Recursive posterior Cramér-Rao lower bound on Lie groups

The recursive posterior Cramer-Rao Lower Bound (CRLB) introduced by Tichavsky et al. (1998) is an efficient tool to estimate the minimum variance of an estimator. Lately, filters and observers on matrix Lie groups have raised interest in control and nonlinear estimation. However, the recursive CRLB applies to variables belonging to the Euclidean space, and a recursive formulation on matrix Lie groups was not introduced yet. Hence, this paper derives a recursive formula of the Fisher information matrix on Lie groups, which reveals to be a natural extension of the information matrix on the Euclidean space. This generic result is applied to nonlinear Gaussian systems on Lie groups and tested on a navigation problem. The proposed recursive CRLB is consistent with state-of-the-art filters and shows a representative behavior with respect to the estimation errors. This paper provides a simple method to recursively compute the minimal variance of an estimator on matrix Lie groups. It enables to assess the behavior of an estimator, which is fundamental to implement robust algorithms.(c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

The paper derives a recursive formula of the Fisher information matrix on Lie groups and applies it to nonlinear Gaussian systems on Lie groups for testing. The proposed recursive CRLB is consistent with state-of-the-art filters and exhibits representative behavior in estimation errors. This paper provides a simple method to recursively compute the minimal variance of an estimator on matrix Lie groups, which is fundamental for implementing robust algorithms.