# An Improved Deep Neural Network Model of Intelligent Vehicle Dynamics via Linear Decreasing Weight Particle Swarm and Invasive Weed Optimization Algorithms

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. DNN Modeling for Vehicle’s Longitudinal-Lateral Dynamics

#### 2.1. Vehicle Dynamics Data Acquisition

#### 2.2. DNN Model of the Vehicle

^{−8}, in this study.

## 3. Optimization Algorithms

#### 3.1. Linear Decreasing Weight Particle Swarm Optimization

#### 3.2. Invasive Weed Optimization

## 4. Numerical Results and Errors

#### 4.1. Numerical Results

- -
- The predicted results of DNNs, LDWPSO-DNNs, and IWO-DNNs for longitudinal responses can fit the results of the vehicle multibody model well. The absolute percentage errors of less than 1% are observed.
- -
- The absolute percentage errors of LDWPSO-DNNs and IWO-DNNs for lateral and longitudinal-lateral responses are smaller than that of DNNs. The results verify that LDWPSO and IWO algorithms improve the prediction accuracy of the DNN model.

#### 4.2. Error Analysis

^{2}. The function MAE represents the average of absolute errors, and directly reflects the actual error of predicted results. The second one, MAPE is one of the widely used metrics for evaluating predictive performance, and can be calculated based on MAE easily. RMSE corresponds to arithmetic square root of MSE, which is more intuitive. For these three error functions, the smaller the value is, the better the predicted results are. Lastly, R

^{2}denotes the coefficient of determination, with a range varying from 0 to 1. The model fits well if the value of R

^{2}is closer to 1. Correspondingly, these four error functions are mathematically expressed as:

^{2}values for LDWPSO-DNN and IWO-DNN models are closer to 1. Furthermore, for the final lateral and longitudinal velocities and yaw angle, the MAPE of LDWPSO-DNN and IWO-DNN models drops sharply. While analyzing the error functions of LDWPSO-DNN and IWO-DNN models, we can observe that most of the indicators for IWO-DNN model are better than the LDWPSO-DNN model.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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

Vehicle mass | 1155 kg |

Wheelbase | 2.8 m |

Centroid height | 0.5373 m |

Tire rolling radius | 0.4673 kg |

Stiffness of front absorber | 40,000 N/m |

Stiffness of rear absorber | 35,000 N/m |

Damping of front absorber | 1800 N/(m/s) |

Damping of rear absorber | 1800 N/(m/s) |

Distance from centroid to front axle | 0.7209 m |

Distance from centroid to rear axle | 2.0791 m |

Final Longitudinal Distance | MAE (m) | MAPE (%) | RMSE (m) | ${\mathbf{R}}^{2}$ |
---|---|---|---|---|

DNN | 0.107476 | 0.078521 | 0.135354 | 0.999990 |

LDWPSO-DNN | 0.071606 | 0.049622 | 0.138596 | 0.999990 |

IWO-DNN | 0.062153 | 0.048728 | 0.135021 | 0.999990 |

Final Lateral Distance | MAE (m) | MAPE (%) | RMSE (m) | ${\mathbf{R}}^{2}$ |
---|---|---|---|---|

DNN | 0.002647 | 9.346083 | 0.006920 | 0.998541 |

LDWPSO-DNN | 0.002158 | 4.741057 | 0.003724 | 0.999577 |

IWO-DNN | 0.001912 | 3.201640 | 0.003467 | 0.999634 |

Final Longitudinal Velocity | MAE (m/s) | MAPE (%) | RMSE (m/s) | ${\mathbf{R}}^{2}$ |
---|---|---|---|---|

DNN | 0.013769 | 0.050521 | 0.018431 | 0.999996 |

LDWPSO-DNN | 0.009383 | 0.032383 | 0.015797 | 0.999997 |

IWO-DNN | 0.007688 | 0.025955 | 0.015518 | 0.999997 |

Final Lateral Velocity | MAE (m/s) | MAPE (%) | RMSE (m/s) | ${\mathbf{R}}^{2}$ |
---|---|---|---|---|

DNN | 0.000524 | 7.754304 | 0.001778 | 0.996591 |

LDWPSO-DNN | 0.000341 | 4.861110 | 0.000791 | 0.999325 |

IWO-DNN | 0.000290 | 3.255596 | 0.000848 | 0.999224 |

Yaw Angle | MAE (rad) | MAPE (%) | RMSE (rad) | ${\mathbf{R}}^{2}$ |
---|---|---|---|---|

DNN | 0.000014 | 6.804836 | 0.000040 | 0.998480 |

LDWPSO-DNN | 0.000010 | 3.879418 | 0.000026 | 0.999348 |

IWO-DNN | 0.000008 | 3.176424 | 0.000016 | 0.999734 |

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**MDPI and ACS Style**

Nie, X.; Min, C.; Pan, Y.; Li, Z.; Królczyk, G.
An Improved Deep Neural Network Model of Intelligent Vehicle Dynamics via Linear Decreasing Weight Particle Swarm and Invasive Weed Optimization Algorithms. *Sensors* **2022**, *22*, 4676.
https://doi.org/10.3390/s22134676

**AMA Style**

Nie X, Min C, Pan Y, Li Z, Królczyk G.
An Improved Deep Neural Network Model of Intelligent Vehicle Dynamics via Linear Decreasing Weight Particle Swarm and Invasive Weed Optimization Algorithms. *Sensors*. 2022; 22(13):4676.
https://doi.org/10.3390/s22134676

**Chicago/Turabian Style**

Nie, Xiaobo, Chuan Min, Yongjun Pan, Zhixiong Li, and Grzegorz Królczyk.
2022. "An Improved Deep Neural Network Model of Intelligent Vehicle Dynamics via Linear Decreasing Weight Particle Swarm and Invasive Weed Optimization Algorithms" *Sensors* 22, no. 13: 4676.
https://doi.org/10.3390/s22134676