Measurement of Railway Track Geometry: A State-of-the-Art Review

Abstract

The worldwide increase in frequency of traffic for passenger trains and the rise of freight trains over the recent years necessitate the more intense deployment of track monitoring and rail inspection procedures. The wheel-rail contact forces, induced by the static axle loads of the vehicle and the dynamic effects of ground-borne vibration coming from the track superstructure, have been a significant factor contributing to the degradation of the railway track system. Measurements of track irregularities have been applied since the early days of railway engineering to reveal the current condition and quality of railway lines. Track geometry is a term used to collectively refer to the measurable parameters including the faults of railway tracks and rails. This paper is aiming to review the characteristics of compact inertial measurement systems (IMUs), their components, installation, the basic measures of the quality of the track using motion sensors, like accelerometers, gyroscopes and other sensing devices mounted on different places of the vehicle. Additionally, the paper briefly discusses the fundamentals of inertial navigation, the kinematics of the translational and rotational train motions to obtain orientation, velocity and position information.

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Cite this article as: Farkas, A. (2019) "Measurement of Railway Track Geometry: A State-of-the-Art Review", Periodica Polytechnica Transportation
Engineering. https://doi.org/10.3311/PPtr.14145
https://doi.org/10.3311/PPtr.14145
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Periodica Polytechnica Transportation Engineering
Measurement of Railway Track Geometry:
A State-of-the-Art Review
András Farkas1*
1InstituteofEntrepreneurshipDevelopment,ÓbudaUniversity,1084Budapest,Tavaszmezőu.15-17,Hungary
* Corresponding author, e-mail: farkas.andras@kgk.uni-obuda.hu
Received: 04 April 2019, Accepted: 13 May 2019, Published online: 22 October 2019
Abstract
Theworldwideincreaseinfrequencyoftracforpassengertrainsandtheriseoffreighttrainsovertherecentyearsnecessitatethe
more intense deployment of track monitoring and rail inspection procedures. The wheel-rail contact forces, induced by the static
axleloads of the vehicleand the dynamic eectsof ground-borne vibration comingfrom the track superstructure,have been a
signicantfactorcontributingtothedegradationoftherailwaytracksystem.Measurementsoftrackirregularitieshavebeenapplied
since the early days of railway engineering to reveal the current condition and quality of railway lines. Track geometry is a term used
to collectively refer to the measurable parameters including the faults of railway tracks and rails. This paper is aiming to review the
characteristics of compact inertial measurement systems (IMUs), their components, installation, the basic measures of the quality of
thetrackusingmotionsensors,likeaccelerometers,gyroscopesandothersensingdevicesmountedondierentplacesofthevehicle.
Additionally,thepaperbrieydiscussesthefundamentalsofinertialnavigation,thekinematicsofthetranslationalandrotationaltrain
motions to obtain orientation, velocity and position information.
Keywords
inertial measurement sensors, train motion kinematics, railway track geometry condition monitoring
1 Introduction
In recent years, considerable research has been done in
many countries related to condition monitoring of railway
track geometry and investigating the performance of rid-
ing quality. Among others, in the paper of Weston et al.
(2015), authors presented an excellent overview about the
current developments of such systems installed on board
of railway coaches running in regular operation all over
the country in Great Britain. A detailed reasearch was
reported from Japan about the use of a vertical displace-
ment measuring system mounted on the bogie and supp-
lemented by an additional sensor from the bogie to the
axlebox to monitor rail geometry (Yazawa and Takeshita,
2002). Jochim and Lademann (2009) in Germany, gave
an overall description about the track geometry characte-
ristics. Ackroyd et al. (2002) reported high accelerations
measured both on the bogie and the body using a condition
sensing system instrumented on Acela trains in the USA.
A stochastic simulation was done in Switzerland, on the
Gotthard line, to investigate the wear propagation process
for both the rails and the wheels (Szabó and Zobory, 1998).
Here, the authors gave estimates of the magnitude and its
distribution of the right/left rails’ wear and wheels’ wear
along the track for straight, curved and transition sec-
tions. Zobory and Péter (1987) proposed a time- and state
dependent, seven degree-of-freedom model with its cor-
responding system of motion equations including several
non-linearities, and examined the behavior of a braking
vehicle on the dynamic effects of track unevennesses by
assuming stationary, stochastic track records. A compre-
hensive overview about the whole topic can be found in
the research report of Nielsen et al. (2013).
The objective of this paper is twofold. One goal is to
discuss the operative aspects of measurement technolo-
gies of railway track geometry by giving an up-to-date
overview based on a state-of-the-art literature review.
Another goal is to point and stimulate towards directions
for further research in this area having utmost impor-
tance for practitioners as well. The organization of the
survey is as follows. The main characteristics of the track
measurement systems will be presented in Section 2.

