The Domain of Attraction of the Desired Path in Vector-field Guided Path Following

TL;DR

This paper analyzes the domain of attraction for vector-field guided path following, proving it is homeomorphic to a product space, which enhances understanding of convergence properties in robotic path planning.

Contribution

It strengthens previous results by showing the domain of attraction is homeomorphic to a specific product space, extending the analysis to higher-dimensional manifolds.

Findings

The domain of attraction is homeomorphic to 5^{n-1} imes \u1d4a^1.

Results extend to k-dimensional compact manifolds for k 5 2.

Provides a topological characterization of the attraction domain.

Abstract

In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the vector field vanishes creates a topological obstruction to global convergence to the desired path and some associated topological analysis has been conducted in our previous work. In this paper, we strengthen the result in our previous work by showing that the domain of attraction of the desired path, which is a compact asymptotically stable one-dimensional embedded submanifold of an $n$-dimensional ambient manifold $\mathcal{M}$, is homeomorphic to $\mathbb{R}^{n-1} \times \mathbb{S}^1$, and not just homotopy equivalent to $\mathbb{S}^1$. This result is extended for a $k$-dimensional compact manifold for $k \ge 2$.