Complexity of Control-Affine Motion Planning

In this paper we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a d...

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Bibliographic Details
Published in:SIAM journal on control and optimization 2015-01, Vol.53 (2), p.816-844
Main Authors: Jean, F., Prandi, D.
Format: Article
Language:English
Subjects:
Approximation
Asymptotic properties
Complexity
Control systems
Controllability
Drift
Estimates
Mathematics
Motion planning
Optimization and Control
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Summary:In this paper we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities. As a byproduct, we are able to treat the long time local controllability problem, giving quantitative estimates on the cost of stabilizing the system near a nonequilibrium point of the drift.
ISSN:0363-0129
1095-7138
DOI:10.1137/130950793