The classification of homotopy classes of bounded curvature paths

TL;DR

This paper establishes necessary and sufficient conditions for when two bounded curvature paths in the Euclidean plane are connected through a deformation that maintains bounded curvature, completing a program initiated by Lester Dubins in 1961.

Contribution

It provides a complete characterization of the homotopy classes of bounded curvature paths in the plane, advancing the understanding of their topological structure.

Findings

Identifies conditions for path homotopy equivalence under curvature constraints

Completes Dubins' program on bounded curvature path classification

Contributes to the theory of curvature-constrained path planning

Abstract

A bounded curvature path is a continuously differentiable piecewise $C^2$ path with bounded absolute curvature that connects two points in the tangent bundle of a surface. In this note we give necessary and sufficient conditions for two bounded curvature paths, defined in the Euclidean plane, to be in the same connected component while keeping the curvature bounded at every stage of the deformation. Following our previous results here we finish a program started by Lester Dubins in 1961.