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Period. Polytech. Transp. Eng.
In Subsection 2.1, the track geometry faults are summa-
rized; in Subsection 2.2, the measuring systems and their
elements will be discussed with special focus on the Inertial
Measurement Units (IMUs); in Subsection 2.3, some types
of these devices are described; in Subsection 2.4, the place-
ment of these intruments on the measuring vehicle will
be analyzed; in Subsection 2.5, the unavoidable errors
emerging in the course of signal processing are discussed;
in Subsection 2.6, the use of the two basic procedures of
track measurement, the continuous in-service vehicles
and the dedicated measurement coaches, are compared; in
Subsection 2.7, the characteristics of accelerometers and
gyroscopes will be given. In Section 3, the kinematics of the
inertial navigation systems are briey summarized; and, in
Section 4, the train motion depending on the track geomet-
ric (and geographic) position will be described.
2 Characteristics of railway track measurements
This section provides the reader a concise overview of the
characteristics of railway track measurement systems.
2.1 Track geometry faults
Railways require to maintain the highest ride comfort and
safety standards for the status of the trains/carriages and
facilitate maintenance planning of the tracks what is called
on-board condition monitoring and control. Integrated track
control and inspection encompass a variety of parameters,
e.g. longitudinal level, track gauge, cant, alignment, twist
- related to the track geometry; then corrugation, squat,
wheel slipping points, dipped joints/welds, cyclic top derail-
ments - grouped into the set of the short rail surface defects;
and, measurements of dynamics and driving comfort para-
meters. As a rule of thumb, if the geometry of the track is
known for each 0.2 m section and a fault can be localized
within 3 m in an absolute position, then such a monitoring
system is well suited to the needs of railway companies in
order to preserve the condition of tracks.
After the initial construction of the track, the geome-
try of the track begins to deviate from its original, called
design geometry. This process is known as track degrada-
tion. Degradation can occur for a number of reasons (Yeo,
2017): Firstly, the vertical geometry of the track tends to
degrade over time (which was originally almost at along
a ~70 m length of track). Secondly, the track stiffness,
which can strongly affect the track’s vertical prole, and
its extreme values can lead to track failures. Deection
typically varies from a fraction of a millimeter to tens
of millimeters in extreme cases. The static axle load of
the train is the load on the wheel pairs as when the train
is stationary, whereas the dynamic axle load of the train
increases as the speed of the train set increases.
Geometric track defects are termed track irregulari-
ties (usually measured in displacement) which originates
in the contact force caused by a continuous interaction
between the wheel and the rail and the sleeper’s deterio-
ration. This is the source of the dynamic excitation which
causes oscillations and vibrations for both the vehicle and
the track. Track irregularities are usually grouped into two
categories: λ > 1 meter’s are said the long (including mid)
wavelength irregularities and λ < 1 meter’s are said the
short wavelength irregularities. By Xing et al. (2015), long
wavelength track irregularities affect railway operation
strongly and impact vehicle stability and comfort. They
are emerging from roadbed deformations due to soil com-
paction of multiple passages of heavy train sets, but they
produce no shocks. The vehicle responses are typically
low-frequency oscillations (ZG Optique, 2016).
When the train passes through short track defects hav-
ing poo r nishing qua lit y, axl e-b ox accele rat ions may pro -
duce 100 g or even more (Molodova et al., 2011). However,
earlier investigations done by Weston et al. (2007) found
that the long wavelength irregularities produce much less
than 1 g axlebox accelerations. Therefore, they claimed
to install sensors on axleboxes which have a low limit on
lower bandwidth. To get usable long wavelength outputs
they should have high linearity, and very low noise as well.
2.2 Measuring devices of track irregularities
The typical form of an Inertial Measurement Unit (IMU)
contains three accelerometers and three gyroscopes and
additionally, depending on the heading requirement, three
magnetometers. An IMU should be mounted into a three
dimensional orthogonal system. Such a conguration (a
full six degree-of-freedom IMU), for a vehicle moving
along the track, makes it possible to provide 3-D position
and velocity information by measuring linear acceleration
and rotation (gimbal rates) for the vehicle in the directions
of each axis. Through the time of the measurements, there
is a high sampling rate, usually at frequencies that are
even higher than 100 Hz. The accelerometers are mounted
with their sensitive axes perpendicular to one another, i.e.
mutually perpendicular. Gyros have specic objectives. A
gyroscope serves as a proper tool for providing accurate
information on the attitude and the heading of the body
and/or the bogie with respect to a predetermined reference
frame (see it in Section 3). Utilizing information that are

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Period. Polytech. Transp. Eng.
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related to linear accelerations and rotation rates, velocity,
and hence position of the vehicle can also be derived.
Nowadays M EMS, Micro-Electro-Mechanical-Systems
have become widely available due to their small sizes and
low costs. They contain inertial measurement sensors (3-D
accelerometers and 3-D gyroscopes) and have a very high
sensitivity. They resist undesired vibration and shock, but
in some cases, they suffer from limited accuracy. MEMS
sensors can be placed into well sheltered and low weight
boxes and then they are assembled onto the axlebox, or the
bogie, or the body of the vehicle.
Such MEMS inertial sensing devices are most often used
in compact Strapdown Inertial Navigation Systems (SINS)
equipped with a receiver used for accommodating remote
sensing signals emitted by satellite navigation systems. A
satellite navigation system also has MEMS accelerometers
and ber-optic gyros. A SINS is usually mounted on the
vehicle frame, or eventually, on the bottom of the carriage.
This complex system supplies fundamental information for
the navigation of the vehicle, i.e. position, velocity, attitude
(heading, bank angle, slope, rotation rates) and relative dis-
placements of the vector of acceleration and the angular
rates with respect to the SINS frame (ZG Optique, 2016).
SINS calculates the railroad track momentum gradient, the
superelevation and crosslevel (positive and negative cant)
and expressed them by numerical terms and optical system
data (ZG Optique, 2016).
For example, a contactless optical system, developed
by the Swiss ZG Optique SA for determining the attitude,
consists of three radiating sources and three receivers (ZG
Optique, 2016). They are installed on the bottom of the
railway carriage body and on the bogie frame. This man-
ner, an optical system without a direct contact is estab-
lished to measure mutual orientation of the vehicle (ZG
Optique, 2016). A satellite receiver is also assembled on
the top of the car body and IMUs are mounted on the bot-
tom of the body and on the carriage frame (ZG Optique,
2016). Nevertheless, to set up such an optical congura-
tion, which, as its usual setting, includes a laser scanner to
monitor and determine the proles of each rail, is expen-
sive, moreover, it is very difcult to keep this optical sen-
sor clear in the dirty operational environment (Xing et al.,
2015). Yet this contactless technique is popular for rail-
ways, because it is effective and quick and is capable of
performing long rail defects with lessened wear.
As a direct monitoring of the track defects a video sys-
tem can be used with digitally sensing video cameras pla-
ced on the body frame and located inside the wheel pairs.
Per iodic track faults signi cantly affe ct the dynam ic load s
coming from the primary suspension of the vehicle and
caused by the wheel-rail interaction. This is true for both
spring-borne and non-spring-borne loading. Such opti-
cal systems can provide very accurate measurements (to
within 0.1 mm) of the track geometry, but could be much
more expensive to maintain.
An odometer is a tool to measure the distance taken by
the vehicle. This device is installed on one of the wheel
pairs. Typically a tachometer measures the rotation speed
of a non-driven axle belonging to a wheel set usually in
revolution/minute. British railway engineers, Weston et
al. (2015:p.1064), have asserted that “The speed of rota-
tion of an unpowered wheel and the odometer is the inte-
gration of the tachometer with respect to time multiplied
by the wheel circumference to convert rotations per unit
time into distance.”. Distance measurement of a traveling
vehicle is done in a direct way applying distance sensors.
It should be noted that it is hard to measure the move-
ments, say, only into the vertical direction, since the ends
of the sensor will surely have certain lateral and longi-
tudinal displacements as well. In this respect, Weston et
al. (2015:p.1066) reported that “The capacitative or the
inductive displacement sensors are more robust but can
only measure small displacements.”.
A complete process control system (PCS) is com-
prised of electronic, optical and computerized units. It is
displayed in Fig. 1 (where their typical components are
described in the caption).
In recent decades, railways have begun to install an
antenna on the roof of the vehicle as a key element of
Fig. 1 “A process cont rol system (PCS) com prises an un interr uptible power
supply (UPS), signal synchronization and transmission board, digital
signal processor (DSP), and hard disk (HD) packaged in a metal casing,
and has USB, Ethernet and other interfaces.” (ZG Optique, 2016:p.2)

