Classical mathematical techniques such as discrete integration, gradient descent optimization, and state estimation (exemplified by the Runge-Kutta method, Gauss-Newton minimization, and extended Kalman filter or EKF, respectively), rely on linear algebra and hence are only applicable to state vectors belonging to Euclidean spaces when implemented as described in the literature. This document discusses how to modify these methods so they can be applied to non-Euclidean state vectors, such as those containing rotations and full motions of rigid bodies. To do so, this document provides an in-depth review of the concept of manifolds or Lie groups, together with their tangent spaces or Lie algebras, their exponential and logarithmic maps, the analysis of perturbations, the treatment of uncertainty and covariance, and in particular the definitions of the Jacobians required to employ the previously mentioned calculus methods. These concepts are particularized to the specific cases of the SO(3) and SE(3) Lie groups, known as the special orthogonal and special Euclidean groups of R3, which represent the rigid body rotations and motions, describing their various possible parameterizations as well as their advantages and disadvantages.

翻译：经典数学方法如离散积分、梯度下降优化和状态估计（分别以Runge-Kutta方法、Gauss-Newton最小化、扩展卡尔曼滤波或EKF为例）依赖于线性代数，因此按文献所述实现时仅适用于属于欧几里得空间的状态向量。本文讨论如何修改这些方法，使其能够应用于非欧几里得状态向量，例如包含刚体旋转和完整运动的状态向量。为此，本文深入综述了流形或李群的概念，及其切空间或李代数、指数映射和对数映射、扰动分析、不确定性与协方差的处理，特别是应用前述微积分方法所需的雅可比矩阵定义。这些概念被具体化为SO(3)和SE(3)李群的特殊情况，即R3中的特殊正交群和特殊欧几里得群，它们代表刚体旋转和运动，描述了其各种可能的参数化方式及其优缺点。