 follows from that for r = n -1 just by permuting l and r. Therefore the relation (4.6.8) is always true and implies [B, B] ⊂ G n f .

4.6.2 Proof of Theorem 4.4.6

The equivalence (i) ⇐⇒ (iii) ⇐⇒ (iv) ⇐⇒ (v) is a direct consequence of Proposition 4.4.5 and the results given by Pasillas-Lépine and Respondek in [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF] (summarized in Theorem 4.4.1 and Remarks following it). To prove Theorem 4.4.6, it is enough to show that (v) =⇒ (ii) =⇒ (iv). For simplicity, we will denote the distribution D Σ associated to Σ by D.

Proof: (v) =⇒ (ii) Note that (C1) ′′ and (C2) ′′ imply that D (i) = D i for i ≥ 0. We start by proving, by induction, that G i+1 f = D i , for 1 ≤ i ≤ n. Obviously, we have G 1 f = D 0 = span {f, g 1 , . . . , g m }.

Assume that G i f = span {f, g j , ad f g j , . . . , ad i-1 f g j , 1 ≤ j ≤ m} = D i-1 .

The condition (C3) ′′ implies that [G, D i ] ⊂ D i , for 1 ≤ i ≤ m. Indeed, (C3) ′′ gives [G, D 0 ] ⊂ D 0 from which we conclude easily (via Jacobi identity) that [G, D (i) ] ⊂ D (i) , but D (i) = D i . We have

D i+1 = D i + [D 0 , D i ] = [f, D i ] + [D 0 , D i ],
which implies

D i+1 = G i+2 f
= span {f, g j , ad f g j , . . . , ad i+1 f g j , 1 ≤ j ≤ m}.

130

We then have G i+1 f = D i = D (i) and, by definition, D (i) = D (i-1) + [D (i-1) , D (i-1) ] which yields [G i f , G i f ] ⊂ G i+1 f . This proves the conditions (OFL1) and (OFL2). The condition (OFL3) coincides with (C3) ′′ while (OFL4) follows from Lemma 4.6.4, where we take B = L n-1 .

(ii) =⇒ (iv) The condition (OFL1) implies that rank G i+1 f = (i + 1)m + 1, for 1 ≤ i ≤ n, in a neighborhood of x 0 . We will show D (i) = G i+1 f by induction. The relations D (0) = G 1 f is obvious. Assume that G i f = span {f, g j , ad f g j , . . . , ad i-1 f g j , 1 ≤ j ≤ m} = D (i-1) , then

D (i) = D (i-1) + [D (i-1) , D (i-1) ] = G i f + [G i f , G i f ] = G i+1 f + span {[ad l f g j , ad s f g k ], 0 ≤ l, s ≤ i -1, 1 ≤ j, k ≤ m}.
Now (OFL2) implies that for 0 ≤ l, s ≤ i -1, 1 ≤ j, k ≤ m,

[ad l f g j , ad s f g k ] ∈ G i+1 f ,
and thus D (i) ⊂ G i+1 f . The inclusion G i+1 f ⊂ D (i) is obvious and thus by an inductive argument, G i+1 f ⊂ D (i) for any i ≥ 0. Therefore the conditions (C1) ′ and (C4) ′ hold and rank D (n-1) = nm + 1. Moreover, the condition (C2) ′ follows from Lemma 4.6.4. Finally, the condition (C3) ′ can be verified directly due to the structure of L n-1 given in the proof of Lemma 4.6.4. In fact, by Lemma 4.6.3, the involutive subdistribution L n-1 is unique and therefore it is given by L n-1 = B = span {g j , ad 1 f g j + b 1 j f, . . . , ad n-1 f g j + b n-1 j f, 1 ≤ j ≤ m}.

It is immediate to see that

D 0 = span {f, g 1 , . . . , g m } ⊂ L n-1 ,
and hence the condition (C3) ′ holds.

131

Conclusions

Nous avons étudié dans ce mémoire la géométrie et la structure des systèmes de contrôle non linéaires qui sont équivalents aux systèmes de contact -une classe de systèmes non holonomes. Nous résumons dans cette partie nos résultats principaux .

