1. Introduction
Track irregularities mainly originate from heavy traffic loads and unexpected ground movement, which excite the wheelset and create unwanted vibrations. These vibrations may affect passenger comfort and cause the degradation of track and train components or, at worst, derailment. Therefore, rail operators and infrastructure managers should regularly monitor and maintain the track irregularities with proper methods to ensure passenger comfort and safety.
Track irregularities, defined as deviations from the original track geometry, are usually measured with a track inspection vehicle equipped with special measurement devices using contact probes, lasers, or optical sensors [
1,
2]. However, using this inspection vehicle with measuring devices is costly and complicated. In addition, the dynamic characteristics of a track inspection vehicle are different from those of a high-speed train in commercial operation, so the dynamic deflection of the track caused by a high-speed train cannot be appropriately measured. As an alternative, a simple and inexpensive device using accelerometers installed on an in-service vehicle has drawn attention because it can be used for daily monitoring of track conditions. Theoretically, displacement can simply be estimated by double integrating acceleration. However, in practice, the integration usually yields unwanted drifts due to the non-zero initial condition of the signal, the direct current (DC) offsets, or the noise due to electrical/mechanical hysteresis in sensors or cables [
3]. To avoid these drawbacks, many researchers have proposed model-based methods. In these methods, system models are required, which can represent the input–output relationship between acceleration and track irregularities.
Several studies have sought models that describe the dependence between acceleration and track irregularities. Kawasaki et al. [
4] presented a method using car body acceleration and an auto-regression model with extra inputs. The properties of the suspension system highly influence the car body acceleration; hence, it is difficult to separate the effect of suspension from the acceleration signals. Weston et al. [
5,
6] used bogie-mounted accelerometers and a gyroscope to monitor track irregularities and proposed a correction using the second-order dynamic model to improve the lateral irregularity. The results of such studies were limited to wavelengths less than 35 m or 70 m and thus were unsuitable for measuring long wavelengths up to 150 or 200 m. Alfi et al. [
7] proposed a method for calculating irregularities with long wavelengths from vehicle acceleration measurements using a model-based identification procedure defined in the frequency domain. However, there was a non-negligible difference in the trend of the power spectral density and in the space diagram for wavelengths up to 120 m, which is essential for passenger comfort during high-speed journeys. Czop et al. [
8] presented an approach to detect track irregularities using axle box accelerometers and the inverse linear parametric vehicle dynamics model. They focused on the relationship between the measured bending moment and axle box vibrations, not displacement, which is essential for making a track maintenance decision. Hidalgo et al. [
9] and Tsunashima et al. [
10] developed Kalman-filter-based techniques that combined a kinematic model and a dynamic model to identify track irregularities. Both works used an accelerometer and a gyroscope to estimate the vertical track irregularities, and relatively acceptable accuracy was obtained. Muñoz et al. [
11] proposed an efficient Kalman-based methodology for monitoring lateral track irregularities using inertial sensors installed on a train in operation. In this study, two accelerometers are utilized to measure the lateral acceleration of the wheelset and the bogie frame. At the same time, a gyroscope is employed to detect the yaw angular velocity of the wheelset.
Lee et al. [
12] proposed a mixed filtering approach using the Kalman, band-pass, and compensation filters for waveband monitoring of lateral and vertical track irregularities, which used accelerometers installed on the axle box and the bogie. The method used Kalman and band-pass filters for displacement estimation from measured acceleration. Compensation filters consisting of finite impulse response models with 40 parameters were used to correct for amplitude and phase difference, which result from the inherent characteristics of the preceding filters and the lateral motion of the wheelset or the bogie with respect to the track. However, the models were expensive and complex because too many parameters were used. This research proposes efficient parametric models with fewer parameters. The parameters are derived from the measured signals obtained using a track geometry inspection system (TGIS). An adaptive Kalman filter algorithm is applied to obtain the unknown parameters of track irregularities with an estimated displacement from acceleration signals as the input and track irregularity signals as the output. Finally, the developed models are used in the analysis of acceleration data measured from the axle box and the bogie of a high-speed train in operation.
This paper is organized as follows: The measurement setup used to obtain acceleration signals and track geometry is described in
Section 2;
Section 3 presents the process of estimating displacement from acceleration signals, while
Section 4 explains the parametric models and the methodology used to estimate the parameters;
Section 5 describes the model section process and validates the selected model; In
Section 6, track irregularities are estimated using acceleration signals obtained from high-speed trains in operation, and the results are compared with the reference irregularities; and the summary and conclusions can be found in
Section 7.
