Singularities of Optimal Control Problems on some Six Dimensional Lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a threedimensional space. A Lie group formulation arises naturally and the vehicles are modelled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically the three-dimensional space forms Euclidean space E, the sphere S and the hyperboloid H. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). The Maximum Principle of optimal control, shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It is then shown that the projections of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.