Observers for kinematic systems with symmetry?

This paper considers the design of nonlinear state observers for finite-dimensional equivariant kinematics of mechanical systems. The observer design problem is approached by lifting the system kinematics onto the symmetry group and designing an observer for the lifted system. Two particular classes of lifted systems are identified, which we term type I and type II systems, that correspond to common configurations of sensor suites for mobile robotics applications. We consider type I systems in detail and define an error signal on the symmetry group using the group structure. We propose an observer structure with a pre-observer or internal model augmented by an equivariant innovation term that leads to autonomous error evolution. A control Lyapunov function construction is used to design the observer innovation that both ensures the required equivariance, and leads to strong convergence properties of the observer error dynamics.