# Genetic Optimization-Based Consensus Control of Multi-Agent 6-DoF UAV System

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## Abstract

**:**

## 1. Introduction

#### 1.1. Multi-Agent Systems

#### 1.1.1. Consensus Control

#### 1.1.2. Formation Control

#### 1.1.3. Optimization Techniques

#### 1.1.4. Genetic Algorithm

- (1)
- Selection: The selection operation forms a mating pool for the next generation by selecting genes (parents) from the existing generation.
- (2)
- Crossover: The crossover process combines two of the parents to form the next generation (children).
- (3)
- Mutation: The mutation forms new children by applying random changes to the parents.

#### 1.2. Literature Survey

#### 1.3. Motivation and Contributions

- Proposing a new consensus controlling scheme based on optimization techniques for a MAS of 6-degree of freedom (DOF) quadrotors.
- Simulating a leader–follower topology in some cases by adding additional terms to control the geometry of the agents, and if these terms are equalized to zero, the agents goes to the consensus case.

#### 1.4. Notations

#### 1.5. Paper Organization

## 2. Preliminaries and Problem Statement

#### 2.1. Quadrotor Mathematical Model

#### 2.2. Graph Theory

- Determining the number of agents.
- Deriving a graph based on agent communication.
- Deriving an adjacency matrix from the graph.
- Converting the adjacency matrix into a Laplacian matrix.
- Using the Laplacian matrix to design the consensus controller.

#### 2.2.1. Adjacency and Degree Matrices

#### 2.2.2. Laplacian Matrix

#### 2.3. Mathematical Representation of Consensus Control

#### 2.3.1. Single-Integrator MAS

#### 2.3.2. Double-Integrator MAS

#### 2.3.3. Higher-Order MAS

#### 2.4. Problem Statement

## 3. Proposed Controlling Scheme Design

#### 3.1. Nonlinear PID (NLPID) Controller

#### 3.2. Improved Active Disturbance Rejection Control (IADRC)

#### 3.2.1. Design of Linear Extended State Observer (LESO)

#### 3.2.2. Design of Improved Tracking Differentiator (ITD)

#### 3.2.3. Design of Nonlinear PD Controller (NLPD)

#### 3.3. Consensus Control for $x-y$ Position Subsystems

#### 3.4. Consensus Control for $z$ and $\psi $ Subsystems

**Remark**

**1.**

## 4. Simulation Results and Discussion

#### 4.1. Achieving Consensus

#### 4.2. Fixed Triangle Formation

#### 4.3. Fixed Formation Tracking Ramp Trajectory

#### 4.4. Leader Orientation-Based Formation Tracking a Helical Trajectory

#### 4.5. Switching Formation Topology

#### 4.6. Fixed Formation Tracking with Exogenous Disturbances

## 5. Conclusions and Future Aspects

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Coordinate Systems

#### Appendix A.2. Euler Angles

#### Appendix A.3. Model Assumptions

- The quadrotor is rigid.
- The quadrotor structure is symmetric.
- The ground effect is neglected.
- The center of gravity origin and the principal axes coincide with the body frame origin and axes.
- Input forces and torques proportional to the squared angular velocity of the rotors.

#### Appendix A.4. Mathematical Model

#### Appendix A.5. Quadrotor Forces and Moments

#### Appendix A.6. Rotor Dynamics Modeling

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**Figure 11.**Consensus in $\psi $-angle (

**a**) $\psi $-angle time response. (

**b**) Error of the $\psi $ angle.

**Figure 16.**Three-dimensional views of leader orientation-based formation tracking a helical trajectory, (

**a**) Agents position at $t=50$ s. (

**b**) Agents position at $t=100$ s. (

**c**) Agents position at $t=150$ s. (

**d**) Agents position at $t=200$ s.

**Figure 19.**Three-dimensional views of switching formation topology, (

**a**) Agents formation at $t=25$ s. (

**b**) Agents formation at $t=50$ s. (

**c**) Agents formation at $t=75$ s. (

**d**) Agents formation at $t=100$ s.

Parameters | Description | Units |
---|---|---|

$\left[x\text{}y\text{}z\right]$ | Linear position vector | m |

$\left[\varphi \text{}\theta \text{}\psi \right]$ | Angular position vector | Rad |

$\left[u\text{}v\text{}w\right]$ | Linear velocity vector | m/s |

$\left[p\text{}q\text{}r\right]$ | Angular velocity vector | Rad/s |

$\left[{I}_{xx}\text{}{I}_{yy}\text{}{I}_{zz}\right]$ | Moment of inertia vector | kg·m^{2} |