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Period. Polytech. Transp. Eng.
their Global Navigation Satellite System (GNSS) in order
to get more accurate information to determine position.
Indeed, measurements have proven that by combining a
tachometer outcome with GPS/GNSS emitted data offers a
remarkable positive change in exploring unbiased position
of track condition data. To obtain absolute position infor-
mation, even using a Kalman ltering technique, however,
remains henceforward a crucial data acquisition task for
railways (Mirabadi et al., 2003).
2.3 Types of measuring devices
As we discussed in the previous section, an IMU is a
frequently used measurement tool of railway compa-
nies which consists of a triaxial accelerometer and a tri-
axial gyroscope. In accelerometers, usually a damped
silicon mass is suspended by a spring where the inow-
ing kinetic energy is converted to an electrical signal by
either a piezoelectric, or a piezoresistive, or a capacitive
part. Accelerometers of piezoelectric mode are composed
of ceramics or are being made of single crystals, while
accelerometers having a capacitive form utilize mainly a
micro-machined sensing element made of silicon. Latter
conguration far surpasses the performance of the other
accelerometers within the range of low frequencies.
A gyroscope is a rotation sensing device which stands
for helping to specify orientation based on Earth’s gravity.
One typical type of gyroscope is made by suspending a
relatively massive rotor assembled onto a spinning axis in
the center of a larger and more stable wheel inside three
rings called gimbals. Gyros have especially high stability
to balancing themselves when the vehicle speed is very
high. An other benecial property of gyroscopes is that
the direction of the high speed rotation axis of their central
rotor remains invariant at high speeds.
The structure of MEMS sensors is extremely simple.
They consist of a cantilever beam supplemented by a proof
mass. It follows from this fact that any effort has been made
towards reducing their purchasing costs has implied less
accuracy in the performance of an IMU as a whole. Still,
the MEMS’s position and orientation estimates are accu-
rate on a short time scale, but suffer from errors (drift, ran-
dom walk) over longer time scales (Mirabadi et al., 2003).
These fundamental facts related to the limited capabilities
of MEMS sensors are often forgotten by railway staffs who
are responsible for track monitoring and inspection.
Some railways employ optical measurement systems,
using either laser scanners or a combination of lasers and
video cameras. Basically these systems are very accurate
in measuring track geometry, but too expensive to main-
tain them in an in-service railway operation because they
require frequent cleaning.
It should also be remarked that even if the orientation
estimate has insignicant errors only, yet it may experi-
ence extraordinarily large bias in the acceleration values.
This fact will magnify errors in the velocity and position
estimation process resulting very poor outputs. From an
engineering point of view, in the lack of use of more effec-
tive rate gyros (FOG or Ring-Laser) or not to adding an
ouside satellite based navigation system, to achieve accu-
rate dead-reckoning is generally impossible. As shown by
a profound experimental research at CH Robotics (2012)
“An angle error of even one degree will cause the esti-
mated velocity to be off by 1.7 m/s after 10 seconds, and
the position to be off by 17.1 meters in the same amount
of time. After a single minute, one degree of angle error
will cause the position estimate to be off by almost a full
kilometer. After ten minutes, the position will be off by 62
kilometers.” (CH Robotics, 2012:p.6).
2.4 Placement of measuring devices on the vehicle
The choice for the right places on the vehicle and the
appropriate type of the measuring devices are a matter of
utmost impor tance. To nd a satisfactory solution to these
problems is dependent on what kind of on-board condition
monitoring and riding quality control system are desired
to implement taking into consideration the type of track
faults intending to be detected as well as the operational
issues of maintanability. Yazawa and Takeshita (2002)
reported that accelerometers mounted on the axlebox are
regularly used on Japanese railway lines on high-speed
trains. However, such an installation of acceleration sen-
sors appears to be infavorable due to the difculties of
their maintenance tasks. Weston et al. (2015) described
their experiences with sensors installed on axleboxes. For
example, under very harsh weather conditions, they can be
frozen, or exposed to a windstorm with a force over 100
mph, or may warm up too strongly from the bearing.
Many authors in this eld of interest believe axlebox
accelerometers are the appropriate manner in helping to
describe the vertical rail prole of track geometry cor-
rectly. Yet, even if we gain a perfect identication of the
proles of the rails, a double integration to derive disp-
lacement is required, which induces some inherent dif-
culties that will be discussed in Subsection 2.5. A slightly
more robust solution is to lay the inertial sensors on the
bogie frame and put optical sensors looking at the rails

