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Distributed Finite-Time Coverage Control of Multi-Quadrotor Systems with Switching Topology^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Graph Theory

#### 2.2. Locational Optimization

**Lemma**

**1**

#### 2.3. Quaternion-Based Rotation

## 3. Problem Formulation

## 4. Finite-Time Control Design

#### 4.1. Translation Control with Fixed Topology

**Lemma**

**2**

**.**Let ${x}_{1},{x}_{2},\dots ,{x}_{n}\ge 0$. Then,

**Lemma**

**3**

**.**Let ${x}_{1},{x}_{2},\dots ,{x}_{n}\ge 0$. Then,

**Lemma**

**4**

**.**The equilibrium point of the scalar system

**Remark**

**1.**

**Theorem**

**1**

**.**Let a group of n agents be connected via a fixed undirected connected graph ${\mathcal{G}}_{n}=({\mathcal{V}}_{n},{\mathcal{E}}_{n})$ with agent dynamics defined in (12) and (11). Let two adjacency matrices corresponding to this graph be denoted by ${A}_{\alpha}=\left[{a}_{ij}^{2{n}_{\mathrm{v}}/{m}_{\mathrm{v}}+{n}_{\mathrm{v}}}\right]\in {\mathbb{R}}^{n\times n}$ and ${A}_{\beta}=\left[{a}_{ij}^{2{q}_{\mathrm{v}}/{p}_{\mathrm{v}}+{q}_{\mathrm{v}}}\right]\in {\mathbb{R}}^{n\times n}$, respectively, with ${m}_{\mathrm{v}},{n}_{\mathrm{v}},{p}_{\mathrm{v}},{q}_{\mathrm{v}}$ being positive odd integers satisfying ${m}_{\mathrm{v}}>{n}_{\mathrm{v}}$ and ${p}_{\mathrm{v}}<{q}_{\mathrm{v}}$. Let the corresponding Laplacians ${\mathcal{L}}_{\alpha}$ and ${\mathcal{L}}_{\beta}$ have the smallest nonzero eigenvalues ${\lambda}_{2}^{\alpha}$ and ${\lambda}_{2}^{\beta}$, respectively. Then, there exist some constants ${\kappa}_{1},{\kappa}_{2}>0$ such that the finite-time coverage problem can be solved by employing the coverage control protocol (15) with settling time expressed as

**Proof.**

#### 4.2. Translation Control with Switching Topology

**Theorem**

**2**

**.**Let a group of n agents have the agent dynamics defined in (12) and (11). Let these agents be connected via a switching connected Delaunay graph ${\mathcal{G}}_{n}\left(t\right)=({\mathcal{V}}_{n},{\mathcal{E}}_{n}\left(t\right))$ for all time $t>0$. Let two adjacency matrices correspond to this graph, denoted by ${A}_{\alpha}=\left[{a}_{ij}^{2{n}_{\mathrm{v}}/{m}_{\mathrm{v}}+{n}_{\mathrm{v}}}\right]\in {\mathbb{R}}^{n\times n}$ and ${A}_{\beta}=\left[{a}_{ij}^{2{q}_{\mathrm{v}}/{p}_{\mathrm{v}}+{q}_{\mathrm{v}}}\right]\in {\mathbb{R}}^{n\times n}$, respectively, where ${m}_{\mathrm{v}},{n}_{\mathrm{v}},{p}_{\mathrm{v}},{q}_{\mathrm{v}}$ are positive odd integers satisfying ${m}_{\mathrm{v}}>{n}_{\mathrm{v}}$ and ${p}_{\mathrm{v}}<{q}_{\mathrm{v}}$. Let the corresponding Laplacians ${\mathcal{L}}_{\alpha}$ and ${\mathcal{L}}_{\beta}$ for every time t have the smallest nonzero eigenvalues for all time $t>0$ be ${\lambda}_{2}^{\alpha *}={min}_{t}{\lambda}_{2}^{\alpha}\left({\mathcal{L}}_{\alpha}\left({\mathcal{G}}_{n}\right)\right)$ and ${\lambda}_{2}^{\beta *}={min}_{t}{\lambda}_{2}^{\beta}\left({\mathcal{L}}_{\beta}\left({\mathcal{G}}_{n}\right)\right)$, respectively. Then, there exist some constants ${\kappa}_{1},{\kappa}_{2}>0$ such that the finite-time coverage problem can be solved by employing the coverage control protocol (30) with settling time given by

**Proof.**

#### 4.3. Rotation Control

**Theorem**

**3**

**.**Let the attitude dynamics of a quadcopter be given by (12) and the error vector between the current and desired attitudes be given by (43). Then, given the control protocol (45), there exist some positive constants ${k}_{\omega}$ such that the equilibrium point of the error vector is finite-time stable with settling time given by

**Proof.**

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Coordinate frame of a quadcopter, adapted from our previous work in [30].

**Figure 2.**Quadcopters on Gazebo simulator, adapted from our previous work in [30].

**Figure 3.**Finite-time coverage control simulation with fixed communication topology and uniform information distribution: (

**a**) Trajectories and optimal Voronoi regions; (

**b**) Objective function convergence, red-dashed line refers to the true objective function, blue line refers to the objective function computed by the agents in every iteration; (

**c**) Trajectory errors; (

**d**) Control inputs. Different colours in Subfigures (

**a**), (

**c**) and (

**d**) refer to different trajectories of the quadcopters.

**Figure 4.**Finite-time coverage control simulation with switching communication topology and diagonal-peak information distribution: (

**a**) Trajectories and optimal Voronoi regions; (

**b**) Objective function convergence, red-dashed line refers to the true objective function, blue line refers to the objective function computed by the agents in every iteration; (

**c**) Trajectory errors; (

**d**) Control inputs. Different colours in Subfigures (

**a**), (

**c**) and (

**d**) refer to different trajectories of the quadcopters.

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**MDPI and ACS Style**

Tnunay, H.; Moussa, K.; Hably, A.; Marchand, N.
Distributed Finite-Time Coverage Control of Multi-Quadrotor Systems with Switching Topology. *Mathematics* **2023**, *11*, 2621.
https://doi.org/10.3390/math11122621

**AMA Style**

Tnunay H, Moussa K, Hably A, Marchand N.
Distributed Finite-Time Coverage Control of Multi-Quadrotor Systems with Switching Topology. *Mathematics*. 2023; 11(12):2621.
https://doi.org/10.3390/math11122621

**Chicago/Turabian Style**

Tnunay, Hilton, Kaouther Moussa, Ahmad Hably, and Nicolas Marchand.
2023. "Distributed Finite-Time Coverage Control of Multi-Quadrotor Systems with Switching Topology" *Mathematics* 11, no. 12: 2621.
https://doi.org/10.3390/math11122621