# Temporal Convolutional Network-Based Axle Load Estimation from Pavement Vibration Data

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## Abstract

**:**

## 1. Introduction

- A pavement vibration acquisition method based on distributed optical vibration sensors (DOVS) is developed;
- A deep learning-based load reconstruction method tailored for pavement vibration data collected by DOVS is proposed.

## 2. Methods

#### 2.1. Data Acquisition

_{i}(n) is the short-time energy of the nth frame, x

_{i}is the vibration signals at channel I, $\omega (t-n)$ is a rectangular windows function expressed as (2), ${\mu}_{i}$ is the mean value of the Gaussian model corresponding to the ith channel, and ${\sigma}_{i}$ is the standard deviation value of the Gaussian model corresponding to the ith channel.

#### 2.2. Data Preprocessing

#### 2.3. Axle Load–Time-History Reconstruction

**X**= [

**x**

^{1},

**x**

^{2}, …,

**x**

^{N}] denote a set of vibration signals, where xi is from the ith vibration sensor, and let

**Y**denote the corresponding load–time-history curve. During the forward propagation step, a subset of

**X**is fed into the TCN. The network then predicts the corresponding load–time-history curve, which is compared with the true load–time-history curve in the training dataset. The difference between them is computed as the error, which is then used to calculate the gradients. The weights of the TCN are updated using the gradients in a process called backpropagation. By repeatedly performing forward propagation followed by backpropagation, the weights of the network are adjusted to minimize the loss.

_{T}filters. There are N

_{B}TCN blocks that use the same number of filters N

_{T}, the same kernel length K

_{T}, and a variable dilation d ∈ {1, 2, 4, …, ${2}^{{N}_{B}}$}. The default parameters are N

_{T}= 32, K

_{T}= 31, and N

_{B}= 12.

#### 2.4. Training Strategies

^{−4}, and we implemented adaptive adjustment of the learning rate. If the validation loss did not decrease for 15 epochs, we reduced the learning rate by a factor of 10.

#### 2.5. Performance Metrics

_{1}is the start time of the one-axis impulse, and T

_{2}is the end time. As there are several axis impulses when a vehicle passes by, T

_{3}is introduced to signify the end time of the last axis impulse.

## 3. Field Testing

#### 3.1. Test Set-Up

#### 3.1.1. Construction on Site

#### 3.1.2. Excitation

#### 3.2. Data Collection

## 4. Results and Discussion

_{T}and TCN blocks N

_{B}were constructed to discuss the influence of the network architectures. Finally, the effect of speed on reconstruction results was discussed.

#### 4.1. Reconstruction Results for the Load–Time History

#### 4.2. Network Architectures

_{T}) and numbers of TCN blocks (N

_{B}). As the network architecture’s impact on the load reconstruction performance of various structures and load types is similar and the loads at Site 2 are more complex and closer to a real-world situation, this study only focuses on Site 2. We constructed and trained these networks with different hyperparameters, and the comparison results are presented in Figure 10.

_{B}, the influence of K

_{T}on recognition performance is diminished. Specifically, the value of K

_{T}-21 results in a smaller PE compared to the values of K

_{T}-31 and K

_{T}-41 under N

_{B}-12 while the opposite trend is observed for MIE. N

_{B}, on the other hand, plays a vital role in reconstructing the impact load–time history, as the network’s fitting ability increases with its depth. However, deeper layers require longer training times. Hence, we select the network with K

_{T}= 31 and N

_{B}= 12 as the optimal architecture, taking both factors into account. Finally, the mean estimation error, computed by averaging the values of PE, MIE, and RMSE, is found to be 10.89%.

