Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 82 (2017) 339–355

http://d0888-32

n CorrE-m

journal homepage: www.elsevier.com/locate/ymssp

Dynamic displacement estimation by fusing LDV and LiDARmeasurements via smoothing based Kalman filtering

Kiyoung Kim, Hoon Sohn n

Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic ofKorea

a r t i c l e i n f o

Article history:Received 31 October 2014Received in revised form25 March 2016Accepted 22 May 2016Available online 30 May 2016

Keyword:Dynamic displacementLDVLiDARKalman filterData fusionSmoothing

x.doi.org/10.1016/j.ymssp.2016.05.02770/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail address: hoonsohn@kaist.ac.kr (H. Sohn).

a b s t r a c t

This paper presents a smoothing based Kalman filter to estimate dynamic displacement inreal-time by fusing the velocity measured from a laser Doppler vibrometer (LDV) and thedisplacement from a light detection and ranging (LiDAR). LiDAR can measure displace-ment based on the time-of-flight information or the phase-shift of the laser beam re-flected off form a target surface, but it typically has a high noise level and a low samplingrate. On the other hand, LDV primarily measures out-of-plane velocity of a moving target,and displacement is estimated by numerical integration of the measured velocity. Here,the displacement estimated by LDV suffers from integration error although LDV canachieve a lower noise level and a much higher sampling rate than LiDAR. The proposeddata fusion technique estimates high-precision and high-sampling rate displacement bytaking advantage of both LiDAR and LDV measurements and overcomes their limitationsby adopting a real-time smoothing based Kalman filter. To verify the performance of theproposed dynamic displacement estimation technique, a series of lab-scale tests areconducted under various loading conditions.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Dynamic displacement is one of the most important measurands that describe the dynamic characteristics of a structure.From measured displacement, not only velocity, acceleration and deflection can be obtained, but also physical parameters ofa structure can be estimated [1–3]. In spite of the usefulness of displacement information, the direct measurement ofdisplacement is often challenging. For example, displacement measurement with a linear variable differential transformer(LVDT), which is one of the most common devices used to measure displacement in field, requires a direct contact of oneend of LVDT with a target structure and the other end with a fixed scaffold or a firm support [4]. This contact nature of LVDTmakes its installation difficult, and the measurement by LVDT can be easily contaminated by support vibrations [5].

Another common practice is to estimate displacement from acceleration measurement. However, the displacement es-timated in this manner often suffers from double integration error, which is not linearly accumulated during integration [6].Mochas [7] proposed a simple formula for the error produced by simple numerical single- and double-integration of ac-celeration. Traditionally, this type of error has been corrected by baseline correction techniques [8]. Traditionally, this type oferror has been corrected by baseline correction techniques [9]. However, the baseline correction techniques are not suitable

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355340

for real-time estimation because the necessary post-processing can be performed only after the completion of data ac-quisition. Furthermore, different baseline correction techniques can result in different displacement values [8,9]. There areongoing efforts to address this non-unique nature of the displacement estimation by adopting displacement reconstructiontechniques based on finite and infinite impulse filters [10] or two-baseline scheme [11]. However, all these techniquescannot estimate displacement properly when the mean value of the acceleration measurement is non-zero, or there arenonlinear or pseudostatic components in displacement.

Recent advancements in noncontact sensing have opened up new opportunities to measure dynamic displacementwithout the complexity of discrete contact sensors such as LVDT and/or accelerometer. For instance, laser Doppler vib-rometer (LDV) already has been solidifying its position as one of the best velocity and displacement measurement sensors[12,13], thanks to its millimeter-level precision and high sampling rate up to 4 MHz. The main issue with LDV is that it doesnot give a reliable displacement estimate when the change rate of displacement is high, and that the noise level of LDV isheavily affected by the speckle pattern of the target surface [14]. A radar sensor has been used as a low frequency dis-placement sensor in recent years [15,16]. The radar sensor can remotely measure dynamic displacement with an accuracy ofless than 0.1 mm and a sampling frequency of up to 200 Hz. The working principle of the radar sensor is basically identicalto that of LDV except that a microwave is used instead of a laser beam. Therefore, the radar sensor shares the same lim-itation of LDV – it cannot reliably measure a sudden change of displacement.

Another example of noncontact sensing is a vision system [17–22]. The vision system can measure in-plane displacementsimultaneously from multiple points by tracing the pixels on the sequential images captured by a high-resolution and high-speed camera. However, the displacement resolution of the vision system is influenced by both the spatial area captured bya camera and the camera's pixel resolution. Furthermore, its measurement quality is heavily affected by ambient lightconditions. GNSS (global navigation satellite system) can also be applied to dynamic displacement measurement [23–26]. Bythe help of real-time kinematic (RTK) technology, the spatial resolution and sampling rate of GNSS have been improved to1–3 cm and 20 Hz, respectively. The limitations of GNSS include (1) performance dependency onweather conditions and thenumber of available satellites and (2) performance degradation in urban areas due to multipath problems [27]. Light de-tection and ranging (LiDAR) is typically considered a distance sensor used for modeling 3D objects [28–30], but it can also beapplied to displacement measurement [31,32]. LiDAR measures a distance to a target point using time-of-flight or phase-shift information of the laser beam reflected back from the target point [33]. Its sampling rate (120 Hz) and precision (3–5 mm) are much better than those of GNSS, but these specifications are still not good enough for many applications thatrequire faster and more precise dynamic displacement measurement. Robotic total station (RTS), which shares the samemeasurement principle as LIDAR but measures one target point at once, has been also studied for dynamic displacementmeasurement [34–36].

To overcome the limitations of the aforementioned approaches, researchers have explored the fusion of measurementsfrom multiple heterogeneous sensors. Moschas and Stiros [37] estimated displacement using a GPS receiver and an ac-celerometer. Since the displacement from GPS is typically highly contaminated with noise, they fused two measurands bydesigning a multi-step digital filter using the acceleration measurement and then applying the filter to the GPS displace-ment measurement. To determine the parameters of the filter, the time duration and frequency range of the actual re-sponses need to be manually selected. Therefore, the technique is not best suited for real-time and autonomous dynamicdisplacement estimation. Furthermore, the estimated displacement has a low sampling rate, inheriting one of the maindrawbacks of GPS.

Chan et al. [38] proposed a data fusion technique based on empirical mode decomposition (EMD) by combining ac-celerometer and GPS measurements. First, the noise level of GPS is estimated using the static displacement measured whileno external excitation is applied to a target structure. Then, an adaptive filter de-noises the GPS displacement using theassessed noise characteristics. Finally, the high frequency component of the double-integrated acceleration is fused with thede-noised GPS displacement using EMD. Since EMD typically introduces a high level of errors at the beginning and end ofthe processed time signal, the record length should be long enough so that the errors at the boundaries can be truncated.Furthermore, because EMD can be performed only after the full-length measurement is acquired, this technique is notattractive for real-time displacement estimation.

Extending their previous study [10] in which displacement is reconstructed using only an acceleration record, Hong et al.[39] derived a new finite impulse response filter fusing acceleration and intermittent displacement measurements. The filterenables estimating dynamic displacement more accurately than their previous work by combining the high frequencycomponents of the acceleration measurement and the low frequency components of the displacement measurement.However, manual adjustment of the filer parameters is necessary, making this technique less attractive for automation.Furthermore, this technique cannot be applied to real-time estimation, since the filter can be designed only after the fre-quency content of the measured acceleration is analyzed through post-processing.

Alternatively, Kalman filters have been employed to obtain the optimal estimate of dynamic displacement by explicitlytaking into account measurement errors when acceleration and intermittent displacement measurements are combined.Smyth and Wu [40] first introduced a multi-rate Kalman filter data fusion technique for displacement estimation. Thetechnique estimates dynamic displacement from a state-space model defined in terms of acceleration and intermittentdisplacement measurement. A series of numerical analyses are performed to demonstrate the effectiveness of the technique.In their state-space model, velocity and displacement are defined as state variables, and the error accumulated due to thedouble integration of the acceleration measurement is corrected only when the intermittent displacement measurement

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 341

becomes available. Therefore, the displacement and velocity in the state-space model evolve without considering the biascomponent in the acceleration record. One issue here is that displacement cannot be properly estimated when the bias islarge. Furthermore, the technique is very sensitive to the noise level of the intermittent displacement measurement sincethe bias correction is solely based on this intermittent displacement measurement. On the contrary, the bias term is ex-plicitly included in the state vector and directly estimated in many engineering problems which deal with biased mea-surements [41,42].

