The Difference between the Left and Right Invariant Extended Kalman Filter

TL;DR

This paper shows that the left- and right-invariant extended Kalman filters are mathematically equivalent when the reset step is included, and that the reset step enhances performance in inertial navigation applications.

Contribution

It demonstrates the equivalence of the two IEKF versions and highlights the importance of the reset step for improved filter performance.

Findings

Left- and right-IEKF algorithms are identical with reset step.

Reset step improves asymptotic performance of IEKF.

Simulation with GNSS-aided inertial navigation confirms equivalence.

Abstract

The extended Kalman filter (EKF) has been the industry standard for state estimation problems over the past sixty years. The Invariant Extended Kalman Filter (IEKF) is a recent development of the EKF for the class of group-affine systems on Lie groups that has shown superior performance for inertial navigation problems. The IEKF comes in two versions, left- and right- handed respectively, and there is a perception in the robotics community that these filters are different and one should choose the handedness of the IEKF to match handedness of the measurement model for a given filtering problem. In this paper, we revisit these algorithms and demonstrate that the left- and right- IEKF algorithms (with reset step) are identical, that is, the choice of the handedness does not affect the IEKF's performance when the reset step is properly implemented. The reset step was not originally proposed as part of the IEKF, however, we provide simulations to show that the reset step improves asymptotic performance of all versions of the the filter, and should be included in all high performance algorithms. The GNSS-aided inertial navigation system (INS) is used as a motivating example to demonstrate the equivalence of the two filters.