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Projective Reeds-Shepp car on S 2 with quadratic cost

Published online by Cambridge University Press:  19 December 2008

Ugo Boscain  and
Francesco Rossi
Ugo Boscain
Affiliation:
LE2i, CNRS UMR5158, Université de Bourgogne, 9 avenue Alain Savary - BP 47870, 21078 Dijon Cedex, France. SISSA, via Beirut 2-4, 34014 Trieste, Italy. boscain@sissa.it; rossifr@sissa.it
Francesco Rossi
Affiliation:
SISSA, via Beirut 2-4, 34014 Trieste, Italy. boscain@sissa.it; rossifr@sissa.it
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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

$J[\gamma]=\int_0^T \left({\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))+K^2_{\gamma(t)}{\g}_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) \right) ~{\rm d}t$

along all smooth curves starting from x with direction η 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.

Keywords

Carnot-Caratheodory distancegeometry of visionlens spacesglobal cut locus
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 2 , April 2010 , pp. 275 - 297
Copyright
© EDP Sciences, SMAI, 2008

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