To tackle the bias problem, a new data fusion technique based on Kalman filter was proposed [43]. In contrast to Smythand Wu [40], the technique explicitly estimates the bias and the integration error in the acceleration measurement bydefining the bias and the integration error as state variables. Then, the displacement is estimated by subtracting the esti-mated integration error from the double-integrated acceleration record. This technique offers two advantages over Smythand Wu [40]: (1) Improvement of the displacement estimation accuracy by considering the bias; and (2) Improved ro-bustness against the displacement noise. In addition, a fixed interval smoothing is adopted to further enhance the esti-mation accuracy. The major drawback of the technique is that two double integration processes, one for the error dynamicsand the other for the displacement, need to be solved, increasing the computational time.

This paper presents a multi-rate data fusion technique using a smoothing based Kalman filter, which estimates dynamicdisplacement in real-time by combining the velocity measured from LDV and the displacement from LiDAR. The proposedtechnique can be applied to measure millimeter level displacements of various civil infrastructures such as bridges,buildings, tunnels and dams at a distance up to 350 m. Also, the proposed technique can cover all the response frequencyrange of civil infrastructure since the sampling rate of LDV is up to 4 MHz. The uniqueness of the study lies in that(1) dynamic displacement is estimated using two fully noncontact sensors contrary to the previous studies, (2) high-ac-curacy, high sampling rate and good SNR can be achieved without integration error, (3) pseudostatic displacement, whichcannot be properly estimated by many conventional techniques, can also be reliably estimated, and (4) dynamic dis-placement with improved accuracy is estimated in real time.

The paper is organized as follows. Section 2 presents the proposed displacement estimation technique, describing howLDV and LiDAR measurements are fused and how the displacement accuracy is enhanced through a real-time smoothingbased Kalman filter. In Section 3, a series of lab-scale tests are performed on a simple cantilever beam to test the effec-tiveness of the proposed technique. The concluding remarks are provided in Section 4.

2. Proposed multi-rate data fusion technique using smoothing based Kalman filters

The proposed technique fuses the velocity measured from LDV and the intermittent displacement measurement fromLiDAR in the framework of a smoothing based Kalman filter to estimate dynamic displacement with a high sampling rate, alow noise level and no integration error. The proposed technique consists of (1) a strategy for fusing data from two fullynoncontact sensors (i.e., LDV and LiDAR), (2) a state-space model that includes both the displacement and the bias in theacceleration record as state variables for improved accuracy and computational efficiency, and (3) a smoothing basedKalman filter that further improves estimation accuracy without compromising the computational time. In Sections 2.1 and2.2, the working principles of LDV and LiDAR are briefly reviewed. Then, the state-space models of the proposed techniqueand the conventional Kalman filter algorithm are described in Section 2.3. A multi-rate data fusion based displacementestimation technique using the smoothing based Kalman filter is proposed in Section 2.4.

2.1. Displacement and velocity measurements using LDV

LDV launches a sinusoidal laser beam to a moving target and measures its dynamic displacement and velocity based onthe interference of an incident laser beam and the reflected beam from a target. The reference and incident beams arelaunched from the identical laser source, but split in the beam splitter inside LDV. The reflected beam is combined with thereference beam to generate interference, and the light intensity of the combined laser beam is captured by the photo-detector inside LDV.

Suppose the light intensities of the reference and reflection beams are u1 and u2 respectively, then the light intensity ofthe combined interference uc detected by the photodetector becomes [44]

( )ω φ= + + + ( )u u u u u t2 cos 1c c m1 2 1 2

where ωc is the frequency shift of the reference beam due to the Bragg cell, and φm is a phase difference between thereference and reflection beams. Two orthogonal signals ui and uq are obtained by (1) removing the DC component +u u1 2

from uc , (2) multiplying ω tcos c and ω tsin c to uc , and (3) filtering out cos2ω tc components using a low-pass filter [45].

φ φ= ( ) = ( ) ( )uA

t uA

t2

cos2

sin 2i m q m

where =A u u2 1 2 and φm is calculated by applying an arctangent function to the ratio of uq to ui:

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355342

( ) ( )( )φ =

( )

⎛⎝⎜⎜

⎞⎠⎟⎟t

u t

u tarctan

3m

q

i

Once φm is calculated, the displacement ( )x t of the moving target is obtained from φ π λ( ) = ⋅ ( )t x t2 2 /m , where λ is thewavelength of the laser beam. Note that φm computed in Eq. (3) is limited to π π(− )2 2 , even in the case that the truedisplacement ( )x t is outside the range. Therefore, the computed ( )φ tm shows discontinuities whenever ( )φ tm intersects with

πn2 . A phase unwrapping algorithm [46] can reconstruct the true phase value by adding πn2 to the calculated one. However,the unwrapping process for displacement estimation can fail in the following cases: (1) for impact loading where ( )φ tm canchange more than π2 within a single time step, the unwrapping algorithm cannot properly reconstruct displacement, (2) theunwrapping algorithm cannot properly find the discontinuities when noise level in ( )φ tm is high. The failure of the un-wrapping process results in the distortion of displacement estimation, which is illustrated in Section 3.

Contrary to the previous displacement estimation, the phase unwrapping algorithm is not necessary in velocity esti-mation since velocity is proportional to φd dt/m not ( )φ tm . Note that a discontinuity in ( )φ tm becomes a spike when φd dt/m isnumerically calculated. This spike can be effectively removed by a de-spiking technique [47], but the technique cannoteliminate all the spikes when the spikes and noise cannot be distinguished due to a low SNR. In spite of this drawback,velocity still can be measured more reliably than displacement, as long as a proper level of SNR is maintained duringvelocity measurement. Therefore, LDV often measures velocity instead of displacement [48–52].

2.2. Displacement measurement using LiDAR

LiDAR scans objects in 3D space and generates their 3D coordinates using a galvanometer and a rotating mechanicalstation. Conventionally, LiDAR has been utilized in the areas of topography [53], geology [54], quality assessment [29],cultural heritage documentation [55,56], reengineering [57] and building information modeling, to name a few. Also, LiDARhas been applied to the measurement of static displacement and deformation [30–32].

Time-of-flight type LiDAR emits a laser pulse to an object surface, and directly measures the travel time of the reflectedlaser pulse using an internal clock. It is preferred for the measurement of large-scale and/or long-distance structures, since ithas a longer sensing range than phase-shift type LiDAR [33]. Time-of-flight type LiDAR has a typical measurement range of50–800 m, but the range can be extended up to 3 km with an auxiliary optical device. The displacement accuracy is around3 mm, and the precision is in the range of 3–5 mm. Some commercial products emit multiple laser pulses simultaneouslyand average their arrival times to increase their precision to enhance accuracy, precision and sampling rate.

Phase-shift type LiDAR emits a sinusoidal laser beam, and the time-of-flight ttr of the laser beam is evaluated by mea-suring the phase-shift of the reflected beam with respect to the emitted beam.

ϕπ

=( )

tf2 4

trm

where ϕ is the phase-shift, and fm is the modulation frequency. One of the advantages of phase-shift type LiDAR is itssubmillimeter-level accuracy [33]. On the other hand, its measurement range is limited to 5 to 30–m since the phase-shiftangle ϕ cannot exceed πf2 m. Currently, the best measurement range that can be achieved by phase-shift LiDAR is 120 massuming at least 90% surface reflectivity condition. Also, its 3 mm precision is comparable to that of time-of-flight typeLiDAR.

Commercial LiDARs are programmed to continuously scan in the horizontal and vertical directions. Therefore, LiDARoriginally measures distance just once per points. This limitation has led the previous studies to focus on static deformationmeasurements. Recently, some manufacturer supports a line-scan mode, where the mechanical station is fixed and only thegalvanometer scans vertically, and dynamic displacement measurement with a sampling rate of 120 Hz can be achieved foreach measurement point along the scan line [58].

2.3. Dynamic displacement estimation by fusing measurements of LDV and LiDAR

Section 2.3 includes two ingredients of conventional Kalman filer: (1) state-space model and (2) recursive Kalman filteralgorithm. As described in Section 1, the previously proposed state-space models for displacement estimation have twodrawbacks: low estimation accuracy [40] and high computational cost [43]. In this section, a new state-space model isproposed for improved displacement accuracy and low computational cost. In addition, the conventional recursive Kalmanfilter algorithm is briefly reviewed based on the proposed state-space model. This conventional Kalman filter is furtheradvanced to a smoothing based Kalman filter in Section 2.4.

