Torsten Reissland, Robert Weigel, Alexander Koelpin, Fabian Lurz
©SHUTTERSTOCK.COM/DENIS BELITSKY
Increasing the use of rail-based passenger transportation, compared to passenger cars, is a key factor in the pursuit of a climate-friendly transport sector. For example, in Germany in 2021, carbon dioxide emissions per person-kilometer produced by long-distance train transportation were 72% lower compared to transportation by passenger cars [1]. To strengthen the competitiveness of rail transport, its reliability and cost from the user’s perspective need to be optimized. One way to achieve this goal is to enable higher levels of automation in rail transport, as is being sought in the automotive sector. And just as in the automotive sector, sensor systems are crucial for a reliable estimation of a vehicle’s condition. This reliable estimation is a necessary prerequisite for enabling automation functions.
In the European Union, a baseline for train functionality is given by the European Rail Traffic Management System (ERTMS), which aims to harmonize the various national rail traffic systems in terms of standards for control, signaling, and communication. One of the two parts of the ERTMS is the European Train Control System (ETCS), which regulates the standardization of signaling and control [2]. To ensure the interoperability of the various systems, minimum requirements for sensors and signaling are formulated here. In this article, we focus on aspects of a train’s speed measurements for which the accuracy of the measurement system is of interest. The ETCS requires an accuracy of $\pm{2}\,{\text{km}}{/}{\text{h}}$ for speeds lower than 30 km/h, which then increases linearly up to $\pm{12}\,{\text{km}}{/}{\text{h}}$ at 500 km/h [3]. This accuracy is the absolute minimum that must be achieved by a rail vehicle speed measurement system. Of course, higher accuracy is always strived for. The challenge of velocity measurements for rail vehicles is the large variety of environments in which they have to function. This can include tunnels, open fields, various track beds, and all kinds of climate conditions. In the first part of this article, we present a survey of true speed-over-ground measurement approaches that have been developed for railway applications or are suitable for them. In that part, systems developed for the automotive area are also looked into, as they could also be used in railway applications. If considering the utilization of automotive sensors in railway applications, some differences regarding the requirements have to be taken into account. The environment of a train is usually harsher when compared to a road. The possibility of stones in the track bed being lifted has to be taken into account, and so the sensor has to be able to withstand such threats. Also, when using the ground under the vehicle for the measurement, the different roughness of asphalt and crushed stones has to be considered. Furthermore, in most countries, trains can reach higher velocities. For example, the Intercity Express 3 in Germany can reach speeds of up to 330 km/h [4]. Therefore, a sensor has to be able to deliver accurate results over a wider range of velocities. Finally, the strong vibrations of the train below the passenger coach can cause disturbances in the measurements.
In the second part, a system developed by our group is presented in more detail. All the presented systems are capable of estimating the true speed over ground of a respective vehicle. Systems that determine only the angular velocity of the wheels are not considered, as a velocity estimated this way might significantly differ from a vehicle’s velocity, due to slip or deviations of the wheel diameter [5].
A well-known possibility for speed determination is global navigation satellite systems (GNSSs), especially GPS [6]. Such systems have been used in rail vehicles for several years in safety-critical applications. An example is the 3InSat system [7] of the European Space Agency, which uses a GNSS for determining a train’s position. Traditionally, this task is carried out using fixed beacons in the track bed, so-called balises [8]. The 3InSat system can also provide the possibility of speed measurements. To assess the accuracy of velocity estimation using a GNSS, several publications on the topic were evaluated. In [9], an accuracy of less than 1.44 km/h is determined in 64% of the measurements in practical tests, which are, however, carried out with bicycles. Here, a commercially available GPS receiver is used, which supports only nondifferential GPS reception. With differential GPS, a reference station is used in addition to the actual receiver to improve the measurement accuracy [10]. For railroad applications, however, this would cause considerable additional infrastructure investment. Extensive studies on the accuracy of position, velocity, and acceleration measurements via GPS can be found in [11]. However, the measurements contained therein take into account only the inaccuracies caused by the satellites. Other factors, such as atmospheric disturbances and nonideal receivers, are not taken into account. The results are thus to be understood as the lower limit of the measurement accuracy. The deviations of the speed measurements in comparison to a reference measurement lie here, depending on the measurement, with a probability of 95% between 0.009 and 0.012 km/h. The largest single deviation is determined to be 0.73 km/h. A current practical assessment of the accuracy of speed measurements using a GNSS can be found in [12]. Here, different algorithms for GNSS-based speed measurement are compared, with the best method achieving a root-mean-square (RMS) value of 0.22 km/h with respect to the horizontal deviation of the speed. The largest observed deviation in this experiment is 0.79 km/h. The tested velocity range is not specified in the paper. However, since the underlying measurement data are taken with a motor vehicle in an urban environment, speeds below 50 km/h can be assumed. In general, it can be stated that GNSS-based systems can be used only if there is sufficient reception of the satellite signals; satellite reception deteriorates in densely built-up or forested areas in particular, and in tunnels, it fails completely. Thus, a GNSS can be used in rail vehicles only as one measuring instrument among several.
