Complexity of control-affine motion planning

TL;DR

This paper investigates the complexity of motion planning for control-affine systems, extending existing concepts from nonholonomic systems to systems with drift, and provides asymptotic estimates for these complexities.

Contribution

It generalizes the notion of motion planning complexity from nonholonomic systems to control-affine systems with drift, including new definitions and asymptotic estimates.

Findings

Defined various complexity measures for control-affine systems

Provided asymptotic estimates for these complexity measures

Extended existing complexity concepts to systems with drift

Abstract

In this paper we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well-understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities.