Projective Reeds-Shepp car on S 2 with quadratic cost

Abstract

Fix two points x,x̄ ∈ S2 and two directions (without orientation) η,η̄ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost J[γ]= ∫T0 (gγ(t)(γ̇(t), γ̇(t))+K2γ(t)g γ(t)(γ̇(t),γ̇(t)) dt along all smooth curves starting from x with direction η and ending in x̄ with direction η̄. Here g is the standard Riemannian metric on S2 and Kγ 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. © EDP Sciences, SMAI, 2008.