The increased use of optical systems, such as lidars and cameras, in cars raises the question of whether they might be an option for the velocity measurement of cars as well as trains. A camera system for determining the intrinsic speed of road vehicles is presented in [13]. The setup essentially consists of a camera that records the background. Here, one makes use of the fact that photos become increasingly blurred as the tangential velocity of the photographed scene increases. Using the Radon transform [14], this blurring is quantified and used to estimate the velocity. The system achieves an RMS deviation value of 0.024 km/h at a velocity of 2.7 km/h. However, this value deteriorates significantly with increasing speed, reaching 5.76 km/h at a speed of 44 km/h. Thus, the system is unsuitable for high speeds. In addition, optical systems are more prone to errors in harsh environments when compared to radar, which, in turn, makes them unattractive for rail applications. In [15], a lidar-based approach for the velocity and pose estimation of cars is proposed. This method makes use of static targets in the point clouds generated by the lidar. These static targets are first determined by the feature-learning network PointNet++ [16]. Using multilayer perceptrons, the relative pose, i.e., the position and orientation of the targets relative to the car, is estimated. Finally, from subsequent poses, the car’s velocity is estimated. This method is simulated and also experimentally evaluated. However, estimation results for the velocity are presented only for the simulation. For the velocity range from 36 to 72 km/h, a mean absolute error of about 6 km/s is determined in the simulation.
In [17], a system is presented that is based on a cross-correlation approach. The cross correlation is carried out on the rail’s permeability, which is measured by a difference inductance sensor (DIS). This sensor is shown in Figure 1. It is composed of two coils mounted at both ends of the sensors, which are placed close to the rails. Since the inductance of the coils is influenced by the nearby ferromagnetic material of the rails, a varying phase shift can be measured at the coils while moving along the rail. These variations are caused by inhomogeneities within the material. The phase shifts from the measurements of both coils can be correlated to estimate the time it takes to cross the distance ${d}$, as in Figure 1. Extensive measurements are available for this system from a field test in which a maximum speed of 97 km/h is achieved. In 80% of the cases, the absolute error in the measurements is less than 3.6 km/h. Therefore, at the current state of development, this method is to be regarded as too inaccurate for use in the velocity measurement of rail vehicles.
Figure 1. A DIS used for the measurement of rail permeability [17].
As a technology for speed measurement in rail vehicles, Doppler radars have been used since about the 1990s [18]. Early works on this topic even date back to the 1970s [19]. These radars transmit an unmodulated continuous-wave (CW) signal and determine a speed via the Doppler shift of the received signal. A photograph of such a radar and a sketch of the operating principle can be seen in Figure 2. As can be seen, the radar receives the reflected signals from different angles and thus with different Doppler shifts. For this reason, such Doppler radars must first be calibrated. In addition, the received Doppler shift depends on the nature of the subsurface, which leads to further uncertainties in velocity determination. Another reason for deviations in this type of Doppler radar are dynamic multiple reflections, which cannot be resolved in a simple CW system. For this reason, modern sensors use two channels with different frequencies and angles of incidence to reduce these uncertainties [20]. Such an approach is used in the DRS05/3 radar sensor from Deuta-Werke. According to the manufacturer, the measurement uncertainty due to track bed conditions is nevertheless specified as up to 1%. The RMS value of the measurement deviations due to noise under ideal conditions is given for the same system as less than 0.4 km/h for speeds below 100 km/h and 0.4% for speeds above [21]. The absolute deviations are thus significant, especially at high speeds.
Figure 2. (a) A Doppler radar for rail vehicles. (b) An indicated operating principle [20].
