Stephen Montgomery-Smith and Cecil Shy, Using Lie derivatives with dual quaternions for parallel robots.
We introduce the notion of the Lie derivative in the context of dual quaternions that represent poses 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 case of robot-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 in the actuators. A large part of our methods is an approximation of the normalization of a pure dual quaternion perturbation of the identity, which shows that it is equal up to the second order to the exponential of the pure dual quaternion.
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