# The invariant extended Kalman filter as a stable observer

arXiv: https://arxiv.org/abs/1410.1465 · HTML: https://mlsanity.com/paper/1410.1465/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2014.

## Built on by (most cited first, 14 of 14)

- [Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras](https://mlsanity.com/paper/2310.04521/) (2023)
- [Legged Robot State Estimation With Invariant Extended Kalman Filter Using Neural Measurement Network](https://mlsanity.com/paper/2402.00366/) (2024)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
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- [DREAM: Domain-aware Reasoning for Efficient Autonomous Underwater Monitoring](https://mlsanity.com/paper/2509.13666/) (2025)
- [GNSS-Inertial State Initialization Using Inter-Epoch Baseline Residuals](https://mlsanity.com/paper/2506.11534/) (2025)
- [Heading Estimation Using Ultra-Wideband Received Signal Strength and Gaussian Processes](https://mlsanity.com/paper/2109.04868/) (2021)
- [The Geometric Observability Index: Influence, Fisher Information, and Weak Observability in SE(3) Pose Estimation](https://mlsanity.com/paper/2602.05582/) (2026)
- [POPL-KF: A Pose-Only Geometric Representation-Based Kalman Filter for Point-Line-Based Visual-Inertial Odometry](https://mlsanity.com/paper/2602.06425/) (2026)
- [K-GMRF: Kinetic Gauss-Markov Random Field for First-Principles Covariance Tracking on Lie Groups](https://mlsanity.com/paper/2603.19601/) (2026)
- [The Transformer as a Polar State Estimator](https://mlsanity.com/paper/2605.11007/) (2026)
- [Hierarchical Variational Kalman Filtering](https://mlsanity.com/paper/2607.00877/) (2026)
- [Learning Forced Multibody Dynamics on Lie Groups](https://mlsanity.com/paper/2607.12627/) (2026)
- [GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometry](https://mlsanity.com/paper/2609.25338/) (2026)

## Abstract (arXiv)

We analyze the convergence aspects of the invariant extended Kalman filter (IEKF), when the latter is used as a deterministic non-linear observer on Lie groups, for continuous-time systems with discrete observations. One of the main features of invariant observers for left-invariant systems on Lie groups is that the estimation error is autonomous. In this paper we first generalize this result by characterizing the (much broader) class of systems for which this property holds. Then, we leverage the result to prove for those systems the local stability of the IEKF around any trajectory, under the standard conditions of the linear case. One mobile robotics example and one inertial navigation example illustrate the interest of the approach. Simulations evidence the fact that the EKF is capable of diverging in some challenging situations, where the IEKF with identical tuning keeps converging.