M Barbero-Liñán

M Sigalotti
email: mario.sigalotti@inria.fr

High order sufficient conditions for tracking

In this paper, we study under which conditions the trajectories of a mechanical control system can track any curve on the configuration manifold. We focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature. The sufficient conditions are expressed in terms of convex cones of vector fields defined through particular brackets of the control vector fields of the system. The tracking control laws obtained by our constructions depend on several parameters. By imposing suitable asymptotic conditions on such parameters, we construct algorithmically one-parameter tracking control laws. The theory is supported by examples of control systems associated with elliptic hovercrafts and ellipsoidal submarines.

Introduction

The tracking problem has gained an increasing interest, mainly because of its applications to robot manipulators as for instance to control the position of underwater vehicles [START_REF] Chambrion | Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws[END_REF] and hovercrafts [START_REF] Pettersen | Underactuated ship tracking control: theory and experiments[END_REF]. The tracking problem appears when a particular trajectory has to be followed by a control system, but there is no control law for the control system that makes this trajectory admissible. Then, the best that can be expected is to find a control law, typically oscillatory, that defines a good enough approximation of the target trajectory.

The mathematical background in the tracking problem includes the averaging theory [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF] as explained, for instance, in [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF]. The averaging techniques transform differential equations difficult to solve into other differential equations whose solutions approximate fairly well the solutions to the first set of equations. This is useful to approximate solutions to differential equations that depend on time or on parameters.

Differential geometry has provided a suitable framework to study in an intrinsic way typical mechanical control systems in engineering as, for instance, underwater submarines, aircraft models, hovercrafts and so on [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF]. Here, we focus on forced affine connection control systems and generalize the sufficient conditions for tracking a trajectory that exist in the literature from a geometric viewpoint [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF]. The existent results can be interpreted as first-order sufficient conditions because only the control vector fields and particular brackets, called symmetric products, between them get involved in the statement of the sufficient conditions. However, our conditions need longer symmetric products and so they are said to be of order higher than two.

In [START_REF] Chambrion | Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws[END_REF] it was observed that the tracking is possible for specific underwater vehicles, even though they do not satisfy the geometric sufficient conditions known in the literature. That motivates our research in order to obtain more general geometric sufficient conditions that ensure the tracking property for a wider range of control systems.

The chances to be able to track a target trajectory are related to some controllability requirement and to the avoidance of "bad" directions. In an informal way, these "bad" directions have to be interpreted as directions that will not make possible to have the starting point in the interior of the reachable set. (We refer to [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF] for an accurate description of the obstructions to controllability in terms of the symmetric products and of particular vector-valued quadratic forms.) The sufficient conditions for being able to track unfeasible trajectories are also related to how nonholonomicity allows to enlarge the set of admissible velocities for the control system. In this regard, constructions of convex cones [START_REF] Bressan | On the controllability of Lagrangian systems by active constraints[END_REF], the above-mentioned vector-valued quadratic forms [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF] and some techniques similar to the ones in [START_REF] Martínez | Motion control algorithms for simple mechanical systems with symmetry[END_REF] have been useful for obtaining the constructions considered here.

The paper is organized as follows. Section 2 contains the necessary background in forced affine connection control systems and in chronological calculus [START_REF] Agrachev | Control Theory from the Geometric Viewpoint[END_REF]. Section 3 defines properly the notion of trackability and reviews the geometric sufficient conditions in the literature [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF]. Sections 4 and 5 contain the main contributions of the paper and some examples to justify the utility of these results.

Notation and preliminaries

Denote by N the set of positive natural numbers and write N 0 for N ∪ {0}. Fix n ∈ N. From now on, Q is a n-dimensional smooth manifold and X(Q) denotes the set of smooth vector fields on Q. All the vector fields are considered smooth, unless otherwise stated. Let τ Q : T Q → Q be the canonical tangent projection, a vector field X on Q defined along τ Q is a mapping X :

T Q → T Q such that τ Q • X = τ Q .

Affine connection control systems

The trajectories γ : I → Q of a Lagrangian mechanical systems on a manifold Q are minimizers of the action functional

A L (γ) = I L(t, γ(t))dt associated with a Lagrangian function L : R×T Q → R.
The solutions to this variational problem must satisfy the well-known Euler-Lagrange equations, d dt

∂L ∂v i - ∂L ∂q i = 0, i = 1, . . . , n, (1)
where (q i , v i ) are local coordinates for T Q. Here we consider controlled Euler-Lagrange equations obtained by modifying the right-hand side on the above equation, as follows:

d dt ∂L ∂v i - ∂L ∂q i = k a=1
u a Y i a , i = 1, . . . , n, with u a : I → R, Y i a : Q → R. When the manifold Q is endowed with the Riemannian structure given by a Riemannian metric g and the Lagrangian function L g (v q ) = 1 2 g(v q , v q ) is considered, the solutions to (1) turn out to be the geodesics of the Levi-Civita affine connection ∇ g associated with the Riemannian metric. (See [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF] for more details and for many examples of mechanical control systems that fit in this description.)

When control forces are added to the geodesic equations we obtain an affine connection control system

∇ g γ(t) γ(t) = k a=1 u a (t)Y a (γ(t)),
with Y a being vector fields on Q.

The notion of affine connection control system can be extended without the need of the Levi-Civita connection.

Definition 2.1. An affine connection is a mapping

∇ : X(Q) × X(Q) -→ X(Q) (X, Y ) -→ ∇(X, Y ) = ∇ X Y,
satisfying the following properties:

1. ∇ is R-linear on X and on Y ; 2. ∇ f X Y = f ∇ X Y for every f ∈ C ∞ (Q); 3. ∇ X f Y = f ∇ X Y + (Xf ) Y , for every f ∈ C ∞ (Q).
(Here Xf denotes the derivative of f in the direction X.)

The mapping ∇ X Y is called the covariant derivative of Y with respect to X. Given local coordinates (q i ) on Q, the Christoffel symbols for the affine connection in these coordinates are given by

∇ ∂ ∂q j ∂ ∂q r = n i=1 Γ i jr ∂ ∂q i .
From the properties of the affine connection, we have

∇ X Y = n i,j,r=1 X j ∂Y i ∂q j + Γ i jr X j Y r ∂ ∂q i , where X = n i=1 X i ∂/∂q i and Y = n i=1 Y i ∂/∂q i . Definition 2.2. A forced affine connection control system (FACCS) is a control mechani- cal system given by Σ = (Q, ∇, Y, Y , U ) where • Q is a smooth n-dimensional manifold called the configuration manifold,
• Y is a time-dependent vector field along the projection τ Q : T Q → Q, measurable and bounded with respect to the time and affine with respect to the velocities,

• Y is a set of k control vector fields on Q, and

• U ⊆ R k . A trajectory γ : I ⊂ R → Q is admissible for Σ if γ : I → T Q
is absolutely continuous and there exists a measurable and bounded control u : I → U such that the dynamical equations of the control system Σ

∇ γ(t) γ(t) = Y (t, γ(t)) + k a=1 u a (t)Y a (γ(t)), (2)
are fulfilled (almost everywhere).

The vector field Y includes all the noncontrolled external forces; e.g., the potential and the non-potential forces. The assumption that Y is affine with respect to the velocities means that, in every local system of coordinates (q i , v i ) on T Q, Y can be written as

Y (t, v q ) = Y 0 (t, q) + n i=1 v i Y i (t, q).
Equation (2) can be rewritten as a first-order control-affine system on T Q,

Υ(t) = Z(Υ(t)) + Y V (t, Υ(t)) + k a=1 u a (t)Y V a (Υ(t)), (3) where
Υ : I → T Q is such that τ Q • Υ = γ,
Z is the geodesic spray associated to the affine connection on Q and Y V a denotes the vertical lift of the vector field Y a (see [START_REF] Abraham | Foundations of Mechanics[END_REF] for more details).

Apart from the usual Lie bracket that provides X(Q) with a Lie algebra structure, the following product of elements in X(Q) associated to ∇ can be introduced.

Definition 2.3. The symmetric product is the map

• : • : X(Q) × X(Q) -→ X(Q) (X, Y ) -→ ∇ X Y + ∇ Y X. It can be proved that [Y V a , [Z, Y V b ]] = Y a : Y b V (4)
(see [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF]).

Chronological calculus

We recall in this section some notion of chronological calculus, which is used later as a tool in the study of the asymptotic behavior of endpoint mappings depending on parameters. For a comprehensive discussion and for the proofs of all results stated in this section see [START_REF] Agrachev | Control Theory from the Geometric Viewpoint[END_REF]. In the sequel all vector fields, autonomous and non-autonomous, are assumed to be complete. The behavior of non-complete vector fields on compact sets can be studied by considering suitable cut-off procedures.

Given a non-autonomous vector field X τ on some manifold M , where τ denotes the time variable and the map (τ, q) → X τ (q) is assumed to be smooth with respect to q and measurable bounded with respect to τ , we denote by -→ exp t 0 X τ dτ the diffeomorphism of M onto itself corresponding to the flow from time 0 to time t of X τ . Hence, -→ exp t 0 X τ dτ (q) is the evaluation at time t of the solution to the non-autonomous Cauchy problem q(τ ) = X τ (q(τ )), q(0) = q.

Any diffeomorphism P : M → M defines an isomorphism AdP of X(M ) through the rule AdP (Y )(q) = P * (Y (P -1 (q))), where P * denotes the pushforward by P .

If P = -→ exp t 0 X τ dτ , then we write AdP = -→ exp t 0 adX τ dτ.
If ad (m) X τ (Y ) = 0 for some m ∈ N and every τ , then

-→ exp t 0 adX τ dτ (Y ) = Y + t 0 [X τ , Y ]dτ + t 0 τ1 0 [X τ2 , [X τ1 , Y ]dτ 2 dτ 1 + • • • + t 0 τ1 0 • • • τm-2 0 adX τm-1 • • • • • adX τ1 Y dτ m-1 • • • dτ 1 . (5)
In the framework of chronological calculus the flow of the sum of two non-autonomous vector fields can be conveniently represented by the following variation formula

-→ exp t 0 (X τ + Y τ )dτ = -→ exp t 0 X τ dτ • -→ exp t 0 -→ exp τ 0 adX s ds Y τ dτ. (6)
Let us recall a useful result for the convergence of flows of non-autonomous vector fields. It states, roughly speaking, that the flows converge if the vector fields converge in integral sense. For further results and a discussion on this kind of convergence from the point of view of ordinary differential equations and control theory, see [START_REF] Biles | Continuous dependence of nonmonotonic discontinuous differential equations[END_REF][START_REF] Liu | Continuous dependence with respect to the input of trajectories of control-affine systems[END_REF]. Lemma 8.10] Let Z j τ , j ∈ N, and Z τ , τ ∈ [0, t 1 ], be non-autonomous vector fields on M , bounded with respect to τ , and let these vector fields have a compact support. If

Lemma 2.4. [2,
t 0 Z j τ dτ → t 0 Z τ dτ, j → ∞, then -→ exp t 0 Z j τ dτ → -→ exp t 0 Z τ dτ, j → ∞,
both convergences being uniform with respect to (t, q) ∈ [0, t 1 ] × M and uniform with all derivatives with respect to q ∈ M .

