Correction of Misalignment Errors in the Integrated GNSS and Accelerometer System for Structural Displacement Monitoring

Qu, Xuanyu; Ding, Xiaoli; Xu, You-Lin; Yu, Wenkun

doi:https://doi.org/10.1155/2023/4919151

Structural Control and Health Monitoring

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4919151 | https://doi.org/10.1155/2023/4919151

Correction of Misalignment Errors in the Integrated GNSS and Accelerometer System for Structural Displacement Monitoring

Xuanyu Qu,^1,2Xiaoli Ding ,^1,2You-Lin Xu,³and Wenkun Yu⁴

Academic Editor: Tzu-Kang Lin

Received06 Jul 2023

Revised27 Nov 2023

Accepted07 Dec 2023

Published19 Dec 2023

Abstract

Structural health monitoring (SHM) systems are widely deployed to monitor the dynamic behaviors of large civil infrastructures such as bridges and tall buildings. Global Navigation Satellite System- (GNSS-) based technologies are often a key component in such an SHM system considering the unique capability of GNSS in determining real-time displacements. GNSS often integrates with an accelerometer to achieve complementary advantages. However, due to the various error sources in GNSS measurements and accelerometer, accuracies of GNSS and accelerometer fusion results often cannot meet the requirements of SHM. We propose to integrate a multi-antenna GNSS and an accelerometer with an unscented multi-rate Kalman filter (UMRKF-MA) to correct the system misalignment errors between the sensors, aiming to produce a much more accurate real-time displacement measurement technology for monitoring large civil infrastructures. Extensive experiments with datasets gathered using a shaking table have indicated that the proposed method was able to improve the accuracy of real-time displacement measurements by up to about 40–65% compared to some existing approaches, and that a 1 mm level of real-time monitoring of displacements could be achieved with the method. The method has also been applied to process a dataset from a real-world long-span bridge when heavy vehicles passed through the bridge in a loading test and significantly improved results were obtained.

1. Introduction

Civil engineering structures such as buildings, long-span bridges, elevated roads, and dams play critical roles in supporting the daily lives of urban residents. Aging and wearing of the structures may lead to structural failures, resulting in financial and life losses. For example, the Sampoong Department Store in Seoul, South Korea, collapsed on 29^th June 1995 and resulted in the loss of over 500 lives [1]. It is vital in many cases to deploy a structural health monitoring (SHM) system to detect structural defects and to warn of pending risks of structural failure.

Global Navigation Satellite System (GNSS) has become an effective technology for structural health monitoring [2–5]. The technology has a unique advantage in obtaining real-time three-dimensional displacements of a structure [6, 7]. The accuracy of the GNSS-based technology is however often unable to meet the requirements of structural health monitoring [8]. GNSS measurements are generally noisier than most other sensors such as accelerometers, survey total stations, and transducers [9]. Additionally, the data sampling rate of GNSS is often lower relatively [10, 11].

GNSS technology has been commonly integrated with other sensors considering the complementary natures of the sensors to improve the overall performance of the fusion systems [12, 13]. For example, GNSS and accelerometers have been integrated in many cases [14, 15] in both postprocessing mode using, e.g., empirical mode decomposition- (EMD-) based [16] or digital filter-based fusion method [17, 18, 19], and real-time mode using, e.g., multi-rate Kalman filter (MRKF) [20–24]. Paziewski et al. [25] integrated accelerometer measurements and displacements derived from asynchronous GNSS observations to detect the dynamic displacements.

Systematic errors may occur when using a standard Kalman filter to deal with nonlinear states, e.g., earthquake-induced vibrations [26]. An unscented Kalman filter has been proposed to model the nonlinear effects in navigation applications [27, 28]. In addition, very limited studies have addressed errors caused by the misalignment between the coordinate frames of GNSS and accelerometers (referred to as misalignment errors hereafter) in an SHM system. The coordinate frame of the GNSS is fixed in space while that of the accelerometer is not, and it varies with the attitude of the structural element that the accelerometer is attached on. Therefore, the three axes of the accelerometer are normally not aligned well with those of the GNSS even though both instruments are attached to the same structural element. The errors become especially significant when the structure rotates due to, e.g., strong winds and heavy vehicle load. The errors can therefore significantly affect the results of the data fusion. A high-pass filter can be adopted considering the fact that the misalignment errors usually exhibit low-frequency characteristics [31, 32]. The technique however may lead to the loss of some useful low-frequency displacement information [31].

