  This paper provides a new observer design methodology for invariant systems
whose state evolves on a Lie group with outputs in a collection of related
homogeneous spaces and where the measurement of system input is corrupted by an
unknown constant bias. The key contribution of the paper is to study the
combined state and input bias estimation problem in the general setting of Lie
groups, a question for which only case studies of specific Lie groups are
currently available. We show that any candidate observer (with the same state
space dimension as the observed system) results in non-autonomous error
dynamics, except in the trivial case where the Lie-group is Abelian. This
precludes the application of the standard non-linear observer design
methodologies available in the literature and leads us to propose a new design
methodology based on employing invariant cost functions and general gain
mappings. We provide a rigorous and general stability analysis for the case
where the underlying Lie group allows a faithful matrix representation. We
demonstrate our theory in the example of rigid body pose estimation and show
that the proposed approach unifies two competing pose observers published in
prior literature.
