High order sufficient conditions for tracking

TL;DR

This paper establishes new high-order sufficient conditions for a mechanical control system to track arbitrary curves, generalizing existing criteria and providing algorithmic control laws with practical examples.

Contribution

It introduces generalized high-order conditions for trajectory tracking in mechanical systems using convex cones of vector fields and brackets, extending prior results.

Findings

Conditions expressed via convex cones of vector fields.

Algorithmic construction of one-parameter tracking control laws.

Applications demonstrated on hovercraft and submarine models.

Abstract

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.