In [22], a system for road vehicles is presented, which is also based on Doppler radar but uses a different measurement geometry and a different approach in signal processing. The radars used here are somewhat more complex since they have to be able to determine the direction of arrival (DOA) for each target. The system also requires at least two radar sensors with different positions and orientations. It takes advantage of the fact that a vehicle is typically surrounded by numerous stationary targets. This can be exploited by first determining the relative velocity vectors as well as the DOA between each sensor and each target. The relation of single sensor’s velocity vector ${(}{v}^{Sx},{v}^{Sy}{)}$ and the observed Doppler velocity ${v}_{i}^{D}$ for a target ${i}$ is presented in Figure 3. This relation depends on the angle ${\Theta}_{i}^{S}$ between the sensor and the target. For stationary targets, there is a clear connection between ${\Theta}_{i}^{S}$ and ${v}_{i}^{D}$. Using this connection, the random sample consensus (RANSAC) algorithm is used to determine outliers and therefore nonstationary targets [23]. A motion model of the vehicle and the environment is then established. This model includes measured Doppler velocities, the estimated DOAs of the targets, the mounting positions and orientations of the sensors, and the velocity vector and yaw rate of the vehicle. The least-squares method is used to determine the most likely direction of motion of the vehicle from the motion model. The method is tested on a vehicle with two Doppler radars and achieves an accuracy of less than 0.18 km/h at relatively low speeds between 20 and 40 km/h. In addition, it is not clear how well the system would work, for example, on open railroad tracks with only a few targets in the vicinity. An advantage when used in road vehicles is the possible estimation of the yaw rate, but this is irrelevant for use in railroad applications. A very similar approach by the same group with comparable results can be found in [24]. Other systems operating on a similar idea are presented in [25] and [26]. The system in [25] uses frequency-modulated CW (FMCW)-radars, which are also capable of distance determination but do not provide angle estimation. Instead, the angular information is obtained by bilaterating the distance measurements of two sensors. The individual sensors are thus simpler in design, but a larger number is required. Only a single measurement is available for this system, which has an error of about 0.25 km/h at a speed of 18 km/h. The systems in [22] and [26] have a relatively low unambiguous maximum speed. If this is exceeded, ambiguities and large errors in the velocity estimation occur. This problem is addressed in [26]. The system consists of radars with the capability of range and DOA estimation; the sensors are therefore rather complex. The signal processing steps of this system are illustrated in Figure 4. They start with a 2D fast Fourier transform, which results in a range–Doppler image. To identify relevant targets, constant false alarm rate detection is employed. Since the unambiguous Doppler range of such a radar is limited, some measures have to be taken to resolve the ambiguities at high velocities. For this, artificially added targets are introduced by duplicating every detected target to all relevant ambiguity ranges. The position of all the targets is then further estimated by employing an angle-of-arrival estimation, and the localized targets are added to a target list. Using the RANSAC algorithm, it is finally possible to filter out the targets in their true ambiguity range. Eventually, from the angle and Doppler information of the stationary targets, the velocity of the measuring vehicle can be determined. With the explained measures, the unambiguous range of 17.3 km/h can be extended by a factor of 35. The associated measurements are performed up to a velocity of about 70 km/h and again reach an RMS error (RMSE) of about 0.18 km/h. The maximum observed deviation is about 0.36 km/h. These results are very promising, so the application of such a system on a rail vehicle seems at least plausible. The only open question is the usability in an environment without highly reflective and distinct targets, such as walls in a tunnel or an open field.
Figure 3. The relation between the actual velocity vector of a sensor and the observed radial velocities ${v}_{i}^{D}$ in [22].
Figure 4. The signal processing steps for the RANSAC-based velocity estimation approach presented in [26]. FFT: fast Fourier transform; CFAR: constant false alarm rate; AAT: artificially added target; AOA: angle of arrival.
A system from another research group that also relies on targets in the environment for velocity estimation is presented in [27]. This involves a rotating FMCW radar that is used to detect distinct targets in the environment. The velocity of the vehicle is then again estimated from the relative velocities of the targets. A median speed error of 0.36 km/h is specified for this system. The test included velocities of up to 36 km/h. This is of a similar order of magnitude to the RMSE values of the previously presented systems. The main drawback of the system is the need for a rotating radar.
