Projective Reeds-Shepp car on $S^2$ with quadratic cost
Abstract
Description
Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $η,\barη$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[γ]=\int_0^T g_{γ(t)}(\dotγ(t),\dotγ(t))+
K^2_{γ(t)}g_{γ(t)}(\dotγ(t),\dotγ(t)) ~dt$$ along all smooth curves starting from $x$ with direction $η$ and ending in $\bar{x}$ with direction $\barη$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_γ$ 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.