The invariant extended Kalman filter as a stable observer

TL;DR

This paper studies the stability of the invariant extended Kalman filter (IEKF) as a deterministic observer on Lie groups, proving local stability under certain conditions and demonstrating its advantages over the standard EKF in robotics and navigation applications.

Contribution

It generalizes the autonomous error property for invariant observers and proves local stability of the IEKF for a broad class of systems, supported by practical examples.

Findings

IEKF maintains convergence where EKF diverges in challenging scenarios.

The autonomous error property is extended to a broader class of systems.

Simulations confirm the stability and robustness of IEKF in real-world applications.

Abstract

We analyze the convergence aspects of the invariant extended Kalman filter (IEKF), when the latter is used as a deterministic non-linear observer on Lie groups, for continuous-time systems with discrete observations. One of the main features of invariant observers for left-invariant systems on Lie groups is that the estimation error is autonomous. In this paper we first generalize this result by characterizing the (much broader) class of systems for which this property holds. Then, we leverage the result to prove for those systems the local stability of the IEKF around any trajectory, under the standard conditions of the linear case. One mobile robotics example and one inertial navigation example illustrate the interest of the approach. Simulations evidence the fact that the EKF is capable of diverging in some challenging situations, where the IEKF with identical tuning keeps converging.