We introduce the notion of the Lie derivative in the context of dual quaternions that represent rigid motions and twists. First we define the wrench in terms of dual quaternions. Then we show how the Lie derivative helps understand how actuators affect an end effector in parallel robots, and make it explicit in the two cases case of Stewart Platforms, and cable-driven parallel robots. We also show how to use Lie derivatives with the Newton-Raphson Method to solve the forward kinematic problem for over constrained parallel actuators. Finally, we derive the equations of motion of the end effector in dual quaternion form, which include the effect of inertia from the actuators.

翻译：我们引入对偶四元数（用于表示刚体运动与旋量）框架下的李导数概念。首先以对偶四元数形式定义力螺旋，进而阐明李导数如何揭示并联机器人中执行器对末端执行器的影响机制，并具体给出斯图尔特平台与索驱动并联机器人两种案例的显式表达。进一步展示如何将李导数与牛顿-拉夫森方法结合，求解过约束并联执行器的正向运动学问题。最后导出末端执行器以对偶四元数形式表达的运动方程，其中包含执行器惯性效应的影响。