Estimation of lateral track irregularity using a Kalman filter. Experimental validation

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in Journal of Sound and Vibration, Vol. 504, on July 2021, available at: https://doi.org/10.1016/j.jsv.2021.116122 Copyright 2021 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Estimation of lateral track irregularity using a Kalman filter. Experimental validation Sergio Mu˜noza,∗, Javier Rosb, Pedro Urdac, Jos´e Luis Escalonac aDepartment of Materials and Transportation Engineering, University of Seville, Spain bDepartment of Mechanical Engineering, Public University of Navarre, Spain cDepartment of Mechanical Engineering, University of Seville, Spain Abstract The aim of this work is the development of a model-based methodology for the estimation of lateral track irregularities from measurements from different sensors mounted on an in-service vehicle: a gyroscope to measure wheelset yaw angular velocity, two accelerometers to measure lateral acceleration of the wheelset and bogie frame, and an encoder to obtain forward velocity of the vehicle. The proposed methodology is based on the Kalman filtering technique, through the use of a highly simplified linear dynamic model of a bogie, capable of capturing the most relevant lateral dynamic behaviour of the entire vehicle. The simplified dynamic model (SM) is based on a vehicle running at variable forward velocity on a track, which comprises straight, curve and transition sections. Finally, the proposed methodology has been experimentally validated through an experimental campaign carried out in a 90 m 1:10 scaled track facility at the University of Seville and an instrumented scaled vehicle. The results of the estimation of the lateral alignment are analysed in the space domain and in the space frequency domain, according to standards. These results are promising, showing a good performance for monitoring lateral alignment on straight and curve tracks, with a very low computational cost. Only in the case of very sharp curves, when continuous flange contact takes place, the estimator is not able to precisely estimate lateral alignment. Keywords: rail vehicle dynamics; track irregularity; Kalman filtering; onboard measurements 1. Introduction Track irregularities are the main source of disturbances for a wheel-rail system, having a direct effect on vehicle dynamics. Consequently, these irregularities have a significant impact on train running safety and stability, as well as on running comfort. According to the EN13848 Standard [1], acceptable limit levels for track irregularities are considered according to their wavelength in three different ranges: D1= [3,25] m , D2= [25,70] m and D3= [70,200] m. Therefore, in the evaluation of quality of the track geometry, irregularities should be analysed taking into account these wavelength ranges, as well as the maximum allowed forward velocity of the vehicle. In this regard, the proper maintenance of railway tracks to ensure the minimum quality standards for both running safety and passenger comfort seems essential in the railway industry. This task has been accomplished by continuous monitoring of track geometry and through the use of track recording ∗Corresponding author: sergiom[email protected] Preprint submitted to Journal of Sound and Vibration November 28, 2022 vehicles (TRV). These specific vehicles, which accurately measure track irregularities using different sets of optical, laser or inertial sensors, have the major inconvenience of being complex and very expensive. In railway administrations that can afford them, the track network that has to be covered is so long that frequency of inspection is low. The alternative to such a high-priced TRV is the use of inexpensive measuring systems, mounted on in-service vehicles for continuous monitoring of track conditions. To this end, simple and robust measuring systems, combined with the development of dynamic model-based filtering techniques, are required to achieve an accurate estimation of track geometry. An extensive review on the use of in-service vehicles for the monitoring of railway tracks can be found in [2]. Regarding the monitoring of vertical irregularities, the simplest methods encountered in bibliography consist of the integration of accelerometer and gyroscope signals to obtain the absolute position of the wheelset and hence track geometry. In [3] for example, by the use of pitch-rate gyroscope sensors mounted on the bogie frame of an in-service vehicle, the vertical curvature of the track centre-line is derived and, through integration, the vertical profile of the track estimated. However, the main drawback of the integration of measured signals is the lack of accuracy and a drift in the obtained results. The use of high-pass filters can partially solve the drift problem, but at the expense of losing information in the low frequency range. In [4], vertical track irregularities are estimated through a sensor fusion algorithm based on complementary filters: the signals from an accelerometer and a gyroscope installed on a bogie are used and the estimation of low frequency irregularity relies on the data from the gyroscope while the high frequency irregularity relies on the data from the accelerometer. Fairly good results are obtained in the estimations, although a loss of accuracy is shown in the case of variable forward velocity of the vehicle. Regarding lateral irregularities, their estimation is much more difficult and challenging: lateral displacement of the wheelset is due to the presence of lateral irregularities of the track and to lateral motion of the wheelset relative to the track, which is related to creep force dynamics and wheel conicity. Furthermore, lateral irregularities, especially lateral alignment, prove to be much more influential in the dynamic behaviour of the vehicle than vertical ones. In [5], lateral alignment of the track is estimated through integration of lateral curvature of the track centre-line, using a procedure analogous to the one used in [3]. In addition to drift and low smoothness problems related to