Equivariant Symmetries for Inertial Navigation Systems (2309.03765v2)

Abstract: This paper investigates the problem of inertial navigation system (INS) filter design through the lens of symmetry. The extended Kalman filter (EKF) and its variants, have been the staple of INS filtering for 50 years; however, recent advances in inertial navigation systems have exploited matrix Lie group structure to design stochastic filters and state observers that have been shown to display superior performance compared to classical solutions. In this work we consider the case where a vehicle has an inertial measurement unit (IMU) and a global navigation satellite system (GNSS) receiver. We show that all the modern variants of the EKF for these sensors can be interpreted as the recently proposed Equivariant Filter (EqF) design methodology applied to different choices of symmetry group for the INS problem. This leads us to propose two new symmetries for the INS problem that have not been considered in the prior literature, and provide a discussion of the relative strengths and weaknesses of all the different algorithms. We believe the collection of symmetries that we present here capture all the sensible choices of symmetry for this problem and sensor suite, and that the analysis provided is indicative of the relative real-world performance potential of the different algorithms.

• The paper presents a unifying framework that reinterprets INS filters as equivariant filters using Lie group structures for improved design.
• It introduces two novel symmetries, the DP-EqF and SD-EqF, which reduce linearization errors and avoid state over-parameterization.
• Monte Carlo simulations and UAV flight tests show that geometric filters outperform classical MEKF in transient response and consistency.

Equivariant Symmetries for Inertial Navigation Systems

The paper under review presents an exploration of inertial navigation system (INS) filter design through the concept of symmetry, specifically focusing on the theory and application of equivariant filters (EqFs). The research seeks to establish a unifying framework that interprets various modern INS filter variants as EqFs applied to distinct symmetry groups, highlighting that the underlying group structure is the primary differentiator among these filter variants.

The authors begin by reviewing classical approaches to INS filtering, including the extended Kalman filter (EKF) and its variants, which have been widely used over the past fifty years. Acknowledging recent advancements, they shift the discussion toward the Lie group structure that underpins stochastic filters and state observers, demonstrating superior performance over traditional methods.

The paper presents six specific symmetry groups corresponding to different INS filter designs. The notable symmetries include the Special Orthogonal group (SO(3)), the Extended Special Euclidean group (SE_2(3)), and variations thereof explored in the context of the Tangent group. By employing the EqF framework, the authors offer a novel perspective on existing algorithms such as the multiplicative extended Kalman filter (MEKF), the Imperfect-Invariant EKF (IEKF), and the Two-Frame Group-Invariant EKF (TFG-IEKF), reinterpretating them via these newly outlined symmetries.

The research introduces two novel symmetries, namely the Direct Position Equivariant Filter (DP-EqF) and the Semi-Direct Bias Equivariant Filter (SD-EqF). These symmetries are designed to provide better integration between navigation states and bias states using a semi-direct product structure without requiring additional state over-parameterization.

The paper provides an in-depth mathematical exposition on the Lie groups, the associated group actions, and how these relate to lifting filters for the equivariant INS system. By reformulating fixed-frame measurements as body-frame relative measurements, the authors achieve a third-order linearization error in the measurement equations, thereby facilitating greater accuracy in filter computations.

A pivotal part of the paper involves the evaluation of filter performance through Monte Carlo simulations and real-world UAV flight data. The simulations reveal that modern geometric filters outperform the classical MEKF, especially regarding transient response and filter consistency. Among the advanced filters, the Tangent Group EqF (TG-EqF) displays superior asymptotic performance and consistency, attributed to its ability to relocate linearization error within the bias states, which are less dynamic compared to the navigation states.

In conclusion, the paper successfully establishes a comprehensive framework that links modern geometric INS filters to symmetry groups. Through rigorous analysis, it posits that symmetries and equivariance offer a promising direction in designing more consistent and accurate stochastic filters for inertial navigation systems. By leveraging the underlying group structures, this approach has the potential to enhance the robustness of INS technologies in real-world applications. Future developments in AI can benefit from exploring such structural and symmetry-based methodologies, potentially leading to advancements in robotics and autonomous navigation systems.