# Reference Point-Free Measurement of Bridge Dynamic Deflection by Fusing Hydraulic Leveling and Accelerometer Signals

Authors: Zhibin Jin; Hongda Xia; Wenbo Ni; Haoyuan Yang
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

Structural health monitoring (SHM) is widely applied to assess the service condition of bridges. Deflection measurements are essential to determine a bridge’s performance, and in particular, dynamic deflection is increasingly required in SHM. However, continuous and high-precision measurements of the absolute dynamic deflection are challenging without a reference point placed at a distance from the bridge. We proposed a reference point-free dynamic deflection measurement system consisting of a level sensor and an accelerometer. The level sensor measures the low-frequency deflection components, while the accelerometer measures the high-frequency deflection components. We redesigned the level sensor from the hydrostatic level sensor and further provided a correction method to achieve the dynamic deflection measurement. A numerical algorithm fuses the signals from the level sensor and accelerometer to obtain the absolute dynamic deflection, and key parameters are calculated. The accuracy of the measurement system was tested in the laboratory, and the sensor results were similar to the ones obtained by the laser test under simple harmonic and random excitations. Field tests conducted on a T-shaped rigid frame bridge with random traffic flow indicate that the system can achieve continuous high-precision deflection measurements of bridges loaded by traffic flow.


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## Body

## 1. Introduction

Bridge deflection measurements are crucial for operational safety and structural damage assessment, as they are highly indicative of the bridge performance. Therefore, both static and dynamic deflection measurements have become an increasingly indispensable part of structural health monitoring (SHM).Static deflection is mostly measured by global positioning systems (GPSs), total station, and hydrostatic leveling systems (HLSs) [1–4] in static load testing and long-term deformation monitoring. These commercial instruments can achieve stable and accurate static deflection measurements, albeit only at certain structure points. The points deflection measurement is often unsatisfactory, and the inclinometer is often employed to reconstruct the static deflection curve of the entire bridge [5, 6].Dynamic deflection is, to some extent, more important than static deflection, because it reflects the bridge stiffness variation and dynamic properties, which are essential for assessing its real-time service condition. Dynamic deflection measurement techniques are sorted according to whether a ground reference point at a distance from the bridge is required.Dynamic deflection measurements with reference points, such as the ground-based interferometric radar, total stations, laser technology, and linear variable differential transformers (LVDTs), are demonstrated measurement methods. The image by the interferometric survey (IBIS-S) system is a ground-based interferometric radar displacement measuring instrument. Therein, the radar transmitter is located at a fixed position on the ground at a distance away from bridges, serving as the reference point. The system can achieve an accuracy of 0.1 mm in the dynamic mode with a sampling frequency of up to 200 Hz [7–9]. Marendić et al. used a robotic total station fixed at a distance from the bridge to measure its deflection with the accuracy of ±2.2 mm at 20 Hz sampling frequency [10]. Zhang et al. [11] achieved 0.2 mm dynamic deflection measurement accuracy based on laser projection and image processing technology. Linear variable differential transformers (LVDTs) have higher measurement accuracy, albeit they are more difficult to install [12]. Recently, vision-based methods have been introduced to measure bridge deformation, most requiring a fixed camera near the bridge. The measurement accuracy of these methods ranges from 0.01 to 1 mm [13–16]. The vision-based system sampling frequency is related to the camera performance, most frequencies being around 60 Hz.Systems with ground reference points exhibit high measurement accuracy (Table1); however, as bridges may span rivers, villages, or other roads, it is not an easy task to establish a fixed reference point outside the bridge. Therefore, there is a demand for reference point-free dynamic deflection measurement.Table 1
Comparison of deflection measurement methods.
MethodGround reference pointApplicable bridge typeHighest sampling rates (Hz)AccuracyTotal stationYesSmall-span bridges20∼2 mmGPS-RTKYesAny100∼1-2 mmInterferometric radar technology (IBIS)YesAny2000.1 mmVision-basedYesAny60∼1 mm∗Acceleration estimationNoSmall-span bridges500<1 mmAcceleration-GPS fusionNoMostly long-span bridges20∼1 cm∗Acceleration-strain fusionNoSmall-span bridges (simply supported)500<1 mmMethods with∗have other limitations that are not convenient to list in the table, whose details are presented in the introduction.The positioning system using satellites does not require a ground reference point, but the accuracy for dynamic measurement only achieves 1 cm [17]. Yoon et al. [18] used an unmanned aerial system to measure the absolute displacement of bridges, whose RMSE is 2.14 mm. Liu et al. [19] employed the connected pipe system (CPS) to monitor the deflection of a long-span suspension bridge. The CPS measures the dynamic deflection of the bridge caused by slowly passing heavy vehicles, whereas the small deflection caused by lighter and faster vehicles cannot be measured accurately. Estimating the displacement using only acceleration data likewise satisfies the requirement of no ground reference point, as the theoretical displacement can be obtained by integrating the acceleration signal twice. However, this method is plagued by drift over time, and several researchers attempted to overcome this problem. Arias-Lara and De-la-Colina [20] compared seven correction algorithms to solve the displacement drift problem, where each algorithm can only be used under specific prerequisites. Converse and Gerald Brady [21] and Chiu [22] corrected the recorded acceleration by baseline correction to reconstruct the displacement. This method was first used in earthquake engineering. The algorithm includes three steps: (1) fitting the baseline of acceleration by the least squares, (2) designing a high-pass filter for acceleration, and (3) decreasing the initial values in velocity on the first integration result. Park et al. [23] used acceleration to estimate the bridge deflection under the moving vehicle load; in this method, the displacement is easily calculated if the initial velocity is known under the zero initial displacement condition. However, the bridge’s zero initial displacement condition is difficult to determine under actual constant traffic flow, limiting the method in continuous measurement. Liu et al. used a sensing system composed of one passive servo electromagnetic induction (PSEMI) velocity sensor and one built-in hardware integrator unit. The measurement errors are less than 10% compared to LVDT [24].Achieving reference point-free dynamic displacement measurements with a single sensor remains a challenge, and fusion methods have been employed as a solution. Ozdagli et al. [25] used tilt and acceleration to obtain displacement. Xu et al. [26] fused the GPS and accelerometer to improve the long-span bridge’s deflection measurement accuracy using a multirate Kalman filter. The modal frequency was determined to be 0.63 Hz; however, the dynamic recognition frequency proved too low for small-span bridges. Cho et al. [27] and Sarwar and Park [28] proposed an accurate displacement estimation method by optimally utilizing acceleration and strain for simply supported bridges. The method requires a definite proportional connection between the strain and deflection at the same location on a bridge, making it not suitable for measuring deflections on bridges without the abovementioned relationship, such as cable-supported bridges.Table1 lists the dynamic deflection measurement methods, including whether a ground reference point at a distance from the bridge is required, the applicable bridge type, the highest sampling rates, and the obtained accuracy.Long-span cable-supported or small-span simply supported bridges have applicable reference point-free methods for deflection measurements. This study focuses on small-span or midspan bridges with a maximum deflection of several millimeters or centimeters. We proposed a reference point-free continuous bridge dynamic deflection measurement system comprising a leveling system and accelerometer using the sensor fusion method. The system includes fixed and moving ends. The fixed end is placed on the pier top as the reference point, while the moving end is placed at the measurement point. Only the relative displacement between both ends by the total station on the bridge is necessary to attain continuous dynamic deflection monitoring without ground reference points. The fundamental contribution of the proposed method is shown in Figure1.Figure 1
The fundamental contribution of the proposed method.The rest of this paper is organized as follows. Section2 introduces the correction method of the hydrostatic leveling systems (HLS) employed for dynamic displacement measurement and further details its design. Section 3 presents the algorithm that fuses the leveling and acceleration signals to reconstruct the displacement. We employed this algorithm to fuse a liquid-level sensor and accelerometer for displacement measurement for the first time. Sections 4 and 5 demonstrate the performance of the measuring system in laboratory and field tests.


