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

Abstract

While previous literature has discussed implications of choosing a left- or right-invariant error formulation, we show that the extended Kalman filter (EKF) on matrix Lie groups produces the same state estimate under both choices when a covariance reset is applied after each measurement update. To establish this result, we derive the continuous-discrete EKF for systems evolving on matrix Lie groups, modeling process noise on the Lie algebra and using a Stratonovich interpretation for the stochastic differential equation. Building on the classical EKF formulation, our approach includes a reparameterization step after each correction, which induces a covariance reset. Monte Carlo simulations on an aided inertial navigation problem confirm the invariance property and demonstrate its failure when the covariance reset is neglected.