A Method for Determination and Compensation of a Cant Influence

applied sciences

Article A Method for Determination and Compensation of a Cant Inﬂuence in a Track Centerline Identiﬁcation Using GNSS Methods and Inertial Measurement

Wladyslaw Koc 1, Cezary Specht 2, Jacek Szmaglinski 1,* and Piotr Chrostowski 1 1 Department of Rail Transportation and Bridges, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, Poland; [email protected] (W.K.); [email protected] (P.C.) 2 Department of Geodesy and Oceanography, Faculty of Navigation, Gdynia Maritime University, 81-225 Gdynia, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-5834-86090

Received: 13 September 2019; Accepted: 12 October 2019; Published: 15 October 2019

Abstract: At present, the problem of rail routes reconstruction in a global reference system is increasingly important. This issue is called Absolute Track Geometry, and its essence is the determination of the axis of railway tracks in the form of Cartesian coordinates of a global or local coordinate system. To obtain such a representation of the track centerline, the measurement methods are developed in many countries mostly by the using global navigation satellite system (GNSS) techniques. The accuracy of this type of measurement in favorable conditions reaches one centimeter. However, some speciﬁc conditions cause the additional supporting measurements with a use of such instruments as tachymetry, odometers, or accelerometers to be needed. One of the common issues of track axis reconstruction is transforming the measured GNSS antenna coordinates to the target position, i.e., to the place between rails on the level of rail heads. The authors in their previous works described the developed methodology, while this article presents a method of determining the correction of horizontal coordinates for measurements in arc sections of the railway track. The presence of a cant causes the antenna’s center to move away from the track axis, and for this reason, the results must be corrected. This article presents a method of calculation of mentioned corrections for positions obtained from mobile satellite surveying with additional inertial measurement. The algorithm presented in the article and its implementation have been illustrated on an example of a complex geometric layout, where cant transitions exist without transition curves in horizontal plane. Such a layout is not preferable due to the additional accelerations and their changes. However, it allows the veriﬁcation of the presented methods.

Keywords: railway track; track geometric layout; track centerline; railway track inventory; GNSS surveying; inertial measurement; cant measurement; Inertial Measurement Unit (IMU)

1. Introduction Nowadays, due to a rapid development of global navigation satellite system (GNSS) [1] the problem of rail routes reconstruction in a global reference system turns out to be increasingly possible. Knowing the coordinates of longitudinal track centerline, it is possible to work with so called Absolute Track Geometry. The essence of the process is determining the axis of railway tracks in the form Cartesian coordinates of a global or local coordinate system. The pioneering studies with a use of satellite measurements on railway tracks using a mobile measuring trolley were conducted in the 1990s at the University of Graz in cooperation with the Plasser & Theurer company [2]. In the early 2000s, a research on the application of precise GNSS measurements to determine track position was

Appl. Sci. 2019, 9, 4347; doi:10.3390/app9204347 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 4347 2 of 16 conducted at Marshall University [3]. Since then, the mobile measurements and analysis of real time kinematic data (RTK) have been developed by researchers and engineers. The current methods being developed for determining coordinates using GNSS techniques allow engineers for an accurate survey of a railway or tram tracks in a practical and economically effective manner [3–9]. In 2009, an interdisciplinary scientific team of the Gdansk University of Technology, Gdynia Maritime University, and Polish Naval Academy joined to this investigations and started to develop a method for mobile measurements of a track centerline [10–16]. The method was called a mobile satellite measurement (MSM) method, which included typical solutions, like differential global positioning system (DGPS) and RTK, but also original measurement data processing algorithms adapted for railway geometry specifics. From the beginning, it was certain that the accuracy of obtained track positions is significantly affected by a horizontal shift of the antennas centres in relation to the track axis. That occurs while the platform runs through circular curves and transition curves. This shift results from the tilting of the measuring platform due to the rail height differences, the so-called cant, which is commonly constructed on the length of these track sections. So, the lack of compensation for the measurement signal results with an inaccuracy in the identification of the track absolute geometry. Although the earlier work of the authors took into account the influence of the track cant on the measurements and the estimation of the track axis coordinates, the corrections have been determined with the knowledge of the track inclination value from independent measurements [17]. This means that the relevant corrections were calculated based on the cant measurements conducted by the methods commonly used in diagnostics and maintenance of the railway track. A subject matter of data fusion of inertial navigation system (INS) and GNSS signals have been discussed for example in papers [3–5,9]. The algorithms currently presented in the literature are often based on the data-fusion process using Kalman filters [4,9]. This data fusion is commonly implemented for combining the GNSS and INS signals, especially when the quality of GNSS is disturbed by terrestrial obstacles or even when the signal is interrupted (i.e., in tunnels or under viaducts). On the other hand, INS is subject to the control of the GNSS signal, especially to avoid the influence of drift. Such redundancy certainly improves reliability of the measurement system [3]. In this paper, the integration of INS and GNSS signals is focused on the previously mention problem, i.e., on calculating the corrections for a track inclination impact on the measured positions. However, the integration of both measurements into the multisensory platform results in an increase of measurement efficiency, and therefore makes the entire process faster and cheaper. Also, a crucial issue in such measurements is to improve the quality of track axis identification in a sense of measurement speed. Therefore, the possibilities to increase the speed in relation to devices pushed along the track, with a speed of about 5 km/h—to a speed of 30–60 km/h, are consequently tested. It is very important to keep a final accuracy in the range of 1–2 cm what is sufficient for problems related to the maintenance or design the geometric layout. So, the research target of the authors is to increase the accuracy along the measurement speed increase, from 5–10 km/h to 30–60 km/h range.

