Motion planning on a class of 6-D Lie groups via a covering map

This paper presents an approach to motion planning for left-invariant kinematic systems defined on the 6-D frame bundles of symmetric spaces of constant cross-sectional curvature. A covering map is used to convert the original differential equation into two coupled equations each evolving on a 3-D Lie group. These lower dimensional systems lend themselves to a minimal global representation that avoid singularities associated with the use of exponential coordinates. Open-loop and closed-loop kinematic control problems are addressed to demonstrate the use of this mapping for analytical and numerical based motion planning methods. The approach is applied to a spacecraft docking problem using two different types of actuation: (i) a fully-actuated continuous low-thrust propulsion system and (ii) an under-actuated single impulsive thruster and reaction wheel system.