Experimental measurement of track irregularities using a scaled track recording vehicle and Kalman filtering techniques

Mechanical Systems and Signal Processing 169 (2022) 108625 Available online 25 November 2021 0888-3270/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp Experimental measurement of track irregularities using a scaled track recording vehicle and Kalman filtering techniques Sergio Muñoz a,∗, Pedro Urda b, José L. Escalonab aDepartment of Materials and Transportation Engineering, University of Seville, Spain bDepartment of Mechanical and Manufacturing Engineering, University of Seville, Spain ARTICLE INFO Communicated by W.-X. Ren Keywords: Track surveying Track recording vehicle Kalman filter Scaled track Track irregularities Experimental validation ABSTRACT The actual trend of railway industry on track surveying is heading up the development of fast automated methods for continuous monitoring of track quality. This paper presents a modelbased methodology for the estimation of lateral and vertical track irregularities from different sensors mounted on a Track Recording Vehicle (TRV): an Inertial Measurement Unit (IMU), a gauge sensor and a position encoder. The proposed methodology, based on the Kalman filtering technique, allows a fast and accurate measurement of track irregularities, without the use of a total station (which usually makes the measurement process very slow). The proposed method has been validated through an experimental campaign carried out on a 90 m long 1:10, scaled track facility and a scaled TRV at the University of Seville. The use of the scaled TRV allows the measurement of the 90 metres long scaled track at an operation velocity of 𝑉= 0.7 m/s in only two minutes. The results of the estimation of track irregularities have been compared with a previous measurement of the mentioned scaled track using manual means, showing a good agreement: with errors lower than 0.7 mm in the short wavelength range D1, which is the most influential in the dynamic behaviour of the vehicle. Additionally, the obtained results have been analysed and compared in the different wavelength ranges, according to standards. 1. Introduction As it is well known in the railway industry, vehicle dynamics is highly influenced by track irregularities. A track with a considerable degree of irregularity reduces the vehicle’s comfort and, in extreme cases, can even jeopardize the passenger safety. 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 track geometry quality, irregularities should be analysed taking into account these wavelength ranges, as well as the maximum allowed forward velocity of a vehicle. Due to the need to ensure the quality level of track geometry, railway operators strive to keep their track infrastructures in the best possible conditions. This requires intensive maintenance work and, most importantly, regular monitoring of track conditions. Surveying a railroad track is a difficult and expensive task, forcing operators to invest a substantial part of their budget in surveying and maintaining the track regularly. Traditionally, the most extended ways to survey a track have been based on the use of the so-called measuring trolleys [2,3]. Normally, it is a device with a T-shape frame which can move along a track pushed by a human operator. These devices are instrumented with an encoder that registers the position of the trolley on the track, a distance sensor that measures the track gauge, a tilt sensor or inclinometer that measures the track cant angle and, in most cases, a prism whose relative position is determined by a total station. The combination of these sensors and the total station allows a ∗Corresponding author. E-mail address: [email protected] (S. Muñoz). https://doi.org/10.1016/j.ymssp.2021.108625 Received 31 March 2021; Received in revised form 4 November 2021; Accepted 6 November 2021 Mechanical Systems and Signal Processing 169 (2022) 108625 2 S. Muñoz et al. precise but slow measurement of the track and its irregularities. New generation devices also include Assisted Global Navigation Satellite System (AGNSS) for absolute positioning on the track with centimetre-level precision [4,5]. Despite their accuracy [6], the main handicap of a measuring trolley lies in its reduced measurement speed, between 100 to 200 metres per hour. In addition, vehicle