2. Materials and Methods

2.1. Track Position Measurements Track axis identification in approach of absolute geometry have been realized with a use of MSM method by using RTK positioning with a run speed around 30-60 km/h. Track positions were recorded using 20 Hz position update rate, depending on a run velocity, in about 0.3 m step [18,19]. Considering the length between measured points, it is possible to increase the ran speed, especially, when the geometric sections are long (tangent and curve sections). It should be emphasized that increased run speed affects the inertial measurements due to a stronger impact of centrifugal effects when curvature on a track exist. In order to analyze the possibilities of the cant distribution estimation on the base of inertial measurements, the authors developed a method for the estimation of this inclination. The algorithm Appl. Sci. 2019, 9, 4347 3 of 16 was adopted to the GNSS track surveying. The measurements were conducted on Gdansk tramway network. As a pulling vehicle, the N8C-MF01 tram was used. The pre-war DWF 300 tram boogies were used as measuring platforms [12]. That solution met the criteria for the track shape restoration because the antennas were fixed relatively close to the track level. The measurement system recorded signals and information about following parameters:

Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 17 Measurement performance date and time, • usedGeodetic as measuring coordinates platforms of the [12 measuring]. That solution point met (plane the criteria coordinates for the track in the shape local restoration coordinate because system), • the2D antennas and 3D positionwere fixed uncertainty relatively close radius, to the track level. • TheNumber measurement of satellites system visible recorded on thesignals horizon, and information about following parameters: • • Measurement performance date and time, Space system geometry, • • Geodetic coordinates of the measuring point (plane coordinates in the local coordinate system), Coordinate determination method (code, phase). • • 2D and 3D position uncertainty radius, • Number of satellites visible on the horizon, 2.2. The• CantSpace Estimation system geometry, Method •The C cantoordinate is one determination of the most important method (code, design phase) parameters. on railway and tram lines. It is defined as a difference of rails elevation in a particular track cross section. Along the circular part of arc the 2.2. The Cant Estimation Method nominal cant value is constant, while in adjacent parts the cant value changes. These sections of differ cant valueThe are cant called is one cant of the transitions most important [20]. On design curves parameters with a cant,on railway it is possible and tram to lines. achieve It is defined higher as speeds, a difference of rails elevation in a particular track cross section. Along the circular part of arc the nominal because a resultant value of centrifugal acceleration is lower. Therefore, even on high speed railways, cant value is constant, while in adjacent parts the cant value changes. These sections of differ cant value the horizontal arcs are designed with a cant and cant transitions between arcs and tangents. are called cant transitions [20]. On curves with a cant, it is possible to achieve higher speeds, because a resultantFrom apoint value of of view centrifugal of presented acceleration measurements, is lower. Therefore cant affects, even the onaccuracy high speed of positioning railways, the track axis inhorizontal the horizontal arcs are planedesigned what with is a shown cant and in cant Figure transitions1. Therefore, between the arcs measured and tangents. coordinates should be correctedFrom by taking a point into of view account of presented a transverse measurements, shift. The cant correction affects the ofaccuracy this shift of positioning is calculated the track according followingaxis in Formula:the horizontal [21]: plane what is shown in Figure 1. Therefore, the measured coordinates should be s corrected by taking into account a transverse shift. The correction2 of this shift is calculated according e D D Xa following Formula: [21]: ∆Y = (1 1 ) + · (1) 2 · − − e e

푒 퐷 퐷 ⋅ 푋푎 where: 훥푌 = ⋅ (1 − √1 − ( )2) + (1) 2 푒 푒 ∆Y – Transverse correction [mm], where: e – Centreline훥푌 – Transverse spacing correction of the rails [mm] [mm],, D – Cant푒 [mm],– Centreline spacing of the rails [mm], Xa – Measuring퐷 – Cant device [mm], mounting height [mm]. 푋푎 – Measuring device mounting height [mm].

FigureFigure 1. Cross-section 1. Cross-section of measured of measured track track with with a cant: a cant e—distance: e – distance between between rails axes;rails axes; D—cant; D – αcant;—cant  – angle. cant angle. The most commonly used method of cant determination is a static measurement using a manual gauge whichThe most is equipped commonly with used a method level or of inclinometer. cant determination Such is a a device static measurement is calibrated using to allow a manual a directly gauge which is equipped with a level or inclinometer. Such a device is calibrated to allow a directly reading the cant in millimeters. The second common method is based on dynamic measurements and is used in measuring trolleys or trains [1,3,4,9,11]. Due to additional accelerations and dynamic effects

Appl. Sci. 2019, 9, 4347 4 of 16 reading the cant in millimeters. The second common method is based on dynamic measurements and is used in measuring trolleys or trains [1,3,4,9,11]. Due to additional accelerations and dynamic effects which affect the measurement during the vehicle run, the effective inertial measurement is relatively difficult. Therefore, to determine the cant, the measuring vehicles for railway track inventory are equipped mainly with advanced gyroscopes. There are two common types of gyros:

Gyrostat, consisting of rotating elements which retain their original position using the gyro eﬀect. • With such a device, it is possible to directly determine the angle of rotation of the gyro casing [22]. Inertial Measurement Unit (IMU), consisting of a group of microelectromechanical systems • (MEMS) of electromechanical accelerometers that determine the rotational speed of the device. The calculation of the integral is required to specify the angle of the IMU casing rotation:

Zt α = ω dt (2) 0

where:

α – Vehicle’s angle of inclination in relation to the plane [rad], t – Time [s], ω – Angular velocity [rad/s].

Gyroscopic measurements have one fundamental disadvantage: the measurement result is dependent on the previous condition of the device, because the angle of rotation is not measured regarding to the global reference system, but to the earlier position of the gyro itself. For this reason, the device must be calibrated before measurement, but the measurement error increases with the test time [23–25]. Therefore, in many standards for railways, the requirements for cant determination are not strict (accuracy within 5 mm) [26–28]. The main disadvantage of a gyroscope is a drift, i.e., a systematic measurement error involving the accumulation of the measurement‘s result and a certain value (oﬀset or bias) [29,30]. The oﬀsets can also change over time. Therefore, the value obtained from the gyroscope should be considered as a resultant value of such components as:

ωg = ω(t) + ωd(t) + W(t) (3) where:

ω – Actual angular velocity [rad/s], ωg – Angular velocity read from the gyroscope [rad/s], ωd – Drift [rad/s], W – White noise [rad/s].

