论文标题
SO（3）和SE（3）说明刚体旋转和运动的代数及其用于离散整合，梯度下降优化和状态估计的应用
The SO(3) and SE(3) Lie Algebras of Rigid Body Rotations and Motions and their Application to Discrete Integration, Gradient Descent Optimization, and State Estimation
论文作者
论文摘要
经典的数学技术，例如离散集成，梯度下降优化和状态估计（以runge-kutta方法为例，分别是高级kuts-newton最小化以及扩展的卡尔曼过滤器或EKF），因此依赖于线性代数，因此仅适用于在Euclidean Space中的属性，仅适用于实现的状态。本文档讨论了如何修改这些方法，以便将它们应用于非欧几里得状态向量，例如包含旋转的旋转和刚体的完整运动的媒介。为此，本文档提供了对歧管或谎言群体的概念的深入审查，以及它们的切线空间或谎言代数，指数和对数的地图，扰动分析，不确定性和协证的治疗，尤其是对先前提到的calculus方法所需的定义。这些概念专门针对SO（3）和SE（3）谎言组的特定情况，称为R3的特殊正交和特殊的欧几里得群，它们代表了刚体的旋转和动作，描述了它们的各种可能的参数化以及其优势和劣势。
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.