Since the misalignment error is caused by the tilt of the structural element, it is theoretically possible to eliminate this misalignment error by determining the rotation angles of a structure. Two different frames can be connected by rotation in the direction of the coordinate axis [32]. Although a gyroscope can be adopted to record the rotation angles in real time, this has rarely been done in practice possibly due to reasons such as additional equipment cost. In this study, we address the misalignment problem in the integrated GNSS and accelerometer system using a multi-antenna (MA) GNSS that is capable of recording the precise attitude information of a structure. Typical applications of an MA-GNSS for attitude determination have been repeatedly demonstrated over the years in, e.g., marine surveying, navigation (e.g., cruise, vehicle, and drones), and seismology [33], but have been rarely considered in an SHM system. Furthermore, the known geometry of an MA-GNSS can be used as a constraint to improve attitude and positioning accuracy (i.e., precision single point positioning (PPP) and relative positioning). For example, Wu et al. [34] proposed a relative positioning model for an MA-GNSS to improve the ambiguity fixing rate and the accuracy of positioning.

We propose a new method that jointly uses a multi-antenna GNSS (MA-GNSS), an accelerometer, and an unscented multi-rate Kalman filter (UMRKF) for correcting the system misalignment errors and for high-accuracy dynamic structural displacement monitoring. The method also eliminates potential nonlinear systematic errors in the system. Data generated with a shaking table that simulates both random and sinusoidal displacements and from a large cable-stayed bridge in Hong Kong during a loading test will be used to validate the effectiveness of the proposed method.

2. Methodology

A multi-antenna GNSS and an accelerometer are combined in this study to achieve high-accuracy displacement measurement of civil infrastructures. The multi-antenna GNSS is deployed to obtain precise displacement and attitude information of a structure where the known distances between the GNSS antennas are used as constraints in the GNSS solutions. A prototype of such an MA-GNSS is given in Section 3.1. The attitude information is used to correct the misalignment error of the accelerometer. The corrected acceleration measurements will then be fused with the displacement measurements derived with the MA-GNSS by the UMRKF to improve displacement measurements.

2.1. Multi-Rate Kalman Filter

The three-dimensional (3D) displacement measurements from the GNSS and the acceleration measurements from the accelerometer data are fused with the multi-rate Kalman filter. Since the sampling rate of the accelerometer is typically much higher than that of the GNSS, the multi-rate Kalman filter updates the state parameters at the data sampling rate of the accelerometer while the GNSS measurements are only used when a new epoch of GNSS measurements is available [20]. The state vector of the Kalman filter iswhere and are the 3D displacement and velocity vectors of a monitoring point, respectively. The dynamic system state is written as [20]where is the transition matrix of the state vector and is a design matrix, defined aswhere and are the identity and zero matrices, respectively, and is the sampling interval of the accelerometer data. is the vector of 3D accelerations; is the noise matrix.where is the covariance matrix of the predicted states, which can be written aswhere is the noise factor of the acceleration measurements.

The observation equation of the MRKF iswhere is the vector of observations that contains the 3D displacements from the GNSS observations; is the design matrix; and is the variance-covariance matrix of the observations and defined aswhere is the noise factor of the GNSS displacement measurements in an SHM system.

2.2. Unscented Multi-Rate Kalman Filter

The unscented multi-rate Kalman filter (UMRKF) is a variation of the MRKF that considers nonlinear state estimation based on the use of the unscented Kalman filter (UKF) [35]. For epoch , a nonlinear state system consisting of random variables with white Gaussian noise and observations with white Gaussian noise can be described aswhere is the nonlinear state transition equation and is the nonlinear observation equation. The steps of the UKF algorithm for estimating the n-dimensional state vector at time are as follows. First, sample points (sigma point set) are obtained first based on the unscented transformation (UT). Second, the predicted state vector and its covariance matrix are then calculated [27]:where is the ^th sigma point of the state vector and is the weight of the sigma point. Third, the predicted observation vector and its covariance as well as the cross-covariance matrix are calculated:where is the ^th sigma point of the predicted observation vector and is the weighted mean of the observation vector. Finally, the estimated state vector , its covariance , and the UKF gain matrix can be updated as [35]

2.3. Correction of Misalignment Errors Based on Multi-Antenna GNSS Observations

Figure 1 shows the two coordinate frames, i.e., the bridge reference frame (BRF, ) and the accelerometer reference frame (ARF, ). The former is defined according to the bridge geometry ( being horizontal and in the direction of the bridge alignment, being horizontal and perpendicular to , and in the vertical direction) while the latter refers to a coordinate system as defined by the three axes of the accelerometer. In general, the two coordinate frames are aligned. However, when the accelerometer experiences tilting or torsion, the ARF changes to .