In the research, a series of tests were performed to check the repeatability of angle measurements. In the case in question, a triaxial MEMS gyroscope was used. The tilt angle of the platform was measured and the obtained calculation results were compared. The tests showed very high convergence of single angular measurement. However, at the same time, rapid accumulation of measurement uncertainty due to the overlapping of measurement errors has been reported. The readout difference of 2 degrees was recorded after only four tilts of the scaling device (tests were made with different rotational speeds). The test results present Figure2. The measured di fference was equivalent to a 52 mm cant. This means that the error is 10 times greater than the permissible value according to the previously mentioned standards [26–28]. Appl. Sci. 2019, 9, 4347 5 of 16 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 17

Figure 2. A graph of the repeatability test of the angle measurement using a gyroscope. Figure 2. A graph of the repeatability test of the angle measurement using a gyroscope. Based on the results of above tests as well as previous authors’ experience, it was assumed to desist Based on the results of above tests as well as previous authors’ experience, it was assumed to desist the gyroscopic measurement due to the probable accumulation of errors during measurement (centrifugal the gyroscopic measurement due to the probable accumulation of errors during measurement accelerations and vehicle vibration). The possibilities for using accelerometric measurements were then (centrifugal accelerations and vehicle vibration). The possibilities for using accelerometric measurements focused on. Such a measurement will be regarded in further considerations as the addition of redundant were then focused on. Such a measurement will be regarded in further considerations as the addition information which allow the verification of indications of other devices to increase the measurement reliability. of redundant information which allow the verification of indications of other devices to increase the 2.3.measurement Theoretical reliability. Assumptions for Cant Calculation

2.3. TheoreticalThe diﬀerence Assumptions between for the Cant rail C heightsalculation is reﬂected in the angle by which a moving vehicle turns. The relationship between cant and the vehicle tilt angle is described by the following Formula: The difference between the rail heights is reflected in the angle by which a moving vehicle turns. The relationship between cant and the vehicleD =tilte anglesin( αis) described by the following Formula: (4) · 퐷 = 푒 ⋅ sin(훼) (4) where: where: D퐷 – C Cantant [mm], [mm], e푒 – C Centrelineentreline spacing spacing of of the the rails rails [mm], [mm], α훼 – V Vehicle’sehicle’s angle of of inclination inclination in in relation relation to to the the plane plane [rad]. [rad]. In a vehicle that is stationary (or is moving along a perfectly straight track in the horizontal plane), the In a vehicle that is stationary (or is moving along a perfectly straight track in the horizontal cant can be measured using the change in the standard gravity component value measured transversely plane), the cant can be measured using the change in the standard gravity component value measured to the car (Figure 3b). transversely to the car (Figure3b). (g(𝑔) ) a푎푥 ==g푔 ⋅sinsin((α훼)) (5(5)) x · where:where: (𝑔) 2 푎(푥g) – Acceleration transverse to the car body [m/s ], a 2 푔x – S Accelerationtandard gravity transverse [m/s2]. to the car body [m/s ], 2 g In–Standard a moving gravity vehicle, [m during/s ]. the movement along a circular curve without cant, an additional transInverse a moving acceleration vehicle, component during the (centrifugal) movement will along appear a circular, as shown curve in Figure without 3a cant,. Where an a additional vehicle is transverseregarded as acceleration a point particle, component the relationship (centrifugal) between will appear, the velocity, as shown curve in Figure radius3a., and Where the acentrifugal vehicle is regardedacceleration as ais pointas follows: particle, the relationship between the velocity, curve radius, and the centrifugal acceleration is as follows: 푣2 (퐶) 2 푎(C푥) =v (6) a = 푅 (6) x R where: where: (퐶) 2 푎푥 – Centrifugal acceleration [m/s ], 푣 – Vehicle’s linear velocity [m/s],

Appl. Sci. 2019, 9, 4347 6 of 16

When a vehicle is moving along a curve with a cant, the above-mentioned states are combined Appl.(Figure Sci. 20193c)., 9, x The FOR measured PEER REVIEW acceleration transverse to the car body will depend on the components6 of 17 derived from the gravity and centrifugal acceleration: 푅 – Curve radius [m]. When a vehicle is moving along a(r )curve with a cant,(c )the above-mentioned states are combined a = g sin(α) a cos(α) (7) (Figure 3c). The measured accelerationx transverse· to − thex car· body will depend on the components derivedGiven from the the gravity small values and centrifugal of the angle acceleration:α (in practice not exceeding 6 degrees), it can be assumed approximately that its cosine is equal(푟) to 1. Such an( assumption푐) simpliﬁes the calculation process. 푎푥 = 푔 ⋅ sin(훼) − 푎푥 ⋅ cos(훼) (7) Basing on Formulas 4–7, the cant can be calculated using the following relationship: Given the small values of the angle 훼 (in practice not exceeding 6 degrees), it can be assumed approximately that its cosine is equal to 1. Such1 an assumption(r) simplifies the calculation process. Basing D = a e v2 e K (8) on Formulas 4–7, the cant can be calculated usingg · thex ·following− · · relationship:

1 (푟) where: 퐷 = ⋅ (푎 ⋅ 푒 − 푣2 ⋅ 푒 ⋅ 훫) (8) 푔 푥 K – Curvature [rad/m] where: 훫 – Curvature [rad/m]

FigureFigure 3. 3.TheThe distribution distribution of of acceleration acceleration in didifferentﬀerent movementmovement patterns. patterns (.a ()a a) casea case of movementof movement along alonga circular a circular curve curve without without a cant, a cant (b) a, ( caseb) a ofcase movement of movement along along a tangent a tangent section section with a with cant, a ( ccant) a case, (c) of a casemovement of movement along aalong circular a circular curve withcurve a with cant. a cant.

In Inthe the presented presented relationship relationship,, the the unknowns unknowns include include the the transverse transverse acceleration, acceleration, the the vehicle vehicle movementmovement velocity velocity,, and and the the curve curve radius radius (in (in general general—curvature). - curvature). The The simultaneous simultaneous determination determination of of thesethese three three values values requires requires the the application application of a of complex a complex measurement measurement technique. technique. In In order order to tosolve solve the the defined deﬁned problem problem for for existing existing tracks tracks,, the the authors authors developed developed an an algorithm algorithm whichwhich uses uses signals signals from from GNSS GNSS and and INS INS measurement measurements.s. The The analytical analytical form form of ofthe the algorithm algorithm and and its its operationoperation will will be beillustrated illustrated in next in next sections. sections. 2.4. Calculation of Ccorrections—Algorithm 2.3. Calculation of Ccorrections - Algorithm The initial step in determining the cant is the rough (automatic) analyses of the obtained The initial step in determining the cant is the rough (automatic) analyses of the obtained measurement signal. Based on the GNSS signal analysis, the track curvature at any given point and measurement signal. Based on the GNSS signal analysis, the track curvature at any given point and the the azimuth of the tangent to the track axis at that place are determined. azimuth of the tangent to the track axis at that place are determined. The track curvature radius can be roughly determined based on an analysis of horizontal versines. The track curvature radius can be roughly determined based on an analysis of horizontal versines. Since this method is aﬀected by numerous uncertainties, it is regarded exclusively as a preliminary Since this method is affected by numerous uncertainties, it is regarded exclusively as a preliminary solution bringing the measured signal closer to the actual track axis shape. To this end, for each of the measured points, a 20 m long virtual chord is introduced (10 m on each side of the point) and the value of the horizontal versine for this point is calculated (Figure 4).