3. Displacement Estimation from Acceleration
As mentioned in the introduction, displacement estimation from noisy acceleration using direct double integration results in unrealistic errors. A discrete state-space model and the Kalman filter were introduced to resolve the error in the previous works [
12,
14]. The following describes the state-space model for displacement estimation from noisy acceleration:
In Equation (3), α is a model parameter () and is the sampling time. In the state-space equations, the state transition matrix is used to update the preceding state, and is the noise-input matrix. At the same time, is the measurement matrix used to map the estimated displacement onto the measured acceleration. The noises and are comprised of zero-mean white Gaussian processes. It is assumed that the initial displacement is zero.
The measured acceleration signals are utilized to estimate the displacement using a Kalman filter algorithm, and its covariance form is described as follows [
15]:
where
is the estimate of
,
is the state error covariance information at step
,
is the auto-covariance of the initially estimated displacement
, and
and
are the auto-covariances of
and
, respectively.
After applying the Kalman filter, the third-order Butterworth band-pass filters are applied to eliminate the short-wavelength effect due to the wheel and the bogie and the long-wavelength effect due to the track’s curves. Block diagrams illustrating the processes are presented in
Figure 2.
6. Track Irregularity Estimation Using Derived Models
To examine the applicability of the derived models, they were used to estimate track irregularities from the signals obtained using the accelerometers installed on the axle box and the bogie of a KTX train in operation. A section of the slab track with notable lateral and vertical track irregularities was selected for comparison. The reference measurement with the TGIS was carried out approximately one year before the measurement tests with the in-service train. The slab track was selected because the variations in time are expected to be smaller than those for the ballasted track. Three measurement tests were carried out within a one-week interval to ensure the reproducibility of the methodology, and the results were compared in the spatial and the wavelength domains as shown in
Figure 11,
Figure 12 and
Figure 13. The maximum train speed in these test campaigns was 300 km/h. The discrepancies were presumed to be a result of the differences in the suspension characteristics and wheel profiles of the KTX train and HSR-350x train.
The estimated and measured lateral track irregularities are compared in
Figure 11a and
Figure 12. The estimated results from the three tests show excellent agreement, confirming that the proposed method can be used for trains in regular operation. The results also show good agreement with the reference irregularity. In the spatial domain, as shown in
Figure 11a, an irregularity is clearly observed near the 0.25 km section from both accelerometers installed on the axle box and the bogie. In the wavelength domain, as shown in
Figure 12, the estimated results show good agreement over all wavelengths except below 4 m and near 10 m.
The estimated and measured vertical track irregularities are compared in
Figure 11b and
Figure 13. The results from the three tests of the bogie-mounted accelerometer show excellent agreement, while those of the axle box-mounted accelerometer show some discrepancies. However, a notable irregularity near the 0.25 km section is clearly observed. Its magnitude is estimated within a tolerable range in both the axle box and the bogie-mounted accelerometers, as shown in
Figure 11b. The wavelength characteristics of the estimated irregularities are shown in
Figure 13. The results obtained from the accelerometers installed on the axle box and the bogie exhibit the same spectral characteristics of the measured irregularity. The results show that bogie-mounted accelerometers estimate the irregularities better. It is presumed that the acceleration signal of the axle box is noisier because the vibration level is much higher.
7. Summary and Conclusions
The parametric models are identified by applying a system identification technique that uses estimated displacement from acceleration as the input and measured track irregularity as the output. The parameters are derived from the acceleration and the track irregularities from a track geometry inspection system (TGIS). The parametric models are set up based on the IIR and/or the FIR, and the adaptive Kalman filter is applied for their estimation. The orders of the parametric models are determined by evaluating the PPMCC and the MSE. The number of parameters can be reduced while improving the performance of the models. In this work, a hybrid IIR/FIR model and a single IIR model are selected for lateral and vertical directions. They are validated by estimating irregularities from the acceleration signals measured by the TGIS.
Finally, track irregularities are estimated using acceleration measured from trains in commercial operation. The results using data obtained from three measurement tests show good agreement, ensuring the methodology’s reproducibility. The estimated irregularities are compared with the reference irregularity in the spatial and wavelength domains. The suggested method can detect the location of irregularities in both the lateral and the vertical directions. It is also demonstrated that the estimated irregularities exhibit the same spectral characteristics as the measured irregularity.
In conclusion, the identified parametric models can be used to predict track irregularities from the accelerometers installed on high-speed trains in commercial operation.