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 article 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 article provides an in-depth review 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, placing special emphasis on the different possible representations, their tangent spaces, the analysis of perturbations, and in particular the definitions of the Jacobians required to employ the previously mentioned calculus methods.

翻译：经典数学技术如离散积分、梯度下降优化和状态估计（分别以龙格-库塔法、高斯-牛顿最小化和扩展卡尔曼滤波或EKF为例），依赖于线性代数，因此在按文献描述实现时仅适用于属于欧几里得空间的状态向量。本文讨论了如何修改这些方法，使其可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，本文深入回顾了SO(3)和SE(3)李群，即R3的特殊正交群和特殊欧几里得群，它们代表了刚体旋转和运动，特别强调了不同可能的表示、它们的切空间、扰动分析，以及特别是应用前述微积分方法所需的雅可比矩阵的定义。