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A geometric approach to the optimal control of nonholnomic mechanical systems

In this paper, we describe a constrained Lagrangian and Hamiltonian formalism for the optimal control of nonholonomic mechanical systems. In particular, we aim to minimize a cost functional, given initial and final conditions where the controlled dynamics is given by nonholonomic mechanical system. In our paper, the controlled equations are derived using a basis of vector fields adapted to the nonholonomic distribution and the Riemannian metric determined by the kinetic energy. Given a cost function, the optimal control problem is understood as a constrained problem or equivalently, under some mild regularity conditions, as a Hamiltonian problem on the cotangent bundle of the nonholonomic distribution. A suitable Lagrangian submanifold is also shown to lead to the correct dynamics. We demonstrate our techniques in several examples including a continuously variable transmission problem and motion planning for obstacle avoidance problems.


Introduction
Although nonholonomic systems have been studied since the dawn of analytical mechanics, there has been some confusion over the correct formulation of the equations of motion (see e.g. [4], [10] and [29] for some of the history). Further it is only recently that their geometric formulation has been understood. In addition, there has been recent interest in the analysis of control problems for such systems. Nonholonomic control systems exhibit distinctive features. In particular, many naturally underactuated systems are controllable, the controllability arising from the nonintegrability of the constraints.
Nonholonomic optimal control problems arise in many engineering applications, for instance systems with wheels, such as cars and bicycles, and systems with blades or skates. There are thus multiple applications in the context of wheeled motion, space or mobile robotics and robotic manipulation. In this paper, we will introduce some new geometric techniques in nonholonomic mechanics to study the case of force minimizing optimal control problems.
Classical nonholonomic constraints which are linear in the velocities can be geometrically encoded by a constant rank distribution D. As we will see, the distribution D will play the role of the velocity phase space. Given a mechanical Lagrangian L = K − V : T Q → R where K and V are the kinetic and potential energy, respectively. and the distribution D, the dynamics of the nonholonomic system is completely determined using the Lagrange-d'Alembert principle [4]. In this paper we will formulate a description in terms of a Levi Civita connection defined on the space of vector fields taking values on D. This connection is obtained by projecting the standard Lie bracket using the Riemannian metric associated with the kinetic energy K (see [3]) and the typical characterization of the Levi-Civita connection (see also [9]). By adding controls in this setting we can study optimal control problems such as the force minimizing problem. Moreover, we can see that the dynamics of the optimal control problem is completely described by a Lagrangian submanifold of an appropriate cotangent bundle and, under some regularity conditions, the equations of motion are derived as classical Hamilton's equations on the cotangent bundle of the distribution, T * D. Although our approach is intrinsic, we also give a local description since it is important for working out examples. For this, it is necessary to choose an adapted basis of vector fields for the distribution. From this point of view, we combine the techniques used previously by the authors of the paper (see [3], [11], [37]). An additional advantage of our method is that symmetries may be naturally analyzed in this setting.
Concretely, the main results of our paper can be summarized as follows: • Geometric derivation of the equations of motion of nonholonomic optimal control problems as a constrained problem on the tangent space to the constraint distribution D. • Construction of a Lagrangian submanifold representing the dynamics of the optimal control problem and the corresponding Hamiltonian representation when the system is regular. • Definition of a Legendre transformation establishing the relationship and correspondence between the Lagrangian and Hamiltonian dynamics. • Application of our techniques to different examples including optimal control of the Chaplygin sleigh, a continuously variable transmission and motion planning for obstacle avoidance problems.

