Equivariant Symmetries for Aided Inertial Navigation

TL;DR

This work develops a unified, geometry-aware framework for inertial navigation state estimation that explicitly accounts for IMU biases via equivariant filtering (EqF). By introducing novel Lie-group symmetries, notably the tangent group on SE_2(3) and related semi-direct products, the authors derive EqF variants with autonomous navigation-state linearization and demonstrate superior accuracy, robustness, and consistency compared to state-of-the-art filters like the IEKF. The dissertation generalizes EqF to vision-aided and multi-sensor INS, including self-calibration of sensor extrinsics and intrinsics, and provides extensive simulations and real-world experiments (e.g., ArduPilot, VINS) to validate reduced linearization error and improved convergence. The framework also furnishes a flexible methodology to transform global-reference measurements into body-referenced forms compatible with the symmetry, enabling broad applicability across measurement types and sensor suites. Overall, the work offers a principled, symmetry-based path to next-generation, reliable navigation estimators for mobile robots and aerial platforms.

Abstract

Respecting the geometry of the underlying system and exploiting its symmetry have been driving concepts in deriving modern geometric filters for inertial navigation systems (INSs). Despite their success, the explicit treatment of inertial measurement unit (IMU) biases remains challenging, unveiling a gap in the current theory of filter design. In response to this gap, this dissertation builds upon the recent theory of equivariant systems to address and overcome the limitations in existing methodologies. The goal is to identify new symmetries of inertial navigation systems that include a geometric treatment of IMU biases and exploit them to design filtering algorithms that outperform state-of-the-art solutions in terms of accuracy, convergence rate, robustness, and consistency. This dissertation leverages the semi-direct product rule and introduces the tangent group for inertial navigation systems as the first equivariant symmetry that properly accounts for IMU biases. Based on that, we show that it is possible to derive an equivariant filter (EqF) algorithm with autonomous navigation error dynamics. The resulting filter demonstrates superior to state-of-the-art solutions. Through a comprehensive analysis of various symmetries of inertial navigation systems, we formalized the concept that every filter can be derived as an EqF with a specific choice of symmetry. This underlines the fundamental role of symmetry in determining filter performance. This dissertation advances the understanding of equivariant symmetries in the context of inertial navigation systems and serves as a basis for the next generation of equivariant estimators, marking a significant leap toward more reliable navigation solutions.

Figures (25)

• Figure 1: Graphical representation of equivariant symmetry $(\mathbf{G}, \phi)$ of a homogeneous space $\mathcal{M}$. A trajectory of a kinematic system $\xi(t) \in \mathcal{M}$ is represented in red. The input of the system $u \in \mathbb{L}$ is represented in orange. The trajectory of the lifted system $X(t) \in \mathbf{G}$ is represented in blue. The dashed light green arrows represent the projection $\xi = \phi_{\mathring{\xi}}(X)$ from $\mathbf{G}$ onto $\mathcal{M}$ given by the transitive action $\phi$. The trajectory of the kinematic system on the homogeneous space, is completely parametrized by the trajectory of the lifted system onto the symmetry group. The dashed light purple arrows represent the action of the Lie group $\mathbf{G}$ on the input space $\mathbb{L}$ that defines equivariance. The dashed light blue lines represent the action of the Lie group $\mathbf{G}$ on the output space $\mathcal{N}$ that defines the equivariance of the output.
• Figure 2: The eqf design methodology applies to any system posed on a homogeneous space. That is, it applies to both invariant and equivariant systems. In particular, the eqf specializes to the well-known iekf when the eqf design methodology is applied to group-affine and invariant systems.
• Figure 3: Graphical representation of the time evolution of a rigid body freely rotating in space, representing the system of interest.
• Figure 4: Representation of the symmetry of the biased attitude system. In black, the core state variables are represented; in orange, the elements of the symmetry group; and in blue, the new elements of the state space after the action of the symmetry group $\phi$ is applied. The left side represents the action in \ref{['bas_phi_direct']}. The right side represents the action in \ref{['bas_phi']}.
• Figure 5: Graphical representation of the system of interest, and the corresponding body-frame direction measurement $y_{1}$ of a fixed direction $\prescript{}{}{\bm{d}}_{1}$, and a fixed spatial direction measurement $y_{2}$ of a time-varying direction $\prescript{}{}{\bm{d}}_{2}$.

Theorems & Definitions (45)

• Lemma 4.2.1
• proof
• Remark 4.2.2
• Lemma 4.2.3
• proof
• Remark 4.2.4
• Theorem 4.2.5
• proof
• Lemma 4.2.6
• proof