Appl. Sci. 2019, 9, 4347 7 of 16 solution bringing the measured signal closer to the actual track axis shape. To this end, for each of the measured points, a 20 m long virtual chord is introduced (10 m on each side of the point) and the value ofAppl. the Sci. horizontal 2019, 9, x FOR versine PEER REVIEW for this point is calculated (Figure4). 7 of 17

FigureFigure 4. AA diagram diagram of of track track curvature curvature calculation calculation using using satellite satellite measurements. measurements.

ConsideringConsidering that the the length length of of a achord chord sections sections 푙1andl1 and 푙2 mayl2 may generally generally be different be diﬀerent from from each each other,other, the the followingfollowing relationship relationship is is used used to tocalculate calculate the thearc arcradius: radius:

p2 푠 = 푅 − √푅 −2 푙1 ⋅ 푙2 (9) s = R R l1 l2 (9) where: − − · where:푠 – Versine [m], 푙1, 푙2– Length of the curve chord part [m]. s – Versine [m], After expanding of this expression into a Maclaurin series and its conversion to calculate the l1radius,, – Length the corresponding of the curve relationship chord part is [m]. expressed by the following Formula: l 2 After expanding of this expression into a Maclaurin series and its conversion to calculate the √(푋 − 푋 )2 + (푌 − 푌 )2 ⋅ √(푋 − 푋 )2 + (푌 − 푌 )2 radius, the corresponding푃(푖−푛) 푃(푎푢푥) relationship푃(푖−푛) is푃 expressed(푎푢푥) by푃( the푎푢푥) following푃(푖+푛) Formula:푃(푎푢푥) 푃(푖+푛) 푅 = +. .. q 2 ⋅ (푋 − 푋 )2 +q(푌 − 푌 )2 √2 푃(푎푢푥) 푃(푖)2 푃(푎푢푥) 푃(푖) 2 2 (XP(i n) XP(aux)) +(YP(i n) YP(aux)) (XP(aux) XP(i+n)) +(YP(aux) YP(i+n)) R = − − q− − · − − + ... 2 2 ((푋 − 푋 )2 + (푌 2− 푌(XP(aux))2)X⋅P(((i)푋) +(YP(aux−)푋YP(i)) )2 + (푌 − 푌 )2) (10) 푃(푖−푛) 푃(푎푢푥) 푃(푖−푛)2 · 푃(푎푢푥) − 2푃(푎푢푥) −푃(푖+푛) 2 푃(푎푢푥) 푃(푖+푛2) . . . + ((XP(i n) XP(aux)) +(YP(i n) YP(2aux)) ) ((XP(aux) XP(i2+n)) +(YP(aux) YP(i+n)) ) (10) ... + − −8 ⋅ ((푋푃(푎푢푥) −−푋푃−(푖)) + (푌푃·(푎푢푥) − 푌−푃(푖)) ) − 8 ((X X )2+(Y Y )2) · P(aux)− P(i) P(aux)− P(i) where: where:푋푝 – The northern coordinate in the Cartesian coordinate system [m], 푌푝 – The eastern coordinate in the Cartesian coordinate system [m], Xp – The northern coordinate in the Cartesian coordinate system [m], 푖 – Measurement index, Y – The eastern coordinate in the Cartesian coordinate system [m], 푛p – Natural number, i 푃(푎푢푥– Measurement) – Auxiliary point. index, n –The Natural first operation number, to be carried out to calculate the auxiliary point coordinates is to calculate the Peq(auxuation) – Auxiliary of the straight point. line passing through the two points that determine the chord and of the straight perpendicular line passing through the point at which the curvature radius is being determined: The ﬁrst operation to be carried out to calculate the auxiliary point coordinates is to calculate the equation of the straight line passing through the two푋푃( points푖−푛) − that푋푃( determine푖+푛) the chord and of the straight 푚푃(푖−푛)푃(푖+푛) = (11) perpendicular line passing through the point at which푌푃(푖−푛 the) − curvature푌푃(푖+푛) radius is being determined:

X 푋푃(푖−푛X) − 푋푃(푖+푛) 푏 = 푋 −P(i n) P(i+n) (12) 푃(푖−푛)푃m(푖P+(푛i )n)P(i+푃n()푖−=푛) − − (11) − YP(i푌푃n)(푖+푛Y)P−(i+푌n푃) (푖−푛) − − The coefficients of the equation of a straight perpendicular line passing through the central point X ( ) X ( + ) are as follows: P i n P i n bP(i n)P(i+n) = XP(i n) − − (12) − − − YP(i+n) YP(i n) −1 − − The coeﬃcients of the equation푚푃 of(푖 a)푃 straight(푎푢푥) = perpendicular line passing through the central(13) point 푚푃(푖−푛)푃(푖+푛) are as follows: 1 1 푏 m=P(i)푋P(aux)+=푌 ⋅− (13) 푃(푖)푃(푎푢푥) 푃(푖) m푃(P푖)(i n)P(i+n) (14) − 푚푃(푖−푛)푃(푖+푛) 1 and the auxiliary point coordinatesbP(i )areP(aux as) follows:= XP(i) + YP(i) (14) · mP(i n)P(i+n) −

