State observers for invariant dynamics on a Lie group

This paper concerns the design of full state observers for state space systems where the state is evolving on a finite dimensional, connected Lie group. Traditional full state observer and filter designs for systems evolving in vector spaces employ the following design principle, going back to the work of Kalman [6] and Luenberger [7] on linear systems. The observer system is designed as a combination of a copy of the system, i.e. a part that can in principle replicate the observed system’s trajectory, plus an innovation term which serves to drive the observer trajectory towards the correct system trajectory in the presence of initialization or measurement errors. We propose an observer design based on a split of the observer dynamics into a synchronous term producing constant error estimates and an innovation term that reduces the error. We show that there is a canonical way of defining error measures in this context. Under mild assumptions, we prove almost global exponential convergence of the resulting observer and demonstrate its utility for a practical pose estimation scenario, a problem that has received strong interest in recent years [1, 2, 8, 9, 10, 11]. Previous general theoretical work in the area has concentrated on invariant observer design [3, 5]. While it is natural to require the same invariance properties of an observer as those obeyed by the observed system, some of the most successful existing observer designs do not share these properties [4]. In this paper, we