Dans le premier chapitre, nous avons étudié la propriété de la platitude des systèmes avec deux contrôles. Premièrement, nous avons montré qu'une structure de Goursat est x-plate en (x 0 , u 0 ), u 0 ∈ U sing (x 0 ), si et seulement si les conditions de régularité dim D (i) (x 0 ) = dim D i (x 0 ), pour 0 ≤ i ≤ n, sont satisfaites. Ceci répond à la question posée par Martin et Rouchon [START_REF] Martin | Feedback linearization and driftless systems[END_REF] (voir aussi [START_REF] Martin | Feedback linearization and driftless systems[END_REF]) pour la x-platitude. Ensuite, nous avons donné des conditions nécessaires et suffisantes vérifiables (Theorème 1.3.2 et 1.3.3) pour qu'une paire de fonctions données (ϕ 1 , ϕ 2 ) forme une x-sortie plate du système Σ avec deux contrôles qui est équivalent au système chaîné (système de contact canonique sur J n (R 1 , R 1 )). Nous avons aussi décrit le lieu singulier de cette x-sorties plate. Notons C n-1 la distribution caractéristique de D (n-1) et g un champ de vecteurs dans D tel que g(x 0 ) ∈ C n-1 (x 0 ). Nous avons montré qu'étant donné une fonction lisse ϕ 1 telle que L c ϕ 1 = 0, ∀ c ∈ C n-1 , et L g ϕ 1 (x 0 ) = 0, il existe toujours une fonction ϕ 2 telle que (ϕ 1 , ϕ 2 ) frome une x-sortie plate de Σ en (x 0 , u 0 ). De plus, ϕ 2 est unique au sens de span {dϕ 1 (x), dϕ 2 (x)} = span {dϕ 1 (x), d φ2 (x)}, pour tous les points x dans un voisinage du point x 0 , où φ2 est une autre fonction telle que (ϕ 1 , φ2 ) est aussi une x-sortie plate. Ces derniers résultats nous permettent de calculer toutes les x-sorties plates pour un système qui est équivalent au système chaîné en résolvant un système d'équations aux dérivées partielles du 1 er ordre. En fin, nous avons appliqué nos résultats au système du robot mobile avec remorques pour décrire toutes ses x-sorties plates. Dans le quatrième chapitre, nous avons étudié le système de contrôle sous la forme Σ : ẋ = f (x)+ m i=1 g i (x)u i , où x ∈ X = R (n+1)m+1 , en supposant que f (x 0 ) = 0 où x 0 est le point autour duquel nous travaillons. Nous avons obtenu des conditions nécessaires et suffisantes vérifiables pour que le système Σ soit linéarisable par bouclage orbital. Toutes ces conditions peuvent être vérifiées directement sur le système original. De plus, nous avons aussi donné une construction de la fonction γ qui décrit le changement de temps, i.e., la transformation orbitale. Il est aussi important de remarquer que notre résultat implique une relation intéressante entre la linéarisation orbitale par bouclage du système Σ et la caractérisation du système de contact sur J n (R 1 , R m ).

Perspectives

A l'issue de cette thèse, certains problèmes liés à nos études demeurent encore non-résolus et indiquent de possibles directions de recherche pour l'avenir :

(1). Dans le premier chapitre, nous avons montré qu'une structure de Goursat est x-plate en (x 0 , u 0 ), u 0 , ∈ U sing (x 0 ), si et seulement si les conditions de regularité dim D (i) (x 0 ) = dim D i (x 0 ), pour 0 ≤ i ≤ n, sont satisfaites. Deux problèmes intéressants se posent : (i) Est-ce que l'hypothèse que la distribution associée D soit une structure de Goursat est nécessaire? Autrement dit, si un système avec deux contrôles est x-plat au point x 0 , est-ce qu'il est toujours équivalent au système chaîné en x 0 ? (ii) Est-ce que ce résultat est aussi vrai si on remplace la x-platitude par la platitude? Autrement dit, est-ce qu'une structure de Goursat est plate en (x 0 , ūk 0 ) si et seulement si les conditions de regularité sont satisfaites en x 0 ?

(2). Dans le chapitre 3, nous avons donné des conditions nécessaires et suffisantes pour qu'une distribution soit équivalente à la distribution de Cartan CC n (R 2 , R m ). Deux problèmes intéressants relatifs aux chapitre 2 et 3 sont : (i) Le premier problème consiste à considérer un système à n-plans qui bouge dans l'espace R m+1 et de voir comment construire un modèle cinématique pour ce système? Est-ce qu'il est localement équivalent au système de contact canonique sur J n (R 2 , R m )? Est-ce que nous pouvons le montrer en utilisant nos résultats dans chapitre 3, caractériser son lieu singulier, étudier la propriété de la platitude et determiner ses x-sorties plates? (ii) Le deuxième problème semble plus difficile à étudier mais il est aussi naturel : comment caractériser la distribution de Cartan dans le cas general? Autrement dit, nous voulons trouver des conditions nécessaires et suffisantes vérifiables pour qu'une distribution soit equivalente à la distribution de Cartan CC n (R k , R m ), pour k > 2.

(3). Considérons le système de contrôle sous la forme Σ : ẋ = f (x)+ m i=1 g i (x)u i , où x ∈ X = R N , pour un certain entier N et x 0 est un point de X. Les problèmes liés au chapitre 4 sont les suivants : (i) Si f (x 0 ) = 0, quand Σ est localement equivalent par bouclage orbital à la forme de Goursat généralisée (Extended Goursat normal form [START_REF] Respondek | Extended Goursat Normal Form: a Geometric Characterization[END_REF]), avec des indices de contrôlabilité differents? (ii) Si f (x 0 ) = 0, i.e., au point d'équilibre, quand Σ est localement linéarisable par bouclage orbital? Il serait souhaitable que toutes conditions puissent être vérifiables directement sur le système original.