## 2. Theory and Techniques for Dynamic Hydraulic Leveling System (DHLS)

The theory behind the correction algorithm for the DHLS is introduced herein based on its dynamic characteristics, and the leveling identification method, based on image recognition, is introduced. The design concerns of the DHLS are addressed, and the design is based on the full-field HLS.


### 2.1. Composition of Leveling System and Fluid’s Forced Oscillation

The system consists of two water containers connected by a communicating pipe at the base. The distance between the two containers isL, and the communicating pipe diameter is d = 2r. Air tubes are installed on containers to prevent foreign objects from falling into them, while ensuring the connection between the fluid and the atmosphere. Because of the principle of communicating vessels, the liquid level in the container tends to maintain the same height at the equilibrium position. The schematic diagram of the system is shown in Figure 2.Figure 2
Schematic diagram for the derivation of forced oscillation of fluid in the pipe.The free oscillation equation has been previously derived in numerous studies, such as [29]. In this section, we derive the forced oscillation equation.In Figure2, the container at the middle of the span moves y(t) relative to the other. Consequently, the liquid in the pipe starts to flow. To obtain the flow state, we take a liquid microelement dl in the pipe for analysis, as shown in Figure 2. For both sides of this microelement, the force difference is given as follows:(1)∆Fp=Fl−Fr=pl,tαp−pl,t+∂pl,t∂ldlαp,where pl,t and pl,t+∂pl,t/∂ldl are pressure on both surfaces of the microelement and αp is the cross-sectional area of the connecting pipe.According to Poiseuille’s Law, the viscous resistance is described as follows:(2)Fc=−8μvtαp⋅dlr2,where vt is the average velocity of profile flow in the connecting pipe and μ is the coefficient of viscosity of the fluid.The massm of the microelement is given as follows:(3)m=ραp⋅dl.The component of gravity force along the bottom pipe ismgsinθ.(4)mgsinθ=ραpg⋅dl∂yt∂l,where ρ is the density of the liquid in the pipe, g is the gravitational acceleration, and θ is the angle between the pipe and the horizontal plane. It can be assumed that sinθ=∂yt/∂l when θ is very small.The inertial forceFI of the microelement is given by the following formula:(5)FI=−ma=−ραp⋅dldvtdt.According to Newton’s second law, the force-balance condition of the microelement can be written as follows:(6)pl,tαp−pl,t+∂pl,t∂ldlαp−8μvtαp⋅dlr2+ραpg⋅dl∂yt∂l−ραp⋅dldvtdt=0.By integrating equation (6) along the x-axis from the left to the right ends of the pipe, the following equation is obtained:(7)−ρgαp⋅2ut−8μvtαp⋅Lr2+ρgαp⋅yt−ραpL⋅dvtdt=0,where u(t) is the vertical position of liquid level relative to the container.Simultaneously, we know that in the containers, the pressure is equal at both liquid surfaces,p1=p2. The average velocity of flow in both the containers is v1t=du/dt=v2t and vt=αc/αpv1t, where αc is the cross-sectional area of the liquid container. Substituting the above conditions, the following equation is obtained:(8)−2u−8μv1Lρgπr4⋅αcαp−σg⋅dv1dt+y=0,where σ=2h+αc/αpL≈αc/αpL (because 2h≪L) and h is the height of the liquid level of the container at equilibrium condition.To simplify equation (8), we move the terms containing excitation y(t) and response u(t) to each side of the equation as follows:(9)d2utdt2+8μρr2dutdt+2gσut=gσyt.Rewriting equation (9) into a standard form composed of ωn and ξ as follows:(10)d2utdt2+2ξωndutdt+ωn2ut=gσyt,where the natural frequency is ωn=2g/2h+Lαc/αp and the damping ratio is ζ=8μ2π2Lαc/ρ2gαp3


### 2.2. Liquid Level Identification Method Based on Image Recognition

The desired dynamic displacement is obtained by correcting the liquid-level change. To obtain direct leveling changes, we completed the liquid-level recognition system (LRS), which uses image recognition technologies with the help of Liu et al. [30] The LRS comprises a camera, shade panel, LED light source, and communicating pipe. The image recognition process is assisted by OpenCV, which provides a real-time optimized computer vision library, tools, and hardware. The image recognition process is illustrated in Figure 3. Its tasks include (1) converting the original color image to grayscale and capturing its target area, (2) performing bilateral filtering on the image to remove noise while protecting the edge information of the image, (3) binarizing the liquid-level edge and background by threshold segmentation and getting the segmented image, (4) extracting the complete edge contour of the liquid level using the Canny edge detection algorithm in OpenCV, and (5) using subpixel edge detection methods to improve the accuracy of the liquid-level edge recognition. We obtain the liquid-level bottom coordinates Cpixelxh,yv of the subpixel edge image by the five steps described above.Figure 3
The process of obtaining subpixel bottom edge coordinates of the liquid level.We used the device shown in Figure4(a) to scale the vertical pixel coordinates yv (given by the image recognition system) to the reality displacement (measured by the laser displacement meter). The relationship between k and yv is shown in the following formula; β^0 and β^1 can be obtained by multiple tests.(11)k=β^0+β^1yv,where β^0=∑i=1nyvi2∑i=1nki−∑i=1nyvi∑i=1nkiyvi/n∑i=1nyvi2−∑i=1nki2 and β^1=n∑i=1nyviki−∑i=1nyvi∑i=1nki/n∑i=1nyvi2−∑i=1nki2, ki and yvi are the result data of each test and n is the total number of tests.Figure 4
Device for liquid-level calibration to displacement (a) drawing for the test device and (b) the location of the lowest line of the concave surface at different liquid levels.
(a)(b)Furthermore, the natural frequency of the liquid level in the DHLS is designed as 0.4 Hz. The damping ratio is 0.068, which filters out high-frequency components. The camera’s sampling frequency is ∼25 Hz. Thus, half of the sampling frequency is ∼12.5 Hz, which is significantly higher than the signal frequency of 0.397 Hz, and thus meets the requirements of Nyquist law.