Another, even more standard, result on the convergence of flows is the following, that we find useful to state as a separate lemma. Its difference from the previous one can be described as follows: if the dependence on time of the Z j τ is prescribed independently on j (it would be enough that they converge uniformly), then the uniform convergence of flows is guaranteed by the uniform convergence of the vector fields, even without any knowledge about the convergence of their derivatives with respect to the state variables. For completeness, we provide a brief proof for it. Lemma 2.5. Let λ 1 , . . . , λ m ∈ L ∞ ([0, t 1 ], R). Let Z j τ , n ∈ N, and Z τ , τ ∈ [0, t 1 ], be non-autonomous vector fields on M of the form

Z j τ = λ 1 (τ )Z j 1 + • • • + λ m (τ )Z j m and Z τ = λ 1 (τ )Z 1 + • • • + λ m (τ )Z m , with Z j i , Z i ∈ X(M ) with compact support. If Z j i → Z i , j → ∞, then -→ exp t 0 Z j τ dτ → -→ exp t 0 Z τ dτ, j → ∞,
both convergences being uniform (the first with respect to the state q and the second with respect to (t, q)).

Proof. Using the variation formula [START_REF] Chambrion | Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws[END_REF] with

X τ = Z τ and Y τ = Z j τ -Z τ , we get -→ exp t 0 Z j τ dτ = -→ exp t 0 Z τ dτ • -→ exp t 0 -→ exp τ 0 adZ s ds (Z j τ -Z τ )dτ. (7)
Since -→ exp

τ 0 Z s ds * , τ ∈ [0, t 1 ]
, is a compact family of operators, then the last diffeomorphism in [START_REF] Gauthier | On the motion planning problem, complexity, entropy, and nonholonomic interpolation[END_REF] converges uniformly to the identity for t ∈ [0, t 1 ] as j goes to infinity.

Tracking problem

The need of trackability appears when one tries to follow a particular trajectory on the configuration manifold, called reference or target trajectory, which is not a solution of the FACCS considered. A trajectory is successfully tracked if there exist solutions to the FACCS that approximate it arbitrarily well.

Consider any distance d : Q × Q → R on Q whose corresponding metric topology coincides with the topology on Q. From now on I will denote the interval [0, t 1 ], with t 1 > 0.

Definition 3.1. A control system Σ is trackable if for every continuous curve γ : I → Q, for every v ∈ T γ(0) Q and for every strictly positive tolerance ǫ, there exist a control u ǫ ∈ L ∞ (I, U ) such that the solution ξ ǫ : I → Q to Σ corresponding to u ǫ and with initial condition ξǫ (0) = v verifies d(γ(t), ξ ǫ (t)) < ǫ for every time t ∈ I. Remark 3.2. Since any continuous curve can be uniformly approximated, with arbitrary precision, by a smooth curve having a prescribed tangent vector at its initial point, then Σ is trackable if and only if every curve γ : I → Q on Q of class C ∞ is trackable for Σ, i.e., for every ǫ > 0, there exist u ǫ ∈ L ∞ (I, U ) such that the solution ξ ǫ : I → Q to Σ corresponding to u ǫ and with initial condition ξǫ (0) = γ(0) verifies d(γ(t), ξ ǫ (t)) < ǫ for every t ∈ I.

In order to give some insights into particular sufficient conditions for tracking, we are going to review a result in the literature.

Theorem 3.3. [5, Theorem 12.26] Let Σ = (Q, ∇, Y, Y , U ) be a FACCS such that • the distribution generated by Y = {Y 1 , . . . , Y k } is regular, that is, it is a subbundle of T Q; • Y a : Y a ∈ span C ∞ (Q) Y for every a ∈ {1, . . . , k}, i.e., Y a : Y a = k b=1 σ ab Y b , where σ ab ∈ C ∞ (Q);
• the distribution Sym (1) (Y ) defined by Sym (1)

(Y ) q = span R Y q + span R { W : Z (q)|W, Z ∈ Y } is the entire tangent bundle T Q. Let γ ref : I → Q be a reference trajectory of class C 3 . Define the functions u ref,a , u ref,bc : I → R, a, b, c ∈ {1, . . . , k}, b < c, as solutions of class C 1 to ∇ γref (t) γref (t) -Y (t, γref (t)) = k a=1 u ref,a (t)Y a (γ ref (t)) + k b,c=1,b<c u ref,bc (t) Y b : Y c (γ ref (t)). Define u slow : I × T Q → R k , u osc : R × I × T Q → R k by u slow,a (t, v q ) = u ref,a (t)+ k b=1 b -1 + k c=b+1 (u ref,bc (t)) 2 4 σ ab (q), u osc,a (τ, t, v q ) = a-1 c=1 ϕ lo(c,a) (τ ) -1 2 k c=a+1 u ref,ac (t)ϕ lo(a,c) (τ ), where lo(a, b) = a-1 j=1 (k -j) + (b -a) for (a, b) ∈ {1, .
. . , k} 2 , a < b, and for i ∈ N

ϕ i : R → R, t → 4πi T cos 2πi T t . ( 8
)
Then γ ref is trackable for Σ and moreover the solutions ξ ǫ of Σ, ǫ > 0, corresponding to the controls

u ǫ : R × T Q -→ U ⊂ R k (t, v q ) -→ u slow (t, v q ) + 1 ǫ u osc t ǫ , t, v q
with initial condition ξǫ (0) = γref (0) are such that d(γ ref (t), ξ ǫ (t)) tends to zero as ǫ goes to zero uniformly with respect to t ∈ I.

Remark 3.4. Observe that under the hypotheses of Theorem 3.3 not only the tracking is guaranteed, but also the tracking control law is given explicitly.

A more general tracking result

The idea of the following construction is to identify, given the set Y , a larger set of control vector fields K 1 such that every trajectory solution of the FACCS obtained by replacing span C ∞ (Q) Y by K 1 can be tracked by solutions of the original FACCS Σ. Repeating the construction on K 1 we obtain an even larger family K 2 and so on. If eventually K l (q) = T q Q for every q ∈ Q for some l ∈ N, then we can show that the system is trackable.

In order to generalize the sufficient conditions for tracking given in Theorem 3.3, we construct the following set of vector fields on Q:

K 0 = span C ∞ (Q) Y , K l = K l-1 -co { Z : Z | Z ∈ L(K l-1 )}, (9)
for l ∈ N, where, for A ⊂ X(Q), L(A) = A ∩ (-A), co(A) denotes the convex hull of A, and A is the closure of A in X(Q) with respect to the topology of the uniform convergence on compact sets. For A ⊂ X(Q) we also write

A(q) = {Y (q) | Y ∈ A} ⊆ T q Q. Proposition 4.1. For every l ∈ N 0 , K l is a convex cone of X (Q). In particular, for every q ∈ Q, K l (q) is a convex cone with vertex at 0 ∈ T q Q.
Proof. The proposition is proved by induction having in mind that the set K l is a convex cone if it contains all conic combinations of elements of K l . Remember that a conic combination of elements of

K l is of the form λ 1 W 1 + • • • + λ r W r with λ i ∈ R ≥0 , W i ∈ K l for every i ∈ {1, . . . , r}.
First, K 0 is subspace of X (Q) and thus it is a convex cone.

The induction step consists of proving that if K i is a convex cone, so is K i+1 . First notice that, if W ∈ L(K i ) and λ ∈ R, then, by the induction hypothesis, λW ∈ L(K i ) and thus λ 2 W : W belongs to { Z : Z | Z ∈ L(K i )}. Hence, K i+1 is the closure of the sum of two convex sets invariant by multiplication by any non-negative scalar. Therefore, K i+1 is itself a convex cone.

Given two functions

α, β : [0, T ] → R, let Λ T (α, β) = 1 2T T 0 τ 0 α(s)ds τ 0 β(s)ds dτ.
We say that a sequence of smooth T -periodic functions ψ j : R → R, j ∈ N, is Λ T -orthonormal and zero-mean if T 0 ψ j (τ ) = 0 for every j ∈ N and Λ T (ψ j , ψ m ) = δ jm for j, m ∈ N, where δ jm denotes the Kronecker delta.

For instance, the sequence defined in [START_REF] Jean | Complexity of nonholonomic motion planning[END_REF], is Λ Torthonormal and zero-mean.

Proposition 4.2. Let Σ = (Q, ∇, Y, Y , R k ) be a FACCS and fix v q ∈ T Q, T > 0, n 1 , . . . , n k ∈ N, and w = (w 1 , . . . , w k ) ∈ L ∞ (I, R k ). For every ǫ > 0 denote by ξ ǫ : I → Q the solution of ∇ ξǫ (t) ξǫ (t) = Y (t, ξǫ (t)) + k a=1 1 ǫ ψ na t ǫ w a (t)Y a (ξ ǫ (t)),
with initial condition ξǫ (0) = v q , where (ψ j ) j∈N is a Λ T -orthonormal and zero-mean sequence. Let also γ : I → Q be the solution of

∇ γ(t) γ(t) = Y (t, γ(t)) - k a,b=1 δ nan b w a (t)w b (t) Y a : Y b (γ(t)), ( 10
)
with initial condition γ(0) = v q . Then there exist C, ǫ 0 > 0 such that d(γ(t), ξ ǫ (t)) < Cǫ for every t ∈ I and every ǫ ∈ (0, ǫ 0 ).

Proof. The proof of this proposition follows from Theorem 9.32 in [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF] by considering the function u a (t/ǫ, t) that appears there as ψ na (t/ǫ)w a (t).

Remark 4.3. Theorem 3.3 follows directly from Proposition 4.2 (and is obtained as such in [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF]). Therefore, some of the hypotheses of Theorem 3.3 can easily be relaxed: first of all the distribution generated by Y need not be a subbundle of T Q.

Moreover, the hypothesis that Sym (1) (Y ) = T Q can be replaced by the requirement that

∇ γref (t) γref (t) -Y (t, γref (t)) ∈ Sym (1) (Y ) γ ref (t)
for every t ∈ I.

Theorem 4.4. Let Σ = (Q, ∇, Y, Y , R k ) be a FACCS. Fix a reference trajectory γ ref : I → Q of class C ∞ . If for every t ∈ I there exists l ∈ N such that K l (γ ref (t)) = T γ ref (t) Q, then γ ref is track- able. Therefore, if for every q ∈ Q there exists l ∈ N such that K l (q) = T q Q, then the control system Σ is trackable.
Proof. First of all, notice that the last part of the statement (the trackability of Σ) directly follows from the first one (the trackability of γ ref ), which is proved below.

