Estimation of the Horizontal Curvature of the Railway Track Axis with the Use of a Moving Chord Based on Geodetic Measurements

Koc, Władysław

doi:10.1061/(ASCE)SU.1943-5428.0000402

Open access

Technical Papers

Jun 20, 2022

Estimation of the Horizontal Curvature of the Railway Track Axis with the Use of a Moving Chord Based on Geodetic Measurements

Author: Władysław Koc, Ph.D., D.Sc. https://orcid.org/0000-0002-4619-7852 [email protected]Author Affiliations

Publication: Journal of Surveying Engineering

Volume 148, Issue 4

https://doi.org/10.1061/(ASCE)SU.1943-5428.0000402

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Abstract

The paper presents the application possibilities of the new method of determining the curvature of the axis of the railway track, called “the method of the moving chord.” Based on the estimation of the existing curvature, it becomes possible to identify the geometric layouts of the track in the horizontal plane. As shown in previous publications, for model layouts, the method of the moving chord shows full compliance of the obtained curvature diagrams constituting the basis for obtaining the corresponding geometric solution. Now, the question of its use for the estimation of the horizontal curvature of the axis of the operated railway track, based on Cartesian coordinates obtained by direct measurements, has been taken up. Measurement data from two geometric layouts of the track with clearly different values of the radius of the horizontal arc and the nature of the curvature present were used. In both cases, the obtained curvature diagrams are less regular and oscillatory in nature, which does not prevent the basic geometrical parameters of the measured layout from being estimated on their basis. The presented adaptation of the moving chord method to the adopted measurement procedure and the practical use of the obtained curvature diagram provide the method in question with appropriate implementation bases. The application of the presented procedure should significantly improve the process of identifying geometric track layouts.

Introduction

From the point of view of the safety of rail transport and the achievable speed of trains, the question of determining and assessing the shape of the track axis in the horizontal plane plays a key role. The purpose of this operation is to determine the basic geometrical parameters of the route:

•

Location and length of straight sections.

•

Position of circular arcs along with their radius and length.

•

Location of transition curves along with their type and length.

On the basis of these data, it becomes possible to simulate the passage of a train on a given geometric layout (and thus to determine the speed of travel), as well as—which seems to be a fundamental issue—to obtain data for the design of the track axis regulation.

The measurement methods used have a very long tradition and although they are subject to various innovations, they are characterized by high labor consumption and the related necessity to incur significant financial outlays. The rules for carrying out measurements are similar in different railway authorities (CEN 2010; DB 2016; FRA 2008; NR 2008; NSW 2016; ÖBB 2004; PKP 2018; SBB 2016).

For example, according to the regulations (PKP 2018) in force in Poland, straight sections are most often measured by the ordinate and abscissa method along the measurement lines, using an appropriate geodetic network. Measurements on the curve include the measurement of horizontal arrows related to the chord defined by the theodolite’s line of sight (the direct method) or marks of the track axis adjustment (indirect method). A more modern measuring system is the total station placed on a trolley, allowing to obtain an accuracy of 1 mm, which additionally allows for measuring the inclination of the track (using an inclinometer) and measuring its width. The position is determined by identing backward to the four points of the matrix. The coordinates are measured at 10-m intervals on straight sections and at 5-m intervals along the arc, with reference to the point on the inside of the right rail track head.

New possibilities in the field of inventory of engineering facilities are created by the development of satellite measurement and increasing the accuracy of measurements based on the Global Navigation Satellite System (GNSS) technique, using active geodetic networks (e.g., networks of reference stations operating in the Real Time Network mode) (Li et al. 2019; Pu et al. 2019; Szwilski et al. 2005). This solution does not require the use of the network of overhead railway points—the measurement systems cooperate with networks of reference stations. Currently, research is underway on the possibility of using systems consisting of satellite receivers mounted on various types of vehicles (Alkan 2019; Chang et al. 2020; Chen et al. 2018; Jiang et al. 2017; Kampczyk 2020; Li et al. 2020; Quan and Lau 2019; Wang et al. 2018; Wu et al. 2020; Zhou et al. 2019).

In Poland, the method of mobile satellite measurements of railway track axis has been developed for over 10 years (Koc and Specht 2010; Koc and Specht 2011; Koc et al. 2012, 2018, 2019a, b; Specht and Koc 2016; Specht et al. 2014, 2017, 2019). This method consists in continuous registration of track axis coordinates using GNSS receivers installed on a moving measurement platform and using the collected measurement data in appropriate calculation algorithms [the analytical method of designing (Koc 2012, 2014, 2015, 2016)].

The main goal of all the measurement methods mentioned is to determine the geometrical parameters of the route. In order to achieve it, appropriate calculation algorithms should be applied after the measurements have been carried out. First, a solution is suggested here, which consists in using the obtained measurement data to determine the existing curvature of the geometric layout. The method currently used for this purpose is based on the curve diagram, which is the most frequently used tool for assigning track points to sections with defined geometry. This diagram—from the model point of view—is identical to the diagram of horizontal arrows related to the chord stretched along the track.

It should be noted, however, that treating the arrow graph as an equivalent of the curvature graph is formally unjustified. This is due to the fact that the values of the arrows are determined in millimeters, and the unit of curvature is rad/m. In addition, it should be taken into account that the drawn graph of arrows shows the values of the measured horizontal arrows, but there are no specific directions in which these arrows are measured. The reference line here is constituted by the directions of the chord setting, which are constantly changing.

The papers (Koc 2020, 2021a, b) present relevant analyzes relating to the proposed new method for determining the curvature of the track axis, referred to as the moving chord method. These analyzes concerned the application of this method for model (i.e., theoretical) geometric layouts. This paper addresses the issue of its use to estimate the horizontal curvature of the axis of an exploited railway track on the basis of Cartesian coordinates obtained by geodetic measurements.

