# Biologically-Inspired Intelligent Flocking Control for Networked Multi-UAS with Uncertain Network Imperfections

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Related Works

#### 1.3. Main Contributions

- The knowledge of system dynamics is not fully or partially required.
- It has the capabilities of overcoming the network-induced delay, handling the uncertainties, and noise/disturbance rejection.
- It is appropriate for real-time implementation due to its low computational complexity (i.e., the developed algorithm is a real-time applicable learning technique).
- It ensures the stability of the system.

## 2. Problem Formulation and Preliminaries

#### 2.1. Flock Modelling

- (i).
- ${u}_{i}^{\alpha}$ is the interaction component between two $\alpha $-agents and is defined as follows:$$\begin{array}{cc}\hfill {u}_{i}^{\alpha}& ={c}_{1}^{\alpha}{\displaystyle \sum _{j\in {N}_{i}^{\alpha}}}{\varphi}_{\alpha}\left({\parallel {q}_{j}\left(t\right)-{q}_{i}\left(t\right)\parallel}_{\sigma}\right){\mathbf{n}}_{i,j}\hfill \\ & +{c}_{2}^{\alpha}{\displaystyle \sum _{j\in {N}_{i}^{\alpha}}}{a}_{ij}\left(q\right)({p}_{j}\left(t\right)-{p}_{i}\left(t\right)),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathbf{n}}_{i,j}& =\frac{{q}_{j}\left(t\right)-{q}_{i}\left(t\right)}{\sqrt{1+\u03f5{\parallel {q}_{j}\left(t\right)-{q}_{i}\left(t\right)\parallel}^{2}}},\hfill \\ \hfill {a}_{ij}\left(q\right)& ={\rho}_{h}({\parallel {q}_{j}\left(t\right)-{q}_{i}\left(t\right)\parallel}_{\sigma}/{r}_{\alpha})\in [0,1],j\ne i,\hfill \end{array}$$
- (ii).
- ${u}_{i}^{\beta}$ is the interaction component between the $\alpha $-agent and an obstacle (named the $\beta $-agent) and is defined as follows:$$\begin{array}{cc}\hfill {u}_{i}^{\beta}& ={c}_{1}^{\beta}{\displaystyle \sum _{k\in {N}_{i}^{\beta}}}{\varphi}_{\beta}\left({\parallel {\widehat{q}}_{i,k}-{q}_{i}\parallel}_{\sigma}\right){\widehat{\mathbf{n}}}_{i,k}\hfill \\ & +{c}_{2}^{\beta}{\displaystyle \sum _{k\in {N}_{i}^{\beta}}}{b}_{i,k}\left(q\right)({\widehat{p}}_{i,k}-{p}_{i}),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\widehat{\mathbf{n}}}_{i,k}& =\frac{{\widehat{q}}_{i,k}-{q}_{i}}{\sqrt{1+\u03f5{\parallel {\widehat{q}}_{i,k}-{q}_{i}\parallel}^{2}}},\hfill \\ \hfill {b}_{i,k}\left(q\right)& ={\rho}_{h}({\parallel {\widehat{q}}_{i,k}-{q}_{i}\parallel}_{\sigma}/{d}_{\beta}).\hfill \end{array}$$
- (iii).
- ${u}_{i}^{\gamma}$ is a goal component that consists of a distributed navigational feedback term and is defined as follows:$$\begin{array}{c}\hfill {u}_{i}^{\gamma}=-{c}_{1}^{\gamma}{\sigma}_{1}({q}_{i}-{q}_{r})-{c}_{2}^{\gamma}({p}_{i}-{p}_{r}),\end{array}$$

**Remark**

**1.**

#### 2.2. Network-Induced Delays

#### 2.3. Brain Emotional Learning-Based Intelligent Controller

#### 2.4. Objectives

## 3. Distributed Intelligent Flocking Control of Networked Multi-UAS Using Emotional Learning

#### 3.1. System Design

#### 3.2. Emotional Signal and Sensory Input Development

#### 3.3. Learning-Based Intelligent Flocking Control

Algorithm 1 : The BELBIC-inspired methodology for distributed intelligent flocking control of networked multi-UAS. |

Initialization: |

Set ${W}_{i}=0$, ${V}_{th}=0$, and ${V}_{i}=0$, for $i=1,\dots ,n$. |

Define $\tau =$ network-induced delay. |

Define $E{S}_{i}=$ Objective function, for $i=1,\dots ,n$. |

for each iteration $t={t}_{s}$ do |

for each agent i do |

Compute $S{I}_{i}={K}_{SI}^{\alpha}{u}_{i}^{\alpha}+{K}_{SI}^{\beta}{u}_{i}^{\beta}+{K}_{SI}^{\gamma}{u}_{i}^{\gamma}$ |