#### 4.3. Sensitivity Analysis of Speed

#### 4.4. Comparison with the Different Axle Load Estimation Methods

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Test setup of the 2 test sites. (

**a**) test set-up at site 1 (plan view); (

**b**) test set-up at site 2 (plan view); (

**c**) accelerated pavement testing equipment scene map of site 1; (

**d**) scene map of site 2.

**Figure 5.**The history of the vibration signal measured from the sensing loop and the mobile load under the wheels. (

**a**) speed = 2 m/s, load level 1; (

**b**) speed = 2 m/s, load level 5; (

**c**) speed = 4 m/s, load level 5; (

**d**) speed = 6 m/s, load level 5.

**Figure 6.**The history of the vehicle load and the vibration signal measured from the sensing loop under vehicles’ wheels. (

**a**) Passenger car; (

**b**) Lorry; (

**c**) Truck-I; (

**d**) Truck-II.

**Figure 7.**Site 1 reconstruction results. (

**a**) speed = 2 m/s, load level 1; (

**b**) speed = 2 m/s, load level 5; (

**c**) speed = 4 m/s, load level 5; (

**d**) speed = 6 m/s, load level 5.

**Figure 10.**Performance of different network architectures for estimating axle loads. (

**a**) PE; (

**b**) MIE; (

**c**) RMSE; (

**d**) training time.

Load Level | No. 1 Axis | No. 2 Axis | No. 3 Axis | No. 4 Axis | No. 5 Axis | No. 6 Axis | Average Axle Load |
---|---|---|---|---|---|---|---|

(kg) | (kg) | (kg) | (kg) | (kg) | (kg) | (kg) | |

1 | 2200 | 1690 | 1880 | 2020 | 2240 | 1750 | 1963.33 |

2 | 5150 | 4770 | 4910 | 5080 | 5250 | 4730 | 4981.67 |

3 | 6180 | 5120 | 5990 | 5830 | 6130 | 5630 | 5813.33 |

4 | 6700 | 6720 | 6900 | 6450 | 6990 | 5820 | 6596.67 |

5 | 7800 | 7080 | 7230 | 7470 | 7000 | 6900 | 7246.67 |

No | Load Level | Loading Speed (m/s) | Percentage of Sample to the Total (%) | Number of Samples |
---|---|---|---|---|

1 | 1, 2, 3, 4, 5 | 2 | 15% | 1004 |

2 | 1, 2, 3, 4, 5 | 4 | 40% | 2676 |

3 | 1, 2, 3, 4, 5 | 6 | 45% | 3010 |

No. | Vehicle Type | Axle Type | Percentage of Sample to the Total (%) | Number of Samples |
---|---|---|---|---|

1 | Passenger car | 1-1 * | 46% | 154 |

2 | Lorry | 1-1 * | 12% | 40 |

3 | Truck-I | 1-1 * | 16% | 53 |

4 | Truck-II | 1-2-2 * | 26% | 87 |

Methods for Axle Load Estimation | Annual Life Cycle Cost ($) | Error | Expected Life (Years) |
---|---|---|---|

DOVS-based (presented) | 1000 | ±11.5% | 15 |

Bending plate | 5000 | ±15% | 4 |

Strip WIM (piezoelectric) | 6000 | ±10% | 6 |

Single load cell | 8000 | ±6% | 12 |

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## Share and Cite

**MDPI and ACS Style**

Bian, Z.; Zeng, M.; Zhao, H.; Guo, M.; Cai, J.
Temporal Convolutional Network-Based Axle Load Estimation from Pavement Vibration Data. *Appl. Sci.* **2023**, *13*, 13264.
https://doi.org/10.3390/app132413264

**AMA Style**

Bian Z, Zeng M, Zhao H, Guo M, Cai J.
Temporal Convolutional Network-Based Axle Load Estimation from Pavement Vibration Data. *Applied Sciences*. 2023; 13(24):13264.
https://doi.org/10.3390/app132413264

**Chicago/Turabian Style**

Bian, Zeying, Mengyuan Zeng, Hongduo Zhao, Mu Guo, and Juewei Cai.
2023. "Temporal Convolutional Network-Based Axle Load Estimation from Pavement Vibration Data" *Applied Sciences* 13, no. 24: 13264.
https://doi.org/10.3390/app132413264