Appl. Sci. 2019, 9, 4347 8 of 16 and the auxiliary point coordinates are as follows:

bP(i)P(aux) bP(i n)P(i+n) YP(aux) = − − (15) mP(i)P(aux) mP(i n)P(i+n) − −

bP(i)P(aux) bP(i n)P(i+n) XP(aux) = mP(i n)P(i+n) − − + bP(i n)P(i+n) (16) − · mP(i)P(aux) mP(i n)P(i+n) − − − where: m – Straight line inclination, b – Absolute term [m]. Then, for each pair of coordinates measured by the satellite method, the time interval between the previous and the next measurement is determined (with an accuracy of 0.01s). Later on, for this time interval, a mean value of the measured acceleration is measured (the acceleration is measured with a frequency of 100 Hz, therefore there are an average of 10 single acceleration measurements per a pair of coordinates). Since half of the analyzed acceleration signal is then introduced into calculations for the next point, the method in question is identical to the analysis of a signal in a time series using a moving average. The measuring set’s movement velocity is also determined based on the analysis of distances between the measured points and on the measurement performance time. The curvature radius, transverse acceleration, and velocity values calculated in this way provide a basis for the calculation of the cant value (according to Formula 8). The next step is the calculation of the value of the necessary shift (correction) of coordinates in the horizontal zone (according to Formula 1) and the shift direction. Given that the coordinates determined by the satellite method are aﬀected by a known uncertainty (determined by the logger), the equation of the straight line perpendicular to the shift direction was calculated using a weighted regression of a speciﬁc number of coordinates (determined by the satellite method) found in the vicinity of this point.

1 mcor(i) = − (17) mh(i)

i+2 dχ2 X 1 = 2 YP(j) XP(j) mh(i) YP(j) bh(i) = 0 (18) dm ( ) · δ2 · · − · − h i j=i 2 (j) − where: χ2 – Random variable chi-squared distribution [m2], j – Measurement index. The corrected coordinates are calculated using the following relationships:

Π X = X + ∆Y cos ( atan m ) (19) Pcor(i) P(i) P(i) · 2 − cor(i) Π Y = Y + ∆Y sin ( atan m ) (20) Pcor(i) P(i) P(i) · 2 − cor(i)

2.5. Advanced Analysis of the Geometric Layout The algorithm for the automatic cant calculation presented above is sensitive to the measurement uncertainties. During the analyses, numerous tests involving the ﬁltering of the measurement signal in the wave number domain were carried out. This enabled a reduction of standard errors in the determination of the track axis coordinates associated with the signal oscillation around its actual position. Appl. Sci. 2019, 9, 4347 9 of 16

In order to calculate the design parameters of the track in horizontal plane and cant variability, a theoretical model based on known and common utilizing types of curves needs to be constructed [31]. For this purpose, SATTRACK program algorithms were used and modiﬁed [12]. Based on the roughly corrected coordinates of the measured track axis, basic geometric parameters are determined by dividing the track into sections of constant and variable curvatures. The analyzed route is then described using parametric equations and its basic characteristic parameters are determined, inter alia: Straight line azimuths, • Route intersection angles, • Curve radii, • Transition curve lengths, • Model of curvature distribution on the transition curve lengths. • Based on the route described in this way, coordinates of characteristic points are calculated, e.g.,: System of polygon vertices, • Circle centres, • Points of tangency between the geometric layout elements. • It is assumed that on a straight section and on the circular part of the curve, the curvature is 1 constant and has a value of κ = 0 for the straight section and κ = R on the circular part of the curve. On transition curves, the curvature is variable and described using the following relationship:

1 κ(l) = g(l) (21) R ·

For the clothoid, function g(l) takes on the following values:

l g(l) = (22) L where: κ(l) – Curvature function, g(l) – Distribution function, l – Distance parameter, along a curve, L – Length of a transition curve. In order to facilitate calculations, it was assumed that the transition curve arc length is approximately equal to the length of its projection. The distance from the start of the transition curve to the measured point located on its length is calculated based on the following relationship:

uv u 1 2 1 2 t XP(i) + YP(i) XP(i) + YP(i) bh mh(i) mh(i) 1 l = (Y · ) + (X + · − X ) (23) (i) TS 1 TS 1 P(i) m Y − m ( ) + m ( ) + 2 − − h(i) P(i) h i mh(i) h i m +1 h(i) ·

The calculations of the shift direction depend on which geometric layout element the measured pair of coordinates is assigned to. For calculations of the shift direction for a point located on the length of a circular curve, the following Formula is used: XP(i) XO m = − (24) cor(i) Y Y O − P(i) for a point located on the length of a straight section:

1 mcor(i) = − (25) mh(str) Appl. Sci. 2019, 9, 4347 10 of 16 and for a point located on the length of a transition curve:

1 mcor(i) = − tan Θ(i) (26) mh(str) ± where Θ—the angle of the tangent to the transition curve arc at a point distant by a certain length from its start. For the clothoid, function Θ(l) takes on the following values:

l2 Θ(l) = (27) 2 L R · · where: l – The location (distance from the start) of the point on the transition curve [m], Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 17 g(l) – Function of variable l, depending on the transition curve type, Lwhere:– Transition curve length [m], TS푙 –– T Thehe location start of the(distance transition from curve the start) (Transition—straight), of the point on the transition curve [m], O푔(푙) –– F Circleunction centre. of variable l, depending on the transition curve type, 퐿 –After Transition performing curve length the calculations, [m], each measured pair of coordinates is assigned additional parameters:푇푆 – The start of the transition curve (Transition - straight), 푂 – Circle centre. Curvature, After• performing the calculations, each measured pair of coordinates is assigned additional parameters: • ShiftCurvature, direction, • • Velocity,Shift direction, • • Acceleration,Velocity, • • CalculatedAcceleration, cant value. • Calculated cant value. The ﬁnal step of the analysis is the restoration of the model course of the cant variability and the The final step of the analysis is the restoration of the model course of the cant variability and the calculation of corrected track axis coordinates (based on Formulas 19-20). The operation principle of calculation of corrected track axis coordinates (based on Formulas 19-20). The operation principle of the the calculation algorithm is presented in the diagram in Figure5. calculation algorithm is presented in the diagram in Figure 5.

