# Stabilization of non-admissible curves for a class of nonholonomic systems

## TL;DR

This paper addresses the problem of stabilizing trajectories of certain nonholonomic systems by proposing an explicit control scheme that ensures exponential convergence to a reference curve, with numerical validation.

## Contribution

It introduces a control design for nonholonomic systems with degree of nonholonomy equal to 1, ensuring exponential convergence to reference curves.

## Key findings

- Trajectories converge exponentially to reference curves.
- Control scheme is explicitly designed for systems with degree of nonholonomy 1.
- Numerical examples validate the theoretical results.

## Abstract

The problem of tracking an arbitrary curve in the state space is considered for underactuated driftless control-affine systems. This problem is formulated as the stabilization of a time-varying family of sets associated with a neighborhood of the reference curve. An explicit control design scheme is proposed for the class of controllable systems whose degree of nonholonomy is equal to 1. It is shown that the trajectories of the closed-loop system converge exponentially to any given neighborhood of the reference curve provided that the solutions are defined in the sense of sampling. This convergence property is also illustrated numerically by several examples of nonholonomic systems of degrees 1 and 2.