Nonholonomic mechanical systems
Constraints on mechanical systems are typically divided into two types: holonomic and nonholonomic, depending on whether the constraint can be derived from a constraint in the configuration space or not. Therefore, the dimension of the space of configurations is reduced by holonomic constraints but not by nonholonomic constraints. Thus, holonomic constraints allow a reduction in the number of coordinates of the configuration space needed to formulate a given problem (see [40]).
We will restrict ourselves to the case of nonholonomic constraints. Additionally, assume that the constraints are given by a nonintegrable distribution D on the configuration space Q. Locally, if we choose local coordinates (q i ), 1 ≤ i ≤ n = dim Q, the linear constraints on the velocities are locally given by equations of the form φ a (q i ,q i ) = µ a i (q)q i = 0, 1 ≤ a ≤ m ≤ n, depending, in general, on configuration coordinates and their velocities. From an intrinsic point of view, the linear constraints are defined by a distribution D on Q of constant rank n − m such that the annihilator of D is locally given by In addition to these constraints, we need to specify the dynamical evolution of the system, usually by fixing a Lagrangian function L : T Q → R. In mechanics, the central concepts permitting the extension of mechanics from the Newtonian point of view to the Lagrangian one are the notions of virtual displacements and virtual work; these concepts were originally formulated in the developments of mechanics in their application to statics. In nonholonomic dynamics, the procedure is given by the Lagrange-d'Alembert principle. This principle allows us to determine the set of possible values of the constraint forces from the set D of admissible kinematic states alone. The resulting equations of motion are d dt where δq i denotes the virtual displacements verifying µ a i δq i = 0 (for the sake of simplicity, we will assume that the system is not subject to nonconservative forces). This must be supplemented by the constraint equations. By using the Lagrange multiplier rule, we obtain d dt The term on the right hand side represents the constraint force or reaction force induced by the constraints. The functions λ a are Lagrange multipliers which, after being computed using the constraint equations, allow us to obtain a set of second order differential equations. Now we restrict ourselves to the case of nonholonomic mechanical systems where the Lagrangian is of mechanical type Here G denotes a Riemannian metric on the configuration space Q representing the kinetic energy of the systems and V : Q → R is a potential function. Locally, the metric is determined by the matrix M = (G ij ) 1≤i,j≤n where G ij = G(∂/∂q i , ∂/∂q j ). Denote by τ D : D → Q the canonical projection of D over Q and Γ(τ D ) the set of sections of τ D which is just the set of vector fields X(Q) taking values on D. If X, Y ∈ X(Q), then [X, Y ] denotes the standard Lie bracket of vector fields.
Definition 2.1. A nonholonomic mechanical system on a manifold Q is given by the triple (G, V, D) where G is a Riemannian metric on Q, specifying the kinetic energy of the system, V : Q → R is a smooth function representing the potential energy and D a non-integrable distribution on Q representing the nonholonomic constraints.
We want to obtain a bracket defined for sections of D. Using the Riemannian metric G we can construct two complementary orthogonal projectors (see [2], [3], [19]). It is clear that this Lie bracket verifies the usual properties of a Lie bracket except the Jacobi identity.
Remark 2.3. From a more differential geometric point of view, D with this modified bracket of sections inherits a skew-symmetric Lie algebroid structure [20,3] where now the bracket of sections of the vector bundle does not satisfy in general the Jacobi identity, as an expression of the nonintegrability of the distribution D.
determined by the following two properties: Then, we determine the Christoffel symbols Γ A BC of the connection ∇ G D by Given local coordinates on Q, (q i ) i = 1, . . . , n; and {e A } a basis of sections . Consider the restricted Lagrangian function : D → R, Definition 2.6 ([3]). A solution of the nonholonomic problem is an admissible curve γ : I → D such that These equations are equivalent to the nonholonomic equations. Locally, these are given byq where (G D ) AB denotes the coefficients of the inverse matrix of (G D ) AB where G D (e A , e B ) = (G D ) AB .
Remark 2.7. Observe that these equations only depend on the coordinates (q i , y A ) on D. Therefore the nonholonomic equations are free of Lagrange multipliers. These equations are equivalent to the nonholonomic Hamel equations (see [11], [37] for example, and reference therein).