The Moving Chord Method

From the definition of curvature, it is necessary to manipulate the angles of the tangent to the geometric layout. Obviously, if you have an analytical record of a given curve, this is not a problem. However, with a real, most often deformed as a result of operation, railway track in mind, the concept appeared not to use tangent but corresponding chords when determining the curvature of the track. In Koc (2020), a new method of determining the curvature is presented, which consists in using the difference of the inclination angles of the chord of a fixed length at a given point (the moving chord method). This theoretical method was verified on selected geometric layouts with fixed parameters. Thanks to the use of the analytical record, the position of the ends of the chord was clearly established in a given case. Fig. 1 shows a schematic diagram of determining the curvature with the proposed method.

It was assumed that when operating with chords of short length (up to 20 m), the straight tangents and the corresponding chords are parallel to each other, and the points of contact project perpendicularly to the center of a given chord. Curvature

k_{i}

occurring at a given point

i

is determined from

k_{i} = \frac{Δ Θ_{i}}{l_{c}}

(1)

where

l_{c}

denotes the length of the chord, and the angle

Δ Θ_{i}

results from the difference in the angles of inclination of adjacent chords converging at point

i

Δ Θ_{i} = Θ_{i / F i} - Θ_{B i / i}

(2)

The use of this procedure requires having the coordinates of a given curve in the |Cartesian system, because the values of the angles

Θ_{B i / i}

and

Θ_{i / F i}

result from the slope coefficients of the lines describing both chords. A positive value of

k_{i}

in Eq. (1) corresponds to the case of convexity of the curve pointing downward, and a negative value to one pointing upward.

Koc (2020) presents the verification of the proposed method of determining the horizontal curvature on a clearly defined elementary geometric layout of tracks, consisting of a circular arc and two symmetrically positioned transition curves (of the same type and length), calculated according to the principles of the analytical design method (Koc 2012). Several geometric cases for different train speeds were considered, and the types of transition curves used and the route turning angles were also differentiated. The obtained curvature diagrams were fully consistent with the diagrams constituting the basis for obtaining the corresponding geometric solution, and this concerned both the circular arc sections and the regions of the transition curves. This is clearly illustrated in Fig. 2, prepared for the following geometric layout: radius of circular arc

R = 500 m

, transition curves in the form of clothoid

l_{T C} = 110 m

, turning angle

α = π / 3 rad

.

Fig. 2. Sample plot of the ordinates of curvature $k (L)$ determined by the moving chord method along the length of the model geometric layout.

It was also noted that the proposed method offers great application possibilities. The practical aspect of the presented considerations may appear when the geometrical characteristics of the track axis determined by measurements are not known and the determination of these characteristics becomes the primary goal. In this situation, the discussed method perfectly corresponds to the assumptions of mobile satellite measurements. These measurements provide the coordinates of the track axis in the rectangular coordinate system in a very large number and in a very short time.

In Koc (2021b), two important detailed issues were taken up: the influence of the chord length on the obtained values of horizontal curvature and the possibility of determining the location of border points between individual geometric elements. The analyzed variants resulted from the type of transition curves used. It was found that the influence of the chord length in the range from 5 to 20 m on the determined curvature values is insignificant. At the same time, attention is drawn to the precision of determining the nature of the curvature and its compliance with the theoretical course on the transition curves.

At the same time, it has been shown that in the moving chord method it is possible to determine the location of the border points between individual geometric elements, and the required chord length must be adapted to the type of transition curve. For example, for a transition curve in the form of clothoid (with linear curvature), a chord with a length of

l_{c} = 2 m

should be used locally, and for a Bloss curve (i.e., a smooth curve) it may be a chord with a length of

l_{c} = 5 m

. In addition, in the case of a clothoid, some inaccuracies should be expected at both ends (i.e., connections with straight sections and circular arc).

In Koc (2021a), not only the problem of determining the curvature in the horizontal plane but also in the vertical plane was considered, indicating the universality of the discussed method. The focus here is on the computational foundations of this method, concerning the shaping of the inclination angles of the moving chord. It was found that for a circular arc in the horizontal plane, the values of the inclination angles of the moving chord depend on the turning angle, the arc radius, and the chord length, while the difference in inclination angles itself depends on the chord length and the arc radius. In the case of a circular arc in the vertical plane, the values of the angles of inclination of the moving chord are much smaller than in the horizontal plane, which is related to the range of the radii of the arcs. As in the horizontal plane, the difference in the angles of the moving chord is determined by the radius of the vertical arc and the chord length.

Adaptation of the Moving Chord Method to the Adopted Measurement Procedure

The moving chord method requires knowledge of the coordinates of the track axis in the Cartesian system. In Koc (2020, 2021a, b), an analytical record of the considered geometric layout was available, so determining the Cartesian coordinates of the ends of the chord directed backward and forward was not a problem. In this situation, it was easy to determine the values of the respective slope angles

Θ_{(i - 1) / i}

and

Θ_{i / (i + 1)}

. When determining the curvature of the track on the basis of measurement data, the coordinates of the track axis are discrete, and it would also be difficult to create an analytical record. Therefore, it is necessary in such a case to adapt the method of the mobile chord to the adopted measurement procedure.

We assume that as a result of measurements—carried out in a traditional way (using total station) or with the use of satellite technology—we have a set of

x_{i}

,

y_{i}

track axis coordinates, determined at relatively small (i.e., several meters) distances. Of course, we are aware that the shape of the measured (real) track differs from the model shape, and moreover, we must take into account the occurrence of a measurement error. In this situation, the character of the determined curvature of the track axis must differ from the curvature of the model layouts considered in Koc (2020, 2021a, b).

All measurement points are numbered

i = 1, 2, \dots, n

. We start the determination of the curvature

k_{i}

from the point

i

, which is located in such a way that it allows to project the virtual chord of length

l_{c}

backward; the end of calculations must come at a point

i

from which a virtual chord of the same length can still be projected forward.