Assume that the velocity ( )x km measured from LDV can be represented as a linear summation of the true velocity ( )x k ,zero-mean Gaussian measurement noise ( )w k0 and bias ( )b k :

( ) ( ) ( ) = + + ( ) ( )x k x k w k b k 5m 0

Then, the true displacement is related to the true velocity based on Taylor's expansion:

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 343

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )+ = + Δ + ¨ Δ + Δ +⋯ = + Δ + Δ ( )x k x k x k t x k t x k t x k x k t e k t112

16 6

2 3

whereΔt is a time interval, and ( )e k is a high-order term error defined as ( ) ( ) ( ) ( )( )= ¨ Δ + Δ + Δ +⋯e k x k t x k t x k t12

16

2 124

4 3 . ( )e kdepends on the measurement parameters such as the time interval and frequency content of the displacement. SubstitutingEq. (5) into (6) leads:

( ) ( ) ( ) ( ) ( )+ = + Δ +ϵ Δ + Δ ( )x k x k x k t k t w k t1 7m

where ( )= − ( )w k w k0 , and the total error ( )ϵ k is defined as ( )− ( )e k b k . Note that the signs of ( )w k and ( )b k do not haveany mathematical or physical significance. ( )ϵ k is assumed to be piecewise constant [59] within two adjacent time steps:

( )ϵ + =ϵ( ) ( )k k1 8

Eq. (8) indicates that ϵ( )k does not have any system dynamics and ϵ( + )k 1 is not updated from ( )ϵ k during the priorprediction step of Kalman filter. However, it should be noted that ϵ( )k is only piece-wise constant and corrected during theposterior correction step using the actual displacement measurement.

To construct a state-space model, ( )x k and ( )ϵ k are defined as state variables.

( ) ( )( )

=ϵ ( )

⎪ ⎪⎪ ⎪⎧⎨⎩

⎫⎬⎭

kx k

kx

9

Note that bold character is used for expressing a vector or matrix hereafter. Assuming that the displacement measured byLiDAR has a zero-mean Gaussian noise ( )v k , the measured displacement can be expressed as:

( ) ( )( ) = + ( )x k x k v k 10m

Combining (Eqs. (7) and 10), and applying Eq. (9) to define ( )x k and ( )ϵ k as state variables, the state-space model formulti-rate data fusion of LDV and LiDAR measurements can be expressed as:

( )( ) ( ) ( ) ( )+ = + + ( )k k x k w kx Ax B1 11m

( ) ( )= + ( ) ( )x k k v kCx 12m

where

= Δ = Δ =⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥ ⎡⎣ ⎤⎦t tA B C1

0 1,

0, 1 0

This state-space model differs from the previous Kalman filter models used for displacement estimation [40,43] in thatboth physical response and error are included in the state vector and estimated simultaneously. The state-space modelproposed by Smyth and Wu [40] does not include error in its state vector, its performance against bias and measurementnoise is not reliable. Because the state-space model proposed by Kim et al. [43] does not include displacement as a statevariable, the technique requires solving an additional recursive integration process for displacement estimation. Instead, thestate-space model is built using bias and integration error as state variable, but the bias is not used in displacement esti-mation. Consequentially, the computational cost is increased. On the other hand, the proposed state-space model can en-hance the displacement estimation accuracy and cut down computational cost by including error variable and eliminating asurplus state variable and an additional recursion process.

The conventional Kalman filter estimates the state vector in (Eqs. (11) and 12) in two steps: prior prediction and posteriorcorrection steps. In the prior prediction step, the Kalman filter predicts the state at the current time step solely based on thesystem dynamics defined in Eq. (11). Since ( )w k is a zero-mean random process, the best conditional estimate of ( )+kx 1given ( )kx , denoted as ( )|^ +k kx 1 , can be expressed as

( ) ( ) ( )^ + | = ^ | + ( )k k k k x kx Ax B1 13m

where ( )|^ k kx is the posterior estimate obtained from Eq. (17) at the previous time step, k. Then, the covariance matrix ofthe expected estimation error in the prior prediction step can be evaluated as:

( ) ( )+ | = | + ( ) ( )k k k k kP AP A Q1 14

where ( )kkP and ( )kQ are defined as

( )( )( ) ( ) ( ) ( ) ( )| = − ^ | − ^ | ( )⎡⎣⎢

⎤⎦⎥k k k k k k k kP x x x xE

15T

( )( ) ( ) ( ) σ= = ( )⎡⎣ ⎤⎦k w k w kQ B B B BE 16

Tw

T2

where σw2 is the variance of ( )w k . Note that [ ( ) ( ) ]w k w kE T becomes σw

2 since ( )w k is a zero-mean random process. Also,

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355344

( )kQ can be treated as constant with respect to time if ( )w k is assumed to be wide-sense stationary. In the proposedtechnique, σw

2 is estimated from the beginning portion of LDV's time signal, where only noise components without targetdynamic responses are present.

In the posterior correction step, the posterior estimate ( )|^ + +k kx 1 1 is obtained as a weighted linear summation of theprior estimate ( )|^ +k kx 1 and the displacement ( )+x k 1m measured by LiDAR.

( )( ) ( ) ( )( ) ( )

^ + | + = ^ + | + ( + ) ( + ) − ^ + |

= − + ^ + | + ( + ) ( + ) ( )⎡⎣ ⎤⎦k k k k k x k k k

k k k k x k

x x K Cx

I K C x K

1 1 1 1 1 1

1 1 1 1 17

m

m

where Kalman gain ( + )kK 1 is a row vector composed of { }ϵK KxT , and determined so that ( )+ | +k kP 1 1 is minimized in

a least square sense:

( )( ) ( ) ( )+ = + | + | + ( )−

k k k k k rK P C CP C1 1 1 18T T 1

where r is the covariance of the measurement noise ( )v k , and it can be treated as the variance of ( )v k , σv2 similar to ( )kQ .

The error covariance of the posterior estimation can be updated from its prior estimate as follows.

( ) ( ) ( ) ( )+ | + = + | − + + | ( )k k k k k k kP P K CP1 1 1 1 1 19

When displacement measurement is not available, only the prior prediction step in Eq. (13) is executed and the in-tegration error is accumulated as the time step passes. The accumulated integration error is corrected only at the posteriorcorrection step in Eq. (17) when the displacement data becomes available from LiDAR.

Although the conventional Kalman filter is more robust against measurement noise than the baseline correction andoptimization based techniques, it still suffers from the following two limitations concerning estimation accuracy.

(1) Eq. (17) shows that best posterior estimate, ( )^ + | +k kx 1 1 , is determined as a weighted averaging of ( + )x k 1m and theprior estimate, ( )^ + |k kx 1 . Considering (Eqs. (10) and 17), the influence of the noise process ( + )v k 1 on ( )|^ + +k kx 1 1 isamplified by ( + )kK 1 . Therefore, the accuracy of the posterior estimation ( )|^ + +k kx 1 1 is heavily affected by the singlerealization of the noise process, ( + )v k 1 .

(2) Since Kalman gain ( )kK is recursively calculated, it takes quite a few time steps before converging to a steady stateestimation value. Therefore, a large error is expected to occur at the beginning of displacement estimation.

2.4. Real-time smoothing based Kalman filter

To address the aforementioned limitations, three smoothing based Kalman filters have been developed [60]: fixed in-terval [41], fixed point [61] and fixed lag smoothing [62]. The smoothing based Kalman filters estimate the current state bymobilizing the measurements in the future time steps as well as in the past time steps (1) to minimize the influence ofnoise, and (2) to improve the measurement accuracy particularly at the beginning of estimation. The smoothing basedKalman filters are basically identical to the conventional Kalman filter, except that additional state variables are introducedin a state-space model including future measurements.

Fixed interval smoothing [41] is a simple averaging of forward and backward Kalman filter estimates. Since the un-certainty of the conventional Kalman filter estimate, expressed by ( )|k kP , decreases in proportion to the elapsed time, theuncertainty can be evenly suppressed by averaging the uncertainties obtained from the forward and backward Kalmanfilters. The reduced uncertainty leads to the enhancement of estimation accuracy and the reduction of noise. However, fixedinterval smoothing is not suitable for real-time estimation, because the displacement estimation at one time step relies onthe measurements from all time steps.

