Equivalence of Left- and Right-Invariant Extended Kalman Filters on Matrix Lie Groups

TL;DR

This paper demonstrates that for continuous-time systems on matrix Lie groups, the extended Kalman filter's state estimate remains invariant whether using left- or right-invariant errors, provided a full-order covariance reset is applied after measurements.

Contribution

It proves the invariance of the EKF on Lie groups under different error definitions when using a full-order covariance reset, enhancing understanding of filter consistency.

Findings

Full-order covariance reset guarantees invariance.

Invariance holds for continuous-time systems on matrix Lie groups.

Reduced-order resets do not preserve invariance.

Abstract

This paper derives the extended Kalman filter (EKF) for continuous-time systems on matrix Lie groups observed through discrete-time measurements. By modeling the system noise on the Lie algebra and adopting a Stratonovich interpretation for the stochastic differential equation (SDE), we ensure that solutions remain on the manifold. The derivation of the filter follows classical EKF principles, naturally integrating a necessary full-order covariance reset post-measurement update. A key contribution is proving that this full-order covariance reset guarantees that the Lie-group-valued state estimate is invariant to whether a left- or right-invariant error definition is used in the EKF. Monte Carlo simulations of the aided inertial navigation problem validate the invariance property and confirm its absence when employing reduced-order covariance resets.