traffic must be interrupted during the measurement process, which implies great costs for the operating company. The actual trend of track surveying methods moves towards the development of fully automated measuring procedures based on the dynamic response of the vehicle throughout the on-board monitoring of the track quality using Track Recording Vehicles (TRV) [7]. The use of automated on-board surveying systems not only allows fast and continuous monitoring of track conditions, but also avoids possible subjectivities in the measurements due to human intervention. The theory behind any automated track surveying method is not trivial. Railway dynamics is governed by highly non-linear equations, making the estimation of track irregularities from the inertial response of the vehicle difficult, specially the calculation of lateral track irregularities. Different approaches for the calculation of track irregularities can be found in literature, an extended review is found in the work of [8,9]. In the work of Yang et al. [10] the authors use a time-dependent intrinsic correlation, unused to this day, to establish a relationship between the dynamic response of a vehicle and track irregularities with promising results. To do so, accelerations are decomposed in a series of intrinsic mode functions with different time scales. A similar approach is presented in Simian’s work [11], where the authors find a correlation between the vehicle’s acceleration and wheel-rail vertical irregularities using a frequency analysis. Real et al. [12] propose a procedure to obtain the vertical rail profile from the vertical acceleration by the use of the Fourier transform and transfer functions in the frequency domain. De Rosa et al. [13] present a frequency-domain based and two time-domain based methods to estimate the lateral and cross alignment based on vehicle dynamics measurements. As an alternative, recent studies try to address railway irregularity measurement on the basis of machine learning and neural computing methods. Kraft et al. [14] have developed a black-box model of railway vehicles using neural networks for track geometry assessment. The use of black-box models trained with recurrent neural networks gives good results compared with multibody models for the assessment of lateral track irregularities. De Rosa et al. [15] have developed a method for the assessment of lateral track irregularities based on machine learning classification algorithms. The method has been developed using numerical simulations. Similarly, Tsunashima [16] has used machine learning classification and car-body sensors for track condition monitoring. This technique has been developed with the help of multibody simulations using the SIMPACK software and field tests. However, the most commonly found methods in the literature for the estimation of track geometry and its irregularities are based on the use of Kalman filtering techniques. Kalman filters have traditionally been applied to other railway problems such as fault diagnosis in suspension systems [17,18], wheel rolling radius estimation [19] or adherence coefficient estimation [20] with success. In the work of Odashima et al. [21] vertical track irregularities are obtained by solving the inverse dynamics problem, using a Kalman filter as integrator and the car-body’s measured acceleration. In the work of Lee et at. [22], 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. Along the same lines, it is the work of Wei et al. [23] that estimates track irregularities in urban lines using a Kalman filter to double integrate the vertical acceleration of the bogie frame. Xiao et al. in [24] propose a filter that estimates vertical track irregularities in railway bridges paying attention to the mechanical behaviour of such structures, very common in high-speed lines. A variation of the Extended Kalman Filter (EKF), the Extended Particle Filter (EPF) is used by Astikwa et al. in [25] to estimate vertical track irregularities using dynamic measurements. Kalman filtering techniques have also been integrated with innovative and quite sophisticated approaches as shown in the work of Jiang et al. [26] where track irregularities are obtained with a millimetre precision level using an instrumented manual trolley and a total station. In a similar way, Gao et al. use a Rauch–Tung–Striebel smoother multisensor