In [28], [29], and [30], another radar-based approach for the velocity estimation of road vehicles can be found. This approach uses simple CW radars without angle estimation. By mixing the received signals of several offset antennas, a comb-like antenna pattern is generated. This principle is described in Figure 5. If a target now moves tangentially to the antennas, a mixed signal is generated whose frequency is proportional to the target velocity. If the antennas are directed toward the ground, this allows a measurement system to be set up to determine the vehicle’s speed. In [29], the evaluation is realized with the help of a neural network. With this approach, an RMS value of the deviations of 0.5 km/h could be achieved at a maximum tested speed of 14 km/h. The behavior of the system at higher speeds is not known. Since this is a system developed for road vehicles, the expected deviations are unknown when the system is being used in a track bed. This uncertainty is due to the difference in the roughness of an asphalt road and a track bed.
Figure 5. (a) The working principle of the system presented in [28]. (b) Mixing the receive signals of the two antennas produces an antenna pattern with a comb-like pattern.
A system with a working principle similar the approach presented in the second part of this article can be found in [31] and [32]. This system is not used to measure self-motion but to measure the velocity of a large target in an industrial environment. It is based on two offset FMCW radars placed parallel to the direction of motion of the target at a defined distance from each other. Both radars are directed as orthogonally as possible to the target. The Fourier-transformed received signals from the radars are then correlated in two dimensions to determine the target velocity in both horizontal directions. The system achieves an RMS deviation of 0.039 km/h for a target velocity of 9.4 km/h. Due to the fact that the system uses antennas with beam steering and a 2D correlation of the range–time images, it is significantly more complex than the system presented here.
Another approach very similar to the one developed by our group but intended to be used in road vehicles can be found in [33]. A block diagram of the system is in Figure 6. The receive antennas are placed with an offset in the direction of movement. Here, a CW radar is used to determine the intrinsic speed of a vehicle by cross correlating two received signals. However, the setup contains only one transmit antenna, which is placed centrally between the two receive antennas. This setup provides valid results when driving on asphalt. As shown in the following, the rough surface of a track bed causes strong disturbances for such a system.
Figure 6. The cross correlation-based approach presented in [33]. HP: high pass.
A system developed by our group, which is a predecessor to the system later examined here, is presented in [34], [35], and [36]. This system is specifically developed for railway applications and has one transmit and two receive antennas, as can be seen in Figure 7. They are facing the track bed and recording its reflectivity. The system employs an FMCW radar in the 24-GHz Industry, Science, Medicine (ISM) band. The function is based on the cross correlation of the two receive signals; this principle is very similar to the older system presented in [33]. Although a proof of concept can be reached with this approach, it exhibits many outliers. The main reason for this is the orientation of the antennas. The reflectivity of the large crushed rocks of which the track bed consists depends on the viewing angle. And since the reflected signals arrive at the receive antennas from two different angles, the similarity of the signals and therefore the correlation peak decreases. This approach works on asphalt, as shown in [33], but fails for coarse ground, such as a track bed.
Figure 7. The radar system used in the measurements found in [35].
Another approach from our group targeting the velocity estimation for railway vehicles is presented in [37]. Here, our group tried to determine the vehicle’s velocity by means of an inverse synthetic aperture radar approach. Here, an FMCW radar without DOA estimation capabilities is employed. The measurements were carried out in a lab environment at very low speeds of up to 11.75 cm/s. To thoroughly examine the capabilities of this method, four different focusing methods were implemented and tested for this approach. And though the method yields very promising accuracies of about 3% for targets with strong reflections, large deviations occur if the targets consist of a more realistic track bed made from crushed rocks. This method was therefore not pursued further.
A system for the estimation of a rail vehicle’s velocity developed by our group was first presented in [38]. Similar to the system in [33], it is based on the cross correlation of two radar signals. But unlike the older system, it employs an FMCW waveform and also consists of two pairs of transmitters and receivers. The measurement principle is illustrated in Figure 8, and the realized system appears in Figure 9. The pairs are mounted at a distance of $\Delta{y}_{R}\,{=}\,{25}{cm}$ from each other. For the sake of simplicity, the two pairs of transmitters and receivers are simply referred to as two radars. The antennas are tilted at an angle ${\alpha}$ toward the ground, as shown in Figure 8. The signal processing chain is provided in Figure 10.
Figure 8. The cross correlation-based radar’s measurement principle. Two radars are moving with the train alongside the tracks while producing images of the ground’s reflectivity.
Figure 9. (a) The closed system as used in the field test. (b) The open system for laboratory measurements.
Figure 10. The signal processing steps. Dotted lines mark provided side information and are not part of the actual signal flow.