Compute $E{S}_{i}={K}_{ES}^{\alpha}{u}_{i}^{\alpha}+{K}_{ES}^{\beta}{u}_{i}^{\beta}+{K}_{ES}^{\gamma}{u}_{i}^{\gamma}$ |

Compute ${A}_{i}={V}_{i}S{I}_{i}$ |

Compute ${A}_{th}={V}_{th}max\left(S{I}_{i}\right)$ |

Compute $O{C}_{i}={W}_{i}S{I}_{i}$ |

Compute $BE{L}_{i}={\sum}_{i}{A}_{i}-{\sum}_{i}O{C}_{i}$ |

Update ${V}_{i}$ |

Update ${V}_{th}$ |

Update ${W}_{i}$ |

end for |

end for |

#### 3.4. Stability Analysis

**Theorem**

**1.**

- I.
- $\left|\left[1-{K}_{v}{\left(S{I}_{l}\right)}^{2}\right]\right|<1,$
- II.
- $\left|\left[1-{K}_{w}{\left(S{I}_{l}\right)}^{2}\right]\right|<1,$

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

## 4. Simulation Results

#### 4.1. Flocking of UGVs in an Obstacle-Free Environment

#### 4.2. Flocking of UASs in an Obstacle-Free Environment

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Non-Adapting Phase

#### Appendix A.2. Main Proof

- I.
- $E{S}_{l}-\widehat{E{S}_{l}}\ge 0,$
- II.
- $E{S}_{l}-\widehat{E{S}_{l}}<0.$

## Appendix B.

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**Figure 3.**Simulation in an obstacle-free environment. (

**Left**) 150 UGVs randomly distributed in a squared area at $t=0$ s. (

**Right**) at $t=40$ s, the 150 UGVs are flocking and have successfully formed a connected network.

**Figure 5.**Mean Square value of the velocities of all UGVs on the x-axis–(

**Left**) and y-axis–(

**Right**) generated by the overall group of agents when flocking in an obstacle-free environment. The BELBIC-based flocking is presented in dot-dashed red, the MCLPA flocking strategy in [36] in dashed green, and the flocking in [3] in solid blue. Notice that the MSE of the BELBIC-based flocking are smaller, and therefore more appropriate to implement in real-robots.

**Figure 6.**Simulation in an obstacle-free environment. (

**Left**) 50 UASs randomly distributed in a squared area at $t=0$ s; (

**Right**) At $t=40$ s, the 50 UASs are flocking and have successfully formed a connected network.

**Figure 8.**Mean Square value of the velocities of all UAVs on the x-axis–(

**Left**) and y-axis–(

**Right**) generated by the overall group of agents when flocking in an obstacle-free environment. The BELBIC-based flocking is presented in dot-dashed red, the MCLPA flocking strategy in [36] in dashed green, and the flocking in [3] in solid blue. Notice that the MSE of the BELBIC-based flocking are smaller, and therefore more appropriate to implement in real-robots.

**Table 1.**Characteristics of the MSE of all flocking strategies for UGVs in the obstacle-free environment.

Flocking in [3] | MCLPA [36] | BELBIC-Based | |
---|---|---|---|

Mean Value on the x-axis | 0.802 | 0.641 | 0.602 |

Standard Deviation on the x-axis | 6.025 × 10^{−5} | 5.566 × 10^{−5} | 0.814 × 10^{−5} |

Mean Value on the y-axis | 0.371 | 0.245 | 0.176 |

Standard Deviation on the y-axis | 6.869 × 10^{−4} | 3.068 × 10^{−4} | 1.324 × 10^{−4} |

**Table 2.**Characteristics of the MSE of all flocking strategies for UAVs in the obstacle-free environment.

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**MDPI and ACS Style**

Jafari, M.; Xu, H.
Biologically-Inspired Intelligent Flocking Control for Networked Multi-UAS with Uncertain Network Imperfections. *Drones* **2018**, *2*, 33.
https://doi.org/10.3390/drones2040033

**AMA Style**

Jafari M, Xu H.
Biologically-Inspired Intelligent Flocking Control for Networked Multi-UAS with Uncertain Network Imperfections. *Drones*. 2018; 2(4):33.
https://doi.org/10.3390/drones2040033

**Chicago/Turabian Style**

Jafari, Mohammad, and Hao Xu.
2018. "Biologically-Inspired Intelligent Flocking Control for Networked Multi-UAS with Uncertain Network Imperfections" *Drones* 2, no. 4: 33.
https://doi.org/10.3390/drones2040033