Observers for systems with invariant outputs

In this paper we introduce a general design approach for observers for left-invariant systems on a Lie group with right-invariant outputs on a manifold. This situation is encountered for example in attitude estimation problems where only partial information of the state can be measured. We propose a gradient-like observer design on the output space which can be lifted to a gradient-like observer on the state space. The observer on the output space converges for generic initial conditions and has local exponential convergence properties. The lifted observer retains the same convergence properties with respect to the fibers of the output map. Our observer constructions are illustrated on the attitude estimation problem for a rigid body.