fusion system to measure track irregularities in [27]. All these methods have the usage of acceleration and angular velocity measurements as estimation model inputs in common. However Weston et al. assure in [28] that track alignment can be determined by analysing standard deviation of the measured yaw angle exclusively. It must be noted that the majority of these works found in literature focus on the calculation of vertical track irregularities, while just a few of them deal with the estimation of lateral track irregularities due to its higher complexity. The different methods described above represent different valid alternatives to obtain track irregularities from the measurements of an instrumented vehicle. Even though the use of artificial intelligence is gaining more visibility in the last years, there are a series of disadvantage in the use of this method. The main disadvantage is the need of a large enough set of data for the training process. These data are not always easy or even feasible to obtain, which results on a reduction of the precision of the estimated result. In addition, any slight change in the test conditions requires a new training process of the network. Despite of that, as it is shown in the literature, artificial intelligence methods are generally precise and computational efficient as long as the training process is performed correctly. The working principle of correlation methods is similar to artificial intelligence methods, with the difference that they do not require a training process. Finally, Kalman filter based method have the disadvantage that they need a kinematic or dynamic model that represents the performance of the vehicle. However, if the model is correctly parameterized, changes in the simulated vehicle can be easily introduced in the model. In addition, if they are correctly implemented, these methods can be computationally very efficient with real time capabilities. As a conclusion, the Kalman filter method can be considered as the most scientific of the three mentioned approaches and, if it is based on a reliable model, this method can offer a great performance. The main goal of the present work is the development of an estimation procedure, based on the Kalman Filter technique, to measure the lateral and vertical track irregularities from different sensors mounted on a dedicated vehicle (Track Recording Vehicle). The proposed method, with the advantage of not needing a total station, allows a fast and accurate measurement of track irregularities. When comparing with previous works in literature, the proposed method obtains very good results in the estimation of both vertical and lateral track irregularities. Especially interesting are the results in the estimation of lateral track Mechanical Systems and Signal Processing 169 (2022) 108625 3 S. Muñoz et al. Fig. 1. Definition of track geometry. irregularities, which are the most difficult type of irregularities to estimate and the most influential one in the dynamic behaviour of the vehicle. The proposed method has been tested with experimental data obtained with an scaled TRV running on a 5-inch gauge experimental track built at the School of Engineering of the University of Seville. As a main contribution, the presented method for the measurement of track irregularities is based exclusively on a kinematic multibody model of the vehicle, unlike other works found in literature that use more sophisticated dynamic models. Despite that, the obtained results demonstrate that the proposed method allows a precise way to obtain track irregularities. The estimated irregularities with the scaled TRV have been compared with a reference measurement obtained with a precise manual method. The obtained results in the estimation of track irregularities 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. This paper is organized as follows: Section 2is devoted to the general definition of track irregularities. Section 3presents the kinematic model of the TRV, while Section 4explains the applied estimation technique in detail. Section 5is devoted to the description of the experimental setup used. The obtained results and their comparison with the reference measurement are presented in Section 6. Summary and conclusions are found in Section 7. 