The basic operation that must first be performed is to establish the numbering of the points delimiting the intervals in which the ends of the virtual chords derived from point

i

are located. For the posterior chord, this interval is marked by points

q_{i}

and

q_{i} + 1

and for the front chord by the points

p_{i} - 1

and

p_{i}

. We determine them in such a way that we successively check the distances between a point

i

and subsequent measurement points, on the contrary and in accordance with the numbering direction. These distances are (for

m = 1, 2, \dots

)

l_{(i - m) / i} = \sqrt{{(x_{i} - x_{i - m})}^{2} + {(y_{i} - y_{i - m})}^{2}}

(3)

l_{i / (i + m)} = \sqrt{{(x_{i} - x_{i + m})}^{2} + {(y_{i} - y_{i + m})}^{2}}

(4)

After each step of the calculations, we check whether the condition

l_{(i - m) / i} \geq l_{c}

for the back chord and

l_{i / (i + m)} \geq l_{c}

for the front chord are satisfied. The first (

i - m

) value for the rear chord that satisfies the applicable condition is denoted

q_{i}

, and satisfying the corresponding condition, the first (

i + m

) value for the front chord as

p_{i}

.

The Angle of Inclination of the Back Chord

Therefore, we know the coordinates of the points

x_{q_{i}}

,

y_{q_{i}}

and

x_{q_{i} + 1}

,

y_{q_{i} + 1}

defining the interval in which the end of the back chord is located, as well as the distances of these points from the point

i

l_{q_{i}} = \sqrt{{(x_{i} - x_{q_{i}})}^{2} + {(y_{i} - y_{q_{i}})}^{2}}

(5)

l_{q_{i} + 1} = \sqrt{{(x_{i} - x_{q_{i} + 1})}^{2} + {(y_{i} - y_{q_{i} + 1})}^{2}}

(6)

These data enable the determination of the coordinates of the other end of the back chord (i.e., the point

B_{i}

). These coordinates result from the following formulas:

x_{B_{i}} = x_{q_{i}} + \frac{l_{q_{i}} - l_{c}}{l_{q_{i}} - l_{q_{i} + 1}} (x_{q_{i} + 1} - x_{q_{i}})

(7)

y_{B_{i}} = y_{q_{i}} + \frac{l_{q_{i}} - l_{c}}{l_{q_{i}} - l_{q_{i} + 1}} (y_{q_{i} + 1} - y_{q_{i}})

(8)

The slope coefficient of the line defining the back chord (passing through the points

B_{i}

and

i

) is

s_{B_{i} / i} = \frac{y_{i} - y_{B_{i}}}{x_{i} - x_{B_{i}}}

(9)

If

s_{B_{i} / i} > 0

, then the chord inclination angle

Θ_{B_{i} / i} \in ⟨ 0, (π / 2) ⟩

is determined by

Θ_{B_{i} / i} = \arctan \frac{y_{i} - y_{B_{i}}}{x_{i} - x_{B_{i}}}

(10)

If

s_{B_{i} / i} < 0

, then the chord inclination angle

Θ_{B_{i} / i} \in ⟨ π / 2, π ⟩

is determined by

Θ_{B_{i} / i} = π + \arctan \frac{y_{i} - y_{B_{i}}}{x_{i} - x_{B_{i}}}

(11)

The Angle of Inclination of the Front Chord

We also know the coordinates of the points

x_{p_{i}}

,

y_{p_{i}}

and,

x_{p_{i} - 1}

,

y_{p_{i} - 1}

defining the interval in which the end of the front chord is located, as well as the distances of this points from the point

i

l_{p_{i}} = \sqrt{{(x_{i} - x_{p_{i}})}^{2} + {(y_{i} - y_{q_{i}})}^{2}}

(12)

l_{p_{i} - 1} = \sqrt{{(x_{i} - x_{p_{i} - 1})}^{2} + {(y_{i} - y_{q_{i} - 1})}^{2}}

(13)

This data allows the determination of the coordinates of the other end of the front chord (i.e., the point

F_{i}

). These coordinates result from the following formulas:

x_{F_{i}} = x_{p_{i} - 1} + \frac{l_{c} - l_{p_{i} - 1}}{l_{p_{i}} - l_{p_{i} - 1}} (x_{p_{i}} - x_{p_{i} - 1})

(14)

y_{F_{i}} = y_{p_{i} - 1} + \frac{l_{c} - l_{p_{i} - 1}}{l_{p_{i}} - l_{p_{i} - 1}} (y_{p_{i}} - y_{p_{i} - 1})

(15)

The slope coefficient of the line delineating the front chord (passing through the points

i

and

F_{i}

) is

s_{i / F_{i}} = \frac{y_{F_{i}} - y_{i}}{x_{F_{i}} - x_{i}}

(16)

If

s_{i / F_{i}} > 0

, then the chord inclination angle

Θ_{i / F_{i}} \in ⟨ 0, π / 2 ⟩

is determined by

Θ_{i / F_{i}} = \arctan \frac{y_{F_{i}} - y_{i}}{x_{F_{i}} - x_{i}}

(17)

If

s_{i / F_{i}} < 0

, then the chord inclination angle

Θ_{i / F_{i}} \in ⟨ π / 2, π ⟩

is determined by

Θ_{i / F_{i}} = π + \arctan \frac{y_{F_{i}} - y_{i}}{x_{F_{i}} - x_{i}}

(18)

The Value of Curvature at Point $i$

The value of curvature at point

i

is determined from

k_{i} = \frac{Θ_{i / F_{i}} - Θ_{B_{i} / i}}{l_{c}}

(19)

A positive value of

k_{i}

in Eq. (19) corresponds to the case of a downward convexity along the length of the curve, and a negative value corresponds to an upward convexity.

Verification of the Proposed Method

In order to verify the method of the moving chord in the operated railway track, measurements of Cartesian coordinates of the track axis were carried out using a satellite antenna in a static manner. These measurements covered two geometric layouts with clearly different values of the radius of the horizontal arc and the nature of the curvature. The distances between the measurement points were approximately 5 m.

