Nonholonomic trajectory optimization and the existence of twisted matrix logarithms

Abstract

The problem studied here is a shortest path problem of the type encountered in sub-Riemannian geometry. It is distinguished by special structures related to its Lie group setting and the Z 2 graded structure on the relevant Lie algebra. In spite of the fact that the first order necessary conditions lead to differential equations that are integrable in terms of elementary functions, in this case there remain questions related to the existence of appropriate values for the parameters which appear. In this paper we treat the problem in some generality but establish the existence of suitable parameter values only in the case of the general linear group of dimension two. © 2008 Springer-Verlag.