# Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Model in the Presence of Static Obstacles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Path-Following Control System

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

_{0}≤ b

_{max}is a bounded bias term. $K,T,{n}_{3}\mathrm{and}{n}_{1}$ are the Nomoto constants. δ is the rudder angle. Notice that, ${n}_{1}=1$ for a stable ship. The parameters can be obtained using the free-running model test in real-time [47]. With the Nomoto model, the sliding surface is defined as

_{d}> 0 is the feedback control gain. η ≥ b

_{max}is a positive design gain [30]. The Lyapunov function can be used to prove that the equilibrium point is globally exponentially stable (GES) (Theorem 4.10 in [29]). The detailed proof can be found in [6,22,48].

_{3}= 0.0669, K = 0.1129. To validate the results, a new ${20}^{\circ}-{20}^{\circ}$ zigzag manoeuvring test was chosen as a test set. This test was not used for the training. The validation result is presented in Figure 5b. The prediction agrees well with the tests. In the training and validation process, the heading angle is the integration of the yaw rate, which is the prediction of the obtained Nomoto model. Therefore, in order to eliminate the accumulated error due to the integration, one step prediction is adopted when calculating the yaw heading.

## 3. Time-Varying Vector Field Guidance Law

_{e}= 0 is uniform semi-global exponential stable (USGES) (Definition 2.7 by Loría and Panteley [31]). In this paper, the function is defined as $\theta (t,{y}_{e})=0.4\left|{y}_{e}\right|+1$ and the function $\mathsf{\Phi}\left(t,{y}_{e}\right)={\left|{y}_{e}\right|}^{0.4\left|{y}_{e}\right|}\ge {e}^{-\frac{2}{5e}}\approx 0.86>0$ is positive and lower-bounded. The cross-track error, ${y}_{e}$, depends on the initial error and then decreases exponentially with time.

## 4. Risk-Based Obstacle Collision Avoidance System

**Definition**

**1.**

## 5. Case Study

#### 5.1. Nonlinear Manoeuvring Model

_{d}= 0.4, η = 1 and λ = 0.1. The time-varying function is chosen as $\theta (t,{y}_{e})=0.4\left|{y}_{e}\right|+1$, which renders the system equilibrium point of the guidance subsystem as USGES.

#### 5.2. Single Static Obstacle

#### 5.3. Multi Static Obstacles

#### 5.4. Collision Avoidance Test Using Ship Model

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**LS-SVM for parameter estimation using 20°−20° zigzag manoeuvring tests: (

**a**) training result; and (

**b**) validation.

**Figure 7.**The proposed collision avoidance system for autonomous surface ships ((

**a**): geometry chart; and (

**b**): the flowchart).

**Figure 8.**The velocity obstacle (VO) with a static obstacle. The green area represents the obstacle. The area surrounded by a dotted elliptical line is the obstacle domain. The area surrounded by a red elliptical line is the conflict position (ConfP).

**Figure 10.**Trajectory of the underactuated ship: (

**a**) round obstacle with collision risk; (

**b**) round obstacle without collision risk; (

**c**) rectangular obstacle with collision risk; and (

**d**) rectangular obstacle without collision risk.

**Figure 11.**Heading angle, surge speed (desired versus true) from the simulations: (

**a**) round obstacle with collision risk; (

**b**) round obstacle without collision risk; (

**c**) rectangular obstacle with collision risk; and (

**d**) rectangular obstacle without collision risk.

**Figure 12.**Rudder angle, cross-track error from the simulations: (

**a**) round obstacle with collision risk; (

**b**): round obstacle without collision risk; and (

**c**) rectangular obstacle with collision risk; (

**d**): rectangular obstacle without collision risk.

**Figure 13.**Path-following simulation with multi static obstacles: (

**a**) round obstacles; and (

**b**) rectangular obstacles.

**Figure 14.**Heading angle, surge speed (desired versus true): (

**a**) multi round obstacle; and (

**b**) multi rectangular obstacle.

**Figure 15.**Rudder angle, cross-track errors: (

**a**) multi round obstacle; and (

**b**) multi rectangular obstacle.

Test a | Test b | Test c | Test d | |
---|---|---|---|---|

Path length (m) | 29.142 | 23.648 | 26.950 | 48.820 |

Computational cost (s) | 1.738 | 1.310 | 1.324 | 2.167 |

Test a | Test b | |
---|---|---|

Path length (m) | 34.075 | 34.576 |

Computational cost (s) | 1.474 | 1.516 |

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**MDPI and ACS Style**

Xu, H.; Hinostroza, M.A.; Guedes Soares, C.
Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Model in the Presence of Static Obstacles. *J. Mar. Sci. Eng.* **2021**, *9*, 652.
https://doi.org/10.3390/jmse9060652

**AMA Style**

Xu H, Hinostroza MA, Guedes Soares C.
Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Model in the Presence of Static Obstacles. *Journal of Marine Science and Engineering*. 2021; 9(6):652.
https://doi.org/10.3390/jmse9060652

**Chicago/Turabian Style**

Xu, Haitong, Miguel A. Hinostroza, and C. Guedes Soares.
2021. "Modified Vector Field Path-Following Control System for an Underactuated Autonomous Surface Ship Model in the Presence of Static Obstacles" *Journal of Marine Science and Engineering* 9, no. 6: 652.
https://doi.org/10.3390/jmse9060652