Projective Reeds-Shepp car on $S^2$ with quadratic cost

TL;DR

This paper solves a geometric optimal control problem on the sphere involving a quadratic cost related to geodesic curvature, revealing complex structures like cusps and stratified cut loci, with applications in mechanics and vision.

Contribution

It provides the first explicit global solution to a projective Reeds-Shepp type problem on $S^2$ with quadratic cost, including analysis of geodesics and cut locus topology.

Findings

Explicit global solution for the problem

Identification of geodesics with cusps

Complex stratified structure of the cut locus

Abstract

Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $\eta,\bar\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\gamma]=\int_0^T g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))+ K^2_{\gamma(t)}g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\eta$ and ending in $\bar{x}$ with direction $\bar\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\gamma$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.