${f}_{t}$ | Total thrust generated by rotors | N |

$\left[{\tau}_{x}\text{}{\tau}_{y}\text{}{\tau}_{z}\right]$ | Control torques | N m |

$\left[{f}_{wx}\text{}{f}_{wy}\text{}{f}_{wz}\right]$ | Wind force vector | N |

$\left[{\tau}_{wx}\text{}{\tau}_{wy}\text{}{\tau}_{wz}\right]$ | Wind torque vector | N m |

$g$ | Gravitational force | m/s^{2} |

$m$ | Total mass | kg |

$c\left(\text{}\right)\equiv \mathrm{cos}\left(\text{}\right),\text{}s\left(\text{}\right)\equiv \mathrm{sin}\left(\text{}\right),$ and $t\left(\text{}\right)\equiv \mathrm{tan}\left(\text{}\right)$ |

Parameter | Value | Description | Unit |
---|---|---|---|

${I}_{xx}$ | $8.5532\times {10}^{-3}$ | Moment of inertia around $x$-axis | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

${I}_{yy}$ | $8.5532\times {10}^{-3}$ | Moment of inertia around $y$-axis | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

${I}_{zz}$ | $1.476\times {10}^{-2}$ | Moment of inertia around $z$-axis | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

$g$ | $9.81\times {10}^{0}$ | Gravitational force | $\mathrm{m}/{\mathrm{s}}^{2}$ |

$m$ | $9.64\times {10}^{-1}$ | Total mass | $\mathrm{kg}$ |

$b$ | $7.66\times {10}^{-5}$ | Thrust coefficient | $\mathrm{N}\xb7{\mathrm{s}}^{2}$ |

$d$ | $5.63\times {10}^{-6}$ | Drag coefficient | $\mathrm{N}\xb7\mathrm{m}\xb7{\mathrm{s}}^{2}$ |

$l$ | $2.2\times {10}^{-1}$ | Motor to center length | $\mathrm{m}$ |

$\mathit{z}$ | $\mathit{\varphi}$ | $\mathit{\theta}$ | $\mathit{\psi}$ | |
---|---|---|---|---|

${\omega}_{0}$ | $300$ | $861.36$ | $671.76$ | $749.05$ |

${b}_{0}$ | $0.5$ | $0.0041$ | $0.005$ | $0.0047$ |

$\mathit{z},\text{}\mathit{\varphi},\text{}\mathit{\theta},\text{}\mathit{\psi}$ | |
---|---|

$\alpha $ | $0.9789$ |

$\beta $ | $2.7936$ |

$\gamma $ | $16.7728$ |

$R$ | $26.5005$ |

$\mathit{z}$ | $\mathit{\varphi}$ | $\mathit{\theta}$ | $\mathit{\psi}$ | |
---|---|---|---|---|

${k}_{11}$ | $32.4800$ | $5.6390$ | $0.51080$ | $0.6993$ |

${k}_{12}$ | $11.4360$ | $0.0764$ | $0.0390$ | $0.2109$ |

${k}_{21}$ | $9.0756$ | $0.7495$ | $0.0660$ | $0.2415$ |

${k}_{22}$ | $0.1518$ | $0.0471$ | $0.0666$ | $0.1024$ |

${\mu}_{1}$ | $0.2814$ | $0.0763$ | $0.5191$ | $0.1272$ |

${\mu}_{2}$ | $0.4234$ | $0.5997$ | $0.7747$ | $0.3760$ |

${\alpha}_{1}$ | $0.9684$ | $0.9591$ | $0.9574$ | $0.9741$ |

${\alpha}_{2}$ | $0.9580$ | $0.9547$ | $1.0034$ | $0.9417$ |

$\mathit{x}$ | $\mathit{y}$ | $\mathit{x}$ | $\mathit{y}$ | ||
---|---|---|---|---|---|

${k}_{11}$ | $3.0735$ | $2.3046$ | ${\mu}_{1}$ | $0.1214$ | $0.1057$ |

${k}_{12}$ | $0.5674$ | $0.1108$ | ${\mu}_{2}$ | $0.9859$ | $0.9744$ |

${k}_{21}$ | $6.0204$ | $4.5269$ | ${\mu}_{3}$ | $0.7656$ | $0.6008$ |

${k}_{22}$ | $2.0076$ | $2.0040$ | ${\alpha}_{1}$ | $0.9106$ | $0.9101$ |

${k}_{31}$ | $2.2409\times {10}^{-4}$ | $2.3828\times {10}^{-4}$ | ${\alpha}_{2}$ | $0.9999$ | $0.9994$ |