By correlating the received data in the range domain across the slow time, a correlation series can be calculated. The peak position of this correlation series corresponds to the temporal shift of both pairs. From the temporal shift and the distance $\Delta{y}_{R}$, the velocity can be determined. The cross correlation is calculated as \begin{align*}&{\phi}_{{S}_{1}{S}_{2}}\left[{{m}_{0},\Delta{m}}\right]\,{=}\, \\ &\quad\frac{\mathop{\sum}\limits_{{n}\,{=}\,{0}}^{{N}{-}{1}}{\mathop{\sum}\limits_{{m}\,{=}\,{0}}^{{M}{-}{1}}{Re}}\left\{{{S}_{1}^{\ast}{[}{n},{m}_{0}\,{+}\,{m}{-}\Delta{m}{]}{S}_{2}{[}{n},{m}_{0}\,{+}\,{m}{]}}\right\}}{\sqrt{\mathop{\sum}\limits_{{n}\,{=}\,{0}}^{{N}{-}{1}}{\mathop{\sum}\limits_{{m}\,{=}\,{0}}^{{M}{-}{1}}{{\left|{{S}_{1}{[}{n},{m}_{0}\,{+}\,{m}{-}\Delta{m}{]}}\right|}^{2}}}\mathop{\sum}\limits_{{n}\,{=}\,{0}}^{{N}{-}{1}}{\mathop{\sum}\limits_{{m}\,{=}\,{0}}^{{M}{-}{1}}{{\left|{{S}_{2}{[}{n},{m}_{0}\,{+}\,{m}{]}}\right|}^{2}}}}}{.} \tag{1} \end{align*}
Here, ${m}$ and ${n}$ denote the indices in slow time and the range domain, respectively; ${m}_{0}$ indicates the start of the comparison data taken into account; ${S}_{1}$ and ${S}_{2}$ describe the receive signals from the two receivers in the range domain; and $\Delta{m}$ is the shift index. The shift index at which the highest correlation is observed is denoted by $\Delta\hat{m}$. From this index, the velocity can be estimated as \[\hat{v}{(}\Delta\hat{m}{)}\,{=}\,\frac{{y}_{R}}{\Delta{\hat{m}}{T}_{\text{RRI}}}{.} \tag{2} \]
A graphical representation of the data to be correlated is in Figure 11. Here, the matrix of the current comparison data is correlated with $\Delta{m}_{\max}$ matrices of the reference data. Therefore, $\Delta{m}_{\max}$ is the maximum shift index, as it corresponds to the lowest velocity that can still be resolved by the approach. Finally, ${N}$ denotes the number of range bins taken into account, and ${M}$ denotes the number of radar snapshots in the slow time domain, which is used in the correlation. These three parameters also determine the computational complexity of the approach.
Figure 11. The data from both channels that are to be correlated. The (a) comparison data (S2) and (b) reference data (S1).
The hardware used in the measurements consists of four main parts. The RF front end contains a BGT24MTR12 transceiver from Infineon Technologies [39], which includes a voltage-controlled oscillator, and an ADF4159 synthesizer from Analog Devices [40]. Together they form the phase-locked loop for the signal generation. This setup is shown in Figure 12. The transceiver also includes the in-phase and quadrature mixers, the low-noise amplifier, and the power amplifier. The FMCW chirps are generated in the 24-GHz ISM band, with a bandwidth of 250 MHz or 1 GHz. The latter bandwidth does not comply with the standard of the ISM band. A triangular waveform with a total duration of ${60}\,{\mu}{s}$ is employed, from which only the rising portion with a duration of ${30}\,{\mu}{s}$ is used in the processing. The ramp repetition interval is therefore ${T}_{\text{RRI}}\,{=}\,{60}\,{\mu}{s}$. The RF front end also contains two ADRF5300 switches from Analog Devices [41] in the transmit path. These can be used to turn off the transmitters. The horn antennas used in the system are 3D printed and exhibit a directivity of 20 dBi. The analog baseband board contains an active high-pass filter, a passive low-pass filter, a variable amplifier for the signal conditioning, and the analog-to-digital converter. A block diagram of this board is provided in Figure 13. The last part of the hardware is the Zynq 7000 SoC ZC706 evaluation board from Xilinx [42]. The system on chip is composed of a Zynq 7000 Kintex-7 field-programmable gate array and two ARM Cortex-A9 processing cores. This part can be used either for the real-time processing of the data or the data acquisition. For the results presented here, the data acquisition was used, and the processing was then carried out using MATLAB.
