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arXiv · 2506.01514

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

Finn G. Maurer·Erlend A. Basso·Henrik M. Schmidt-Didlaukies·Torleiv H. Bryne
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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.

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Finn G. Maurer, Erlend A. Basso, Henrik M. Schmidt-Didlaukies, Torleiv H. Bryne. 2025-06-02. Equivalence of Left- and Right-Invariant Extended Kalman Filters on Matrix Lie Groups. https://arxiv.org/abs/2506.01514

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