numerical integration, the proposed method cannot take into account lateral displacement of the wheelset relative to the track. In [6], a Kalman filter is used as a kind of integrator of lateral acceleration of the wheelset to obtain lateral displacements and, subsequently, a set of compensation filters are used in the corresponding wavelength bands to predict lateral irregularities. A model-based unknown input identification filter is used in [7]. Here, a linearized lateral dynamic model of a bogie with two wheelsets has been used, together with the use of the H- infinity theory to maximize sensitivity towards lateral displacement of the wheelset. More recently, in [8], the authors propose three different model-based methods to estimate both lateral track alignment and cross-level irregularities: 1) pseudo-inversion of the vehicle’s frequency response function (FRF) matrix, 2) unknown input estimation using a deterministic observer and 3) unknown input estimation using a linear Kalman filter as a stochastic observer. They use a very complex linear dynamic model of a railway vehicle composed of one car body, two bogies and four wheelsets. In the proposed model with 17 degrees 2 of freedom, relative motion between the wheelset and the track has been taken into account, considering the effect of creepage forces acting at the wheel-rail contact. The proposed methodologies have been validated through the use of numerical experiments based on a rich non-linear multibody model. Quite good results in the estimation are obtained with all three methods, especially with the Kalman filter approach. The main drawback of these methods is the complexity of the dynamic model used (17 degrees of freedom) and the high number of sensors installed on the vehicle (36 accelerometers). It is noticeable that this is one of the few references in which real data is used for validation: the FRF approach has been run using real measurement data from a track recording vehicle (TRV). Although the results are promising, a degradation in performance is evident in comparison with the validation using virtual sensor synthesised data. Even though there are several references dealing with the estimation of track irregularities, published literature on the estimation of lateral alignment is relatively scarce and presents a series of important inconveniences, which are commented next. Firstly, most of these works only focus on tangent (straight) line track segments. Secondly, despite several authors having demonstrated promising results, there appears to be a lack of profound analysis of these results: the results obtained should be thoroughly analysed in different wavelength ranges, according to standards, and a systematic analysis to test accuracy and robustness of the proposed technique must be performed. Finally, one remarkable inadequacy of most works in literature is the lack of rigour in the validation of the proposed estimation technique, since they have not been experimentally validated: most of them have been validated through the use of virtual experiments, obtained by complex numerical models. It must be clearly stated that, even though numerical tools to obtain virtual experiments can be a useful tool to analyse performance and robustness of an estimator, any estimator is suggested to be finally validated through the use of real experiments, if they are available. Regarding experimental validation, the use of scaled vehicles seems to be a useful tool for validation purposes with reduced operational costs. Some interesting examples of scaling design applied to the railroad field can be found in [9, 10, 11]. It must be noted that the use of scaled vehicles comprises inconveniences related to geometry scaling and causes design problems that should be given due consideration. In this regard, a thorough scaling strategy for the design of a 1:10 scaled railway vehicle developed by the railway research group of the University of Seville can be found in [12]. The procedure followed in this work shows that the scaled vehicle is a safe and economic tool for validation purposes (computational models, wheel-rail contact approaches or new running safety criteria). However, the great reduction in size that the length scaling factor of 1:10 involves, avoids a complete similarity achievement, the dynamic behaviour of the prototype not being directly extended to a full-scaled system. The main objective of this work is to estimate lateral alignment of a track from measurements of inertial sensors mounted on an in-service vehicle. This estimation aims for the vehicle to track relative motion, that is, the vehicle is expected not to follow track geometry. To this end, a model-based numeric procedure based on Kalman filtering tecniques has been developed in this work, under the following assumptions: the vehicle is running at variable forward velocity on a track (which can comprise straight, curve and transition sections) with no gauge variation, the wheelset has a conical profile and there is 3 no flange contact. Finally, the most important contribution of the present work is the experimental validation of the proposed methodology, through an experimental campaign carried out in the 1:10 scaled track facility at the University of Seville, consisting of a 90 m long scaled track and an instrumented scaled vehicle [12]. The obtained results in the estimation of lateral alignment have been analysed in the space frequency domain, according to standards [1]. In this respect, even though there are three wavelength ranges considered in the standard (D1, D2 and D3), when evaluating the acceptable limits of track irregularities, the long wavelength range (D3) is not considered, since this range is not directly related with safety, but with vehicle rolling quality. Consequently, only the first two wavelength ranges (D1 and D2) will be considered in the present work. The paper is organised as follows: Section 2 presents the standard definition of track irregularities. Section 3 describes the dynamic modelling on which the estimation procedure is based. In Section 4, the estimation technique used in this work is presented, together with the Kalman filter algorithm on which it is based. In Section 5, the main features of the experimental work, scaled track and instrumented vehicle, are presented. In Section 6, the results of the lateral alignment estimation are presented and discussed, validating the proposed estimation technique experimentally and evaluating its performance in real conditions. Finally, Section 7 provides conclusions and a summary. 