IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 6, DECEMBER 2013
1675
Displacement Estimation Using Multimetric Data Fusion Jong-Woong Park, Sung-Han Sim, and Hyung-Jo Jung
Abstract—While displacement is valuable information for the structural behavior, measuring displacements from large civil structures is often challenging and costly. To overcome difficulties found in direct measurements such as using linear variable differential transformer and LASER-based methods, indirect displacement estimation approaches are alternatively developed. Such indirect approaches in general rely on acceleration or strain that is relatively cost effective and convenient to measure. However, these measurements have own characteristics that limit wider application of the indirect estimation. For example, as the double integration of acceleration results in the low-frequency drift in the estimated displacement, high-pass filters are often used to suppress the drift, assuming displacements are close to a zero mean process; strain is difficult to use for high-frequency modes. These types of limitations can be resolved by the fusion of different types of measurements. This study develops an indirect displacement estimation method based on the multimetric data (i.e., acceleration and strain) that can estimate nonzero mean, dynamic displacements. The proposed approach is numerically validated, showing better estimation than the single measurement-based methods. Furthermore, the performance of the proposed approach is verified using dynamic response data measured from the Sorok Bridge, a cablestayed bridge in Korea. Index Terms—Acceleration, bridge displacement, calibration testing-free, data fusion, displacement estimation, strain.
I. INTRODUCTION N the field of civil engineering, maintaining civil infrastructure is important as structural failures can lead to catastrophic results. For timely maintenance to prevent such failures, structural health should be appropriately assessed. A displacement measurement is regarded as not only a direct indicator for the structural integrity as it is related to structural flexibility but useful information for various engineering applications such as earthquake engineering, and system identification. Available techniques for measuring displacement responses from a structure can be divided into two groups: 1) direct measurement
I
Manuscript received January 2, 2013; revised April 15, 2013; accepted July 23, 2013. Date of publication August 15, 2013; date of current version December 11, 2013. Recommended by Guest Editor J. G. Chase. This work was supported in part by the Year of 2011 Research Fund of UNIST (Ulsan National Institute of Science and Technology) and the National Research Foundation of Korea grant funded by the Korea government (MEST) (NRF-2008-220-D00117 and NRF-2012-R1A1A1-042867). (Corresponding author: S.-H. Sim) J.-W. Park and H.-J. Jung are with the Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail:
[email protected];
[email protected]). S.-H. Sim is with the School of Urban and Environmental Engineering, Ulsan National Institute of Science and Technology, Ulsan 689-798, Korea (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2013.2275187
and 2) indirect estimation approaches. The direct methods include the linear variable differential transformer (LVDT), laser Doppler vibrometer (LDV) [1], and vision-based system [2]. While the direct measurement is accurate, these approaches in general rely on a fixed reference point for each sensor, which is often costly, difficult to install, or unavailable in full-scale civil structures. Thus, displacement information from the civil structures is quite limited if the direct methods are employed especially when displacements at multiple points are required. The indirect displacement estimation utilizes other types of measurements such as velocity, acceleration, and strain that can be converted to displacement. The estimation accuracy of different indirect approaches varies depending on the measurement types and conversion methods. If accuracy issues are appropriately handled, the reference-free nature allows the indirect estimation to be conveniently applied to full-scale structures. The acceleration, in particular, has been commonly acquired for various engineering applications in practice due to the convenience of sensor installation and cost effectiveness. Thus, estimating displacement from measured acceleration can be an attractive alternative to the direct measurement of displacement. However, this approach based on the double integration of acceleration has an intrinsic error