# Optimum k-Nearest Neighbors for Heading Synchronization on a Swarm of UAVs under a Time-Evolving Communication Network

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## Abstract

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## 1. Introduction

- Cohesion, which means that the members of the group should stay together. This is achieved if the elements try to reduce the distance between them.
- Separation, the distance between elements should be enough to ensure that collisions between flockmates do not occur.
- Alignment, the members of the group have to match their speed vectors. This, has two requirements, they have to maintain similar linear velocity to stay in the flock, and the elements need to synchronize their orientation, in order to follow the same direction of the group.

## 2. Preliminaries

#### Graph Theory

**Example 1.**

**Example 2.**

## 3. Heading Synchronization

## 4. Optimum k-NN Neighborhood Size for a Group of UAVs

## 5. Discussion

## 6. Conclusions

- There exists a bulk of literature on consensus, that relies on the simple P-like consensus algorithm, some of them combining it with another control technique to improve it benefits. This algorithm is efficient only if the communication graph is connected, or at least contains a spanning tree.
- A simple asymptotic P-like controller, was successfully employed to achieve heading synchronization on a group of up to 100 UAVs.
- For the different scenarios, the statistical evidence shows that seven neighbors were enough to cope with the problem, as observed in nature with flocks of European starlings.
- A small neighborhood, combined with a simple heading synchronization based on graph theory, is recommended to be employed with simple UAVs, especially for the swarm robotics community.
- Further analysis on the communication scheme should be made to implement this solution.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Initial positions and orientations of the 10 UAVs. Each drone is depicted as a different color.

**Figure 7.**Final positions and orientations of the 10 UAVs. Each drone is depicted as a different color.

**Figure 8.**RMS heading errors for the experiment varying the size of the neighborhood over a 100-UAV swarm.

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**MDPI and ACS Style**

Martínez-Clark, R.; Pliego-Jimenez, J.; Flores-Resendiz, J.F.; Avilés-Velázquez, D.
Optimum k-Nearest Neighbors for Heading Synchronization on a Swarm of UAVs under a Time-Evolving Communication Network. *Entropy* **2023**, *25*, 853.
https://doi.org/10.3390/e25060853

**AMA Style**

Martínez-Clark R, Pliego-Jimenez J, Flores-Resendiz JF, Avilés-Velázquez D.
Optimum k-Nearest Neighbors for Heading Synchronization on a Swarm of UAVs under a Time-Evolving Communication Network. *Entropy*. 2023; 25(6):853.
https://doi.org/10.3390/e25060853

**Chicago/Turabian Style**

Martínez-Clark, Rigoberto, Javier Pliego-Jimenez, Juan Francisco Flores-Resendiz, and David Avilés-Velázquez.
2023. "Optimum k-Nearest Neighbors for Heading Synchronization on a Swarm of UAVs under a Time-Evolving Communication Network" *Entropy* 25, no. 6: 853.
https://doi.org/10.3390/e25060853