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：我们介绍在对偶四元数（表示刚体运动与旋量）语境下李导数的概念。首先，我们将力旋量定义为对偶四元数的形式。接着，我们展示李导数如何帮助理解并联机器人中驱动器对末端执行器的影响，并分别在Stewart平台和索驱动机器人两种情形下作出明确阐述。我们还展示了如何将李导数与牛顿-拉夫逊方法结合，以解决过约束并联驱动器的正运动学问题。最后，我们推导出末端执行器对偶四元数形式的运动方程，其中包含了驱动器惯性的影响。