2. Definition of track geometry Track geometry is defined as the superposition of the design geometry and the irregularities, as it can be seen in Fig. 1. The design geometry is defined by the track centre line and the cant angle. Track frame (TF) ⟨𝑋𝑡, 𝑌 𝑡, 𝑍𝑡⟩is not a single frame but a field defined for each value of the arc-length coordinate along the horizontal projection of the track centre line, 𝑠. The absolute position vector of the TF with respect to inertial and global frame (GF) ⟨𝑋, 𝑌 , 𝑍⟩is a function of arc-length 𝑠, as follows: 𝐑𝑡(𝑠)=⎡⎢⎢⎣ 𝑅𝑡 𝑥(𝑠) 𝑅𝑡 𝑦(𝑠) 𝑅𝑡 𝑧(𝑠)⎤⎥⎥⎦ (1) Orientation matrix 𝐀𝑡(𝑠)of the TF with respect to the GF is defined through the Euler angles: yaw 𝜓𝑡(ℎ𝑒𝑎𝑑𝑖𝑛𝑔 angle), pitch 𝜃𝑡(vertical slope) and roll 𝜑𝑡(𝑐𝑎𝑛𝑡 𝑎𝑛𝑔𝑙𝑒). The yaw angle can have an arbitrary value but the roll and pitch are small angles. A small-angle approximation of orientation matrix 𝐀𝑡(𝑠)is given by: 𝐀𝑡(𝑠)≃⎡⎢⎢⎣ c𝜓𝑡−s𝜓𝑡𝜑𝑡s𝜓𝑡+𝜃𝑡c𝜓𝑡 s𝜓𝑡c𝜓𝑡𝜃𝑡s𝜓𝑡−𝜑𝑡c𝜓𝑡 −𝜃𝑡𝜑𝑡1⎤⎥⎥⎦ (2) Twist curvature 𝜌𝑡𝑤, vertical curvature 𝜌𝑣and horizontal curvature 𝜌ℎof the track are the space derivatives of roll angle 𝜑𝑡, pitch angle 𝜃𝑡yaw angle 𝜓𝑡with respect to the arc-length 𝑠, respectively. As for irregularities, Fig. 1 shows the displacement of the rail heads due to irregularity in a cross-section of the track (𝑌𝑡−𝑍𝑡 plane). Irregularity vectors 𝑟𝑙𝑖𝑟 (𝑙𝑖𝑟, left rail irregularity) and 𝑟𝑟𝑖𝑟 (𝑟𝑖𝑟, right rail irregularity) describe the displacement of the rail Mechanical Systems and Signal Processing 169 (2022) 108625 4 S. Muñoz et al. Fig. 2. (a) Track recording vehicle CAD. (b) Isolated instrumented axle CAD. centre lines with respect to their design positions. The components of these vectors in the TF are functions of 𝑠, given by:  𝐫𝑙𝑖𝑟 =⎡⎢⎢⎣ 0 𝑦𝑙𝑖𝑟 𝑧𝑙𝑖𝑟⎤⎥⎥⎦ , 𝐫𝑟𝑖𝑟 =⎡⎢⎢⎣ 0 𝑦𝑟𝑖𝑟 𝑧𝑟𝑖𝑟⎤⎥⎥⎦ (3) The lateral and vertical irregularities of a track are usually defined by four well-known variables in the railway industry: Alignment: 𝜉𝑎𝑙 =(𝑦𝑙𝑖𝑟 +𝑦𝑟𝑖𝑟)∕2 Vertical profile: 𝜉𝑣𝑝 =(𝑧𝑙𝑖𝑟 +𝑧𝑟𝑖𝑟)∕2 Gauge variation: 𝜉𝑔𝑣 =𝑦𝑙𝑖𝑟 −𝑦𝑟𝑖𝑟 Cross level: 𝜉𝑐𝑙 =𝑧𝑙𝑖𝑟 −𝑧𝑟𝑖𝑟 (4) These irregularities can be divided into two different types: global irregularities as the variation of the track centre line (alignment and vertical profile), and relative irregularities as the variation of the relative position between both rails (gauge variation and cross-level). 3. Multibody model of the TRV 3.1. Description of the TRV The vehicle is equivalent to a track trolley, but including an electric drive system. Fig. 2(a) shows the Computational Assisted Design (CAD) of the scaled Track Recording Vehicle. It is composed of two parts: the drive system, consisting of the vehicle’s body and traction wheels, and the instrumented axle located in the front-end of the vehicle, which performs the measurements. Both parts are connected through a spherical joint that decouples the rotations of both bodies. Fig. 2(b) shows the CAD of the instrumented axle. It includes two carriages which can slide along the guide rail. A couple of lateral guide wheels installed on both sides of the instrumented axle and a tensile spring that connects both carriages make it possible for the instrumented axle to perfectly follow the track geometry. During its operation, the left carriage where the IMU is installed is locked while the right one moves freely along the guide rail following the track irregularity. The linear variable differential transformer sensor (LVDT), that is attached to the right carriage (see Fig. 2(b)) measures the variation on the track gauge. The instrumented axle also includes a precision encoder that measures the travelled distance along the track and a beacon detector that register the position of a set of magnetic beacons distributed along the track and whose position was previously measured using a total station. Using the encoder and the position of the beacons the vehicle can be accurately located along the track. Fig. 3 shows a simplified model of the kinematics of the instrumented axle. This figure shows the track frame moving along the track centre line with forward velocity 𝑉and forward acceleration  𝑉, and irregularity vectors 𝑟𝑙𝑖𝑟 and 𝑟𝑟𝑖𝑟 describing the displacement of the rail centre lines with respect to their design positions. The instrumented axle is assumed to comprise two bodies (left and right carriages in Fig. 2(b)) connected with a prismatic joint. The following assumptions have been taken: •The left body, which includes the installed