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

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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 fnancial 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 efective technology for structural health monitoring [2][3][4][5].Te technology has a unique advantage in obtaining real-time three-dimensional displacements of a structure [6,7].Te 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].
Systematic errors may occur when using a standard Kalman flter to deal with nonlinear states, e.g., earthquakeinduced vibrations [26].An unscented Kalman flter has been proposed to model the nonlinear efects 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.Te coordinate frame of the GNSS is fxed 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.Terefore, 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.Te errors become especially signifcant when the structure rotates due to, e.g., strong winds and heavy vehicle load.Te errors can therefore signifcantly afect the results of the data fusion.A high-pass flter can be adopted considering the fact that the misalignment errors usually exhibit low-frequency characteristics [31,32].Te 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 diferent 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 multiantenna (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 fxing rate and the accuracy of positioning.
We propose a new method that jointly uses a multiantenna GNSS (MA-GNSS), an accelerometer, and an unscented multi-rate Kalman flter (UMRKF) for correcting the system misalignment errors and for high-accuracy dynamic structural displacement monitoring.Te 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 efectiveness of the proposed method.

Methodology
A multi-antenna GNSS and an accelerometer are combined in this study to achieve high-accuracy displacement measurement of civil infrastructures.Te 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.Te attitude information is used to correct the misalignment error of the accelerometer.Te corrected acceleration measurements will then be fused with the displacement measurements derived with the MA-GNSS by the UMRKF to improve displacement measurements.

Multi-Rate Kalman
Filter.Te three-dimensional (3D) displacement measurements from the GNSS and the acceleration measurements from the accelerometer data are fused with the multi-rate Kalman flter.Since the sampling rate of the accelerometer is typically much higher than that of the GNSS, the multi-rate Kalman flter 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].Te state vector x k of the Kalman flter is where d k and _ d k are the 3D displacement and velocity vectors of a monitoring point, respectively.Te dynamic system state is written as [20] x where ϕ k−1 is the transition matrix of the state vector and ψ k−1 is a design matrix, defned as where I 3×3 and 0 3×3 are the 3 × 3 identity and zero matrices, respectively, and τ a is the sampling interval of the accelerometer data.u k−1 is the vector of 3D accelerations; w k−1 is the noise matrix.
2 Structural Control and Health Monitoring where Q k−1 is the covariance matrix of the predicted states, which can be written as where q is the noise factor of the acceleration measurements.Te observation equation of the MRKF is where L k is the vector of observations that contains the 3D displacements from the GNSS observations; H k−1 is the design matrix; and R k−1 is the variance-covariance matrix of the observations and defned as where r is the noise factor of the GNSS displacement measurements in an SHM system.

Unscented Multi-Rate Kalman Filter.
Te unscented multi-rate Kalman flter (UMRKF) is a variation of the MRKF that considers nonlinear state estimation based on the use of the unscented Kalman flter (UKF) [35].For epoch k, a nonlinear state system consisting of random variables x k with white Gaussian noise w k and observations Z k with white Gaussian noise e k can be described as where f is the nonlinear state transition equation and h is the nonlinear observation equation.Te steps of the UKF algorithm for estimating the n-dimensional state vector x at time k are as follows.First, 2n + 1 sample points (sigma point set) are obtained frst based on the unscented transformation (UT).Second, the predicted state vector x k and its covariance matrix P x k are then calculated [27]: where ( x k ) i is the i th sigma point of the state vector and ω i is the weight of the sigma point.Tird, the predicted observation vector L k and its covariance P L k as well as the crosscovariance matrix P x k L k are calculated: where (  L k ) i is the i th sigma point of the predicted observation vector and L k is the weighted mean of the observation vector.Finally, the estimated state vector  x k , its covariance P  x k , and the UKF gain matrix K k can be updated as [35]

