Spatial group error and synchrony for tracking control of left-invariant kinematic systems on Lie groups

Trajectory tracking for underactuated systems has been heavily studied for several decades. When the system statespace is a Lie group, the group multiplication can be used to define a global error. In this paper, we consider the spatial, or right-invariant, group error in the design of a tracking controller for left-invariant systems. This choice of error is shown to exhibit synchrony; that is, the error kinematics depend linearly on the control input difference. In particular, if the actual system is driven by the desired control signal, then the error is constant. This property is used to propose a simple nonlinear tracking control scheme that is globally stable and locally exponentially stable for a class of persistently exciting trajectories for left-invariant systems on Lie-groups. We explore the example system of a mobile robot and show that in this particular case, the proposed control scheme is almost-globally asymptotically stable.