135 ABSTRACT: In the first part of this Ph.D. thesis, we characterize all flat outputs and their singular loci of any 2-input driftless control system which is equivalent to the chained form.

G
Then we apply that result to the n-trailer system in order to calculate all its flat outputs. In the second part, we establish a new model of the n-bar system in (m + 1)-dimensional space.

With the help of this model, we show that the system is locally equivalent to the m-chained form (canonical contact system on J n (R, R m )) and also describe its singular locus. Furthermore, we analyze its flatness property and determine its minimal flat outputs. In the third part, we give necessary and sufficient conditions for a distribution to be locally equivalent to the Cartan distribution for surfaces. Finally, in the fourth part, we give necessary and sufficient verifiable conditions for a multi-input affine control system to be orbital feedback linearizable. Key-words: Control system, nonholonomic system, flatness, flat output, contact system, Cartan distributon, n-bar system, orbital feedback linearization

P 1 P 2 P n- 2 PFigure 1 :

 1221 Figure 1: le système à n-barres dans R m+1

  où L est une sousdistribution involutive de corang deux dans D(n-1) . L'existence de L est garantie par la condition (iii).Toutes ces conditions (i) -(iv) sont vérifiables directement sur la distribution D. Premièrement, le rang d'Engel peut être vérifié directement d'après la définition. Deuxièmement, si les conditions (i) -(iii) sont satisfaites, la condition (iv) peut être vérifiée algébriquement parce que la sous-distribution L, si elle existe, est unique et caclulable (voir Chapitre 3 pour les détails). Le theorème suivant montre une relation entre le rang d'Engel et l'existence d'une sous-distribution involutive de corang k et répond à la question : quand une distribution D contient une sous-distribution involutive L qui soit de corang k dans D? Theorem 0.0.8 Soit m ≥ 3 et D une distribution de rang k + mk, définie sur une variété M de dimension m + k + mk, telle que D (1) = T M. Si la distribution caractéristique de D est réduite à {0}, alors les conditions suivantes sont équivalentes: (i) Le rang de Cartan de D ⊥ est constant et égal à k; (ii) Le rang d'Engel de D ⊥ est constant et égal à k; (iii) Il existe une sous-distribution F de D de corang k telle que [F , F ] ⊂ D et il n'existe aucune autre sous-distribution de D vérifiant cette propriété et de corang plus petit que k. (iv) Il existe une sous-distribution involutive L qui est de corang k dans D et il n'existe aucune autre sous-distribution involutive D de corang plus petit que k.

Theorem 1 . 3 . 3 (

 133 Characterization of flat outputs, second version) Consider a driftless 2-input smooth control system Σ defined on a manifold M of dimension n + 2 whose associated distribution D satisfies rank D

4 .Figure 1 . 2 :

 412 Figure 1.2: the rolling disk

  are necessary and sufficient for local feedback equivalent to the chained form.

  for any z = (x, y) in an open and dense subset O ′ of O and by Corollary 1.4.2 and the form of C n-1 , we conclude that

1. 4 . 3

 43 Proof of Theorem 1.3.2

1 Figure 2 . 1 :

 121 Figure 2.1: the rigid bar in R 3

  n-bar system to the m-chained form 2.3.1 Characterization of Cartan distribution CC n (R, R m ) Consider an arbitrary distribution D. The derived flag of D is the sequence of modules of vector fields D (0) ⊂ D (1) ⊂ • • • defined inductively by

3 . 1 .

 31 Let a distribution D of rank m + 1, with m ≥ 1, satisfy the items (i) and (ii) of Theorem 2.3.1. The regular locus of D, denoted by Reg(D), is the subset of M consisting of points at which D is equivalent to the Cartan distribution CC n (R, R m ) at 0 ∈ R (n+1)(m+1) . It can be proved that Reg(D) is an open and dense subset of M; its complement, called the singular locus of D, will be denoted by Sing(D). In the case m ≥ 2, since L n-1 is unique we clearly have Reg(D) = {p ∈ M : D(p) ⊂ L n-1 (p)} and Sing(D) = {p ∈ M : D(p) ⊂ L n-1 (p)}. 2.3.2 Main result: equivalence of the n-bar system to the m-chained form Consider two driftless systems

0,

  . . . , Φ * d Xm+1 n-2 , Φ * dx 1 n-1 , . . . , Φ * dx m+1 n-1 are always independent on Q and

Theorem 3 . 3 . 4

 334 Assume that m ≥ 3. Let D be a distribution of constant rank k + mk defined on a manifold M of dimension m + k + mk such that D (1) = T M. If the characteristic distribution of D satisfies C(D) = 0, then the following conditions are equivalent: (i) The Cartan rank of D ⊥ is constant and equals k; (ii) The Engel rank of D ⊥ is constant and equals k; (iii) There exists a subdistribution F of D of corank k such that [F , F ] ⊂ D and there does not exist any subdistribution of D with that property of a corank smaller than k; (iv) There exists an involutive subdistribution L that is of corank k in D and there does not exist any involutive subdistribution of D of a corank smaller than k.