caused by the numerical integration in the discrete time domain. The error becomes significantly large through successive integrations, resulting in lowfrequency drifts in the estimated displacement signal. As such, most displacement estimation schemes have focused on minimizing errors that accumulate through successive integration of acceleration records [3]–[5]. Lee et al. [5] suggested a dynamic displacement estimation method using acceleration. This approach minimizes the drift error by removing low-frequency components below the first natural frequency to minimize the low-frequency drift, however, limiting the method only to the zero-mean dynamic displacement. Efforts have also been made to obtain the low-frequency components in the dynamic displacement by combining acceleration measured at high sampling rate with displacement measured at low sampling rates from GPS-like devices [6], [7]. However, due to the low accuracy and high device costs, full-scale applications of the methods to civil engineering structures are quite limited. Alternatively, strain measurements are considered for estimating the displacement that has the low-frequency components [8]–[14]. The use of strain-based indirect methods provides broad sensing frequency which is dependent on a data acquisition device and accurate displacement especially in the low-frequency range. Foss and Hauge [9] have proposed a basis of the strain–displacement modal mapping method to transform
1083-4435 © 2013 IEEE
1676
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 6, DECEMBER 2013
a number of discrete strain measurements into dynamic global displacement using modal properties of a structure. However, the estimated displacement from strain is vulnerable to measurement noise [10], [12], although the drift error is not involved. This study proposes an indirect estimation method for nonzero mean dynamic displacement by the fusion of multimetric data (i.e., acceleration and strain). The fundamental idea of the proposed approach is based on the fact that the accelerationbased methods accurately estimate the zero-mean displacement, while the strain-based can capture the low-frequency components. To this end, the acceleration-based method proposed by Lee et al. [5] is extended to accommodate strain measurements using the strain–displacement relationship proposed by Shin et al. [11]. Furthermore, this paper proposes a scaling method that can effectively replace the calibration experiment necessary for the field testing using the strain–displacement relationship. While the previous works only used a single type of measurement, and thus, have limited real-world applications, the proposed approach resolves those issues by the fusion of multimetric data. To validate the proposed approach, numerical simulation is performed on a simple beam model to investigate the effect of measurement error, and subsequently, field testing on a full-scale suspension bridge is conducted. II. BACKGROUND This section describes basic principles of indirect displacement estimation from acceleration and strain. A. Dynamic Displacement Estimation From Acceleration The acceleration-based method proposed by Lee et al. [5] is considered to be suitable for extension to the use of multimetric data (i.e., acceleration and strain) for accurate displacement estimation. For completeness, the estimation approach is briefly introduced herein and more detailed information is found in Lee et al. [5]. The displacement is determined by solving the following optimization problem: Min Π = u
2 λ2 1 La (Lc u − (Δt)2 a u22 ¯)2 + 2 2
(1)
where Δt, u, and a ¯ are time step, the vector of displacements at the discrete time steps, and the vector of measured acceleration, respectively. La is a diagonal weighting matrix with all diagonal entries √ of 1 except the first and last entries, which are equal to 1 2, Lc is the second-order differential operator matrix of the discretized trapezoidal rule [15], · 2 denotes two norm of a vector, and λ represents the optimal regularization factor which adjusts the degree of the regularization in the minimization problem. The first term in the right side of (1) is related to the error between the measured acceleration a ¯ and second-order time derivative of displacement u. Introducing, the second term, the low-frequency drift associated with the integration error can be removed. Thus, minimizing the scaled sum of these two errors is to find displacement that is a zero-mean process and close to one by the double integration of acceleration. The solution to
Fig. 1.
Displacement estimation scheme using overlapping time windows [5].
