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.

翻译：尊重底层系统的几何结构并利用其对称性，一直是推导现代惯性导航系统几何滤波器的核心驱动力。尽管现有方法已取得一定成功，但惯性测量单元偏差的显式处理仍具挑战性，这揭示了当前滤波器设计理论中存在的空白。为填补这一空白，本论文基于新兴的等变系统理论，旨在解决并克服现有方法的局限性。目标是识别包含IMU偏差几何处理的新型惯性导航系统对称性，并利用这些对称性设计在精度、收敛速度、鲁棒性和一致性方面超越现有最优方案的滤波算法。本论文运用半直积规则，首次引入惯性导航系统的切群作为能够恰当处理IMU偏差的等变对称性。在此基础上，我们证明了可以推导出具有自主导航误差动力学的等变滤波器算法。所得滤波器性能显著优于当前最优方案。通过对惯性导航系统多种对称性的系统分析，我们形式化地论证了每种滤波器均可通过特定对称性选择推导为等变滤波器，这揭示了对称性在决定滤波器性能方面的根本性作用。本论文深化了对惯性导航系统等变对称性的理解，为新一代等变估计器的开发奠定了基础，标志着向更可靠导航解决方案迈出的重要跨越。