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Period. Polytech. Transp. Eng.
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(Weston et al., 2015). Laterally mounted axle-box accele-
rometers can be used to measure short wavelength lateral
defects such as poor rail alignment at switches and crossi-
ngs (Ward et al., 2011).
Roberts et al. (2004) have written how they started to
experiment with the placement of sensors onto different
locations on the vehicle, in the mid-2000s. They raised
a seemingly strange question as to whether the explicit
exploration of geometry the track is explicitly needed at
all. In this context, Roberts et al. (2004) have asserted
that “If the bogie follows a smooth trajectory down the
track, then the track geometry must be reasonably good.
Conversely, poor track geometry almost always leads
to something detectable in the trajectory of the bogie.”.
Some experiments have been carried out to investigate
the motion of the bogie to ignore the primary suspension
(Weston et al., 2007).
Mounting a single IMU directly onto the bogie frame
results in a much simpler installation process. A bogie-
mounted IMU gives acceptable measurement results, par-
ticularly when the exact geometry does not need to be
reconstructed (Weston et al., 2007). It is suitable for mea-
surement of mid-wavelength (1-3 m) geometry features.
Instrumentation mounted in this way is better suited to
in-service installation, as it is less obtrusive and easier to
remove and re-attach when the bogie requires maintenance.
In the sequel, we will take over several ndings have
been reported in the outstanding work of Yeo (2017).
Using a bogie-mounted IMU, the very short wavel-
ength geometry features are ltered out by the primary
suspension. Some short wavelength defects are ltered out
by the dynamic effects of the bogie, which could result in
some loss of delity. The point of measurement is before
the secondary airbag suspension between the bogie and
the coach of the train, allowing much shorter wavelength
measurements to be recorded than if the IMU were moun-
ted on board the coach. The ride quality, however, is best
measured from on board the car itself, as this is where pas-
sengers would be sitting or standing (Yeo, 2017).
The IMU itself is usually mounted on top of the bogie,
using four tapped holes provided for the addition of extra
equipment to the bogie (Yeo, 2017). The positioning of the
IMU on the bogie is displayed in Fig. 2.
In order to reconstruct the geometry of the track, a bet-
ter approach when using inertial sensors, seems to be to
install the instrumentation on the bogie frame. It may come
up also the top of the axlebox of the vehicle. These place-
ments move the measurement point to below the secondary
suspension. Often the primary suspension is very stiff,
meaning that measurements taken on the bogie are close
to those experienced at axlebox level. Instrumentation of
the axlebox appears to be the best way of detecting very
short wavelength features such as corrugation and squats.
Bogie instrumentation is appropriate for providing measu-
rements of mid-wavelength geometry features, as the pri-
mary suspension and the bogie itself provide a natural low-
pass ltering of the accelerations experienced (Yeo, 2017).
If linear acceleration is used to measure vertical rail
prole, then, a sensor on the bogie over the axlebox
should be placed, and the vertically sensed displacement
to the axlebox from the accelerometer. This combination
is an example of censor fusion. King (2004) argued that
“Moving the accelerometer from the axlebox to the bogie
isolates the accelerometer from the worst impulsive accel-
erations, and reduces the range with which the accelerom-
eter has to contend. However, the displacement transducer
is then vulnerable.” (King, 2004:p.18).
Sensor combination contributes to the improvement of
measurement accuracy considerably. As an other example
for censor fusion, Trehag et al. (2010) reported that GNSS
data combined with yaw rate sensor and tachometer data
to measure curvature produces a completely bias-free out-
put, where curvature is computed as the ratio between the
yaw rate and the speed of the vehicle. Nevertheless, the
speed of the train impacts the result, since, at low speed,
the yaw rate gyro is very sensitive to drifting offset phe-
nomenon. which affects the curvature estimate. A curva-
ture information for the long-run can be obtained from the
GNSS data that may modify the offset error at low speeds
provided that a correct mathematical model was setting up
by the analyst. These investigations require to apply both
linear and nonlinear ltering (Trehag et al., 2010).
Fig. 2 IMU positioning on the bogie (Yeo, 2017:p.48)

