# Formation Control of Unmanned Surface Vehicles Using Fixed-Time Non-Singular Terminal Sliding Mode Strategy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Problem Formulation

#### 2.1. Preliminaries

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

#### 2.2. Problem Formulation

## 3. Design of Formation Control Strategy

#### 3.1. Design of Tracking Control Subsystem and Stability Analysis

**Theorem**

**1.**

**Proof of Theorem 1.**

#### 3.2. Design of Formation Control Subsystem and Stability Analysis

**Theorem**

**2.**

**Proof of Theorem 2.**

## 4. Simulation and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Parameter | Value | Parameter | Value |
---|---|---|---|

${m}_{11}$ | $m-{X}_{\dot{u}}$ | ${c}_{23}\left(\nu \right)$ | ${m}_{11}u$ |

${m}_{22}$ | $m-{Y}_{\dot{v}}$ | ${d}_{11}\left(\nu \right)$ | $-{X}_{u}-{X}_{\left|u\right|u}\left|u\right|-{X}_{uuu}{u}^{2}$ |

${m}_{23}$ | $m{x}_{g}-{Y}_{\dot{r}}$ | ${d}_{22}\left(\nu \right)$ | $-{Y}_{v}-{Y}_{\left|v\right|v}\left|v\right|$ |

${m}_{32}$ | $m{x}_{g}-{N}_{\dot{v}}$ | ${d}_{23}\left(\nu \right)$ | $-{Y}_{r}-{Y}_{\left|v\right|r}\left|v\right|-{Y}_{\left|r\right|r}\left|r\right|$ |

${m}_{33}$ | ${I}_{z}-{N}_{\dot{r}}$ | ${d}_{32}\left(\nu \right)$ | $-{N}_{v}-{N}_{\left|v\right|v}\left|v\right|-{N}_{\left|r\right|v}\left|r\right|$ |

${c}_{13}\left(\nu \right)$ | ${m}_{11}-{m}_{23}r$ | ${d}_{33}\left(\nu \right)$ | $-{N}_{r}-{N}_{\left|v\right|r}\left|v\right|-{N}_{\left|r\right|r}\left|r\right|$ |

Parameters | Values | Parameters | Values | Parameters | Values |
---|---|---|---|---|---|

m | 23.8000 | ${Y}_{v}$ | −0.8612 | ${X}_{\dot{\mu}}$ | −2.0 |

${I}_{z}$ | 1.7600 | ${Y}_{\left|v\right|v}$ | −36.2823 | ${Y}_{\dot{v}}$ | −10.0 |

${x}_{g}$ | 0.460 | ${Y}_{r}$ | 0.1079 | ${Y}_{\dot{r}}$ | 0.0 |

${X}_{\mu}$ | −0.7225 | ${N}_{v}$ | 0.1052 | ${N}_{\dot{v}}$ | 0.0 |

${X}_{\left|\mu \right|\mu}$ | −1.3274 | ${N}_{\left|v\right|v}$ | 5.0437 | ${N}_{\dot{r}}$ | −1.0 |

${X}_{\mu \mu \mu}$ | −5.8664 |

Parameters | Values | Parameters | Values |
---|---|---|---|

$m,p$ | 7 | $n,q$ | 9 |

${\lambda}_{2},{\lambda}_{i,2}$ | 3 | ${a}_{1}$ | 9 |

${\lambda}_{1},{\lambda}_{i,1}$ | 2 | ${a}_{2}$ | 5 |

${\lambda}_{0},{\lambda}_{i,0}$ | 0.03 | ${b}_{1}$ | 5 |

$\alpha $ | 3 | ${b}_{2}$ | 7 |

$\beta $ | 3 | ${Z}_{i}$ | $diag(27,27,27)$ |

Parameters | Values | Parameters | Values |
---|---|---|---|

${\eta}_{d}\left(0\right)$ | ${[-4,-2,\pi /4]}^{\mathrm{T}}$ | ${\nu}_{d}\left(0\right)$ | ${[1/2,0,0]}^{\mathrm{T}}$ |

${\eta}_{0}\left(0\right)$ | ${[-2,-3,0]}^{\mathrm{T}}$ | ${\nu}_{0}\left(0\right)$ | ${[0,0,0]}^{\mathrm{T}}$ |

${\eta}_{1}\left(0\right)$ | ${[0,-3,0]}^{\mathrm{T}}$ | ${\nu}_{1}\left(0\right)$ | ${[0,0,0]}^{\mathrm{T}}$ |

${\eta}_{1}\left(0\right)$ | ${[-6,1,0]}^{\mathrm{T}}$ | ${\nu}_{2}\left(0\right)$ | ${[0,2,0]}^{\mathrm{T}}$ |

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**MDPI and ACS Style**

Er, M.J.; Li, Z.
Formation Control of Unmanned Surface Vehicles Using Fixed-Time Non-Singular Terminal Sliding Mode Strategy. *J. Mar. Sci. Eng.* **2022**, *10*, 1308.
https://doi.org/10.3390/jmse10091308

**AMA Style**

Er MJ, Li Z.
Formation Control of Unmanned Surface Vehicles Using Fixed-Time Non-Singular Terminal Sliding Mode Strategy. *Journal of Marine Science and Engineering*. 2022; 10(9):1308.
https://doi.org/10.3390/jmse10091308

**Chicago/Turabian Style**

Er, Meng Joo, and Zhongkun Li.
2022. "Formation Control of Unmanned Surface Vehicles Using Fixed-Time Non-Singular Terminal Sliding Mode Strategy" *Journal of Marine Science and Engineering* 10, no. 9: 1308.
https://doi.org/10.3390/jmse10091308