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Iterated Invariant Extended Kalman Filter (IterIEKF)

Sven Goffin, Axel Barrau, Silvère Bonnabel, Olivier Brüls, Pierre Sacré

TL;DR

The paper develops the Iterated Invariant EKF (IterIEKF), a Gauss-Newton-based refinement of the IEKF update for state estimation on Lie groups, and analyzes its behavior in the noise-free limit where measurements define an observed set. It proves local and global compatibility properties, showing that the iteration keeps the distribution within the observed set and effectively reduces the update to a reduced subspace, with the same properties extending to an adapted LG-IterEKF. An engineering application to extended-pose estimation of a crane hook with an IMU (state in $SE_{2}(3)$) demonstrates that IterIEKF and the adapted LG-IterEKF outperform IEKF and EKF under high measurement accuracy, while requiring only a couple of GN iterations in practice. The work provides theoretical guarantees for invariant filtering with iterative measurement updates and shows practical real-time viability for high-precision inertial navigation tasks.

Abstract

We study the mathematical properties of the Invariant Extended Kalman Filter (IEKF) when iterating on the measurement update step, following the principles of the well-known Iterated Extended Kalman Filter. This iterative variant of the IEKF (IterIEKF) systematically improves its accuracy through Gauss-Newton-based relinearization, and exhibits additional theoretical properties, particularly in the low-noise regime, that resemble those of the linear Kalman filter. We apply the proposed approach to the problem of estimating the extended pose of a crane payload using an inertial measurement unit. Our results suggest that the IterIEKF significantly outperforms the IEKF when measurements are highly accurate.

Iterated Invariant Extended Kalman Filter (IterIEKF)

TL;DR

The paper develops the Iterated Invariant EKF (IterIEKF), a Gauss-Newton-based refinement of the IEKF update for state estimation on Lie groups, and analyzes its behavior in the noise-free limit where measurements define an observed set. It proves local and global compatibility properties, showing that the iteration keeps the distribution within the observed set and effectively reduces the update to a reduced subspace, with the same properties extending to an adapted LG-IterEKF. An engineering application to extended-pose estimation of a crane hook with an IMU (state in ) demonstrates that IterIEKF and the adapted LG-IterEKF outperform IEKF and EKF under high measurement accuracy, while requiring only a couple of GN iterations in practice. The work provides theoretical guarantees for invariant filtering with iterative measurement updates and shows practical real-time viability for high-precision inertial navigation tasks.

Abstract

We study the mathematical properties of the Invariant Extended Kalman Filter (IEKF) when iterating on the measurement update step, following the principles of the well-known Iterated Extended Kalman Filter. This iterative variant of the IEKF (IterIEKF) systematically improves its accuracy through Gauss-Newton-based relinearization, and exhibits additional theoretical properties, particularly in the low-noise regime, that resemble those of the linear Kalman filter. We apply the proposed approach to the problem of estimating the extended pose of a crane payload using an inertial measurement unit. Our results suggest that the IterIEKF significantly outperforms the IEKF when measurements are highly accurate.
Paper Structure (15 sections, 10 theorems, 30 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 15 sections, 10 theorems, 30 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Iterated Invariant EKF (IterIEKF)
  3. The IEKF equations
  4. The IterIEKF equations
  5. Theoretical properties of the IterIEKF
  6. Properties of considered noise-free measurements
  7. Compatibility of the IterIEKF with the measurements
  8. Consequences of the compatibility property
  9. Extending the theory to the iterated EKF on Lie groups
  10. Application of engineering interest
  11. Conclusion
  12. The IterIEKF derivation
  13. Standard case: noisy measurements
  14. Special case: noise-free measurements
  15. Adaptation and extension to the LG-IterEKF

Key Result

Proposition 1

Starting from the prior eq:concentrated_gaussian, with $l=k-1$, supposed to encode the state distribution at time $k$ conditional on past information $\mathbf{y}_0,\ldots,\mathbf{y}_{k-1}$, the MAP estimate in the light of latest measurement $\mathbf{y}_k$ is given by $\hat{\raisebox{1pt}{$\boldsymb where $\mathbf{z}_k$ is the innovation from eq:IEKF_equations, $\|\boldsymbol{\eta}\|^2_{\Xi}:=\bol

Figures (2)

  • Figure 1: Evolution of the average and standard deviation of the estimation error norm.
  • Figure 2: Histogram of the number of iterations required at the update stage of the iterated filters, computed over 500 runs.

Theorems & Definitions (19)

  • Proposition 1
  • proof
  • Definition 1
  • Definition 2
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Proposition 3
  • Theorem 1
  • ...and 9 more