# Formation Control for Second-Order Multi-Agent Systems with Collision Avoidance

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Graph Theory

#### 2.2. Saturating and Switching Functions

**Definition 1.**

- $\varphi \left(x\right)=0\iff x=0$;
- $-r\le \varphi \left(x\right)\le r$ for some $r>0$;
- $x\varphi \left(x\right)>0$, $\forall x\ne 0$;
- $0<\frac{\partial \varphi \left(x\right)}{\partial x}<{M}_{1}<\infty $.

**Definition 2.**

- $\psi \left(x\right)=1$ if $x\le a$;
- $\psi \left(x\right)=0$ if $x\ge b$;
- $0<\psi \left(x\right)<1$ if $a<x<b$;
- $-\infty <\frac{\partial \psi \left(x\right)}{\partial x}\le 0$.

#### 2.3. Input-to-State Stability

**Definition 3.**

**Lemma 1 ([39]).**

**Lemma 2 ([39]).**

## 3. Problem Statement

#### 3.1. Control Objective

- (i)
- The agents reach the desired relative positions, that is,$$\underset{t\to \infty}{lim}({z}_{i}\left(t\right)-{z}_{i}{\left(t\right)}^{*})=0,\phantom{\rule{1.em}{0ex}}i=1,\dots ,N;$$
- (ii)
- There are no collisions among agents. Even more, the agents remain at some predefined distance d from each other, that is, $\parallel {z}_{i}\left(t\right)-{z}_{j}\left(t\right)\parallel \ge d$, $\forall t\ge 0$, $i\ne j$;
- (iii)
- Once the agents achieve the desired formation, the geometrical pattern does not move from its current location any more, i. e., ${lim}_{t\to \infty}{v}_{i}\left(t\right)\to 0$, $\forall i\in N$.

#### 3.2. Position Error Dynamics

## 4. Control Design

#### 4.1. Formation Control Strategy

**Theorem 1.**

**Proof.**

#### 4.2. Formation Control Strategy with Collision Avoidance

#### 4.3. Reduced System

**Remark 1.**

#### 4.4. General System

**Remark 2.**

**Theorem 2.**

**Proof.**

## 5. Simulations

#### 5.1. Desired Formations

#### 5.2. Simulation Results and Discussions

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ISS | Input-to-State Stable |

GAS | Globally Asymptotically Stable |

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**Figure 5.**Switching function to turn on/off the repulsive vector fields. A function which satisfy conditions given in Section 2, with the same behaviour, could also be used instead of the proposed one.

**Figure 6.**Distancesbetween any pair of agents from 0 [s] to 100 [s]. The dashed line indicates the minimum allowed distance.

Parameter | Value |
---|---|

$\mu $ | 1 |

$\epsilon $ | 0.6 |

$\lambda $ | 1 |

D | 2.8 |

d | 2 |

a | 10 |

b | 2.4 |

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**MDPI and ACS Style**

Flores-Resendiz, J.F.; Avilés, D.; Aranda-Bricaire, E.
Formation Control for Second-Order Multi-Agent Systems with Collision Avoidance. *Machines* **2023**, *11*, 208.
https://doi.org/10.3390/machines11020208

**AMA Style**

Flores-Resendiz JF, Avilés D, Aranda-Bricaire E.
Formation Control for Second-Order Multi-Agent Systems with Collision Avoidance. *Machines*. 2023; 11(2):208.
https://doi.org/10.3390/machines11020208

**Chicago/Turabian Style**

Flores-Resendiz, Juan Francisco, David Avilés, and Eduardo Aranda-Bricaire.
2023. "Formation Control for Second-Order Multi-Agent Systems with Collision Avoidance" *Machines* 11, no. 2: 208.
https://doi.org/10.3390/machines11020208