Figure 12. The structure of the front end. The signal names refer to the naming conventions in [39] and [40].
Figure 13. The most relevant parts of the analog baseband board. Dashed lines refer to digital signals.
As already indicated, there is a lower limit ${v}_{\min}$ for the velocity measurement [38]. This limit can be calculated by inserting $\Delta{m}_{\max}$ in (2) as \[{v}_{\min}\,{=}\,\frac{{y}_{R}}{\Delta{m}_{\max}{T}_{\text{RRI}}}{.} \tag{3} \]
Since the maximum shift index $\Delta{m}_{\max}$ cannot be increased indefinitely, ${T}_{\text{RRI}}$ has to be increased at low velocities. This is done by using only every ${N}_{\text{th}}{-}{\text{th}}$ snapshot in the processing; ${N}_{\text{th}}$ is called here the thinning factor. At high velocities, on the other hand, the limited number of snapshots per time interval leads to a limited resolution of the velocity. Without the snapshot thinning, the relative resolution is defined as \begin{align*}{v}_{\text{res}}\,&{=}\,\frac{\hat{v}{(}\Delta\hat{m}{)}{-}\hat{v}{(}\Delta\hat{m}\,{+}\,{1}{)}}{\frac{1}{2}\left({\hat{v}{(}\Delta\hat{m}{)}\,{+}\,\hat{v}{(}\Delta\hat{m}\,{+}\,{1}{)}}\right)} \tag{4} \\ &{=}\,\frac{2}{{2}\frac{\Delta{y}_{R}}{{vT}_{\text{RRI}}}\,{+}\,{1}}{.} \tag{5} \end{align*}
To incorporate the snapshot thinning, one has to multiply ${T}_{\text{RRI}}$ by ${N}_{\text{th}}$. Therefore, one has to find a proper value of ${N}_{\text{th}}$, depending on the current velocity. For determining this value, a relative minimum velocity resolution is introduced and set to 1%. The velocity at which this resolution is reached is called ${v}_{1\%}$. Now, ${N}_{\text{th}}$ is set so that the actual velocity is as close as possible to the mean value between ${v}_{\min}$ and ${v}_{1\%}$ [38]. In the implementation, the last measured velocity is used as an approximation of the actual velocity. The relative velocity resolution ${v}_{\text{res}}$ is displayed in Figure 14 for two different ramp repetition intervals.
Figure 14. Relative velocity resolutions for the realized system and the two modes used.
Running the two radars at the same time can lead to performance degradation caused by mutual interference and channel coupling. Here, mutual interference refers to signals that are, for example, received by receiver 1 and transmitted by transmitter 2, while channel coupling refers to the direct coupling between the receive channels. The decoupling problem is well known in the literature, and there are many approaches to solve it. In [43], an approach for uniform linear arrays is presented. However, for the system presented here, one can make use of the cross correlation-based approach for which the system is developed. The decoupling approach employed here is presented in [44]. It is based on the determination of decoupling matrices. The ideal decoupling matrix for the range bin ${n}$ is defined as \[{D}{[}{n}{]}\,{=}\,\mathop{\text{argmax}}\limits_{\tilde{\text{D}}[n]}\mathop{\sum}\limits_{{m}_{0}}{{\phi}_{{S}_{\tilde{\text{D}}1}{S}_{\tilde{\text{D}}2}}}{[}{m}_{0},\Delta\hat{m}{]} \tag{6} \] with \begin{align*}\left({\begin{array}{c}{{S}_{\tilde{\text{D}}1}{[}{n},{m}{]}}\\{{S}_{\tilde{\text{D}}2}{[}{n},{m}{]}}\end{array}}\right)\,{=}\,\tilde{\text{D}}{[}{n}{]}\left({\begin{array}{c}{{S}_{1}{[}{n},{m}{]}}\\{{S}_{2}{[}{n},{m}{]}}\end{array}}\right){.} \tag{7} \end{align*}
Here, ${S}_{\tilde{\text{D}}1}$ and ${S}_{\tilde{\text{D}}2}$ describe the decoupled receive signals in the range domain. For the determination of the decoupling matrices, $\Delta\hat{m}$ has to be known. For this reason, the determination of these matrices is carried out using lab recordings with a constant known velocity. For the determination of the decoupling matrices, the method of steepest ascent is employed.