Optimal control of nonholonomic mechanical systems
The purpose of this section is to study optimal control problems for a nonholonomic mechanical systems. We shall assume that all the considered control systems are controllable, that is, for any two points q 0 and q f in the configuration space Q, there exists an admissible control u(t) defined on the control manifold U ⊆ R n such that the system with initial condition q 0 reaches the point q f at time T (see [4,13] for more details).
We will analyze the case when the dimension of the input or control distribution is equal to the rank of D. If the rank of D is equal to the dimension of the control distribution, the system will be called a fully actuated nonholonomic system.
or, equivalently, where u A are the control inputs.
Locally, the equations may be written aṡ Given a cost function the optimal control problem consists of finding an admissible curve γ : I → D solution of the fully actuated nonholonomic problem given initial and final boundary conditions on D and minimizing the functional where γ is an admissible curve. We define the submanifold D (2) of T D by and we can choose coordinates ( Therefore, D (2) is locally described by the constraints on T Ḋ Observe now that our optimal control problem is alternatively determined by a smooth function L : The following diagram summarizes the situation: To derive the equations of motion for L we can use standard variational calculus for systems with constraints defining the extended Lagrangian L, Example: continuously variable transmission (CVT). We want to study the optimal control of a simple model of a continuously variable transmissions, where we assume that the belt cannot slip (see [38] for more details). The shafts are attached to spiral springs that are fixed to a chasis. The belt between the two cones is translated along the shafts in accordance with the coordinate x, thus providing a varying transmission ratio. The belt is kept in a plane perpendicular to the shafts, so that the belt keeps a constant length. The variables θ 1 and θ 2 denote the angular deflections of the shafts. m denotes the mass of the belt slider, J 1 > 0 is the inertia about the center of mass of the driving pulley and J 2 > 0 is the inertia about the center of mass of the driven pulley. The configuration space is S 1 × S 1 × R and the configuration is given by The control inputs are denoted by u 1 and u 2 . The first one corresponds to a force applied perpendicular to the center of mass of the belt slider and the second one is the torque applied about the the center of mass of the driving pulley. Also, we assume that x < 1 (which correspond to assuming that the gear ratio is finite).
The belt imposes a constraint given by the no slip condition and is expressed in differential form by Therefore the constraint distribution D is given by The Lagrangian is metric on Then the Lagrangian L : The projection map P : Let q = (θ 1 , θ 2 , x) be coordinates on the base manifold S 1 × S 1 × R and take the basis {X 1 , X 2 } of vector fields on S 1 × S 1 × R. This basis induces adapted coordinates (θ 1 , θ 2 , x, y 1 , y 2 ) ∈ D in the following way: Given the vector fields X 1 and X 2 generating the distribution D we obtain the relations for q ∈ S 1 × S 1 × R Then, Each element e ∈ D q is expressed as a linear combination of these vector fields: Therefore, the vector subbundle τ D : D → S 1 × S 1 × R is locally described by the coordinates (θ 1 , θ 2 , x; y 1 , y 2 ); the first three for the base and the last two, for the fibers. Observe that and, in consequence, D is described by the conditions (admissibility conditions): as a vector subbundle of T Q where y 1 and y 2 are the adapted velocities relative to the basis of D defined before. The nonholonomic bracket is given by . Observe now, The restricted Lagrangian function in these new adapted coordinates is rewritten as The Euler-Lagrange equations, together with the admissibility conditions, for this Lagrangian areẏ Now, we add controls in our picture. Therefore the controlled Euler-Lagrange equations are now The optimal control problem consists of finding an admissible curve satisfying the previous equations given boundary conditions on D and minimizing the functional J(θ 1 , θ 2 , x, y 1 , y 2 , u 1 , u 2 ) = 1 2 T 0 u 2 1 + u 2 2 dt, for the cost function C : D × U → R given by C(θ 1 , θ 2 , x, y 1 , y 2 , u 1 , u 2 ) = 1 2 (u 2 1 + u 2 2 ).
This optimal control problem is equivalent to the constrained optimization problem determined by the lagrangian L : D (2) → R given by Here, D (2) is a submanifold of the vector bundle T D over D defined by where the inclusion i D (2) : D (2) → T D, is given by the map The equations of motion for the extended Lagrangian The resulting system of equations for the optimal control problem of the continuously variable transmission is difficult to solve explicitly and from this observation it is clear that it is necessary to develop numerical methods preserving the geometric structure for these mechanical control systems. The construction of geometric numerical methods for this kind of optimal control problem is a future research topic as we remark in Section 6.

