This article is an exhaustive revision of concepts and formulas related to
quaternions and rotations in 3D space, and their proper use in estimation
engines such as the error-state Kalman filter.
The paper includes an in-depth study of the rotation group and its Lie
structure, with formulations using both quaternions and rotation matrices. It
makes special attention in the definition of rotation perturbations,
derivatives and integrals. It provides numerous intuitions and geometrical
interpretations to help the reader grasp the inner mechanisms of 3D rotation.
The whole material is used to devise precise formulations for error-state
Kalman filters suited for real applications using integration of signals from
an inertial measurement unit (IMU).

本文对于三维空间内的四元数和旋转相关的概念和公式进行详尽的修订，并探讨了它们在估计引擎 (如误差状态 Kalman 滤波器) 中的正确使用。论文深入研究了旋转群及其李群结构，使用四元数和旋转矩阵进行公式推导，并对旋转扰动、导数和积分进行特别关注。论文的整体材料用于设计适用于 IMU 信号集成的实际应用的误差状态 Kalman 滤波器的精确公式。