Using Lie derivatives with dual quaternions for parallel robots

TL;DR

This paper introduces a novel approach using Lie derivatives with dual quaternions to analyze and solve kinematic and dynamic problems in parallel robots, enhancing understanding and computational methods.

Contribution

It develops a dual quaternion-based Lie derivative framework for parallel robot kinematics and dynamics, including actuator effects and forward kinematic solutions.

Findings

Effective use of Lie derivatives with dual quaternions for parallel robot analysis

Application to Stewart Platforms and cable-driven robots

Derivation of equations of motion including actuator inertia

Abstract

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.