3.2.
Example: the Chaplygin sleigh. We want to study the optimal control of the so-called Chaplygin sleigh (see [4]) introduced and studied in 1911 by Chaplygin [14], [40] and more recently by A. Ruina [42] (see also [17] and [18]). The sleigh is a rigid body moving on a horizontal plane supported at three points, two of which slide freely without friction while the third is a knife edge which allows no motion orthogonal to its direction as show in Figure 3.
We assume that the sleigh cannot move sideways. The configuration space of this dynamical system is the group of Euclidean motions of the two-dimensional plane R 2 , SE(2), which we parameterize with coordinates (x, y, θ) since an element A ∈ SE(2) is represented by the matrix   cos θ − sin θ x sin θ cos θ y 0 0 1   with x, y ∈ R and θ ∈ S 1 .
θ and (x, y) are the angular orientation of the sleigh and position of the contact point of the sleigh on the plane, respectively. Let m be the mass of the sleigh and JI + ma 2 is the inertia about the contact point, where I is the moment of inertia about the center of mass C and a is the distance from the center of mass to the knife edge. The configuration space will be identified with R 2 ×S 1 with coordinates q = (x, y, θ) ∈ R 2 × S 1 . The control inputs are denoted by u 1 and u 2 . The first one corresponds to a force applied perpendicular to the center of mass of the sleigh and the second one is the torque applied about the vertical axis.
The constraint is given by the no slip condition and is expressed in differential form by ω = sin θ dx − cos θ, dy.
Therefore the constraint distribution D is given by That is, the distribution is given by the span of the vector fields The Lagrangian is metric on Q where the matrix associated with the metric G is Then the Lagrangian L : T (R 2 × S 1 ) → R is given by the kinetic energy of the body, which is a sum of the kinetic energy of the center of mass and the kinetic energy due to the rotation of the body where x C = x + a cos θ,y C = y + a sin θ.
Let q = (x, y, θ) be coordinates on the base manifold R 2 × S 1 and take the basis {X 1 , X 2 } of vector fields of D. This basis induces adapted coordinates (x, y, θ, y 1 , y 2 ) ∈ D in the following way: Given the vector fields X 1 and X 2 generating the distribution we obtain the relations for q ∈ R 2 × S 1 Then, Each element e ∈ D q is expressed as a linear combination of these vector fields: e = y 1 X 1 (q) + y 2 X 2 (q), q ∈ R 2 × S 1 .
Therefore, the vector subbundle τ D : D → R 2 × S 1 is locally described by the coordinates (x, y, θ; y 1 , y 2 ); the first three for the base and the last two, for the fibers. Observe that The restricted Lagrangian function in the new adapted coordinates is given by (x, y, θ, y 1 , y 2 ) = 1 2m Therefore, the equations of motion are Now, by adding controls in our picture, the controlled Euler-Lagrange equations are written as The optimal control problem consists on finding an admissible curve satisfying the previous equations given boundary conditions on D and minimizing the functional J(x, y, θ, y 1 , y 2 , u 1 , u 2 ) = 1 2 T 0 (u 2 1 + u 2 2 ) dt, for the cost function C : D × U → R given by C(x, y, θ, y 1 , y 2 , u 1 , u 2 ) = 1 2 (u 2 1 + u 2 2 ).