### 2.3. Correction Method for DHLS

The amplitude-frequency characteristicDω of the DHLS is given as follows:(12)Dω=UωYω=11−β22+2ξβ2,where Yω is maximum amplitudes of the liquid container’s displacement and Uω is twice of fluid level measured by DHLS under harmonic excitation with different frequencies. Only the amplitude-frequency characteristic is considered, and the effect of phase difference is ignored. β=ω/ωn, where ωn is the natural frequency of HLS and ω is excitation frequency.Suppose we can design a filter whose transfer function is1/Dω and which can reconstruct the bridge deflection Yω from the measured liquid-level difference Uω, as shown in the following equation:(13)Yω=Uω·1Dω.The simple moving average (SMA) filter is computationally effective in the time domain. Here, the filter function1/Dω is approximated by the moving average method. Assuming the window length of the moving average filter is 2tm, its pulse-response function in the time domain reads as follows [31]:(14)Ht=12tm,−tm,tm,0,esel.The response function in the frequency domain is the fourier transform (FT) of its pulse-response function in the time domain. Thus, the frequency response functionHω of the moving average method is as follows:(15)Hω=∫−tmtmHtejωtdt=sinωtmωtm≈ωtmωtm−ωtm36ωtm,where sinωt is turned into the Taylor Expansion to facilitate the derivation, in the last expression of equation (15).To obtain the optimaltm make Hω appropriate 1/Dω. We construct the following function equation (15), which minimizes the square difference between the frequency response function of SMA (Hω) and HLS 1/Dω. In the last expression of equation (15), ω is discretized into frequency points ω=ωii=1,2,…,n.(16)Jtm=mintmHω−1Dω22=mintm∑i=1nωitmωitm−ωitm36ωitm−1−ωi2ωn22+2ξωiωn22.In equation (16), f∗22 denotes the square of the 2-Norm (square root of the integral of f∗ squared over the interval). Then, equation (16) simplifies as follows:(17)Jtm=mintm∑i=1n−ωi26tm2−1−ωi2ωn22+2ξωiωn2+12.Equation (17) is expanded into matrix form ωi=n∆ω with n=1,2,…,i as the discrete interval.(18)Jtm=mintm−∆ω26tm2−1−∆ω2ωn22+2ξ∆ωωn2−1−2∆ω26tm2−1−2∆ω2ωn22+2ξ2∆ωωn2−1⋮−ωn26tm2−1−ωn2ωn22+2ξωnωn2−1T−∆ω26tm2−1−∆ω2ωn22+2ξ∆ωωn2−1−2∆ω26tm2−1−2∆ω2ωn22+2ξ2∆ωωn2−1⋮−ωn26tm2−1−ωn2ωn22+2ξωnωn2−1.Letxωi=−ωi2/6,X=xω1,xω2…xωiT and gωi=1−ωi2/ωn22+2ξωi/ωn2+1,G=gω1,gω2…gωiT. The complex formula in equation (18) is replaced with xωiandgωi, such that(19)Jtm=mintm∑i=1mxωi·tm2−gωi2=mintmXtm2−GTXtm2−G.Finally, the length of the moving average window2tm is determined by letting the derivative of Xtm2−GTXtm2−G equal to zero.(20)tm=XTX−1XTG.


### 2.4. Design Example and Laboratory Test for the DHLS

The length of the connecting pipeL and the area ratio αc/αp are the main factors affecting the measurement performance of DHLS. The length of the connecting pipe is determined by the span of the bridge to be measured. In this design solution, assuming that the bridge span is 30 m, we determined that the length of the connecting pipe is 15 m. The area ratio adjusts the natural frequency of the liquid oscillation after determination of the pipe length (equation (10)). According to the correction in the previous section, the suitable measuring frequency range of DHLS is 0 to ωn. Meanwhile, the displacement measured by DHLS must be fused with the acceleration. We determine ωn by the applicable frequency range of the accelerometer. The microelectro-mechanical system (MEMS) accelerometer selected in this study has a minimum measurement frequency of 0.4 Hz under the premise of ensuring product measurement accuracy. Finally, by substituting ωn=0.4Hz and L=15m into equation (10), the area ratio αc/αp=1/4.7 can be obtained.We conducted experiments to test the designed DHLS system. The experimental setup includes a shaking table, an LDM, and a liquid-level measurement system, as shown in Figure5. The device is also used to verify the fusion algorithm in the following section, explaining the installation of the accelerometer.Figure 5
The experimental device of the proposed displacement measurement system.The data required for this experiment include the amplitude of the liquid-level oscillation in the fixed end and deck displacement of the shaking table at different excitation frequencies. The deck displacementy is measured by LDM, erected vertically above the shaking table. The liquid-level measurement system yields the liquid-level change u at the fixed end. To obtain the amplitude-frequency characteristics (1/Dω) of DHLS, we measure the ratio of Yω and Uω under harmonic excitation at an interval of 0.05 Hz between 0 and 0.4 Hz.The experimental measurement ofωn is 0.397 Hz, which is consistent with the expected result. Then, we identify the frequency response function (Hω) of SMA. Figure 6 shows the comparison of measured DHLS amplitude-frequency characteristics and the theoretical value of SMA.Figure 6
Comparison of amplitude-frequency characteristics of DHLS and SMA.


