1 results
Projective Reeds-Shepp car on S 2 with quadratic cost
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- Journal:
- ESAIM: Control, Optimisation and Calculus of Variations / Volume 16 / Issue 2 / April 2010
- Published online by Cambridge University Press:
- 19 December 2008, pp. 275-297
- Print publication:
- April 2010
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- Article
- Export citation
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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.
