  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.
