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

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.

Hybrid Nonlinear Observers for Inertial Navigation Using Landmark Measurements

“…In this subsection we provide a different version of the hybrid observer H o 1 using variable gains relying on the solution of a continuous Riccati equation. Let us define the following gain map P K : R 5×5 → se 2 (3) inspired by [19], such that for all A 1 ∈ R 3×3 , a 2 , · · · , a 5 ∈ R 3 and a 6 , · · · , a 9 ∈ R, one has…”

“…The map γ is defined in (24) and the flow and jump sets F o , J o are defined in (19) and (20), respectively. The gain map P K is given by (26 uses variable matrix gains K v , K p to be designed later in this subsection.…”

GPS-denied relative motion estimation for fixed-wing UAV using the variational pose estimator

“…One adverse consequence of these unstable estimation and control schemes is that they end up taking longer to converge compared to stable schemes with the same initial conditions and same initial transient behavior. Attitude observers and filtering schemes on SO(3) and SE(3) have been reported in, e.g., [4], [14], [15], [17], [18], [19], [24], [27], [31], [32]. These estimators do not suffer from kinematic singularities like estimators using coordinate descriptions of attitude, and they do not suffer from the unstable unwinding phenomenon encountered by continuous estimators using unit quaternions.…”

A DCM Based Attitude Estimation Algorithm for Low-Cost MEMS IMUs

“…They also show valuable aspects of a practical implementation of a filter. Finally, the observability problem for systems, in which the measurement of system input is corrupted by an unknown constant bias, is tackled in [39].…”