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European Journal of Applied Sciences – Vol. 11, No. 3

Publication Date: June 25, 2023

DOI:10.14738/aivp.113.14933

Koc, W. (2023). Verification of the Theoretical Basis for Determining the Curvature of the Track Axis Using the Moving Chord

Method. European Journal of Applied Sciences, Vol - 11(3). 577-597.

Services for Science and Education – United Kingdom

Verification of the Theoretical Basis for Determining the

Curvature of the Track Axis Using the Moving Chord Method

Wladyslaw Koc

Faculty of Civil and Environmental Engineering,

Gdansk University of Technology, Gdansk, Poland

ABSTRACT

The paper indicates the possibility of a new approach to the problem of identifying

the measured railway route. The currently used methods of identifying the track

axis have been characterized, which consist in generating an optimized geometrical

layout in the tested area by minimizing the occurring deviations of this layout from

the measurement points. The proposed new approach uses the obtained

measurement data to determine the existing curvature of the geometric layout. This

paper presents a short review of the current activities related to the determination

of the curvature of model geometrical layouts and the operated railway track using

a new method of determining the curvature, known as the moving chord method.

The demonstrated high efficiency of this method should not, however, obscure the

fact that the assumptions adopted in it apply – generally speaking – only to a certain,

limited range of geometric parameters. The presented analysis showed that the

adopted assumptions could be applied to the railway track, because it uses large

radii of circular arcs (of the order of several hundred or several thousand meters)

and transition curves with a very mild change in curvature. However, the condition

must be met that in the calculation procedure appropriate lengths of the moving

chord will be used, increasing as the radius of the circular arc increases. Using the

test geometric layout, it was clearly demonstrated that the discretization of data

used in the moving chord method does not affect the accuracy of the obtained

curvature values at individual measurement points.

Keywords: Railway track, Curvature of the track axis, The moving chord method,

Verification of theoretical foundations.

INTRODUCTION

During many years of operation, the shape of the existing railway lines is gradually changing

and begins to deviate from its originally planned geometric layout. Deviations of the track axis

in the horizontal plane may turn out to be particularly unfavorable in terms of passenger

comfort and operational safety. In order to eliminate them, it is necessary to determine the

current geometric shape of the railway route. This is done on the basis of the designated

Cartesian coordinates of the track axis points in the appropriate national spatial reference

system. In Poland, for plane coordinates, the PL-2000 system [1] is in force, created on the basis

of a mathematically unambiguous assignment of points on the GRS 80 reference ellipsoid [2] to

appropriate points on the plane according to the Gauss-Krüger mapping theory [3].

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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 3, June-2023

The currently used measurement methods are similar in different railway administrations [4–

12]. In classical geodetic techniques, distances and angles are measured using total stations in

relation to the spatial geodetic network. Further possibilities are provided by stationary

satellite measurements based on the Global Navigation Satellite System (GNSS) technique. This

solution does not require the use of a network of the railway ground network points – the

measurement systems use the so-called active geodetic networks (e.g., networks of reference

stations operating in Real Time Network (RTK) mode [13–15]). Mobile satellite measurement

methods are also being introduced, in which, apart from GNSS receivers, Inertial Navigation

System (INS) devices [16] are used as supporting devices, and vision methods such as

Terrestrial Laser Scanning (TLS) [17]. Research is underway on the possibility of using systems

consisting of satellite receivers mounted on various types of vehicles [19–23].

Determining the coordinates of the track axis makes it possible to visualize a given railway

route, giving a general orientation about its location. However, since the purpose of the

measurements is to determine the geometric parameters (i.e., identification) of the measured

route, appropriate calculation algorithms should be used. In the case under consideration (i.e.,

considering the horizontal plane), the analysis is based on the determined values of the plane

eastern coordinates Yi and the northern Xi coordinates of a given measurement point in the PL- 2000 system. However, the proper identification of the track axis is provided by the appropriate

graphs referring to the length parameter L. Therefore, in order to create the possibility of

further analysis, it is necessary to go to the linear system, i.e. on the basis of Yi and Xi

coordinates, determine the distance (Li variable) of individual measurement points from the

selected starting point O (Y0, X0) (i.e. point i0).