4 . 2 ,

 42 then there exists a local change of coordinates z = ψ(z) such that the codistribution ψ * Ĩ is locally given, in z = (x, p)-coordinates, by the Bryant normal form (3.4.1). Define the local coordinates change Ψ : z = ψ(z) y = ỹ and we have immediately that the pull-back Ψ * takes I into the canonical form (3.4.1) in (z, ỹ) = (x, p, y)-coordinates. So in order to prove Theorem 3.4.3, it remains to verify that Ĩ satisfies the conditions of Theorem 3.4.2, that is, (1): Ĩ(1) = 0, (2): rank E Ĩ = k and (3): rank R( Ĩ) = m + k + mk.

Lemma 3 . 5 . 4

 354 Let a distribution D satisfy the conditions (i)-(iv) of Theorem 3.3.2.

and dω ∧ π 1 ∧

 1 • • • ∧ π k ≡ 0 mod I (3.5.5) for all ω ∈ I. The relation (3.5.5) can be rewritten, for the ω i 's that span I, as dω i = k j=1 η i j ∧ π j mod I. (3.5.6)

  e., F is involutive. Put L = F , then L is involutive and has corank k in D. Now we will prove the uniqueness of L. Suppose that there exists another involutive subdistribution L in D of corank k and L = L. Observe that rank (L ∩ L) is locally constant on an open and dense submanifold M of M and constant on each connected component of M. Choose one such component. Clearly, we have mkk ≤ rank (L ∩ L) ≤ mk -1.

+ 1 ≤ 1 ,

 11 , g k } = span { ∂ ∂x 1 , . . . , ∂ ∂x mk, g 1 , . . . , g r , g r+1 , . . . , g k } with g j = ∂ ∂y j j ≤ k. Since C(D) = 0, clearly D satisfies the assumption of Lemma 3.5.3, with r = mk, l = k and s = m + k. Therefore, at any q ∈ M , we must have rank ∂G ∂x (q) = mk, where G = g r+1 1 , . . . , g k+m r+1 , . . . , g k+m r+1 , . . . , g k+1 k , . . . , g k+m k ⊤ .

where the functions b l i are given by b

  where D is equivalent to the Cartan distribution CC n (R, R m ). The proof of this result follows immediately form that of Proposition 4.4.5, given in the next subsection. (C1) D Σ is locally equivalent to the Cartan distribution CC n (R, R m ); (C2) C(D

Lemma 4 . 6 . 2

 462 If the control-affine system Σ, defined by (4.2.6) satisfies the conditions (OFL1) -(OFL3) of Theorem 4.3.1 and, moreover, there exists a subdistribution B of G

1 , 1 i, for 1 ≤ 1 i, 1 ≤ 1 f g i + bn- 1 i• 1 i) 1 i) -bn- 1 i(L ad f g 1 1 i(x 0 ) • ad 2 f g 1 1 i-L ad n- 1 f 1 i) -bn- 1 i(L ad f g 1

 111111111111111111 1) h0 = f + b0 • ad f g 1 ,where ( b1 2 , . . . , b1 m , b2 1 , . . . , b2 m , . . . , bn-1 . . . , bn-1 m , b0 ) coincides with the first row of B. Now we discuss the value of the function b0 at point x 0 . If b0 (x 0 ) = 0, notice that b0 is just the element that lies in the first row and (n -1)m-th column of the matrix B and then, clearly, the first (n -1)m rows of B are linearly independent. In other words,rank B 1 B -1 2 = (n -1)m,which contradicts the assumption that rank B 1 = (n -1)m -1. If b0 (x 0 ) = 0, let us consider the vector fields h0 and hni ≤ m. On one hand, since h0 ⊂ H ⊂ B and hn-1 i ⊂ H ⊂ B, we have, for 1 ≤ i ≤ m h(x 0 ) = [ h0 , hn-1 i ](x 0 ) ⊂ [B, B](x 0 ) ⊂ G n f (x 0 ). (4.6.1)On the other hand, calculating the Lie bracket of h0 and hni ≤ m, we haveh(x 0 ) = [ h0 , hn-1 i ](x 0 ) = [f + b0 • ad f g 1 , ad n-ad f g 1 ](x 0 ) = ad n f g i (x 0 ) + bn-1 i (x 0 ) • ad 2 f g 1 (x 0 ) + (L f bn-(x 0 ) • ad f g 1 (x 0 ) + b0 (x 0 )[ad f g 1 , ad n-1 f g i ](x 0 ) -(L ad n-1 f g i b0 )(x 0 ) • ad f g 1 (x 0 ) +( b0 (L ad f g 1 bn-b0 ))(x 0 ) • ad f g 1 (x 0 ) = ad n f g i (x 0 ) + bn-(x 0 ) + η(x 0 ) • ad f g 1 (x 0 ),where η = L f bng i b0 + b0 (L ad f g 1 bn-b0 )

