# A continuation method for motion-planning problems

ESAIM: Control, Optimisation and Calculus of Variations (2005)

- Volume: 12, Issue: 1, page 139-168
- ISSN: 1292-8119

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topChitour, Yacine. "A continuation method for motion-planning problems." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2005): 139-168. <http://eudml.org/doc/90786>.

@article{Chitour2005,

abstract = {
We apply the well-known homotopy continuation method to address the
motion planning problem (MPP) for smooth driftless control-affine
systems. The homotopy continuation method is a Newton-type procedure
to effectively determine functions only defined implicitly. That
approach requires first to characterize the singularities of a
surjective map and next to prove global existence for the solution of
an ordinary differential equation, the Wazewski equation. In the
context of the MPP, the aforementioned singularities are the abnormal
extremals associated to the dynamics of the control system and the
Wazewski equation is an o.d.e. on the control space called the Path
Lifting Equation (PLE). We first show elementary facts
relative to the maximal solution of the PLE such as local existence and
uniqueness. Then we prove two general results, a finite-dimensional
reduction for the PLE on compact time intervals and a
regularity preserving theorem. In a second part, if the Strong Bracket
Generating Condition holds, we show, for
several control spaces, the global existence of the solution of the PLE,
extending a previous result of H.J. Sussmann.
},

author = {Chitour, Yacine},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Homotopy continuation method; path following;
Wazewski equation; sub-Riemannian geometry; nonholonomic
control systems; motion planning problem.; homotopy continuation method; Wazewski equation; nonholonomic control systems; motion planning problem},

language = {eng},

month = {12},

number = {1},

pages = {139-168},

publisher = {EDP Sciences},

title = {A continuation method for motion-planning problems},

url = {http://eudml.org/doc/90786},

volume = {12},

year = {2005},

}

TY - JOUR

AU - Chitour, Yacine

TI - A continuation method for motion-planning problems

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2005/12//

PB - EDP Sciences

VL - 12

IS - 1

SP - 139

EP - 168

AB -
We apply the well-known homotopy continuation method to address the
motion planning problem (MPP) for smooth driftless control-affine
systems. The homotopy continuation method is a Newton-type procedure
to effectively determine functions only defined implicitly. That
approach requires first to characterize the singularities of a
surjective map and next to prove global existence for the solution of
an ordinary differential equation, the Wazewski equation. In the
context of the MPP, the aforementioned singularities are the abnormal
extremals associated to the dynamics of the control system and the
Wazewski equation is an o.d.e. on the control space called the Path
Lifting Equation (PLE). We first show elementary facts
relative to the maximal solution of the PLE such as local existence and
uniqueness. Then we prove two general results, a finite-dimensional
reduction for the PLE on compact time intervals and a
regularity preserving theorem. In a second part, if the Strong Bracket
Generating Condition holds, we show, for
several control spaces, the global existence of the solution of the PLE,
extending a previous result of H.J. Sussmann.

LA - eng

KW - Homotopy continuation method; path following;
Wazewski equation; sub-Riemannian geometry; nonholonomic
control systems; motion planning problem.; homotopy continuation method; Wazewski equation; nonholonomic control systems; motion planning problem

UR - http://eudml.org/doc/90786

ER -

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