# Research on Black-Box Modeling Prediction of USV Maneuvering Based on SSA-WLS-SVM

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## Abstract

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## 1. Introduction

- (1)
- To obtain maneuvering data samples that meet the condition of regression processing.
- (2)
- Determine the black-box regression model and train the model using the dataset obtained above to obtain the nonlinear relationship between the input and output samples.
- (3)
- To predict the output of the test set using the trained black-box model and to compare the output with the actual results to verify the generalization of the model.

## 2. Description of a Ship Motion Prediction Problem

**ρ**(⋅) is a non-linear vector function;

**x**is the input sample matrix composed of input sample vectors ${\mathit{x}}_{\mathit{i}}$, $\mathit{x}={\left[{\mathit{x}}_{\mathbf{1}},\cdots {\mathit{x}}_{\mathit{i}},\cdots ,{\mathit{x}}_{\mathit{N}}\right]}^{T}$; ${\mathit{x}}_{\mathit{i}}$ is the input sample vector, ${\mathit{x}}_{\mathit{i}}=\left[{u}_{i},{v}_{i},{r}_{i},{\delta}_{i}\right]$;

**y**is the output sample matrix composed of the output sample vector, $\mathit{y}={\left[{\mathit{y}}_{\mathbf{1}},\cdots {\mathit{y}}_{\mathit{i}},\cdots ,{\mathit{y}}_{\mathit{N}}\right]}^{T}$; and ${\mathit{y}}_{\mathit{i}}$ is the output sample vector, ${\mathit{y}}_{\mathit{i}}=\left[{\dot{u}}_{i},{\dot{v}}_{i},{\dot{r}}_{i}\right]$. The superscript T represents the transpose of the matrix. $i$ represents the input or output vector corresponding to the ${i}_{th}$ data point, and $i=1,2,\cdots ,N$, $N$ is the number of sample points in sample set S. $u$ is the longitudinal velocity, $v$ is the lateral velocity, and $r$ is the steering angular speed.

## 3. Black-Box Model Training Data Acquisition

- (1)
- A 3-DOF MMG motion simulation in typical scenes was conducted;
- (2)
- A test platform was established for test verification (including filtering of test data);
- (3)
- The test date was compared with the MMG simulation date to verify the accuracy of the MMG model;
- (4)
- The MMG model was used to produce training data.

#### 3.1. MMG Modeling for Simulation Purposes

#### 3.1.1. Static Hydrodynamic (Moment) Modeling

#### 3.1.2. Calculation Object

#### 3.1.3. Compute Grid

#### 3.1.4. Calculation Result of Direct Navigation Resistance

#### 3.1.5. Coupled Hydrodynamics of Oblique Motion and Circling Motion

#### 3.1.6. Hydrodynamic Derivative

#### 3.2. Test Platform Construction

#### 3.3. Test Data Processing

#### 3.4. Validity Verification of Simulation Samples

- (1)
- Test platform data acquisition

- (2)
- Simulation data acquisition

## 4. The SSA-WLS-SVM Algorithm

- (1)
- The sample set $S=\left\{\left({x}_{i},{y}_{i}\right),\left({x}_{i},{y}_{i}\right)\in {\mathit{R}}_{\mathit{N}},i=1,2,\cdots ,N\right\}$ given by WLS-SVM is taken as the benchmark, the regularization parameter $C$ and kernel parameter $K$ of WLS-SVM were taken as the optimization objects of the SSA algorithm. The sparrow population size, iteration times, and initial safety threshold are determined to initialize the SSA optimization algorithm.
- (2)
- Initial or subsequent updated values of $C$ and $K$ are taken as optimization parameters of the LS-SVM algorithm, and the LS-SVM Lagrange multiplier ${\alpha}_{i}$ is weighted to obtain error variables ${e}_{i}={\alpha}_{i}/C$. According to Equation (16), the standard variance ${e}_{i}$ is taken to calculate $\widehat{S}$; then, the weighted coefficient ${v}_{i}$ is determined based on ${e}_{i}$ and $\widehat{S}$, and Equation (13) is solved to obtain the WLS-SVM model corresponding to the $C$ and $K$ values.
- (3)
- The RMSEs of both the predicted value $\widehat{\mathit{y}}$ from the WLS-SVM model and the actual sample value $\mathit{y}$ are used to calculate the adaptive value of each sparrow.
- (4)
- Update the position of the sparrow particles based on Equations (21)–(23) to obtain the fitness value of the sparrow population, and save the optimal individual position and global optimal position in the population.
- (5)
- Check whether the termination conditions have been met or whether the maximum number of update iterations has been reached. If yes, the loop will end and the optimal individual solution, which determines a set of optimal parameters of WLS-SVM, will be returned; otherwise, steps (2)–(4) will continue.
- (6)
- Take the optimal particle value output using the SSA algorithm as the regularization parameter $C$ and kernel parameter $K$ in WLS-SVM. Step (2) is repeated to calculate the WLS-SVM model corresponding to the optimal regularization parameter $C$ and kernel parameter $K$.

## 5. Model Training

#### 5.1. Design and Pre-Processing of Datasets

#### 5.2. Model Training

#### 5.3. Model Generalization Verification

## 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.**Wave shape (Fourier coefficient:

**left**: 0.2,

**middle**: 1,

**right**: 1.6). Different colors represent different wave heights, which gradually decrease from the tail to the distance.