2. Definition of track irregularities Lateral and vertical irregularities of a track are usually defined by four well-known irregularities variables in the railway industry: track gauge variation, ξg, lateral alignment, ξa, cross-level, ξcl, and vertical profile, ξvp. These variables are defined as follows: ξg= (ulr y−urr y), ξa= (ulr y+urr y)/2 ξcl = (ulr z−urr z), ξvp = (ulr z+urr z)/2 (1) where ulr y,urr y,ulr zand urr zare lateral (’y’) and vertical (’z’) deviation of the left (’lr’) and right (’rr’) rail cross-sections from their ideal position, see Fig. 1. 2Lr Y Z t trr u lr u Figure 1: Definition of track irregularities These irregularities can be divided into two different types: global irregularities as the variation of the track centre-line (lateral alignment and vertical profile), and relative irregularities as the variation of the relative position between both rails (track gauge variation and cross-level). 4 It is important to observe that the definition of the irregularities used in this work is based on knowledge of design geometry or, in other words, the position of the track frame. This design geometry may not be available in a real scale TRV. In this case, the definition of irregularities must be related to an estimated design geometry, which can be obtained from inertial sensor measurements and an odometry system. 3. Dynamic modeling With the aim of estimating lateral alignment of the track through a Kalman filter, a very simplified dynamic model (SM) of the vehicle has been developed in this work. The following is assumed: the vehicle is running at variable forward velocity on a track (which can consist of straight, transition and curve sections) with no gauge variation and the wheelset has a conical profile and there is no flange contact. A schematic plan view of the SM is presented in Fig. 2: the different bodies and elements of the SM are shown in the drawing on the left, while an arbitrary position of the system is shown in the one on the right. The SM only models one wheelset and a suspended frame, representing dynamic interaction of the wheelset with the rest of the vehicle. The SM uses two generalised coordinates for the wheelset (lateral displacement, y, and yaw rotation, ψ), and one generalised coordinate for the suspended frame (lateral displacement, yf). All generalised coordinates are referred to a Track Frame hXt, Y ti, which moves along an irregularity-free track centre line with the same forward velocity as the vehicle. Both bodies, the wheelset and the suspended frame, are connected by longitudinal and lateral primary suspension elements. kycy ls kx cx Suspended frame Wheelset l r0 a s Y yf x (s) y kx cx cy ky V Xt t Xt Yt y Figure 2: Plan view of Simplified Model The set of generalized coordinates of the simplified vehicle model is therefore: q=yψyfT(2) 5 The equations of motion of lateral dynamics associated with this model are: M ¨q +bCs+Ccc˙q +bKs+Kccq=Qc,0+QF orIn (3) where M,Csand Ksare constant mass, suspension damping and suspension stiffness matrices associated with lateral dynamics, respectively; Ccand Kcare damping and suspension matrices associated with tangential contact forces acting on the wheelset in the lateral direction; vector Qc,0contains the terms that appear in the generalized contact forces when the lateral coordinates and velocities are zero; and QF orIn is the vector of generalised inertia forces due to forward motion. These matrices and vectors are obtained using symbolic computation: Lagrange Equations are used to find inertia, elastic and damping generalised forces, while the principle of virtual work is used to find contact tangential generalised forces. Wheel-rail contact at the treads is modelled with knife-edge contact constraints [13], and tread tangential contact forces are calculated following Kalker’s creep linear theory [14]. The calculation of flange contact forces has been excluded from this study, assuming there is no flange contact. It is important to note that, non-linear generalized suspension forces and contact forces have been linearized with respect to generalized coordinates and their first time-derivatives. Finally, the vector of generalised inertia forces can easily be calculated through the following expressions: QF orIn =      −mρhV2 −I[ρh˙ V+ρ0 hV2] −mfρhV2      (4) where mand Iare the mass and yaw moment of inertia of the wheelset, mfis the mass of the suspended frame, Vand ˙ Vare the forward velocity and acceleration of the wheelset, and ρhand ρ0 h are the horizontal curvature of the track and its space derivative. The symbolic calculation of all these matrices and vectors can be found in detail in [15]. It is important to note that the Simplified Model (SM) used in the present work is a very simplified dynamic model of a single wheelset. Several comments about the SM must be made. First, the lateral and vertical dynamics of the wheelset has been considered to be weakly coupled and, as consequence, only the lateral dynamics of the wheelset has been used. The influence of vertical dynamics on the lateral one is the vertical loads transmitted through the primary suspension, which affects the creep contact forces through Kalker’s linear creep coefficients. This influence has been proved not to be important and, in consequence, Kalker’s coefficients have been assumed to be constant in the present work. The second comment is related to the dynamic behaviour of the whole vehicle. Track irregularities can be seen as a sequence of time-delayed inputs in the different wheelsets of the vehicle and, as a consequence, affecting the dynamic behaviour of the whole vehicle in a very specific manner. The Simplified