We define the rotation angles around the y-, x-, and z-axes as pitch (), roll (), and yaw (), respectively. are connected to the BRF and the ARF:where and denote the 3D displacements in x-, y- and z-directions, respectively. The subscripts and represent the ARF and BRF, respectively. is an orthogonal transformation matrix. . and are cosine and sine operators, respectively. In general, the rotation angles of a civil engineering structure are small. Therefore equation (13) can be replaced with

Equation (12) can then be written as

We assume that the ARF and BRF of an SHM system are initially in fine alignment. The accelerometer data can be converted from ARF to BRF and integrated with the GNSS measurements based on equations (12)–(15). The observation equation for the acceleration measurements can then be obtained as shown in the following equation [14, 31]:where, and denote the 3D acceleration measurements in the BRF and ARF, respectively. is the vector of acceleration residuals that are assumed to be zero-mean white Gaussian noise. is the acceleration due to gravity.

It can be seen from the above discussion that rotation angles are needed to calculate the 3D accelerations in the BRF. We propose to use a multi-antenna GNSS to determine the rotation angles. Rotation matrix (see equation (15)) can be estimated based on the least squares principle [36]:where = and = are the coordinates of the ith GNSS antenna in the BRF and ARF, respectively, represents the number of a GNSS antenna, and is a positive weighting parameter.

2.4. Integration of Multi-Antenna GNSS and Accelerometer Measurements Based on UMRKF

The UMRKF is used to fuse the multi-antenna GNSS and accelerometer measurements to achieve high-accuracy measurements of displacements. The state vector in equation (1) is first extended into the following:where, , and are the displacements, velocities, and rotation angles about x-, y-, and z-axes, respectively. The transition matrices and in equation (2) accordingly becomewhere and are identity and zero matrices, respectively; we assume that the rotation angles vary over time and behave as random-walk variables:where denote the process noise. The observations from the MA-GNSS and the observation matrix arewhere is the observation from the MA-GNSS including the 3D displacements and three Euler angles and is the design matrix. The overall framework of the proposed method is shown in Figure 2.

3. Experiments with Data from a Shaking Table

3.1. Experimental Setup

Several simulation tests were investigated using a shaking table set up on the terrace garden of Block Z on the campus of the Hong Kong Polytechnic University. As shown in Figure 3, a shaking table was used with a four-antenna GNSS (Trimble R12 and Leica GR25) and a MEMS (micro-electromechanical system) accelerometer (model: TE Connectivity 4630a-005) to simulate and record the displacements. The shaking table was capable of generating several types of displacements in horizontal direction with a peak-to-peak displacement range of 150 mm. To record the actual displacements of the shaking table, a laser transducer (model: Keyence LK-503) was deployed parallel to the shaking direction of the shaking table, and a signal reflector was fixed on the shaking table edge to obtain the high-accuracy (0.01 mm) displacements of the shaking table.

The MA-GNSS consisted of two sets of Trimble R12 geodetic receivers (ANT2 and ANT3 in Figure 3(a)), one set of Leica GR25 GNSS geodetic receivers (ANT1 in Figure 3(a)), and one set of on-board Septentrio GNSS receiver (ANT4 in Figure 3(a)). The distances between ANT1, ANT2, and ANT3 were 1.2 m and antenna ANT4 was mounted on the center. The GNSS antennas were mounted on a rigid triangular platform. All the other sensors used in the study were synchronized to the GPS time. The ARF of the accelerometer was initially aligned with the BRF and is shown in Figure 3(a) (blue lines). The GNSS reference station (model: Trimble Net R9) was set up about 50 m away from the shaking table to provide correction information.

