María Barbero-Liñán

High-order sufficient conditions for configuration tracking of affine connection control systems

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.

Isotropic submanifolds and the inverse problem for mechanical constrained systems

We give a new characterization of the inverse problem of the calculus of variations that is easily extended to constrained systems, both in the autonomous and non-autonomous cases. The transition from unconstrained to constrained systems is given by passing from Lagrangian submanifolds to isotropic ones. If the constrained system is variational we use symplectic techniques to extend these isotropic submanifolds to Lagrangian ones and describe the solutions of the constrained system as solutions of a variational problem without constraints. Mechanical examples such as the rolling disk are provided to illustrate the main results.

A geometric framework for discrete Hamilton-Jacobi equation

In classical mechanics the Hamilton-Jacobi Equation is useful to integrate partially or completely Hamiltons equations [2]. Recent developments have provided this theory with an intrinsic formulation, see for instance [3]. Another branch in mechanics that has been studied from a geometric viewpoint is discrete lagrangian and hamiltonian mechanics [5, 6]. In this contribution we aim to mingle those two theories to describe the discrete Hamilton-Jacobi Equation. This has already started to be studied in the literature [7], but not intrinsically. We will show here that the use of Lagrangian submanifolds [8] creates the natural setting to describe geometrically the discrete Hamilton-Jacobi equation.

Lie algebroids and optimal control: abnormality

Candidates to be solutions to optimal control problems, called extremals, are found using Pontryagin’s Maximum Principle [9]. This Principle gives necessary conditions for optimality and, under suitable assumptions, starts a presymplectic constraint algorithm in the sense given in [3]. This procedure, first considered in optimal control theory in [6], can be adapted to characterize the different kinds of extremals [1].In this paper, we describe the constraints given by the algorithm for the so‐called abnormal extremals for optimal control problems defined on Lie algebroids [4, 7, 8]. The peculiarity of the abnormal extremals is their independence on the cost function to characterize them. In particular, we are interested in how useful the geometry provided by the Lie algebroid is to study the constraints obtained in the optimal control problems for affine connection control systems. These systems model the motion of different types of mechanical systems such as rigid bodies, nonholonomic systems and robot...

Configuration tracking for mechanical systems by kinematic reduction and fast oscillating controls

We propose sufficient conditions for a mechanical control system to track by admissible trajectories any curve on the configuration manifold. We adopt the affine connection control systems framework, extending conditions for tracking known in the literature by a combination of averaging procedures by oscillating controls with the notion of kinematic reduction.

New high order sufficient conditions for configuration tracking

In this paper, we propose new conditions guaranteeing that 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 new results are proved by a combination of averaging procedures by highly oscillating controls with the notion of kinematic reduction.