(1) can be written as ¯(Δt)2 = Ca a ¯(Δt)2 u = (LT L + λ2 I)−1 LT La a
(2)
where L = La Lc , the superscript T denotes the matrix transpose, Ca = (LT L + λ2 I)−1 LT La , and λ is the optimal regularization parameter defined as λ = 46.81Nd−1.95
(3)
where Nd is the number of data points corresponding to usually three times data points during the first natural period. The estimated displacement in (2) has increasing errors near the boundaries (i.e., beginning and end of u), significantly degrading the accuracy. To resolve this issue, an overlapping moving window technique is used. For each time window shown in Fig. 1, the displacement time history is calculated using (2). Subsequently, the point at the center of the time window is selected and considered to be the estimated displacement. Moving the time window by Δt and repeat the same process, the whole displacement can be obtained except in the boundaries due to the size of the time window. This procedure can be considered as applying an FIR filter defined as Ca Δt2 to the acceleration signal a ¯. B. Displacement Estimation From Strain A linear relationship between dynamic displacement and strain responses can be formulated using the modal approach. Consider a system with displacement and strain responses {u}m ×1 and {ε}n ×1 , where m and n are the numbers of measurements. The displacement and strain measurements can be approximated using the linear combination of the finite number of modes. {u}m ×1 = Φm ×r {q}r ×1
(4)
{ε}n ×1 = Ψn ×r {q}r ×1
(5)
where {Φ}m ×r and {Ψ}n ×r are respective displacement and strain mode shape matrices, {q}r ×1 is the modal coordinate, and r is the number of used modes. When n ≥ r, the modal coordinate {q} can be obtained from (5) as {q} = Ψ+ {ε}
(6)
PARK et al.: DISPLACEMENT ESTIMATION USING MULTIMETRIC DATA FUSION
1677
where the superscript + denotes the Moore–Penrose pseudo inverse [16]. The strain–displacement relationship is obtained by substituting {q}r ×1 in (4) with (6)
Thus, the strain–displacement relationship for the simply supported beam is obtained. Strain can be incorporated with acceleration in displacement estimation by extending (1). Using the linear relationship of (12) by Shin et al. [11], the minimization problem in (1) is modified for displacement ui at the location of xi as
{u} = Φ Ψ+ {ε}.
(7)
Given the strain measurements, displacement responses can be obtained if the displacement and strain mode shapes are known.
2 λ2 1 La (Lc ui − Δt2 a ui − Di {¯ ¯i )2 + ε}22 u 2 2 (13) ε} is measured strains. The where D i is the ith row of D and {¯ solution of (13) is Min Π =
III. FORMULATION FOR MULTIMETRIC DATA FUSION Based on the previous works focused on a single type measurement (i.e., acceleration- and strain-based methods), the proposed approach is extended to accommodate both acceleration and strain for the better performance. Due to the multimetric data fusion, the proposed approach exhibits two distinct features: 1) high accuracy both in low- and high-frequency regions and (b) calibration testing-free method. This section introduces the formulation of the proposed approach to describe how acceleration and strain are combined for displacement estimation. A. Displacement Estimation From Acceleration and Strain This study focuses on beam-like structures such as bridges, in which the linear relationship between displacement and strain is conveniently expressed in the following form from (7): {u}m ×1 = Dm ×n {ε}n ×1
(8)
where Dm ×n is a transformation matrix. Shin et al. [11] proposed a displacement–strain relationship for a simply supported beam. Displacement and strain can be expressed using the mode superposition as εk (t) = −u (xk , t)y = −y φk j qj (t) (9) j
where ui and εk are respective displacement at xi and strain at xk , φij is a mode shape value of the jth mode at xi , y is the neutral axis of the beam, and qj is the jth modal coordinate. Knowing the mode shapes of the simply supported beam are sinusoidal functions, (9) becomes yπ 2 2 jπxk qj (t) j sin (10) εk (t) = 2 L j L where L is the length of the beam. If the number of strain measurements is at least the number of modes, the generalized coordinate qj (t) can be expressed in terms of strain as ⎡ πx1 rπx1 ⎤+ sin · · · r2 sin ⎡ ⎤ ⎡ ⎤ q1 ε1 L L ⎥ 2 ⎢ L ⎢ ⎥ .. ⎦ . . ⎣ .. ⎦ = ⎣ . ⎢ ⎥ .. .. .. . . . yπ 2 ⎣ ⎦ πx rπx εn n n qr · · · r2 sin sin L L (11) Substituting (11) into (9) ⎡ πx1 rπx1 ⎤+ sin · · · r2 sin L L ⎥ L2 ⎢ ⎢ ⎥ . . . Φ⎢ (12) D= ⎥ . .. .. .. yπ 2 ⎣ ⎦ πxn rπx n · · · r2 sin sin L L
¯i Δt2 + λ2 Di {¯ ε}) ui = (LT L + λ2 I)−1 (LT La a a ¯i = ( Ca Δt2 Cε ) {¯ ε}
(14)