Contrary to fixed interval smoothing, fixed point [61] and fixed lag smoothing [62] utilize only a small fraction of futuremeasurements nearest the current time step. In other words, ( )x km and ( )x km up to time step k are used to estimate

|^ ( − )k N kx , which denotes the estimate of ( )−k Nx at time step k. In fixed lag smoothing, the dimension of state vector in Eq.(9) changes from ×2 1 to ( + ) ×N2 1 1, since additional states from ( )|^ − − +k N k Nx 1 to ( )|^ −k N kx are also estimated in onestate-space model. Therefore, the dimension of the transition matrix A in Eq. (11) increases to ( + )× ( + )N N2 1 2 1 . On theother hand, the dimension of the state vector is always fixed to ×4 1 in fixed point smoothing, and an additional recursiveprocess is executed instead.

When the noise level is high, a large value of N is required to reduce the uncertainty caused by the noise [63]. Then, thecomputational costs of fixed lag and fixed point smoothing increase proportionally to N2 and N , respectively. Therefore, fixedpoint smoothing is adopted in this study, because of the high noise level in LiDAR measurement. Note that the estimationaccuracy of two smoothing algorithms is exactly the same. The detailed comparison of the two smoothing algorithms isprovided in Section 3.4.

Fixed point smoothing is originally designed to estimate a state at a fixed time step, but can be modified to estimatestates at multiple time points by shifting the target time step sequentially as new LDV velocity measurement becomesavailable. In fixed point smoothing, an additional state vector ( )− kxk N is introduced. Here, ( )− kxk N denotes ( − )k Nx pro-pagated to time step k with an identity transition matrix and no process noise. Therefore, ( )− kxk N should have the dynamics

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 345

of

( ) ( ) ( ) ( ) ( )− = − = − + =⋯ = − = ( )− − − −k N k N k N k kx x x x x1 1 20k N k N k N k N

Then, the augmented state vector of fixed point smoothing is defined as:

( )( )

( )=( )−

⎪ ⎪⎪ ⎪⎧⎨⎩

⎫⎬⎭

kk

kx

x

x 21k Naug

Combining (Eqs. (20) and 21) with the state-space model in (Eqs. (11) and 12), an augmented state-space model isconstructed for fixed point smoothing:

( )( ) ( )

( ) ( )

( )( )

=( − )

( − )+ − + −

=( )

( )+

( )

− −

−

⎧⎨⎩⎫⎬⎭

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩⎫⎬⎭

⎡⎣⎢

⎤⎦⎥

⎡⎣ ⎤⎦⎧⎨⎩

⎫⎬⎭

k

k

k

kx k w k

x kk

kv k

xx

AI

xx

B

Cx

x

00

11 0

1 1

022

k N k Nm

mk N

k

where I is an 2�2 identity matrix. Applying the state-space model in Eq. (22) into the conventional Kalman filter definedin (Eqs. (13)–19) leads to the following:

( )^ ( | )

^ ( − | )=

^( − | − )^( − | − )

+ −^( − | − )^( − | − ) ( )

⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎜ ⎡⎣ ⎤⎦

⎧⎨⎩

⎫⎬⎭

⎞

⎠⎟⎟

k k

k N k

k k

k N kx k

k k

k N k

x

x

AI

x

xK C

x

x

00

1 1

10

1 1

1 23maug

Here, ( )^ − |k N kx is identical to ^ ( )− kxk N , which denotes the estimate of ( )− kxk N in Eq. (20). Kaug is an augmented Kalmangain, which can be obtained by substituting the augmented state-space model defined in Eq. (22) into Eq. (18).

( )^ − |k N kx is estimated recursively using Eq. (23). First, ( )^ − | −k N k Nx is estimated from conventional Kalman filter, andthen, setting the initial augmented state vector as{ }( ) ( )^ − | − ^ − | −k N k N k N k Nx x

T, ( )^ − | − +k N k Nx 1 is estimated from

the following equation:

( )^ ( − + | − + )

^( − | − + )=

^( − | − )^( − | − )

+ −^( − | − )^( − | − ) ( )

⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

⎫⎬⎭

⎛

⎝⎜⎜ ⎡⎣ ⎤⎦

⎧⎨⎩

⎫⎬⎭

⎞

⎠⎟⎟

k N k N

k N k N

k N k N

k N k Nx k

k N k N

k N k N

x

x

AI

x

xK C

x

x

1 1

1

00

024

maug

Now, |^ ( − − + )k N k Nx 1 is estimated from the second row of the augmented state vector in the left-hand side of Eq. (24).

Repeating this recursion N times, |^ ( − )k N kx is finally estimated. After completing the recursion process, the state at the nexttime step of LDV velocity measurement is estimated. Similarly, the initial augmented state vector becomes

{ }( ) ( )^ + − | + − ^ + − | + −k N k N k N k Nx x1 1 1 1T

and another recursion process is executed to estimate

|^ ( + − + )k N kx 1 1 .Note that dynamic displacement is estimated using Eq. (23) only when LiDAR displacement measurement is available.

When only LDV velocity is measured, the displacement is estimated as follows:

^( | )^ ( − | )

=^( − | − )^( − | − ) ( )

⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

⎫⎬⎭

k k

k N k

k k

k N k

x

x

AI

x

x

00

1 1

1 25

3. Experimental test

3.1. Test setup

A series of lab-scale tests are conducted on a cantilever beam shown in Fig. 1 to validate the effectiveness of the proposedtechnique. A beam of dimensions of 1 m�10 cm�6 mm is made of mild steel, whose density is 7850 kg/m3 and Young'smodulus is 210 GPa. The fundamental resonance frequency of the beam is found to be 5.12 Hz from a preliminary free-vibration test in the frequency domain.

Polytec PSV-400 is used to measure the out-of-plane velocity and displacement from the tip of the beamwith a samplingrate of 1280 Hz. This LDV is placed 1.5 m away from the cantilever beam, and the horizontal and vertical incident angle ofthe laser beam with respect to the beam is set to 90° and 68.5°, respectively. RIEGL VZ-400 3D terrestrial scanner is used forintermittent displacement measurement with a sampling rate of 120 Hz. This LiDAR is installed 1 m away from the LDV, andthe distance from the cantilever beam is 1.8 m. The horizontal incident angle of its laser beamwith respect to the cantileverbeam surface is calculated as 56.31°. Also, Optex FA CD4-350 laser displacement sensor with a sampling rate of 1280 Hz isused to obtain reference (ground truth) displacement for comparison. This laser displacement sensor is installed about

LDV

LiDAR

Cantilever beam

Laser displacement sensor

Data acquisition

system

Measurement point

Fig. 1. Test setup for the lab scale tests. LDV and LiDAR are placed 1.5 m and 1.8 m away from the measurement point, respectively, and laser displacementsensor is used for reference displacement measurement.

Table 1Four different loading types applied to the cantilever beam.

Case # Loading type

1 Impact by an impact hammer2 Manual pulling and release for free vibration3 Very slow manual pulling and pushing for pseudostatic

loading4 Hold the beam by hands for mimicking highly damped

vibration after impact by an impact hammer

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355346

30 cm away from the tip of the cantilever beam and supported by a rigid frame.The cantilever beam is excited by four different loading cases as summarized in Table 1. The first loading is an impact

applied by PCB Piezotronics 08C03 impact hammer with a metallic tip. This loading is selected to demonstrate the limitationof the unwrapping process employed in LDV based displacement measurement as discussed in Section 2. Second, a freevibration test is conducted by pulling and then releasing the tip of the cantilever beam manually. This loading case isselected to evaluate the accuracy of the proposed Kalman filter technique for one of the most common vibration types inreal applications. Third, a pseudostatic loading is applied to the beam by manually pulling and pushing the beam very slowlyto highlight the advantage of the proposed technique over the baseline correction and other displacement reconstruction

-50

0

50

-50

0

50

-50

0

50

0 5 10 15 20 25 30-50

0

50

Time (s)

Dis

plac

emen

t (m

m)

LiDAR displacementReference displacement

Case 1

Case 2

Case 3

Case 4

Fig. 2. Comparison of the displacement measured by LiDAR and the reference displacement measured by the laser displacement sensor for 4 loading cases.

-50

0

50

-200-100

0100

-50

0

50

0 5 10 15 20 25 30-100

-500

50

Time (s)

Dis

plac

emen

t (m

m)

Integration of LDV velocityReference displacement

Case 1

Case 2

Case 3

Case 4

Fig. 3. Comparison of the displacement obtained by integrating LDV velocity measurement and the reference displacement measured by the laser dis-placement sensor for 4 loading cases.