${k}_{32}$ | $5.5135\times {10}^{-7}$ | $6.8046\times {10}^{-7}$ | ${\alpha}_{3}$ | $0.9140$ | $0.9137$ |

$x$ | $y$ | ||

${k}_{x}$ | ${\mathcal{K}}_{x}$ | ${k}_{y}$ | ${\mathcal{K}}_{y}$ |

$4.0259$ | $0.9798$ | $3.9732$ | $1.0631$ |

$z$ | $\psi $ | ||

${\mathcal{K}}_{z1}$ | ${\mathcal{K}}_{z1}$ | ${\mathcal{K}}_{\psi 1}$ | ${\mathcal{K}}_{\psi 2}$ |

$\mathrm{14,169.9808}$ | $427.2904$ | $13.8067$ | $0.5709$ |

Consensus Control | |||
---|---|---|---|

ITAE1 | ITAE2 | ITAE3 | |

$x$ | $1.076761$ | $1.009015$ | $1.879853$ |

$y$ | $1.203705$ | $1.177282$ | $2.106002$ |

$z$ | $0.838854$ | $0.838702$ | $0.001418$ |

$\psi $ | $0.005238$ | $0.005962$ | $0.001104$ |

OPI | $0.845325$ |

Initial States | ||||
---|---|---|---|---|

$\mathit{x}$ | $\mathit{y}$ | $\mathit{z}$ | $\mathsf{\psi}$ | |

Leader | $0.001$ | $0.001$ | $0.001$ | $0$ |

Follower 1 | $0.5$ | $0.5$ | $0.8$ | $\pi /4$ |

Follower 2 | $-0.5$ | $-0.5$ | $0.4$ | $-\pi /4$ |

State | Reference Trajectory | Time (s) |
---|---|---|

$x$ | $\mathrm{cos}\left(0.01\pi t\right)$ | $[25,{t}_{f}$ *] |

$y$ | $\mathrm{sin}\left(0.01\pi t\right)$ | $[25,{t}_{f}$ *] |

$z$ | $0.2t$ | $[0,{t}_{f}$ *] |

$\psi $ | $0$ | $[0,{t}_{f}$ *] |

Leader | Follower 1 | Follower 2 | |
---|---|---|---|

${\mathsf{\varrho}}_{x}$ | 0 | $-\mathrm{cos}\left({\theta}_{L}\right)+\mathrm{sin}\left({\theta}_{L}\right)$ | $-\mathrm{cos}\left({\theta}_{L}\right)+\mathrm{sin}\left({\theta}_{L}\right)$ |

${\mathsf{\varrho}}_{y}$ | 0 | $\mathrm{cos}\left({\theta}_{L}\right)-\mathrm{sin}\left({\theta}_{L}\right)$ | $-\mathrm{cos}\left({\theta}_{L}\right)-\mathrm{sin}\left({\theta}_{L}\right)$ |

${\theta}_{L}={\mathrm{tan}}^{-1}\frac{{y}_{L}}{{x}_{L}}$ |

State | Reference Trajectory | Time (s) |
---|---|---|

$x$ | $-u\left(t-50\right)$ | $[110,{t}_{f}$ *] |

$z$ | $u\left(t\right)$ | $[0,{t}_{f}$ *] |

$y,\psi $ | $0$ | $[0,{t}_{f}$ *] |

Leader | Follower 1 | Follower 2 | |
---|---|---|---|

${\mathsf{\varrho}}_{x}$ | 0 | $u\left(t-25\right)+u\left(t-50\right)$ | $u\left(t-25\right)$ |

${\mathsf{\varrho}}_{y}$ | 0 | $1-u\left(t-50\right)$ | $-1+u\left(t-50\right)+u\left(t-75\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Najm, A.A.; Ibraheem, I.K.; Azar, A.T.; Humaidi, A.J.
Genetic Optimization-Based Consensus Control of Multi-Agent 6-DoF UAV System. *Sensors* **2020**, *20*, 3576.
https://doi.org/10.3390/s20123576

**AMA Style**

Najm AA, Ibraheem IK, Azar AT, Humaidi AJ.
Genetic Optimization-Based Consensus Control of Multi-Agent 6-DoF UAV System. *Sensors*. 2020; 20(12):3576.
https://doi.org/10.3390/s20123576

**Chicago/Turabian Style**

Najm, Aws Abdulsalam, Ibraheem Kasim Ibraheem, Ahmad Taher Azar, and Amjad J. Humaidi.
2020. "Genetic Optimization-Based Consensus Control of Multi-Agent 6-DoF UAV System" *Sensors* 20, no. 12: 3576.
https://doi.org/10.3390/s20123576