The dynamic response of a structure induced by different loadings, e.g., temperature, crossing winds, traffic flows, and earthquakes, may exhibit different waveforms [37]. Two typical types of displacement signals were simulated in the experiments, i.e., a random signal and a 1 Hz sinusoidal signal with an amplitude of 1.9 cm. The displacements simulated by the shaking table were measured by the MA-GNSS, the accelerometer, and the laser transducer, with the sampling rate of 20 Hz, 200 Hz, and 200 Hz, respectively. A GNSS reference station was set up on the roof top of the same building to enable RTK (real-time kinematic) GNSS data processing. The details of RTK parameters are shown in Table 1.

In the experiments, the shaking table was first kept static to obtain the reference coordinates. The motion signals were then uploaded onto the shaking table system. The shaking table was tilted for some small angles in the northwest direction by manually adjusting one of the leveling screws located on the corner of the platform to generate the misalignment errors. The tilt angle of the platform was first raised slowly to a certain angle (less than 2°) and then lowered until it was close to the initial level. Six calculated schemes were used for comparison: Scheme 1: displacements from GNSS observations only Scheme 2: displacements from accelerometer measurements by double integration after a high-pass filter Scheme 3: displacements from GNSS and accelerometer data fusion based on the EMD model (EMD hereafter) proposed by Chan et al. [16] Scheme 4: displacements from GNSS and accelerometer data fusion based on conventional MRKF (CMRKF) Scheme 5: displacements from multi-antenna GNSS and accelerometer data fusion based on conventional MRKF (CMRKF-MA) Scheme 6: displacements from multi-antenna GNSS and accelerometer data fusion based on unscented MRKF (UMRKF-MA)

It should be noted that as demonstrated in Chan et al. [16], to obtain the fusion results using Scheme 3, another static GNSS monitoring station is needed to mitigate the background noise of the GNSS. In this study, a high-pass Butterworth filter [39] is only applied in Scheme 2, with a cutoff frequency of 0.05 Hz to mitigate the low-frequency effects.

3.2. Experiment with Sinusoidal Displacements

Sinusoidal displacements in the horizontal direction were simulated by the shaking table and recorded by the MA-GNSS, the accelerometer (ACC), and the laser transducer (True). Displacements were then calculated based on the six computational schemes mentioned above. The displacements from the GNSS data only and the EMD fusion solutions were 20 Hz and those from the other four solutions were 200 Hz. The calculated displacements were compared with the laser measurements (black lines and used as the truth). The errors in the displacements, i.e., differences between the laser measurements and the calculated displacements from the six computation schemes, are shown in Figure 4 where the errors (blue lines) were shifted by -5 cm for visualization clarity. The RMSE (root-mean-square error) and the peak errors were calculated to evaluate the performances of the different methods as shown in Table 2.

As shown in Figure 4, the displacements could be captured by all the methods except for using only the ACC measurements. The GNSS data-only solutions (Figure 4(a)) had a good agreement with the laser measurements, and the RMSE and peak error were 1.5 mm and 4.2 mm, respectively. There was a systematic shift in the ACC solutions (Figure 4(b)) for the period of sinusoidal displacements (about 65 s to 200 s), resulting in RMSE and peak errors of 8.1 mm and 18.3 mm, respectively. This can be attributed to the misalignment errors induced by tilting the shaking table. The EMD results (Figure 4(c)) achieved an RMSE of 1.4 mm. The possible reason for the good performance was that the misalignment errors were filtered out in the denoising process. However, the EMD fusion algorithm requires an additional high-precision GNSS observation station nearby to eliminate the environmental noise (e.g., multi-path), which makes this algorithm impossible for online processing.

The displacements derived with CMRKF (Figure 4(d)) were significantly contaminated by the misalignment errors, resulting in a systematic shift with the RMSE being 2.5 mm. The proposed technique effectively mitigated the impact of the misalignment errors (Figures 4(e) and 4(f)). The UMRKF-MA solutions achieved the best performance, with the RMSE and the peak error lowered to 1.0 mm and 2.5 mm, respectively.

Welch’s power spectral density (PSD) [40] of the displacements from the different computational schemes and laser measurements are shown in Figure 5. For the low-frequency band (0.01–0.2 Hz, zone 1), comparable PSD results can be observed among the different solutions and the laser measurements, except for the ACC solutions that deviated more from the other results. All the methods could capture the 1 Hz sinusoidal displacements accurately. When the frequency is higher than 1 Hz, the GNSS-only solutions show the poorest accuracy. For frequency band 2–10 Hz, the EMD solutions show the best agreement with the reference PSD curve, since in processing the data, the noise in both the GNSS and accelerometer measurements was suppressed. The CMRKF-MA method performed better than the CMRKF method although their curves were similar in the 2–10 Hz band. The EMD and GNSS-only methods failed to detect the vibration signals of over 10 Hz, agreeing with the Nyquist frequency principle. The UMRKF-MA method offered the most consistent PSD results with the reference data across the whole frequency range, especially for the higher frequency band (zone 4: 2–100 Hz), indicating the outstanding performance of the method in both the temporal and frequency domains.