IMU, is assumed to keep point Qin contact with the centre line of the left rail. •The right body is assumed to keep point Pin contact with the centre line of the right rail. •The axle of the left body is perpendicular to the centre line of the left rail. Mechanical Systems and Signal Processing 169 (2022) 108625 5 S. Muñoz et al. Fig. 3. Kinematics of the instrumented axle. •The IMU is assumed to be installed in a vertical plane that contains points Pand Qat distance 𝑑𝑦in the lateral direction and 𝑑𝑧in the vertical direction with respect to point Q. •The three Euler angles that define the orientation of the IMU with respect to the TF (𝜑𝑖𝑚𝑢, 𝜃𝑖𝑚𝑢 and 𝜓𝑖𝑚𝑢) are supposed to be small. The first two assumptions are reasonable due to the mechanical design of the instrumented axle. The spring that connects the two main bodies of the instrumented axle and the horizontal and vertical wheels attached to them that roll over the rails guarantee that there exist two non-material points Pand Qthat fulfil these kinematic constraints. As regards the third assumption, since the left and right rail threads are not parallel, the instrumented axle cannot be perpendicular to both lines simultaneously, but the relative angle of the axle with respect to the perpendicular line is very small. Therefore, the assumption of perpendicularity of the left body is somehow arbitrary (the right body could be selected as perpendicular) but the resulting inaccuracies are very small. The fourth assumption is not true. However, it is well known from rigid body kinematics that in a rigid body that does not rotate, the acceleration of all points is the same. If the body rotates, the difference in acceleration between two points is proportional to their relative distance. Because the distance between the IMU and the line that connects Pand Qis also small, the measurement of the IMU can be considered accurate. Anyway, the inaccuracy due to this assumption can be corrected if the angular velocity and acceleration of the axle are known. However, in this paper we have consider that this correction is not needed for the sake of simplicity of the resulting equations. In the last assumption, the angles that are assumed to be small are the relative angles of the IMU with respect to the TF. It must be note that the absolute angles of the track or the absolute angles of the IMU may become large in curved tracks, but the relative angles remain small even with small radius of curvature of the track. Using the kinematics of an irregular track shown in Fig. 3 and the simplifying assumptions stated above, the following relationships can be established: 𝑟𝑖𝑚𝑢 𝑦=𝑦𝑙𝑖𝑟 +𝐿𝑟−𝑑𝑦 𝑟𝑖𝑚𝑢 𝑧=𝑧𝑙𝑖𝑟 +𝑑𝑧 𝜑𝑖𝑚𝑢 = (𝑧𝑙𝑖𝑟 −𝑧𝑟𝑖𝑟)∕2𝐿𝑟 𝜃𝑖𝑚𝑢 = − 𝑑𝑧𝑙𝑖𝑟 𝑑𝑠 = −𝑧𝑙𝑖𝑟′ 𝜓𝑖𝑚𝑢 =𝑑𝑦𝑙𝑖𝑟 𝑑𝑠 =𝑦𝑙𝑖𝑟′ 𝑑𝑃 𝑄 = 2𝐿𝑟+𝑦𝑙𝑖𝑟 −𝑦𝑟𝑖𝑟 (5) where 𝑟𝑖𝑚𝑢 𝑦and 𝑟𝑖𝑚𝑢 𝑧are the non-zero components of the position vector of the IMU with respect to the TF, and 2𝐿𝑟is the nominal gauge (distance between both rails without irregularities). For the fourth and fifth expressions in Eq. (5), it assumed that local axis 𝑋𝑖𝑚𝑢 is tangent to the 3D curve of the left rail at all instants (see left rail lateral and vertical projections shown in Fig. 3). For the last expression, the distance between points P and Q, is assumed to be equal to its horizontal projection. Mechanical Systems and Signal Processing 169 (2022) 108625 6 S. Muñoz et al. Finally, by time-derivation of Eq. (5), the following expressions can be obtained, which will be useful later: 𝑟𝑖𝑚𝑢 𝑦= 𝑉 𝑦𝑙𝑖𝑟′+𝑉2𝑦𝑙𝑖𝑟′′ 𝑟𝑖𝑚𝑢 𝑧= 𝑉 𝑧𝑙𝑖𝑟′+𝑉2𝑧𝑙𝑖𝑟′′ 𝜑𝑖𝑚𝑢 =𝑉(𝑧𝑙𝑖𝑟′−𝑧𝑟𝑖𝑟′)∕2𝐿𝑟  𝜃𝑖𝑚𝑢 = −𝑉 𝑧𝑙𝑖𝑟′′ 𝜓𝑖𝑚𝑢 =𝑉 𝑦𝑙𝑖𝑟′′ (6) It is important to note that, although the IMU’s trajectory follows more closely the left rail irregularity, the assumed kinematic model accounts for the effect of the 3D geometry of the left and the right irregular rails. Therefore, there is no reason to assume that the lack of symmetry of the instrumented axle can induce measurements more related to the left rail irregularities. 