As the reference trajectory γ ref is defined on a compact set and γ ref is continuous, then Imγ ref is a closed compact set. By hypothesis, for every t ∈ I there exists l ∈ N such that

K l (γ ref (t)) = T γ ref (t) Q. So there exist n + 1 vector fields in K l whose conic combinations at γ ref (t)
i ∈ N such that K l i (γ ref (t)) = T γ ref (t) Q for every t ∈ γ -1 ref (U t i ). Then l = max{l 1 , . . . , l r } satisfies K l (γ ref (t)) = T γ ref (t) Q for every t ∈ I.
Moreover, there exists a partition of unity subordinated to the finite open subcover {U t i } i=1,...,r (see [START_REF] Lee | Introduction to Smooth Manifolds[END_REF]) that allows us to define a finite set of smooth global vector fields Z a in K l such that

∇ γref (t) γref (t) -Y (t, γref (t)) = N l a=1 λ a (t)Z a (γ ref (t)) (11) for t ∈ I, N l ∈ N, where λ a : I → [0, +∞) is of class C ∞ .
We introduce for a curve on Q the notion of having a regular parameterization on K i . A curve γ : I → Q admits a regular parameterization on

K i if there exist N i ∈ N, λ 1 , . . . , λ N i ∈ C(I) and Z 1 , . . . , Z N i ∈ K i such that λ a (t) ≥ 0 for every t ∈ I and a ∈ {1, . . . , N i } and ∇ γ(t) γ(t) -Y (t, γ(t)) = N i a=1 λ a (t)Z a (γ(t)).
Fix ǫ > 0 and let ξ l = γ ref . Then ξ l admits a regular parameterization on K l . We are going to prove by induction that there exists a finite sequence {ξ 0 , ξ 1 , . . . , ξ l } of curves on Q such that each ξ i satisfies ξi (0) = γref (0), admits a regular parameterization on K i and d(ξ i (t), ξ i+1 (t)) < ǫ/l for every i = 0, . . . , l -1 and every t ∈ I. The induction step claims that: for i = 0, . . . , l -1 if there exists ξ i+1 : I → Q admitting a regular parameterization on K i+1 , then there exists ξ i : I → Q with ξi+1 (0) = ξi (0) admitting a regular parameterization on K i and satisfying d(ξ i (t), ξ i+1 (t)) < ǫ/l for every t ∈ I.

Let us prove the induction step for i. Let Z 1 , . . . , Z N i+1 ∈ K i+1 be the vector fields that determine the regular parameterization of ξ i+1 . Here we split the proof in two steps, considering first the special case:

1. Z a = F a -G a with F a ∈ K i and G a ∈ co { Z : Z | Z ∈ L(K i )} for every a = 1, . . . , N i+1 ,
and then the general case

2. Z a ∈ K i -co { Z : Z | Z ∈ L(K i )}.
Let us study case 1: ξ i+1 admits the parameterization

∇ ξi+1 (t) ξi+1 (t) -Y (t, ξi+1 (t)) = N i+1 a=1 λ a (t)(F a -G a )(ξ i+1 (t)). ( 12
)
Each G a is given by

Na b=1 α a,b G a,b : G a,b where G a,b ∈ L(K i ) and α a,b ≥ 0. Then (12) becomes ∇ ξi+1 (t) ξi+1 (t) -Y (t, ξi+1 (t)) = N i+1 a=1 λ a (t) F a (ξ i+1 (t)) - Na b=1 λ a (t)α a,b G a,b : G a,b (ξ i+1 (t)) .
(13) The dynamics described in [START_REF] Milne-Thomson | Theoretical hydrodynamics[END_REF] are of the same form as in [START_REF] Lee | Introduction to Smooth Manifolds[END_REF] as long as we take Y = {G a,b } a=1,...,N i+1 ;b=1,...,Na , and

δ n ab n a ′ b ′ = δ aa ′ δ bb ′ , w a,b = λ a α a,b .
Then by Proposition 4.2 the solutions to ( 13) can be approximated by solutions to

∇ ξǫ i i (t) ξǫ i i (t) -Y (t, ξǫ i i (t)) = N i+1 a=1 λ a (t)F a (ξ ǫ i i (t)) + Na b=1 1 ǫ i ψ n ab t ǫ i w a,b (t)G a,b (ξ ǫ i i (t)) (14)
being (a, b) → n ab any injective map from N × N to N and (ψ j ) j∈N a Λ T -orthonormal and zero-mean sequence. More precisely, there exist C i , ǫ i,0 > 0 such that d(ξ ǫ i i (t), ξ i+1 (t)) < C i ǫ i for every t ∈ I, ǫ i ∈ (0, ǫ i,0 ). In particular, we can choose ǫ i such that C i ǫ i < ǫ/l.

The finite linear combination of elements in K i (ξ ǫ i i (t)) for every t ∈ I on the right-hand side of ( 14) does not necessarily satisfy the nonnegativeness of the coefficients. However, G a,b ∈ L(K i ), so -G a,b ∈ L(K i ). Then we can rewrite the coefficients of G a,b as follows:

ψ n ab t ǫ i w a,b (t)G a,b = max 0, ψ n ab t ǫ i w a,b (t)G a,b + max 0, -ψ n ab t ǫ i w a,b (t)(-G a,b ).
Thus all the coefficients are continuous and nonnegative. We can conclude that ξ ǫ i i admits a regular parameterization on K i . We define then ξ i = ξ ǫ i i and the induction step has been proved for i in the case 1.

Let us turn to case 2. We recall that the closure appearing in ( 9) is considered with respect to the topology of the uniform convergence on compact sets. Then there exist two sequences F j a ∈ K i and

G j a ∈ co { Z : Z | Z ∈ L(K i )} such that F j a -G j a
converges uniformly to Z a on a neighborhood of the curve ξ i+1 . For every j ∈ N let ξ j i+1 : I → Q be the solution to

∇ ξj i+1 (t) ξj i+1 (t) -Y (t, ξj i+1 (t)) = N i+1 a=1 λ a (t)(F j a -G j a )(ξ j i+1 (t)) (15)
satisfying ξj i+1 (0) = ξi+1 (0). Then, thanks to Lemma 2.5, ξ j i+1 converges to ξ i+1 uniformly on I as j tends to infinity. Take  large enough such that

d(ξ  i+1 (t), ξ i+1 (t)) < ǫ 2l
for every t ∈ I. We can apply to ξ  i+1 the same reasoning as in the first case. Then solutions to [START_REF] Pettersen | Underactuated ship tracking control: theory and experiments[END_REF] are approximated by solutions to [START_REF] Munnier | Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalisms[END_REF]

replacing ξ ǫ i i by ξ ,ǫ i i , F a by F  a and G a,b by G  a,b . In other words, there exist C i , ǫ i,0 > 0 such that d(ξ ,ǫ i i (t), ξ  i+1 (t)) < C i ǫ i for every t ∈ I, ǫ i ∈ (0, ǫ i,0 )
. Again, ǫ i can be chosen in such a way that C i ǫ i < ǫ 2l and we define

ξ i = ξ ,ǫ i i . Thus, d(ξ i (t), ξ i+1 (t)) ≤ d(ξ ,ǫ i i (t), ξ  i+1 (t)) + d(ξ  i+1 (t), ξ i+1 (t)) < ǫ l
and ξ i admits a regular parameterization on K i . Hence, the induction step has been proved for i.

After the induction, we end up with a curve ξ 0 on Q admitting a regular parameterization on K 0 and such that

d(ξ 0 (t), γ ref (t)) ≤ l-1 i=0 d(ξ i (t), ξ i+1 (t)) < l ǫ l = ǫ for every t ∈ I. (Recall that ξ l = γ ref .) Moreover, by compactness of I, ǭ = ǫ -max t∈I d(ξ 0 (t), γ ref (t)) > 0. As K 0 = span C ∞ (Q) Y , we conclude from Lemma 2.5 that there exist θ a,j ∈ C ∞ (Q), a ∈ {1, . . . , N 0 }, j ∈ {1, . . . , k}, such that the solution to ∇ ξ(t) ξ(t) -Y (t, ξ(t)) = N 0 a=1 k j=1 λ a (t)θ a,j (ξ(t))Y j (ξ(t))
with initial condition ξ(0) = ξ0 (0) = γref (0) satisfies d(ξ(t), ξ 0 (t)) < ǭ for every t ∈ I. Thus, d(ξ(t), γ ref (t)) < ǫ for every t ∈ I.

Since ξ is an admissible trajectory for Σ, we conclude that γ ref is trackable for Σ with the tracking control law given by u j (t) =

N 0 a=1 λ a (t)θ a,j (ξ(t)), j = 1, . . . , k. Corollary 4.5. Let Σ = (Q, ∇, Y, Y , R k ) be a FACCS.
Define the following set of vector fields on Q for l ∈ N:

H 0 = span C ∞ (Q) Y , H l = H l-1 -co { Z : Z | Z ∈ L(H l-1 )} . ( 16
) Fix a smooth reference trajectory γ ref : I → Q. If for every t ∈ I there exists l ∈ N such that H l (γ ref (t)) = T γ ref (t) Q, then γ ref is trackable. Therefore, if for every q ∈ Q there exists l ∈ N such that H l (q) = T q Q, then the control system Σ is trackable.
Proof. Let us prove by induction that

H i ⊆ K i i ∈ N 0 .
It is trivial that H 0 ⊆ K 0 (see ( 9) and ( 16)).

The claim now is that if

H i ⊆ K i , then H i+1 ⊆ K i+1 . By definition, an element in H i+1 is of the form F -G with F ∈ H i and G ∈ co { Z : Z | Z ∈ L(H i )}. Since H i ⊆ K i , then F ∈ K i and L(H i ) ⊆ L(K i ). So G ∈ co { Z : Z | Z ∈ L(K i )}. We can conclude that H i+1 ⊆ K i+1 .
By hypotheses, for every t ∈ I there exists l ∈ N such that

H l (γ ref (t)) = T γ ref (t) Q. As H l ⊆ K l , we have K l (γ ref (t)) = T γ ref (t) Q.
The hypotheses of Theorem 4.4 are satisfied, so the result holds.

Remark 4.6. If one follows the proof of Theorem 4.4 under the stronger hypotheses of Corollary 4.5, then each step of the induction procedure is of the type considered in case 1. This will be important in the next section, where we will turn such procedure in an algorithmic construction.

Corollary 4.7. Let Σ = (Q, ∇, Y, Y , R k ) be a FACCS.
Define the following sets of vector fields for l ∈ N,

Z 0 =Y , Z l =Z l-1 ∪ { Z a : Z b | Z a , Z b ∈ Z l-1 }. ( 17
)
If there exists l ∈ N such that span R Z l (q) = T q Q for all q ∈ Q and for each i ∈ {0, . . . , l -1}, for each

Z ∈ Z i , Z : Z ∈ span C ∞ (Q) Z i , then the system Σ is trackable.
Proof. Let us prove by induction that

span C ∞ (Q) Z i ⊆ K i , i ∈ N 0 .
Once this inclusion is proved Theorem 4.4 guarantees the trackability of the system.

It is trivial by definition that span

C ∞ (Q) Z 0 ⊆ K 0 .
Assume that span C ∞ (Q) Z i ⊆ K i and let us prove the inclusion for i. Since K i+1 is a convex cone by Proposition 4.1 and span [START_REF] Bressan | On the controllability of Lagrangian systems by active constraints[END_REF] and to the hypotheses on the symmetric products of elements of Z i ,

C ∞ (Q) Z i ⊂ K i+1 , it is enough to prove that α Z a : Z b belongs to K i+1 for Z a , Z b ∈ Z i and α ∈ C ∞ (Q). Thanks to
αZ a : Z b ∈ α Z a : Z b + span C ∞ (Q) Z i , αZ a : αZ a , Z b : Z b ∈ span C ∞ (Q) Z i .
Hence the symmetric product

αZ a -Z b : αZ a -Z b belongs to -2α Z a : Z b + span C ∞ (Q) Z i . Thus, α Z a : Z b belongs to span C ∞ (Q) Z i -1 2 αZ a - Z b : αZ a -Z b , which is contained in K i+1 .
Remark 4.8. The proof above actually shows that, under the assumptions of Corollary 4.7,

span C ∞ (Q) Z i ⊆ H i . It is easy to check that, in addition, H i = span C ∞ (Q) Z i .
Remark 4.9. The main interest of Corollary 4.7 is that its hypotheses are formulated in terms of a finite set of vector fields, in contrast with the infinite family of vector fields considered in Theorem 4.4 and Corollary 4.5.