Correction of Misalignment Errors Based on Multi-
Antenna GNSS Observations.Figure 1 shows the two coordinate frames, i.e., the bridge reference frame (BRF, O − x b y b z b ) and the accelerometer reference frame (ARF, O − x a y a z a ).Te former is defned according to the bridge geometry (x b being horizontal and in the direction of the bridge alignment, y b being horizontal and perpendicular to x b , and z b in the vertical direction) while the latter refers to a coordinate system as defned 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 O − x a′ y a′ z a′ .We defne the rotation angles around the y-, x-, and zaxes as pitch (p), roll (r), and yaw (y), respectively.p r y   T are connected to the BRF and the ARF: Structural Control and Health Monitoring where x a y a z a   T and x b y b z b   T denote the 3D displacements in x-, y-and z-directions, respectively.Te subscripts a and b represent the ARF and BRF, respectively.C a b is an orthogonal transformation matrix.
and S(•) are cosine and sine operators, respectively.In general, the rotation angles of a civil engineering structure are small.Terefore 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 fne alignment.Te accelerometer data can be converted from ARF to BRF and integrated with the GNSS measurements based on equations ( 12)- (15).Te observation equation for the acceleration measurements can then be obtained as shown in the following equation [14,31]: where, a x ′ a y ′ a z ′   T and a x a y a z   T denote the 3D acceleration measurements in the BRF and ARF, re- is the vector of acceleration residuals that are assumed to be zero-mean white Gaussian noise.g is the acceleration due to gravity.
It can be seen from the above discussion that rotation angles p r y   T 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 C b a (see equation (15)) can be estimated based on the least squares principle [36]: where B i � x b,i y b,i z b,i   T and A i � x a,i y a,i z a,i   T are the coordinates of the ith GNSS antenna in the BRF and ARF, respectively, i � 1, 2, . . ., m represents the number of a GNSS antenna, and W i is a positive weighting parameter.

Integration of Multi-Antenna GNSS and Accelerometer
Measurements Based on UMRKF.Te UMRKF is used to fuse the multi-antenna GNSS and accelerometer measurements to achieve high-accuracy measurements of displacements.Te state vector x k in equation ( 1) is frst extended into the following: where , and p r y   T are the displacements, velocities, and rotation angles about x-, y-, and z-axes, respectively.Te transition matrices ϕ k−1 and ψ k−1 in equation ( 2) accordingly become where I 3×3 and 0 3×3 are 3 × 3 identity and zero matrices, respectively; we assume that the rotation angles vary over time and behave as random-walk variables: where ε p ε r ε y   T denote the process noise.Te observations from the MA-GNSS and the observation matrix are where z k is the observation from the MA-GNSS including the 3D displacements and three Euler angles and H k−1 is the design matrix.Te overall framework of the proposed method is shown in Figure 2. Te 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)).Te distances between ANT1, ANT2, and ANT3 were 1.2 m and antenna ANT4 was mounted on the center.Te GNSS antennas were mounted on a rigid triangular platform.All the other sensors used in the study were synchronized to the GPS time.Te ARF of the accelerometer was initially aligned with the BRF and is shown in Figure 3(a) (blue lines).Te GNSS reference station (model: Trimble Net R9) was set up about 50 m away from the shaking table to provide correction information.

Experiments with Data from a Shaking Table
Te dynamic response of a structure induced by diferent loadings, e.g., temperature, crossing winds, trafc fows, and earthquakes, may exhibit diferent 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.Te 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.Te details of RTK parameters are shown in Table 1.
In the experiments, the shaking table was frst kept static to obtain the reference coordinates.Te motion signals were then uploaded onto the shaking table system.Te 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.Te tilt angle of the platform was frst 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 flter 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 flter [39] is only applied in Scheme 2, with a cutof frequency of 0.05 Hz to mitigate the low-frequency efects.

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.Te displacements from the GNSS data only and the EMD fusion solutions were 20 Hz and those from the other four solutions were 200 Hz.Te calculated displacements were compared with the laser measurements (black lines and used as the truth).Te errors in the displacements, i.e., diferences 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.Te RMSE (root-mean-square error) and the peak errors were calculated to evaluate the performances of the diferent 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.Te 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.
Tere 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.Tis can be attributed to the misalignment errors induced by tilting the shaking table.Te EMD results (Figure 4(c)) achieved an RMSE of 1.4 mm.Te possible reason for the good performance was that the misalignment errors were fltered 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.
Te displacements derived with CMRKF (Figure 4(d)) were signifcantly contaminated by the misalignment errors, resulting in a systematic shift with the RMSE being 2.5 mm.Te proposed technique efectively mitigated the impact of the misalignment errors (Figures 4(e) and 4(f )).Te 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 diferent computational schemes and laser measurements are shown in Figure 5.For the lowfrequency band (0.01-0.2 Hz, zone 1), comparable PSD results can be observed among the diferent solutions and the laser measurements, except for the ACC solutions that deviated more from the other results.All the methods could  Structural Control and Health Monitoring 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.Te CMRKF-MA method performed better than the CMRKF method although their curves were similar in the 2-10 Hz band.Te EMD and GNSS-only methods failed to detect the vibration signals of over 10 Hz, agreeing with the Nyquist frequency principle.Te UMRKF-MA method ofered 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.) to obtain the reference coordinates, and then random displacements were generated.Te shaking table was also tilted manually when the displacements were simulated.Te 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.Te errors in the displacements relative to the laser measurements and the statistics are also given in the plots and in Table 3. Te data resolutions of the GNSS and the EMD fusion solutions were 20 Hz and 200 Hz for the other solutions.
Te 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 flter.Te ACC solutions (Figure 6(b)) still had high peak error and RMSE, 21.7 mm and 12.2 mm, respectively.Te tilt angle of the shaking table impacted signifcantly the CMRKF solutions (Figure 6(d)), resulting in high peak error and RMSE, 32.6 mm and 12.7 mm, respectively.Te denoising capability of the EMD helped reduce the efect 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 diferent computation schemes are given in Figure 7. Te results show that the diferent computational schemes produced largely similar PSDs in the low frequency (lower than 0.1 Hz) and high frequency (higher than 15 Hz) bands.Te major diferences in the PSDs between the diferent 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 efectiveness of the proposed method of using the MA-GNSS data to correct the misalignment errors.

