# Attitude Synchronization of a Group of Rigid Bodies Using Exponential Coordinates

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Notation

#### 2.2. Kinematic and Dynamics

#### 2.3. Graph Theory

## 3. Attitude Synchronization

**Proposition**

**1.**

**Proof.**

## 4. Simulations

## 5. Conclusions

## Author Contributions

## Funding

**A1-S-31628**and

**1030**“Collective behaviors of unmanned vehicles”.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Swarm of rigid bodies with four elements. The rigid body’s attitude is obtained by projecting the body frame’s axes (${\Sigma}_{\mathcal{B}}$) with the inertia frame’s axes (${\Sigma}_{\mathcal{I}}$).

**Figure 2.**Relation between the exponential coordinates $\mathit{\xi}$ and the rotation matrix $R\in SO\left(3\right)$. The identity element ${I}_{3}$ on $SO\left(3\right)$ is mapped to the origin $\mathit{\xi}=\mathbf{0}$ in the exponential coordinates.

**Figure 3.**Time evolution of the exponential coordinates ${\mathit{\xi}}_{i}\left(t\right)$ with $i=1,2,3,4$; the dashed line denotes the desired attitude ${\mathit{\xi}}_{\mathrm{d}}\left(t\right)$.

**Figure 4.**Time evolution of the attitude tracking error ${\tilde{\mathit{\xi}}}_{i}\left(t\right)$ (rad) with $i=1,2,3,4$.

**Figure 5.**Time evolution of the relative attitude error ${\mathit{\xi}}_{i}\left(t\right)-{\mathit{\xi}}_{j}\left(t\right)$ (rad) for all $i,j\in \mathcal{N}$.

**Figure 6.**Time evolution of the rotation angle ${\theta}_{i}\left(t\right)$ with $i=1,2,3,4$, and the dashed line denotes the desired rotation angle ${\theta}_{\mathrm{d}}\left(t\right)$.

**Figure 7.**Time evolution of rigid bodies’ angular velocity ${\mathit{\omega}}_{i}\left(t\right)$ with $i=1,2,3,4$, and the dashed line denotes the desired angular velocity ${\mathit{\omega}}_{\mathrm{d}}\left(t\right)$.

Index | 1 | 2 | 3 | 4 |
---|---|---|---|---|

${\theta}_{i}\left(0\right)$ | 0° | 45° | 125° | 170° |

${\mathit{n}}_{i}\left(0\right)$ | arbitrary | $\frac{1}{\sqrt{2}}\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$ | $\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$ | $\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right]$ |

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**MDPI and ACS Style**

Sidón-Ayala, M.; Pliego-Jiménez, J.; Cruz-Hernandez, C.
Attitude Synchronization of a Group of Rigid Bodies Using Exponential Coordinates. *Entropy* **2023**, *25*, 832.
https://doi.org/10.3390/e25060832

**AMA Style**

Sidón-Ayala M, Pliego-Jiménez J, Cruz-Hernandez C.
Attitude Synchronization of a Group of Rigid Bodies Using Exponential Coordinates. *Entropy*. 2023; 25(6):832.
https://doi.org/10.3390/e25060832

**Chicago/Turabian Style**

Sidón-Ayala, Miguel, Javier Pliego-Jiménez, and César Cruz-Hernandez.
2023. "Attitude Synchronization of a Group of Rigid Bodies Using Exponential Coordinates" *Entropy* 25, no. 6: 832.
https://doi.org/10.3390/e25060832