The invariant extended Kalman filter as a stable observer (1410.1465v4)

Abstract: 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.

• The paper demonstrates that the IEKF guarantees local stability by decoupling error dynamics from the state trajectory.
• It introduces a log-linear property that simplifies the nonlinear error evolution and improves convergence analysis over traditional EKF.
• Empirical results in mobile robotics and inertial navigation underscore IEKF's robustness and superior performance compared to standard methods.

An Analysis of the Invariant Extended Kalman Filter as a Stable Observer

The exploration of non-linear observers has long been a challenging area within system dynamics, primarily due to the complexity involved in guaranteeing convergence. This paper presents an in-depth examination of the Invariant Extended Kalman Filter (IEKF) when applied to systems with continuous dynamics and discrete observations on Lie groups. The authors, Axel Barrau and Silvère Bonnabel, contribute to the field by addressing the limitations of traditional Extended Kalman Filters (EKF) and proposing the IEKF as a robust alternative.

Key Concepts and Methodology

The central premise of the paper is the notion that for left-invariant systems on Lie groups, the estimation error can be characterized by an autonomous differential equation. This insight offers a pathway to achieve local stability for the IEKF irrespective of the state trajectory, distinguishing it from standard EKF that lacks such rigor in non-linear settings due to its reliance on linear approximation around the estimated trajectory.

The paper generalizes the setting by identifying a class of systems defined by specific algebraic properties that satisfy this autonomy criterion. This class includes systems where the error dynamics can be expressed through simplified linear-like equations despite the inherently non-linear system trajectory. A significant theoretical contribution is the formal characterization of this class, broadening the scope of application of the IEKF beyond traditional left-invariant systems.

Theoretical Contributions

• Independence from State Trajectory: The IEKF's framework guarantees that for a broad set of systems, the error dynamics are independent of the specific trajectory of the system, achieving a theoretical stability for the filter. This is pivotal as it bypasses the traditional limitations where the linear approximations depend critically on the state estimation proximity.
• Log-Linear Property: The authors elucidate a “log-linear” property where the nonlinear error evolution mirrors linear differential dynamics when expressed in logarithmic coordinates. This property aids in maintaining stability and simplifies the convergence analysis.
• Empirical Evaluation: Two representative applications—mobile robotics and inertial navigation—exemplify the theory. Through simulations, the IEKF demonstrates superior performance over the EKF, particularly in scenarios where the EKF is prone to divergence due to inappropriate linearization.

Practical and Theoretical Implications

1. Robustness in Real-World Applications: The enhanced stability properties of the IEKF make it an appealing alternative in engineering domains requiring reliable navigation and control solutions, such as autonomous vehicles and robotic systems.
2. Guaranteed Local Convergence: IEKF achieves local convergence guarantees, which are typically hard to obtain for non-linear systems without stringent assumptions.
3. Future Research Directions: The exploration of the invariant filtering approach paves the way for further expansion into more complex dynamics and different Lie group structures, potentially enriching the array of systems benefitting from these methodologies.

Conclusion

This research represents a significant technical development within the domain of state estimation in non-linear systems, particularly emphasizing the applicability and robustness of IEKF. The findings emphasize the importance of leveraging geometric structures inherent to Lie groups, optimizing the observer performance beyond standard EKFs. Thus, IEKF's methodology positions it as a favorable choice for systems where traditional EKF may falter, offering stability and efficiency gains grounded in a strong theoretical framework. As such, it invites further exploration into practical implementations and system-specific adaptations to expand its utility across diverse applications.