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：我们引入了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，我们展示了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，并在斯图尔特平台和缆索驱动并联机器人两种情形中进行了明确阐述。我们还展示了如何将李导数与牛顿-拉夫森方法结合，以求解过约束并联驱动器正运动学问题。最后，我们推导了末端执行器以对偶四元数形式表示的运动方程，其中包含了来自驱动器的惯性效应。