# Using Lie Derivatives with Dual Quaternions for Parallel Robots

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## Abstract

**:**

## 1. Introduction

## 2. Notation

## 3. Dual Quaternions to Represent Wrenches

## 4. The Normalization of a Dual Quaternion

- If $\eta $ is a unit dual quaternion, then $\widehat{\eta}=\eta $.
- Normalization preserves multiplication, that is, if ${\eta}_{1}$ and ${\eta}_{2}$ are two dual quaternions, then$$\widehat{{\eta}_{1}{\eta}_{2}}={\widehat{\eta}}_{1}{\widehat{\eta}}_{2}.$$

## 5. Notation for Three by Three Matrices

## 6. Lie Derivatives

- If $g\left(\eta \right)$ is linear in $\eta $, then$${\mathcal{L}}_{\theta}g\left(\eta \right)=g\left(\eta \theta \right).$$
- The product rule: if * is any binary operator which is bilinear over the real numbers, such as the product of real numbers, the inner product, the cross product, or the dual quaternion product, then$${\mathcal{L}}_{\theta}({g}_{1}*{g}_{2})={g}_{1}*\left({\mathcal{L}}_{\theta}{g}_{2}\right)+\left({\mathcal{L}}_{\theta}{g}_{1}\right)*{g}_{2}.$$
- The chain rule:$${\mathcal{L}}_{\theta}\left(h({g}_{1},{g}_{2},\cdots ,{g}_{m})\right)=\sum _{i=1}^{m}\frac{\partial}{\partial {g}_{i}}h({g}_{1},{g}_{2},\cdots ,{g}_{m}){\mathcal{L}}_{\theta}{g}_{i}.$$
- Let $\tilde{\mathit{s}}$ be a constant position vector, and $\tilde{\mathit{n}}$ be a constant direction. Let $\mathit{s}$ and $\mathit{n}$ be their corresponding values with respect to the moving frame. Then$$\begin{array}{cc}\hfill {\mathcal{L}}_{\theta}\mathit{s}& =2\left[\begin{array}{cc}\mathsf{S}\left(\mathit{s}\right)& -\mathsf{I}\end{array}\right]\theta ,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{L}}_{\theta}\mathit{n}& =2\left[\begin{array}{cc}\mathsf{S}\left(\mathit{n}\right)& \mathsf{0}\end{array}\right]\theta .\hfill \end{array}$$

## 7. Applications to Parallel Robots

## 8. Second Lie Derivatives

## 9. The Examples of a Stewart Platform, and a Cable-Driven Parallel Robot

#### Singularity Analysis for Stewart Platforms

## 10. Forward Kinematics

#### 10.1. Forward Kinematics for Stewart Platforms

#### 10.2. Forward Kinematics for Over-Constrained Parallel Robots

#### 10.3. Results of Simulations for Forward Kinematics

## 11. Dynamics of the End Effector

**Theorem 1.**

- ${\mathsf{M}}_{e}\dot{\mathit{w}}$ and ${m}_{e}\dot{\mathit{v}}$ are inertial resistance to change of angular and translational velocities.
- ${m}_{e}\mathit{w}\times \mathit{v}$ is the centripetal force required to rotate and move at the same time.
- $\mathit{w}\times \left({\mathsf{M}}_{e}\mathit{w}\right)$ is the precession torque (so that if the moment of inertia is not isotropic, then the body spins in a counter-intuitive manner, see, for example, [35]).
- $${\mathrm{\Lambda}}^{T}{\mathsf{M}}_{0}\left({\mathcal{L}}_{\phi}\mathrm{\Lambda}\right)\phi +{\mathrm{\Lambda}}^{T}{\mathsf{M}}_{0}\mathrm{\Lambda}\alpha =\mathsf{T}{\mathit{f}}_{0}$$$${\mathit{f}}_{0}={\mathsf{M}}_{0}{\mathcal{L}}_{\phi}\left(\mathrm{\Lambda}\phi \right)+{\mathsf{M}}_{0}\mathrm{\Lambda}\alpha .$$
- ${m}_{e}\mathit{g}$ is the force due to gravity.
- All terms containing ${\mathit{r}}_{0}$ are corrections required since the center of gravity isn’t necessarily the same as the origin of the moving frame of reference. They could be derived by first finding the equations of motion when ${\mathit{r}}_{0}=0$, and then applying Equations (27) and (28).

#### Numerical Verification of the Dynamics Equations

## 12. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proofs

**Lemma A1.**

**Proof of Lemma A1.**

**Lemma A2.**

**Proof.**

**Proof that Definition**(41)

**implies Equation**(43).

**Proof that Equation**(47)

**implies Equation**(48).

**Proof of Equations**(53)

**and**(54).

**Proof of Equation**(62).

**Proof of Equation**(64).

**Proof of Equation**(65).

**Proof of Equation**(67).

**Proof of Equation**(69).

**Justification of Equation**(80).

**Justification of Equation**(83).

**Theorem A1.**

**Proof.**

**Lemma A3.**

**Proof.**

**Proof of Theorem 1.**

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Montgomery-Smith, S.; Shy, C.
Using Lie Derivatives with Dual Quaternions for Parallel Robots. *Machines* **2023**, *11*, 1056.
https://doi.org/10.3390/machines11121056

**AMA Style**

Montgomery-Smith S, Shy C.
Using Lie Derivatives with Dual Quaternions for Parallel Robots. *Machines*. 2023; 11(12):1056.
https://doi.org/10.3390/machines11121056

**Chicago/Turabian Style**

Montgomery-Smith, Stephen, and Cecil Shy.
2023. "Using Lie Derivatives with Dual Quaternions for Parallel Robots" *Machines* 11, no. 12: 1056.
https://doi.org/10.3390/machines11121056