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3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Edición Especial Special Issue Abril 2020
EXTENDED KALMAN FILTER FOR ESTIMATION OF
CONTACT FORCES AT WHEEL-RAIL INTERFACE
Khakoo Mal
PhD Scholar, Department of Electronic Engineering,
Mehran University of Engineering and Technology, Jamshoro, (Pakistan).
E-mail: 17phdiict05@students.muet.edu.pk ORCID: https://orcid.org/0000-0002-5754-0441
Imtiaz Hussain
Associate Professor, Electrical Engineering.
DHA Sua University. Karachi, (Pakistan).
E-mail: imtiaz.hussain@dsu.edu.pk ORCID: https://orcid.org/0000-0002-7947-9178
Bhawani Shankar Chowdhry
Professor Emeritus.
Mehran University of Engineering and Technology. Jamshroo, (Pakistan).
E-mail: bhawani.chowdhry@faculty.muet.edu.pk ORCID: https://orcid.org/0000-0002-4340-9602
Tayab Din Memon
Associate Professor, Department of Electronics.
Mehran University of Engineering and Technology. Jamshoro, (Pakistan).
E-mail: tayabdin82@gmail.com ORCID: https://orcid.org/0000-0001-8122-5647
Recepción:
20/01/2020
Aceptación:
15/04/2020
Publicación:
30/04/2020
Citación sugerida Suggested citation
Mal , K., Hussain, I., Chowdhry, B. S., y Memon, T. D. (2020). Extended Kalman lter for estimation
of contact forces at wheel-rail interface. 3C Tecnología. Glosas de innovación aplicadas a la pyme. Edición
Especial, Abril 2020, 279-301. http://doi.org/10.17993/3ctecno.2020.specialissue5.279-301
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ABSTRACT
The wheel-track interface is the most signicant part in the railway dynamics because the
forces produced at wheel-track interface governs the dynamic behavior of entire vehicle.
This contact force is complex and highly non-linear function of creep and aected with
other railway vehicle parameters. The real knowledge of creep force is necessary for reliable
and safe railway vehicle operation. This paper proposed model-based estimation technique
to estimate non-linear wheelset dynamics. In this paper, non-linear railway wheelset is
modeled and estimated using Extended Kalman Filter (EKF). Both wheelset model and
EKF are developed and simulated in Simulink/MATLAB.
KEYWORDS
Railway dynamics, Wheel-rail interface, Model-based estimation, Extended Kalman Filter.
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1. INTRODUCTION
The main element of any study of rolling stock behavior is the wheel-track interaction
patch (Simon, 2006). All the forces which help and direct the railway vehicle transmit via
this narrow area of contact and knowing of the nature of these forces is most important for
any investigation of the generic railway vehicle behavior (Melnik & Koziak, 2017).
The Wheel-track condition information can be detected in real time to provide traction
and braking control schemes for re-adhesion. For example, in Charles, Goodall and Dixon
(2008) an indirect technique based on Kalman Filter (KF) is proposed for the estimation
of low adhesion with wheel-track prole by using conicity and wheel-rail contact forces.
A method using Kalman lter has also been introduced in Mei, Yu and Wilson (2008)
and Hussain and Mei (2009) to identify the slip after evaluating the torsional frequencies
in the axle of wheelset. Two indirect monitoring schemes using a bank of Kalman lters
are proposed for (i) wheel slip detection and, (ii) real time contact condition and adhesion
estimation in Hussain and Mei (2010, 2011). In Hussain, Mei and Ritchings (2013) and
Ward, Goodall and Dixon (2011), the development of techniques based on Kalman-Bucy
lter proposed for the estimation of wheel-track interface conditions in real time to predict
the track and wheel wear, the development of rolling contact fatigue and any regions of
adhesion variations or low adhesion.
