# MF-IEKF: A Multiplicative Federated Invariant Extended Kalman Filter for INS/GNSS

## TL;DR

This paper introduces a new filter, MF-IEKF, that improves navigation accuracy by combining different error models for INS/GNSS systems.

## Contribution

The novel MF-IEKF integrates left- and right-invariant errors to enhance estimation accuracy and convergence under large misalignment.

## Key findings

- MCL-IEKF improves state estimation accuracy compared to classical multiplicative left- or right-IEKF.
- MF-IEKF achieves faster convergence in position and velocity under large misalignment angles compared to MCL-IEKF.
- Experiments show MF-IEKF outperforms ML1-IEKF in navigation accuracy and robustness.

## Abstract

What are the main findings?
Based on the two left-invariant errors on the SE2(3) Lie group, a corrected error on the Lie algebra is proposed for the multiplicative corrected left-invariant extended Kalman filter (MCL-IEKF).Leveraging the decoupling of position and attitude in the right-invariant observation matrix of GNSS, a multiplicative federated IEKF (MF-IEKF) is developed by combining left- and right-invariant errors.

Based on the two left-invariant errors on the SE2(3) Lie group, a corrected error on the Lie algebra is proposed for the multiplicative corrected left-invariant extended Kalman filter (MCL-IEKF).

Leveraging the decoupling of position and attitude in the right-invariant observation matrix of GNSS, a multiplicative federated IEKF (MF-IEKF) is developed by combining left- and right-invariant errors.

What are the implications of the main findings?
MCL-IEKF improves state estimation accuracy compared to classical multiplicative left- (right-)IEKF.Under large misalignment angle conditions, MF-IEKF achieves faster convergence in position and velocity compared to the MCL-IEKF.

MCL-IEKF improves state estimation accuracy compared to classical multiplicative left- (right-)IEKF.

Under large misalignment angle conditions, MF-IEKF achieves faster convergence in position and velocity compared to the MCL-IEKF.

The integration of an inertial navigation system (INS) with the Global Navigation Satellite System (GNSS) is crucial for suppressing the error drift of the INS. However, traditional fusion methods based on the extended Kalman filter (EKF) suffer from geometric inconsistency, leading to biased estimates—a problem markedly exacerbated under large initial misalignment angles. The invariant extended Kalman filter (IEKF) embeds the state in the Lie group SE2(3) to establish a more consistent framework, yet two limitations remain. First, its standard update fails to synergize complementary error information within the left-invariant formulation, capping estimation accuracy. Second, velocity and position states converge slowly under extreme misalignment. To address these issues, a multiplicative federated IEKF (MF-IEKF) was proposed. A geometrically consistent state propagation model on SE2(3) is derived from multiplicative IMU pre-integration. Two parallel, mutually inverse left-invariant error sub-filters (ML1-IEKF and ML2-IEKF) cooperate to improve overall accuracy. For large-misalignment scenarios, a short-term multiplicative right-invariant sub-filter is introduced to suppress initial position and velocity errors. Extensive Monte Carlo simulations and KITTI dataset experiments show that MF-IEKF achieves higher navigation accuracy and robustness than ML1-IEKF.