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This paper provides a new observer design methodology for invariant systemswhose state evolves on a Lie group with outputs in a collection of relatedhomogeneous spaces and where the measurement of system input is corrupted by anunknown constant bias. The key contribution of the paper is to study thecombined state and input bias estimation problem in the general setting of Liegroups, a question for which only case studies of specific Lie groups arecurrently available. We show that any candidate observer (with the same statespace dimension as the observed system) results in non-autonomous errordynamics, except in the trivial case where the Lie-group is Abelian. Thisprecludes the application of the standard non-linear observer designmethodologies available in the literature and leads us to propose a new designmethodology based on employing invariant cost functions and general gainmappings. We provide a rigorous and general stability analysis for the casewhere the underlying Lie group allows a faithful matrix representation. Wedemonstrate our theory in the example of rigid body pose estimation and showthat the proposed approach unifies two competing pose observers published inprior literature.

Observers for invariant systems on Lie groups with biased input measurements and homogeneous outputs