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Period. Polytech. Transp. Eng.
Similarly to track geometry monitoring, the integration
of axlebox, bogie- and body- mounted measuring devices
can help to reveal the current state of the structural com-
ponents of the bogie and to identify the absolute speed of
the moving train (Charles et al., 2008).
In the course of a series of experiments Weston et al.
(2007) applied a bogie-mounted pitch rate gyroscope to
derive the average vertical alignment and axlebox mounted
accelerometers to compute the defects wavelength. This
setting made it possible to analyze from 5 m to 70 m long
wavelengths with high precision. In general, if we wish
to monitor short wavelengths, then axlebox accelerome-
ters should be used. Due to the fact that a long wavelength
irregularity produces less than 1 g accelerations the axle-
box accelerator sensor must have a low limit on lower band-
width, high linearity and emitting extremely low noise
effects in order to obtain reliable long wavelength output.
According to Xing et al. (2015) “Assuming that the lead-
ing and trailing wheel-sets follow the vertical track geome-
try and the primary suspension is innitely stiff, the bogie
pitch is determined by the vertical positions of the leading
and trailing wheel-sets divided by the bogie wheelbase.”
(Xing et al., 2015:p.219). Based on some geometric con-
siderations (e.g. spatial ltering) they stated that the bogie
pitch cannot sense wavelength irregularities whose lengths
are just equal to the bogie wheelbase (Xing et al., 2015).
Originally in Destek (1974), and later in Zobory and
Zábori (1996), then in Zobory (2015), the authors expressed
their belief that to look upon railway tracks as being
described by pure geometry alone would be an outdated
and professionally inappropriate view. Such an approach
ignores the system dynamics aspects, i.e. the railway
induced ground-borne vibrations result from the interac-
tion between the moving vehicle, the tracks superstruc-
ture and the subsoil. As a part of condition monitoring, the
outcome of a particular track qualifying process is highly
dependent on the type of the running train, the speed of
the train, the type of track, the different loads, the prop-
erties of primary and secondary suspension, the unsprung
masses, the axle distances, the supporting stiffness, the
damping and inertial properties of the track system. At any
cross-section of the track the location of rail proles can be
xed in an unloaded state only. Otherwise, the recurrent
loading caused by a regular railway trafc can be thought
of a time dependent stochastic process which transmits this
static setting into a probability distribution.
To eliminate the inuence of vehicle parameters, Zobory
et al. (1998) have started to construct a specic measuring
car equipped with a “measuring wheelset – carrier frame –
track” subsystem which is discoupled from the underframe
dynamically, in order to absolve the carrier frame from the
parasitic vibrations of the vehicle which is excited by track
unevennesses. The carrier frame was loaded by four nearly
constant vertical forces by four congruent air-springs. This
conguration is serving as compressed actuators between
the carrier frame and the underframe of the measuring car.
The layout of the vertical dynamic model of the measur-
ing wheel-set together with the measuring frame as being
the major structural elements of this alternative measuring
approach is shown in Fig. 3.
The primary goal of establishing this track measuring
car was to supply vertical acceleration data for a developed
simulation based identication procedure to determine the
inhomogeneities in the vertical track stiffness along the
longitudinal length direction of the track. Authors used a
wavelet-backed approximation to modeling the unknown
stiffness variations along the track length. They showed
experimentally that this approach corresponds well to
track stiffness irregularities that can be observed in rail-
way operation practice (there are some essential peaks at
random distance from each other, while the intermediate
track sections are disturbed only by a mild narrow band-
width with low stiffness variations).
This arrangement ensures the transduction of the longitu-
dinal stiffness functions of the track to the vertical accelera-
tion function measured by the measuring frame acceleration
sensors mounted over the axlebox as it is seen in Fig. 3.
The longitudinally valid vertically sensed track stiffness
parameters are then derived as a minimization of the time
integral of the vectorial difference norm square of the mea-
sured and the simulated acceleration responses. The esti-
mated optimal values of the vertical stiffness parameters
can then be derived from a properly formed least-squares
problem, whereas the vertical displacement functions can
Fig. 3 The vertical dynamical model of the measuring wheel-set and the
measuring frame connected with it. (Zobory and Zábori, 2018)

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Period. Polytech. Transp. Eng.
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be determined later on by two successive numerical inte-
grations with respect to time using the acceleration moving
average functions (for the sake of comparison).
2.5 Errors emerging during signal processing
In the phase of signal processing the accumulating errors
from the continuous measurement process give rise to a
peculiar issue called a drift. Drift is basically dependent
on the movement of the vehicle. In addition, the sources of
drift in orientation are not the same as those of in velocity
(Fasel , 2017).
In order to calculate orientation, position and velocity,
the measured accelerations and angular velocities have to
be integrated. Double integration of acceleration to obtain
position is unavoidable. Many times occurs in practice
that a zero mean acceleration error is assumed, and the
double integration with respect to time results in a given
position as its mean were zero. Indeed, this mathematical
operation that adds to position the integration of a random
walk, is not a controllable error and it grows fast. Only by
applying a continuous external sensing to bound this type
of error can prevent us from this undesired phenomenon
(Dissanayake et al., 2001).
Double integration might only be stabilized by adding
a high-pass lter. Unfortunately, latter method can only
moderately reduce the problem of offset drift (Weston et
al., 2015). The different techniques of double integration of
the signals extracted from inertial sensors have attracted
immense interest in the related literature. We refer to the
excellent paper of Naganuma et al. (2008) to nd details
of this problem.
Lee et al. (2012) proposed a ltering approach includ-
ing Kalman lter, band-pass lter, compensation lter and
phase correction to estimate the track irregularities. Still,
for the time being, there is no perfect method for extracting
a usable signal from an accelerometer by double integrating
it, because their scope is rather limited. Similarly, the noise
generated by an accelerometer (and also which comes from
an A/D converter through signal processing) more or less
affects the ultimate results as well (Weston et al., 2015).
In summary, Fig. 4 provides an overview of the steps
of the computations about the above discussed processing
issues. Observe here the closed-loop feedbacks of the cor-
rection process on both of the two branches starting from
the measurement data obtained for acceleration and angu-
la r velocity.
Gravity is generally greater than the different types
of other accelerations hit the vehicle moving along the
track. Eventhough, tiny errors in the attitude may result
in large drifts in determining velocity and thus position.
The errors in attitude are usually due to false estimation
procedures or they are originated in temperature com-
pensation. Measured acceleration is valid referring to the
local geographic frame of the inertial sensors, however,
it contains gravity. Hence, physical measures have to be
given with respect to a global reference frame and grav-
ity must be removed. Even if this correction is done in a
proper way, 1-2 degrees of errors in orientation are often
remain (Fasel, 2017). Imbalanced gravity removal means
that on one axis too much gravity is removed, while a cer-
tain amount of gravity is augmented on the other axis. In
contrast to orientation drift, the drift in velocity is even
greater in magnitude (Fasel, 2017).
To eliminate two types of gyro output errors, the one is a
Gaussian noise caused by environmental vibration or elect-
romagnetic radiation, the other can be an installation error
when the sensors’s sensitive axis is not completely perpen-
dicular to the plane of the train running line, Xing et al.
(2015) proposed a mix-ltering approach to obtain a bias
free bogie pitch rate, containing time-space domain trans-
formation, double integration, baseline correction and a
recursive least squares based adaptive compensation lter.
2.6 Continuous measurement vs. dedicated measuring
coaches
For a long time past it is well known that a regular track
monitoring and track inspection are the main guarantees
in achieving high standards for safety and developing an
efcient maintenance optimization policy in railway traf-
c. Measurements of the railroad track could be made by
the instrumentation of an in-service vehicle, specically
on a multiple-unit passenger train. A key advantage of this
operational manner is that track measurements might be
done even in an hour-to-hour basis without interrupting
the scheduled operation of railway lines having large traf-
c, and thus, surveying the track heavily. Gathered data
are sent to big database centers where a frequent data
processing supports to detect the type and geographic
Fig. 4 “Overview of the computation steps required for tracking the
sensor’s position over time.”. (Fasel, 2017:p.22)

