# Projective Reeds-Shepp car on S2 with quadratic cost

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

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

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topBoscain, 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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