# Formation Control of Mobile Robots Based on Pin Control of Complex Networks

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## Abstract

**:**

^{®}Waffle Pi robots. The Turtlebot3

^{®}Waffle Pi is a differential mobile robot with the Robot Operating System (ROS). It has a light detection and ranging (LiDAR) sensor used to compute the collision avoidance control law term. Tests show favorable results on different formations testing on various groups of robots, each composed of a different number of robots. From this work, implementation on other devices can be derived, as well as trajectory tracking once in formation, among other applications.

## 1. Introduction

^{®}robots.

## 2. Mathematical Background

#### 2.1. Complex Networks and Pinning Control

#### 2.2. Differential Drive Robots

## 3. Proposed Scheme for Formation Control of Differential Robots

## 4. Simulation and Physical Implementations

^{®}Waffle Pi robots. Such implementation in both scenarios was held using Python and ROS (Robot Operating System) environments.

#### 4.1. Experiment Description

^{®}Waffle Pi robots. Among the hardware elements of a Turtlebot3

^{®}Waffle Pi robot, it has a Raspberry Pi 3, actuators XL430-W20, LiDAR sensor 360 laser Distance Sensor LDS-01, IMU with gyroscope 3 axis, and accelerometer 3 axis. The robot pose is calculated by the robot giving an odometry ROS message with its position and orientation. It is important to note that the accuracy of this measurement varies in function of the terrain conditions and wear of the robot equipment.

#### 4.1.1. Simulation Test: 4 Robots

#### 4.1.2. Simulation Test: 6 Robots

#### 4.1.3. Simulation Test: 12 Robots

#### 4.2. Experiment Results

^{®}Waffle Pi robots. The results of tests with 4 and 6 robots are compared to the experimental implementation on the physical hardware. The results of the 12 robot tests were left at simulation level.

^{®}Core™ i7-4770 CPU @3GHz under Ubuntu 16.04 operative system.

^{®}Waffle Pi Robots. First, we present the results for the 4-robot test, where Figure 16 shows the trajectory of the robots.

## 5. Results Discussion

#### Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Gazebo 7 of the Open Source Robotics Foundation, used in simulations with four Turtlebot3

^{®}Waffle Pi robots aligned at the beginning of the test. The red, green, and blue lines indicate the X, Y, and Z axes, respectively.

**Figure 22.**Movement of the robots in the plane in the comparison tests without collision avoidance: (

**a**) Scheme proposed in this research; (

**b**) scheme proposed in [21].

**Figure 23.**Trajectories along iterations in the comparison tests without collision avoidance: (

**a**) Scheme proposed in this paper; (

**b**) scheme proposed in [21].

**Figure 24.**Movement of the robots in the plane in the comparison tests with collision avoidance but without an end point reference: (

**a**) Scheme proposed in this paper; (

**b**) scheme proposed in [17].

**Figure 25.**Trajectories along iterations in the comparison tests with collision avoidance but without an end point reference: (

**a**) Scheme proposed in this paper; (

**b**) scheme proposed in [17].

j | 1 | 2 | 3 | 4 | ||||
---|---|---|---|---|---|---|---|---|

i | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} |

1 | 0 | 0 | −D | 0 | 0 | D | −D | D |

2 | D | 0 | 0 | 0 | D | D | 0 | D |

3 | 0 | −D | −D | −D | 0 | 0 | −D | 0 |

4 | D | −D | 0 | −D | D | 0 | 0 | 0 |

j | 1 | 2 | 3 | 4 | 5 | 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

i | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} |

1 | 0 | 0 | −D | 0 | 0 | D | −D | D | 0 | 2D | −D | 2D |

2 | D | 0 | 0 | 0 | D | D | 0 | D | D | 2D | 0 | 2D |

3 | 0 | −D | −D | −D | 0 | 0 | −D | 0 | 0 | D | −D | D |

4 | D | −D | 0 | −D | D | 0 | 0 | 0 | D | D | 0 | D |

5 | 0 | −2D | −D | −2D | 0 | −D | −D | −D | 0 | 0 | −D | 0 |

6 | D | −2D | 0 | −2D | D | −D | 0 | −D | D | 0 | 0 | 0 |

j | 1 | 2 | 3 | 4 | 5 | 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | |

i | ||||||||||||

1 | 0 | 0 | −D | 0 | −2D | 0 | $\frac{D}{2}$ | D | $\frac{-D}{2}$ | D | $-3\frac{D}{2}$ | D |

2 | D | 0 | 0 | 0 | −D | 0 | $3\frac{D}{2}$ | D | $\frac{D}{2}$ | D | $\frac{-D}{2}$ | D |

3 | 2D | 0 | −D | 0 | 0 | 0 | $5\frac{D}{2}$ | D | $3\frac{D}{2}$ | D | $\frac{D}{2}$ | D |

4 | $\frac{-D}{2}$ | −D | $-3\frac{D}{2}$ | −D | $-5\frac{D}{2}$ | −D | 0 | 0 | −D | 0 | −2D | 0 |