8|Farkas
Period. Polytech. Transp. Eng.
position of track faults. Such Track Recording Coaches
(TRCs), in order to monitor tracks, are in use today e.g.,
in Great Britain and belonging to the Network Rail’s New
Measurement Train (NMT), (Yeo, 2017). The vehicles are
capable of measuring all aspects of the track geometry at
speeds of up to 140 mph. Positioning is done through a
combination of GPS, wheel tachometers and by detecting
track-mounted electromagnets in the Automatic Warning
System (AWS), as a part of the British signaling system.
If an Unattended Geometry Measurement System
(UGMS) is used, then additional sensors can be installed,
preferably axlebox-mounted accelerometers, to inspect some
specic elements of the tracks like switches and crossings,
and focusing on to monitor corrugation and joints. This way,
the speed of the vehicle is normalizing accelerations and it
permits to calculate versines which are mostly independent
of the speed of the vehicle (Weston et al., 2015).
If an UGMS is not used, then a bogie-mounted IMU –
when the primary suspension is stiff – is a perfect solution
to monitoring the track, even if the gauge and the twist
cannot be sensed. But adding an axlebox accelerometer
helps to measure short wavelength data as well. Using
such a setting, the vertical acceleration measured on the
bogie together with a displacement to the axlebox ensures
a rst-rate class quality measurement of the vertical rail
prole (Weston et al., 2015).
Dedicated measurement vehicles have limitations.
They can be expensive to run, requiring specialist crews
to operate, as well as being expensive to build initially.
Consequently, a railway operating company may only
have access to a small number of measurement vehicles.
This, coupled with high trafc, causing limited availabi-
lity on some track, means that measurements may only
be possible every one or two months at best in some areas
(Ye o, 2 017).
2.7 Accelerometer/Gyroscope characteristics
An accelerometer measures linear acceleration and grav-
ity. In case of an accelerometer, the magnitude of the sig-
nal is biased by gravity and a difcult double integration
with respect to time is required to derive displacement
plus it has a low signal-to-noise ratio. Contrary to these,
in case of a gyroscope, the magnitude of the signal is unbi-
ased, information pertains to bandwidth and frequency
available to the extent of zero frequency, a one time
integration is sufcient to obtain angular displacement
and there is a high signal to noise ratio. Accelerometers
must have a range greater than ±100 m/s2 for bogie, and
±10 m /s2 for body (Yeo, 2017). Gyros must have a range
greater than ±10 ○/s for bogie, and ±1.25 ○/s for body
(Yeo, 2017). Very often, railways apply a zero setting
during the initial calibration of accelerometers and gyros-
copes in order to avoid offset as a harmful behavior at
these inertial sensors. Despite these efforts, the offset
tends to drift overtime, mainly because of the changes in
temperature. Another inuencing factors lend themselves
in the variations of supply voltage, or other troublesome
external impacts, or materials aging.
There is a linear relationship between the acceleration,
brought about the static and dynamic effects of the whe-
el-rail interaction, and the square of the vehicle speed. The
acceleration is proportionally related to the amplitude of
the irregularities in the track and it is inverse proportionally
related to the square of the wavelength. As concerns some
particular measurement data, Weston et al. (2015) reported
that “A sinusoidal vertical geometrical irregularity with an
amplitude of 10 mm and a wavelength of 50 m gives an
acceleration of 0.32 m/s2 at 45 m/s, but gives 0.0032 m/s2 at
4.5 m/s.” (Weston et al., 2015:p.1027). They found the ver-
tical acceleration measured by an axlebox sensor may even
exceed 100 g, especially when a running wheel-pair hits a
badly aligned rail joint (Weston et al., 2015).
The observation of long wavelength geometry from an
axlebox sensor is almost impossible. We are often facing
with this case at low vehicle speeds, e.g. when an in-ser-
vice vehicle slows down for stopping. Grassie (1996)
describes the axlebox-mounted accelerometers as being
suitable devices to sensing vertical displacements to con-
clude from the vertical rail prole. But he mentions also
that this measurement is challenging, as there are very
strict requirements against the design of such sensors with
respect to offset and drift for long wavelengths.
In a spectacular contrast to the previous considerations,
placing an accelerometer sensor on the bottom of the body
of the vehicle bears much less difculty, because here, the
typical acceleration values become less than 1 g. Hence, the
long wavelength geometry induced by small accelerations
are more simple to measure on the body. On the bogie, the
vertical accelerations are usually only around or less than
10 g. Furthermore, whatever the choice is for the placement
of the acceleration sensors, the values of the measurements
are speed dependent. It can be said that the faster is the
movement of the train, the greater the acceleration values
are likely to occur at the same section of the track.
Regarding the use of the gyroscope sensors, several
authors expressed that they perform very effectively on