Examples

Let us consider some examples of mechanical systems for which the above results guarantee the trackability, but Theorem 3.3 could not guarantee it.

Hovercraft

Consider an elliptic hovercraft moving on the surface of a fluid, identified with R 2 . The configuration manifold is Q = S 1 × R 2 with local coordinates (θ, x 1 , x 2 ) where θ is the attitude and (x 1 , x 2 ) is the position of the center of symmetry of the hovercraft. Let ω and (v 1 , v 2 ) be the standard angular and linear velocity, respectively, of the hovercraft with respect to a body-fixed coordinate frame attached at the center of symmetry of the body and whose axes coincide with those of the ellipse. Assume that the center of mass according to that body-fixed coordinate frame is on the horizontal axis and is different from the center of symmetry. Then the added inertia matrix is the following 3 × 3 symmetric matrix:

M =   a 0 c 0 e 0 c 0 e   ,
with a, c, e > 0 (see [START_REF] Lamb | Hydrodynamics. Reprint of the 1932[END_REF][START_REF] Milne-Thomson | Theoretical hydrodynamics[END_REF] for more details). Denote the corresponding impulse vector by (Π, P 1 , P 2 ) that is related to the velocities through the inertia matrix M as follows

  Π P 1 P 2   = M   ω v 1 v 2   .
The dynamics of the systems governed by the Kirchhoff equations in dimension 2 with two controls are

  θ ẋ ẏ  =   1 0 0 0 cos θ -sin θ 0 sin θ cos θ     ω x y   ,   ω v1 v2   = M -1 P • v ⊥ ωP ⊥ +   u 1 0 u 2  
where w ⊥ = (-w 2 , w 1 ) denotes the rotation by π/2 of a vector w = (w 1 , w 2 ) in R 2 . The control vector fields are Y 1 = (1, 0, 0) and Y 2 = (0, 0, 1). They correspond to the external torque, usually called yaw, and the external force, usually called surge, applied to the body. Let Y = {Y 1 , Y 2 }.

The drift term Z appearing in (3) is, for the mechaincal system considered here, the vector field corresponding to the uncontrolled Kirchhoff's equations (notice that Y = 0). Using (4) we can compute Y 1 : Y 1 = (0, 2c/e, 0). The sufficient conditions for tracking given by Theorem 3.3 are not satisfied because

Y 1 : Y 1 / ∈ span C ∞ (Q) {Y 1 , Y 2 }.
However, due to Corollary 4.5 tracking is possible because

H 1 (q) = T q Q ∀ q ∈ Q. Indeed, for w 1 , w 2 ∈ C ∞ (Q), w 1 Y 1 + w 2 Y 2 : w 1 Y 1 + w 2 Y 2 = 1 e (0, 2cw 2 1 + 2ew 1 w 2 , 0)+ ∈ span C ∞ (Q) Y 2 2 i,j=1 w i Y i (w j )Y j ,
where Y i (w j ) denotes the Lie derivative of w j with respect to Y i . In particular, taking as w 1 any nonzero constant function, we get

span C ∞ (Q) (0, 1, 0) ⊂ H 0 -co { Z : Z | Z ∈ L(H 0 )} .
So we conclude that H 1 (q) = T q Q for all q ∈ Q.

Submarine

Let us apply Corollary 4.7 to determine the trackability of a particular control system describing the motion of a submarine. The system corresponds to the case γ = 0 considered in [START_REF] Chambrion | Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws[END_REF]. It models a neutrally buoyant ellipsoid vehicle immersed in a infinite volume fluid that is inviscid, incompressible and whose motion is irrotational. The dynamics are obtained through Kirchhoff equations [START_REF] Lamb | Hydrodynamics. Reprint of the 1932[END_REF] and have a particularly simple form due to some symmetry assumption on the distribution of mass (see [START_REF] Chambrion | Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws[END_REF] for details and also [START_REF] Munnier | Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalisms[END_REF] for general overview of control motion in a potential fluid).

Consider the coordinates (ω, v) for the angular and linear velocity of the ellipsoid with respect to a body-fixed coordinate frame. Then the impulse (Π, P ) of the system is given by

Π P = M ω v
where, under the symmetry assumptions mentioned above,

M = diag(J 1 , J 1 , J 3 , M 1 , M 2 , M 3 )
with M 1 = M 2 , where diag(J 1 , J 1 , J 3 ) is the usual inertia matrix and M 1 , M 2 , M 3 take into account the mass of the submarine and the added masses due to the action of the fluid.

The configuration manifold Q for this problem is the Special Euclidean group or the group of rigid motions SE(3), which is homeomorphic to R 3 × SO(3). Let (r, A) ∈ SE(3) be the position and the attitude of the ellipsoid. Denote by S : R 3 → so(3) the linear bijection between R 3 and the linear algebra so(3) of SO(3) such that

S(x 1 , x 2 , x 3 ) =   0 -x 3 x 2 x 3 0 -x 1 -x 2 x 1 0  
The dynamics of the controlled system are given by Due to Theorem 1.2 in [START_REF] Chambrion | Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws[END_REF], we know that system (18)-( 19) is trackable. This case cannot be recovered from Theorem 3.3 because Sym (1) (Y ) = T Q.

dr dt = Av, dA dt = AS(ω), (18)
Indeed, it can be computed that

Y 1 : Y 2 = 0, Y 2 : Y 3 = M 3 M 1 ∂ ∂P 1 , Y 1 : Y 3 = - M 3 M 2 ∂ ∂P 2 .
However, this case is covered by Corollary 4.7 with l = 2, because

0 = Y 1 : Y 1 = Y 2 : Y 2 = Y 3 : Y 3 = = Y 2 : Y 3 : Y 2 : Y 3 = Y 1 : Y 3 : Y 1 : Y 3 , Y 2 : Y 3 : Y 1 : Y 3 = - M 2 3 J 3 1 M 1 - 1 M 2 ∂ ∂Π 3 .
Thus,

Z : Z = 0 for every Z ∈ Z 2 and T q Q = span R Z 2 (q) = span R {Y 1 (q), Y 2 (q), Y 3 (q), Y 2 : Y 3 (q), Y 1 : Y 3 (q), Y 2 : Y 3 : Y 1 : Y 3 (q)} for all q ∈ Q.
The model studied in [START_REF] Chambrion | Tracking control for an ellipsoidal submarine driven by Kirchhoff's laws[END_REF] can therefore be handled with the techniques proposed here. In particular, we can obtain for it one-parameter tracking control laws, as explained in the next section.

One-parameter tracking control laws

The aim of this section is to provide an algorithmic implementation of the results obtained in the previous one about the existence of controls yielding tracking. This will be done separately under the hypotheses of Corollaries 4.5 and 4.7, using two different algorithms. The first one is based on the procedure proposed in the proof of Theorem 4.4, while the second one exploits the construction proposed in [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF] and recalled in Theorem 3.3.

In both cases we will consider a reference trajectory γ ref : I → Q, which is assumed to be of class C ∞ .

A simple, albeit crucial, fact that will be used several times in the following sections is stated in the lemma below.

Lemma 5.1. If f : R × I → R, (τ, s) → f (τ, s), is smooth on R × I and T -periodic with respect to τ , then t 0 f (s/ǫ, s)ds = t 0 f (s)ds + O(ǫ f ∞ ) + O(ǫ ∂ 2 f ∞ )
for ǫ close to zero, where f (s) = (1/T ) T 0 f (τ, s)dτ and ∂ 2 denotes the partial derivative with respect to the second variable.

Case H

As noticed in Remark 4.6, the hypotheses of Corollary 4.5 guarantee that every step of the induction argument proposed in the proof of Theorem 4.4 falls in the framework of case 1 (see page 6). Hence, starting from a parameterization

∇ γref (t) γref (t) -Y (t, γref (t)) = N l a=1 λ a (t)Z l a (γ ref (t))
of γ ref : I → Q, with Z l a ∈ H l and λ a smooth and non-negative on I for every a = 1, . . . , N l , we can construct algorithmically a l-parameter family of admissible trajectories ξ ǫ 1 ,...,ǫ l of Σ with ǫ 1 , . . . , ǫ l > 0 such that ξ l = γ ref , the uniform limit

ξ ǫ 1 ,...,ǫ l-i i = lim ǫ l →0 lim ǫ l-1 →0 • • • lim ǫ l-i+1 →0 ξ ǫ 1 ,...,ǫ l (20)
exists for every i = 1, . . . , l and every ǫ 1 , . . . , ǫ l-i > 0 and satisfies

∇ ξǫ 1 ,...,ǫ l-i i (t) ξǫ 1 ,...,ǫ l-i i (t) -Y (t, ξǫ 1 ,...,ǫ l-i i (t)) ∈ H i (ξ ǫ 1 ,...,ǫ l-i i (t)). ( 21
)
We also write ξ ǫ 1 ,...,ǫ l 0 for ξ ǫ 1 ,...,ǫ l . It is important to notice that the order of the limits in (20) cannot in general be reversed.

Let us recall to which extent the construction is algorithmic. Fix any injective map j : N × N → N.

By backward recursion on

i, if ξ ǫ 1 ,...,ǫ l-i i satisfies ∇ ξǫ 1 ,...,ǫ l-i i (t) ξǫ 1 ,...,ǫ l-i i (t) -Y (t, ξǫ 1 ,...,ǫ l-i i (t)) = N i a=1 λ ǫ 1 ,...,ǫ l-i a (t)Z i a (ξ ǫ 1 ,...,ǫ l-i i (t)) (22) with λ ǫ 1 ,...,ǫ l-i a ∈ C(I, [0, +∞)) and Z i a ∈ H i , then ξ ǫ 1 ,...,ǫ l-i+1 i-1
is defined as the solution to

∇ ξǫ 1 ,...,ǫ l-i+1 i-1 (t) ξǫ 1 ,...,ǫ l-i+1 i-1 (t) -Y (t, ξǫ 1 ,...,ǫ l-i+1 i-1 (t)) = N i a=1 λ ǫ 1 ,...,ǫ l-i a (t)F i-1 a (ξ ǫ 1 ,...,ǫ l-i+1 i-1 (t))+ 1 ǫ l-i+1 N i a b=1 ψ j(a,b) t ǫ l-i+1 λ ǫ 1 ,...,ǫ l-i a (t)α i-1 a,b G i-1 a,b (ξ ǫ 1 ,...,ǫ l-i+1 i-1 (t)) (23)
with ξǫ 1 ,...,ǫ l-i+1

i-1

(0) = γref (0), where

Z i a = F i-1 a - N i a b=1 α i-1 a,b G i-1 a,b : G i-1 a,b and F i-1 a ∈ H i-1 , G i-1 a,b ∈ L(H i-1 ), α i-1 a,b ≥ 0. Recall that (ψ j ) j∈N is a Λ T -orthonormal and zero- mean sequence, for some T > 0. Each λ ǫ 1 ,...,ǫ l-i+1 a (•) is either equal to some λ ǫ 1 ,...,ǫ l-i ã (•) or is of the form λ ǫ 1 ,...,ǫ l-i ã (•)α i-1 ã,b ǫ l-i+1 max υψ j(ã,b) • ǫ l-i+1
, 0 with υ equal to 1 or -1.

