给出四种全局惯性导航滤波器的数学公式，便于对比与实现。
Derivations of Error-State Kalman Filter Kinematics for Globally Applicable Aided Inertial Navigation Systems
- 基于李群对称性构建不变误差滤波器，提升全球导航精度。
- 推导出四种经典与不变滤波器的误差状态方程，适配不同传感器配置。
- 适合做高精度导航系统研发或算法对比的研究者参考。
全球导航系统需要能够处理地球曲率、自转及重力变化的状态估计算法，这些因素在局部导航中通常可忽略。经典误差状态卡尔曼滤波（ESKF）的误差动态依赖于轨迹；而不变ESKF利用李群对称性表示误差，可使误差传播对轨迹无关，适用于群仿射系统。选择标准滤波（位置速度误差在导航帧中加性定义）、左不变滤波（误差在机体帧表示）或右不变滤波（误差在导航/世界帧表示），取决于系统动力学和传感器配置。本文给出了四种适用于全局辅助惯性导航系统的经典与不变ESKF的数学公式，旨在作为系统性参考，用于算法比较与实现。
原文摘要 · Abstract (English)
Global navigation systems require state estimation algorithms that handle Earth's curvature, Earth's rotation, and gravitational variations. These factors can typically be neglected in local navigation algorithms for robots, drones, etc. In classical error-state Kalman Filtering (ESKF) the error state dynamics are trajectory-dependent. Invariant ESKFs utilize Lie Group symmetries to represent the error, which can render error propagation trajectory-independent for group-affine systems. Choosing between a standard filter (where position and velocity errors are defined additively in the navigation frame), a left-invariant filter (where errors are represented in the body frame) and a right-invariant filter (where errors are represented in the navigation/world frame) depends on system dynamics and sensor configuration. This note presents the mathematical formulas for four classical and invariant ESKFs for globally applicable aided inertial navigation systems. It is intended to serve as a systematic reference for comparison and implementation.
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