# Adaptive Non-Strict Trajectory Tracking Control Scheme for a Fully Actuated Unmanned Surface Vehicle

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## Abstract

**:**

## 1. Introduction

## 2. Kinematics and Dynamic Models of USV

## 3. Control Strategy

#### 3.1. Desired Region Description and Error Dynamics

#### 3.2. Control Law Formulation

#### 3.3. Stability Analysis

- Case 1:
- If $\dot{\delta \mathit{\eta}}>0,\text{}\delta \mathit{\eta}0$, or $\dot{\delta \mathit{\eta}}<0,\text{}\delta \mathit{\eta}0$, then $\delta \mathit{\eta}$ will evidently converge to zero gradually.
- Case 2:
- If $\dot{\delta \mathit{\eta}}\ge 0\text{}\mathrm{and}\text{}\delta \mathit{\eta}\ge 0$, then ${[\mathbf{\U0001d4db}\left(t\right)\dot{\delta \mathit{\eta}}+\dot{\mathbf{\U0001d4db}}\left(t\right)\delta \mathit{\eta}+\epsilon {\Xi}_{e}]}^{T}\ge 0$. ${\mathit{\Omega}}_{2}\le 0$ does not hold if and only if $-\left({\mathrm{exp}}^{\left|{\mathit{K}}_{s}\delta \mathit{\eta}\right|}-1-\mathit{\varpi}\right)sgn\left(\delta \mathit{\eta}\right)\ge 0$, that is $\delta \mathit{\eta}\in \left[0,{\mathit{K}}_{s}^{-1}\mathrm{ln}\left(1+\mathit{\varpi}\right)\right]$.
- Case 3:
- If $\dot{\delta \mathit{\eta}}\le 0\text{}\mathrm{and}\text{}\delta \mathit{\eta}\le 0$, then ${[\mathbf{\U0001d4db}\left(t\right)\dot{\delta \mathit{\eta}}+\dot{\mathbf{\U0001d4db}}\left(t\right)\delta \mathit{\eta}+\epsilon {\Xi}_{e}]}^{T}\le 0$. ${\mathit{\Omega}}_{2}\le 0$ does not hold if and only if $-\left({\mathrm{exp}}^{\left|{\mathit{K}}_{s}\delta \mathit{\eta}\right|}-1-\mathit{\varpi}\right)sgn\left(\delta \mathit{\eta}\right)\le 0$; that is, $\delta \mathit{\eta}\in \left[-{\mathit{K}}_{s}^{-1}\mathrm{ln}\left(1+\mathit{\varpi}\right),0\right]$.

## 4. Simulation Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Results of the traditional RC method. (

**a**) Trajectory of the USV using the traditional RC method; (

**b**) Zoom view of the results from −10 to 40 m in the X direction; (

**c**) Zoom view of the results from 790 to 805 m in the X direction; (

**d**) Zoom view of the results from 225 to 298 m in the X direction.

**Figure 5.**Results of the NTTC method. (

**a**) Trajectory of the USV using the NTTC method; (

**b**) Zoom view of the results from −10 to 40 m in the X direction; (

**c**) Zoom view of the results from 790 to 805 m in the X direction; (

**d**) Zoom view of the results from 225 to 298 m in the X direction.

**Figure 10.**Velocities of the USV. (

**a**) Surge velocities of the USV using the RC and NTTC method; (

**b**) Sway velocities of the USV using the RC and NTTC method; (

**c**) Yaw velocities of the USV using the RC and NTTC method.

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

$m/\mathrm{kg}$ | 23.8000 | ${x}_{g}/\mathrm{m}$ | 0.0460 |

${I}_{z}/(\mathrm{kg}\xb7{\mathrm{m}}^{2})$ | 1.7600 | ${X}_{u}$ | −0.7225 |

${X}_{\left|u\right|u}$ | −1.3274 | ${X}_{uuu}$ | −5.8664 |

${Y}_{v}$ | −0.8612 | ${Y}_{\left|v\right|v}$ | −36.2823 |

${Y}_{r}$ | 0.1079 | ${N}_{v}$ | 0.1052 |

${N}_{\left|v\right|v}$ | 5.0437 | ${X}_{\dot{u}}$ | −2.0 |

${Y}_{\dot{v}}$ | −10.0 | ${Y}_{\dot{r}}$ | −0.0 |

${N}_{\dot{v}}$ | −0.0 | ${N}_{\dot{r}}$ | −1.0 |

Method | RMS | ||
---|---|---|---|

${\mathit{\tau}}_{\mathit{u}}/\mathit{N}$ | ${\mathit{\tau}}_{\mathit{v}}\text{}/\mathit{N}$ | ${\mathit{\tau}}_{\mathit{r}}\text{}/\text{}\left(\mathit{N}\mathit{m}\right)$ | |

RC Method | 6.41 | 3.71 | 1.26 |

NTTC Method | 4.65 | 3.45 | 1.14 |

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**MDPI and ACS Style**

Wang, J.; Liu, J.-y.; Yi, H.; Wu, N.-l. Adaptive Non-Strict Trajectory Tracking Control Scheme for a Fully Actuated Unmanned Surface Vehicle. *Appl. Sci.* **2018**, *8*, 598.
https://doi.org/10.3390/app8040598

**AMA Style**

Wang J, Liu J-y, Yi H, Wu N-l. Adaptive Non-Strict Trajectory Tracking Control Scheme for a Fully Actuated Unmanned Surface Vehicle. *Applied Sciences*. 2018; 8(4):598.
https://doi.org/10.3390/app8040598

**Chicago/Turabian Style**

Wang, Jian, Jing-yang Liu, Hong Yi, and Nai-long Wu. 2018. "Adaptive Non-Strict Trajectory Tracking Control Scheme for a Fully Actuated Unmanned Surface Vehicle" *Applied Sciences* 8, no. 4: 598.
https://doi.org/10.3390/app8040598