The approach used so far to the problem of identifying the measured route is about generating

an optimized geometrical layout in the studied area by minimizing the deviations of this layout

from the measurement points, while meeting the appropriate maintenance and operational

requirements. The traditional solution to a design problem is largely based on human

experience and requires overcoming many difficulties [14,24–28]. To achieve this, many works

attempt to automate the discussed process. There are two variants of the procedure: geometric

identification and alignment optimization. These variants may be implemented separately or

simultaneously.

In terms of geometric identification, an elementary approach is possible, such as the use of

discrete approximation methods using splines [29–31] to determine the geometric extent of

the various system components. There are also a whole range of other methods that have been

presented in works [32–39].

When it comes to alignment optimization, the research methods used vary greatly. Many of

them concern transition curves. In paper [40], railway horizontal curves were fitted using the

creeping Nelder-Mead simplex method. In [41] horizontal railway curves were reproduced

using the Lagrange multiplier method. Papers [42,43] present a general method of curve fitting

based on the theory of maximum likelihood estimation and fitting railway transition curves

using the Levenberg-Marquardt non-linear optimization algorithm. In paper [44], particle

swarm optimization was proposed in order to reconstruct existing railway lines. Other applied

alignment optimization methods are presented in [45–51].

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Koc, W. (2023). Verification of the Theoretical Basis for Determining the Curvature of the Track Axis Using the Moving Chord Method. European

Journal of Applied Sciences, Vol - 11(3). 577-597.

URL: http://dx.doi.org/10.14738/aivp.113.14933

In the works mentioned above, geometric identification and alignment optimization were

studied separately. In fact (in engineering practice) both of these variants are interrelated. This

means that the geometric identification results are used as input for automatic alignment

optimization, which in turn – doing the opposite – also changes the identification results.

Therefore, in works [13,36,52,53], several methods of combining these variants in an iterative

process of track axis reconstruction were proposed. Most of these methods first identify and

reconstruct straight line segments (as tangents to curves) and then fit curved line segments.

The possibility of a complete departure from the presented methods of fitting the hypothetical

model layout to the points of the track axis, determined by direct measurements, makes it

possible to use the obtained measurement data to determine the existing curvature of the

geometric layout. For this purpose, an appropriate method for determining the curvature in the

railway track had to be developed. Papers [54–56] present relevant analyzes relating to the

proposed new method of determining the curvature of the track axis, referred to as the "moving

chord method". They concerned the application of this method for model geometric layouts

(described with mathematical equations). Papers [57–60] address the issue of its use for the

estimation of the horizontal curvature of the axis of the operated railway track on the basis of

Cartesian coordinates obtained by direct measurements.

The demonstrated high efficiency of the moving chord method in determining the curvature of

the railway track should not obscure the fact that it is based on certain assumptions which, in

general, apply only to a certain, limited range of geometrical parameters. This paper tries to

clarify these issues by verifying the theoretical basis of the proposed method for determining

the curvature of the track axis.

DETERMINATION OF THE CURVATURE OF THE RAILWAY TRACK AXIS BY THE MOVING

CHORD METHOD

General Assumptions

From the definition of curvature, it is necessary to operate with the angles of inclination of the

tangent to the geometric layout. The measure of curvature κ of road route is the ratio of the

angle ΔΘ, by which the direction of the longitudinal axis of the vehicle changes after passing a

certain arc, to the length of this arc Δl (Fig. 1). The curvature of the curve K at point M is the

limit approached by the ratio of the acute angle ΔΘ included between the tangents to the curve

K at points M and M1 to the length of the arc Δl, when point M1 follows the curve K to point M.

0

lim

l

d

l dl



 →

 

= =



(1)