We choose (ψ j ) j∈N as follows: we require ψ 1 to be positive in (0, T /2) and to annihilate, together with all its derivatives, at 0 and T /2. We also require it to satisfy

ψ 1 T 2 -t = ψ 1 (t), t ∈ 0, T 2 ,
and we extend it by

ψ 1 (t) = -ψ 1 t - T 2 , t ∈ T 2 , T ,
and by T -periodicity over R. Finally we normalize ψ 1 in such a way that Λ T (ψ 1 , ψ 1 ) = 1. Then we define ψ j by

ψ j (t) = 2 j ψ 1 (2 j t).
Such choice of (ψ j ) j∈N is motivated by the property that, for every choice of j ∈ N, l ∈ N 0 and υ ∈ {-1, 1}, the function t → 2 l max {υψ j (t) , 0} is smooth. In particular, by backward recursion, each λ ǫ 1 ,...,ǫ i a is smooth and is the product of functions of the type 2 l 1 ǫ h max υψ j

• ǫ h , 0 and of 2 m √ λ b for some m ∈ N 0 and some b ∈ {1, . . . , N l }.

An important consequence of this factorization, which will be exploited in the proof of Theorem 5.2, is that the derivatives of λ ǫ 1 ,...,ǫ i a with respect to time can be bounded by a finite constant depending explicitly on ǫ 1 , . . . , ǫ l-1 .

The smoothness of λ ǫ 1 ,...,ǫ i a , moreover, allows us to consider λ ǫ 1 ,...,ǫ i a Z l-i a as a smooth vector field on the extended manifold R × Q and, similarly, λ ǫ 1 ,...,ǫ l-i a (Z i a ) V as a smooth vector field on the extended manifold R × T Q.

Summing up, the trajectories of the l-parameter family ξ ǫ 1 ,...,ǫ l 0 = ξ ǫ 1 ,...,ǫ l are driven by a l-parameter family of control laws u ǫ 1 ,...,ǫ l ∈ C ∞ (I, R k ) depending smoothly on ǫ 1 , . . . , ǫ l . The construction of Theorem 4.4 can be summarized as follows:

given ǫ > 0, if 0 < ǫ l ≪ ǫ l-1 ≪ • • • ≪ ǫ 1 ≪ 1 ( 24
)
then d(ξ ǫ 1 ,...,ǫ l (t), γ ref (t)) < ǫ for every t ∈ I. Our aim is here to quantify the relations in (24). More precisely, we introduce l -1 functions η 2 , . . . , η l : (0, +∞) → (0, +∞) and we look for asymptotic conditions on their convergence to zero at zero such that

lim ǫ→0 d(ξ ǫ,η 2 (ǫ),η 3 •η 2 (ǫ),...,η l •••••η 2 (ǫ) (t), γ ref (t)) = 0
(25) uniformly with respect to t ∈ I. Let ηi = η i • • • • • η 2 for i = 2, . . . , l and define η1 as the identity on (0, +∞). We say that ǫ → u η1 (ǫ),...,η l (ǫ) is a one-parameter tracking control law for γ ref if (25) holds true.

Theorem 5.2. Let Σ = (Q, ∇, Y, Y , R k ) be a FACCS. Let H i , i ∈ N 0 ,
be defined as in [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF]. Fix a reference trajectory γ ref ∈ C ∞ (I, Q) and assume that there exists l ∈ N such that

∇ γref (t) γref (t) -Y (t, γref (t)) ∈ H l (γ ref (t))
for every t ∈ I. Construct ξ ǫ 1 ,...,ǫ l , u ǫ 1 ,...,ǫ l and ηi as above. If η i : (0, +∞) → (0, +∞) satisfies lim sup ǫ→0 η i (ǫ)/ǫ 3 < ∞ for every i = 2, . . . , l, then ǫ → u η1 (ǫ),...,η l (ǫ) is a one-parameter tracking control law for γ ref .

Proof. The first step of the proof consists in estimating the order with respect to ǫ of the L ∞ -norm of the time-dependent parameters appearing in the parameterization (22). We write ǫ i for ηi (ǫ). Denoting by C any constant not depending on the ǫ j , it is easy to check by backward induction on i = 0, . . . , l that

λ ǫ 1 ,...,ǫ i a ∞ ≤ C 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 • • • √ ǫ i-1 ǫ i .
Exploiting the factorization of λ ǫ 1 ,...,ǫ i a described above we get, in addition,

d m dt m λ ǫ 1 ,...,ǫ i a ∞ ≤ C 2 i √ ǫ 1 2 i-1 √ ǫ 2 • • • 4 √ ǫ i-1 ǫ i m+ 1 2 for m ∈ N 0 .
Consider the extended system on R × T Q associated with (3) where the time is the new variable. In the following computations we write, using the notation introduced in (23),

X e = (1, Z + Y V ), Φ i a = (0, (F i a ) V ), Γ i a,b = (0, (G i a,b ) V ),
all seen as vector fields on R × T Q. We also define γ e ref (0) = (0, γref (0)) and, given a smooth function λ : I → R, we write λ V to denote a smooth function on R × T Q such that λ V (t, v) = λ(t) for every t ∈ I and every v ∈ T Q. In particular, we define

θ i a,b = λ ǫ 1 ,...,ǫ i a α l-i-1 a,b V , θi a,b = d dt λ ǫ 1 ,...,ǫ i a α l-i-1 a,b V .
Then, applying ( 5) and ( 6), (t, ξǫ 1 ,...,ǫ l 0

(t)) = -→ exp t 0 (X e + N 0 a=1 λ ǫ 1 ,...,ǫ l a (s)(0, (Z 0 a ) V ))ds γ e ref (0) = -→ exp t 0 X e + N 1 a=1 λ ǫ 1 ,...,ǫ l-1 a (s)Φ 0 a + 1 ǫ l N 1 a=1 N 1 a b=1 ψ j(a,b) ( s ǫ l ) λ ǫ 1 ,...,ǫ l-1 a (s)α 0 a,b Γ 0 a,b ds γ e ref (0) = -→ exp t 0 1 ǫ l N 1 a=1 N 1 a b=1 ψ j(a,b) ( s ǫ l )θ l-1 a,b Γ 0 a,b ds • -→ exp t 0 X e + N 1 a=1 λ ǫ 1 ,...,ǫ l-1 a (s)Φ 0 a + N 1 a=1 N 1 a b=1 s 0 1 ǫ l ψ j(a,b) ( s 0 ǫ l )ds 0 (θ l-1 a,b [Γ 0 a,b , X e ] -θl-1 a,b Γ 0 a,b ) -N 1 a,a ′ =1 N 1 a b=1 N 1 a ′ b ′ =1 s 0 1 ǫ l ψ j(a,b) ( s 1 ǫ l ) s 1 0 1 ǫ l ψ j(a ′ ,b ′ ) ( s 0 ǫ l )ds 0 ds 1 θ l-1 a,b θ l-1 a ′ ,b ′ (0, G 0 a,b : G 0 a ′ ,b ′ V ) dsγ e ref (0)
. From now on, let us denote by V any vertical flow, i.e., any flow on R × T Q preserving the base point on Q.

Notice that, by construction,

N 1 a=1 λ ǫ 1 ,...,ǫ l-1 a (s)Φ 0 a + -N 1 a,a ′ =1 N 1 a b=1 N 1 a ′ b ′ =1 Λ T (ψ j(a,b) , ψ j(a ′ ,b ′ ) ) λ ǫ 1 ,...,ǫ l-1 a (s)λ ǫ 1 ,...,ǫ l-1 a (s)α 0 a,b α 0 a ′ ,b ′ (0, G 0 a,b : G 0 a ′ ,b ′ V ) = N 1 a=1 λ ǫ 1 ,...,ǫ l a (s)(0, (Z 1 a ) V ).
Then, (t, ξǫ 1 ,...,ǫ l 0

(t)) = V • -→ exp t 0 X e + N 1 a=1 λ ǫ 1 ,...,ǫ l-1 a (s)(0, (Z 1 a ) V ) + N 1 a=1 N 1 a b=1 s 0 1 ǫ l ψ j(a,b) ( s 0 ǫ l )ds 0 (θ l-1 a,b [Γ 0 a,b , X e ] -θl-1 a,b Γ 0 a,b ) + N 1 a,a ′ =1 N 1 a b=1 N 1 a ′ b ′ =1 (Λ T (ψ j(a,b) , ψ j(a ′ ,b ′ ) ) - s 0 1 ǫ l ψ j(a,b) ( s 1 ǫ l ) s 1 0 1 ǫ l ψ j(a ′ ,b ′ ) ( s 0 ǫ l )ds 0 ds 1 ) (0, G 0 a,b : G 0 a ′ ,b ′ V )θ l-1 a,b θ l-1 a ′ ,b ′ dsγ e ref (0).
Applying iteratively the same computation as above, one ends up with

(t, ξǫ 1 ,...,ǫ l 0 (t)) = V • -→ exp t 0 X e + λ a (s)(0, (Z l a ) V ) + T (s) ds γ e ref (0)
, where T (s) is a sum of terms of the form ζ(s)V where V is a vector field on R × T Q independent of the ǫ j , while ζ : I → R is smooth, depends on the ǫ j and is of one of the following four types:

ζ 1 (s) = s 0 1 ǫ i ψ j(a,b) ( s 0 ǫ i )ds 0 λ ǫ 1 ,...,ǫ i-1 a (s)α l-i a,b , ζ 2 (s) = - s 0 1 ǫ i ψ j(a,b) ( s 0 ǫ i )ds 0 d ds λ ǫ 1 ,...,ǫ i-1 a (s)α l-i a,b , ζ 3 (s) = 1 1+δ bc 2(Λ T (ψ j(a,b) , ψ j(a ′ ,b ′ ) )) - s 0 1 ǫ i ψ j(a,b) ( s 0 ǫ i )ds 0 s 0 1 ǫ i ψ j(a ′ ,b ′ ) ( s 0 ǫ i )ds 0 λ ǫ 1 ,...,ǫ i-1 a (s)λ ǫ 1 ,...,ǫ i-1 a ′ (s)α l-i a,b α l-i a ′ ,b ′ for i = 1, . . . , l, or ζ 4 (s) = - s 0 1 ǫ j ψ j(a,b) ( s 0 ǫ j )ds 0 s 0 1 ǫ i ψ j(a ′ ,b ′ ) ( s 0 ǫ i )ds 0 λ ǫ 1 ,...,ǫ j-1 a (s)λ ǫ 1 ,...,ǫ i-1 a ′ (s)α l-j a,b α l-i a ′ ,b ′ for i > j, i, j ∈ {1, . . . , l}.
According to Lemma 2.4, the theorem is proved if we show that, for every ζ of one of the four types introduced above, t 0 ζ(s)ds converges to zero uniformly with respect to t ∈ I as ǫ goes to zero. This can be done by applying Lemma 5.1. Taking for instance

f (τ, s) = τ 0 ψ j(a,b) (s 0 )ds 0 λ ǫ 1 ,...,ǫ i-1 a (s)α l-i a,b ,
and ǫ = ǫ i leads to

t 0 ζ 1 (s)ds ≤ C ǫ i 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 • • • 4 √ ǫ i-2 ǫ 3 2 i-1 . Similarly, t 0 ζ 2 (s)ds ≤ C ǫ i 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 • • • 4 √ ǫ i-2 ǫ 5 2 i-1 . Taking f (τ, s) = τ 0 ψ j(a,b) (s 0 )ds 0 τ 0 ψ j(a ′ ,b ′ ) (s 0 )ds 0 λ ǫ 1 ,...,ǫ i-1 a (s)λ ǫ 1 ,...,ǫ i-1 a ′ (s)α l-i a,b α l-i a ′ ,b ′ we obtain t 0 ζ 3 (s)ds ≤ C ǫ i 2 i-2 √ ǫ 1 2 i-3 √ ǫ 2 • • • √ ǫ i-2 ǫ 2 i-1
.

