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

TL;DR

This paper introduces a novel observer design methodology for invariant systems on Lie groups with biased input measurements, addressing the challenge of non-autonomous error dynamics and providing a unified approach for pose estimation.

Contribution

It develops a new observer design framework for Lie group systems with input bias, extending beyond specific cases to a general setting with stability guarantees.

Findings

The proposed observer handles biased inputs effectively.

Unified existing pose observers under a common framework.

Rigorous stability analysis for systems with matrix-representable Lie groups.

Abstract

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.