Model only takes into account the leading wheelset and the effect of the rest of the vehicle is modelled through the primary suspension parameters. Despite this simplification, a proper estimation of the equivalent parameters has proved to be sufficient for our estimation purposes. The complete set of parameters that characterises the SM is given by vector: p= [m I l lsα r0mff11 f22 f23 f33 kxcxkycy]T(5) 6 where m,Iand lare the mass, the yaw moment of inertia and half-width of the wheelset; lsthe length to the primary suspension (0y0component of the local position vector of the primary suspension); αthe nominal conicity of the wheel tread; r0the rolling radius of the wheels when the wheelset is centered on the track; mfthe mass of the suspended frame; fij Kalker’s linear creep coefficients (which are assumed to be constant across the profile); and kx,cx,kyand cythe primary suspension parameters. The longitudinal position of the vehicle along the track is defined using arc-length coordinate s, which is assumed to be an input in the simulation. The instantaneous value of sis used to find track geometry inputs ρhand ρ0 h, as well as to compute Vand ˙ V. Out of the four irregularities defined in Eq. (1), only lateral alignment is considered in the SM, hereinafter referred to as ξ. Since the gauge variation is assumed to be zero in this work, lateral alignment is defined as ξ=ξa=ulr y=urr y. This lateral alignment is a function of the arc-length coordinate, ξ=ξ(s). The considerable simplification of the vehicle model proposed with the SM should be sufficient to describe the stability and guidance dynamics of the wheelset due to the lateral track irregularity. This model will be valid for our purpose only if it can adequately describe the lateral dynamic response of the wheelset running on a track with irregularities. Therefore, before using the proposed SM in the Kalman filter, the model has to be identified. The set of equivalent parameters can be divided into two subsets: the parameters that can directly be obtained from the real train, pfix, and the parameters to be identified by optimisation, popt. Set pfix is composed of the first eleven parameters of vector p, and can directly be obtained from the real train: the first six parameters (m, I, l, ls, α, r0) correspond to inertial and geometrical properties of the wheelset. The seventh parameter, the mass of the suspended frame (mf), can approximately be calculated as the mass supported by the wheelset in a static equilibrium position: that is, a quarter of the mass of the car body plus half the mass of one bogie frame. The next four parameters (f11, f22, f23, f33) are the creep coefficients, which depend on normal contact force, the size and shape of the contact patch, the curvatures of the surfaces at the contact points and the elastic properties of the bodies in contact. In this work, the creep coefficients are assumed to be constant and their values have been calculated following Kalker’s creep linear theory [14], considering the magnitude of the normal contact force as the weight supported by each wheel at the static equilibrium position. Set popt consist of the last four parameters (kx,cx,ky and cy), corresponding to the properties of the primary suspension of the SM, which cannot directly be taken from the real suspension elements of the train, but must be identified by optimisation, through the following procedure. First, the simulation of the vehicle running along a track with irregularities has been carried out using a complete and non-linear model of the vehicle (CM), previously proposed by the authors [15]. For this simulation, the real parameters of the vehicle have been used. Second, the same simulation by the Simplified Model (SM) has been carried out, using the equivalent parameters to be identified, popt, in this case. Finally, the set of parameters popt has been obtained by optimisation, in such a way that similar dynamic behaviour of the leading wheelset is predicted by both models. For the evaluation of dynamic behaviour, the most relevant dynamic variables of the leading wheelset have been chosen: y,ψand ym. It is important to note that this procedure has been performed with different track geometries and associated irregularities. The different track design geometries have been generated including different track sections (straight, transitions and curves). The associated track irregularities 7 have been generated randomly from power spectral density functions (PSD) of real tracks, after having been scaled. Therefore, the obtained set of equivalent parameters is valid for any track. 4. Estimation technique The main objective of this work is to estimate the lateral alignment of a track, ξ, from the measurements from inertial sensors mounted on an in-service vehicle. To this end, a model-based numeric procedure based on Kalman filtering tecniques has been used. As commented, the estimation technique is based on a very simplified dynamic model (SM), presented in Section 3. The estimation technique, which is graphically summarised in Fig. 3, is explained as follows. KALMAN FILTER Experimental data V (t) ÿmeas (t) ymeas (t) ÿmeas (t) Track design geometry [ r (s) , r’ (s) ] f xest (s) hh Figure 3: Estimation procedure As it can be observed, the estimation procedure is based on a Kalman filter, whose design will be explained later. The filter has two different kinds of inputs. On one hand, the curvature of the ideal track centre line (ρhand ρ0 h) as a function of arc-length coordinate s. Notice that, the definition of the track centre line is needed in the estimation technique to calculate the vector of generalised inertial forces, QF orIn, through Eq. 4. On the other hand, the Kalman filter needs the experimental data of the dynamic response of the vehicle as an input: the vehicle’s forward velocity (V), acceleration in the lateral direction (¨ymeas) and yaw angular velocity ( ˙ ψmeas) of the wheelset, and acceleration in the lateral direction (¨yf meas) of the suspended frame. All these data must be acquired from one encoder and two inertial sensor units (IMU) installed in the vehicle, and synchronized through a data acquisition system. From all these data inputs, making use of the Kalman filter, the lateral alignment of the track can be estimated. 