# Fixed-Time Average Consensus of a Dynamic Event-Triggered Mechanism in a System of Multiple Underactuated Autonomous Underwater Vehicles Based on a Body Frame Spherical Coordinate System

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## Abstract

**:**

## 1. Introduction

- 1.
- The new consensus control objective for a system of multiple underactuated AUVs based on a BF spherical coordinate system is proposed, which is more adaptive to the kinematic and kinetic characteristics of underactuated AUVs. The tracking error kinematics equation is constructed in a BF spherical coordinate system to reduce the accumulated error that results from the transformation between different coordinate systems.
- 2.
- The average consensus strategy is proposed to design a consensus controller for a system of multiple underactuated AUVs. Under the new control objective, the expression between the actuator control variable and the control objective can be constructed, which simplifies the controller design. Moreover, the angle consensus for a system of multiple underactuated AUVs has practical physical significance. Under this control strategy, the states of AUVs can achieve a consensus faster than in the leader–follower strategy.
- 3.
- The dynamic event-triggered mechanism is adopted to reduce the unnecessary triggers resulting from the decrease of the system operation-triggered threshold, which reduces the communication requirements and energy consumption in the consensus controller.

## 2. Mathematical Model and Preliminaries

#### 2.1. Graph Theory

**Lemma 1**

#### 2.2. Fixed-Time Control Stability Theory

#### 2.3. Useful Lemmas

**Lemma 2**

**Lemma 3**

**Lemma 4.**

#### 2.4. The AUV Model

**Remark 1.**

#### 2.5. Average Consensus

**Definition 1.**

**Theorem 1.**

**Proof of Theorem 1.**

#### 2.6. Problem Description

**Definition 2.**

**Remark 2.**

## 3. Multiple Underactuated AUVs System Fixed-Time Average Consensus Controller Design Based on Dynamic Event Triggering Mechanism

#### 3.1. Tracking Error Kinematic Equations in BF Spherical Coordinate System

**Remark 3.**

#### 3.2. Fixed-Time Average Consensus Controller Design

**Theorem 2.**

**Remark 4.**

**Proof of Theorem 2.**

**Remark 5.**

**Remark 6.**

#### 3.3. Dynamic Event-Triggering Mechanism

**Theorem 3.**

**Proof of Theorem 3.**

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Triggering instants of control signals. (

**a**) Event-triggered time in ${\tau}_{u}$ controllers. (

**b**) Event-triggered time in ${\tau}_{q}$ controllers. (

**c**) Event-triggered time in ${\tau}_{r}$ controllers.

Variables | Values |
---|---|

m | 2146.3 kg |

${X}_{\dot{u}}$ | −0.00158 |

${X}_{u\left|u\right|}$ | −0.00686 |

${Y}_{v}$ | −0.06973 |

${Y}_{\dot{v}}$ | −0.08154 |

${Y}_{v\left|\dot{v}\right|}$ | −0.1844 |

${N}_{r}$ | −0.0261 |

${N}_{\dot{r}}$ | −0.00168 |

${N}_{r\left|r\right|}$ | −0.000092 |

${I}_{x}$ | 542 |

${I}_{y}$ | 7580 |

${I}_{z}$ | 7620 |

Initial Position | x(m) | y(m) | z(m) |
---|---|---|---|

AUV1 | −30 | 30 | 5 |

AUV2 | 30 | 30 | 6 |

AUV3 | 0 | 15 | 7 |

AUV4 | 30 | 0 | 8 |

AUV5 | −30 | 0 | 9 |

Initial Position | Event1 | Event2 | Event3 | Time Sample |
---|---|---|---|---|

AUV1 | 4 | 374 | 40 | 400 |

AUV2 | 24 | 374 | 133 | 400 |

AUV3 | 14 | 374 | 100 | 400 |

AUV4 | 5 | 374 | 29 | 400 |

AUV5 | 5 | 374 | 68 | 400 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Zhang, A.; Qin, H.; Jiang, S.
Fixed-Time Average Consensus of a Dynamic Event-Triggered Mechanism in a System of Multiple Underactuated Autonomous Underwater Vehicles Based on a Body Frame Spherical Coordinate System. *J. Mar. Sci. Eng.* **2023**, *11*, 385.
https://doi.org/10.3390/jmse11020385

**AMA Style**

Wang Z, Zhang A, Qin H, Jiang S.
Fixed-Time Average Consensus of a Dynamic Event-Triggered Mechanism in a System of Multiple Underactuated Autonomous Underwater Vehicles Based on a Body Frame Spherical Coordinate System. *Journal of Marine Science and Engineering*. 2023; 11(2):385.
https://doi.org/10.3390/jmse11020385

**Chicago/Turabian Style**

Wang, Zhuo, Ao Zhang, Hongde Qin, and Shukai Jiang.
2023. "Fixed-Time Average Consensus of a Dynamic Event-Triggered Mechanism in a System of Multiple Underactuated Autonomous Underwater Vehicles Based on a Body Frame Spherical Coordinate System" *Journal of Marine Science and Engineering* 11, no. 2: 385.
https://doi.org/10.3390/jmse11020385