Finally, with

f (τ, s) = s/ǫ j 0 ψ j(a,b) (s 0 )ds 0 τ 0 ψ j(a ′ ,b ′ ) (s 0 )ds 0 λ ǫ 1 ,...,ǫ j-1 a (s)λ ǫ 1 ,...,ǫ i-1 a ′ (s)α l-j a,b α l-i a ′ ,b ′ we get t 0 ζ 4 (s)ds ≤ C ǫ i 2 j-1 √ ǫ 1 2 j-2 √ ǫ 2 ••• 4 √ ǫ j-1 √ ǫ j-1 1 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 ••• 4 √ ǫ i-2 ǫ 3 2 i-1
.

Notice that the upper bound for t 0 ζ 2 (s)ds is the one growing faster as ǫ goes to zero. Indeed, in order to compare it with the one for

t 0 ζ 3 (s)ds it suffices to notice that 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 ••• 4 √ ǫ i-2 ǫ 5 2 i-1 2 i-2 √ ǫ 1 2 i-3 √ ǫ 2 ••• √ ǫ i-2 ǫ 2 i-1 = ǫ i-1 ǫ i-2 4 ǫ i-2 ǫ i-3 • • • 2 i-2 ǫ 2 ǫ 1 2 i-1 √ ǫ 1
converges to zero as ǫ goes to zero.

Hence, for each ζ as above, there exists i = 1, . . . , l such that

t 0 ζ(s)ds ≤ C ǫ i 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 • • • 4 √ ǫ i-2 ǫ 5 2
i-1

and we are left to notice that

ǫ i 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 ••• 4 √ ǫ i-2 ǫ 5 2 i-1 = ǫ i ǫ 3 i-1 ǫ i-1 ǫ i-2 4 ǫ i-2 ǫ i-3 • • • 2 i-2 ǫ 2 ǫ 1 2 i-1 √ ǫ 1
tends to zero as ǫ goes to zero.

Remark 5.3. The hypothesis lim sup ǫ→0 η i (ǫ)/ǫ 3 < ∞ has been chosen, in the statement of Theorem 5.2, because of its simplicity. However, we can weaken it by requiring that lim sup ǫ→0 η i (ǫ)/ǫ 5 2 +a < ∞ with a = √ 5

2 -1 ≃ 0.12. Indeed, with this choice of a,

ǫ a j 2 j √ ǫ 1 2 j-1 √ ǫ 2 ••• 4 √ ǫ j-1 = ǫ j ǫ 5 2 +a j-1 a ǫ a j-1 2 j-1 √ ǫ 1 2 j-2 √ ǫ 2 ••• 4 √ ǫ j-2 1 2
, so that

ǫ i 2 i-1 √ ǫ 1 2 i-2 √ ǫ 2 ••• 4 √ ǫ i-2 ǫ 5 2 i-1 = ǫ i ǫ 5 2 +a i-1 ǫ i-1 ǫ 5 2 +a i-2 a ǫ i-2 ǫ 5 2 +a i-3 a 2 • • • ǫ 2 ǫ 5 2 +a 1 a 2 i-3 ǫ a 2 i-2 1 .
Hence, each t 0 ζ(s)ds goes to zero uniformly with respect to t ∈ I as ǫ tends to zero.

Case Z

Analogously to Section 5.1, the aim is to provide an algorithmic implementation of the results obtained in Section 4 about the existence of controls yielding tracking but this time under the hypotheses of Corollary 4.7. Instead of adopting the algorithmic scheme on which the proof of Theorem 4.4 is based, as done in Section 5.1, we rely here on the iteration of the scheme proposed in [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF] and recalled in Theorem 3.3 (see also Remark 4.3). The advantage is that, under the more restrictive hypotheses of Corollary 4.7, we can base the iteration scheme on the Λ T -orthonormal and zero-mean sequence defined in (8) using trigonometric functions, which is more convenient for numerical implementation than the sequence (ψ j ) j∈N constructed in the previous section.

We start from a parameterization

∇ γref (t) γref (t) -Y (t, γref (t)) = N l a=1 λ a (t)Z l a (γ ref (t)) (26) of γ ref : I → Q, with Z l
a ∈ Z l and λ a smooth on I for every a = 1, . . . , N l . (For the definition of Z l , see (17).) As in Section 5.1, we can construct algorithmically a l-parameter family of admissible trajectories ξ ǫ 1 ,...,ǫ l of Σ with ǫ 1 , . . . , ǫ l > 0 such that ξ l = γ ref , the uniform limit in (20) exists for every i = 1, . . . , l and every ǫ 1 , . . . , ǫ l-i > 0, and, instead of (21), it satisfies

∇ ξǫ 1 ,...,ǫ l-i i (t) ξǫ 1 ,...,ǫ l-i i (t) -Y (t, ξǫ 1 ,...,ǫ l-i i (t)) ∈ span R (Z i (ξ ǫ 1 ,...,ǫ l-i i (t))).
The algorithm works by applying, at each step, the construction of Theorem 3.3 with Z i as Y (see Remark 4.3). Then by backward recursion on i, the solutions to

∇ ξǫ 1 ,...,ǫ l-i i (t) ξǫ 1 ,...,ǫ l-i i (t) -Y (t, ξǫ 1 ,...,ǫ l-i i (t)) = N i a=1 λ ǫ 1 ,...,ǫ l-i a (t)Z i-1 a (ξ ǫ 1 ,...,ǫ l-i i (t)) + N i b,c=1,b<c λ ǫ 1 ,...,ǫ l-i b,c (t) Z i-1 b : Z i-1 c (ξ ǫ 1 ,...,ǫ l-i i (t)) for Z i-1 a , Z i-1 b , Z i-1 c ∈ Z i-1 are trackable by solu- tions to ∇ ξǫ 1 ,...,ǫ l-i+1 i-1 (t) ξǫ 1 ,...,ǫ l-i+1 i-1 (t) -Y (t, ξǫ 1 ,...,ǫ l-i+1 i (t)) = N i-1 a=1 u ǫ 1 ,...,ǫ l-i
slow,a (t) The construction of u slow and u osc is such that

+ 1 ǫ l-i+1 u ǫ 1 ,...,ǫ l-i osc,a (t/ǫ l-i+1 , t) Z i-1 a (ξ ǫ 1 ,...,ǫ l-i+1 i-1 (t)) (27
N i-1 a=1 u ǫ 1 ,...,ǫ l-i slow,a (t)Z i-1 a - N i-1 b,c=1 Λ T (t, u ǫ 1 ,...,ǫ l-i osc,b , u ǫ 1 ,...,ǫ l-i osc,c ) Z i-1 b : Z i-1 c = N i-1 a=1 λ ǫ 1 ,...,ǫ l-i a (t)Z i-1 a + N i-1 b,c=1,b<c λ ǫ 1 ,...,ǫ l-i b,c (t) Z i-1 b : Z i-1 c
(see [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF]). As in Section 5.1, our aim is here to quantify the relations in (24). We will use the same notations for η i and ηi as in Section 5.1. The notion of being a one-parameter tracking control law for γ ref is again defined through (25).

Theorem 5.4. Let Σ = (Q, ∇, Y, Y , R k ) be a FACCS. Let Z i , i ∈ N 0 , be defined as in (17) satisfying the same hypotheses as in Corollary 4.7. Fix a reference trajectory γ ref ∈ C ∞ (I, Q) and assume that there exists l ∈ N such that

∇ γref (t) γref (t) -Y (t, γref (t)) ∈ Z l (γ ref (t))
for every t ∈ I. Construct ξ ǫ 1 ,...,ǫ l , u ǫ 1 ,...,ǫ l and ηi as above. If η i : (0, +∞) → (0, +∞) satisfies lim sup ǫ→0 η i (ǫ)/ǫ 4 < ∞ for every i = 2, . . . , l, then ǫ → u η1 (ǫ),...,η l (ǫ) is a one-parameter tracking control law for γ ref .

Proof. The first step of the proof consists in estimating the order with respect to ǫ of the L ∞ -norm of the time-dependent parameters appearing in the parameterization (27). Once more we write ǫ i for ηi (ǫ). By induction it can be proved that

∂ m 2 u ǫ 1 ,...,ǫ i osc,a ∞ ≤ C(ǫ 1 • • • ǫ i-1 ǫ m+1 i ) -1
for every m ∈ N 0 .

Consider, as in the proof of Theorem 5.2, the extended system on R × T Q associated with (3). In the following computations we write

V i a = (0, (Z i a ) V ), W i b,c = (0, Z i b : Z i c V ).
We also write u ǫ 1 ,...,ǫ i osc,a (σ) to denote (u ǫ 1 ,...,ǫ i osc,a (σ, •)) V , i.e., u ǫ 1 ,...,ǫ i osc,a (σ) is the smooth function

on R × T Q such that u ǫ 1 ,...,ǫ i osc,a (σ)(t, v) = u ǫ 1 ,...,ǫ i osc,a (σ, t).
Then, applying ( 5) and ( 6), (t, ξǫ 1 ,...,ǫ l 0

(t)) = -→ exp t 0 X e + N 0 a=1 u ǫ 1 ,...,ǫ l-1 slow,a (s)V 0 a + 1 ǫ l N 0 a=1 u ǫ 1 ,...,ǫ l-1 osc,a (s/ǫ l , s)V 0 a ds γ e ref (0) = -→ exp t 0 1 ǫ l N 0 a=1 u ǫ 1 ,...,ǫ l-1 osc,a (s/ǫ l )V 0 a ds • -→ exp t 0 X e + N 0 a=1 u ǫ 1 ,...,ǫ l-1 slow,a (s)V 0 a + N 0 a=1 s 0 1 ǫ l u ǫ 1 ,...,ǫ l-1 osc,a (s 0 /ǫ l )ds 0 [V 0 a , X e ] -N 0 a=1 s 0 1 ǫ l ∂ 2 u ǫ 1 ,...,ǫ l-1 osc,a (s 0 /ǫ l )ds 0 V 0 a -N 0 b,c=1 s 0 1 ǫ l u ǫ 1 ,...,ǫ l-1 osc,b (s 1 /ǫ l ) s 1 0 1 ǫ l u ǫ 1 ,...,ǫ l-1 osc,c (s 0 /ǫ l )ds 0 ds 1 W 0 b,c ds γ e ref (0)
Denoting by V any vertical flow, we have

