# Formation Control of Multiple Underactuated Surface Vessels with a Disturbance Observer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Tracking Strategy

#### 2.1. Dynamics Model of the USV

#### 2.2. Path Tracking Strategy Design

## 3. Controller System Design

#### 3.1. Disturbance Observer Design

#### 3.2. Controller Design

#### 3.2.1. Kinematics Controller

#### 3.2.2. Heading Angle Control

#### 3.2.3. Surge Control

## 4. Simulation and Analysis

_{11}, 0.9m

_{22}, 0.9m

_{33}, 1.1d

_{11}, 1.1d

_{22}, and 1.1d

_{33}. Among the external disturbances acting on the USV, the effect of waves on navigation control is considered dominant. To more accurately represent the real external environmental disturbance encountered, sea state 4 is simulated as the external disturbance of the USV, and the uncertainty of the model is added to the external disturbance as a constant disturbance. The unknown dynamic disturbance is expressed as follows:

^{2}, 0.1 m/s

^{2}, and 0.5°/s

^{2}, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Estimation of external disturbances with different methods. (

**a**) Disturbance observed value ${\widehat{w}}_{eu}$. (

**b**) Error of the ${\widehat{w}}_{eu}$ with different methods. (

**c**) Disturbance observed value ${\widehat{w}}_{er}$. (

**d**) Error of the ${\widehat{w}}_{er}$ with different methods.

**Figure 4.**Path tracking error curves. (

**a**) Longitudinal tracking errors. (

**b**) Lateral tracking errors.

Parameters | Value |
---|---|

Overall length (m) | 5.2 |

m_{11} (kg) | 2403 |

m_{22} (kg) | 3350 |

m_{33} (kg) | 24,896 |

d_{11} (kg/s) | 24.5 + 50|u| |

d_{22} (kg/s) | 580 + 450|v| |

d_{33} (kg/s) | 1420 + 1250|r| |

Parameters | Value | Parameters | Value |
---|---|---|---|

k_{er} | 10 | k_{u} | 2.5 |

k_{eu} | 10 | ${\lambda}_{\phi r}$ | 0.5 |

k_{ev} | 10 | k_{τr} | 0.2 |

Δ(m) | 50 | k_{x} | 0.5 |

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## Share and Cite

**MDPI and ACS Style**

Sun, Z.; Sun, H.; Li, P.; Zou, J.
Formation Control of Multiple Underactuated Surface Vessels with a Disturbance Observer. *J. Mar. Sci. Eng.* **2022**, *10*, 1016.
https://doi.org/10.3390/jmse10081016

**AMA Style**

Sun Z, Sun H, Li P, Zou J.
Formation Control of Multiple Underactuated Surface Vessels with a Disturbance Observer. *Journal of Marine Science and Engineering*. 2022; 10(8):1016.
https://doi.org/10.3390/jmse10081016

**Chicago/Turabian Style**

Sun, Zhiyuan, Hanbing Sun, Ping Li, and Jin Zou.
2022. "Formation Control of Multiple Underactuated Surface Vessels with a Disturbance Observer" *Journal of Marine Science and Engineering* 10, no. 8: 1016.
https://doi.org/10.3390/jmse10081016