Figure 5. Block diagram of the cant calculation and coordinate correction algorithm. Figure 5. Block diagram of the cant calculation and coordinate correction algorithm. 3. Results 3. Results In order to verify the presented methodology for the correction of coordinates obtained by the MSMIn method, order to a verify series ofthe both presented satellite methodology and accelerometric for the measurementscorrection of coordinates were carried obtained out as wellby the as MSadditionalM method, reference a series measurements of both satellite using and a accelerometric manual track gauge.measurements A tram linewere fragment carried out consisting as well as of additionala curve with reference a cant andmeasurements the adjoining using straight a manual sections track was gauge. selected. A tram line fragment consisting of a curveFor with the a cant measurement and the adjoining of accelerations, straight sections an MPU-6500 was selected. triaxial MEMS accelerometer was used. ThisFor accelerometer the measurement is comprised of accelerations, of microscopic an MPU movable-6500 triaxial elements MEMS which accelerometer deﬂect under was used. the action This accelerometer is comprised of microscopic movable elements which deflect under the action of acceleration in the corresponding direction. The device determines the change in the system’s capacitance and, based on that, calculates the deflection of the element and the associated acceleration on a particular axis. The rotation of the device casing results in a change in the direction of standard gravity’s action on the accelerometer and, consequently, enables the calculation of the rotation angle. The cant on the curve and on the adjoining straight sections was then measured using a manual track gauge. On the straight sections, the measurements were taken every 10 m. The restored lengths of the curve and ramps are shown in Figure 6. The measurement is marked in blue while the identified linear estimation of the cant variability is in green. It was found that the surveyed layout comprised of two cant transitions and a section with a constant cant value. The cant transitions were not located on transition curves that were not detected on the surveyed geometric layout.

Appl. Sci. 2019, 9, 4347 11 of 16 of acceleration in the corresponding direction. The device determines the change in the system’s capacitance and, based on that, calculates the deﬂection of the element and the associated acceleration on a particular axis. The rotation of the device casing results in a change in the direction of standard gravity’s action on the accelerometer and, consequently, enables the calculation of the rotation angle. The cant on the curve and on the adjoining straight sections was then measured using a manual track gauge. On the straight sections, the measurements were taken every 10 m. The restored lengths of the curve and ramps are shown in Figure6. The measurement is marked in blue while the identiﬁed linear estimation of the cant variability is in green. It was found that the surveyed layout comprised of two cant transitions and a section with a constant cant value. The cant transitions were not located on transitionAppl. Sci. 2019 curves, 9, x FOR that PEER were REVIEW not detected on the surveyed geometric layout. 11 of 17

Figure 6. Cant on the curve measured using a manual track gauge.

An automatic measurementmeasurement of cant was then taken using the algorithm described above. Measurements were carried out at at various various velocities velocities of of the the measuring measuring vehicle. vehicle. On On a properly properly maintained track, a a change in the vehicle velocity had no significantsignificant effecteffect on the results ofof the conducted analyses.analyses. A Ann analysis result for the highest of the recorded velocities, i.e.,i.e. 60 60 km/h, km/h, is is shown. shown. A graph of of the the variability variability of of the the recorded recorded acceleration acceleration and and the the restored restored cant cant is shown is shown in Figure in Figure 7. On7. Onstraight straight sections, sections, the theestimated estimated cant cant oscillates oscillates around around 0, 0,and and on on the the cant cant transitions transitions (located (located on the straight sections sections of of the the track) track) an an evident evident increase increase occurs occurs in the in theacceleration acceleration as well as well in the in calculated the calculated cant. cant.At the Atconnection the connection between between the straight the straightline and linethe curve and the (at the curve point (at of the tangenc pointy of which tangency is also which a point is alsoof the a super point- ofelevation the super-elevation ramp end), a ramprapid change end), a in rapid the direction change in of the acceleration direction ofaction acceleration occurred. action After occurred.passing this After point, passing the calculated this point, cant the calculated value remained cant value at a remained relatively at constant a relatively level. constant An opposite level. Ansituation opposite occurred situation at occurred the transition at the transitionfrom the curvefrom the into curve the into straight the straight line and line on and the on second the second cant canttransition transition.. It can be be noted noted that that both both the the restored restored value value of of the the cant cant on on the the curve curve and and the the cant cant variability variability on the on thelength length corresponds corresponds to the to values the values measured measured using usingthe manual the manual track gauge. track gauge. A mean A value mean of value the cant of the on cantthe circular on the circularpart of the part curve, of the measured curve, measured based on based the measurement on the measurement of accelerations, of accelerations, differs dibyff ers2 mm by 2(for mm a velocity (for a velocity of 53 km/h) of 53 from km/h) the from value the measured value measured using the using track the gauge. track The gauge. oscillation The oscillation of the restored of the restoredcant results cant from results thefrom GNSS the measurement GNSS measurement errors and errors the andinduced the induced vibrations vibrations of the measuring of the measuring vehicle vehicleitself. itself. The acceleration variability during a passage of the measuring car was then calculated. In order to better illustrate the possibilities offered by the presented methodology, a passage at a velocity of 60 km/h was selected. Figure8 shows a graph of acceleration variability at the time of running along the curve (only the section between the start of the first cant transitions and the end of the second one is shown).

Figure 7. A graph of the automatically restored cant. The measured acceleration is marked in green, the calculated cant in red, and the measured cant in blue.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 17

Figure 6. Cant on the curve measured using a manual track gauge.

An automatic measurement of cant was then taken using the algorithm described above. Measurements were carried out at various velocities of the measuring vehicle. On a properly maintained track, a change in the vehicle velocity had no significant effect on the results of the conducted analyses. An analysis result for the highest of the recorded velocities, i.e. 60 km/h, is shown. A graph of the variability of the recorded acceleration and the restored cant is shown in Figure 7. On straight sections, the estimated cant oscillates around 0, and on the cant transitions (located on the straight sections of the track) an evident increase occurs in the acceleration as well in the calculated cant. At the connection between the straight line and the curve (at the point of tangency which is also a point of the super-elevation ramp end), a rapid change in the direction of acceleration action occurred. After passing this point, the calculated cant value remained at a relatively constant level. An opposite situation occurred at the transition from the curve into the straight line and on the second cant transition. It can be noted that both the restored value of the cant on the curve and the cant variability on the length corresponds to the values measured using the manual track gauge. A mean value of the cant on the circular part of the curve, measured based on the measurement of accelerations, differs by 2 mm (for a velocity of 53 km/h) from the value measured using the track gauge. The oscillation of the restored Appl. Sci.cant2019 results, 9, 4347 from the GNSS measurement errors and the induced vibrations of the measuring vehicle12 of 16 itself.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 17

The acceleration variability during a passage of the measuring car was then calculated. In order to better illustrate the possibilities offered by the presented methodology, a passage at a velocity of 60 km/h was selected. Figure 8 shows a graph of acceleration variability at the time of running along the Figure 7. A graph of the automatically restored cant. The measured acceleration is marked in green, curveFigure (only 7.the A graphsection of between the automatically the start restored of the first cant. cant The transitionsmeasured acceleration and the end is markedof the second in green, one is the calculated cant in red, and the measured cant in blue. shown).the calculated cant in red, and the measured cant in blue.