4.1. Design of the Kalman filter As previously explained, the estimation technique is based on the well-known Kalman filter algorithm [16], which is briefly defined as follows. The state vector includes the generalized coordinates, q, their derivatives, ˙q, and lateral irregularity, ξ, as: x= [q ˙q ξ]T=hyψyf˙y˙ ψ˙yfξiT (6) Lateral irregularity, ξ, included in the augmented state vector, is thus described by the typical random walk equation. However, the related treatments of this problem in literature [17, 18, 19] has shown that 8 6. Results In this section, the proposed estimation technique is validated and its performance analysed. To do so, a complete campaign of six experiments has been carried out, involving different conditions: three different forward speed levels (1.5 m/s, 2.0 m/s and 2.5 m/s) and both directions of travel (forward and backward). In the present work, only one of the experiments will be presented as representative: the vehicle moving forward on the track at constant speed V= 2.0 m/s, for a total length of 85 m. It must be pointed out that, even though the speed set-point is a fixed value, the real forward speed is not always constant during the whole experiment: during the starting and braking stages, the speed experiences an increasing and decreasing curve, respectively; besides, during the rest of the journey, the speed remains almost constant, with certain oscillations due to motor control. As previously explained in Section 4.2, the Kalman filter needs an estimation of the sensor noise variance, σsensors, for the prior calculation of the covariance matrices, Qand R. The noise variance has been taken from the data sheet of inertial measurement units. Therefore, a value of (0.005)2m2/s4for the accelerometers (¨ymeas and ¨yf meas) and (0.0005)2rad2/s2for the gyroscope ( ˙ ψmeas) are considered. For the lateral irregularity included as a virtual sensor (ξmeas), a value of 25 mm2has been set, which is the order of the variance of lateral irregularity. 6.1. Identification of the equivalent parameters Previously to the use of the Kalman filter, the identification of the equivalent parameters of the SM has been carried out. Table 1 shows the entire set of parameters for the SM, obtained from the parameters of the real vehicle. Parameter Description Value Units mWheelset mass 4.315 kg IWheelset yaw moment of inertia 0.0588 kg.m2 lHalf width of the wheelset 0.0668 m lsLength to the primary suspension 0.0968 m αNominal conicity 0.1 r0Rolling radius of the wheels 0.0635 m mfMass of the suspended frame 8.7375 kg f11 Longitudinal creep coeficient 2.27e4 N f22 Lateral creep coeficient 2.33e4 N f23 Lateral spin creep coeficient 3.43 N.m f33 Spin creep coeficient 0 N.m2 kxLongitudinal suspension stiffness 7012 N/m cxLongitudinal damper coefficient 2.14 N.s/m kyLateral suspension stiffness 12556 N/m cyLateral damper coefficient 20 N.s/m Table 1: Equivalent parameters of the SM 15 6.2. Input data from measurements In this section, the input data from the experiment under study (forward direction and V= 2 m/s) are presented. Firstly, the forward speed profile obtained from the encoder installed in the vehicle is presented in Fig. 7. As observed, the forward speed experiences an initial steep increasing slope until reaching the set-point value, when certain oscillations are observed before stabilization. Similar behaviour is observed at the end of the ride, when the vehicle brakes. 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 V ( m / s ) D i s t a n c e ( m ) Figure 7: Forward speed profile Secondly, the experimental data from the inertial sensors installed in the vehicle (in the wheelset and in the bogie frame) are presented in Fig. 8. 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 - 4 - 2 0 2 4 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 - 4 - 2 0 2 4 . . .. .ym e a s ( m / s 2) D i s t a n c e ( m ) Ψm e a s ( r a d / s ) D i s t a n c e ( m ) y f m e a s ( m / s 2) D i s t a n c e ( m ) Figure 8: Data from inertial sensors: a) ¨ymeas, b) ˙ ψmeas and c) ¨yf meas In view of the experimental data from the IMU’s, several aspects can be highlighted. Regarding the information from the accelerometers, the level of acceleration in bogie frame, ¨yf meas, is lower than the one in the wheelset, ¨ymeas, as a result of the primary suspension. Furthermore, some impulsive lateral accelerations can be appreciated in the wheelset along the ride. These impulses are due to impacts between the rail and the flange of the wheel, which have been measured by the accelerometer not only 16 in the lateral direction but also in the vertical one (not included in this work). As for the yaw angular velocity, ˙ ψmeas, it clearly detects the open curve (23 ∼47 m) and the sharp curve (63 ∼73 m). 6.3. Estimation of lateral irregularity Once the track centre line has been measured and the equivalent parameters of the SM of the vehicle estimated, the prediction of lateral irregularity has been carried out through the proposed Kalman filter algorithm. To this end, the experimental data acquired during the experiment under study, presented in Section 6.2, have been used as input. Figure 9 shows the comparison between the estimated lateral alignment and the reference one (previously measured and presented in Fig. 5), for the whole track length. It should be noted that both profiles, estimated and real, have been filtered with a Butterworth bandpass filter in the range of interest, according to standards [1]: frequencies corresponding to a wavelength between 0.3 and 7 m, for the 1:10 scaled track. 