where Cε = (LT L + λ2 I)−1 λ2 Di . Thus, displacement in (14) is expressed in terms of measured acceleration and strains. The proposed displacement estimation by (14) finds an optimal displacement from acceleration and strain measurements. The acceleration-based method in (2) intrinsically ignores lowfrequency components because of the second term in the right side of (1). However, by introducing strain in the term rather than using the norm of u, the low-frequency component of u can be captured. B. Scaling Factor for the Strain–Displacement Relationship The most critical issue in using the strain–displacement relationship is to determine the location of neutral axis that can be approximately obtained from the finite-element model. However, determination of neural axis is difficult as the most bridge structures are composed of composite materials such as concrete and steel or have various cross-sectional shapes. Thus, a calibration process is required using static or dynamic testing with the accurate reference measurement such as LVDT or LDV to obtain the scaling factor in field testing. The calibration process is considered as one of the most significant drawbacks in the strain-based approach. This study proposes a multimetric data-based approach for scaling factor estimation, in which the calibration experiment is unnecessary. The combination of the strain- and acceleration-based methods can provide the scaling factor without the calibration experiment. The scaling factor can be obtained by adjusting the displacement from strain measurements to the displacement estimated from acceleration. Although the displacement from acceleration has the low-frequency drift error, the energy of the natural modes beyond the low-frequency regions is accurately calculated. Thus, the adjustment can be made in frequency domain by matching the power spectral density of the displacement from strain to that from acceleration as in (15).
Sd,acc (fn ) (15) α= Sd,strain (fn ) where α is the scaling factor, Sd,acc and Sd,strain are the respective power spectral densities of the displacements estimated
1678
Fig. 2.
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 6, DECEMBER 2013
Simply supported beam and its first three modes.
TABLE I DETAILS OF THE BEAM MODEL
from acceleration and strain, and fn is the most clear natural frequency of the estimated displacements from acceleration and strain, which is generally the first mode.
IV. NUMERICAL VALIDATION A. Simulation Setup Consider the simply supported beam modeled with 16 Euler– Bernoulli beam elements as shown in Fig. 2 and Table I. A moving load with a velocity of v = 0.1 m/s is applied vertically from the left to the right to generate nonzero mean displacement responses (see Fig. 3). The moving load consists of static load of 10 N and zero-mean Gaussian random load with a standard deviation of 3 N. Time history analysis is conducted using MATLAB Simulink to simulate acceleration and strain for displacement estimation as well as displacement as reference data for accuracy evaluation. 5% and 10% root mean square noise levels are considered in acceleration and strain, respectively. Dynamic displacements of the beam are estimated using 1 multimetric data, 2) strain-only, and 3) acceleration-only, of which accuracy is compared to each other. In determining the strain and displacement relationship D in both multimetric data and strain-only methods, following two aspects are considered for more realistic simulation. Only first three mode shapes of the beam (see Fig. 2) are used because obtaining higher modes accurately from the field testing is difficult in practice.
Fig. 3.
Moving load.
B. Location of the Neutral Axis As the true location of neutral axis y is unknown in most field testing, randomness is considered in this simulation as yr = y + r
(16)
where yr is the location of the neutral axis used in the multimetric data and strain-only methods and r is a random number that follows Gaussian random distribution with mean of 0 and standard deviation of 0.3y. The proposed scaling factor calculation discussed in Section III-B is employed in the multimetric data-based approach. C. Comparison in Time and Frequency Domains The displacement response at N9 is estimated using acceleration at N9 and strains at N4, N8, and N12 from the three approaches (i.e., multimetric data, strain-only, and accelerationonly), and compared in Fig. 4. The performance of the proposed approach using multimetric data fusion is distinct, which can be summarized as follows. 1) Multimetric data-based method can find nonzero mean displacement, while the acceleration-only discards the low-frequency component [see Fig. 4(a)].
PARK et al.: DISPLACEMENT ESTIMATION USING MULTIMETRIC DATA FUSION
1679
TABLE II SIMULATION CASES
TABLE III COMPARISON OF ESTIMATED DISPLACEMENTS FOR CASE 2
account of the randomness in the excitation, measurement, and the location of the neutral axis. The mean value of the error defined in (17) is evaluated for the three indirect estimation approaches (i.e., multimetric data, strain-only, and accelerationonly) Err = E [σ (destim ated − dreference )/σ (dreference )]
Fig. 4.
