Projective Reeds-Shepp car on S2 with quadratic cost

Boscain, Ugo; Rossi, Francesco

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 275-297.

Fix two points $x,\overline{x}\in S^2$ and two directions (without orientation) $\eta ,\overline{\eta }$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J\left[\gamma \right]={\int }_0^T\left({}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))+K_{\gamma \left(t\right)}^2{}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))\right)\mathrm{d}t$ along all smooth curves starting from x with direction η and ending in $\overline{x}$ with direction $\overline{\eta }$. Here g is the standard riemannian metric on S² 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.

Classification : 49J15, 53C17
Mots clés : Carnot-caratheodory distance, geometry of vision, lens spaces, global cut locus

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