The equations of motion for the extended Lagrangian
L(x, y, θ, y 1 , y 2 ,ẋ,ẏ,θ,ẏ 1 ,ẏ 2 , λ) The first two equations can be integrated as λ 1 = c 1 and λ 2 = c 2 where c 1 and c 2 are constants and differentiating the equation for λ 3 with respect to the time and substituting into the third equation, the problem is reduced to solve the system ...
Integrating these equations and using the admissibility conditions we obtain constants of integration c i , i = 3, . . . , 8 and the equations Therefore the controls u 1 and u 2 are Remark 3.2. A similar optimal control problem was studied also [9]. The authors have also used the theory of affine connections to analyze the optimal control problem of underactuated nonholonomic mechanical systems. The main difference with our approach is that in our paper we are working on the distribution D itself. We impose the extra condition λ 1 = λ 2 = 0 to obtain explicitlly the controls minimizing the cost function. Usually, there is prescribed an initial boundary condition on D and a final boundary condition on D. For the Chaplygin sleigh we impose conditions (x(0), y(0), θ(0), y 1 (0), y 2 (0)) and (x(T ), y(T ), θ(T ), y 1 (T ), y 2 (T )). Heuristically, observe that if we transform these conditions into initial conditions we will need to take the initial condition (x(0), y(0), θ(0), y 1 (0), y 2 (0),ẏ 1 (0),ẏ 2 (0), λ 1 (0), λ 2 (0), λ 3 (0)) and it is not necessary that some of the multipliers are zero from the very beginning.
Let the obstacle be circular in the xy-plane, located at the point (x C , y C ) = (0.5, 0.5). For llustrative purposes, we use a simple inverse square law for the navigation function. Let V (x, y) given by where the parameter κ is introduced to control the strength of the potential function.
Appending the potential into the cost functional (4) the optimal control problem is equivalent to solve the constrained optimization problem determined by L : .

The equations of motion for the extended Lagrangian
L(x, y, θ, y 1 , y 2 ,ẋ,ẏ,θ,ẏ 1 ,ẏ 2 , λ) We solve the earlier boundary value problem for several values of κ. Starting with κ = 0, which corresponds to a zero potential function, we incremente κ until the potential field was strong enough to prevent the sleigh from interfering with the obstacle. We try with κ = 0, 0.01, 0.1, 0.25, and 0.5 for T =1. The result is shown in Fig. 4. Note that for κ = 0.25 and 0.5 the sleigh avoids the obstacle, and as one may anticipate, as κ increases, the total control effort and therefore, the total cost J = 1 2 1 0 (u 2 1 + u 2 2 + V (x, y))dt increases. For example, J = 17.0242 when κ = 0.25 and J = 18.4634 when κ = 0.5. Hence, we select κ = 0.25 since it corresponds to a trajectory that avoids the obstacle with the least possible cost (of all five tried in this simulation). The trajectories profile is shown in Figures 5,  6 and 7. This example illustrate how our approach can be used with the method of navigation functions of optimal motion generation for obstacle avoidance.

Lagrangian submanifolds and nonholonomic optimal control problems
In this section we study the construction of Lagrangian submanifold representing intrinsically the dynamics of the optimal control problem and the corresponding Hamiltonian representation when the system is regular. In the regular case, the definition of a particular Legendre transformation give rise the relationship and correspondence between the Lagrangian and Hamiltonian dynamics. 4.1. Lagrangian submanifolds. In this subsection we will construct Lagrangian submanifolds that are interesting for our purposes in the study of the geometry of optimal control problems of controlled mechanical systems (see [33,49]). A distinguished symplectic manifold is the cotangent bundle T * Q of any manifold Q. If we choose local coordinates (q i ), 1 ≤ i ≤ n, then T * Q has induced coordinates (q i , p i ). Denote by π Q : T * Q → Q the canonical projection of the cotangent bundle defined by π Q ( q ) = q, where q ∈ T * q Q. Define the Liouville 1-form or canonical 1-form θ Q ∈ Λ 1 (T * Q) by (θ Q ) , X = , T π Q (X) , where X ∈ T T * Q , ∈ T * Q.
In local coordinates we have that θ Q = p i dq i . The canonical two-form ω Q on T * Q is the symplectic form ω Q = −dθ Q (that is ω Q = dq i ∧ dp i ). Now, we will introduce some special Lagrangian submanifolds of the symplectic manifold (T * Q, ω Q ). For instance, the image Σ λ = λ(Q) ⊂ T * Q of a closed 1-form λ ∈ Λ 1 Q is a Lagrangian submanifold of (T * Q, ω Q ), since λ * ω Q = −dλ = 0. We then obtain a submanifold diffeomorphic to Q and transverse to the fibers of T * Q. When λ is exact, that is, λ = df , where f : Q → R, we say that f is a generating function of the Lagrangian submanifold Σ λ = Σ f (see [49]).
A useful extension of the previous construction is the following result due to W.Tulczyjew: [48]). Let Q be a smooth manifold, τ Q : T Q → Q its tangent bundle projection, N ⊂ Q a submanifold, and f : N → R. Then is a Lagrangian submanifold of T * Q.
Taking f as the zero function, for example, we obtain the following Lagrangian submanifold which is just the conormal bundle of N :

