This paper presents an equivariant filter (EqF) transformation approach for visual--inertial navigation. By establishing analytical links between EqFs with different symmetries, the proposed approach enables systematic consistency design and efficient implementation. First, we formalize the mapping from the global system state to the local error-state and prove that it induces a nonsingular linear transformation between the error-states of any two EqFs. Second, we derive transformation laws for the associated linearized error-state systems and unobservable subspaces. These results yield a general consistency design principle: for any unobservable system, a consistent EqF with a state-independent unobservable subspace can be synthesized by transforming the local coordinate chart, thereby avoiding ad hoc symmetry analysis. Third, to mitigate the computational burden arising from the non-block-diagonal Jacobians required for consistency, we propose two efficient implementation strategies. These strategies exploit the Jacobians of a simpler EqF with block-diagonal structure to accelerate covariance operations while preserving consistency. Extensive Monte Carlo simulations and real-world experiments validate the proposed approach in terms of both accuracy and runtime.

翻译：本文提出一种面向视觉-惯性导航的等变滤波（EqF）变换方法。通过建立具有不同对称性的等变滤波之间的解析联系，该方法实现了系统化的一致性设计与高效实现。首先，我们形式化了从全局系统状态到局部误差状态的映射，并证明该映射可诱导任意两个等变滤波误差状态之间的非奇异线性变换。其次，我们推导了关联线性化误差状态系统及其不可观子空间的变换法则。这些结果产生了一个通用的一致性设计原理：对于任意不可观系统，可通过变换局部坐标图，合成具有状态独立不可观子空间的一致等变滤波，从而避免启发式对称性分析。第三，为减轻因维持一致性所需的非块对角雅可比矩阵带来的计算负担，我们提出了两种高效实现策略。这些策略利用具有块对角结构的简化等变滤波的雅可比矩阵，在保持一致性的同时加速协方差运算。大量蒙特卡洛仿真与真实世界实验从精度与运行时间两方面验证了所提方法的有效性。