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 - 1 2 - 8 - 4 0 4 8 1 2 S e c t i o n 3 S e c t i o n 2 S e c t i o n 1 A m p l i t u d e ( m m ) D i s t a n c e ( m ) R e a l E s t i m a t i o n 9 5 % C . I Figure 9: Lateral alignment estimation of the whole track, filtered in D1-D2 range (λ= 0.3 - 7 m) In light of the results, a relatively good agreement of the estimated and the real lateral alignment is observed across the entire length of the scaled track. The highest disagreement is encountered in the transition segments, between Section 1, 2 and 3, where a slight phase delay is observed. Additionally, an important disagreement is also observed in the estimation along the sharp curve (Section 3), which will be discussed in a latter analysis. In order to eliminate the phase delay obtained by the Kalman filter, a Smoother Kalman Filter has also been used in this work. Even though the Smoother Kalman Filter (a forward-backward version of the Kalman Filter), usually offers better performance in estimation, avoiding the phase delay, no significant improvements have been encountered in this case. Consequently, these results have not been included. Furthermore, in order to add more information about the lateral alignment estimation, the 95% confidence interval of the prediction has been calculated and included in the figure. This confidence interval has been obtained by projecting the error covariance matrix, P+ k, to the estimated values. As it can be observed, the width of the confidence interval has a reduced and constant value (±0.1 mm) along the whole length of the scaled track. In order to numerically evaluate the results achieved with the proposed Kalman filter estimator, an 17 accuracy index has been calculated. In this work, two different accuracy indices have been used: J=rms (ξest −ξreal) Jrel =rms (ξest −ξreal) rms (ξreal) (30) The first one is absolute accuracy index J, calculated as the root mean square value (rms) of the difference between the estimated and the real lateral irregularity. This index Jhas length units and is particularly useful and intuitive for measuring the agreement of the estimation with the real data. The second one is relative accuracy index Jrel, which corresponds to the non-dimensional value of J. This index has no dimensions and completes the information of the absolute accuracy index. Furthermore, the relative accuracy index has a clear advantage: it is scale-independent, so it can be used to compare errors in scaled tracks and in real-scale tracks. Therefore, the accuracy indices for the estimations, according to different wavelength ranges (whole range, D1 and D2), are shown in Table 2. In addition to the results obtained in the analysis of the whole track length (85 m), Table 2 also shows the results from the analysis of the different track sections under study, which will be analysed later. Whole range D1 range D2 range Whole track 1.89 / 1.08 0.77 / 1.31 1.57 / 0.92 Section 1 0.52 / 0.98 0.34 / 1.03 0.45 / 0.86 Section 2 0.61 / 0.72 0.50 / 1.02 0.45 / 0.53 Section 3 2.28 / 1.32 1.34 / 1.65 2.17 / 1.30 Table 2: Accuracy indices, J(in mm) / Jrel (in different wavelength ranges) From the analysis of the whole track length, it can be seen that the values of J= 1.89 mm and Jrel = 1.08 are achieved in the estimation when the whole spectrum is considered. At first sight, an absolute error of 1.89 mm may look excessive in the estimation of the lateral alignment. However, as previously pointed out, a significant amount of this error corresponds to the phase delay between the real and estimated alignment, along the transition segments. This fact is reflected in the analysis of the error by ranges in Table 2: much higher values in the low frequency range (D2), J= 1.57 mm, than the ones in the high frequency range (D1), J= 0.77 mm. Another significant amount of the error in the estimation corresponds to bad performance of the estimator in the sharp curve (Section 3), which will be discussed later. Finally, in order to test the efficiency of the proposed Kalman filtering method, the computing time to simulate the case under study has been calculated. The algorithm has been developed in Matlab R2016a with a computer with an Intel Core i7 CPU 2600 3.4 GHz processor. Only 20.2 sof computation time has been required to simulate the total time of the case under study, 45.9 s, corresponding to 85 m of track length. In these simulations, the temporal discretisation is ∆t= 0.002 s, corresponding to the acquisition rate of the sensors (500 Hz). The computational time can even be improved significantly if the Kalman 18 filter is implemented using a low-level programming language like Fortran or C/C++. Consequently, the proposed algorithm is particularly appropriate for real-time applications. To carry out a more in-depth analysis, the different segments in which stationary motion is reached (Sections 1-3 in Fig. 9) will be separately studied in subsequent subsections. 6.3.1. Section 1: straight segment The first segment under study is part of the initial straight segment (Section 1), corresponding to the track length from 2 to 20 m. Note that the first two meters of the straight segment have been eliminated in the analysis, since stationary motion with constant velocity has not yet been reached in this starting stage, as it can be seen in Fig. 7. The results of the estimation of lateral alignment along the straight segment corresponding to Section 1, is presented in Fig. 10. In the upper subplot, two irregularity profiles are compared in the space domain, for the 18 m track-length under study (notice that the 95% confidence interval of the prediction has been included). The lower subplot of Fig. 10 shows both profiles in the space frequency domain, obtained by the FFT. 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 - 2 0 2 0 . 1 1 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 A m p li t u d e ( m m ) λ ( m ) A m p li t u d e ( m m ) D i s t a n c e ( m ) R e a l E s t i m a t i o n 9 5 % C . I D 2 (λ = 2 . 5 - 7 m ) R e a l E s t i m a t i o n D 1 (λ = 0 . 3 - 2 . 