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 six-dimensional 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 three-dimensional Lie group. These lower dimensional systems lend themselves to a minimal global representation that avoids 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 motion planning methods. The approach is applied to a spacecraft docking problem using two different types of actuation: 1) a fully actuated continuous low-thrust propulsion system; and 2) an underactuated single impulsive thruster and reaction wheel system.