## 2.1. Composition of Leveling System and Fluid’s Forced Oscillation

The system consists of two water containers connected by a communicating pipe at the base. The distance between the two containers isL, and the communicating pipe diameter is d = 2r. Air tubes are installed on containers to prevent foreign objects from falling into them, while ensuring the connection between the fluid and the atmosphere. Because of the principle of communicating vessels, the liquid level in the container tends to maintain the same height at the equilibrium position. The schematic diagram of the system is shown in Figure 2.Figure 2
Schematic diagram for the derivation of forced oscillation of fluid in the pipe.The free oscillation equation has been previously derived in numerous studies, such as [29]. In this section, we derive the forced oscillation equation.In Figure2, the container at the middle of the span moves y(t) relative to the other. Consequently, the liquid in the pipe starts to flow. To obtain the flow state, we take a liquid microelement dl in the pipe for analysis, as shown in Figure 2. For both sides of this microelement, the force difference is given as follows:(1)∆Fp=Fl−Fr=pl,tαp−pl,t+∂pl,t∂ldlαp,where pl,t and pl,t+∂pl,t/∂ldl are pressure on both surfaces of the microelement and αp is the cross-sectional area of the connecting pipe.According to Poiseuille’s Law, the viscous resistance is described as follows:(2)Fc=−8μvtαp⋅dlr2,where vt is the average velocity of profile flow in the connecting pipe and μ is the coefficient of viscosity of the fluid.The massm of the microelement is given as follows:(3)m=ραp⋅dl.The component of gravity force along the bottom pipe ismgsinθ.(4)mgsinθ=ραpg⋅dl∂yt∂l,where ρ is the density of the liquid in the pipe, g is the gravitational acceleration, and θ is the angle between the pipe and the horizontal plane. It can be assumed that sinθ=∂yt/∂l when θ is very small.The inertial forceFI of the microelement is given by the following formula:(5)FI=−ma=−ραp⋅dldvtdt.According to Newton’s second law, the force-balance condition of the microelement can be written as follows:(6)pl,tαp−pl,t+∂pl,t∂ldlαp−8μvtαp⋅dlr2+ραpg⋅dl∂yt∂l−ραp⋅dldvtdt=0.By integrating equation (6) along the x-axis from the left to the right ends of the pipe, the following equation is obtained:(7)−ρgαp⋅2ut−8μvtαp⋅Lr2+ρgαp⋅yt−ραpL⋅dvtdt=0,where u(t) is the vertical position of liquid level relative to the container.Simultaneously, we know that in the containers, the pressure is equal at both liquid surfaces,p1=p2. The average velocity of flow in both the containers is v1t=du/dt=v2t and vt=αc/αpv1t, where αc is the cross-sectional area of the liquid container. Substituting the above conditions, the following equation is obtained:(8)−2u−8μv1Lρgπr4⋅αcαp−σg⋅dv1dt+y=0,where σ=2h+αc/αpL≈αc/αpL (because 2h≪L) and h is the height of the liquid level of the container at equilibrium condition.To simplify equation (8), we move the terms containing excitation y(t) and response u(t) to each side of the equation as follows:(9)d2utdt2+8μρr2dutdt+2gσut=gσyt.Rewriting equation (9) into a standard form composed of ωn and ξ as follows:(10)d2utdt2+2ξωndutdt+ωn2ut=gσyt,where the natural frequency is ωn=2g/2h+Lαc/αp and the damping ratio is ζ=8μ2π2Lαc/ρ2gαp3


## 2.2. Liquid Level Identification Method Based on Image Recognition

The desired dynamic displacement is obtained by correcting the liquid-level change. To obtain direct leveling changes, we completed the liquid-level recognition system (LRS), which uses image recognition technologies with the help of Liu et al. [30] The LRS comprises a camera, shade panel, LED light source, and communicating pipe. The image recognition process is assisted by OpenCV, which provides a real-time optimized computer vision library, tools, and hardware. The image recognition process is illustrated in Figure 3. Its tasks include (1) converting the original color image to grayscale and capturing its target area, (2) performing bilateral filtering on the image to remove noise while protecting the edge information of the image, (3) binarizing the liquid-level edge and background by threshold segmentation and getting the segmented image, (4) extracting the complete edge contour of the liquid level using the Canny edge detection algorithm in OpenCV, and (5) using subpixel edge detection methods to improve the accuracy of the liquid-level edge recognition. We obtain the liquid-level bottom coordinates Cpixelxh,yv of the subpixel edge image by the five steps described above.Figure 3
The process of obtaining subpixel bottom edge coordinates of the liquid level.We used the device shown in Figure4(a) to scale the vertical pixel coordinates yv (given by the image recognition system) to the reality displacement (measured by the laser displacement meter). The relationship between k and yv is shown in the following formula; β^0 and β^1 can be obtained by multiple tests.(11)k=β^0+β^1yv,where β^0=∑i=1nyvi2∑i=1nki−∑i=1nyvi∑i=1nkiyvi/n∑i=1nyvi2−∑i=1nki2 and β^1=n∑i=1nyviki−∑i=1nyvi∑i=1nki/n∑i=1nyvi2−∑i=1nki2, ki and yvi are the result data of each test and n is the total number of tests.Figure 4
Device for liquid-level calibration to displacement (a) drawing for the test device and (b) the location of the lowest line of the concave surface at different liquid levels.
(a)(b)Furthermore, the natural frequency of the liquid level in the DHLS is designed as 0.4 Hz. The damping ratio is 0.068, which filters out high-frequency components. The camera’s sampling frequency is ∼25 Hz. Thus, half of the sampling frequency is ∼12.5 Hz, which is significantly higher than the signal frequency of 0.397 Hz, and thus meets the requirements of Nyquist law.