Lemma 4 . 6 . 3

 463 Let D be a distribution such that D and D(1) have constant rank d 0 and d 1 , respectively. Put r 0 = d 1d 0 and assume that D contains a subdistribution B ⊂ D that has constant corank one in D and satisfies [B, B] ⊂ D.(i) If r 0 ≥ 2, then B is unique. (ii) If r 0 ≥ 3, then B is involutive.In the case of D = G n f the existence of B as above is described by the following Lemma 4.6.4 If the control-affine system Σ, defined by (4.2.6), for m ≥ 2, satisfies the conditions (OFL1) -(OFL3) of Theorem 4.3.1, then the following conditions are equivalent: (i) There exists a subdistribution B of G n f such that corank (B ⊂ G n f ) = 1 and [B, B] ⊂ G n f ; (ii) The functions T k,l

f

  and G n f are of constant rank and rank G n+1 f rank G n f = m. Then if m ≥ 3, by Lemma 4.6.3, the subdistribution B ⊂ G n f given in item (i) of Lemma 4.6.4 is involutive and unique. Therefore Lemma 4.6.4 implies that if Σ, for m ≥ 3, satisfies the conditions (OFL1)-(OFL3) of Theorem 4.3.1, then the condition (OFL4) holds if and only if there exists an involutive subdistribution B of G n f which satisfies corank (B ⊂ G n f ) = 1. Proof: (i) =⇒ (ii): Applying Lemma 4.6.2 we have B = G ⊕ H = span {g 1 , . . . , g m } ⊕ span {h j i , 1 ≤ i ≤ m, 1 ≤ j ≤ n -1}

( 4 . 6 . 2 ) 1 f g i b l j -L ad l f g j b n- 1 i+ b n- 1 iL f b l j -b l j L b n- 1 i.

 4621111 where η = L ad n-On one hand, the definition (4.3.1) of the differential forms ω 1 , . . . , ω m implies thatω k ([h n-1 i , h l j ]) ≡ 0, 1 ≤ k ≤ m, 1 ≤ l ≤ n -1, since [h n-1 i , h l j ] ∈ [B, B] ⊂ G n f .On the other hand, we have

, 1 ≤

 1 .6.3) gives ω k ([h n-1 i , h l j ]) = T k,l i,j = 0. If k = i, since ω k (ad n f g i ) = ω k (ad n f g k ) = 1, then the equation (4.6.3) gives ω k ([h n-1 k , h l j ]) = T k,l k,jb l j = 0,and thus T k,l k,j = b l j , for any 1 ≤ k ≤ m and hence we conclude that T r,l r,j = b l j = T k,l k,j , for any 1 ≤ r, j, k ≤ m. (4.6.4)In the case l = n -1, if i = k = j then clearly ω k (ad n f g i ) = ω k (ad n f g j ) = 0 and from the equation (4.6.3) we get T k,n-1 i,j = 0.If k = i and i = j, then clearly ω k (ad n f g i ) = ω k (ad n f g k ) = 1 and ω k (ad n f g j ) = 0. Therefore it follows from the equation (4.6.3j = r, j = k ≤ m. (4.6.6) If k = j and i = j, then T k,n

  Dans le deuxième chapitre, nous avons étudié le système à n-barres dans l'espace R m+1 qui généralise le système du robot mobile avec remorques sur le plan. Nous avons introduit un modèle cinématique de ce système sans utiliser les variables d'angle. Les résultats principaux sont les suivants: (1) Le système à n-barres dans R m+1 est localement équivalent au système m-chaîné (i.e., système de contact canonique surJ n (R 1 , R m ))en tous les points réguliers que nous avons caractérisés. Nous avons aussi montré que les lieux singuliers sont essentiellement différents entre les deux cas m = 1 et m ≥ 2. De plus, le système à n-barres est contrôllable globalement dans l'espace de configuration Q = R m+1 × (S m ) n . (2) Le système à n-barres R m+1 , pour m ≥ 2, est x-plat en tous les points réguliers. De plus, les coordonnées cartésiennes du point source P 0 = (x 1 0 , x 2 0 , • • • , x m+1 0 ) de la première barre --→ P 0 P 1 forment la seule x-sortie plate minimale. Contrairement au cas m = 1, où la sortie plate minimale n'est pas unique. Dans le troisième chapitre, nous avons donné des conditions nécessaires et suffisantes pour qu'une distribution D soit équivalente à la distribution de Cartan CC n (R 2 , R m ). Toutes ces conditions sont vérifiables sur la distribution D. Pour montrer notre théorème, nous avons proposé une forme normale de Bryant étendue qui généralise la caractérisation de la distribution de Cartan CC 1 (R k , R m ). Nous avons aussi étudié le problème suivant : quand une distribution D contient une sous-distribution involutive qui est de corang k dans D. Nous avons obtenu le résultat suivant : si D est une distribution de rang k + mk définie sur une variété M de dimension m + k + mk, pour m ≥ 3, telle que D (1) = T M et C(D) = 0, alors D contient une sous-distribution involutive de corang k dans D si et seulement si le rang d'Engel de D ⊥ est constant et égal à k.