5 m ) Figure 10: Lateral alignment estimation in section 1, filtered in the whole range (λ= 0.3 - 7 m) In view of the figure, a good agreement of the estimated and the real lateral alignment is observed across the entire length of the spatial profile. The results in the space frequency domain complement the information obtained, showing a good prediction of lateral alignment throughout the whole space frequency range, which is divided into two ranges according to standards: D1 (λ= 0.3 - 2.5 m) and D2 (λ= 2.5 - 7 m). The good performance of the estimator in this section is confirmed by the analysis presented in Table 2, which shows accuracy indices of J= 0.52 mm and Jrel = 0.98. For a more in-depth analysis, the results obtained have been divided into two different ranges, by filtering them into the corresponding bandpass limits (i.e. D1 and D2). Therefore, results have been plotted in Fig. 11, where the comparison between estimated and real lateral alignment are presented in two different ranges, in the space domain. 19 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 - 2 - 1 0 1 2 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 - 2 - 1 0 1 2 R a n g e D 1 ( λ = 0 . 3 - 2 . 5 m ) A m p l i t u d e ( m m ) D i s t a n c e ( m ) A m p l i t u d e ( m m ) D i s t a n c e ( m ) R e a l E s t im a t i o n R a n g e D 2 ( λ = 2 . 5 - 7 m ) R e a l E s t im a t i o n Figure 11: Lateral alignment estimation in section 1, filtered in different ranges: D1 (λ= 0.3 - 2.5 m), D2 (λ= 2.5 - 7 m) Again, a good agreement is obtained in both wavelength ranges, especially good in the short wavelength range (D1), as confirmed in Table 2: J= 0.34 mm for D1 range and J= 0.45 mm for D2 range. 6.3.2. Section 2: open curve The second segment under study refers to the open curve of the track, corresponding to the track length from 23 to 47 m. Notice that, along the whole curve, the forward velocity remains constant, as observed in Fig. 7. The results of the estimation of lateral alignment along Section 2 are presented in Fig.12. Similarly as in the previous case, the comparison in the space domain is presented in the upper subplot (with the 95% confidence interval of the prediction included), whereas the comparison in the space frequency domain is shown in the the lower subplot. 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 - 4 - 2 0 2 4 0 . 1 1 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 A m p li t u d e ( m m ) λ ( m ) A m p li t u d e ( m m ) D i s t a n c e ( m ) R e a l E s t i m a t i o n 9 5 % C . I D 2 (λ = 2 . 5 - 7 m ) R e a l E s t i m a t i o n D 1 (λ = 0 . 3 - 2 . 5 m ) Figure 12: Lateral alignment estimation in section 2, filtered in the whole range (λ= 0.3 - 7 m) 20 From the results shown in the figure, a good agreement between the estimated and the real lateral alignment is observed along the entire length of the spatial profile. The same good agreement is observed in the results plotted in the space frequency domain throughout the whole frequency range (λ= 0.3 - 7 m). The good performance of the estimator is confirmed by the analysis presented in Table 2, which shows accuracy indices of J= 0.61 mm and Jrel = 0.72. The obtained results have been split into two different ranges, D1 and D2, by filtering them into the corresponding bandpass limits. Therefore, results are presented in Fig. 13, where both estimated and real lateral alignment are compared in the space domain, for both ranges, D1 and D2. 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 - 2 - 1 0 1 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 - 2 - 1 0 1 2 R a n g e D 1 ( λ = 0 . 3 - 2 . 5 m ) D i s t a n c e ( m ) A m p l i t u d e ( m m ) A m p l i t u d e ( m m ) D i s t a n c e ( m ) R e a l E s t im a t i o n R a n g e D 2 ( λ = 2 . 5 - 7 m ) R e a l E s t im a t i o n Figure 13: Lateral alignment estimation in section 2, filtered in different ranges: D1 (λ= 0.3 - 2.5 m), D2 (λ= 2.5 - 7 m) As a result, a good agreement is obtained in both wavelength ranges, especially in range D2 (see Table 2). In this case, we can see the importance of using both accuracy indexes, Jand Jrel. Even though the absolute accuracy index, J, is similar for both ranges, D1 and D2 (J= 0.5 mm and 0.45 mm), the relative accuracy index is quite different for both ranges (Jrel = 1.02 and 0.53), giving complementary and very useful information: a better performance of the predictor in range D2. The good performance of the estimator in the open curve is due to an essential aspect: along the open curve, there is hardly any flange contact in the wheels, which is one of the main drawbacks of the estimator (flange contact is not included in the simplified dynamic model, SM). Therefore, the linear lateral dynamic model works well. 6.3.3. Section 3: sharp curve Finally, the third segment of interest is analysed, corresponding to the sharp curve, from a track length of 63 to 73 m. Similarly to the previous sections, the constant forward velocity is kept constant along the curve (see Fig. 7). The results of the estimation of lateral alignment along Section 3 are shown in Fig.14: in the space domain in the upper subplot (with the 95% confidence interval of the prediction included) and in the space frequency domain in the lower subplot. 21 6 4 6 6 6 8 7 0 7 2 - 8 - 4 0 4 8 0 . 1 1 0 . 0 0 . 4 0 . 8 1 . 2 1 . 6 2 . 0 2 . 4 A m p li t u d e ( m m ) λ ( m ) A m p li t u d e ( m m ) D i s t a n c e ( m ) R e a l E s t i m a t i o n 9 5 % C . I D 2 (λ = 2 . 5 - 7 m ) R e a l E s t i m a t i o n D 1 (λ = 0 . 3 - 2 . 5 m ) Figure 14: Lateral alignment estimation in section 3, filtered in the whole range (λ= 0.3 - 7 m) Unlike the previous cases, not very good results are observed in the prediction of the lateral alignment along the sharp curve. The bad performance of the estimator in this section is confirmed by the analysis presented in Table 2, showing accuracy indices of J= 2.28 mm and Jrel = 1.32. As can be seen in the space frequency domain plot (lower plot of Fig.14), the predicted lateral alignment is overestimated for very low wavelengths (λ= 0.3 - 0.5 m) and very long wavelengths (λ= 5 - 7 m). The results of the estimation have been divided into the two different ranges, D1 and D2, and presented in Fig. 15 in the space domain. 6 4 6 6 6 8 7 0 7 2 - 4 - 2 0 2 4 6 4 6 6 6 8 7 0 7 2 - 8 - 4 0 4 8 R a n g e D 1 ( λ = 0 . 3 - 2 . 5 m ) D i s t a n c e ( m ) A m p l i t u d e ( m m ) A m p l i t u d e ( m m ) D i s t a n c e ( m ) R e a l E s t im a t i o n R a n g e D 2 ( λ = 2 . 