## 2.3. Correction Method for DHLS

The amplitude-frequency characteristicDω of the DHLS is given as follows:(12)Dω=UωYω=11−β22+2ξβ2,where Yω is maximum amplitudes of the liquid container’s displacement and Uω is twice of fluid level measured by DHLS under harmonic excitation with different frequencies. Only the amplitude-frequency characteristic is considered, and the effect of phase difference is ignored. β=ω/ωn, where ωn is the natural frequency of HLS and ω is excitation frequency.Suppose we can design a filter whose transfer function is1/Dω and which can reconstruct the bridge deflection Yω from the measured liquid-level difference Uω, as shown in the following equation:(13)Yω=Uω·1Dω.The simple moving average (SMA) filter is computationally effective in the time domain. Here, the filter function1/Dω is approximated by the moving average method. Assuming the window length of the moving average filter is 2tm, its pulse-response function in the time domain reads as follows [31]:(14)Ht=12tm,−tm,tm,0,esel.The response function in the frequency domain is the fourier transform (FT) of its pulse-response function in the time domain. Thus, the frequency response functionHω of the moving average method is as follows:(15)Hω=∫−tmtmHtejωtdt=sinωtmωtm≈ωtmωtm−ωtm36ωtm,where sinωt is turned into the Taylor Expansion to facilitate the derivation, in the last expression of equation (15).To obtain the optimaltm make Hω appropriate 1/Dω. We construct the following function equation (15), which minimizes the square difference between the frequency response function of SMA (Hω) and HLS 1/Dω. In the last expression of equation (15), ω is discretized into frequency points ω=ωii=1,2,…,n.(16)Jtm=mintmHω−1Dω22=mintm∑i=1nωitmωitm−ωitm36ωitm−1−ωi2ωn22+2ξωiωn22.In equation (16), f∗22 denotes the square of the 2-Norm (square root of the integral of f∗ squared over the interval). Then, equation (16) simplifies as follows:(17)Jtm=mintm∑i=1n−ωi26tm2−1−ωi2ωn22+2ξωiωn2+12.Equation (17) is expanded into matrix form ωi=n∆ω with n=1,2,…,i as the discrete interval.(18)Jtm=mintm−∆ω26tm2−1−∆ω2ωn22+2ξ∆ωωn2−1−2∆ω26tm2−1−2∆ω2ωn22+2ξ2∆ωωn2−1⋮−ωn26tm2−1−ωn2ωn22+2ξωnωn2−1T−∆ω26tm2−1−∆ω2ωn22+2ξ∆ωωn2−1−2∆ω26tm2−1−2∆ω2ωn22+2ξ2∆ωωn2−1⋮−ωn26tm2−1−ωn2ωn22+2ξωnωn2−1.Letxωi=−ωi2/6,X=xω1,xω2…xωiT and gωi=1−ωi2/ωn22+2ξωi/ωn2+1,G=gω1,gω2…gωiT. The complex formula in equation (18) is replaced with xωiandgωi, such that(19)Jtm=mintm∑i=1mxωi·tm2−gωi2=mintmXtm2−GTXtm2−G.Finally, the length of the moving average window2tm is determined by letting the derivative of Xtm2−GTXtm2−G equal to zero.(20)tm=XTX−1XTG.


## 2.4. Design Example and Laboratory Test for the DHLS

The length of the connecting pipeL and the area ratio αc/αp are the main factors affecting the measurement performance of DHLS. The length of the connecting pipe is determined by the span of the bridge to be measured. In this design solution, assuming that the bridge span is 30 m, we determined that the length of the connecting pipe is 15 m. The area ratio adjusts the natural frequency of the liquid oscillation after determination of the pipe length (equation (10)). According to the correction in the previous section, the suitable measuring frequency range of DHLS is 0 to ωn. Meanwhile, the displacement measured by DHLS must be fused with the acceleration. We determine ωn by the applicable frequency range of the accelerometer. The microelectro-mechanical system (MEMS) accelerometer selected in this study has a minimum measurement frequency of 0.4 Hz under the premise of ensuring product measurement accuracy. Finally, by substituting ωn=0.4Hz and L=15m into equation (10), the area ratio αc/αp=1/4.7 can be obtained.We conducted experiments to test the designed DHLS system. The experimental setup includes a shaking table, an LDM, and a liquid-level measurement system, as shown in Figure5. The device is also used to verify the fusion algorithm in the following section, explaining the installation of the accelerometer.Figure 5
The experimental device of the proposed displacement measurement system.The data required for this experiment include the amplitude of the liquid-level oscillation in the fixed end and deck displacement of the shaking table at different excitation frequencies. The deck displacementy is measured by LDM, erected vertically above the shaking table. The liquid-level measurement system yields the liquid-level change u at the fixed end. To obtain the amplitude-frequency characteristics (1/Dω) of DHLS, we measure the ratio of Yω and Uω under harmonic excitation at an interval of 0.05 Hz between 0 and 0.4 Hz.The experimental measurement ofωn is 0.397 Hz, which is consistent with the expected result. Then, we identify the frequency response function (Hω) of SMA. Figure 6 shows the comparison of measured DHLS amplitude-frequency characteristics and the theoretical value of SMA.Figure 6
Comparison of amplitude-frequency characteristics of DHLS and SMA.


## 3. Fusion Hydraulic Leveling and Accelerometer Signals for Deflection Measurement

Theoretically, the displacement signal is obtained after integrating the acceleration signal twice. Accelerometers are known to have large errors in low-frequency acceleration measurements, and it may produce a drift in the displacement reconstruction. To address this, we apply the corrected hydraulic leveling measurement signal as a supplement to eliminate displacement drift and low-frequency errors from acceleration signals.Acceleration is the second-order differential of displacement, as indicated in the following formula:(21)at=d2utdt2=a¯t+εa,where at is the real acceleration, expressed in terms of displacement through a second-order ordinary differential equation; ut is the real displacement; a¯t is the measured acceleration; and εa is measurement error.Displacement is the double integral of acceleration:(22)ut=∫∫a¯t+εadtdt=u¯t+U0+V0t+12εat2,where u¯t is the displacement obtained by a double integral of measured acceleration and U0, V0, and εa can be considered as the initial displacement, velocity, and acceleration, respectively.If the initial state is identified, we easily obtain the reconstructed displacement. The following formula is a vector form used to identify the initial state for discrete measured data.(23)ut1ut2ut3⋮utn=000⋯0100⋯0210⋯0⋮⋮⋮⋱0n−1n−2n−3⋯10a¯t1a¯t2⋮a¯tn−1·∆t2+U0111⋮1+V0123⋮n∆t+12εa122232⋮n2∆t2.The start time of this string of measured data ist0, the sampling interval is ∆t, the total time is T=n∆t, and the measured acceleration is a¯ti.To obtainU0, V0, and εa, we must fuse the corrected displacement (Ul) measured by DHLS. Specifically, we subtract Ul=ult1ult2ult3⋯ultnT from equation (23) and obtain the displacement residual ∆U as follows:(24)∆U=000⋯0100⋯0210⋯0⋮⋮⋮⋱0n−1n−2n−3⋯10a¯t1a¯t2⋮a¯tn−1·∆t2+U0111⋮1+V0∆t123⋮n+12εa∆t2122232⋮n2−ult1ult2ult3⋮ultn.By collecting unknown variables, we obtained the following equation:(25)∆U=100⋯0210⋯0321⋯0⋮⋮⋮⋱0nn−1n−2⋯10a¯t1a¯t2⋮a¯tn−1−Ul+1∆t12∆t212∆t222∆t213∆t322∆t2⋮⋮⋮1n∆tn22∆t2U0V0εa.We rewrite the above equation into a simple form as follows:(26)∆U=La¯−Ul+Cx¯,where L=100⋯0210⋯0321⋯0⋮⋮⋮⋱0nn−1n−2⋯1, a¯=0a¯t1a¯t2⋮a¯tn−1, Ul=ult1ult2ult3⋮ultn, and C=1∆t1/2∆t212∆t22/2∆t213∆t32/2∆t2⋮⋮⋮1n∆tn2/2∆t2 are already known, and only the initial state vector x¯=U0V0εa is unknown.The initial state recognition is transformed into parameter optimization by the following formula:(27)Min∏∆U=La¯−Ul+Cx¯22=La¯−Ul+Cx¯TLa¯−Ul+Cx¯.To obtain the solution of the unknown initial state vectorx¯ in equation (27), we let the derivative of La¯−Ul+Cx¯TLa¯−Ul+Cx¯ equal to zero. Then, x¯ can be represented as follows:(28)x¯=U0V0εaT=CTC−1−CTLa¯−Ul.Finally, we obtain the reconstruction displacement by substituting the initial stateU0,V0,εa into equation (23). The proposed displacement reconstruction method is shown in Figure 7.Figure 7
Process of the displacement reconstruction.