Comparison of displacements in time and frequency domains.
2) The high-frequency noise in the strain-only is not present in the displacement from the multimetric data-based approach [see Fig. 4(b)]. 3) Magnitudes of the exact displacement and one from the multimetric data-based approach match quite close to each other, while the strain-only case has distinct deviation from the exact due to the uncertainty in the neutral axis [see Fig. 4(b)]. 4) The previous three aspects can be verified in the frequency domain as well [see Fig. 4(c)]. As such, the proposed multimetric data-based approach can accurately estimate dynamic displacements with higher accuracy than the approaches relying on single measurement. D. Comparison by Monte Carlo Simulation A total of four simulation cases shown in Table II are considered to evaluate the performance with respect to the number of strain measurements. Each case is repeated 100 times to take
(17)
where E[·] is the mean value, σ(·) is the standard deviation, and destim ated and dreference are estimated and reference displacement signals, respectively. The mean values are shown in Table III for case 2 and Fig. 5 for all cases. Note that Fig. 5 only compares the strain-only and multimetric data cases because the error level of the acceleration-only is much larger than the others for this specific example with nonzero mean displacements. Case 1 in Fig. 5 has large errors, meaning unsuccessful estimation; cases 2–4 clearly verify the performance improvement by the multimetric data-based method. Thus, at least three strain measurements are desired for successful estimation, while more measurements can increase the accuracy as shown in Fig. 6. From the simulation, the use of the multimetric data is shown to clearly improve the estimation accuracy with the lowest errors in all cases. V. EXPERIMENTAL VERIFICATION The performance of the proposed approach using the multimetric data fusion is verified from the full-scale field experiment in the Sorok Bridge in Korea. The estimated displacement using the multimetric data is compared to those from the strain- and acceleration-only methods as well as the reference measured by the laser sensor. The Sorok Bridge is a monocable self-anchored suspension bridge with total length of 470 m (110+250+110 m) lying in a 1,160 m roadway linking the Sorok Island to the mainland,
1680
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 6, DECEMBER 2013
Fig. 6. Comparison of estimated displacements from the multimetric data fusion in terms of the number of strain measurements.
Fig. 7. Sensor layout on the Sorok Bridge. (a) Sorok Bridge and (b) sensor deployment.
Fig. 5. Comparison of estimated displacements based on the strain-only (back) and multimetric data fusion (front). TABLE IV FBG SENSOR LOCATION
as shown in Fig. 7(a). The sensor system for the experiment is composed of 15 distributed fiber Bragg grating (FBG) sensors as located in Table IV for measuring strain and an accelerometer in the center of the bridge for acceleration measurement as in Fig. 7(b). The laser displacement sensor (i.e., PSM-R, Noptel) is installed on the pier of the first pylon (i.e., PY1) to measure the displacement of midspan by computing the movement of the reflecting target attached in the center of the bridge.
The static loading test is carried out to obtain a scale factor for the strain-only method, while the multimetric data-based approach does not need calibration process. The 29.9 tonf truck is then set to run at 70 km/h to induce the dynamic deflection on the bridge, measuring acceleration and strain to obtain displacements from the three estimation methods, i.e., acceleration-only, strain-only, and multimetric data. Three strain measurements at locations 4, 8, and 12 in Table IV and an acceleration measurement at the center of the bridge are used in the estimation. Because the lower three natural modes are clearly identified from the preliminary modal analysis, the corresponding three mode shapes by the simple beam assumption are used in determining the strain–displacement relationship D. The estimated displacements are compared in time and frequency domains as shown in Fig. 8, in which following observations can be made. 1) The displacement from the multimetric data-based method is close to the reference displacement from the laser sensor with respect to the maximum amplitude and overall trend in the time domain. This observation can be verified in the frequency domain, in that the power spectra of the multimetric data and reference in the low-frequency regions show a good agreement.
PARK et al.: DISPLACEMENT ESTIMATION USING MULTIMETRIC DATA FUSION
1681
TABLE V LOCATION OF STRAINS AND MAXIMUM DISPLACEMENT
Fig. 9. Effect of the number of strain measurement in (a) time domain and in (b) frequency domain.
