Using Lie derivatives with dual quaternions for parallel robots
This work addresses modeling challenges for engineers and researchers working with parallel robots, but it appears incremental as it builds on existing dual quaternion methods.
The paper tackles the problem of modeling and controlling parallel robots by introducing Lie derivatives in the context of dual quaternions to represent rigid motions and twists, resulting in explicit formulations for Stewart Platforms and cable-driven robots, and deriving equations of motion that include actuator inertia effects.
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.