## 4. Experiments

We reconstructed the displacement by fusing the hydraulic leveling accelerometer signals. To this end, we redesigned the hydraulic leveling system in Section2 for the dynamic displacement measurement. The experimental test device in this section is the same as shown in Figure 5. To achieve continuous displacement measurement, the system also includes data processing, storage, and wireless transmission parts. The sensors have different sampling frequencies, and data must be synchronized before applying the proposed algorithm for displacement reconstruction. We added a timestamp to each acquired data during the measurement, and the data format of the measured acceleration acctai and liquid-level Leveltai signal is shown as follows:(29)acctai=ta1a1ta2a2⋮⋮tanan,LeveltLi=tL1L1tL2L2⋮⋮tLmLm.Then, we fit a cubic spline difference function to the liquid level acquisition data. Finally, we inserted the timetai of the acceleration data acctai into the function to obtain the liquid level after the difference as given by equation (30). In this manner, synchronization between both approaches is achieved and adopted.(30)Leveltai=ta1L1ta2L2⋮⋮tanLn.A series of tests are conducted to verify whether the proposed displacement measurement system and matching algorithm can meet expectations. The experiment separately measures the harmonic, nonharmonic, and impact displacements.


### 4.1. Harmonic Excitation

We test the accuracy of this system under single-frequency harmonic excitation between 0 and 3 Hz. Figure8 shows the comparison of the proposed system’s displacement measuring results with the LDM. The displacement measured by the proposed system is the same as that of the LDM at 0.2, 0.5, 1.0, and 3.0 Hz.Figure 8
Comparison of displacement measured by the proposed system with LDM (the estimate is the displacement measured by the proposed system and the reference is measured by the LDM).We employed RMSE to assess the measurement accuracy as follows:(31)RMSE=∑i=1Nrefi−esti2N,where refi is the data point measured by the LDM and esti is the data point measured by the proposed system. Furthermore, we collected several measurements at different frequencies between 0 and 3.0 Hz.Figure9 shows the RMSE of this displacement measurement system at different frequencies. The RSME is below 0.9 mm for all measured frequencies between 0.11 and 3.08 Hz (the frequency interval below 1.0 Hz is 0.1 Hz, between 1.0 and 2.0 Hz is 0.25 Hz, and between 2.0 and 3.0 Hz is 0.5 Hz). We concluded that this measurement system has high accuracy for harmonic displacement excitation.Figure 9
RMSE at different frequencies.


### 4.2. Nonharmonic Excitation

We test the displacement measurement accuracy under the nonharmonic excitation. According to different frequency concentration ranges, excitations are sorted into four types as shown in Table2.Table 2
Settings of nonharmonic displacement excitation test conditions.
CaseDetailsAExcitation occurs mainly in low frequencyBExcitation occurs mainly in high frequencyCExcitation frequency includes low frequency (<0.4 Hz) and high frequency (>0.4 Hz), and displacement excitation amplitude of low frequency is largerDExcitation frequency includes low frequency (<0.4 Hz) and high frequency (>0.4 Hz), and displacement excitation amplitude of the high frequency is largerFigure10 shows the displacement measured by the proposed system and LDM when the excitation occurs mainly at low frequencies. In Figure 10, the estimate is the displacement measured by the proposed system, the reference is the displacement measured by the LDM, and the measured results of the LDM are considered as the real displacement (as in Figures 10–13). The frequency components with large amplitudes can be observed mainly in 0–0.4 Hz. The RMSE of the system under this excitation is 0.64 mm, and Figure 10 shows that the error is mainly concentrated in 0–0.2 Hz.Figure 10
Absolute displacement and spectrum of experimental case A.Figure11 shows the comparison of the displacements obtained by the proposed system and LDM with the excitation mainly in the high-frequency range. The frequency components with large amplitudes are observed mainly in the range 2.0–2.5 Hz. The RMSE at this excitation is 0.61 mm. Figure 11 indicates that the error is mainly concentrated in the range 0–1.0 Hz.Figure12 shows the case when the amplitude of the low frequency is larger. In this situation, the RMSE of the system under this excitation is 0.35 mm. We can see that the error is small among the measured frequency range in this case.Figure13 illustrates the case with larger high frequency amplitude. In this situation, the RMSE of the system under this excitation is 0.55 mm.


### 4.3. Impact Excitation

Figure14 compares the displacement measurement results of the proposed system with the LDM under impact loads. The RSME of this displacement measurement system is 0.54 mm.Figure 11
Absolute displacement and spectrum of experimental case B.Figure 12
Absolute displacement and spectrum of experimental case C (amplitude of low frequency is larger).Figure 13
Absolute displacement and spectrum of experimental case D (amplitude of high frequency is larger).Figure 14
Absolute displacement of experimental condition under impact excitation.Table3 contains the comparison of the RMSE of different cases and displacement reconstruction methods. The proposed method is introduced in Section 3. The method for DHLS is introduced in Section 2. The acceleration estimation involves a baseline correction algorithm with three steps: (1) fitting the baseline of acceleration by the least squares, (2) filtering the acceleration to remove the remaining errors and noise with a high-pass filter, and (3) decreasing the initial values in velocity on the first integration result [21, 22].Table 3
Summary of RMSE (mm) for different methods and cases.
IIIIIIIVVProposed system0.640.610.350.550.54Acceleration estimation4.533.572.983.282.44DHLS3.192.595.742.850.76I–IV correspond to case A–D in nonharmonic excitation;V presents impact excitation.Table3 shows that the RMSE of III (case C) is the lowest, which means the system has higher measurement accuracy with larger amplitudes at low frequency. Furthermore, the proposed system exhibits a lower RMSE than other methods, indicating higher measurement accuracy for each experiment.We conclude that the system achieves continuous measurement and is also suitable for impact displacement.


