## 3 Experimental Results
In this experiment, the photos were only acquired at arbitrary moments compared to the measurement using the laser tracker, so the deformation at the time is indicated as a dot in Figure 8. In this figure, the \(X\), \(Y\) and \(Z\) axes are negative values for the left, downward, and camera directions concerning the center of the reflector, respectively, and the opposite directions are positive values. When pointing only to the measurement time-step of the proposed method in Figure 8, the measurement of the laser tracker shows almost no displacement in the \(X\) and \(Z\) directions, while the proposed method showed displacement of about 3 mm. However, in the \(Y\) direction, the difference between the proposed method and the laser tracker is less than 1 mm. As shown in Figure 8, due to the third passenger train having the heaviest weight and the fastest speed among the three types of trains, the passenger train's deflection (\(Y\)-axis displacement) by the laser tracker was the largest. As illustrated in Table 2, the ROP photogrammetry technique exhibits an error of approximately 2 mm. Therefore, the correlation between train weight and speed could not be identified within this error range. This study also extracted and compared only the same time-step points among the laser tracker measurements as the proposed method's measurements, as shown in Figure 9. In the \(Y\) direction, the deformation shapes are substantially similar between the proposed method and laser tracker results, while it is slightly different in the \(X\) direction. Specifically, it shows a relatively large error in the \(Z\) direction. These causes are the low performance of the camera including the calibrated IOPs, target matching errors of feature points on the bridge surface used to determine ROPs, and incomplete time synchronization, even though stereo images from both cameras were taken simultaneously by a remote control. Nevertheless, it was observed that the displacement forms of the proposed method are similar to the results of the laser tracker in the \(X\)- and \(Z\)-axes enlarged to a 0.1 mm scale. Therefore, it seems to be the case that the displacement values of the proposed method are exaggerated because the real size of one pixel is about 1 mm, and the ROPs scale error is caused through the photogrammetric process as mentioned above, but the displacement directivity is relatively accurate. Figure 8: Displacement results of a reflector position in 3D directions by the laser tracker and the proposed method. (**a**) First train; (**b**) second train; (**c**) third train. Table 3 lists the discrepancies between the proposed method and the laser tracker measurement at the same time-step points in the 3D axes. The RMSEs for all types of trains were approximately 0.3-2.3 mm. The absolute maximum error of the proposed method did not exceed 1.5 mm in either the \(X\) or \(Y\) directions, whereas an error of about 3.5 mm occurred in the \(Z\) direction. These results have a similar tendency to the accuracy of the scale-adjusted ROPs presented in Table 2. In this implementation, the one-pixel size of the images is about 1 mm. Figure 9: Enlarged deformation curve in the 3D axes’ directions. (**a**) First train; (**b**) second train; (**c**) third train. Figure 10 shows the 3D displacement time-steps measured using the proposed method at six target locations. Twelve other targets were also measured, but there was no significant difference in the results, which are shown in Figure 10, so they are omitted in this article. Both the top and the bottom in Figure 10 show movements similar to the \(Y\)-axis sag and the \(X\)-\(Z\)-axis twist shown in Figure 9. The three-axis displacement of the six targets also occurred in a similar manner to the laser point results, and the third passenger train exhibited the largest fluctuation range. The sagging of the bridge by the three types of trains of up to approximately 4 mm occurred in the \(Y\)-direction. At the same time, torsion up to about 4 mm also occurred in the \(X\)-\(Z\) axes.