## 2 Identification Method
### Identification of the Transverse Position  
As shown in Figure 1, stress measurement points are placed on the lower edges of multiple U-ribs within a lane, and the actual number of measurement points varies with the lane width and the U-rib size. Without a loss of generality, six stress measurement points are placed here (N1 to N6), and the strain response of all measurement points is collected when the vehicle drives over the segmental model at different transverse positions. Due to the local force characteristics of the orthotropic bridge deck, the length of strain influence line of the measurement points on the U-rib is approximately the spacing between the front span and the back span of the diaphragm; thus, the entire strain response when the vehicle travels on the three-span spacing of the diaphragm is considered. Meanwhile, the response beyond the range is very small and has little effect on the following analysis. Firstly, the area of the strain response for each measurement point can be calculated based on the estimated vehicle speed, as shown in the following equation:  
\[S_{ij}=v\times\int_{t_{1}}^{t_{2}}\epsilon_{ij}(t)dt, \tag{3}\]  
where \(\epsilon_{ij}\) is the strain response of measurement point \(j\) when the vehicle passes the transverse position \(i\); \(S_{ij}\) is the area of the strain response \(\epsilon_{ij}\); \(t_{1}\) is the moment when the first axle of the vehicle travels to the start point of the strain influence line (section A-A, as shown in Figure 2); and \(t_{2}\) is the moment when the last axle of the vehicle travels to the end point of the strain influence line (section D-D, as shown in Figure 2).  
For the transverse position \(i\) of a vehicle through the model, the strain response area of all measurement points constitutes a vector \(\textbf{B}_{i}=[S_{i1},S_{i2},\cdots,S_{i6}]\), which will vary with the transverse position \(i\) of the vehicle. Before identifying the vehicle's gross weight, the strain response area vectors under vehicle load at different transverse positions  
Figure 3: Cross-correlation function of measurement points N7 and N8.  
are calibrated to obtain the standard strain response area matrix \(\textbf{B}^{C}=[\textbf{B}_{1}^{C},\textbf{B}_{2}^{C},\cdots,\textbf{B}_{n}^{C}]\), where \(n\) is the number of transverse positions. We can obtain the vectors in the strain response area at refined transverse positions by interpolating between the strain response areas at different positions.  
Then, the strain response area vector \(\textbf{B}_{T}=[S_{T1},S_{T2},\cdots,S_{T6}]\) is measured under a vehicle with an unknown transverse position. A cosine similarity \(CI\) index is introduced to estimate the transverse position by comparing vector \(\textbf{B}_{T}\) with the standard value \(\textbf{B}_{i}^{C}\), and is expressed as follows:  
\[CI(i)=\cos(\textbf{B}_{T}\cdot\textbf{B}_{i}^{C})=\frac{\textbf{B}_{T}\cdot \textbf{B}_{i}^{C}}{|\textbf{B}_{T}|\cdot|\textbf{B}_{i}^{C}|}, \tag{4}\]  
The closer the \(CI\) is to 1, the more similar the two vectors are. By finding the vector \(\textbf{B}_{k}^{C}\) when the \(CI\) index is closest to 1, we can estimate the vehicle's transverse position \(k\), which corresponds to the standard value \(\textbf{B}_{k}^{C}\).