Asymptotics of motion planning complexity for control-affine systems

TL;DR

This paper investigates the asymptotic behavior of the complexity involved in approximating nonadmissible curves within control-affine systems satisfying Hörmander's condition, focusing on tubular approximation complexities in specific distribution cases.

Contribution

It provides explicit asymptotic equivalences and constants for approximation complexities in control-affine systems with co-rank one distributions, including special cases with singularities.

Findings

Asymptotic equivalences for approximation complexity are derived.

Explicit constants are provided for generic cases.

Special analysis for 3D distributions with singularities.

Abstract

In this paper, we study the complexity of the approximation of nonadmissible curves for nonlinear control-affine systems satisfying the strong H{\"o}rmander condition. Focusing on tubular approximation complexities, we provide asymptotic equivalences, with explicit constants, for all generic situations where the distribution, i.e., the linear part of the control system, is of co-rank one. Namely, we consider curves in step 2 distributions and any dimension. In the 3 dimensional case, we also consider the case of distributions with Martinet-type singularities that are crossed by the curve at isolated points.