3.2. Kinematics of the IMU For the kinematic description of the left body, to which the IMU is attached, the following coordinates are used: 𝐪𝑖𝑚𝑢 =[𝑠𝑖𝑚𝑢 𝑟𝑖𝑚𝑢 𝑦𝑟𝑖𝑚𝑢 𝑧𝜑𝑖𝑚𝑢 𝜃𝑖𝑚𝑢 𝜓𝑖𝑚𝑢]𝑇(7) where 𝑠𝑖𝑚𝑢 is the arc length of the track frame that follows the body motion,  𝐫𝑖=[0𝑟𝑖 𝑦𝑟𝑖 𝑧]𝑇is the local position vector of the IMU frame with respect to the TF resolved in the TF, and 𝜑𝑖𝑚𝑢, 𝜃𝑖𝑚𝑢 and 𝜓𝑖𝑚𝑢 are the three Euler angles that define the orientation of the IMU with respect to the TF. Assuming that the three Euler angles are small, the following kinematic linearization of the IMU to the TF rotation matrix is obtained: 𝐀𝑡,𝑖𝑚𝑢 ≃⎡⎢⎢⎣ 1 −𝜓𝑖𝑚𝑢 𝜃𝑖𝑚𝑢 𝜓𝑖𝑚𝑢 1 −𝜑𝑖𝑚𝑢 −𝜃𝑖𝑚𝑢 𝜑𝑖𝑚𝑢 1⎤⎥⎥⎦ (8) The absolute acceleration of the IMU frame, projected in the TF, is given by [29]:   𝐑𝑖𝑚𝑢 =  𝐑𝑡+  𝐫𝑖𝑚𝑢 +( 𝜶𝑡+ 𝝎𝑡 𝝎𝑡) 𝐫𝑖𝑚𝑢 + 2  𝝎𝑡  𝐫𝑖𝑚𝑢 (9) And the absolute angular velocity of the IMU frame resolved in the IMU frame, under the small-angles assumption, is given by [29]:  𝝎𝑖𝑚𝑢 = 𝝎𝑡+ 𝝎𝑡,𝑖𝑚𝑢 =(𝐀𝑡,𝑖𝑚𝑢)𝑇 𝝎𝑡+ 𝝎𝑡,𝑖𝑚𝑢 (10) As it can be found in [29], the terms associated with the TF are given by:   𝐑𝑡=⎡⎢⎢⎣  𝑉 𝜌ℎ𝑉2 −𝜌𝑣𝑉2⎤⎥⎥⎦ , 𝝎𝑡=⎡⎢⎢⎣ 𝜌𝑡𝑤𝑉 𝜌𝑣𝑉 𝜌ℎ𝑉⎤⎥⎥⎦ , 𝜶𝑡=⎡⎢⎢⎣ 𝜌𝑡𝑤  𝑉 𝜌𝑣 𝑉 𝜌ℎ 𝑉+𝜌′ℎ𝑉2⎤⎥⎥⎦ (11) 3.3. IMU signals The IMU has a 3D gyroscope and a 3D accelerometer. The 3D accelerometer signals include the three components of the absolute acceleration in the IMU frame, plus the three components of the gravity vector, which is assumed to act in the absolute 𝑍direction: 𝐚𝑎𝑐𝑚 =(𝐀𝑡,𝑖𝑚𝑢)𝑇  𝐑𝑖𝑚𝑢 +(𝐀𝑡𝐀𝑡,𝑖𝑚𝑢)𝑇[0 0 𝑔]𝑇(12) where 𝐚𝑎𝑐𝑚 is an array that includes the quantities measured by the 3D accelerometer. The 3D gyroscope provides three digital signals that measure the three components of the IMU absolute angular velocity in the body frame, as follows: 𝝎𝑔𝑦𝑟 = 𝝎𝑖𝑚𝑢 (13) where 𝝎𝑔𝑦𝑟 is an array that includes the quantities measured by the 3D gyroscope. 3.4. Simplifying assumptions In order to simplify previous equations, the following assumptions have been taken: •The pitch and roll angles of the TF with respect to the GF are considered null (𝜃𝑡=𝜑𝑡=0), which leads to: 𝜌𝑣=𝜌𝑡𝑣 =0. This is common practice in the railway field, which considers that the design geometry has neither a vertical slope nor a cant angle. In this case, both the real vertical slope and the real cant angle are included as a part of the irregularities. •The relative displacement of the IMU frame with respect to the TF is small, so it can be assumed that the relative centripetal and Coriolis acceleration are negligible in comparison with the second derivatives of  𝐫𝑖𝑚𝑢. This assumption is equivalent to eliminating the third and fourth terms of the right-hand side of Eq. (9). Mechanical Systems and Signal Processing 169 (2022) 108625 7 S. Muñoz et al. Fig. 4. Estimation procedure. •The relative orientation of the IMU with respect to the TF is so small that the components of the absolute angular velocity of the TF are the same when projected to the TF or when projected to the IMU frame. This assumption is equivalent to making 𝐀𝑡,𝑖𝑚𝑢 equal to the identity matrix in Eq. (10). As demonstrated in [29], and taking into account the simplifying assumptions, the signal obtained from the accelerometer and the gyroscope presented in Eqs. (12) and (13) can be written as: [𝑎𝑎𝑐𝑚 𝑦 𝑎𝑎𝑐𝑚 𝑧]=[𝑟𝑖𝑚𝑢 𝑦+𝑔𝜑𝑖𝑚𝑢 +𝜌ℎ𝑉2 𝑟𝑖𝑚𝑢 𝑧+𝑔](14) ⎡⎢⎢⎢⎣ 𝜔𝑔𝑦𝑟 𝑥 𝜔𝑔𝑦𝑟 𝑦 𝜔𝑔𝑦𝑟 𝑧 ⎤⎥⎥⎥⎦ =⎡⎢⎢⎣ 𝜑𝑖𝑚𝑢  𝜃𝑖𝑚𝑢 𝜓𝑖𝑚𝑢 +𝜌ℎ𝑉⎤⎥⎥⎦ (15) Additionally, the equation related to the gauge variation must be included in the measurements of the Kalman filter, as it will described in the next section: 𝜉𝑚𝑒𝑎𝑠 𝑔𝑣 =𝑑𝑃 𝑄 − 2𝐿𝑟=𝑦𝑙𝑖𝑟 −𝑦𝑟𝑖𝑟 (16) where the nominal gauge (2𝐿𝑟) has been subtracted from the distance between points P and Q measured by the LVDT sensor (last expression of Eq. (5)). 4. Irregularity estimation technique The main objective of this work is to estimate the lateral and vertical irregularities of a track from the measurements from the different sensors mounted on the TRV. To this end, a model-based numeric procedure based on Kalman filtering techniques has been used. As commented, the estimation technique is based on the kinematic model of the front mechanism of the vehicle, presented in Section 3. The estimation technique, which is graphically summarized in Fig. 4, is explained next. 