3.3. Experiment with Random Displacements

Similar to the test with the simulated sinusoidal displacements, the shaking table first remained static for a short period of time (about 60 s) to obtain the reference coordinates, and then random displacements were generated. The shaking table was also tilted manually when the displacements were simulated. The data sampling rates were 20 Hz for the GNSS and 200 Hz for the accelerometer and the laser transducer, respectively.

Figure 6 shows the displacement time series computed based on the six computational schemes described above. The errors in the displacements relative to the laser measurements and the statistics are also given in the plots and in Table 3. The data resolutions of the GNSS and the EMD fusion solutions were 20 Hz and 200 Hz for the other solutions.

The accuracy of the GNSS-only solution (Figure 6(a)) in the random displacement test was slightly lower than that in the sinusoidal displacement test, with the RMSE being 2.4 mm compared with 1.5 mm in the case of the sinusoidal displacement test. Part of the low-frequency errors in the accelerometer measurements was removed by the high-pass filter. The ACC solutions (Figure 6(b)) still had high peak error and RMSE, 21.7 mm and 12.2 mm, respectively. The tilt angle of the shaking table impacted significantly the CMRKF solutions (Figure 6(d)), resulting in high peak error and RMSE, 32.6 mm and 12.7 mm, respectively. The denoising capability of the EMD helped reduce the effect of the misalignment errors and achieve an accuracy that is compatible with that of the GNSS data-only solutions (RMSE = 2.7 mm). After correcting the misalignment errors of the accelerometer with the MA-GNSS observations, both CMRKF-MA and UMRKF-MA produced improved results. Compared to the results of CMRKF, the RMSE and peak error of CMRKF-MA were reduced by about 83% and 73% to 2.2 mm and 9.1 mm, respectively. UMRKF-MA further improved the accuracy by about 35%.

Welch’s PSDs of the computed displacements from the different computation schemes are given in Figure 7. The results show that the different computational schemes produced largely similar PSDs in the low frequency (lower than 0.1 Hz) and high frequency (higher than 15 Hz) bands. The major differences in the PSDs between the different computational schemes are in the 2–10 Hz frequency band, where the PSD of the UMRKF-MA is similar to that of the EMD and is better than that of the CMRKF and CMRKF-MA. For frequency bands higher than 10 Hz, the CMRKF-MA and UMRKF-MA performed the best, indicating the effectiveness of the proposed method of using the MA-GNSS data to correct the misalignment errors.

4. Tests with Data from a Long-Span Bridge under Heavy Vehicle Loading

Stonecutters Bridge (SCB) is a long-span cable-stayed bridge in Hong Kong that was constructed during 2004–2012 and is the third-longest cable-stayed bridge in the world. As shown in Figure 8(a), SCB is supported by eight piers, and two 298-meter-tall cable towers, and has a main span of 1080 meters and two side spans of 289 meters. A long-term SHM system, including GNSS, accelerometers, and thermometers, was deployed to monitor the health conditions of the bridge. The SHM data from the SCB under vehicle loadings were used to demonstrate the advantage of the proposed method. The SHM system has eight pairs of colocated GPS receivers and accelerometers installed on the towers and the deck of the bridge as shown in Figure 8(b). The raw displacements (denoted as GPS solutions hereinafter) were calculated using the RTK technique and multi-GNSS data (GPS: L1/L2 and GLONASS: G1/G2) from Leica GRX1230GG GNSS receivers acquired at a sampling rate of 10 Hz. The distance between reference (at the Nam Wan end of the bridge) and monitoring stations is about 1.5 km.

