Optimal under-actuated kinematic motion planning on the ϵ-group

A global motion planning method is described based on the solution of minimum energy-type curves on the frame bundle of connected surfaces of arbitrary constant cross sectional curvature ϵ. Applying the geometric framing of Pontryagin's principle gives rise to necessary conditions for optimality in the form of a boundary value problem. This arbitrary dimensional boundary value problem is solved using a numerical shooting method derived from a general Lax pair solution. The paper then specializes to the 3-dimensional case where the Lax pair equations are integrable. A semi-analytic method for matching the boundary conditions is proposed by using the analytic form of the extremal solutions and a closed form solution for the exponential map. This semi-analytical approach has the advantage that an analytic description of the control accelerations can be derived and enables actuator constraints to be incorporated via time reparametrization. The method is applied to two examples in space mechanics: the attitude control of a spacecraft with two reaction wheels and the spacecraft docking problem.