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 kind of inputs. On one hand, the curvature of the ideal track centre line (𝜌ℎ) 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 accelerations and angular velocities measured by the IMU, which are due to the movement of the TF with respect to the GF. On the other hand, the Kalman filter needs the experimental data of the dynamic response of the mechanism as inputs: the vehicle forward velocity (𝑉), measurements from the accelerometer (𝑎𝑎𝑐𝑚 𝑦and 𝑎𝑎𝑐𝑚 𝑧) and from the gyroscope (𝜔𝑔𝑦𝑟 𝑥,𝜔𝑔𝑦𝑟 𝑦and 𝜔𝑔𝑦𝑟 𝑧), and gauge variation (𝜉𝑚𝑒𝑎𝑠 𝑔𝑣 ). All these data are acquired from one encoder and an inertial sensor unit (IMU) installed in the vehicle, and synchronized through a data acquisition system, at the same sampling rate (250 Hz in the present work). From all these data inputs, and making use of the Kalman filter, the four track irregularities can be estimated. Mechanical Systems and Signal Processing 169 (2022) 108625 8 S. Muñoz et al. 4.1. Design of the Kalman filter As previously explained, the estimation technique is based on the well-known Kalman filter algorithm [30], which is briefly defined as follows. The state vector includes the lateral and vertical displacement of both rails with respect to their ideal position, and their first and second spatial derivatives, as: 𝐱=[𝑦𝑙𝑖𝑟 𝑦𝑟𝑖𝑟 𝑦𝑙𝑖𝑟′𝑦𝑟𝑖𝑟′𝑦𝑙𝑖𝑟′′ 𝑦𝑟𝑖𝑟′′ 𝑧𝑙𝑖𝑟 𝑧𝑟𝑖𝑟 𝑧𝑙𝑖𝑟′𝑧𝑟𝑖𝑟′𝑧𝑙𝑖𝑟′′ 𝑧𝑟𝑖𝑟′′]𝑇(17) The measurement vector includes accelerometer and gyroscope signals measured by the IMU, and the gauge variation measured by the LVDT distance sensor, plus two additional measurements of the lateral and vertical displacement of the left rail: 𝐳𝑚𝑒𝑎𝑠 =[𝑎𝑎𝑐𝑚 𝑦𝑎𝑎𝑐𝑚 𝑧𝜔𝑔𝑦𝑟 𝑥𝜔𝑔𝑦𝑟 𝑦𝜔𝑔𝑦𝑟 𝑧𝜉𝑚𝑒𝑎𝑠 𝑔𝑣 𝑟𝑖𝑚𝑢 𝑦𝑟𝑖𝑚𝑢 𝑧]𝑇(18) Most of the components of this measurement vector are directly obtained from the IMU mounted on the vehicle. However, the measurement of the lateral and vertical displacement of the left rail, 𝑟𝑖𝑚𝑢 𝑦and 𝑟𝑖𝑚𝑢 𝑧, are not real measurements but virtual sensors, which in practice always provide constant values. The inclusion of virtual sensors in the measurement vector has been shown to offer an improvement in the performance of the Kalman filter in previous works [31]. It has two purposes. On one hand, it makes the system observable. On the other hand, from the practical point of view, the virtual sensor avoids a drift in the prediction of lateral and vertical irregularities. The inclusion of a virtual sensors with constant measurements can be seen as a highly disturbing procedure. To avoid this, it is very important an adequate selection of the variance associated with the Gaussian error of these virtual sensors, which must be carefully chosen to avoid the drift while keeping a good accuracy. If the value of the covariance is high, the virtual sensors have little influence in the results, being weak their effect on eliminating the drift. On the other hand, if the value of the covariance is small, the effect of the virtual sensor is strong, flattening the curve of the estimated irregularities. Consequently, the covariance of the noise associated with the virtual sensors must coincide with the covariance of the expected irregularities. This solution has been proven to give good results in previous works [31] The equation of the Kalman filter in a discrete form can be expressed as follows: 𝐱𝑘=𝐅𝑥𝑘−1 +𝐯𝑘(19) where subscript krepresents discrete time, and 𝐅is the state transition matrix, which can be obtained as: 𝐅=[𝐅𝐿0 0𝐅𝑉];𝐅𝐿=𝐅𝑉=⎡⎢⎢⎢⎢⎢⎢⎣ 1 0 𝑉 𝛥𝑡 000 0 1 0 𝑉 𝛥𝑡 0 0 0 0 1 0 𝑉 𝛥𝑡 0 0 0 0 1 0 𝑉 𝛥𝑡 0 0 0 0 1 0 0 0 0 0 0 1 ⎤⎥⎥⎥⎥⎥⎥⎦ (20) Notice that the system model is just an integrator and state transition matrix 𝐅depends on the forward velocity, 𝑉, and the time increment, 𝛥𝑡. In this case, term 𝐯𝑘is the process noise, which can be modelled as a zero-mean Gaussian noise with covariance matrix 𝐐. The measurement process can be expressed through the following equation: 𝐳𝑘=𝐇𝐱𝑘+𝐆+𝐰𝑘(21) where subscript krepresents discrete time, 𝐇is the measurement transition matrix and 𝐆is an input matrix. These matrices can be obtained by substituting Eqs. (5)–(6) for Eqs. (14)–(16), obtaining: 𝐻= ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 0 0  𝑉0𝑉20𝑔∕2𝐿𝑟−𝑔∕2𝐿𝑟0 0 0 0 0 0 0 0 0 0 0 0  𝑉0𝑉20 0 0 0 0 0 0 0 0 𝑉∕2𝐿𝑟−𝑉∕2𝐿𝑟0 0 0 0 0 0 0 0 0 0 0 0 −𝑉0 0 0 0 0 𝑉0 0 0 0 0 0 0 1 −1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22) 𝐆=[𝜌ℎ𝑉2, 𝑔 , 0,0, 𝜌ℎ𝑉 , 0,(𝐿𝑟−𝑑𝑦), 𝑑𝑧]𝑇(23) In this case, term 𝐰𝑘is the measurement noise, which can be modelled as a zero-mean Gaussian noise with covariance matrix 𝐑. 5. Experimental setup This section is devoted to the experimental setup that has been used to experimentally validate the proposed estimation algorithm, main contribution of this work. Firstly, the main features of the experimental scaled track are presented and, secondly, the final assembly of the scaled TRV and its instrumentation are also explained in detail. Additionally, the methodology proposed in the present work has been compared with an alternative methodology previously presented by the authors [32], which also makes use of the same TRV but with a different configuration. Mechanical Systems and Signal Processing 169 (2022) 108625 9 S. Muñoz et al. Fig. 5. (a) Aerial view of the scaled track. (b) Plan view of the scaled track. Fig. 6. (a) Scaled track and metallic benches. (b) Adjustable mechanisms. 