Projective Reeds-Shepp car on S2 with quadratic cost

Ugo Boscain; Francesco Rossi

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 2, page 275-297
  • ISSN: 1292-8119

Abstract

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Fix two points x , x ¯ S 2 and two directions (without orientation) η , η ¯ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost J [ γ ] = 0 T γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) + K γ ( t ) 2 γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t along all smooth curves starting from x with direction η and ending in x ¯ with direction η ¯ . Here g is the standard Riemannian metric on S2 and K γ 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.

How to cite

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Boscain, Ugo, and Rossi, Francesco. "Projective Reeds-Shepp car on S2 with quadratic cost." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 275-297. <http://eudml.org/doc/250766>.

@article{Boscain2010,
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 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. },
author = {Boscain, Ugo, Rossi, Francesco},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Carnot-Caratheodory distance; geometry of vision; lens spaces; global cut locus; Carnot-Carathéodory distance},
language = {eng},
month = {4},
number = {2},
pages = {275-297},
publisher = {EDP Sciences},
title = {Projective Reeds-Shepp car on S2 with quadratic cost},
url = {http://eudml.org/doc/250766},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Boscain, Ugo
AU - Rossi, Francesco
TI - Projective Reeds-Shepp car on S2 with quadratic cost
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/4//
PB - EDP Sciences
VL - 16
IS - 2
SP - 275
EP - 297
AB - 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 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.
LA - eng
KW - Carnot-Caratheodory distance; geometry of vision; lens spaces; global cut locus; Carnot-Carathéodory distance
UR - http://eudml.org/doc/250766
ER -

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