(t, ξǫ 1 ,...,ǫ l 0 (t)) = V • -→ exp t 0 X e + N 0 a=1 λ ǫ 1 ,...,ǫ l-1 a (s)V 0 a + N 0 b,c=1, b<c λ ǫ 1 ,...,ǫ l-1 b,c (s)W 0 b,c + s 0 1 ǫ l N 0 a=1 u ǫ 1 ,...,ǫ l-1 osc,a (s 0 /ǫ l , s)ds 0 [V 0 a , X e ] -N 0 a=1 s 0 1 ǫ l ∂ 2 u ǫ 1 ,...,ǫ l-1 osc,a (s 0 /ǫ l , s)ds 0 V 0 a + N 0 b,c=1 (Λ T (s, u ǫ 1 ,...,ǫ l-1 osc,b , u ǫ 1 ,...,ǫ l-1 osc,c
)-

s 0 1 ǫ l u ǫ 1 ,...,ǫ l-1 osc,b (s 1 /ǫ l , s) s 1 0 1 ǫ l u ǫ 1 ,...,ǫ l-1 osc,c
(s 0 /ǫ l , s)

ds 0 ds 1 )W 0 b,c ds γ e ref (0).
Noticing that

N 0 a=1 λ ǫ 1 ,...,ǫ l-1 a (s)V 0 a + N 0 b,c=1 (λ ǫ 1 ,...,ǫ l-1 b,c (s)W 0 b,c = N 1 â=1 u ǫ 1 ,...,ǫ l-2 slow,â (s) + 1 ǫ l-1 u ǫ 1 ,...,ǫ l-2 osc,â (s/ǫ l-1 , s) V
(t)) = V • -→ exp t 0 X e + λ a (s)V l a + T (s) ds γ e ref (0), where T (s) is a sum of terms of the form ζ(s)V where V ∈ X (R × T Q) is independent of the ǫ j , while ζ ∈ C ∞ (I, R
) is of one of the following four types:

ζ 1 = s 0 1 ǫ i u ǫ 1 ,...,ǫ i-1 osc,a (s 0 /ǫ i , s)ds 0 , ζ 2 = - s 0 1 ǫ i ∂ 2 u ǫ 1 ,...,ǫ i-1 osc,a (s 0 /ǫ i , s)ds 0 , ζ 3 = 1 1+δ bc 2Λ T (s, u ǫ 1 ,...,ǫ i-1 osc,b , u ǫ 1 ,...,ǫ i-1 osc,c ) - s 0 1 ǫ i u ǫ 1 ,...,ǫ i-1 osc,b (s 0 /ǫ i , s)ds 0 s 0 1 ǫ i u ǫ 1 ,...,ǫ i-1 osc,c
(s 0 /ǫ i , s)ds 0 , for i = 1, . . . , l, and

ζ 4 = - s 0 1 ǫ j u ǫ 1 ,...,ǫ j-1 osc,b (s 0 /ǫ j , s)ds 0 s 0 1 ǫ i u ǫ 1 ,...,ǫ i-1 osc,c
(s 0 /ǫ i , s)ds 0 , for i > j, i, j ∈ {1, . . . , l}.

We are left to prove that every t 0 ζ(s)ds converges to zero uniformly with respect to t as ǫ goes to zero (Lemma 2.4).

Applying Lemma 5.1 with

f (τ, s) = τ 0 u ǫ 1 ,...,ǫ i-1 osc,a (s 0 , s)ds 0 and ǫ = ǫ i leads to t 0 ζ 1 (s)ds ≤ Cǫ i (ǫ 1 • • • ǫ i-2 ǫ 2 i-1 ) -1 . Similarly, t 0 ζ 2 (s)ds ≤ C(ǫ 1 • • • ǫ i-2 ǫ 3 i-1 ) -1 ǫ i . Taking f (τ, s) = τ 0 u ǫ 1 ,...,ǫ i-1 osc,b (s 0 , s)ds 0 τ 0 u ǫ 1 ,...,ǫ i-1 osc,c
(s 0 , s)ds 0 , we obtain

t 0 ζ 3 (s)ds ≤ C(ǫ 1 • • • ǫ i-2 ) -2 ǫ -3 i-1 ǫ i .
Finally, with

f (τ, s) = s/ǫ j 0 u ǫ 1 ,...,ǫ j-1 osc,b (s 0 , s)ds 0 τ 0 u ǫ 1 ,...,ǫ i-1 osc,c
(s 0 , s)ds 0 , we have

t 0 ζ 4 (s)ds ≤ C(ǫ 1 • • • ǫ j-2 ǫ j-1 ) -1 (ǫ 1 • • • ǫ i-2 ǫ 2 i-1 ) -1 ǫ i . Hence, each ζ satisfies t 0 ζ(s)ds ≤ Cǫ i (ǫ 1 • • • ǫ i-2 ) 2 ǫ 3 i-1 = Cǫ i ǫ 4 i-1 ǫ i-1 (ǫ 1 • • • ǫ i-2 ) 2
and it is easy to prove by recurrence that ǫ i-1 (ǫ 1 • • • ǫ i-2 ) -2 tends to zero as ǫ goes to zero. Remark 5.5. As in Remark 5.3, the hypothesis lim sup ǫ→0 η i (ǫ)/ǫ 4 < ∞ in the statement of Theorem 5.4 can be weakened by requiring that lim sup ǫ→0 η i (ǫ)/ǫ 3+a < ∞ with a = √ 3 -1 ≃ 0.73. Indeed, with this choice of a,

ǫ a i ǫ 2 1 ǫ 2 2 • • • ǫ 2 i-1 = ǫ i ǫ 3+a i-1 a ǫ a i-1 ǫ 2 1 • • • ǫ 2 i-2
, so that

ǫ i ǫ 2 1 •••ǫ 2 i-2 ǫ 3 i-1 = ǫ i ǫ 3+a i-1 ǫ i-1 ǫ 3+a i-2 a ǫ i-2 ǫ 3+a i-3 a • • • ǫ 2 ǫ 3+a 1 a ǫ a 1 .
Hence, each t 0 ζ(s)ds goes to zero uniformly with respect to t ∈ I as ǫ tends to zero.

Under special assumptions the relations in (24) that have been quantified in Theorem 5.4 can be reduced up to lim ǫ→0 η i (ǫ)/ǫ 2 = 0 as stated in the following corollary and illustrated in the numerical simulation included in Section 5.2.1. The required assumptions are stated in terms of the number of steps in the algorithm and of the coefficients providing the parametrization of the reference trajectory.

Corollary 5.6. Let Σ = (Q, ∇, Y, Y , R k ) be a FACCS. Let Z i , i ∈ N 0 , be defined as in (17) and assume that span R Z 2 (q) = T q Q for all q ∈ Q and that for each i ∈ {0, 1} and each Z ∈ Z i , Z : Z (q) ∈ span R (Z i (q)). Fix a reference trajectory γ ref ∈ C ∞ (I, Q) such that the coefficients λ a associated with γ ref as in ( 26) are constant functions. Construct ξ ǫ 1 ,ǫ 2 , u ǫ 1 ,ǫ 2 and ηi as above. If η 2 : (0, +∞) → (0, +∞) satisfies lim ǫ→0 η 2 (ǫ)/ǫ 2 = 0, then ǫ → u ǫ,η 2 (ǫ) is a one-parameter tracking control law for γ ref .

Proof. The hypotheses of the corollary and the expressions of the controls appearing in Theorem 3.3 guarantee that the oscillatory controls are as follows:

u osc,a (τ, t) ∈ span R {ϕ j 0 (τ )},

u ǫ 1
osc,a (τ, t) ∈ span R {ϕ j 0 (τ ), 1 ǫ 1 ϕ j 1 (t/ǫ 1 )ϕ j 2 (τ )}, where j 0 , j 1 , j 2 vary in N.

Thus the integrals of all the terms T that appear in the proof of Theorem 5.4 only contain product of trigonometric functions, more specifically cosines and sines and converge to zero uniformly with respect to t as ǫ goes to zero if η 2 is such that lim ǫ→0 η 2 (ǫ)/ǫ 2 = 0.

Numerical simulation: submarine

In this section, we illustrate the method to obtain a one-parameter control law in a concrete situation that fulfills the assumptions in Corollary 5.6. The algorithm described in Section 5.2 has been implemented with Scilab.

For our example, we have the submarine presented in Section 4.1.2 whose dynamics are given by ( 18) and (19). We consider the same kind of inertia matrix as in Section 4.1.2, taking

J 1 = 1, J 3 = 3, M 1 = 1, M 2 = 2, M 3 = 3.
We recall that in the case under consideration Theorem 3.3 cannot be applied. However, our method provides a one-parameter control law that solves the tracking problem. The trajectory to be tracked is given by r(t) = (-t, -t, -t), A 33 (t) = 1, t ∈ [0, 1] with initial condition r(0) = (0, 0, 0), A(0) = Id, being Id the identity 3 × 3 matrix, Π(0) = (0, 0, 0), P (0) = (-1, -2, -3). Thus there are degrees of freedom in the attitude of the submarine, but the target position of the center of the submarine is fully determined. .

In this implementation we take η 2 (ǫ) = ǫ 2.5 because the considered reference trajectory satisfies the hypotheses of Corollary 5.6.

Conclusions

The previously known sufficient conditions for tracking were given in terms of finite sets of vector fields, as reviewed in Section 3. Here we have constructed a sequence of infinite family of vector fields that defines a sequence of convex cones suitable for characterizing trackability (Theorem 4.4 and Corollary 4.5). Different convex cones, ( 9) and ( 16), have been considered. Under additional assumptions, using the cones in ( 16) and a particular sequence of finite families of vector fields, it is possible to recover the sufficient conditions for tracking already known in the literature [START_REF] Bullo | Geometric Control of Mechanical Systems. Modeling, analysis and design for simple mechanical control[END_REF], see Corollary 4.7. However, our constructions not only recover the previously known results, but they also extend them, as shown in Section 4.1.

The sequence of families of vector fields in Corollaries 4.5 and 4.7 are also suitable for constructing a one-parameter tracking control law (Theorems 5.2 and 5.4). It remains as future work to generalize the construction of one-parameter tracking control laws when the sets [START_REF] Lamb | Hydrodynamics. Reprint of the 1932[END_REF], that include the closure, are considered.

Another future research line is the study of the complexity for the control-affine systems considered in this work. Apart from tracking non-admissible trajectories, one could impose more requirements on the solution to the tracking problem, as for instance, to save energy. The complexity provides a good tool to formulate this kind of problems and, so far, has been only studied for control-linear systems [START_REF] Gauthier | On the motion planning problem, complexity, entropy, and nonholonomic interpolation[END_REF][START_REF] Jean | Complexity of nonholonomic motion planning[END_REF].

  The control vector fields areY 1 = ∂/∂Π 1 , Y 2 = ∂/∂Π 2 and Y 3 = ∂/∂P 3 .They correspond to a linear acceleration along one of the three axes of the submarine and to two angular accelerations around the other two axes.

) where u ǫ 1

 1 ,...,ǫ l-i slow,a and u ǫ 1 ,...,ǫ l-i osc,a are constructed as in Theorem 3.3. Notice thatu ǫ 1 ,...,ǫ l-i slow,a (t) + 1 ǫ l-i+1 u ǫ 1 ,...,ǫ l-i osc,a (t/ǫ l-i+1 , t)plays the role of a λǫ 1 ,...,ǫ l-i+1 â (t) or a λ ǫ 1 ,...,ǫ l-i+1 α, β : [0, T ] × I → R, define Λ T as Λ T (t, α, β) = s,t)ds τ 0 β(s, t)ds dτ.

Figure 1 :

 1 Figure 1: The control law u ǫ,η 2 (ǫ) 1.

Figure 2 :

 2 Figure 2: The control law u ǫ,η 2 (ǫ) 2.

Figure 3 :

 3 Figure 3: The control law u ǫ,η 2 (ǫ) 3

First, we 1 , u ǫ 1 ,ǫ 2 2 and u ǫ 1 ,ǫ 2 3

 122 compute the one-parameter control laws u ǫ 1 ,ǫ 2 as described in the proof of Theorem 5.4. Then we fix ǫ = 1/39 ≈ 0.0256, so that η 2 (ǫ) ≈ 0.0001. The corresponding control laws are represented in Figures1, 2 and 3. By construction, the controls are highly oscillatory.