The First Geometric Layout

The obtained set of

x_{i}

,

y_{i}

track axis coordinates for the first geometric layout made it possible to draw an appropriate graph, which is shown in Fig. 3. The estimated maximum measurement error was

\pm 25 mm

. The ordinate of the plot at the first measurement point (for

x = 0

) is

y = 92.820 m

.

Fig. 3. Chart of Cartesian coordinates of the axis of the measured railway track for the first geometric layout.

The basis for identifying the track axis in the horizontal plane is the estimation of its horizontal curvature along its length. Therefore, to create the possibility of further analysis, go to the linear system [i.e., determine the distances (the variable

L

expressed in m) of individual measurement points from the selected starting point

i_{0}

]. In the considered case, it was assumed that the starting point (i.e.,

L = 0

) has Cartesian coordinates

x_{0} = 211.644 m

,

y_{0} = 149.530 m

.

The distance between two consecutive measuring points is

Δ l_{i / i + 1} = \sqrt{{(x_{i + 1} - x_{i})}^{2} + {(y_{i + 1} - y_{i})}^{2}}

(20)

The linear coordinate

L_{i}

[i.e., the distance from the point

(x_{0}, y_{0})

] is determined from

L_{i} = \sum_{i = 0}^{n - 1} Δ l_{i / i + 1}

(21)

Eq. (21) uses the coordinates of all

n

points located on straight and arc segments.

Using this aforementioned procedure (after taking the chord length

l_{c} = 20 m

), the horizontal curvature of the geometric layout shown in Fig. 3 was determined. The obtained graph of the curvature

k (L)

is shown in Fig. 4. In this case, the curvature is positive, which corresponds to the presence of a downward convexity in the curvilinear segment.

Fig. 4. Graph of the curvature $k (L)$ of the axis of the measured railway track obtained using the moving chord method for the first geometric layout.

As you can see, the obtained

k (L)

plot consists of two types of elements:

•

Sections oscillating around a horizontal course that decribe a curvature of a fixed value (zero on straight sections of the track and nonzero on circular arcs).

•

Segments oscillating around a linear wave (i.e., straight lines inclined to the

L

axis) that describe the variable curvature appearing on the transition curves.

On the basis of this graph, it is possible to determine the value of the radius of the circular arc and the lenghts of the transition curves, as well as the location of the characteristic points (lying at the connections of straight sections with the transition curves and transition curves with a circular arc).

It is assumed that the curvature of the track in straight sections is equal to zero, and the disturbances occurring there in the curvature diagram are the result of the existing deformations and measurement error. The arithmetic mean

\bar{k_{C A}}

is determined from the selected range of curvature values that undoubtedly belong to the circular arc; its reciprocal is given by the value of the radius

R ≅ \frac{1}{\bar{k_{C A}}}

(22)

The value obtained by Eq. (22) should be properly rounded (to full meters) and then used in the further calculation procedure. In the case under consideration, the mean value

\bar{k_{C A}} = 0.0008332428 rad / m

was obtained, with standard deviation

σ_{k} = 0.0000719681 rad / m

. Using Eq. (22), it corresponds to the value of the radius

R = 1,200.13045 m

.

There are transition curves on both sides of the circular arc. At the end of the left-hand straight section in Fig. 3 is the beginning of the

T C 1

transition curve (i.e.,

B T C 1

point), and at the beginning of the right-hand line is the beginning of the

T C 2

transition curve (i.e.,

B T C 2

point). The ends of both curves (i.e., the

E T C 1

and

E T C 2

points) mark the beginning and the end of the circular arc, respectively.

In order to determine the linear coordinates of the mentioned characteristic points, it is necessary to determine the lines of the least squares describing the regions of the

k (L)

plot with variable curvature values. Fig. 5 shows an enlarged fragment of the curvature plot in Fig. 4 including the

T C 1

transition curve.

For the selected range of curvature values belonging to the

T C 1

transition curve (i.e., connected to the straight line on the left side in Fig. 3), we obtain

k = a_{1} + b_{1} L

(23)

For the beginning of this curve (

B T C 1

point), the value of curvature

k = 0

; hence, its linear coordinate is

L_{B T C 1} = - \frac{a_{1}}{b_{1}}

(24)

And for the end of the curve (

E T C 1

point), the value of the curvature

k = \bar{k_{C A}}

; therefore, its linear coordinate is equal to

L_{E T C 1} = \frac{\bar{k_{C A}} - a_{1}}{b_{1}}

(25)

The length of the transition curve connected with the straight line on the left in Fig. 3 result directly from the values of the determined coordinates

L_{B T C 1}

and

L_{E T C 1}

l_{T C 1} = L_{E T C 1} - L_{B T C 1} = \frac{\bar{k_{C A}}}{b_{1}}

(26)

The further calculation procedure will take place in the

x

,

y

coordinate system; therefore, the appropriate Cartesian coordinates of the designated characteristic points should be determined. For a linear coordinate

L_{B T C 1}

, find such an interval of measurement points

⟨ i, i + 1 ⟩

that

L_{B T C 1} \in ⟨ L_{i}, L_{i + 1} ⟩

. The abscissa

x_{B T C 1}

and the ordinate

y_{B T C 1}

can be now determined from the following formulas:

x_{B T C 1} = x_{i} + \frac{x_{i + 1} - x_{i}}{L_{i + 1} - L_{i}} (L_{B T C 1} - L_{i})

(27)

y_{B T C 1} = y_{i} + \frac{y_{i + 1} - y_{i}}{L_{i + 1} - L_{i}} (L_{B T C 1} - L_{i})

(28)

Similarly, the abscissa

x_{E T C 1}

of the end of the

T C 1

transition curve as well as the corresponding ordinate

y_{E T C 1}

are determined.

The curvature equation for the

T C 1

transition curve shown in Fig. 5 is as follows:

k = 0.00000428498 L - 0.0008361

where

L

is expressed in m and

k

in

rad / m

.