  Considérons deux systèmes affines Σ et Σ tels que les distributions de contrôle G et G soient de rang constant. Alors ils sont équivalents par bouclage si et seulement s'il existe un diffeomorphisme ϕ tel que

	Ã = ϕ * A,
	et sont équivalents par bouclage orbital s'il existe un diffeomorphisme ϕ et une fonction
	γ(•) = 0 tels que
	Ã = ϕ

* (γA), où γA = γf + G. Soit D est une distribution de rang constant sur X. Pour chaque distribution affine A ⊂ D, où A = f + G est telle que corang (G ⊂ D) = 1 et f (x) ∈ G(x), nous l'associons au système de contrôle Σ A . Nous avons les deux résultats suivants qui décrivent la relation entre la linéarisation par bouclage orbital du système (0.0.6) et la caractérisation de la distribution de Cartan CC n (R 1 , R m ). Proposition 0.0.10 Soit D une distribution de rang constant dans laquelle on choisit une distribution affine

  Theorem 4.3.1 is actually the implication (i) ⇔ (ii) of the more general Theorem 4.4.6 (where other characterizations of orbital feedback linearization are given) and therefore the proof of Theorem 4.3.1 follows form the proof of Theorem 4.4.6 given in Section 4.6. Remark 1. The following generalization of (OFL3) is necessary for orbital feedback linearization (see Lemma 4.6.1 in Section 4.6.1)

	2,n-l 2,1 .

  and the corresponding control systems Σ 2 , with f replaced by γ 2 f , as well as Σ 3 , with f replaced by γ 3 f , become feedback linearizable. This illustrates an interesting phenomenon that the linearizing time re-scaling function is not unique and may lead to different control systems (in fact, Σ 1 , Σ 2 and Σ 3 are not mutually locally equivalent via a diffeomorphism φ in the state space) which are, of course, feedback equivalent to each other.

	Example 4.5.2 (Rigid bar moving in R 3 )

  Clearly, both Theorem 4.3.1 and Proposition 4.4.5 are contained in Theorem 4.4.6 which gives a complete characterization of the orbital feedback linearization of the system Σ defined by (4.2.6). So it is enough to prove Theorem 4.4.6 which we will do in this section. The proof will be based on some lemmata given in Subsection 4.6.1.

	4.6 Proof of Theorem 4.4.6
	4.6.1 Useful results

Lemma 4.6.1 If the system Σ satisfies the conditions (OFL1) -(OFL3) of Theorem 4.3.1, then we have

  ÉOM ÉTRIE ET CLASSIFICATION DES SYST ÈMES DE CONTACT: APPLICATIONS AU CONTR ÔLE DES SYST ÈMES M ÉCANIQUES NON HOLONOMES R ÉSUM É: Dans la première partie de cette thèse, nous caractérisons complètement toutes les sorties plates et leurs lieux singuliers pour un système avec deux contrôles qui est équivalent au système chaîné. Nous appliquons aussi ce résultat au système du robot mobile avec des remorques pour calculer toutes ses sorties plates. Dans la deuxième partie, nous présentons un nouveau modèle pour le système à n-barres dans l'espace de dimension m + 1. Nous montrons que ce système est localement équivalent au système m-chaîné (système de contact sur J n (R, R m )) et caractérisons aussi ses lieux singuliers. Ensuite, nous analysons sa propriété de platitude et donnons ses sorties plates minimales. Dans la troisième partie, nous donnons des conditions nécessaires et suffisantes pour qu'une distribution soit équivalente à la distribution de Cartan pour des surfaces. Finalement, dans la quatrième partie, nous donnons des conditions nécessaires et suffisantes vérifiables pour qu'un système multi-entrées soit linéarisable par bouclage orbital. Mots-clés: Système de contrôle, système nonholonome, platitude, sortie plate, système de contact, distributon de Cartan, système à n-barres, linearisation par bouclage orbital GEOMETRY AND CLASSIFICATION OF CONTACT SYSTEMS: APPLICATIONS TO CONTROL OF NONHOLONOMIC MECHANICAL SYSTEMS

Remerciements

0 , . . . , P

= τ j (P 0 , Ṗ0 , P

0 , . . . , P

), for some functions τ j . In this way, the entire state and all controls v i , for 1 ≤ i ≤ n + m + 1, are expressed as functions of the coordinates of P 0 = (x 1 0 , x 2 0 , . . . , x m+1

0

) and their derivatives up to order n + 1. The n-bar system Γ has m + 1 controls while the system ∆ has n + m + 1 controls. So there must be n relations between controls of ∆ when restricted to Q = {x ∈ X : Ψ(x) = 0}. We will see this below and at the same time we will clarify the problem of singularities. Clearly, in order that the above computations hold, all the controls v i , for 1 ≤ i ≤ n, cannot vanish. It is sufficient, however, to assume that the control v 1 is nonzero since around any point x satisfying Ψ(x) = 0 and the regularity condition m+1 j=1 (x j ix j i-1 )(x j i+1x j i ) = 0, the condition v 1 = 0 implies that v i = 0, for 2 ≤ i ≤ n. In fact, differentiating the constraint

