Pose Regulation via the Dual Unitary Group: An Application to Spacecraft Rendezvous

Six-degree-of-freedom (pose) control is often developed on the dual-quaternions since they provide an efficient representation for implementation and a unified translational and rotational approach to control design. In this article we present a control defined on the dual 2×2 unitary groups [DU(2)] which are isomorphic to the dual-quaternions. This formulation brings us to a natural setting for the error kinematics in the form of left-invariant control systems defined on matrix Lie groups. Moreover, it is shown how this setting leads us to define geometric sliding surfaces that allow the resulting pose trajectory to be defined analytically. Furthermore, a chattering-free reaching law is used to drive the states to reach the sliding surface in finite time. It is shown that once the system states reach the geometrically defined sliding surface, the solutions of the states can be obtained as an analytic function of time, initial conditions, and the tuning parameters. This analytic form enables control in the presence of path constraints by analytically selecting an appropriate desired attitude. This approach is applied in simulation to perform a deep-space rendezvous maneuver.