Figure 8. A graph of acceleration variability at the time of running along the curve. Figure 8. A graph of acceleration variability at the time of running along the curve. It is evident that this track segment can be divided into five sections characterized by different It is evident that this track segment can be divided into five sections characterized by different accelerationacceleration variability variability parameters: parameters: Red• andRed and blue blue color—the color - the cant cant transition transition locatedlocated on a a straight straight section section.. In a In model a model fashion, fashion, it is it is • characterizedcharacterized by by a constant a constant increase increase inin acceleration,acceleration, while while in inreality, reality, given given the thetram tram car design, car design, strong oscillations occur at the end of the cant transition, strong oscillations occur at the end of the cant transition, • Yellow and purple color – a rapid change in acceleration associated with the change in curvature Yellow and purple color—a rapid change in acceleration associated with the change in curvature • at the point, determined by the length of the car’s rigid base, at• theG point,reen color determined – a circular by part the of lengththe curve of characterized the car’s rigid by base,constant curvature and cant. In a model Greenfashion, color—a acceleration circular should part of have the curvea constant characterized value as well by. H constantowever, the curvature vibrations and induced cant. Inon athe model • fashion,cant accelerationtransition are clearly should detected. have a constant value as well. However, the vibrations induced on the cantBased transition on the acceleration are clearly value detected. regression in the indicated intervals and the analysis of the restored cant variability, a model of cant on the curve’s length was constructed. A graph of identified cant Basedvariability on the corresponds acceleration to that value measured regression using in a the track indicated gauge with intervals a relatively and thehigh analysis accuracy. of the restoredParameters cant variability, of the restored a model cant variability of cant on are the shown curve’s in Table length 1. was constructed. A graph of identified cant variability corresponds to that measured using a track gauge with a relatively high accuracy. Parameters of the restoredTable cant 1. A variability comparison areof the shown restored in cant Table variability1. parameters. Track section geometry Manual track gauge MEMS Differences 1st cant traTablensition 1. A length comparison of the restored46.99 cantm variability46.80 parameters. m 0.19 m 0.40 % 2ndTrack cant Section transition Geometry length Manual Track47.11 Gauge m MEMS44.23 m Di2.88fferences m 6.11 % Length of the curve with constant cant 77.54 m 78.58 m 1.04 m 1.34 % 1st cant transition length 46.99 m 46.80 m 0.19 m 0.40% 1st2nd cant cant transition transition lengthsteepness 47.112.68 m mm/m 44.232.71 m mm/m 2.880.03 m mm/m 6.11%1.23 % Length2nd of cant the curve trans withition constant steepness cant 77.542.67 m mm/m 78.582.87 m mm/m 1.040.20 m mm/m 1.34%7.33 % 1st cantMean transition value steepnessof cant 2.68 mm126.00/m mm 2.71 mm126.48/m mm 0.03 mm0.48/m mm 1.23%0.38 % 2nd cant transition steepness 2.67 mm/m 2.87 mm/m 0.20 mm/m 7.33% StandardMean value deviation of cant of cant 126.00 mm3.53 mm 126.4812.20 mm mm 0.488.67 mm mm 0.38%245.80 % Standard deviation of cant 3.53 mm 12.20 mm 8.67 mm 245.80% The last step of the analysis was to introduce corrections to the measured track axis coordinates. Figure 9 shows a fragment of the measured curve. The measured coordinates that were characterized by a constant shift in relation to the model location of the track axis (green line) due to the measuring vehicle’s tilt are marked in red. The coordinates corrected by the value of ΔY are marked in blue. As can be seen, they coincide with the expected location of the track axis.

Appl. Sci. 2019, 9, 4347 13 of 16

The last step of the analysis was to introduce corrections to the measured track axis coordinates. Figure9 shows a fragment of the measured curve. The measured coordinates that were characterized by a constant shift in relation to the model location of the track axis (green line) due to the measuring vehicle’s tilt are marked in red. The coordinates corrected by the value of ∆Y are marked in blue. Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 17 Appl.As can Sci. be2019 seen,, 9, x FOR they PEER coincide REVIEW with the expected location of the track axis. 13 of 17

Figure 9. A comparison of the location of measured and corrected coordinates and the theoretical FigureFigure 9. AA comparison comparison of of the the location location of of measured measured and and corrected corrected coordinates coordinates and and the the theoretical location of the track axis. locationlocation of of the the track track axis. axis. 4. Discussion 4. Discussion In this section,section, the eeffectsffects of the algorithm operation have been presented andand discussed.discussed. In order In this section, the effects of the algorithm operation have been presented and discussed. In order to assessassess the eeffectivenessffectiveness of the analysis, didifferencesfferences between identifiedidentified layout and the reference one to assess the effectiveness of the analysis, differences between identified layout and the reference one from thethe measurementmeasurement were analyzed.analyzed. Figure 10 shows eeffectffect ofof tracktrack layoutlayout estimation.estimation. The signalsignal from the measurement were analyzed. Figure 10 shows effect of track layout estimation. The signal clearly shows the presence of super-elevationsuper-elevation ramps which are located on straight sections (an evident clearly shows the presence of super-elevation ramps which are located on straight sections (an evident linear increase in in delta delta differences) differences) and and of of cant cant with with a a constant constant value value on on the the curve. curve. linear increase in delta differences) and of cant with a constant value on the curve. Knowing that this layout has no transition curves, it becomes obvious that this about 130 mm Knowing that this layout has no transition curves, it becomes obvious that this about 130 mm shift inKnowing the central that this part layout of layout has no should transition be corrected. curves, it Figurebecomes 11 obvious shows thethat ethisffect about of correction. 130 mm shift in the central part of layout should be corrected. Figure 11 shows the effect of correction. An shiftAn improvement in the central in part layout of layout estimation should can be be corrected. noticed. DeviationsFigure 11 shows between the estimation effect of correction. and measured An improvement in layout estimation can be noticed. Deviations between estimation and measured improvementpositions for the in correctedlayout estimation coordinates can are be relativelynoticed. Deviations evenly distributed between on estimation the length and of the measured section positions for the corrected coordinates are relatively evenly distributed on the length of the section posi(straighttions sectionsfor the corrected are marked coordinates in black and are red,relatively while theevenly circular distributed curve is on marked the length in green). of the section (straight sections are marked in black and red, while the circular curve is marked in green). (straight sections are marked in black and red, while the circular curve is marked in green).

