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.

This letter re-visits the problem of visual-inertial navigation system (VINS) and presents a novel filter design we dub the multi state constraint equivariant filter (MSCEqF, in analogy to the well known MSCKF). We define a symmetry group and corresponding group action that allow specifically the design of an equivariant filter for the problem of visual-inertial odometry (VIO) including IMU bias, and camera intrinsic and extrinsic calibration states. In contrast to state-of-the-art invariant extended Kalman filter (IEKF) approaches that simply tack IMU bias and other states onto the $\mathbf{SE}_2(3)$ group, our filter builds upon a symmetry that properly includes all the states in the group structure. Thus, we achieve improved behavior, particularly when linearization points largely deviate from the truth (i.e., on transients upon state disturbances). Our approach is inherently consistent even during convergence phases from significant errors without the need for error uncertainty adaptation, observability constraint, or other consistency enforcing techniques. This leads to greatly improved estimator behavior for significant error and unexpected state changes during, e.g., long-duration missions. We evaluate our approach with a multitude of different experiments using three different prominent real-world datasets.