4.2.
Lagrangian submanifold description of nonholonomic mechanical control problems. Next, we derive the equations of motion representing the dynamics of the optimal control problem .
Given the function L : D (2) → R, following Theorem (4.1), when N = D 2 ⊂ T D we have the Lagrangian submanifold Σ L ⊂ T * T D. Therefore, L : D (2) → R generates a Lagrangian submanifold Σ L ⊂ T * T D of the symplectic manifold (T * T D, ω T D ) where ω T D is the canonical symplectic 2-form on T * T D.
The relationship between these spaces is summarized in the following diagram: Definition 4.2. Let D be a non-integrable distribution, T D its tangent bundle and D (2) the subbundle of T D defined on (1). A second-order nonholonomic system is a pair (D (2) , Σ L ) where Σ L ⊂ T * T D is the Lagrangian submanifold generated by L : D (2) → R.
Consider local coordinates (q i , y A ,q i ,ẏ A ) on T D. These coordinates induce local coordinates (q i , y A ,q i ,ẏ A , µ i , µ A , γ i , γ A ) on T * T D. Therefore, locally, the system is characterized by the following set of equations on T * T D  In the case of the Chaplygin sleigh local coordinates on T * T D will be given by (x, y, θ, y 1 , y 2 ,ẋ,ẏ,θ,ẏ 1 ,ẏ 2 , µ x , µ y , µ θ , µ 1 , µ 2 , γ x , γ y , γ θ , γ 1 , γ 2 ), where (x, y, θ, y 1 , y 2 ,ẋ,ẏ,θ,ẏ 1 ,ẏ 2 ) are local coordinates on T D. The Lagrangian submanifold of T * T D is described by the equations µ x = 0, µ y = 0, After a straightforward computation one can check easily that these equations are equivalent with those obtained in the Lagrangian formalism.

4.3.
Legendre transformation and regularity condition. We define the map Ψ : T * T D → T * D as Definition 4.5. Define the Legendre transform associated with a second-order nonholonomic system (D (2) , Σ L ) as the map FL : Σ L → T * D given by FL = Ψ•i Σ L .
In local coordinates, it is given by The following diagram summarizes the situation We say that the second-order nonholonomic system (D (2) , From the local expression of FL we can observe that from a direct application of the implicit function theorem we have:  Observe that if the Lagrangian L : D (2) → R is determined from an optimal control problem and its expression is given by (2) then the regularity of the matrix for the cost function.