5 - 7 m ) R e a l E s t im a t i o n Figure 15: Lateral alignment estimation in section 3, filtered in different ranges: D1 (λ= 0.3 - 2.5 m), D2 (λ= 2.5 - 7 m) The poor results in the estimation is clearly observed in the figure, and can be confirmed in Table 2: J= 1.34 mm for D1 range, and J= 2.17 mm for D2 range. The bad results in the estimation in the sharp curve can be explained as follows. Along the whole sharp curve, there is continuous flange contact 22 between the front right wheel and the rail, due to the small radius of the curve. Consequently, since the SM model does not model the flange contact, the main assumptions of the proposed Kalman filter estimator are not fulfilled, which causes the estimator to not work correctly, overestimating the level of lateral alignment. 7. Conclusions In this work, a simple measuring system combined with a dynamic model-based Kalman filter estimator has been proposed to be used on in-service vehicles for continuous monitoring of track geometry and estimation of lateral alignment. The proposed numeric technique is based on the Kalman filtering method, using the measurement from different sensors mounted on an in-service vehicle (one gyroscope, two accelerometers and one encoder), through the use of a highly simplified linear dynamic model (SM), which is based on a vehicle running at variable forward velocity on a track, which can be comprised of straight and curve sections. The proposed methodology has been experimentally validated in a 1:10 scaled track facility in the University of Seville, comprised of a 90 m long scaled track and an instrumented scaled vehicle. The results in the estimation of the lateral alignment have been first analysed in the space domain, and subsequently, in the space frequency domain, for the different wavelength ranges considered in the standards. In this regard, only the two first ranges (D1 and D2) have been considered in the present work, since the third one (D3) is not directly related with safety. For an in depth analysis, three different sections of the whole track (a straight segment, an open curve and a sharp curve) in which the permanent regime is maintained, have been separately studied. The results obtained with the proposed method are promising, showing a good performance for monitoring lateral alignment on straight and curve tracks. Additionally, the efficiency of the proposed estimator has been checked, showing a very low computational cost, which makes it especially appropriate for real-time applications. When analysing the results in the different sections of interest, several conclusions can be drawn. The estimation in the straight segment and in the open curve is quite accurate, with errors around 0.5 - 0.6 mm. Only in the case of very sharp curves, when continuous flange contact takes place, the estimator is not able to precisely estimate the lateral alignment, showing errors up to 2.28 mm. The bad performance of the estimator in sharp curves is related to the continuous flange contact which takes place along the curve, phenomenon that has not been modelled in the simplified dynamic model used by the estimator. Since continuous flange contact implies that the wheelset follows closely the geometry of the outer rail, this property can be used in future works to extend the validity of the proposed approach. Acknowledgements This research was supported by the Conserjer´ıa de Econom´ıa, Conocimiento, Empresas y Universidad de la Junta de Andaluc´ıa under the project reference US-1257665. This support is gratefully acknowledged. 23 References [1] CEN European Committee for Standardization, Railway Applications - Track - Track Geometry Quality - IS EN 13848, Tech. rep. (2009). [2] P. Weston, C. Roberts, G. Yeo, E. Stewart, Perspectives on railway track geometry condition monitoring from in-service railway vehicles, Vehicle System Dynamics 53 (2015) 1063–1091. [3] P. Weston, C. S. Ling, C. Roberts, C. J. Goodman, P. Li, R. M. Goodall, Monitoring vertical track irregularity from in-service railway vehicles, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 221 (1) (2007) 75–88. [4] J. L. Escalona, Vertical track geometry monitoring using inertial sensors and complementary filters, in: Proceedings of IDETC/CIE 2016, ASME, Charlotte, NC, USA, 2016. [5] P. Weston, C. S. Ling, C. J. Goodman, C. Roberts, P. Li, R. M. Goodall, Monitoring lateral track irregularity from in-service railway vehicles, Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 221 (1) (2007) 89–100. [6] J. Seok Lee, S. Choi, S.-S. Kim, C. Park, Y. Guk Kim, A mixed filtering approach for track condition monitoring using accelerometers on the axle box and bogie, IEEE T. Instrumentation and Measurement 61 (2012) 749–758. [7] X. Wei, S. Guo, H. Liu, Estimation of lateral track irregularity based on an unknown input identification filter, in: International Conference on Automatic Control and Artificial Intelligence (ACAI 2012), Xiamen, China, 2012, pp. 999–1004. [8] A. De Rosa, S. Alfi, S. Bruni, Estimation of lateral and cross alignment in a railway track based on vehicle dynamics measurements, Mechanical Systems and Signal Processing 116 (2019) 606–623. [9] A. Jaschinski, On the application of similarity laws to a scaled railway bogie model, Ph.D. thesis, TU Delft, Delft University of Technology (1990). [10] A. Jaschinski, H. Chollet, S. Iwnicki, A. Wickens, J. Von W¨urzen, The application of roller rigs to railway vehicle dynamics, Vehicle System Dynamics 31 (5-6) (1999) 345–392. [11] R. Illingworth, Railway Wheelset Lateral Excitation by Track Irregularities, Vehicle System Dynamics 6 (2-3) (1977) 144–147. doi:10.1080/00423117708968525. [12] J. F. Aceituno, R. Chamorro, D. Garc´ıa-Vallejo, J. L. Escalona, On the design of a scaled railroad vehicle for the validation of computational models, Mechanism and Machine Theory 115 (2017) 60–76. doi:10.1016/j.mechmachtheory.2017.04.015. [13] J. Escalona, J. F. Aceituno, P. Urda, O. Balling, Railway multibody simulation with the knifeedge-equivalent wheel–rail constraint equations, Multibody System Dynamics 48 (2019) 373–402. doi:10.1007/s11044-019-09708-x. 24