# Formation and Flocking Control Algorithms for Robot Networks with Double Integrator Dynamics and Time-Varying Formations

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Problem Statement

**Formation**.- The time-varying formation problem consists of steering all the robots of the team to a certain position to form a desired time-varying geometric shape or pattern while the velocity of each robot converges to the rate of change of the formation. The formation control objective can be defined as$$\begin{array}{c}\hfill \underset{t\to \infty}{lim}{\mathit{p}}_{ij}\left(t\right)-{\mathsf{\delta}}_{ij}\left(t\right)=\mathbf{0}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\underset{t\to \infty}{lim}{\mathit{v}}_{i}\left(t\right)={\dot{\mathsf{\delta}}}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall i,j\in \mathcal{N}\end{array}$$
**Flocking**.- In this collective behavior, all the robots follow a common reference velocity while maintaining a desired formation; thus, the flocking control objective is to achieve$$\begin{array}{c}\hfill \underset{t\to \infty}{lim}{\mathit{p}}_{ij}\left(t\right)-{\mathsf{\delta}}_{ij}\left(t\right)=\mathbf{0}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\underset{t\to \infty}{lim}{\mathit{v}}_{i}\left(t\right)={\mathit{v}}^{\mathrm{d}}\left(t\right)+{\dot{\mathsf{\delta}}}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall i,j\in \mathcal{N}\end{array}$$

#### 2.2. Graph Theory

## 3. Formation and Flocking Controllers

**Proposition**

**1.**

- (i)
- time-varying formation if ${\mathit{v}}^{\mathrm{d}}\left(t\right)=\mathit{0}$;
- (ii)
- flocking behavior in the sense of (3).

**Proof.**

## 4. Time-Varying Formation and Flocking Controllers without Velocity Measurements

**Proposition**

**2.**

- (i)
- time-varying formation if ${\mathit{v}}^{\mathrm{d}}\left(t\right)=\mathit{0}$;
- (ii)
- flocking as defined in (3);
- (iii)
- ${\widehat{\mathit{v}}}_{i}\left(t\right)\to {\mathit{v}}_{i}\left(t\right)$ as $t\to \infty $.

**Proof.**

## 5. Flocking Control with Partial Information

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

## 6. Numerical Results

#### 6.1. Time-Varying Formation Control

#### 6.2. Flocking Control

## 7. Conclusions

## Author Contributions

## Funding

**A1-S-31628**and

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

## Acknowledgments

**A1-S-31628**and

**1030**“Collective behaviors of unmanned vehicles”. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. Special thanks to César Cruz Hernández, who helped us during the elaboration of this work.

## Conflicts of Interest

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**Figure 1.**Block diagram of the proposed control for formation and flocking. In the green block, the external controller is included, while the orange block contains the internal controller; the values of the terms ${a}_{i1}$, ${a}_{ij}$ and ${a}_{iN}$ will change depending on the used graph.

**Figure 3.**Time evolution of the quantities $\left|\left|{\mathit{e}}_{{v}_{x}}\right|\right|$ and $\left|\left|{\mathit{e}}_{{v}_{y}}\right|\right|$.

**Figure 4.**Time evolution of the formation error ${p}_{ij}\left(t\right)-{\mathsf{\delta}}_{ij}\left(t\right)$ in the first simulation, (

**a**) x [$\mathrm{m}$] and (

**b**) y [$\mathrm{m}$].

**Figure 5.**Estimated velocity of each robot, (

**a**) ${\widehat{\mathit{v}}}_{x}$ [$\mathrm{m}/\mathrm{s}$] and (

**b**) ${\widehat{\mathit{v}}}_{y}$ [$\mathrm{m}/\mathrm{s}$].

**Figure 6.**Trajectory of the robots in the plane during the first simulation; □ denotes the initial position of the robots and ◯ denotes the final position of the robots.

**Figure 7.**Estimated desired velocities generated by the distributed observer (24), (

**a**) estimated velocity in the x coordinate, (

**b**) estimated velocity in the y-component. The dotted line represents the desired velocity profile.

**Figure 8.**Estimated velocity of each robot in the flocking simulation, (

**a**) ${\widehat{\mathit{v}}}_{x}$ [$\mathrm{m}/\mathrm{s}$] and (

**b**) ${\widehat{\mathit{v}}}_{y}$ [$\mathrm{m}/\mathrm{s}$].

**Figure 9.**Time evolution of the distance error ${\mathit{p}}_{ij}\left(t\right)-{\mathsf{\delta}}_{ij}\left(t\right)$ in the flocking simulation, (

**a**) ${\mathit{p}}_{x}$ [$\mathrm{m}$] and (

**b**) ${\mathit{p}}_{y}$ [$\mathrm{m}$].

**Figure 10.**Trajectory of the robots in the plane during the flocking simulation; □ denotes the initial position of the robots and ◯ denotes the final position of the robots.

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**MDPI and ACS Style**

Montañez-Molina, C.; Pliego-Jiménez, J.; Martínez-Clark, R.
Formation and Flocking Control Algorithms for Robot Networks with Double Integrator Dynamics and Time-Varying Formations. *Entropy* **2023**, *25*, 834.
https://doi.org/10.3390/e25060834

**AMA Style**

Montañez-Molina C, Pliego-Jiménez J, Martínez-Clark R.
Formation and Flocking Control Algorithms for Robot Networks with Double Integrator Dynamics and Time-Varying Formations. *Entropy*. 2023; 25(6):834.
https://doi.org/10.3390/e25060834

**Chicago/Turabian Style**

Montañez-Molina, Carlos, Javier Pliego-Jiménez, and Rigoberto Martínez-Clark.
2023. "Formation and Flocking Control Algorithms for Robot Networks with Double Integrator Dynamics and Time-Varying Formations" *Entropy* 25, no. 6: 834.
https://doi.org/10.3390/e25060834