Projective Reeds-Shepp car on S2 with quadratic cost
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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 $J\left[\gamma \right]={\int }_{0}^{T}\left({}_{\gamma \left(t\right)}\left(\stackrel{˙}{\gamma }\left(t\right),\stackrel{˙}{\gamma }\left(t\right)\right)+{K}_{\gamma \left(t\right)}^{2}{}_{\gamma \left(t\right)}\left(\stackrel{˙}{\gamma }\left(t\right),\stackrel{˙}{\gamma }\left(t\right)\right)\right)\phantom{\rule{3.33333pt}{0ex}}\mathrm{d}t$ 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.

DOI: 10.1051/cocv:2008075
Classification: 49J15,  53C17
Keywords: Carnot-caratheodory distance, geometry of vision, lens spaces, global cut locus
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title = {Projective {Reeds-Shepp} car on {S2} with quadratic cost},
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Boscain, Ugo; Rossi, Francesco. Projective Reeds-Shepp car on S2 with quadratic cost. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 275-297. doi : 10.1051/cocv:2008075. http://archive.numdam.org/articles/10.1051/cocv:2008075/

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