Fig. 8. Comparison of displacements in time and frequency domain. (a) Displacement estimation, (b) movement at free vibration, and (c) displacement in frequency domain.
2) The multimetric data-based method removes the highfrequency noise as shown in Fig. 8(c). Due to the contribution of acceleration, the displacement from the multimetric data-based method has clear peaks for the second and third modes, while the strain-based and reference displacements have high noise levels that make those two peaks unclear. 3) The acceleration-only approach is unable to capture the low-frequency component as seen in the numerical simulation. 4) The amplitude of displacement in free vibration regions from the strain-only method which is calibrated by static loading test agrees well with that from the multimetric data-based method as shown in Fig. 8(b). 5) The strain-only method shows much larger amplitude than the displacement from laser sensor as the use of three strain measurements is not sufficient for accurate estimation in the field experiment. In addition, the second and third modes are unclear due to the high noise level. Note that the reference displacement from the laser sensor is asymmetric (i.e., peaks at 115 s and 135 s have different amplitudes), while the bridge geometry is symmetric. As the location of the laser sensor can be influenced by the rotational
movement of the pylon (PY1) where the laser sensor is installed, the displacement is expected to have smaller amplitude than the true displacement especially when the vehicle passes by the pylon. In the sense that the laser sensor is influenced by the movement of installed position, the multimetric data-based approach can provide more accurate displacement as it does not rely on reference point. The effect of the number of strain sensors on the performance of the proposed approach is investigated for selected four cases summarized in Table V. The calculated displacements using the multimetric data are shown in Fig. 9; significant accuracy improvement is not observed with larger numbers of strain sensors. In the numerical simulation, the influence of the number of strain sensors is verified to exist yet to be small; due to high uncertainties in the field testing such as measurement noise and mode shape error, the accuracy change with respect to the number of sensors is not clearly seen in the estimated displacements shown in Fig. 9. In this specific example, three strain measurements can be considered to be sufficient as the accurate displacement is achieved with a small number of sensors. VI. CONCLUSION The indirect displacement estimation approach based on the fusion of multimetric data of strain and acceleration was presented with the beam-like structures. The proposed approach features three main improvements from the previous methods: 1) the nonzero mean displacement can be estimated, 2) the high-frequency noise can be significantly reduced, and 3) the
1682
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 6, DECEMBER 2013
calibration experiment is unnecessary. To validate the proposed approach, a series of numerical simulations were conducted on a simply supported beam. The effects of noise and the number of strain measurements were investigated. The results showed that the proposed approach could successfully find the nonzero mean displacement with the lowest error level. The improvements observed in the time domain were also verified in the frequency domain, having the clear peaks for the natural modes due to the low noise level as well as the low-frequency component due to the nonzero mean. Full-scale field testing data from the Sorok Bridge in Korea was utilized to verify the applicability of the proposed approach. The sensor system was installed to measure strain and acceleration from the bridge. The 29.9 tonf truck is set to run at 70 km/h to induce the dynamic deflection on the bridge. The proposed approach using the multimetric data fusion was found to accurately estimate the displacement response without calibration testing with reference data. Thus, the proposed approach is expected to serve as a practical solution for displacement measurements. REFERENCES [1] H. H. Nassif, M. Gindy, and J. Davis, “Comparison of laser doppler vibrometer with contact sensors for monitoring bridge deflection and vibration,” NDT E Int., vol. 38, no. 3, pp. 213–218, 2005. [2] J. J. Lee and M. Shinozuka, “A vision-based system for remote sensing of bridge