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 347

techniques that use only acceleration measurement. Finally, a highly damped vibration is mimicked by exerting an impactusing the impact hammer and holding the beam by hands after the impact to add additional damping to the beam. This typeof vibration also produces sudden displacement changes, and can be observed during impact events against massiveconcrete structures such as outer shells of nuclear power plants and piers of long-span bridges. For each loading case,velocity and displacement measurements are repeated five times.

3.2. Displacement estimation using either LiDAR or LDV alone

Fig. 2 compares the displacement measured by LiDAR with the reference displacement measured by laser displacementsensor. As for LiDAR displacement, a spatial averaging is employed to reduce noise. Nevertheless, the LiDAR displacementhas a higher noise level than the reference displacement. Here, the sampling rate of LiDAR displacement is limited to 120 Hzwhile the reference displacement is sampled at 1280 Hz.

Fig. 3 shows the displacement estimated by integrating LDV velocity measurement for all loading cases. Thanks to thehigh precision of LDV (0.1 μm/s), the noise level in the estimated displacement is very low. The integration error is apparentfor all cases, and linearly increases over time for cases 1, 2 and 4. On the other hand, the integration error for case 3 does notlinearly accumulate over time as pointed out in Section 3. This implies the bias in this loading case cannot be assumed aslinear. The existing baseline correction or displacement reconstruction techniques can handle only linearly accumulatingbias, but not this type of fluctuating integration errors.

Fig. 4 compares the displacement measured by LDV and the reference displacement. Note that the LDV displacement inFig. 4 is obtained using the unwrapping process described in Section 2.1. When the rate of displacement change is slow (thebeginning of case 2 and the entire region of case 3), the displacement estimated from LDV shows a good match with the

-50

0

50

-100

-50

0

50

-50

0

50

4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10-50

0

50

100

Time (s)

Dis

plac

emen

t (m

m)

LDV displacementReference displacement

Case 1

Case 2

Case 3

Case 4

Fig. 4. Comparison of the displacement measured by LDV and the reference displacement measured by the laser displacement sensor (close-up for 4.5 s to10 s). In Case 1, 2 and 4, LDV fails to properly reconstruct the displacement through unwrapping process.

-50

0

50

0 5 10 15 20 25 30-50

0

50

(a)

(b)

(c)

Time (s)

Dis

plac

emen

t (m

m) -60

-40-20

02040 Estimate

Reference

Fig. 5. Displacements estimated for loading case 2 by applying the conventional Kalman filter to the state-space models proposed by (a) Smyth [40],(b) Kim et al. [43] and (c) this study, and reference displacement measured by laser displacement sensor.

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355348

reference displacement. However, when there are sudden displacement changes (cases 1, 2 and 4), the unwrapping processfails to properly reconstruct the displacement. The test results suggest that LDV may have difficulties in measuring dis-placements subject to sudden changes.

3.3. Displacement estimation using the conventional Kalman filter for data

In Fig. 5, the displacement corresponding to loading case 2 is estimated by applying the conventional Kalman filterdescribed in (Eqs. (13)–19) to the state-space models developed by the references [40,43] and this study. The state-spacemodel proposed by Smyth and Wu [40] fails to properly estimate the displacement (Fig. 5(a)), while the other two modelsestimate the displacement properly (Fig. 5(b) and (c)). As shown in Table 2, the RMS errors between the estimates andreference displacement are 2.11, 0.22 and 0.21 mm for the state-space models of Smyth andWu, Kim et al., and the proposedtechnique, respectively.

The improved accuracy of the proposed technique and the one by Kim et al. comes from the inclusion of the bias term intheir state-space models. Recall that the posterior estimate ( )^ + | +k kx 1 1 is a weighted average of the prior estimate

( )^ + |k kx 1 and LiDAR displacement measurement ( )x km as shown in Eq. (17). Since the noise level of LiDAR measurement istypically high, Kalman gain K becomes small and the contribution of ( )x km to ( )^ + | +k kx 1 1 diminishes. In the case of thestate-space model of Smyth and Wu [40], its prior estimate is obtained by integrating velocity measurement and the in-tegration error accumulates over time. If the contribution of the prior estimate becomes larger than that of ( )x km due to thehigh noise level of ( )x km , much of the integration error is still reflected on the posterior estimate. This leads a permanentoffset error in displacement estimate as shown in Fig. 5(a). However, the proposed technique explicitly considers the biasterm into the state-space model, and the bias is subtracted from the velocity measurement. This approach effectivelyeliminates the residual integration error from displacement estimate.

Although the estimation accuracies of the proposed technique and the one by Kim et al. [43] are almost identical, theproposed technique is computationally more efficient than that of Kim et al. They approach requires two separate recursiveintegration processes for bias error and displacement. First, the bias is estimated and integrated to estimate integration errorin its state-space model. Then, the integration error is subtracted from the integration of velocity measurement. Therefore,two individual recursive integration processes are required, and the estimated bias is not directly used in the displacementestimation. However, the proposed technique requires only one integration process since the technique estimates the bias,subtracts the bias from velocity measurements and then integrates the resultant velocity to estimate displacement in itsstate-space model without a surplus state variable. Using MATLAB in 64 bit computer environment, the proposed techniqueeliminates one redundant integration process from the technique of Kim et al., and decreases the computational time by38.7%.

Table 2RMS errors between estimates of the three state-space models with conventional Kalman filter and reference displacement measurements for 4 loadingcases (unit: mm).

Loading case Smyth and Wu[40]

Kim et al.[43]

Proposedtechnique

1 0.62 0.23 0.222 6.63 0.23 0.203 0.48 0.21 0.214 0.69 0.22 0.22Average 2.11 0.22 0.21

-50

0

50

-50

0

50

-50

0

50

0 5 10 15 20 25 30-50

0

50

Time (s)

Dis

plac

emen

t (m

m)

Case 1

Case 2

Case 3

Case 4

EstimateReference

Fig. 6. Displacements estimated for all loading cases by applying the conventional Kalman filter to the state-space model developed by this study.

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 349

Fig. 6 illustrates that the proposed technique successfully estimates the displacements of all loading cases. Note that thedisplacement of loading case 3 does not fulfill the assumption of the traditional baseline correction and displacementreconstruction techniques, since the bias is far from linear as illustrated in Fig. 3(c), and the mean value of the displacementis non-zero (-4.7 mm). However, the proposed technique shows a good agreement with the reference displacement.

Fig. 7 illustrates the drawbacks of the proposed technique as described at the end of Section 2.3 when the conventionalKalman filter is used with the state-spate model developed in this study. A large overshooting of ϵK is observed for the initial2 s in Fig. 7(a). Subsequently, this transient Kalman gain results in the inaccurate estimation of total error ϵ (Fig. 7(b)) anddisplacement (Fig. 7(c)). Furthermore, the total error, ϵ, which is measured in terms of velocity, is accumulated in the finaldisplacement estimate, and acts as a permanent drift error when the transient state is excessively long.

Another issue in the conventional Kalman filter is the deterioration of the displacement estimation accuracy because ofits dependency on single realization of the noise process ( )v k in LiDAR's displacement. Even after ϵK converges to its steadystate, ϵ continues to fluctuate as shown in Fig. 8(a). This variation of ϵ is mainly caused by ( )v k in the LiDAR displacementshown in Fig. 8(c). Fig. 8(b) illustrates that ( )Δϵ k defined as ( )ϵ −ϵ( − )k k 1 varies in a pattern similar to ( )v k , substantiatingthe fact that a single realization of ( )v k significantly affects the posterior estimate.

3.4. Displacement estimation using the proposed smoothing-based Kalman filter for data fusion

The displacement estimation accuracy is enhanced by applying the smoothing based Kalman filter introduced in Section2.4. The comparison of ′ϵK s obtained from the proposed smoothing based and conventional Kalman filters in Fig. 9(a) showsthat ϵK of the smoothing based Kalman filter converges much faster than that of the conventional Kalman filter, and ϵ is alsostabilized more quickly for the proposed smoothing based Kalman filter. In particular, the effect of the smoothing is pre-eminent at the initial stage of the displacement estimation. The faster convergence rate leads the improved accuracy indisplacement estimation (Fig. 9(c)). With smoothing, the RMS error of the estimated displacement is reduced by 37.2% onaverage. Note that =N 256, Δ =t 1/1280 s and ΔN t¼0.2 s are used in this example.