## 4.1. Harmonic Excitation

We test the accuracy of this system under single-frequency harmonic excitation between 0 and 3 Hz. Figure8 shows the comparison of the proposed system’s displacement measuring results with the LDM. The displacement measured by the proposed system is the same as that of the LDM at 0.2, 0.5, 1.0, and 3.0 Hz.Figure 8
Comparison of displacement measured by the proposed system with LDM (the estimate is the displacement measured by the proposed system and the reference is measured by the LDM).We employed RMSE to assess the measurement accuracy as follows:(31)RMSE=∑i=1Nrefi−esti2N,where refi is the data point measured by the LDM and esti is the data point measured by the proposed system. Furthermore, we collected several measurements at different frequencies between 0 and 3.0 Hz.Figure9 shows the RMSE of this displacement measurement system at different frequencies. The RSME is below 0.9 mm for all measured frequencies between 0.11 and 3.08 Hz (the frequency interval below 1.0 Hz is 0.1 Hz, between 1.0 and 2.0 Hz is 0.25 Hz, and between 2.0 and 3.0 Hz is 0.5 Hz). We concluded that this measurement system has high accuracy for harmonic displacement excitation.Figure 9
RMSE at different frequencies.


## 4.2. Nonharmonic Excitation

We test the displacement measurement accuracy under the nonharmonic excitation. According to different frequency concentration ranges, excitations are sorted into four types as shown in Table2.Table 2
Settings of nonharmonic displacement excitation test conditions.
CaseDetailsAExcitation occurs mainly in low frequencyBExcitation occurs mainly in high frequencyCExcitation frequency includes low frequency (<0.4 Hz) and high frequency (>0.4 Hz), and displacement excitation amplitude of low frequency is largerDExcitation frequency includes low frequency (<0.4 Hz) and high frequency (>0.4 Hz), and displacement excitation amplitude of the high frequency is largerFigure10 shows the displacement measured by the proposed system and LDM when the excitation occurs mainly at low frequencies. In Figure 10, the estimate is the displacement measured by the proposed system, the reference is the displacement measured by the LDM, and the measured results of the LDM are considered as the real displacement (as in Figures 10–13). The frequency components with large amplitudes can be observed mainly in 0–0.4 Hz. The RMSE of the system under this excitation is 0.64 mm, and Figure 10 shows that the error is mainly concentrated in 0–0.2 Hz.Figure 10
Absolute displacement and spectrum of experimental case A.Figure11 shows the comparison of the displacements obtained by the proposed system and LDM with the excitation mainly in the high-frequency range. The frequency components with large amplitudes are observed mainly in the range 2.0–2.5 Hz. The RMSE at this excitation is 0.61 mm. Figure 11 indicates that the error is mainly concentrated in the range 0–1.0 Hz.Figure12 shows the case when the amplitude of the low frequency is larger. In this situation, the RMSE of the system under this excitation is 0.35 mm. We can see that the error is small among the measured frequency range in this case.Figure13 illustrates the case with larger high frequency amplitude. In this situation, the RMSE of the system under this excitation is 0.55 mm.


## 4.3. Impact Excitation

Figure14 compares the displacement measurement results of the proposed system with the LDM under impact loads. The RSME of this displacement measurement system is 0.54 mm.Figure 11
Absolute displacement and spectrum of experimental case B.Figure 12
Absolute displacement and spectrum of experimental case C (amplitude of low frequency is larger).Figure 13
Absolute displacement and spectrum of experimental case D (amplitude of high frequency is larger).Figure 14
Absolute displacement of experimental condition under impact excitation.Table3 contains the comparison of the RMSE of different cases and displacement reconstruction methods. The proposed method is introduced in Section 3. The method for DHLS is introduced in Section 2. The acceleration estimation involves a baseline correction algorithm with three steps: (1) fitting the baseline of acceleration by the least squares, (2) filtering the acceleration to remove the remaining errors and noise with a high-pass filter, and (3) decreasing the initial values in velocity on the first integration result [21, 22].Table 3
Summary of RMSE (mm) for different methods and cases.
IIIIIIIVVProposed system0.640.610.350.550.54Acceleration estimation4.533.572.983.282.44DHLS3.192.595.742.850.76I–IV correspond to case A–D in nonharmonic excitation;V presents impact excitation.Table3 shows that the RMSE of III (case C) is the lowest, which means the system has higher measurement accuracy with larger amplitudes at low frequency. Furthermore, the proposed system exhibits a lower RMSE than other methods, indicating higher measurement accuracy for each experiment.We conclude that the system achieves continuous measurement and is also suitable for impact displacement.


## 5. Field Test

To assess the performance of the displacement measurement system and algorithm, a T-shaped rigid frame bridge with a span of 36 m was selected for the field test. The field test was conducted to measure the midspan deflection.


### 5.1. Device for Field Test

We mounted one liquid container of the measurement system at the midspan (moving end). The displacement at the pier position is assumed to be sufficiently small to omit, therefore another container was installed at the pier position (fixed end). Figure15 shows the bridge condition and location of the sensors.Figure 15
Configurations of measurement systems during the field test: (a) drawing of the test device in the field, (b) field-test bridge, (c) location of each sensor, (d) LDM and target under the bridge at the midspan, (e) construction of proposed system, and (f) system installation location on the bridge.
(a)(b)(c)(d)(e)(f)The main environmental condition affecting the measurements is the temperature. The system also employs image recognition, such that lighting must also be considered at night. The temperature affects the water volume, changing the water level in the tube. The level sensor measures the difference between the two water levels. As the water levels of the two ends from temperature change simultaneously, the measured water level difference remains unaffected by the temperature change. A LED light tape was placed to illuminate the tubes and enable measurement of the water level at night (Figure15(e)).