The mutual interference, however, cannot be easily compensated. Therefore, a so-called time-division multiplexing (TDM) mode is introduced in which the transmitters are run in an alternating fashion. For doing so, the already mentioned switches are used. Although the interference is avoided this way, this measure doubles the effective ramp repetition interval to a minimum of ${T}_{\text{RRI}}\,{=}\,{120}\,{\mu}{s}$. The effect on the relative velocity resolution can be seen in Figure 14. The resolution limit of 1% is reached at a velocity of 75 km/h. To solve this issue, it is sufficient to simply interpolate the correlation series ${\phi}_{{\text{S}}_{1}{\text{S}}_{2}}$ by a small factor. The practical effects of this measure and its theoretical limitations are presented in [45]. At the tested velocities, no difference regarding the velocity resolution is observable anymore between the two modes. For the distinction of the TDM mode, the mode in which both radars transmit continuously is referred to as the continuous mode (CM).
Since the wavelength of the transmitted signals is only about 1.25 cm, the two channels should be built as symmetrical as possible to avoid phase differences between the two channels; thus, the mounting of the system under the train has to be precise. This precise mounting is quite challenging in practice; a phase difference is therefore unavoidable. This would lead to a serious performance degradation if no measure for this issue were employed. The measure that is taken to overcome this problem in the system is to first do a calibration run after the system is mounted. From this run, the phase difference can be calculated and easily corrected. If such a calibration is not wanted or possible for some reason, there is also a second option for this problem. By using only the signal amplitudes in the calculation of the cross correlation, the influence of the phase differences can be eliminated. This type of correlation is denoted here as amplitude correlation, whereas the original type is referred to as phase correlation. The drawback of using only the amplitude is a smaller difference between the correlation peak and the values outside the correlation peak. At a low signal-to-noise ratio, this could lead to higher deviations in the measured velocities. However, in the test runs with the setup presented here, no significant differences in the results were observed [46]. This step is also shown in Figure 10.
The calculation of the velocities from the correlation series is highly nonlinear. This can lead to strong outliers in the measurements, which would lead to strong deviations if linear postprocessing methods were applied. Some examples for these outliers can be seen in Figure 15. Therefore, a plausibility check is introduced to determine the outliers and to eliminate them by extrapolating from the last correct measurements. These last measurements are also used to check the plausibility based on the largest velocity changes expected for the train. To make sure that the last measurements were indeed correct, a locked state is introduced. This locked state is activated as soon as the velocity measurements are mutually plausible for a certain amount of time. In the practical test, 0.5 s turned out to be sufficiently long to achieve a lock in the correct velocity range. After the plausibility check and, where necessary, the extrapolation, the velocity values are low-pass filtered and decimated to reach an update rate of about 10 Hz. The original update rate corresponds to the ramp repetition interval after thinning and is therefore much higher than necessary for this application.
Figure 15. Raw measurements of the phase correlation and reference values of a stepped run, with ${B}\,{=}\,{0}{.}{25}\,{GHz}$, ${\alpha}\,{=}\,{20}^{\circ}$, ${M}\,{=}\,{256}$, and $\Delta{m}_{max}\,{=}\,{256}$, without interpolation and with activated decoupling.
From the flowchart in Figure 10, it can be seen that there is one more step before the final velocity values are available. The velocity estimation assumes a constant velocity during the time in which the correlated data are obtained. In practice, this is, of course, not the case. Therefore, during acceleration, some deviations have to be expected. While at constant velocity each snapshot position of the first radar can be unambiguously assigned to the nearest snapshot position of the second radar, this is obviously no longer the case during accelerated motion. Thus, the position of the peak of the correlation series does not necessarily correspond to the actual velocity. However, by numerical analysis of a signal model, which is not presented here, this deviation can be calculated for various values of ${M}$, the relevant velocity range, and the realistic acceleration range. The largest deviations in the measurements, which are caused by this effect, are about 0.5 km/h and are therefore not negligible. In the implementation, these deviations are precalculated and then used in the compensation algorithm.