The linear coordinates of the beginning of the curve [based on Eq. (24)] and its end [based on Eq. (25)] are:

L_{B T C 1} = 195.134 m

,

L_{E T C 1} = 389.591 m

. Thus, the length of the transition curve

l_{T C 1} = 194.457 m

determined from Eq. (26).

Among the measurement points, the linear coordinate

L_{B T C 1} \in ⟨ 194.998; 200.024 ⟩ m

. This corresponds to the Cartesian coordinates

x \in ⟨ 400.000; 404.830 ⟩ m

and

y \in ⟨ 200.000; 201.294 ⟩ m

. On this basis—using Eqs. (27) and (28)—the Cartesian coordinates of the

B T C 1

point can be determined as

x_{B T C 1} = 400.131 m

,

y_{B T C 1} = 200.035 m

.

The linear coordinate of the end of the curve

L_{E T C 1} \in ⟨ 385.008; 389.996 ⟩ m

. This corresponds to the Cartesian coordinates

x \in ⟨ 582.190; 586.905 ⟩ m

and

y \in ⟨ 253.747; 255.410 ⟩ m

. On this basis—using Eqs. (27) and (28)—it is possible to determine the Cartesian coordinates of the

E T C 1

point as

x_{E T C 1} = 586.522 m

,

y_{E T C 1} = 255.275 m

.

Fig. 6 shows an enlarged fragment of the curvature plot in Fig. 4 including the

T C 2

transition curve connecting the circular arc with the right-hand straight section in Fig. 3.

Fig. 6. Enlarged fragment of the curvature plot in Fig. 4 including the $T C 2$ transition curve.

For the selected range of curvature values belonging to the

T C 2

transition curve (i.e., connected to the straight section on the right in Fig. 3), we obtain

k = a_{2} + b_{2} L

(29)

For the beginning of this curve (

B T C 2

point), the value of curvature

k = 0

; hence, its linear coordinate is

L_{B T C 2} = - \frac{a_{2}}{b_{2}}

(30)

And for the end of the curve (

E T C 2

point), the value of curvature

k = \bar{k_{C A}}

; therefore, its linear coordinate is equal to

L_{E T C 2} = \frac{\bar{k_{C A}} - a_{2}}{b_{2}}

(31)

The length of the transition curve connected with the straight section on the right in Fig. 3 results directly from the values of the determined coordinates

L_{B T C 2}

and

L_{E T C 2}

l_{T C 2} = L_{B T C 2} - L_{E T C 2} = - \frac{\bar{k_{C A}}}{b_{2}}

(32)

The further calculation procedure will take place in the

x

,

y

orthogonal coordinate system; therefore, the appropriate Cartesian coordinates of the designated characteristic points should be determined. This is done in a similar way to the

T C 1

transition curve.

The determined equation of curvature for the transition curve shown in Fig. 6 is as follows:

k = - 0.00000415252 L + 0.006857

The linear coordinates of the beginning of the curve [based on Eq. (30)] and its end [based on Eq. (31)] are as follows:

L_{B T C 2} = 1,651.309 m

,

L_{E T C 2} = 1,450.649 m

. Thus, the length of the curve [determined from Eq. (32)] is

l_{T C 2} = 200.660 m

.

Among the measurement points, the linear coordinate

L_{B T C 2} \in ⟨ 1,650.007; 1,655.014 ⟩

m. This corresponds to the Cartesian coordinates

x \in ⟨ 1,371.532; 1,372.827 ⟩

m and

y \in ⟨ 1,170.375; 1,175.204 ⟩

m. On this basis, it is possible to determine the Cartesian coordinates of the

B T C 2

point as

x_{B T C 2} = 1,371.869 m

,

y_{B T C 2} = 1,171.631 m

.

The linear coordinate of the end of the curve

L_{E T C 2} \in ⟨ 1,450.033; 1,455.009 ⟩

m. This corresponds to the Cartesian coordinates

x \in ⟨ 1, 314.310; 1, 315.998 ⟩

m and

y \in ⟨ 978.802; 983.508 ⟩

m. Based on this, the Cartesian coordinates of the

E T C 2

point can be determined as

x_{E T C 2} = 1,314.519 m

,

y_{E T C 2} = 979.385 m

.

The list of the determined geometrical parameters of the measured layout is presented in Table 1.

Table 1. List of the determined parameters of the first geometric layout

Layout element	Radius, $R$ (m)	Length, $l_{T C}$ (m)	Coordinate, $L$ (m)	Abscissa, $x$ (m)	Ordinate, $y$ (m)
$T C 1$ transition curve	—	194.457	—	—	—
The beginning of $T C 1$	—	—	195.134	400.131	200.035
The end of $T C 1$	—	—	389.591	586.522	255.275
Circular arc	1,200.130	—	—	—	—
$T C 2$ transition curve	—	200.660	—	—	—
The end of $T C 2$	—	—	1,450.649	1,314.519	979.385
The beginning of $T C 2$	—	—	1,651.309	1,371.869	1,171.631

The numerical data in Table 1 fully identify the measured geometrical layout, which is shown in Fig. 3. They were determined by estmating the horizontal curvature made by the moving chord method. The new method of curvature estimation has proved to be useful here.

The Second Geometric Layout

The obtained set of track axis coordinates for the second geometric layout made it possible to draw a graph

y (x)

, which is shown in Fig. 7. The estimated maximum measurement error was

\pm 20 mm

. The ordinate of the plot at the first measurement point (for

x = 0

) is

y = - 12.504 m

.

Fig. 7. Chart of Cartesian coordinates of the axis of the measured railway track for the second geometric layout.

