Adding an integrator for output regulation of systems with matrix Lie-group states

This paper considers the problem of tracking reference trajectories for systems defined on matrix Lie Groups. Both the reference system (exosystem) and controlled system have states on the same matrix Lie group, with the exosystem having constant velocity. The measurements are associated with a group action on a homogeneous space of the state space and can be thought of as measured partial relative state information. We look for a controller depending only on the relative state errors and the local state of the controlled system, that is a control that is independent of the exogenous variables. The proposed design embeds an integral estimate of the unknown exosystem velocity as a dynamic state in the controller. The approach is motivated by a range of robotics applications posed on classical Lie-groups SO(3), SE(3), SL(3), although we develop a general result for kinematic systems on arbitrary matrix Lie-groups. In the specific case of SO(3), namely systems defined on the Lie-group of orthogonal rotations, we go further by presenting an `error feedback' controller for systems modeled by kinematics and dynamics equations.