The field measurements for the system were carried out on a rail vehicle test ring in Germany, for which a map illustration is given in Figure 16. The ring has a length of 6.1 km and a typical track bed composed of rubble and railroad ties, as can be seen in Figure 17. During the tests, two velocity profiles were driven; the stepped profile was already detailed in Figure 15. The second profile, referred to as the constant profile, includes an accelerating phase until a velocity of about 80 km/h is reached as well as a constant velocity phase and a decelerating phase at the end. For the angle ${\alpha}$, three options, with ${\alpha}\,{=}\,{20}^{\circ}$, ${\alpha}\,{=}\,{40}^{\circ}$, and ${\alpha}\,{=}\,{60}^{\circ}$, were tested. Also, the two different bandwidths and modes were tested. Most of the runs were also carried out twice to prove repeatability. Here, not all the results for the combinations of configurations are presented but only the most promising ones.
Figure 16. The test ring [47]. (Source: OpenStreetMap.)
Figure 17. The system mounted below the train, with a tilt angle of ${\alpha}\,{=}\,{40}^{\circ}$.
To give a comprehensive impression of the results, a box plot is included in Figure 18 for one of the datasets. Often, the length of the whiskers in box plots is limited to 1.5 times the interquartile range. In the plot shown here, this limitation is increased to three times the interquartile range. This somewhat limits the number of outliers in the plots, which is otherwise caused by the non-Gaussian distribution of the deviations. The plot shows a failure of the approach in only three cases: datasets 9, 12, and 13. This can happen if too many measurements fail and the system doesn’t reach the locked state. However, most measurement runs were successful, and some resulted in especially low deviations, with maximum deviations of below 0.5 km/h. Surprisingly, a higher bandwidth and therefore range resolution does not lead to more accurate results. This is especially important, as the ISM band at 24 GHz is limited to a bandwidth of 250 MHz. A possible explanation for this is the occurrence of multipath propagation among the crushed rocks. Since the system with a bandwidth of 1 GHz has a higher range resolution, differences in the path lengths might have a larger impact on the correlation results. These differences could arise if both radars take their snapshots at slightly shifted locations. Due to the multipath propagation, these small deviations might have a significant impact on the path length that the signals experience. Since the stepped profile contains overall higher velocities, the deviations are larger in general. This can be discovered from the larger boxes in the box plot for these measurement runs. For the tilt angle, the options ${\alpha}\,{=}\,{20}^{\circ}$ and ${\alpha}\,{=}\,{40}^{\circ}$ seem to be viable, while the option ${\alpha}\,{=}\,{60}^{\circ}$ leads to significantly higher deviation. With such a high tilt, only a small portion of the track bed is in the focus of the antennas, and therefore, are a large portion of the reflective pattern cannot be used in the correlation.
Figure 18. Deviations of the results for M = 256, $\Delta{m}_{max}\,{=}\,{256}$, an interpolation factor of six, with activated decoupling and phase correlation. Here, C denotes the constant profile, while S denotes the stepped profile.
The configuration with ${B}\,{=}\,{250}\,{\text{MHz}}$ and ${\alpha}\,{=}\,{20}^{\circ}$ in CM is considered here as the overall configuration for comparison. These settings lead to very good repeatable results with moderate complexity in the investigations. The mean values of the corresponding RMSE of the two runs with this configuration are calculated as the final RMSE values of the deviations. The maximum deviation is the largest single deviation of both runs. In addition to the deviations, the corresponding speed ranges are also given in the comparison. The requirements of the ETCS are clearly met in all cases, as shown in Table 1.
Table 1. A summarized comparison of the accuracies of the cross-correlation approach and three other systems.
The most accurate results using an optical system were obtained with the camera-based system presented in [13]. At very low velocities, this system reaches a very good absolute accuracy, but at moderate velocities, it provides the lowest accuracy of the systems presented in this table. The results from [12] are used as comparative values for GNSS-based systems since this is a recent publication and the data were obtained in a realistic measurement environment. The DRS05/3 sensor is used as a representative of Doppler radars since it is already used in railroad operations [21]. The deviations are calculated here from the product data for a speed of 100 km/h since no concrete measurement results are available. As a representative of the radar systems that use the RANSAC algorithm, we compared the system from [26] since the best results are achieved here, even for comparatively high speeds. The other systems presented either have a significantly worse accuracy than these systems or are not suitable for comparison due to the low velocities tested. The final results are summarized as an overview in Table 1. It can be seen that the approach presented here performs best in terms of the RMS value of the determined velocity deviations over the entire velocity range. The method presented in [26] achieves a somewhat lower maximum deviation and a somewhat higher RMS value. However, it is unclear to what extent this method is suitable for use in rail vehicles and the associated measurement environment.
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Digital Object Identifier 10.1109/MMM.2023.3314320