5 | $\frac{D}{2}$ | −D | $\frac{-D}{2}$ | −D | $-3\frac{D}{2}$ | −D | D | 0 | 0 | 0 | −D | 0 |

6 | $3\frac{D}{2}$ | −D | $\frac{D}{2}$ | −D | $\frac{-D}{2}$ | −D | −2D | 0 | D | 0 | 0 | 0 |

j | 7 | 8 | 9 | 10 | 11 | 12 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | |

i | ||||||||||||

1 | $-5\frac{D}{2}$ | D | D | 2D | 0 | 2D | −D | 2D | −2D | 2D | −3D | 2D |

2 | $-3\frac{D}{2}$ | D | 2D | 2D | D | 2D | 0 | 2D | −D | 2D | −2D | 2D |

3 | $\frac{-D}{2}$ | D | 3D | 2D | 2D | 2D | D | 2D | 0 | 2D | −D | 2D |

4 | −3D | 0 | $\frac{D}{2}$ | D | $\frac{-D}{2}$ | D | $-3\frac{D}{2}$ | D | $-5\frac{D}{2}$ | D | $-7\frac{D}{2}$ | D |

5 | −2D | 0 | $3\frac{D}{2}$ | D | $\frac{D}{2}$ | D | $\frac{-D}{2}$ | D | $-3\frac{D}{2}$ | D | $-5\frac{D}{2}$ | D |

6 | −D | 0 | $5\frac{D}{2}$ | D | $3\frac{D}{2}$ | D | $\frac{D}{2}$ | D | $\frac{-D}{2}$ | D | $-3\frac{D}{2}$ | D |

j | 1 | 2 | 3 | 4 | 5 | 6 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | |

i | ||||||||||||

7 | $5\frac{D}{2}$ | −D | $3\frac{D}{2}$ | −D | $\frac{D}{2}$ | −D | −3D | 0 | 2D | 0 | D | 0 |

8 | −D | −2D | −2D | −2D | −3D | −2D | $\frac{-D}{2}$ | −D | $-3\frac{D}{2}$ | −D | $-5\frac{D}{2}$ | −D |

9 | 0 | −2D | −D | −2D | −2D | −2D | $\frac{D}{2}$ | −D | $\frac{-D}{2}$ | −D | $-3\frac{D}{2}$ | −D |

10 | D | −2D | 0 | −2D | −D | −2D | $3\frac{D}{2}$ | −D | $\frac{D}{2}$ | −D | $\frac{-D}{2}$ | −D |

11 | 2D | −2D | D | −2D | 0 | −2D | $5\frac{D}{2}$ | −D | $3\frac{D}{2}$ | −D | $\frac{D}{2}$ | −D |

12 | 3D | −2D | 2D | −2D | D | −2D | $7\frac{D}{2}$ | −D | $5\frac{D}{2}$ | −D | $3\frac{D}{2}$ | −D |

j | 7 | 8 | 9 | 10 | 11 | 12 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | T_{x} | T_{y} | |

i | ||||||||||||

7 | 0 | D | $-7\frac{D}{2}$ | D | $5\frac{D}{2}$ | D | $3\frac{D}{2}$ | D | $\frac{D}{2}$ | D | $\frac{-D}{2}$ | D |

8 | $-7\frac{D}{2}$ | 0 | 0 | 0 | −D | 0 | −2D | 0 | −3D | 0 | −4D | 0 |

9 | $-5\frac{D}{2}$ | 0 | −D | 0 | 0 | 0 | −D | 0 | −2D | 0 | −3D | 0 |

10 | $-3\frac{D}{2}$ | 0 | −2D | 0 | D | 0 | 0 | 0 | −D | 0 | −2D | 0 |

11 | $\frac{-D}{2}$ | 0 | −3D | 0 | 2D | 0 | D | 0 | 0 | 0 | −D | 0 |

12 | $\frac{D}{2}$ | 0 | −4D | 0 | 3D | 0 | 2D | 0 | D | 0 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Rios, J.D.; Ríos-Rivera, D.; Hernandez-Barragan, J.; Pérez-Cisneros, M.; Alanis, A.Y.
Formation Control of Mobile Robots Based on Pin Control of Complex Networks. *Machines* **2022**, *10*, 898.
https://doi.org/10.3390/machines10100898

**AMA Style**

Rios JD, Ríos-Rivera D, Hernandez-Barragan J, Pérez-Cisneros M, Alanis AY.
Formation Control of Mobile Robots Based on Pin Control of Complex Networks. *Machines*. 2022; 10(10):898.
https://doi.org/10.3390/machines10100898

**Chicago/Turabian Style**

Rios, Jorge D., Daniel Ríos-Rivera, Jesus Hernandez-Barragan, Marco Pérez-Cisneros, and Alma Y. Alanis.
2022. "Formation Control of Mobile Robots Based on Pin Control of Complex Networks" *Machines* 10, no. 10: 898.
https://doi.org/10.3390/machines10100898