Besides the initial portion of the signal, the filtering also improves the estimation accuracy of the remaining part of thesignal as shown in Fig. 10. Fig. 10(a) shows that the fluctuation of ϵ is reduced using smoothing. Furthermore, Fig. 10

0 0.5 1 1.5 2 2.5 3 3.5

Dis

plac

emen

t(m

m)

Tota

l err

or(m

m/ s

)K

alm

an g

ain

Time (s)

(a)

(b)

(c)

0

0.02

0.04

0.06

-1

-0.5

0

0.5

-0.5

0

0.5

1EstimateReference

4 4.5 5

∋

K

∋

Fig. 7. Initial estimation error of conventional Kalman filter for loading case 3 with the conventional Kalman filter and the state-space model proposed bythis study: (a) Kalman gain ϵK , (b) total error ϵ and (c) estimated displacement x .

1.268

1.269

1.270

1.271

-1-0.5

00.5

1

-5

0

5

10

x10-3

20.4 20.5 20.6 20.7 20.8Time (s)

20.7520.6520.55

Tota

l err

or

(m

m/s

)

(a)

Noi

se in

LiD

AR

m

easu

rem

ent (

mm

)(b)

(c)

20.45

(mm

/s)

Fig. 8. The dependency of conventional Kalman filter accuracy on noise process in LiDAR measurement: (a) Total error ϵ estimated by applying theconventional Kalman filter to the state-space model proposed by this study, (b) the change of ϵ after each posterior correction step, and (c) the noise ( )v k inthe LiDAR displacement measurement (loading case 4, zoomed-in for 20.4–20.8 s).

0 0.5 1 1.5 2 2.5 3 3.5

Dis

plac

emen

t(m

m)

Tota

l err

or(m

m/s

)K

alm

an g

ain

Time (s)

(a)

(b)

(c)

0

0.02

0.04

0.06

-1

-0.5

0

0.5

-0.5

0

0.5

1

4 4.5 5

With smoothingWithout smoothingReference displacement

∋

K

∋

Fig. 9. Comparison of initial estimation error of conventional and smoothing-based Kalman filter for loading case 3: (a) Kalman gain ϵK , (b) total error ϵ and(c) estimated displacement x and reference displacement measurement ( ΔN t ¼0.2 s).

1.266

1.268

1.27

1.272

-1-0.5

00.5

1

-0.1-0.05

00.05

0.1

Tota

l err

or

(m

m/s

)

(a)

Dis

plac

emen

t es

timat

e (m

m)

(b)

(c)

(mm

/s)

20.4 20.5 20.6 20.7 20.8Time (s)

20.7520.6520.5520.45

With smoothingWithout smoothingReference displacement

Fig. 10. Comparison of the dependencies of conventional and smoothing based Kalman filter accuracy on noise process in LiDAR measurement: (a) Totalerrorϵ, (b) Δϵ(k)¼ ( )ϵ −ϵ( − )k k 1 , and (c) ( )v k (loading case 4, Δ =N t 0.2 s, zoomed-in for 20.4–20.8 s).

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355350

(b) shows that the variance of Δϵ is reduced from 2.03�10�6 to 1.74�10�6, which corresponds to 14.1% reduction. As aresult, the displacement estimated by the proposed smoothing based Kalman filter shows a better agreement with thereference displacement than the conventional one. The RMS error is reduced by 26.3% within 5–30 s. Note that estimationaccuracy can be further improved if measurement noise ( )v k can be further suppressed. It is known that the smoothingbased Kalman filter can improve estimation accuracy by 80% using only 10 or less future measurements when measurementnoise is sufficiently small [63].

To offer a guideline for the selection of ΔN t , the relation between the displacement estimation accuracy and the timedelay is studied. Fig. 11 shows ϵK and ϵ for loading case 1 as ΔN t increases from 0 to 1 s. The plots of ϵK and ϵ are simply

0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

0

0.01

0.02

0.03

0.04

0.05

Kal

man

gai

nTo

tal e

rror

(mm

/s)

Time (s)

NΔt = 0 sNΔt = 0.4 sNΔt = 0.8 s

(b)

(a)

∋∋

K

Fig. 11. Change of smoothing-based Kalman filter accuracy at the beginning of estimation with respect to ΔN t: (a) Kalman gain ϵK and (b) total error ϵestimated using various time delays ΔN t (loading case 1, Δ =t 1/1280 s).

0.120.140.160.18

0.20.220.24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.120.140.160.18

0.20.220.24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

RM

S e

rror (

mm

)

NΔt (s) NΔt (s)

RM

S e

rror

(mm

)

(a) (b)

(d)(c)

Fig. 12. The relationship between the average RMS error of smoothing-based Kalman filter and time delay ΔN t for (a) loading case 1, (b) case 2, (c) case3 and (d) case 4.

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 351

shifted to the left without any shape changes as ΔN t increases. This indicates that the uncertainty of the smoothing basedKalman filter at time step k is identical to that of the conventional Kalman filter at time step +k N . Therefore, the smoothingbased Kalman filter converges to the steady state faster than its conventional counterpart by N time steps.

Fig. 12 shows that the average RMS error decreases as ΔN t increases. Here, the RMS error is defined between the re-ference displacement and the displacement estimated by the proposed smoothing based Kalman filter. When ΔN t¼0, theproposed smoothing based Kalman filter is reduced to the conventional Kalman filter. It seems that ϵK and ϵ converge tocertain values mostly within 0.2–0.3 s of time delay but no longer than 0.5 s. The proposed technique enhances the dis-placement estimation accuracy by 36.4% on average compared to the conventional Kalman filter when 0.5 s of time delay isapplied, and by 32.0% for 0.3 s of time delay.

3.5. Fixed point vs. Fixed lag smoothing

The final investigation in this paper is designated to the comparison of the computational costs of the fixed pointsmoothing and fixed lag smoothing. For the same ΔN t value, the two smoothing filters always provide identical results interms of ( )x k , ( )ϵ k and RMS error. Therefore, the estimation results of the fixed lag smoothing are omitted in the paper, andonly a brief overview of the algorithm is provided in the Appendix. The only difference between two smoothing algorithmsis computational cost. While fixed lag smoothing requires a larger memory space due to its larger matrix dimension, no

0 20 40 60 80 100 1200

1

2

3

4

5

Fixed pointFixed lag

N

Com

puta

tion

time

(s)

Fig. 13. Comparison of the computation time of fixed lag and fixed point smoothing filters with respect to time delay N (The average computation times forall 4 loading cases are provided).

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355352

recursion process is need. On the other hand, fixed point smoothing requires only a small memory space by employing anadditional recursive process. In Fig. 13, the computation times of two algorithms implemented in MATLAB under 64 bitcomputing environment are compared. Here, the average computation times for all 4 load cases are provided. The com-putation time of fixed lag smoothing is less than that of fixed point smoothing when N is less than 32, because of the smallincrease in matrix size. However, the computational time of fixed lag smoothing increases proportionally to N2, while that offixed points smoothing increases in the order of N . In conclusion, when the memory space is limited and a long time delay isnecessary due to a low SNR, fixed point smoothing can be a preferred option. For example, the time delay of 0.2 sec used inthis study corresponds to =N 256. This implies that fixed point smoothing is computationally more effective than fixed lagsmoothing. On the other hand, fixed lag smoothing can be a more rational choice when more precise displacement sensor isused instead of LiDAR.

4. Conclusion

The paper proposes a noncontact dynamic displacement estimation technique by fusing LDV velocity and LiDAR dis-placement measurements through smoothing based Kalman filter. The proposed method can estimate dynamic displace-ment with minimizes integration error, high sampling rate and low noise level in real-time by taking advantages of LDV andLiDAR measurements and effectively diminishing their drawbacks. The experiments using a cantilever beam demonstratethat the proposed state-space model can reduce the estimation error to 9.5% level of the estimation error of the techniqueproposed by Smyth and Wu, and reduce the computation time by 38.7% compared to that of Kim et al. Smoothing furtherenhances the accuracy up to 36.4% compared to the conventional Kalman filter using the proposed state-space model. Thesuperiority of the proposed technique comes from the robustness against displacement measurement noise of LiDAR, and arapid convergence of Kalman gain. Although the proposed technique enhances the accuracy and computational time fordisplacement estimation, further improvement in the following aspects are warranted: (1) further reduction of the com-putational time by avoiding smoothing and (2) further advancement of the proposed technique for a time-varying bias inacceleration measurement. In addition, since the proposed algorithm can be applied to any combinations of acceleration,velocity and displacement measurements, other sensing combinations such as GPS-RTK, RTS (robotic total station) andgeophone are also possible depending on users’ specific applications.

