Optimal Trajectory Tracking of Nonholonomic Mechanical Systems: a geometric approach

We study the tracking of a trajectory for a nonholonomic system by recasting the problem as an optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a non-holonomic system and the desired reference trajectory evolving on the distribution which defines the nonholonomic constraints. We propose a geometric framework since it describes the class of nonlinear systems under study in a coordinate-free framework. Necessary conditions for the existence of extrema are determined by the Pontryagin Minimum Principle. A nonholonomic fully actuated particle is used as a benchmark example to show how the proposed method is applied.

I. Introduction

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. The earliest work on control of nonholonomic systems is by R. W. Brockett in [9]. A. M. Bloch [2], [3] has examined several control theoretic issues which pertain to both holonomic and nonholonomic systems in a very general form. The seminal works about stabilization in nonholonomic control systems were done by A. M. Bloch, N. H. McClamroch, and M. Reyhanoglu in [3], [5], [6], [7], and more recently by A. Zuryev [27].