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Dubins' problem is intrinsically three-dimensional

Published online by Cambridge University Press:  15 August 2002

D. Mittenhuber
D. Mittenhuber*
Affiliation:
([email protected])
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Abstract

In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constantand prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higherdimensions (cf. [15]). 
Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if then-dimensional problem always reduces to the three-dimensional case. In this paper we will prove that this is true in the euclidean as well as inthe noneuclidean case. At first glance one might consider this a trivial problem, but we will also give an example showing that this is not thecase. 


Keywords

Noneuclidean Dubins' problemlength-minimizing curves with bounded curvaturevariational problems on Lie groupsSerret-Frenet differential systemmaximum principle.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 3 , 1998 , pp. 1 - 22
Copyright
© EDP Sciences, SMAI, 1998

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