As before, the estimation of the curvature of the track axis in the horizontal plane was carried out in a linear system. In the considered case, it was assumed that the starting point (i.e.,

L = 0

) has Cartesian coordinates

x_{0} = 6.470 m

,

y_{0} = 11.644 m

. Assuming the chord length

l_{c} = 20 m

, the horizontal curvature of the geometric layout shown in Fig. 7 was determined. The obtained graph of the curvature

k (L)

is shown in Fig. 8. In this case, the curvature is negative, which corresponds to the appearance of a convexity directed upward in the curvilinear segmant.

Fig. 8. Graph of the curvature $k (L)$ of the axis of the measured railway track obtained using the moving chord method for the second geometric layout.

On the basis of this graph, it is possible to determine the value of the radius of the circular arc by Eq. (22). In the case under consideration, the mean value

\bar{k_{C A}} = - 0.0020866355 rad / m

was obtained, with standard deviation

σ_{k} = 0.0000540623 rad / m

. This corresponds to the radius value

R = 479,2403885 m

.

At the end of the left-hand straight section in Fig. 7 is the

B T C 1

point, and at the beginning of the right-hand straight section is the

B T C 2

point. The ends of both transition curves (i.e., the

E T C 1

and

E T C 2

points) mark the beginning and the end of the circular arc, respectively. Fig. 9 shows an enlarged fragment of the curvature plot, in which the region covering the

T C 1

transition curve (with variable curvature values) is described by the least squares line.

Fig. 9. Enlarged fragment of the curvature plot in Fig. 8 including the $T C 1$ transition curve.

For the curvature values belonging to the

T C 1

transition curve (i.e., connected to the straight section on the left in Fig. 7), the following equation was obtained:

k = - 0.0000171975 L + 0.003332

.

The linear coordinates of the beginning of the curve [based on Eq. (24)] and its end [based on Eq. (25)] are:

L_{B T C 1} = 193.726 m

,

L_{E T C 1} = 315.060 m

. Thus, the length of the transition curve

l_{T C 1} = 121.334 m

determined from Eq. (26).

Among the measurement points, the linear coordinate

L_{B T C 1} \in ⟨ 190.015; 195.006 ⟩

m. This corresponds to the Cartesian coordinates

x \in ⟨ 55.646; 56.940 ⟩

m and

y \in ⟨ 195.170; 200.000 ⟩

m. On this basis—using Eqs. (27) and (28)—the Cartesian coordinates of the

B T C 1

point can be determined as

x_{B T C 1} = 56.608 m

,

y_{B T C 1} = 198.761 m

.

The linear coordinate of the end of the curve

L_{E T C 1} \in ⟨ 315.016; 320.000 ⟩

m. This corresponds to the Cartesian coordinates

x \in ⟨ 92.774; 94.684 ⟩

m and

y \in ⟨ 314.438; 319.058 ⟩

m. On this basis, the Cartesian coordinates of the

E T C 1

point are

x_{E T C 1} = 92.791 m

,

y_{E T C 1} = 314.478 m

.

Fig. 10 shows an enlarged fragment of the curvature plot in Fig. 8 including the

T C 2

transition curve connecting the circular arc with the right-hand straight section in Fig. 7.

Fig. 10. Enlarged fragment of the curvature plot in Fig. 8 including the $T C 2$ transition curve.

For the marked range of curvature values belonging to the

T C 2

transition curve, we obtain

k = 0.0000165356 L - 0,013574

.

The linear coordinates of the beginning of the curve [based on Eq. (30)] and its end [based on Eq. (31)] are as follows:

L_{B T C 2} = 820.894 m

,

L_{E T C 2} = 694.703 m

. Thus, the length of the transition curve is

l_{T C 2} = 126.191 m

, determined from Eq. (32).

Among the measurement points, the linear coordinate

L_{B T C 2} \in ⟨ 820.015; 825.012 ⟩ m

. This corresponds to the Cartesian coordinates

x \in ⟨ 463.288; 468.117 ⟩

m and

y \in ⟨ 611.759; 613.054 ⟩

m. On this basis the Cartesian coordinates of the

B T C 2

point are:

x_{B T C 2} = 464.137 m

,

y_{B T C 2} = 611.986 m

.

The linear coordinate of the end of the curve

L_{E T C 2} \in ⟨ 689.993; 695.000 ⟩ m

. This corresponds to the Cartesian coordinates

x \in ⟨ 344.588; 349.178 ⟩ m

and

y \in ⟨ 572.979; 574.963 ⟩ m

. The Cartesian coordinates of the

E T C 2

point are:

x_{E T C 2} = 348.906 m

,

y_{E T C 2} = 574.845 m

.

The list of the determined geometrical parameters of the second measured layout is presented in Table 2.

Table 2. List of the determined parameters of the second geometric layout

Layout element	Radius, $R$ (m)	Length, $l_{T C}$ (m)	Coordinate, $L$ (m)	Abscissa, $x$ (m)	Ordinate, $y$ (m)
$T C 1$ transition curve	—	121.334	—	—	—
The beginning of $T C 1$	—	—	193.726	56.608	198.761
The end of $T C 1$	—	—	315.060	92.791	314.478
Circular arc	479.240	—	—	—	—
$T C 2$ transition curve	—	126.191	—	—	—
The end of $T C 2$	—	—	694.703	348.907	57.845
The beginning of $T C 2$	—	—	820.894	464.137	611.986

The numerical data in Table 2 fully identify the measured geometrical layout, which is shown in Fig. 7.

Summary of Verification Carried Out

The identification of both exemplary geometric layouts was carried out by estimating the horizontal curvature made by the moving chord method. The new method of curvature estimation has proved to be fully useful here. The obtained diagrams of horizontal curvature for the measured railway track (Figs. 4–6 and 8–10) clearly differ from the diagrams for model layouts presented in Koc (2020, 2021a, b). They are oscillatory in nature, which results from the track deformation and measurement error. However, this did not prevent the basic geometrical parameters of the measured layout from being estimated.