Acknowledgment

This research was supported by the Fire Fighting Safety & 119 Rescue Technology Research and Development Programfunded by the Ministry of Public Safety and Security (MPSS-fire safety-2015-72).

Appendix A. Fixed lag smoothing

Fixed lag smoothing eliminates the recursion process necessary for the displacement estimation in Eq. (23) by in-troducing a larger state vector. The augmented state vector in fixed lag smoothing is defined as

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 353

( )( )( )

( )

( ) =

⋮

( )

−

−

−

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

k

k

k

k

k

x

x

x

x

x A-1

k

k

k N

aug

1

2

Based on Eq. (20), the following relations can be established:

( ) ( )

( )

= − = ( − )

( ) = ( − )⋮

= ( − ) ( )

− −

− −

− −

k k k

k k

k k

x x x

x x

x x

1 1

1

1 A-2

k k

k k

k N k N

1 1

2 2

Combining Eqs. (A-1) and (A-2) with the state-space model in (Eqs. (11) and 12) creates an augmented state-space modelfor fixed lag smoothing.

( )

( )( )( )

( )

( )( )( )

( )

( ) ( )

( )( )( )

( )

( )

⋮

=

⋯⋯⋯

⋮ ⋱ ⋱ ⋱ ⋮⋯

−

−

−⋮

−

+⋮

− + −

( ) = ⋯

⋮

+

( )

−

−

−

−

−

− −

−

−

−

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡⎣ ⎤⎦

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

k

k

k

k

k

k

k

k

x k w k

x k

k

k

k

k

v k

x

x

x

x

AI

I

I

x

x

x

x

B

C

x

x

x

x

0 0 00 0 0

0 0 0

0 0 0

1

1

1

1

00

0

1 1

0 0

A-3

k

k

k N

k

k

k N

m

m

k

k

k N

k

1

2

2

3

1

1

2

where I is a 2�2 identity matrix. Using this augmented model, the best estimate of the states in a recursive Kalman filteralgorithm can be expressed as:

( )( )( )

( )

( )( )( )

( )

( )( )( )

( )

|

|

|

|

|

|

|

|

|

|

|

|

^

^ −^ −

⋮^ −

=

⋯⋯⋯

⋮ ⋱ ⋱ ⋱ ⋮⋯

^ − −^ − −^ − −

⋮^ − + −

+ − ⋯

^ − −^ − −^ − −

⋮^ − + − ( )

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪⎪

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪⎪

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎡⎣ ⎤⎦

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪⎪

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

k k

k k

k k

k N k

k k

k k

k k

k N k

x

k k

k k

k k

k N k

x

x

x

x

AI

I

I

x

x

x

x

K C

x

x

x

x

1

2

0 0 00 0 0

0 0 0

0 0 0

1 1

2 1

3 1

1 1

0 0

1 1

2 1

3 1

1 1 A-4

maug

In this algorithm, ( )|^ −k N kx is directly estimated without any additional recursion that is required in fixed pointsmoothing. The prior estimate of fixed lag smoothing is expressed as

( )( )( )

( )

( )( )( )

( )

|

|

|

|

|

|

|

|

^

^ −^ −

⋮^ −

=

⋯⋯⋯

⋮ ⋱ ⋱ ⋱ ⋮⋯

^ − −^ − −^ − −

⋮^ − + − ( )

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪⎪

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪⎪

k k

k k

k k

k N k

k k

k k

k k

k N k

x

x

x

x

AI

I

I

x

x

x

x

1

2

0 0 00 0 0

0 0 0

0 0 0

1 1

2 1

3 1

1 1 A-5

References

[1] G. Di Massa, R. Russo, S. Strano, M. Terzo, System structure identification and adaptive control of a seismic isolation test rig, Mech. Syst. Signal Process.40 (2013) 736–753.

[2] J. Kim, K. Kim, H. Sohn, Data-driven physical parameter estimation for lumped mass structures from a single point actuation test, J. Sound Vib. 332(2013) 4390–4402.

[3] J. Kim, K. Kim, H. Sohn, In situ measurement of structural mass, stiffness, and damping using a reaction force actuator and a laser Doppler vibrometer,Smart Mater. Struct. http://dx.doi.org/10.1088/0964-1726/22/8/085004.

[4] F. Moreu, H. Jo, J. Li, R.E. Kim, S. Cho, A. Kimmle, S. Scola, H. Le, B.F. Spencer Jr., J.M. LaFav, Dynamic assessment of timber railroad bridges using

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355354

displacements, J. Bridge Eng. 20 (2014) 04014114.[5] W.J. Merkle, J.J. Myers, Use of the total station for loading test of retrofitted brideges with limited access, in: Proceeding of SPIE, Smart Structures and

Materials 2004: Sensors and Smart Structures Technologies for Civil, Mechanical and Aerospace Systems, 5391, 2004, pp. 687–694.[6] Y.K. Thong, M.S. Woolfson, J.A. Crowe, B.R. Hayes-Gill, D.A. Jones, Numerical double integration of acceleration measurement in noise, Measurement 36

(2004) 73–92.[7] S. Stiros, Errors in velocities and displacements deduced from accelerographs: an approach based on the theory of error propagation, Soil Dyn. Earthq.

Eng. 28 (2008) 415–420.[8] D.M. Moore, Effect of baseline corrections on response spectra for two recodings of the 1999 Chi-Chi, Taiwan, earthquake, U.S. Geological Survey Open-

File Report 99-545, 1999.[9] D.M. Moore, Effect of baseline corrections of displacement and response spectra for several recordings of the 1999 Chi-Chi, Taiwan, earthquake, Bull.

Seismol. Soc. Am. 91 (2001) 1199–1211.[10] Y.H. Hong, H. Kim, H.S. Lee, Reconstruction of dynamic displacement and velocity from measured accelerations using the variational statement of an

inverse problem, J. Sound Vib. 329 (2010) 4980–5003.[11] W.D. Iwan, M.A. Moser, C. Peng, Some observations on strong motion earthquake measurement using a digital accelerograph, Bull. Seismol. Soc. Am.

75 (1985) 1225–1246.[12] N.A. Halliwell S.A. Kember S.J. Rothberg, Laser vibrometry for impact measurements, in: V.I. Babitsky (Ed.), Dynamics of vibro-impact Systems:

Proceedings of the Euromech colloquium, Springer, 1999, pp. 279–288.[13] P. Castellini, P. Martarelli, E.P. Tomasini, Laser Doppler Vibrometry: development of advanced solutions answering to technology's needs, Mech. Syst.

Signal Process. 20 (2006) 1265–1285.[14] S. Rothberg, Numerical simulation of speckle noise in laser vibrometry, Appl. Opt. 45 (2006) 4523–4533.[15] C. Negulescu, G. Luzi, M. Crosetto, D. Raucoules, A. Roullé, D. Monfort, L. Pujades, B. Colas, T. Dewez, Comparison of seismometer and radar mea-

surements for the modal identification of civil engineering structures, Eng. Struct. 51 (2013) 10–22.[16] G. Luzi, M. Crosetto, M. Cuevas-González, A radar-based monitoring of the Collserolar tower (Barcelona), Mech. Syst. Signal Process. 49 (2014)

234–248.[17] A.M. Wahbeh, J.P. Caffrey, S.F. Masri, A vision-based approach for the direct measurement of displacements in vibrating systems, Smart Mater. Struct.

12 (2003) 785–794.[18] K. Kamiyama, K. Vlack, T. Mizota, H. Kajimoto, N. Kawakami, S. Tachi, Vision based sensor for real-time measuring of surface traction field, IEEE

Comput. Graph. Appl. 25 (2005) 68–75.[19] J.J. Lee, M. Shinozuka, A vision based system for remote sensing of bridge displacement, NDTE Int. 39 (2006) 425–431.[20] J.W. Park, J.J. Lee, H.J. Jung, H. Myung, Vision-based displacement measurement method for high-rise building structures using partitioning approach,

NDTE Int. 43 (2010) 642–647.[21] S.W. Kim, N.S. Kim, Multi-point displacement response measurement of civil infrastructures using digital image processing, Procedia Eng. 14 (2011)

195–203.[22] H. Yang, T. Takaki, I. Ishii, Real-time multidirectional modal parameter estimation of beam-shaped objects using high-speed stereo vision, in: Pro-

ceedings of the IEEE, Sensors, Taipei, Taiwan, October 2012, pp. 1–4.[23] S. Nakamura, GPS measurement of Wind-induced suspension bridge girder displacement, ASCE J. Struct. Eng. 126 (2000) 1413–1419.[24] P. Breuer, T. Chmielewski, P. Górski, E. Konopka, Application of GPS technology to measurements of displacements of high-rise structures due to weak

wind, J. Wind Eng. Ind. Aerodyn. 90 (3) (2000) 223–230.[25] Y. Tamura, M. Matsui, L.C. Pagnini, R. Ishibashi, A. Yoshida, Measurement of wind-induced response of buildings using RTK-GPS, J. Wind Eng. Ind.

