# Optimal Trajectory Tracking of Nonholonomic Mechanical Systems: a geometric approach

## TL;DR

This paper presents a geometric optimal control approach for trajectory tracking in nonholonomic systems, minimizing position and velocity errors while respecting system constraints, demonstrated on a fully actuated particle example.

## Contribution

It introduces a coordinate-free geometric framework and applies Pontryagin's Minimum Principle to derive optimal trajectories for nonholonomic systems.

## Key findings

- Effective trajectory tracking achieved with the proposed method
- Framework applicable to a class of nonlinear nonholonomic systems
- Validated on a fully actuated particle example

## Abstract

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 nonholonomic system and the desired reference trajectory evolving on the distribution which defines the nonholonomic constraints. We prepose 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.