Farkas
Period. Polytech. Transp. Eng.
|9
the bogie, but they are much less benecial on the axle-
box (Weston et al., 2015). Gyros measuring yaw and roll
rates have been employed successfully for long ago (Lewis
and Richards, 1988), but other authors claimed in favor
of using bogie-mounted pitch rate gyros on the in-service
vehicles (Weston et al., 2015). Gyroscopes can provide
valuable measurement data at low vehicle speeds compa-
red to the conventionally used accelerometers, particularly
in the case of frequent station stops at the regular in-ser-
vice train sets (Weston et al., 2007; Yeo, 2017).
3 Kinematical background of inertial navigation and
measurement systems
In this section, we present a concise description about
the physical background of the inertial navigation prob-
lem. In this framework, the required measurements are
made by accelerometers and gyroscopes which devices
are installed on a dedicated or in-service vehicle. The
sensed transitional and rotational motions are used to
obtain the position of the vehicle. The totality of this data
set comprises both of these two types of measurements
and enables the users to determine the true motion of the
vehicle within a properly chosen inertial frame of refer-
ence, and thus, to calculate its position. Our brief discus-
sion about the essential functions that an inertial naviga-
tion system must perform will follow the excellent book
of Titterton and Weston (1997). Firstly, it is necessary to
dene an adequate reference frame for any inertial naviga-
tion and measurement system. Each of these frames repre-
sents an orthogonal axis set. This triaxial coordinate sys-
tem is interpreted as a right-handed one.
Consider a xed environment without acceleration and
rotation. To estimate the constituent members of the accel-
eration with respect to a space-xed reference system, the
measured elements of the specic force and the gravitatio-
nal estimates should be summed. Let r denote the column
vector (position vector) of an arbitrarily chosen point P
tied to origin O of the reference frame. Now, let the accel-
eration of P be dened in the i-frame (inertial frame with
its origin at the centre of the Earth and axes which are
non-rotating) and denoted by the index i, as described by:
arfgCf g
i
i
ii
b
ib i
d
d
==+= +
2
2
t, (1)
where triaxial accelerometers will provide us the measure
of the specic force, f that acts at point P. In Eq. (1), g
represents the so called mass attraction gravitation vector.
The specic force is usually given in a b-frame (in a body
xed axis set, denoted by f b which is an orthogonal coor-
dinate system aligned with the roll, pitch and yaw axes of
the vehicle within the established navigation system). The
specic force should be pre-multiplied by the direction
cosine matrix,
C
b
i. With proper numerical integration of
the navigation equation represented by Eq. (1), the velocity
and position of the vehicle can be derived. The velocity of
point P with respect to the to the i-frame is produced by
the rst integral and given as
v
r
i=d
di
t, (2)
whilst a suitable next integration yields the position of
point P in the same reference frame. The 3×3 sized direc-
tion cosine matrix, Cb
i, whose columns are unit vectors in
body frame which were then projected along the reference
axes and may be calculated from the gyroscopes’ mea-
surements for the angular rates, is seen in Eq. (3):

CC C=
b
i
b
i
ib
b
b
i
=
−
−
−










Ω
0
0
0
rq
rp
qp
,
(3)
where the second factor, Ω
ib
b
of the matrix product is com-
posed of the elements of the vector,
ω
ib
b=
[]
pqr,,  rep-
resenting the turn rate of the body with respect to the
i-frame, as directly measured by the gyros.
In this system, it is required to calculate and use the
speed of the vehicle with respect to the Earth, called
ground speed in the i-frame axes, which is denoted by the
symbol
v
e
i (Titterton and Weston, 1997):
d
d
e
i
eb
ib
ie
i
ie
i
ie
ii
vvCfv
rg
i
tie
i
== −−


+×××

ωωω
. (4)
In Eq. (4), the rst member,
Cf
b
ib
, represents the spe-
cic force acceleration; whilst the second one in this
expression is the Coriolis acceleration given in the form
of a vector product induced by its velocity over the surface
of the rotating Earth; nally, the third member in Eq. (4)
describes the centripetal acceleration impacting due to the
rotating Earth which cannot make to be separately distin-
guishable from the gravitational acceleration which arises
through mass attraction, g. Due to their small effects in the
b-frame, the latter two accelerations are usually neglected
from Eq. (4) (Titterton and Weston, 1997).
Gyroscopic sensors are used to instrument a reference
co-ordinate frame within a vehicle which is free to rotate
about any direction. The calculated attitude of the vehicle
may be stored as a set of numbers in a computer within

10|Farkas
Period. Polytech. Transp. Eng.
the vehicle. The stored attitude is updated as the vehicle
rotates using the measurements of the turn rates provided
by the gyroscopes (Titterton and Weston, 1997). Positive
rotations about each axis are interpreted as spinnings into
clockwise directions when looking along the axis from the
origin. A comprehensive block diagram representation of
the inertial frame mechanization system is exhibited in
Fig. 5 (Titterton and Weston, 1997:p.29).
The rate of change of the direction cosine matrix,
C
b
n
with time given in an n-frame (the navigation frame is a
local geographic frame which has its origin at the loca-
tion of the navigation system, i.e. point P, and axes alig-
ned with the directions of north, east and the local verti-
cal down) yields, in a limiting sense, as Δt→0, and ψ, ϕ
and θ are the (small) Euler rotation angles through which
the b-frame has rotated over the time interval Δt about its
yaw, pitch and roll axes (see in Fig. 6), respectively:

CC C
b
n
b
n
nb
b
b
n
==
−
−
−










Ω
0
0
0
ω
ω
ω
ω
ω
ω
zy
zx
yx
.
(5)
Each rotation in Eq. (5) is then separated into three
distinct direction cosine matrices, entries of which are
composed of appropriate sine and cosine functions of the
Euler angles. For more details see Titterton and Weston
(1997). Due to the very small angle rotations here, these
sines and cosines become zero and one, respectively, in
the skew symmetric matrix given in Eq. (5). Hence, in
terms of Euler rotations, the following approximate direc-
tion cosine matrix may be obtained, which relates from the
body to the reference axes:
Cb
n
≅−
−










−1
1
1
ψθ
ψ
φ
φ
θ
.
(6)
According to the gimbal suspension analogy (a pivoted
support that allows the rotation of an object about a single
axis), ψ, ϕ and θ are the gimbal angles and, 