5.1. Experimental scaled track This unique facility is located on the rooftop of the School of Engineering at the University of Seville as it is shown in Fig. 5(a). It is a 90-m long 5-inch-wide scaled track whose design geometry consists of a combination of tangent sections, transitions and constant curvature sections, like any other railway track. The different track segments of this particular track are depicted in Fig. 5(b), where different types of section have been highlighted in different colours. The lengths of the different track segments are summarized in Table 1. The rails have been machined in three metre long pieces of AISI-304 stainless steel, reproducing a scaled version of an UIC-54 standard profile. The different sections are not welded with each other, but they are mechanically linked using special pieces. Both rails fit in a series of mechanisms that simulate a real track sleeper. The mechanisms are rigidly attached to a set of metallic benches leaning on the ground (see Fig. 6(a)). These benches were precisely located and levelled using a total station before the rails were installed. The mechanisms that support the rails give a unique value to the scaled track: they have been designed to allow the manual introduction of track irregularities (see Fig. 6(b)). With the correct actions on each mechanism, track gauge, cant angle and relative height between both rails can be modified precisely. The cant angle in the different curve sections has been changed manually in order to cancel the centrifugal force effects of a potential vehicle that moves along the track at a forward speed of 1.5 m/s. Taking into account that the scaled reduction of this velocity corresponds to 15 m/s on a full scaled track, which is an average maximum velocity for many metropolitan trains. Mechanical Systems and Signal Processing 169 (2022) 108625 16 S. Muñoz et al. Fig. 13. Track irregularities estimation by both methods under study, filtered in the D2 range (𝜆=2.5–7 m). The results obtained with the proposed method are promising, showing a good performance in measuring track irregularities on straight and curved tracks. A very accurate estimation of relative irregularities (gauge variation and cross level) have been obtained, with errors around 0.15 mm. The highest disagreement was encountered in the estimation of global irregularities (lateral alignment and vertical profile), with errors around 1.5–2.75 mm. However, part of these errors is due to the phase delay between the real and estimated irregularities along the transition segments, especially accused in the long wavelength range (D2), which is less influential on dynamic behaviour of a vehicle. When analysing the short wavelength range (D1), these errors were reduced to around 0.4–0.6 mm. In comparison with the alternative method proposed by the authors (TRV +TS), similar results are obtained in the estimation of track irregularities, but with the advantage of not needing a total station. This makes the proposed methodology very attractive in terms of simplicity and speed in the measurement of track irregularities. Especially promising are the results obtained with the proposed method, when comparing with alternative methods and technologies in literature, which are normally not able to measure all four types of irregularities at the same time, and hardly obtain errors lower than 0.6 mm in the estimation of track irregularities. 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. Mechanical Systems and Signal Processing 169 (2022) 108625 17 S. Muñoz et al. CRediT authorship contribution statement Sergio Muñoz: Term, Conceptualization, Methodology, Software, Validation, Writing – review & editing. Pedro Urda: Methodology, Validation, Investigation, Resources, Writing – review & editing. José L. Escalona: Conceptualization, Methodology, Writing – review & editing, Project administration. 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