Monitoring data during a heavy vehicle loading test were collected at a sampling rate of 10 Hz for the GNSS data and 50 Hz for the accelerometers. The bridge was temporarily closed to the public during the test and a convoy of trucks with a total weight of nearly 500 tons first assembled at the west tower (Figure 8(a)) and then moved together eastwards along the bridge (Figure 8(b)) to generate the loadings to the bridge. A similar test was then done when the vehicles moved westwards. Figure 9 shows the 3D displacement and acceleration signatures at the GPS-2/ACC-2 station (Figure 8(b)) during the test. The most significant displacements occurred at the station in the vertical direction, with the maximum vertical displacement reaching −100 mm and −150 mm, respectively, during the two time periods of vehicle loadings. The vehicle induced a longitudinal displacement of less than 20 mm and no obvious transversal displacement. The acceleration response showed a similar pattern although also indicated some acceleration in the transversal direction of the bridge.

(a)

(b)

Figure 10 shows the transverse displacements derived from data of GPS-2, the fusion of GPS-2 and ACC-2 data with MRKF, and the fusion of GPS-2, GPS-3, and ACC-2 data with UMRKF-MA. It should be noted that we have assumed that the structure between GPS-2 and GPS-3 was a rigid body and a two-antenna MA system was formed by the two antennas when using the UMRKF-MA model. It can be found in Figure 10 that the displacements computed with UMRKF-MA agree well with those computed with GPS data alone, but the noise level in the former was much lower. The displacements from the MRKF were significantly different from the other two solutions over the time periods of vehicle loading as highlighted with the red dashed rectangles. The MRKF model may have overestimated the transverse displacements by up to over 10 mm as shown in Figures 10(b) and 10(c). The likely reason for the results was that when the vehicle loading was applied on one side of the bridge deck, the loading caused the bridge deck to tilt about the x-axis (or a line parallel to the x-axis), resulting in the misalignment between the BRF and the ARF. As shown in Figure 11, when the bridge deck was tilted about the x-axis, part of the accelerations in the vertical direction was transformed into the transverse direction, affecting the computed transverse displacements (Figures 9 and 10).

(a)

(b)

(c)

We have calculated the tilt angles of the bridge deck and the displacement differences between the MRKF-derived and the GPS-measured results and between the MRKF-derived and UMRKF-MA-calculated results, respectively, as shown in Figure 12. For the computation of the former, the MRKF-derived displacements were first downsampled to 10 Hz, the same as that of the GPS measurements. The difference in the displacements between the MRKF-derived results and the results of the other two solutions had a strong correlation with the calculated tilt angle, which again verified that the misalignment error in the MRKF displacements was due to the tilt of the bridge deck.

(a)

(b)

5. Conclusions

We have proposed a new method for integrating data from a multi-antenna GNSS and an accelerometer with an unscented multi-rate Kalman filter (UMRKF-MA) to correct the system misalignment errors between the sensors and to produce much higher accuracy real-time displacement measurements for monitoring structural health conditions of large civil infrastructures. Experimental results with displacement data simulated with a shaking table and from a loading test of a large cable-stayed bridge indicated that the misalignment errors have significant effects on the fusion results, resulting in the overestimation of displacement with a magnitude of nearly 2 cm. The proposed method was effective in correcting the misalignment errors and that the accuracy of real-time displacement measurements could be improved by up to about 40% over using a standard RTK GNSS approach and by up to about 65% over integrating the multi-antenna GNSS and accelerometer data using conventional multi-rate Kalman filter method without considering the misalignment errors, and that about 1 mm level of real-time monitoring of displacements could be realized using the new method. We recommend strongly to consider the misalignment errors between accelerometers and GNSS when integrating the observations in monitoring all major civil infrastructures that may experience significant displacements to maximize the accuracy and reliability of such structural health monitoring systems.

Data Availability

The datasets used in this study are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

XYQ designed the research, developed the approach, performed the experiment, analyzed the data, and wrote the manuscript. XLD designed the research, analyzed the data, and wrote and revised the manuscript. YLX and WKY analyzed the data and revised the manuscript. All authors reviewed the manuscript.

Acknowledgments

This research was jointly supported by the Research Grants Council (RGC) of the Hong Kong Special Administrative Region (PolyU 152164/18E, PolyU 152233/19E, and PolyU 152318/22E) and the Innovative Technology Fund (ITP/019/20LP). The bridge monitoring dataset used in the study was provided by the Highways Department, Hong Kong SAR Government, China. The first author is grateful to Chinese National Rail Transit Electrification and Automation Engineering Technology Research Centre (Hong Kong Branch) for the PhD studentship provided.

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Copyright © 2023 Xuanyu Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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