, for 1 ≤ j ≤ m + 1, into the above equation, by a simple calculation we get

Recall that around any regular point q 0 , we have always that m+1 j=1 (x j 1 -x j 0 )(x j 2 -x j 1 ) = 0. Therefore, the condition v 1 = 0 implies that v 2 = 0 and similarly, it can be shown that

The above equations show, first, that v 1 = 0 is equivalent to

Secondly, they show that there exist n relations between the controls v i , 1 ≤ i ≤ n + m+1, of ∆ implying that the n-bar system possesses, indeed, m+1 controls. Moreover, the condition v 1 = 0 (which yields v i = 0, 1 ≤ i ≤ n) implies that the instantaneous velocity Ṗ0 of the point P 0 is nonzero (and, consequently, the instantaneous velocities Ṗi of the points P i , 0 ≤ i ≤ n -1, are nonzero). Therefore the condition (2) holds.

= (-1) m (x

By a similar computation as above, we get

1 , Φ * dX 0 are independent on the configuration space Q. Therefore, (2.6.1) is true and item (c) holds.

Proofs

Useful results

In this section we give a series of results that we will use in the subsequent sections when proving our theorems. Lemma 3.5.1 Let D be a constant rank distribution on a manifold M. Assume that both D and its derived distribution D (1) are of constant rank and let C 0 and C 1 denote, respectively, their characteristic distributions. Then C 0 ⊂ C 1 .

Proof: In order to prove this lemma, it is sufficient to show that for any g ∈ C 0 , we have also g ∈ C 1 . Assume that f is an arbitrary vector field of D (1) . Since (1) , which implies that [g, f ] ∈ D (1) . Lemma 3.5.2 Let E and F be two involutive distributions on a manifold M. If they satisfy the following conditions

), in a neighborhood of any point in M, such that

The proof of this lemma (simultaneous rectification of involutive distributions) can be found, e.g., in [START_REF] Jakubczyk | On linearization of control systems[END_REF]. For completeness we give below another proof.

Chapter 4

Orbital feedback linearization for multi-input control systems

Introduction

Feedback linearization is a powerful tool for nonlinear control systems and has attracted a lot of research in recent years. Following the work of Brockett [START_REF] Brockett | Feedback invariants for nonlinear systems[END_REF] who solved the state feedback linearization for single-input systems under a restricted feedback, Jakubczyk and Respondek [START_REF] Jakubczyk | On linearization of control systems[END_REF] and Hunt and Su [START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF] gave geometric necessary and sufficient conditions for linearizing multi-input affine control systems under change of coordinates and general feedback which modifies both the drift and the control vector fields.

In the theory of dynamical systems two natural equivalence relations are considered: equivalence under diffeomorphisms and orbital equivalence. In the former case, the diffeomorphism establishing the equivalence maps trajectories into trajectories, understood as parameterized curves. In the latter case, equivalent systems have also the same trajectories but considered as non-parameterized curves (that is, the same trajectories up to a diffeomorphism and a time re-scaling). Observe that any two feedback equivalent systems have the same families of trajectories, up to a diffeomorphism, although their parametrization with respect to controls is different. It is then natural to ask when two control systems have the same families of trajectories, up to a reparametrization with respect to controls and a time re-scaling (that is, when they are orbital feedback equivalent). The linearization of a control system under feedback and time re-scaling was studied for the first time by Sampei and Furuta [START_REF] Sampei | On time scaling for nonlinear systems: applications to linearization[END_REF]. In their approach, solving a system of PDE's is necessary in order to verify feedback linearizability conditions and to find the time re-scaling function. Respondek developed in [START_REF] Respondek | Orbital Feedback Linearization of Single-input Nonliear Control Systems[END_REF], for the orbital feedback linearization of a single-input system, necessary and sufficient conditions, which can be conveniently verified upon the original system. and define on R (n+1)m+1 a distribution of rank m + 1

called the Cartan distribution. Formal similarity of the t-augmented Brunovsky canonical form Λ Br t and of the vector fields spanning CC n (R, R m ) rise a natural question: what are relations between control systems that are orbital feedback equivalent to Λ Br t and distributions equivalent to CC n (R, R m ). This section is devoted to various aspects of that question.

Consider an arbitrary distribution D. The derived flag of D is the sequence of modules of vector fields D (0) ⊂ D (1) ⊂ • • • defined inductively by

The Lie flag of D is the sequence of modules of vector fields

In general, the derived and Lie flags are different, though for any point x in the underlying manifold the inclusion D i (x) ⊂ D (i) (x) holds, for i ≥ 0.

A characteristic vector field of a distribution D is a vector field f that belongs to D and satisfies [f, D] ⊂ D. The characteristic distribution of D, which will be denoted by C(D), is the module spanned by all its characteristic vector fields. It follows directly from the Jacobi identity that the characteristic distribution is always involutive but, in general, it need not be of constant rank. Let L be a subdistribution contained in D, we denote by corank (L ⊂ D) the corank of L in D, that is the value of rank D-rank L.