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 298meter-tall cable towers, and has a main span of 1080 meters   Structural Control and Health Monitoring GLONASS: G1/G2) from Leica GRX1230GG GNSS receivers acquired at a sampling rate of 10 Hz.Te 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.Te bridge was temporarily closed to the public during the test and a convoy of trucks with a total weight of nearly 500 tons frst 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.Te most signifcant 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.Te vehicle induced a longitudinal displacement of less than 20 mm and no obvious transversal displacement.Te acceleration response showed a similar pattern although also indicated some acceleration in the transversal direction of the bridge.
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.Te displacements from the MRKF were signifcantly diferent from the other two solutions over the time periods of vehicle loading as highlighted with the red dashed rectangles.Te MRKF model may have overestimated the transverse displacements by up to over 10 mm as shown in Figures 10(b) and 10(c).Te 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, afecting the computed transverse displacements (Figures 9 and 10).
We have calculated the tilt angles of the bridge deck and the displacement diferences between the MRKF-derived and the GPS-measured results and between the MRKFderived and UMRKF-MA-calculated results, respectively, as shown in Figure 12.For the computation of the former,  Structural Control and Health Monitoring the MRKF-derived displacements were frst downsampled to 10 Hz, the same as that of the GPS measurements.Te diference 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 verifed that the misalignment error in the MRKF displacements was due to the tilt of the bridge deck.

Conclusions
We have proposed a new method for integrating data from a multi-antenna GNSS and an accelerometer with an unscented multi-rate Kalman flter (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 signifcant efects on the fusion results, resulting in the overestimation of displacement with a magnitude of nearly 2 cm.Te proposed method was efective 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 flter 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 signifcant displacements to maximize the accuracy and reliability of such structural health monitoring systems.

Figure 1 :
Figure 1: Accelerometer body frame (x a , y a , z a ) and bridge reference frame (x b , y b , z b ).

Scheme 3 :
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)

Figure 2 :
Figure 2: Framework of the proposed method for integrated use of multi-antenna GNSS and accelerometer observations.

Figure 3 :
Figure 3: Experimental setup for simulating and recording various types of displacements.(a) Shaking table with a multi-antenna GNSS and an accelerometer and the bridge reference frame (blue lines).(b) Shaking table signal generator.(c) Laser transducer.

Figure 4 :
Figure 4: Displacement solutions and their errors for the test with simulated sinusoidal displacements.(a) GNSS data only.(b) Accelerometer data only after a high-pass flter.(c) GNSS and accelerometer data fusion with EMD.(d) GNSS and accelerometer data fusion with CMRKF.(e) GNSS and accelerometer data fusion with CMRKF-MA.(f ) GNSS and accelerometer data fusion with UMRKF-MA.

Figure 5 :
Figure 5: PSD results of displacements computed from the MA-GNSS, the accelerometer (ACC) measurements, the EMD-based fusion algorithm (EMD), the conventional Kalman flter (CMRKF), CMRKF-MA, and UMRKF-MA based on data from sinusoidal displacement simulation.Te PSD results of the laser measurements are also given as the truth for comparison.

Figure 6 :
Figure 6: Displacements computed and their errors for the test with the simulated random displacements.(a) GNSS data only.(b) Integration from accelerometer data after a high-pass flter.(c) GNSS and accelerometer data fusion with EMD.(d) GNSS and accelerometer data fusion with CMRKF.(e) GNSS and accelerometer data fusion with CMRKF-MA.(f ) GNSS and accelerometer data fusion with UMRKF-MA.

Table 1 :
Key parameters of GNSS RTK positioning.

Table 2 :
Statistics of errors of diferent solutions (unit: mm).

Table 3 :
Statistics of displacement errors based on the six computation schemes for the random displacement test (unit: mm).