### 5.2. Results of the Field Test

Traffic was not blocked on the bridge during the whole test process, and vehicles could pass freely. An LDM was also arranged at the midspan position under the bridge to measure the absolute displacement. The confirmation measurement lasted for ten minutes. We verified the accuracy of the proposed system by comparing the measurement data with the LDM.We considered 13 min of displacement data measured by the LDM and the proposed system, shown in Figure16. In order to show the details, we ploted the measured data range 300–450 s in another window. The large waveforms are the deformation caused by vehicles passing the bridge and small waveforms are the high-frequency vibrations associated with the modal frequency of the bridge. The big waveforms are basically the same between proposed system and laser (LDM), but there are errors in small waves. Furthermore, we determined the source of measurement error through spectral analysis. Herein, the RMSE of the proposed system is 0.46 mm. Figure 17 shows the spectrogram of the displacement component measured by LDM and the proposed system. The maximum error is 0.6-0.7 mm, mainly caused by the DHLS measurement error, in the frequency range below 0.4 Hz. We further observe that at approximately 2.5 Hz, there is a maximum error of 0.3 mm, caused by the LDM measurement error (as there is a traffic lane under the field-test bridge, and the passing vehicles vibrate the bracket where LDM is installed).Figure 16
Absolute displacement using proposed approach in field test.Figure 17
Spectrogram of displacement component of measured data.


### 5.3. Practicality of the System

The camera and accelerometer used in the system are industrial-grade products with good durability. The camera can work in the temperature range of −10–65°C, and the accelerometer is in the range of −40–85°C. The system is arranged in a box with dustproof and waterproof functions (shown in Figure18(a)), which makes the sensor suitable for working in field conditions.Figure 18
(a) Dustproof and waterproof box and (b) the components arranged in the box in advance.
(a)(b)The entire measurement system in the box can be premanufactured at the factory. The communicating pipe is needed to be linked to the box through the predrilled holes (shown in Figure18(b)) when the system is installed on the bridge. The whole installation process is convenient and expeditious. After installation, the door of the box is closed to protect the measuring system.While completing the new system, the cost issue was also considered. Since price of the system is very limited (less than $100 when purchased in China), the cost for maintenance and replacement of system is acceptable for practical application.


## 5.1. Device for Field Test

We mounted one liquid container of the measurement system at the midspan (moving end). The displacement at the pier position is assumed to be sufficiently small to omit, therefore another container was installed at the pier position (fixed end). Figure15 shows the bridge condition and location of the sensors.Figure 15
Configurations of measurement systems during the field test: (a) drawing of the test device in the field, (b) field-test bridge, (c) location of each sensor, (d) LDM and target under the bridge at the midspan, (e) construction of proposed system, and (f) system installation location on the bridge.
(a)(b)(c)(d)(e)(f)The main environmental condition affecting the measurements is the temperature. The system also employs image recognition, such that lighting must also be considered at night. The temperature affects the water volume, changing the water level in the tube. The level sensor measures the difference between the two water levels. As the water levels of the two ends from temperature change simultaneously, the measured water level difference remains unaffected by the temperature change. A LED light tape was placed to illuminate the tubes and enable measurement of the water level at night (Figure15(e)).


## 5.2. Results of the Field Test

Traffic was not blocked on the bridge during the whole test process, and vehicles could pass freely. An LDM was also arranged at the midspan position under the bridge to measure the absolute displacement. The confirmation measurement lasted for ten minutes. We verified the accuracy of the proposed system by comparing the measurement data with the LDM.We considered 13 min of displacement data measured by the LDM and the proposed system, shown in Figure16. In order to show the details, we ploted the measured data range 300–450 s in another window. The large waveforms are the deformation caused by vehicles passing the bridge and small waveforms are the high-frequency vibrations associated with the modal frequency of the bridge. The big waveforms are basically the same between proposed system and laser (LDM), but there are errors in small waves. Furthermore, we determined the source of measurement error through spectral analysis. Herein, the RMSE of the proposed system is 0.46 mm. Figure 17 shows the spectrogram of the displacement component measured by LDM and the proposed system. The maximum error is 0.6-0.7 mm, mainly caused by the DHLS measurement error, in the frequency range below 0.4 Hz. We further observe that at approximately 2.5 Hz, there is a maximum error of 0.3 mm, caused by the LDM measurement error (as there is a traffic lane under the field-test bridge, and the passing vehicles vibrate the bracket where LDM is installed).Figure 16
Absolute displacement using proposed approach in field test.Figure 17
Spectrogram of displacement component of measured data.


## 5.3. Practicality of the System

The camera and accelerometer used in the system are industrial-grade products with good durability. The camera can work in the temperature range of −10–65°C, and the accelerometer is in the range of −40–85°C. The system is arranged in a box with dustproof and waterproof functions (shown in Figure18(a)), which makes the sensor suitable for working in field conditions.Figure 18
(a) Dustproof and waterproof box and (b) the components arranged in the box in advance.
(a)(b)The entire measurement system in the box can be premanufactured at the factory. The communicating pipe is needed to be linked to the box through the predrilled holes (shown in Figure18(b)) when the system is installed on the bridge. The whole installation process is convenient and expeditious. After installation, the door of the box is closed to protect the measuring system.While completing the new system, the cost issue was also considered. Since price of the system is very limited (less than $100 when purchased in China), the cost for maintenance and replacement of system is acceptable for practical application.


## 6. Conclusions

This study provides a reference point-free dynamic deflection measurement system for bridges. The proposed system is composed of a DHLS and an accelerometer, and the absolute displacement is reconstructed by fusing the two signals.The results of this study are listed as follows:(1)
The correction method for DHLS based on the level forced oscillation equation is proposed. The DHLS performance is tested in the laboratory, obtaining a RMSE below 0.41 mm.(2)
The signal fusion algorithm mainly consists of the following steps: Step 1: Displacement reconstruction equation composed of measured acceleration and initial motion conditions; Step 2: Subtraction of the liquid level measured displacement from the reconstructed displacement and obtaining the residual; Step 3: Obtaining the initial state of motion under the condition of minimum displacement residual; and Step 4: Substituting the initial state into the equation to obtain the reconstruction displacement. The experiment result demonstrates that the fusion algorithm improves the adaptation and reduces the displacement drift caused by the acceleration integral.(3)
The field test shows that the proposed system achieves continuous high-precision deflection measurement. Furthermore, the system can be optimized by improving the measurement accuracy of DHLS, which is the main contributor to deflection reconstruction errors.



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*Source: 3203261-2023-02-09.xml*