**Figure 6.**Coupled ripple of oblique motion and circumferential motion (

**left**: dimensionless steering angular speed is 0.4, drift angle is 0 degrees;

**middle**: dimensionless steering angular speed is 0.4, drift angle is −10 degrees;

**right**: dimensionless steering angular speed is 0.8, drift angle is −10 degrees).

**Figure 7.**Test Platform Architecture Diagram (

**a**) Establishment of the test coordinate system; (

**b**) Schematic diagram of data transmission on the hardware platform.

**Figure 9.**Comparison results of different data processing methods (

**a**) Comparison results of the resultant speed. (

**b**) Comparison results of the position.

**Figure 10.**Comparison of the prediction and experiment in the 30° turning circle. (

**a**) Speed comparison. (

**b**) Track comparison.

**Figure 12.**Comparison of the prediction with SSA-WLS-SVM and simulation in dataset 1. (

**a**) Longitudinal acceleration. (

**b**) Longitudinal velocity. (

**c**) Lateral acceleration. (

**d**) Lateral velocity. (

**e**) Steering angular acceleration. (

**f**) Steering angular speed. (

**g**) Rudder angle. (

**h**) Track of USV.

**Figure 13.**Comparison of the prediction with SSA-WLS-SVM and simulation in dataset 2. (

**a**) Longitudinal acceleration. (

**b**) Longitudinal velocity. (

**c**) Lateral acceleration. (

**d**) Lateral velocity. (

**e**) Steering angular acceleration. (

**f**) Steering angular speed. (

**g**) Rudder angle. (

**h**) Track of USV.

**Figure 14.**Comparison of the prediction with SSA-WLS-SVM and simulation of the 25° turning maneuvering motion. (

**a**) Longitudinal acceleration. (

**b**) Longitudinal velocity. (

**c**) Lateral acceleration. (

**d**) Lateral velocity. (

**e**) Steering angular acceleration. (

**f**) Steering angular speed. (

**g**) Rudder angle. (

**h**) Track of USV.

Project | Value |
---|---|

Hall | |

${L}_{pp}\left(m\right)$ | 1.5 |

$B\left(m\right)$ | 0.444 |

$d\left(m\right)$ | 0.107 |

${C}_{B}$ | 0.395 |

${x}_{G}\left(m\right)$ | −0.12 |

${z}_{G}\left(m\right)$ | −0.3 |

${I}_{z}\left(kg\xb7{m}^{2}\right)$ | 3.947 |

${I}_{x}\left(kg\xb7{m}^{2}\right)$ | 0.35 |

Propeller | |

${D}_{p}\left(m\right)$ | 0.05 |

${Z}_{P}\left(\mathrm{number}\right)$ | 5 |

Rudder | |

${A}_{R}$ $\left({m}^{2}\right)$ | 0.001675 |

${H}_{R}\left(m\right)$ | 0.05 |

Hydrodynamic Derivative | Calculated Value |
---|---|

${X}_{0}^{\prime}$ | −0.0026 |

${X}_{u}^{\prime}$ | −0.000700 |

${X}_{uu}^{\prime}$ | 0.002500 |

${X}_{uuu}^{\prime}$ | −0.001800 |

${X}_{vv}^{\prime}$ | 0.002744 |

${X}_{rr}^{\prime}$ | −0.002807 |

${X}_{vr}^{\prime}$ | 0.01247 |

${Y}_{v}^{\prime}$ | −0.01471 |

${Y}_{vvv}^{\prime}$ | 0.112200 |

${N}_{v}^{\prime}$ | −0.006399 |

${N}_{vvv}^{\prime}$ | −0.006952 |

${Y}_{r}^{\prime}$ | 0.003013 |

${Y}_{rrr}^{\prime}$ | −0.000467 |

${N}_{r}^{\prime}$ | −0.001708 |

${N}_{rrr}^{\prime}$ | −0.000261 |

${Y}_{vvr}^{\prime}$ | −0.005687 |

${Y}_{vrr}^{\prime}$ | −0.013020 |

${N}_{vvr}^{\prime}$ | −0.015600 |

${N}_{vrr}^{\prime}$ | −0.001047 |

Datasets | Details of Dataset |
---|---|

Training sets | random maneuvering dataset No.1 |

Test sets | random maneuvering dataset No.2 |

Generalized set | random maneuvering dataset No.2 |

25° turning dataset No.3 |

Type of Test | RMSE | CC | ||||
---|---|---|---|---|---|---|

u | v | r | u | v | r | |

random maneuver set1 | 4.92 × 10^{−3} | 5.43 × 10^{−3} | 1.07 × 10^{−2} | 0.99383 | 0.99696 | 0.99382 |

random maneuver set2 | 4.25 × 10^{−3} | 7 × 10^{−3} | 1.08 × 10^{−2} | 0.9880 | 0.9915 | 0.9878 |

25° turning circle maneuver set | 2.15 × 10^{−3} | 1.19 × 10^{−3} | 1.64 × 10^{−3} | 0.9989 | 0.9991 | 0.9990 |

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## Share and Cite

**MDPI and ACS Style**

Song, L.; Hao, L.; Tao, H.; Xu, C.; Guo, R.; Li, Y.; Yao, J.
Research on Black-Box Modeling Prediction of USV Maneuvering Based on SSA-WLS-SVM. *J. Mar. Sci. Eng.* **2023**, *11*, 324.
https://doi.org/10.3390/jmse11020324

**AMA Style**

Song L, Hao L, Tao H, Xu C, Guo R, Li Y, Yao J.
Research on Black-Box Modeling Prediction of USV Maneuvering Based on SSA-WLS-SVM. *Journal of Marine Science and Engineering*. 2023; 11(2):324.
https://doi.org/10.3390/jmse11020324

**Chicago/Turabian Style**

Song, Lifei, Le Hao, Hao Tao, Chuanyi Xu, Rong Guo, Yi Li, and Jianxi Yao.
2023. "Research on Black-Box Modeling Prediction of USV Maneuvering Based on SSA-WLS-SVM" *Journal of Marine Science and Engineering* 11, no. 2: 324.
https://doi.org/10.3390/jmse11020324