Computational Algebra Support for the Chen-Fliess-Sussmann Differential Equation

Introduction

A motion planning task in robotics and related domains is to steer an object from its initial location $\pmb{q}_0=\pmb{q}(0)$ to a goal point $\pmb{q}_f=\pmb{q}(T)$ [3, 7]. The task is particularly difficult when the number of controls is smaller than the dimensionality of a state space. In this paper the class of driftless nonholonomic systems is considered. The systems are described by

$$ \dot{\pmb{q}}=\sum_{i=1}^m \pmb{g}_i(\pmb{q}) u_i= \pmb{G}(\pmb{q}) \pmb{u}, \;\;\;\; \dim \pmb{q}=n > m=\dim \pmb{u}, (13.1) $$

where $\pmb{q}$ is a configuration, $\pmb{g}_i$, i = 1, ...,m, are generators (smooth enough vector fields) of system (13.1), and $\pmb{u}$ denote controls.