Figure 4 :

 4 Figure 4: Evolution of the position of the center of the submarine with respect to time. The target trajectory is the non-oscillating curve.

Figure 5 :

 5 Figure 5: Evolution of A 33 entry of the attitude matrix with respect to time. The target trajectory is the non-oscillating curve.Then we integrate the dynamics of the system using the numerical integrator stiff included in Scilab. As a result, Figures4 and 5show that the target trajectory, corresponding with the non-oscillating line, is tracked by the oscillating curve. The error of the approximation, computed by the supremum distance, is d((r ref (t), A 33,ref (t)), (r(t), A 33 (t))) ≈

  give the whole tangent space T γ ref (t) Q. The smoothness of these vector fields guarantees that this is still true in an open neighborhood U t of γ ref (t). In this way we construct an open cover of Imγ ref . As Imγ ref is compact, there exists a finite open subcover of {U t } t∈I given by U t 1 , . . . , U tr . For each t i there exists a different l

Acknowledgements

The first author acknowledges the financial support of Comissionat per a Universitats i Recerca del Departament d'Innovació, Universitats i Empresa of Generalitat de Catalunya in the final preparation of this paper.


Guillaume Charles

Serge Corbel

Marie-Christiane Carré

Thibault Roques-Carmes

Orfan Zahraa

Design of a micro-channel reactor for decomposition of organic pollutants in waste water treatment

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Methods, such as photochemical treatments, were envisaged for detoxification of the water and air. Among them, photocatalysis is efficient with the advantage of oxidizing most contaminants of water (fertilizers, pesticides, dyes, ...) until their complete destruction (mineralization) [START_REF] Avila | Gas-phase photo-assisted mineralization of volatile organic compounds by monolithic titania catalysts[END_REF] with no selective transfer of the pollutants and products on the surface of adsorbent. It uses the inexpensive and available UV-A light as sources of irradiation as well as the water and oxygen as oxidizing agents [START_REF] Hermann | Heterogeneous photocatalysis: fundamentals and applications to the removal of various types of aqueous pollutants[END_REF]. The simplest processes are the fluidized bed TiO 2 (this poses problems of cost particularly during the stage of separation after purification) or the tubular reactor with catalyst deposited on the walls of reactor, suffering sometimes of limitation of transfer of matter or non homogeneous irradiation. Among the possible solutions for intensification of photocatalytic reactions there is the use of micro-reactors [START_REF] Van Gerven | A review of intensification of photocatalytic processes[END_REF]. The design and manufacture of micro-reactors by the process of stereolithography, authorizes the fabrication of complex forms with a great flexibility. The first step of the research has concerned the design and manufacture of a catalytic support whose characteristics are:

-a capacity to overcome the mass transfer, -the largest ratio of surface on volume in order to obtain a great contact of the pollutant with the catalyst, -the most important homogeneous irradiation on the whole surface of the structure.

The second step has focused on the study of the influence of geometric characteristics of the support (depth, width, length) on the photocatalytic activity.

EXPERIMENTAL

The micro-channel reactor with the geometry shown on the Figure 2 has been made in epoxy resin (SI 30) by a self-made stereolithography apparatus using a UV Nd-YAG laser [START_REF] Corbel | Micro-structured reactors designed by stereolithography and characterized by fluorescent probes[END_REF].

Design of a micro-channel reactor for decomposition of organic pollutants in waste water treatment ABSTRACT: Photocatalytic micro-channel reactor was built by using stereolithography process. A reactor with a micro-channel as a support of TiO 2 photocatalyst was designed in order to reduce dimensions while improving the efficiency. Photocatalytic activity of the micro-reactor at various flow rates was evaluated by the inlet and outlet concentrations of salicylic acid as a model of pollutant. Influence of the initial pollutant concentration, the irradiation intensity on the rate of degradation of the pollutant was studied. A model with the kinetic parameters for the given support geometry was applied to reactor design and scale-up. The surface of the channel was coated with a TiO 2 thin film by filling it with an acidic aqueous suspension of TiO 2 (4g/L; adjusted to pH 3 by addition of nitric acid). After drying at room temperature, the surface load was 20 mg/cm 2 , this value being above the minimal load generally required for complete light absorption

The experimental setup used for the photocatalytic tests is shown on Figures 3 and4. A flow of water containing the pollutant is injected with a syringe at a low speed into the microstructured reactor. An UV fluorescent lamp (Philips, light power 3W, 56 cm long, λ max ≈ 365 nm) located at 1 cm above the reactor. The incident light power was recorded by a UV radiometer whose spectral response is centered at 365 nm with a half band-width of 12nm. The value of 2.8 mW/cm 2 was fixed at the surface of the catalyst. A flexible tube at the outlet of the reactor allows the collect of the irradiated solution samples. In order to validate the micro-structured reactor in a photocatalytic reaction, we have chosen salicylic acid (SA) as model molecule previously studied in the laboratory [START_REF] Ould-Mame | Photocatalytic degradation of salicylic acid on fixed TiO 2 -kinetic studies[END_REF]. Its concentration after photoreaction was measured by HPLC with conditions already described. Briefly, the HPLC system is equipped with a variable Shimadzu SPD-20A UV-Vis detector, a Rheodyne Model 7725 injector, a loop size of 20 μL and a Shimadzu LC-20AT pump. Reverse-phase chromatographic analysis using a Lichrosorb RP-18 column (250 x 4.6 mm i.d., particle size 5 μm, Supelco-Inc) was carried out at 25°C, under isocratic conditions (mobile phase consisted of a mixture of methanol and water containing 0.1% volumic of phosphoric acid; 70:30, v/v; 1mL/min). The column effluent was monitored at λ = 295 nm. Data acquisition was performed using a Varian System (Star Data Acquisition Workstation, System Control Version 5-52).

RESULTS AND DISCUSSION

Photocatalytic degradation of salicylic acid

The efficiency of the monolithic photocatalyst was carried out by estimating the degradation rate R in salicylic acid per volume of reactor (mg/min/cm 3 ) at different flow rates and inlet concentration of salicylic acid.

dV dX C Q R 0 0 = ( 1
)
where Q 0 = flow rate in mL/min ; dX = experimental conversion of salicylic acid during its contact with the catalyst into the apparent elementary reactor volume dV (cm 3 ).

The results of photocatalytic degradation carried out with the micro-reactor described previously, will be presented as. the variation of the outlet concentration of salicylic acid C out , obtained at variable flow rate Q 0, for a constant inlet concentration C 0 .

The value of inlet concentration is 10 mg/L in salicylic acid, and among the geometries tested, we have characterized the performance of the reactor with a 0.5 mm depth channel. In photocatalytic studies, the kinetics of photocatalytic degradation is generally represented by the Langmuir-Hinshelwood (LH) model. The reaction rate R is defined by the following relationship [START_REF] Zahraa | Kinetics of atrazine degradation by photocatalytic process in aqueous solution[END_REF].

Modelling photocatalytic reaction with Langmuir-Hinshelwood model (LH)

KC KC k R + = 1 (2)
Where K is the adsorption constant (L.mg -1 ); C, the concentration of salicylic acid in the liquid-phase (mg.L -1 ) and k, an apparent kinetic constant (mg.min -1 .L -1 ). It is also noticeable that the rate is expressed per unit of reactor apparent volume. The importance of substrate pre-adsorption on a given photocatalyst can be evaluated by the use of a LH kinetic model, considering that the adsorption of reaction intermediates and products is not significant. In our case, the disappearance of the pollutant during the photocatalytic process has been followed. Then, if photoreactor considered close to a Plug Flow Reactor (PFR), and a constant flow rate (permanent regime) therefore the mass balance through it is given by the following equation:

0 0 C Q R dV dX = (3)
with X conversion within the section volume dV of the reactor, where dX = -dC/ C 0 , then :

R = dV dC Q 0 - (4)
from of equations ( 2) and (4) it can be found:

dV dC Q 0 - = KC KC k + 1 ( 5
)
Where dV is the elementary volume of the reactor, Q 0 , the volumetric flow rate. The integration of Eq. ( 5) is possible only when there is no limitation by mass transfer, no difference of concentration between bulk flow and the catalyst surface [START_REF] Furman | Influence of the geometry of a monolithic support on the efficiency of photocatalyst for air cleaning[END_REF][START_REF] Dingwang | External and internal mass transfer effect on photocatalytic degradation[END_REF]. By integrating equation (5) between the reactor inlet and outlet, it follows:

( )

0 0 0 1 ln 1 Q V C C k C C kK out out - = - +         (6)
Eq. ( 6) can be written introducing the pollutant conversion X = 1-C out /C 0 :

( ) X C Q V k X C X kK 0 0 0 1 1 ln 1 - = - - (7)
A plot of -V/(Q 0 C 0 X) versus ln(1 -X)/(C 0 X) should be linear and will allow the determination of two constants k and K. From this optimization, the values of k and K obtained are respectively 2.66 10 -5 mmol.min -1 .cm -2 and 25 mmol -1 .mL.

Effect of the contact time

The effect of contact time on salicylic acid (SA) outlet concentrations was investigated in the range of 0.06 min to 6.33 min. Figure 6 shows that pollutant outlet concentration decreases according to the contact time τ = V/Q 0 . It has to be noticed that, the effect of the contact time of reactant was studied by varying the volumetric flow rate with a constant concentration of SA.

Figure 7 shows that SA conversion increased first rapidly as the contact time increased in the first minutes then slowly until 6.0 minutes. This behavior has been obtained at approximately high concentration where the rate is independent of concentration (zero order kinetic). It can be noticed that the photocatalytic degradation rate is independent of contact time (or volumetric flow rate) in the first part of the curve.

Influence of the light Intensity

The light flux has been varied by changing the fluorescent lamp and irradiation at two power intensities: 2.8 mW/cm 2 and 0.3 mW/cm 2 (Figure 7). The influence of the irradiation is clear and the conversion increases a little bet at higher irradiation.

CONCLUSION

The photocatalytic activity of a micro-reactor has been tested in this study. The experiments have been performed in order to determine, how the geometry influences reaction rate, mass transfer rate and finally the photocatalytic efficiency. It has been shown that the geometry influences the kinetic constant k. As a consequence, the perspective is the development of a geometry having a great ratio surface to volume. At the same time, a very good accessibility of the photons to the catalytic surface must be achieved.

Besides, the optical properties of the polymer show that the support is not inert since it absorbs the light and probably leads to a decrease of the reactivity. That is why the use of another material transparent to the UV light, and able to promote favorable interaction with the pollutant will be developed in the future.

Figure 1 .

 1 Figure 1. Schematic view of the micro-reactor designed by CAD system; a) Front view; b) side view; c) perspective view.

Figure 2 .

 2 Figure 2. Photographic view of the polymeric micro-reactor.

Figure 3 .

 3 Figure 3. Schematics of the experimental set-up for the degradation in the micro-reactor.

Figure 4 .

 4 Figure 4. Experimental setup for Photocatalytic reaction with a micro-reactor and syringe used for the injection of the aqueous salicylic acid solution.

Figure 5 .

 5 Figure 5. Determination of k and K constants from equation (7).

Figure 6 .

 6 Figure 6. Effect of contact time on the outlet concentration of salicylic Acid.

Figure 7 .

 7 Figure 7. Influence of the Irradiation parameter on the efficiency of the salicylic acid degradation.

Acknowledgements

This work was supported in part by grants of the "Region de Lorraine".