ψφ
θ
, and
are the gimbal rates which are directly related to the body
rates,
ωω ω
xy z
, and .
4 Notions and description of train motions
A rail-road track is usually represented by a single para-
meter, s associated with the change of the general posi-
tion coordinate of the moving train set over the time inter-
val Δt. The well-known kinematic motion equation can be
expressed as:
ss st st=+
++
…
0
2
1
2
, (7)
producing the displacement (distance) into the longitu-
dinal direction on the effects of the velocity of the train
which is the rst time derivative of s and the train accel-
eration or deceleration (as a result of traction or brakes) as
the second time derivative of the general parameter s.
To study the motion of a train on the track properly, the
angular coordinates should be introduced. The deviation
of the track position from geographic north is called head-
ing and denoted by ψ(s). A change in heading is termed
as a curvature of the track, dψ(s)/ds, which can be exp-
ressed by the reciprocal of the radius for a given segment
of the track (Heirich et al., 2011). Moving in a curvature
results in lateral acceleration as a response to the front
wheel pair input. Irregularities, on the effect of centrifu-
gal forces may have a large impact on the riding quality
of the performance of the running vehicle, if the velocity
is high (v>250 km/h) and the radius is relatively small
(R<5,000 m), (Yi, 2018:p.311).
Turning in a circle requires a vehicle to have a centrip-
etal acceleration inwards on the turn. To avoid serious
physical balancing problems, the track is banked inwards,
i.e. the inner rail is lower than the outer rail (supereleva-
tion/cant). The bank angle, θ(s) is the lateral inclination of
the track. The change in the bank, dθ(s)/ds is called a bank
rate. The horizontal inclination, simply saying a slope, ϕ(s)
in the longitudinal direction of the track can be ascending
or descending. The change in the altitude level, dϕ(s)/ds is
termed a pitch rate. For more details about these geometric
interpretations, see e.g. Heirich et al. (2011).
Railway track routes are designed as a combination
of their basic elements: straight lines, circles and transi-
tion curves (clothoids, S-shaped curves, etc.). Geometric
design of the transition segments attempts to achieve a lin-
ear increase of curvature (Heirich et al., 2011).
Fig. 5 Block diagram representation of an iner tial navigation system
(Titterton and Weston, 1997:p.29)

Farkas
Period. Polytech. Transp. Eng.
|11
There have some specic elements of the railway tracks.
They are the switches, crossings, rail joints, etc. which
cause strong lateral and vertical jerk. In addition, they may
result in a sudden growth in the turn rate and both in the
lateral and the vertical accelerations. It is interesting to note
here that there are over 300,000 of such units within the
networks of the EU 28 countries (Capacity 4 Rail, 2015).
Using a three dimensional coordinate system (b-frame)
for the train (or for a structural element of it, i.e. the bogie),
the different rotational motions around the corresponding
translational axes are expressed by the angular veloci-
ties and can be measured by gyroscopic sensors, denoted
by
ωω ω
yawpitc
hr
oll
, and . An other coordinate system, the
n-frame stands for measuring the attitude of the train
frame by rotation angles between the axes with reference
to gravity and geographical north of the Earth. This ref-
erence frame has a horizontal plane which is perpendic-
ular to gravity. Rotation angles to the train frame are: ψ
for the heading of the train (rotate through angle ψ about
reference z-axis), ϕ for slope angle (rotate through angle ϕ
about a new y-axis) and θ for bank angle (rotate through
angle θ about a new x-axis), as they are displayed in Fig. 6.
A rotation around an axis is taken to be positive, if it
happens in clockwise direction while looking along the
axis from the origin, as it is indicated in Fig. 6. Observe
also that the z-axis is pointing downwards. It follows that
ψ=ϕ=θ=0 means that the vehicle is positioned to the north
direction and is standing perpendicular to gravity. It should
be noted that one should keep the order of the rotations.
The movements experienced by an IMU have to be
interpreted in three dimensions. As of in the six degree-
of-freedom model depicted in Fig. 6, the bogie may move
in 3 perpendicular axes and may rotate about those axes.
The geometry frame is oriented with the bogie of
the IMU so that axis x leads always in parallel to the
track. The IMU is mounted on the bogie such that each
of its faces are always parallel to one of the three axes.
Therefore, each of the three accelerometers directly mea-
sures the accelerations pointing into the predened axes,
x, y and z, and each of the three gyroscopes directly mea-
sure the changes in the rotation angles, ψ, ϕ and θ . The
measured accelerations will vary depending on the actual
placement of the IMU on the bogie. The same is not true
for the angular velocities, which will remain invariant
in each axis when measured at different locations on the
bogie by the gyros (Yeo, 2017).
5 Conclusions
In this overview, the most recent ndings in condition
monitoring of railway track geometry have been col-
lected and reviewed based on a comprehensive literature
research. The kinematics of inertial navigational systems
and basic train motions on the effects of the different geo-
metric and geographic train positions have also been sum-
marized. In conclusion, this survey has shown that there is
not any universally applicable conguration of a railway
track condition monitoring system which would incorpo-
rate the required measurements of track defects of every
kind. A choice for the actual instrumentation of such a
system in case of a given railway network is a matter of
engineering and economic rationale.
Future research should address more focus on condi-
tion monitoring issues of high speed railway lines like
the SNCF TGV Iris 320 dedicated track recording train,
which is equipped by pantograph, laser diodes and trans-
versal linear, and longitudinal matrix cameras to offer
simultaneous two axis recording.
Which clearly observable at leading railway companies
means that a really effective track geometry measurement
system should be composed of a six degree-of-freedom
inertial measurement system (IMU) including closed
loop congured accelerometers with possibly high per-
formance operating parameters that are installed on the
bogie, plus, sometimes, it would be expedient to mount
displacement transducers between the bogie and the axle
boxes. Eventually, an optical attitude determination sys-
tem might also be added. Most European and Far-Eastern
railway companies strive to employ in-service vehicles.
The obtained signals should be high-pass ltered and have
a baseline and phase correction to eliminate the errors
emerging in the double integration process to derive more
accurate orientation, velocity, and position information.
Fig. 6 Interpretation of a six degree-of-freedom model applied to a
vehicle bogie (Yeo, 2017:p.52)

12|Farkas
Period. Polytech. Transp. Eng.
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