Hamiltonian formalism.
Assume that the system is regular. Then if we denote by p i = γ i and p A = ∂L ∂ẏ A we can writeẏ A =ẏ A (q i , y A , p A ). Define the Hamiltonian function H : T * D → R by where α ∈ T * D is a one-form on D, and π T * T D | Σ L : Σ L → D (2) is the projection locally given by π T * T D | Σ L (q i , y A ,ẏ A , γ i ) = (q i , y A ,ẏ A ). Locally the Hamiltonian is given by where we are using Below we will see that the dynamics of the nonholonomic optimal control problem is determined by the Hamiltonian system given by the triple (T  *  D, ω D , H) where ω D is the standard symplectic 2−form on T * D.
The dynamics of the optimal control problem for the second-order nonholonomic system is given by the symplectic hamiltonian dynamics determined by the dynamical equation Therefore, if we consider the integral curves of X H , there are of the type t → (q i (t),ẏ A (t),ṗ i (t),ṗ A (t)); the solutions of the nonholonomic Hamiltonian system is specified by the Hamilton's equations on T * Ḋ From equation (6) it is clear that the flow preserves the symplectic 2−form ω D . Moreover, these equations are equivalent to equations given in (3) using the identification between the Lagrange multipliers with the variables p i and the relation Remark 4.8. We observe that in our formalism the optimal control dynamics is deduced using a constrained variational procedure and equivalently it is possible to apply the Hamilton-Pontryagin's principle (see [21] for example), but, in any case, this "variational procedure" implies the preservation of the symplectic 2-form, and this is reflected in the Lagrangian submanifold character. Moreover, in our case, under the regularity condition, we have seen that the Lagrangian submanifold shows that the system can be written as a Hamiltonian system (which is obviously symplectic). Additionally, we use the Lagrangian submanifold Σ L as a way to define intrinsically the Hamiltonian side since we define the Legendre transformation using the Lagrange submanifold Σ L . However there exist other possibilities. For instance, in [1] (Section 4.2) the authors defined the corresponding momenta for a vakonomic system. Using this procedure the momenta are locally expressed as follows where L is an arbitrary extension of L to T D and f j =q j − ρ j A y A = 0 are the constraint equations. A simple computation shows that both are equivalent, but our derivation is more intrinsic and geometric, that is, independent of coordinates or extensions and without using Lagrange multipliers. 4.5. Example: continuously variable transmission (CVT) (cont'd). Now, we continue the example of the optimal control problem for a continuously variable transmission that we considered in Section 3.1. Recall that the constraint distribution for the CVT is given by D ⊂ T (S 1 × S 1 × R)

Conclusions and future research
In this section we summarize the contributions of our work and discuss future research. 5.1. Conclusions: In this paper we study optimal control problems for a class of nonholonomic mechanical systems. We have given a geometrical derivation of the equations of motion of a nonholonomic optimal control problem as a constrained problem on the tangent space to the constraint distribution. We have seen how the dynamics of the optimal control problem can be completely described by a Lagrangian submanifold of an appropriate cotangent bundle and under some mild regularity conditions we have derived the the equations of motion for the nonholonomic optimal control problem as a classical set of Hamilton's equations on the cotangent bundle of the constraint distribution. We have introduced the notion of Legendre transformation in this context to establish the relationship between the Lagrangian and Hamiltonian dynamics. We applied our techniques to different examples: optimal control of a continuously variable transmission, Chaplygin sleigh and to optimal planning for obstacle avoidance problems.

5.2.
Future research: Construction of geometric and variational integrators for optimal control problems of nonholonomic mechanical systems. In this paper we have seen that an optimal control problem of a nonholonomic system may be viewed as a Hamiltonian system on T * D. One can thus use standard methods for symplectic integration such as symplectic Runge-Kutta methods, collocation methods, Störmer-Verlet, symplectic Euler methods, etc.; developed and studied in [26], [27], [28], [43], [44], e.g., to simulate nonholonomic optimal control problems.
Also, we would like to build variational integrators as an alternative way to construct integration schemes for these kinds of optimal control problems following the results given in Section 3. Recall that in the continuous case we have considered a Lagrangian function L : D (2) → R. Since the space D (2) is a subset of T D we can discretize the tangent bundle T D by the cartesian product D × D. Therefore, our discrete variational approach for optimal control problems of nonholonomic mechanical systems will be determined by the construction of a discrete Lagrangian L d : D (2) d → R where D (2) d is the subset of D × D locally determined by imposing the discretization of the constraintq i = ρ i A (q)y A , for instance we can consider Now the system is in a form appropriate for the application of discrete variational methods for constrained systems (see [34] and references therein).