Figure 10. A graph of the measured track axis deviations in relationrelation to the designed shape. Figure 10. A graph of the measured track axis deviations in relation to the designed shape.

Appl. Sci. 2019, 9, 4347 14 of 16 Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 17

Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 17 Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 17

FigureFigure 11.11. AA graphgraph ofof thethe measuredmeasured tracktrack axisaxis deviationsdeviations inin relationrelation toto thethe designeddesigned shapeshape followingfollowing Figure 11. A graph of the measured track axis deviations in relation to the designed shape following thethe correctioncorrectionFigure 11. of Aof coordinatesgraphcoordinates of the measured due due to to the the track occurring occurring axis deviations cant. cant. in relation to the designed shape following the correction of coordinates due to the occurring cant. the correction of coordinates due to the occurring cant. ApartApart fromfrom thethe clearclear decreasedecrease inin thethe uncertaintyuncertainty ofof thethe routeroute coursecourse estimationestimation thatthat isis visiblevisible whilewhile Apart from the clear decrease in the uncertainty of the route course estimation that is visible while comparingcomparingApart also from Figures the clear 1212 and and decrease 13, 13 ,the thein meanthe mean uncertainty values values of of ofthe the the differences route diﬀ erencescourse and estimation and their their standard that standard is visible deviations deviations while also comparing also Figures 12 and 13, the mean values of the differences and their standard deviations also also clearlycomparing decreased. also Figures However, 12 and 13, a shiftthe mean of the values mean of the value differences on the sectionsand their standard with cant deviations in relation also to the clearlyclearly decreased decreased. However,. However, a shift a shift of of the the mean mean valuevalue onon the the sections sections with with cant cant in relation in relation to the to sections the sections clearly decreased. However, a shift of the mean value on the sections with cant in relation to the sections sectionswith withno withcant no cantcan no cant canstill still canbe seen.be still seen. beRelatively Relatively seen. Relatively higher higher valuesvalues higher maymay values be be correlated correlated may be correlatedwith with deformations deformations with deformations of existing of existing of with no cant can still be seen. Relatively higher values may be correlated with deformations of existing existingtrack trackaxis track axisalong axisalong the along thecurve curve the or curveorcan can be be or the the can effect effect be the ofof measurementmeasurement eﬀect of measurement quality quality. . quality. track axis along the curve or can be the effect of measurement quality.

Figure 12. Absolute values of the mean matching error: blue bars – before the correction, red bars – FigureFigure 12. 12.Absolute Absolute values values of the the mean mean matching matching error error:: blue bars blue – before bars—before the correction, the correction, red bars – red following the correction, 1 – straight line 1; 2 – super-elevation ramp 1; 3 – circular curve; 4 – super- bars—followingFigurefollowing 12. Absolute the thecorrection values correction,, 1of – thestraight 1—straight mean line matching 1; 2 line– super 1;error- 2—super-elevationelevation: blue barsramp – 1 ;before 3 – circular ramp the 1;correction, curve; 3—circular 4 – super red curve;-bars – elevation ramp 2; 5 – straight line 2. 4—super-elevationfollowingelevation the ramp correction 2; ramp 5 – straight, 2;1 5—straight– straight line 2. line line 1; 2. 2 – super-elevation ramp 1; 3 – circular curve; 4 – super- elevation ramp 2; 5 – straight line 2.

FigureFigure 13. Values 13. Values of theof the matching matching error error standard standard deviations:deviations: blue blue bars bars—before – before the thecorrection, correction, red red Figure 13. Values of the matching error standard deviations: blue bars – before the correction, red bars—followingbars – following the the correction); correction); 1—straight 1 – straight lin linee 1, 1, 2 – 2—super-elevation super-elevation ramp ramp 1; 3 – 1;circular 3—circular curve; 4 curve;– bars – following the correction); 1 – straight line 1, 2 – super-elevation ramp 1; 3 – circular curve; 4 – 4—super-elevationsuper-elevation rampramp 2; 5 5—straight – straight line line 2. 2. super-elevation ramp 2; 5 – straight line 2. Figure 13. Values of the matching error standard deviations: blue bars – before the correction, red

bars – following the correction); 1 – straight line 1, 2 – super-elevation ramp 1; 3 – circular curve; 4 – super-elevation ramp 2; 5 – straight line 2.

Appl. Sci. 2019, 9, 4347 15 of 16

5. Conclusions The Mobile Satellite Measurement method enables the efficient and effective reconstruction of the railway and tram tracks geometry. Its supplementation by including measurements concerning the cant, presented in the paper, enable a further enhancement of measurement reliability and accuracy. As has been demonstrated, it is possible to measure cant using accelerometry. Thanks to the use of standard gravity variability, this is a certain method as the obtained results are not determined by the previous location of the device or its indications. For this reason, no measurement result drift occurs and there is no need to perform precision device calibrations (a measurement can be started on a track with any given cant or twist). In this particular case, even the use of standard, easily available measuring devices allowed obtaining satisfactory results. The use of a more complex measuring system would have yielded results of a probably significantly higher quality. In further research, it will be advisable to create a common logger for satellite and accelerometric measurements in order to avoid the need for signal synchronization in the time domain which, by assumption, introduces an additional uncertainty due to measuring device clock errors.

Author Contributions: Conceptualization, W.K., C.S., J.S., P.C.; methodology, W.K., C.S., J.S., P.C.; software, J.S., P.C.; validation, J.S., P.C.; formal analysis, J.S., P.C.; investigation, C.S., W.K., J.S.; resources, C.S., J.S.; data curation, C.S., J.S., P.C.; writing—original draft preparation, J.S., P.C.; visualization, J.S., P.C.; supervision, W.K., C.S. Funding: This research received no external funding. Acknowledgments: The authors would like to thank the Municipal Transport Company GAiT (Gdansk, Poland) for their support in the field research. Conflicts of Interest: The authors declare no conflict of interest.

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