The motions of mechanisms can be described in terms of screw coordinates by means of an exponential mapping. The product of exponentials (POE) describes the configuration of a chain of bodies connected by lower pair joints. The kinematics is thus given in terms of joint screws. The POE serves to express loop constraints for mechanisms as well as the forward kinematics of serial manipulators. Besides the compact formulations, the POE gives rise to purely algebraic relations for derivatives wrt. joint variables. It is known that the partial derivatives of the instantaneous joint screws (columns of the geometric Jacobian) are determined by Lie brackets the joint screws. Lesser-known is that derivative of arbitrary order can be compactly expressed by Lie brackets. This has significance for higher-order forward/inverse kinematics and dynamics of robots and multibody systems. Various relations were reported but are scattered in the literature and insufficiently recognized. This paper aims to provide a comprehensive overview of the relevant relations. Its original contributions are closed form and recursive relations for higher-order derivatives and Taylor expansions of various kinematic relations. Their application to kinematic control and dynamics of robotic manipulators and multibody systems is discussed.

翻译：机构运动可通过指数映射用螺旋坐标描述。乘积指数（POE）描述了由低副关节连接的刚体链的位形，其运动学由关节螺旋给出。POE不仅可用于表达机构的闭环约束和串联机械臂的正向运动学，其紧凑的公式形式还导出了关于关节变量的纯代数导数关系。已知关节螺旋（几何雅可比矩阵的列向量）的偏导数由关节螺旋的李括号决定，但鲜为人知的是任意阶导数均可通过李括号紧凑表达。这对机器人和多体系统的高阶正向/逆向运动学及动力学具有重要意义。相关关系虽有报道却分散于文献中且未被充分认知。本文旨在系统综述这些重要关系，其原创贡献包括：各类运动学关系的高阶导数闭式与递推公式及其泰勒展开式，并讨论了它们在机器人操控臂与多体系统运动控制及动力学中的应用。