Symmetries in vakonomic dynamics: applications to optimal control

2008

This paper studies the optimal motion control of mechanical systems through a discrete geometric approach. At the core of our formulation is a discrete Lagrange-d'Alembert-Pontryagin variational principle, from which are derived discrete equations of motion that serve as constraints in our optimization framework. We apply this discrete mechanical approach to holonomic systems with symmetries and, as a result, geometric structure and motion invariants are preserved. We illustrate our method by computing optimal trajectories for a simple model of an air vehicle flying through a digital terrain elevation map, and point out some of the numerical benefits that ensue.

Proceedings of Institute of Mathematics of National …, 2004

The role of symmetry is well studied in physics and economics, where many great contributions have been made. With the help of Emmy Noether's celebrated theorems, a unified description of the subject can be given within the mathematical framework of the calculus of variations. It turns out that Noether's principle can be understood as a special application of the Euler-Lagrange differential equations. We claim that this modification of Noether's approach has the advantage to put the role of symmetry on the basis of the calculus of variations, and in a key position to give answers to some fundamental questions. We will illustrate our point with the interplay between the concept of invariance, the theory of optimality, Tonelli existence conditions, and the Lipschitzian regularity of minimizers for the autonomous basic problem of the calculus of variations. We then proceed to the general nonlinear situation, by introducing a concept of symmetry for the problems of optimal control, and extending the results of Emmy Noether to the more general framework of Pontryagin's maximum principle. With such tools, new results regarding Lipschitzian regularity of the minimizing trajectories for optimal control problems with nonlinear dynamics are obtained.

arXiv preprint math/0604072, 2006

We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether's first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2, 3, 5, 8).

2002

This paper presents a computationally efficient method for deriving coordinate representations for the equations of motion and the affine connection describing a class of Lagrangian systems. We consider mechanical systems endowed with symmetries and subject to nonholonomic constraints and external forces. The method is demonstrated on two robotic locomotion mechanisms known as the snakeboard and the roller racer. The resulting coordinate representations are compact and lead to straightforward proofs of various controllability results.

European Journal of Control, 2002

We obtain a generalization of E. Noether's theorem for the optimal control problems. The generalization involves a one-parameter family of smooth maps which may depend also on the control and a Lagrangian which is invariant up to an addition of an exact differential.

Applied Mathematics E-Notes, 2003

We extend the second Noether theorem to optimal control problems which are invariant under symmetries depending upon k arbitrary functions of the independent variable and their derivatives up to some order m. As far as we consider a semi-invariance notion, and the transformation group may also depend on the control variables, the result is new even in the classical context of the calculus of variations.

We treat the vakonomic dynamics with general constraints within a new geometric framework which will be appropriate to study optimal control problems. We compare our formulation with Vershik-Gershkovich one in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description wich enables us to develop an algorithm of comparison between the solutions of both dynamics. Some examples illustrating the theory are treated.

Journal of Dynamical and Control Systems

The goal of this paper is to describe a method to solve a class of time optimal control problems which are equivalent to finding the sub-Riemannian minimizing geodesics on a manifold M. In particular, we assume that the manifold M is acted upon by a group G which is a symmetry group for the dynamics. The action of G on M is proper but not necessarily free. As a consequence, the orbit space M/G is not necessarily a manifold but it presents the more general structure of a stratified space. The main ingredients of the method are a reduction of the problem to the orbit space M/G and an analysis of the reachable sets on this space. We give general results relating the stratified structure of the orbit space, and its decomposition into orbit types, with the optimal synthesis. We consider in more detail the case of the so-called K − P problem where the manifold M is itself a Lie group and the group G is determined by a Cartan decomposition of M. In this case, the geodesics can be explicitly calculated and are analytic. As an illustration, we apply our method and results to the complete optimal synthesis on SO(3) .

1998

This is the second article in the series that began in . Jacobi curves were defined, computed, and studied in that paper for regular extremals of smooth control systems. Here we do the same for singular extremals. The last section contains a feedback classification and normal forms of generic single-input affine in control systems on a 3-dimensional manifold.

Journal of Dynamical and Control Systems, 1997

2018

This paper provides a new perspective on the structure of kinematic systems with complete symmetry. These systems naturally occur as models for mechanical systems with symmetry, for example flying or submersible robots. The configuration space of such systems is a homogeneous space of the symmetry Lie group, and it is well known that their kinematics can be lifted to equivariant kinematics on the symmetry group thus allowing global state observer constructions. We provide explicitly checkable sufficient differential-algebraic conditions on the symmetry that will lead to a lifted system in the form of standard left or right invariant kinematics on the symmetry group. Previously known conditions for one of these two cases required finding a velocity lift map with particular properties for which there was no general construction known.

1998

2004

We obtain a generalization of Noether's invariance principle for optimal control problems with equality and inequality state-input constraints. The result relates the invariance properties of the problems with the existence of conserved quantities along the constrained Pontryagin extremals. A result of this kind was posed as an open question by Vladimir Tikhomirov, in 1986.

This work was intended as an attempt to pose a better definition for Lagrangian systems and their symmetries. Symmetries and infinitesimal symmetries of these sys- tems are defined and then, we proceed with the study of constants of motion in these systems and find their relations to the symmetries of the system. All these definitions are done in the cases time-independent and time-dependent Lagrangians.

8th IFAC Symposium on Nonlinear Control Systems, 2010

In this paper, we investigate a generalization of the infinite time horizon linear quadratic regulator (LQR) for systems evolving on the special orthogonal group SO(3). Using Pontryagin's Maximum Principle, we derive the necessary conditions for optimality and the associated Hamiltonian equations. For a special class of weighting matrices, we show that the optimal feedback can be computed explicitly and we prove that the non differentiable value function is the viscosity solution of an appropriate Hamiltn-Jacobi-Bellman equation on SO(3). For arbitrary positive definite weighting matrices, numerical simulations allow us to explore the relationship between the optimal trajectories and weighting matrices, and in particular to highlight nontrivial non differentiability properties of the value function.