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：我们引入李导数在对偶四元数中的概念，对偶四元数用于表示刚体运动与螺旋运动。首先，我们以对偶四元数形式定义力旋量。接着，我们展示李导数如何帮助理解并联机器人中执行器对末端效应器的影响，并在斯图尔特平台和缆索驱动并联机器人这两种情形中加以明确。我们还展示了如何结合李导数与牛顿-拉夫森方法，求解过约束并联执行器的正运动学问题。最后，我们推导出末端效应器运动方程的对偶四元数形式，其中包含了来自执行器惯性的影响。