The following theorem, given by Pasillas-Lépine and Respondek [START_REF] Pasillas-Lépine | Contact systems and corank one involutive subdistributions[END_REF], [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF], characterizes distributions which are equivalent to the Cartan distribution CC n (R, R m ). Theorem 4.4.1 A rank m+1 distribution D, with m ≥ 2, on a manifold X of dimension (n+1)m+1 is locally, at x 0 ∈ X, equivalent to the Cartan distribution CC n (R, R m ) if and only if the following conditions hold around x 0 , for 0 ≤ j ≤ n.

(B1) Each distribution D (j) is of rank (j + 1)m + 1 and contains an involutive subdistribution L j such that corank (L j ⊂ D (j) ) = 1; (B2) rank D (j) = rank D j .

Remark. One can prove that, for 0 ≤ j ≤ n -2, we have L j = C(D (j+1) ), the characteristic distribution of D (j+1) . It follows (see [START_REF] Pasillas-Lépine | Canonical Contact Systems for Curves: A Survey[END_REF] for details) that the condition (B1) can be weaken to

From control-affine systems to distributions

To any control-affine system Σ we will attach a distribution spanned by all vector fields corresponding to all controls: Definition 4.4.3 Consider a given control-affine system Σ defined by (4.2.6). The distribution D Σ associated with Σ is defined as

Let D Σ and DΣ be the distributions associated to Σ and Σ, respectively, i.e., D Σ = span {f }+G Σ and DΣ = span { f}+ GΣ . Clearly the orbital feedback equivalence of Σ and Σ implies the equivalence of the distributions D Σ and DΣ . But the former is apparently stronger because it also implies the equivalence of the control distributions G Σ and GΣ .

The pairs (D, G) and ( D, G) of distributions on X and X, respectively, are called equivalent if there exists a diffeomorphism ϕ : X → X such that

We have the following characterization of orbital feedback equivalence. Proposition 4.4.4 Assume that for control-affine systems Σ and Σ, the control distributions G Σ and GΣ are of constant rank and f (x) ∈ G Σ (x), for any x ∈ X and f (x) ∈ GΣ (x), for any x ∈ X. Then Σ and Σ are orbital feedback equivalent if and only if the pairs (D Σ , G Σ ) and ( DΣ , GΣ ) are equivalent.

Proof: As we have already observed, necessity is obvious. To prove sufficiency, denote by ϕ a diffeomorphism that establish the equivalence of (D Σ , G Σ ) and ( DΣ , GΣ ).

The assumption ϕ * G Σ = GΣ implies, because of the constant rank of G Σ and GΣ , the existence of functions β j i such that (4.2.3) holds. Choose any vector fields f ∈ A and f ∈ Ã such that f (x) ∈ G Σ (x) and f (x) ∈ GΣ (x). It follows that ϕ * (span {f }) = span { f} mod GΣ , and thus there exists a nonvanishing function γ such that ϕ * (γf ) = f mod GΣ . Hence (4.2.2) holds which proves feedback equivalence of Σ and Σ. This leads to a new, with respect to Theorem 4.3.1, characterization of orbital feedback linearizable systems. Proposition 4.4.5 Consider a control-affine system Σ = (f, G Σ ) and let D Σ be its associated distribution. Then Σ is locally orbital feedback equivalent to the t-augmented Brunovsky canonical form Λ Br t , with all controllability indices equal n + 1, if and only if D Σ satisfies the following conditions:

Put γ = µ 0 0 and α j = µ j 0 , for 1 ≤ j ≤ m, then we get

which implies that Σ is orbital feedback equivalent to the t-augmented Brunovsky canonical form with time re-scaling function γ = µ 0 0 .

All considerations presented in this section, together with Theorem 4.3.1 lead to the following complete characterization of the orbital feedback linearizability of the control-affine system Σ: Theorem 4.4.6 Consider the control-affine system Σ, defined by (4.2.6). Let D Σ be the distribution associated to Σ and G Σ = span {g 1 , . . . , g m } be the control distribution spanned by the control vectors. Then the following conditions are equivalent, locally around an arbitrary point x 0 ∈ X:

(i) Σ is orbital feedback equivalent to a t-augmented linear system Λ t , with all controllability indices equal to n + 1. (ii) Σ satisfies the conditions (OFL1) -(OFL4) of Theorem 4.

Σ ] ⊂ D

Σ .

The equivalence (i) ⇔ (ii) is just Theorem 4.3.1 whereas (i) ⇔ (iii) is just Proposition 4.4.5. The conditions (OFL1) -(OFL4) are expressed directly in terms of the control system Σ. On the other hand, the conditions of (iii), (iv) and (v)

and hence the statement of Lemma 4.6.2 follows. In order to finish the proof, it remains to show that rank B 1 can not be less than (n -1)m.

Suppose that rank B 1 (x 0 ) < (n -1)m, then clearly we have rank B 1 (x 0 ) = (n -1)m -1. We can assume, for simplicity, that the matrix B 2 obtained by neglecting the first row of B is invertible, i.e.,