Aerodyn. 90 (2002) 1783–1793.[26] W. –S. Chan, Y. –L. Xu, X. –L. Ding, Y. –L. Xiong, W. –J. Dai, Assessment of dynamic measurement accuracy of GPS in three directions, ASCE J. Surv. Eng.

132 (2006) 108–117.[27] M.S. Grewal, L.R. Weill, A.P. Andrews, Global Positioning Systems, Inertial Navigation, and Integration, John Wiley & Sons, Inc., Hoboken, NJ, 2007.[28] R. Staiger, Terrestrial laser scanning—Technology, systems and applications, in: Proceedings of the 2nd FIG Regional Conference, Marrakech, Morocco,

December 2003.[29] M.K. Kim, H. Sohn, C. -C. Chang, Automated dimensional quality assessment of precast concrete panels using terrestrial laser scanning, Autom. Constr.

45 (2014) 163–177.[30] O. Monserrat, M. Crosetto, Deformation measurement using terrestrial laser scanning data and least squares 3D surface matching, ISPRS J. Photo-

gramm. Remote Sens. 63 (2008) 142–154.[31] S.J. Gordon D.D. Lichti M.P. Stewart J. Franke, Modeling point clouds for precise structural deformation measurement, in: Proceedings of the XXth

International Society for Photogrammetry and Remote Sensing Congress, Istanbul, Turkey, July 2004, pp. 1014–1019.[32] H.S Park, H.M Lee, H. Adeli, I Lee, A new approach for health monitoring of structures: terrestrial laser scanning, Computer-Aided Civil Infrastruct. Eng.

22 (2007) 19–30, http://dx.doi.org/10.1111/mice.2007.22.issue-1.[33] F. Blais, Review of 20 years of range sensor development, J. Electron. Imaging 13 (2004) 231–240.[34] F. Moschas, S. Stiros, High accuracy measurement of deflections of an electricity transmission line tower, Eng. Struct. 80 (2014) 418–425.[35] F. Moschas, S. Stiros, Three dimensional dynamic deflections and natural frequencies of a stiff footbridge, based on measurements of collocated

sensors, Struct. Control Health Monit. 21 (2014) 23–42.[36] S. Stiros, P. Psimoulis, Response of a historical short-span railway bridge to passing trains: 3-D deflections and dominant frequencies derived from

Robotic Total Station (RTS) measurements, Eng. Struct. 45 (2012) 362–371.[37] F. Moschas, S. Stiros, Measurement of the dynamic displacements and of the modal frequencies of a short-span pedestrian bridge using GPS and an

accelerometer, Eng. Struct. 33 (2011) 10–17.[38] W.S. Chan, Y.L. Xu, X.L. Ding, W.J. Dai, An integrated GPS-accelerometer data processing technique for structural deformation monitoring, J. Geod. 80

(2006) 705–719.[39] Y.H. Hong, S.G. Lee, 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. 38 (2013) 460–481.[40] A. Smyth, M. Wu, Multi-rate Kalman filtering for the data fusion of displacement and acceleration response measurement in dynamic system

monitoring, Mech. Syst. Signal Process. 21 (2007) 706–723.[41] S.G. Tzafestas, On optimum distributed-parameter filtering and fixed-interval smoothing for colored noise, IEEE Trans. Autom. Control 17 (1972)

448–458.[42] D. Haessig, B. Friedland, Separate-bias estimation with reduced-order Kalman filters, IEEE Trans. Autom. Control 43 (1998) 983–987.[43] J. Kim, K. Kim, H. Sohn, Autonomous dynamic displacement estimation from data fusion of acceleration and intermittent displacement measurements,

Mech. Syst. Signal Process. 42 (2014) 194–205.[44] C.B. Scruby, L.E. Drain, Laser Ultrasonics: Techniques and Application, Adam Hilger, Bristol, UK, 1990.[45] M. Johansmann, G. Siegmund, M. Pineda, Targeting the limits of laser Doppler vibrometry, 2005, in: Proceedings of International Disk Equipment and

Materials Association, Tokyo, Japan, 2005, pp. 1-12.[46] J.M. Tribolet, A new phase unwrapping algorithm, IEEE Trans. Acoust. Speech Signal Process. 25 (1977) 170–177.[47] V. Aranchuck, A.K. Lal, C.F. Hess, J.M. Sabatier, R.D. Burgett, I. Aranchuk, W.T. Mayo Jr., Speckle noise in a continuous scanning multi-beam laser Doppler

vibrometer for acoustic landmine detection, in: Proceeding of SPIE, Detection and Remediation Technologies for Mines and Minelike Targets XI, vol.6217, Orlando, FL, April 2006, http://dx.doi.org/10.1117/12.668783.

K. Kim, H. Sohn / Mechanical Systems and Signal Processing 82 (2017) 339–355 355

[48] S.J. Rothberg, N.A. Halliwell, Vibration measurements on rotating machinery using laser Doppler velocimetry, J. Vib. Acoust. 116 (1994) 326–331.[49] B. Stanbridge, D.J. Ewins, Modal testing using a scanning laser Doppler vibrometer, Mech. Syst. Signal Process. 13 (1999) 225–270.[50] M. Martarelli, D.J. Ewins, Continuous laser Doppler vibrometry and speckle noise occurrence, Mech. Syst. Signal Process. 20 (2006) 2277–2289.[51] D.M. Siringoringo, Y. Fujino, Noncontact operational modal analysis of structural members by laser Doppler vibrometer, Computer-Aided Civil In-

frastruct. Eng. 24 (2009) 249–265.[52] M.S. Allen, M.W. Sracic, A new method for processing impact excited continuous-scan laser Doppler vibrometer measurements, Mech. Syst. Signal

Process. 24 (2010) 721–735.[53] G. Teza, A. Galgaro, N. Zaltron, R. Genevois, Terrestrial laser scanner to detect landslide displacement fields: a new approach, Int. J. Remote Sens. 28

(2007) 3425–3446.[54] A. Abellán, J.M. Vilaplana, J. Martinez, Application of a long-range terrestrial laser scanner to a detailed rockfall study at Vall de Núria (Eastern

Pyrenees, Spain), Eng. Geol. 88 (2006) 136–148.[55] N. Yastikli, Documentation of cultural heritage using digital photogrammetry and laser scanning, J. Cult. Herit. 8 (2007) 423–427.[56] M. Murphy, E. McGovern, S. Pavia, Parametric vector modelling of laser and image surveys of 17th century classical architecture in Dublin, in:

Proceeding of the 8th International Symposium on Virtual Reality, Archaeology and Cultural Heritage, Brighton, UK, 2007, pp. 79–84.[57] S. Pu, G. Vosselman, Knowledge based reconstruction of building models from terrestrial laser scanning data, ISPRS J. Photogramm. Remote Sens. 64

(2009) 575–584.[58] K. Kim, J. Kim, Dynamic displacement measurement of a vibratory object using a terrestrial laser scanner, Meas. Sci. Technol. 26 (4) (2015) 045002.[59] T. Rajbhandary X.–P. Zhang, Piecewise constant modeling and Kalman filter tracking of systematic market risk, in: Proceeding of the IEEE Global

Conference on Signal and Information Processing, Austin, TX, 2013, p. 1144.[60] J.S. Meditch, Stochastic Optimal Linear Estimation and Control, MacGraw-Hill, New York, 1969.[61] H.E. Rauch, Solutions to the linear smoothing problem, IEEE Trans. Autom. Control AC-8 (1963) 371–372.[62] J.B. Moore, Discrete-time fixed-lag smoothing algorithms, Automatica 9 (1973) 163–173.[63] D. Simon, Optimal State Estimation – Kalman, H1, and Nonlinear Approaches, John Wiley & Sons Inc, Hoboken, NJ, 2006.