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Using Lie derivatives with dual quaternions for parallel robots

Stephen Montgomery-Smith, Cecil Shy

TL;DR

The Lie derivative helps understand how actuators affect an end effector in parallel robots, and is made explicit in the two cases case of Stewart Platforms, and cable-driven parallel robots.

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.

Using Lie derivatives with dual quaternions for parallel robots

TL;DR

The Lie derivative helps understand how actuators affect an end effector in parallel robots, and is made explicit in the two cases case of Stewart Platforms, and cable-driven parallel robots.

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.
Paper Structure (18 sections, 5 theorems, 142 equations, 2 figures)

This paper contains 18 sections, 5 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Dual quaternions to represent wrenches
  4. The normalization of a dual quaternion
  5. Notation for three by three matrices
  6. Lie derivatives
  7. Applications to parallel robots
  8. Second lie derivatives
  9. The examples of a Stewart Platform, and a cable-driven parallel robot
  10. Singularity analysis for Stewart Platforms
  11. Forward kinematics
  12. Forward kinematics for Stewart Platforms
  13. Forward kinematics for over-constrained parallel robots
  14. Results of simulations for forward kinematics
  15. Dynamics of the end effector
  16. ...and 3 more sections

Key Result

Theorem 1

If the kinetic energy satisfies equation ke with equation M example holding, and the potential energy $v$ is calculated in the usual manner from the mass of the end effector in a constant gravitational field whose value is $\bm g$ measured with respect to the moving frame, then the equation of motio where and

Figures (2)

  • Figure 1: Schematic of a cable-driven parallel robot.
  • Figure 2: The pulley and attached cable

Theorems & Definitions (20)

  • Theorem 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma approx normalize']}
  • Lemma 2
  • proof
  • proof : Proof that Definition \ref{['lie diff defn']} implies Equation \ref{['lie diff defn 2']}
  • proof : Proof that Equation\ref{['decomp theta']} implies Equation \ref{['decomp partial theta']}
  • proof : Proof of Equations \ref{['lie deriv vector 1']} and \ref{['lie deriv vector 2']}
  • proof : Proof of Equation \ref{['T=L^T']}
  • proof : Proof of Equation \ref{['second lie deriv']}
  • ...and 10 more