The Difference between the Left and Right Invariant Extended Kalman Filter

2025-07-06

AI Summary

A long-standing debate in robotics concerns whether left- and right-invariant extended Kalman filters (IEKFs) are equivalent and whether their chirality must match the measurement model’s. Method: This paper proves that, on Lie groups, left- and right-IEKFs are mathematically equivalent when a proper state reset step is incorporated—grounded in group-affine system theory and validated via simulations in a GNSS-aided inertial navigation framework. Results: The reset mechanism eliminates chirality-induced discrepancies and substantially improves the asymptotic estimation performance of IEKF. Crucially, the work refutes the common misconception that IEKF chirality must be selected based on the measurement model, establishing the reset step as an essential design element for ensuring consistency and robustness of invariant filtering. This provides both a unifying theoretical perspective and practical guidance for state estimation on Lie groups.

Technology Category

• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Comparing left and right Invariant Extended Kalman Filters (IEKF) performance
• Demonstrating equivalence of left- and right- IEKF with reset step
• Improving asymptotic performance in GNSS-aided inertial navigation systems

Innovation

• Uses left and right Invariant Extended Kalman Filter
• Demonstrates equivalence of left and right IEKF
• Proposes reset step to improve filter performance