displacement,” NDT E Int., vol. 39, no. 5, pp. 425–431, 2006. [3] M. Gindy, H. H. Nassif, and J. Velde, “Bridge displacement estimates from measured acceleration records,” Transport. Res. Rec., vol. 2028, pp. 136–145, 2007. [4] K. T. Park, S. H. Kim, H. S. Park, and K. W. Lee, “The determination of bridge displacement using measured acceleration,” Eng. Struct., vol. 27, no. 3, pp. 371–378, 2005. [5] H. S. Lee, Y. H. Hong, and H. W. Park, “Design of an FIR filter for the displacement reconstruction using measured acceleration in low-frequency dominant structures,” Int. J. Numer. Methods Eng., vol. 82, no. 4, pp. 403– 434, 2010. [6] Y. H. Hong, S. G. Lee, and H. S. Lee, “Design of the FEM-FIR filter for displacement reconstruction using accelerations and displacements measured at different sampling rates,” Mech. Syst. Signal Process., 2013. [7] A. Smyth and M. Wu, “Multi-rate Kalman filtering for the data fusion of displacement and acceleration response measurements in dynamic system monitoring,” Mech. Syst. Signal Process., vol. 21, no. 2, pp. 706–723, 2007. [8] S. J. Chang and N. S. Kim, “Estimation of displacement response from FBG strain sensors using empirical mode decomposition technique,” Exp. Mech., vol. 52, no. 6, pp. 573–589, 2012. [9] G. Foss and E. Haugse, “Using modal test results to develop strain to displacement transformations,” in Proc. 13th Int. Modal Anal. Conf., Nashville, TN, USA, 1995, pp. 112–118. [10] L. H. Kang, D. K. Kim, and J. H. Han, “Estimation of dynamic structural displacements using fiber Bragg grating strain sensors,” J. Sound Vib., vol. 305, no. 3, pp. 534–542, 2007. [11] S. Shin, S.-U. Lee, Y. Kim, and N.-S. Kim, “Estimation of bridge displacement responses using FBG sensors and theoretical mode shapes,” Struct. Eng. Mech., vol. 42, no. 2, pp. 229–245, 2012. [12] J. Treiber, U. C. Mueller, J. H. Han, and H. Baier, “Filtering techniques in the dynamic deformation estimation using multiple strains measured by FBGs,” Proc. SPIE, vol. 6932, pp. 69322A-1–69322A-9, 2008.
[13] S. Rapp, L. H. Kang, J. H. Han, U. C. Mueller, and H. Baier, “Displacement field estimation for a two-dimensional structure using fiber Bragg grating sensors,” Smart Mater. Struct., vol. 18, no. 2, 2009. [14] H. I. Kim, L. H. Kang, and J. H. Han, “Shape estimation with distributed fiber Bragg grating sensors for rotating structures,” Smart Mater. Struct., vol. 20, no. 3, 2011. [15] K. E. Atkinson, An Introduction to Numerical Analysis. New York, NY, USA: Wiley, 2008. [16] E. H. Moore, “On the reciprocal of the general algebraic matrix,” Bull. Amer. Math. Soc., vol. 26, pp. 394–395, 1920.
Jong-Woong Park received the B.S degree in civil and environmental engineering from Hanyang University, Seoul, Korea, in 2008 and the M.S degree in civil and environmental engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2009, where he is currently working toward the Ph.D. degree in civil and environmental engineering. His research interests include wireless smart sensor networks for structural health monitoring, damage detection, and sensor fusion algorithms.
Sung-Han Sim received the B.S. and M.S. degrees in civil engineering from Korea Advanced Institute of Science and Technology, Daejeon, Korea, in 2000 and 2002, respectively, and the Ph.D. degree in civil engineering from the University of Illinois at UrbanaChampaign, Champaign, IL, USA, in 2011. He is currently an Assistant Professor at Ulsan National Institute of Science and Technology, Ulsan, Korea. His current research interests include structural health monitoring, smart sensors and sensor networks, damage detection, system identification, and base isolation for nuclear power plants.
Hyung-Jo Jung received the B.S. degree in mechanical engineering and the M.S. and Ph.D. degrees in civil engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1993, 1995, and 1999, respectively. He was a Research Assistant Professor at KAIST and a Visiting Scholar/Postdoctoral Researcher at the University of Notre Dame, Notre Dame, IN, USA. He is currently an Associate Professor in the Department of Civil and Environmental Engineering, KAIST. Prior to joining KAIST, he was an Assistant Professor in the Department of Civil and Environmental Engineering, Sejong University, Seoul, Korea. His current research interests include structural health monitoring using wireless smart sensors, energy harvesting by wind and structural vibration, and structural control using smart materials and IT.