It can be assumed that in the corresponding model solution, there is a circular arc with a radius of 1,000 m in the first geometrical layout, and in the second layout, an arc with a radius of 480 m. This means that there is a much lower value of curvature in the first layout than in the second one. Therefore, the graph of curvature in Fig. 4 can be a bit misleading and the oscillation overexposed compared to Fig. 8 because these drawings have a different vertical scale. The estimation of the radius in the second geometry is more precise. For this case, the standard deviation is

σ_{k} = 5,406 E - 5 rad / m

, while for the first geometry it is

σ_{k} = 7,197 E - 5 rad / m

.

The verified method made it possible to determine the location of individual geometric elements along the length of the entire layout (Tables 1 and 2). At the same time, a certain degree of differentiation in the length of the transition curves was demonstrated. In the first geometric layout,

l_{T C 1} = 194,457 m

and

l_{T C 2} = 200,660 m

were obtained, and in the second one,

l_{T C 1} = 121,334 m

and

l_{T C 2} = 126,191 m

. This probably follows from the assumed (simmetrical) model solution; as it should be assumed, there should be, respectively, 200 m and 120 m here.

Conclusions

Identification of track geometric layouts in the horizontal plane is most often based on the estimation of the existing horizontal curvature. The knowledge of this curvature allows to determine the location of straight sections and sections located in the arc, as well as to determine the appropriate geometric parameters. In engineering practice, the identification of the nature of the horizontal curvature in a given geometric layout of the track is most often carried out indirectly—on the basis of measured arrows from the chord stretched along the track. The drawn chart of arrows shows the values of the measured horizontal arrows, but there are no specific directions in which this arrows are measured. The reference line here is constituted by the directions of the chord setting, which are constantly changing.

From the definition of curvature, it is necessary to manipulate the angles of the tangent to the geometric layout. In the real railway track reconstructed on the basis of measurements, it is very difficult to determine the position of straight tangents. However, the situation is completely different with regard to stretched measuring chords, the position of which is always unambiguously determined. Therefore, a concept emerged that when determining the curvature of the track, not to use tangent but corresponding chords. In this way, the idea of a new method of determining the curvature of the track axis was born, which consists in using the difference of the inclination angles of the chord of a fixed length at a given point (the moving chord method).

The verification of the proposed method for determining the curvature on a clearly defined elementary track geometric layout, carried out in Koc (2020, 2021a, b), showed full compliance of the obtained curvature diagrams with the graphs constituting the basis for obtaining the corresponding geometric solution. This applied to both the circular arc sections and the regions of transition curves. This study adresses the issue of using the moving chord method to estimate the horizontal curvature of the axis of the railway track in operation, based on Cartesian coordinates obtained by direct measurements. Measurement data from two geometric layouts of the track with clearly different values of the radius of the horizontal arc and the nature of the curvature present were used. In both cases, the obtained curvature plots differ significantly from the plots for model layouts; they are less regular, oscillating in nature, which results from the track deformation and measurement error. However, this did not prevent the basic geometrical parameters of the measured layout from being estimated.

The adaptation of the moving chord method to the adopted measurement procedure presented in this paper and the way of using the obtained curvature diagram provide the method in question with appropriate application bases. The implementation of the presented procedure should significantly improve the process of identyfying geometric layouts of the track in the horizontal plane.

Notation

The following symbols are used in this paper:

$a_{1}$: intercept in the equation of curvature for the $T C 1$ transition curve;
$a_{2}$: intercept in the equation of curvature for the $T C 2$ transition curve;
$B T C 1$: the beginning of $T C 1$ transition curve;
$B T C 2$: the beginning of $T C 2$ transition curve;
$B_{i}$: point marking the end of the back chord;
$b_{1}$: the slope factor of the line in the equation of curvature for the $T C 1$ transition curve;
$b_{2}$: the slope factor of the line in the equation of curvature for the $T C 2$ transition curve;
$E T C 1$: the end of $T C 1$ transition curve;
$E T C 2$: the end of $T C 2$ transition curve;
$F_{i}$: point marking the end of the front chord;
$i$: number of the measuring point (the point where the curvature is determined);
$i_{0}$: starting point number;
$k$: curvature;
$k_{i}$: value of the determined curvature at point $i$ ;
$\bar{k_{C A}}$: the average value of the determined curvature of the circular arc;
$L$: linear coordinate;
$L_{B T C 1}$: the linear coordinate of the beginning of $T C 1$ transition curve;
$L_{B T C 2}$: the linear coordinate of the beginning of $T C 2$ transition curve;
$L_{E T C 1}$: the linear coordinate of the end of $T C 1$ transition curve;
$L_{E T C 2}$: the linear coordinate of the end of $T C 2$ transition curve;
$L_{i}$: linear coordinate of a point of a given geometric layout;
$L_{i + 1}$: the linear coordinate of the next measurement point after the point $i$ ;
$l_{c}$: chord length used;
$l_{(i - m) / i}$: the current chord length when determining the interval for the end of the back chord;
$l_{i / (i + m)}$: the current chord length when determining the interval for the end of the front chord;
$l_{p_{i}}$: distance of the further border of the interval for the end of the forward chord from point $i$ ;
$l_{p_{i} - 1}$: distance of the nearest border of the interval for the end of the forward chord from point $i$ ;
$l_{q_{i}}$: distance of the further border of the interval for the end of the backward chord from point $i$ ;
$l_{q_{i} + 1}$: distance of the nearest border of the interval for the end of the backward chord from point $i$ ;
$l_{T C 1}$: the length of $T C 1$ transition curve;
$l_{T C 2}$: the length of $T C 2$ transition curve;
$M_{B_{i} / i}$ , $M_{i / F_{i}}$: marked points on the curve;
$m$: a number used to determine the interval for the ends of both chords;
$n$: number of measurement points;
$p_{i}$: the number of the measuring point defining the interval for the end of the forward chord;
$q_{i}$: the number of the measuring point defining the interval for the end of the backward chord;
$R$: radius of a circular arc;
$s_{B_{i} / i}$: the slope factor of the line defining the back chord;
$s_{i / F_{i}}$: the slope factor of the line defining the front chord;
$T C 1$: transition curve connected to a straight line on the left side of the geometric layout;
$T C 2$: transition curve connected to a straight line on the right side of the geometric layout;
$x$: abscissa of a point of a given geometric layout in the rectangular coordinate system;
$x_{0}$: abscissa of the starting point in the rectangular coordinate system;
$x_{B_{i}}$: abscissa of the point marking the end of the back chord;
$x_{B T C 1}$: abscissa of the beginning of $T C 1$ transition curve;
$x_{B T C 2}$: abscissa of the beginning of $T C 2$ transition curve;
$x_{E T C 1}$: abscissa of the end of $T C 1$ transition curve;
$x_{E T C 2}$: abscissa of the end of $T C 2$ transition curve;
$x_{F_{i}}$: abscissa of the point marking the end of the front chord;
$x_{i}$: abscissa of the point $i$ in the rectangular coordinate system;
$x_{i + 1}$: abscissa of the next point after the point $i$ in the rectangular coordinate system;
$x_{i - m}$: abscissa of the point preceding point $i$ in the rectangular coordinate system;
$x_{i + m}$: abscissa of a point after point $i$ in the rectangular coordinate system;
$x_{p_{i}}$: abscissa of the measurement point defining the boundary of the interval for the end of the forward chord further away from the point $i$ ;
$x_{p_{i} - 1}$: abscissa of the measurement point defining the boundary of the interval for the end of the forward chord, less distant from the point $i$ ;
$x_{q_{i}}$: abscissa of the measurement point defining the boundary of the interval for the end of the backward chord further away from the point $i$ ;
$x_{q_{i} + 1}$: abscissa of the measurement point defining the boundary of the interval for the end of the backward chord, less distant from the point $i$ ;
$y$: ordinate of a point of a given geometric layout in the rectangular coordinate system;
$y_{0}$: ordinate of the starting point in the rectangular coordinate system;
$y_{B T C 1}$: ordinate of the beginning of $T C 1$ transition curve;
$y_{B T C 2}$: ordinate of the beginning of $T C 2$ transition curve;
$y_{E T C 1}$: ordinate of the end of $T C 1$ transition curve;
$y_{E T C 2}$: ordinate of the end of $T C 2$ transition curve;
$y_{B_{i}}$: ordinate of the point marking the end of the back chord;
$y_{F_{i}}$: ordinate of the point marking the end of the front chord;
$y_{i}$: ordinate of the point $i$ in the rectangular coordinate system;
$y_{i + 1}$: ordinate of the next point after the point $i$ in the rectangular coordinate system;
$y_{i - m}$: ordinate of the point preceding point $i$ in the rectangular coordinate system;
$y_{i + m}$: ordinate of a point after point $i$ in the rectangular coordinate system;
$y_{p_{i}}$: ordinate of the measurement point defining the boundary of the interval for the end of the forward chord further away from the point $i$ ;
$y_{p_{i} - 1}$: ordinate of the measurement point defining the boundary of the interval for the end of the forward chord, less distant from the point $i$ ;
$y_{q_{i}}$: ordinate of the measurement point defining the boundary of the interval for the end of the backward chord further away from the point $i$ ;
$y_{q_{i}}$: ordinate of the measurement point defining the boundary of the interval for the end of the backward chord, less distant from the point $i$ ;
$Δ l_{i / i + 1}$: distance between two consecutive measuring points;
$Δ Θ$: angle between two tangents to the curve;
$Δ Θ_{i}$: difference in the angles of inclination of the chords converging at point $i$ ;
$Θ$: slope angle of the tangent to the curve;
$Θ_{B_{i} / i}$: angle of inclination of the back chord; and
$Θ_{i / F_{i}}$: angle of inclination of the front chord.

Data Availability Statement

Cartesian coordinate values of the considered geometric layouts of the track that support the findings of this study are available from the author upon reasonable request.

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Information & Authors

Information

Published In

Journal of Surveying Engineering

Volume 148 • Issue 4 • November 2022

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Oct 8, 2021

Accepted: Apr 8, 2022

Published online: Jun 20, 2022

Published in print: Nov 1, 2022

Discussion open until: Nov 20, 2022

Authors

Affiliations

Władysław Koc, Ph.D., D.Sc. https://orcid.org/0000-0002-4619-7852 [email protected]

CEng.

Professor, Dept. of Civil and Environmental Engineering, Gdańsk Univ. of Technology, Narutowicza 11/12, Gdańsk 80-233, Poland. ORCID: https://orcid.org/0000-0002-4619-7852. Email: [email protected]

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Wladyslaw Koc, The Procedure of Identifying the Geometrical Layout of an Exploited Railway Route Based on the Determined Curvature of the Track Axis, Sensors, 10.3390/s23010274, 23, 1, (274), (2022).
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Estimation of the Horizontal Curvature of the Railway Track Axis with the Use of a Moving Chord Based on Geodetic Measurements

Abstract

Introduction

The Moving Chord Method

Adaptation of the Moving Chord Method to the Adopted Measurement Procedure

The Angle of Inclination of the Back Chord

The Angle of Inclination of the Front Chord

The Value of Curvature at Point $i$

Verification of the Proposed Method

The First Geometric Layout

The Second Geometric Layout

Summary of Verification Carried Out

Conclusions

Notation

Data Availability Statement

References

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Abstract

Introduction

The Moving Chord Method

Adaptation of the Moving Chord Method to the Adopted Measurement Procedure

The Angle of Inclination of the Back Chord

The Angle of Inclination of the Front Chord

The Value of Curvature at Point i

Verification of the Proposed Method

The First Geometric Layout

The Second Geometric Layout

Summary of Verification Carried Out

Conclusions

Notation

Data Availability Statement

References

Information

Published In

Copyright

History

Authors

Affiliations

Metrics

Citations

Download citation

Cited by

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The Value of Curvature at Point $i$