However, due to nonlinear nature of wheel-rail dynamic behavior, Kalman-Bucy lter
is dicult to use for entire operating conditions. A method using Heuristic non-linear
contact model and Kalker’s linear theory is proposed in Anyakwo, Pislaru and Ball (2012)
for modeling and simulation of dynamic behavior of wheel-track interaction in order to
discover the shape of interaction patch and for obtaining the tangential interaction forces
generated in wheel-rail interaction area. On the basis of measurement of traction motor’s
parameters, (i) creep forces can be predicted by means of Kalman lter between roller and
wheel (Zhao, Liang & Iwnicki, 2012) and (ii) slip-slide is detected and estimated by using
Extended Kalman Filter (EKF) (Zhao & Liang, 2013).
A system based on two dierent processing methods, i.e., model-based approach using
Kalman-Bucy lter and non-model based using direct data analysis, is presented for on-
board indirect detection of low adhesion condition in Hubbard et al. (2013a, 2013b).
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However, the technique using yaw acceleration as a normalization method provides only a
rough estimate and introduces a huge delay to obtain an estimate. A model-based technique
using Unscented Kalman Filter (UKF) is proposed by Zhao et al. (2014) for estimation
of creep, creep forces as well as friction coecient from the behavior of traction motor.
However estimators seem unreliable in some critical track conditions, hence still work is
needed to monitor these wheel-rail parameters more eectively in real time.
A system based on the principles of synergetic control theory is proposed in Radionov
and Mushenko (2015) to estimate adhesion moment in wheel-track contact point. Two-
dimensional inverse wagon model based on acceleration is developed in Sun, Cole and
Spiryagin (2015) for evaluation and monitoring of wheel-rail contact dynamics forces. The
results at higher speed are agreeable, however improvement in the model is further needed
to reduce the error at all expected speeds. Another technique using multi-rate EKF state
identication is presented in Wang et al. (2016) for detection of slip velocity by merging
the multi-rate technique and Extended Kalman lter technique to identify the load torque
of traction motor. On the basis of tting non-linear model, EKF can also be applied to
identify the wheel-track interaction forces and moments that takes into account the interface
nonlinearities (Strano & Terzo, 2018).
After reviewing the literature on condition monitoring of railway wheelset dynamics, it
is observed that the problem to analyze wheelset conditions and update them to desired
situation still needs to be improved in order to accomplish the expectation of railway vehicle
to be really high speed, high comfort, more safer and economical means of transport across
the world.
In this paper, Extended Kalman lter is designed for non-linear railway wheelset model
to estimate lateral velocity and yaw rate of wheelset as well as creep and creep force.
Polach formulae for creep force and friction coecient are used in modeling of non-
linear wheelset. Both modeling of non-linear wheelset and designing of EKF are done in
Simulink/MATLAB.
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2. MODELING OF NON-LINEAR WHEELSET
The motion of a railway vehicle is directed by interaction forces produced at wheel-
track contact, which change non linearly with respect to creepage and are aected by the
unpredictable variations in the adhesion conditions (Hussain, 2012). A single solid-axle
wheelset shown in Figure 1 is taken for modeling and estimation of wheel-rail conditions.
Figure 1. Railway wheelset [captured by author during eld visit].
The creepages (the relative speed of the wheel to rail) of right and left wheels of wheels in
longitudinal direction are expressed in following equations.
(1)
(2)
The main objective of this paper is to develop a state of art technique to detect the changes
in wheel-rail contact conditions. The term
in equations (1) and (2) does not involve
lateral and yaw dynamics, hence can be excluded in simplied longitudinal creep equations
because only yaw and lateral dynamics are sucient for detecting these changes. Further
, so the simplied creep equations used in above model become as:
(3)
(4)
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The creepages in lateral direction are expressed as:
(5)
While in equations (6) total creep of the wheels is depicted.
(6)
As the wheel-rail contact forces govern railway vehicle’s dynamics are creep forces and
are the function of creeps. The adhesion coecient is the ratio of tangential force that
is creep force to normal force and hence is also a function of creep. Figure 2 illustrates a
classic nonlinear change of the adhesion coecient with respect to creepage for all track
conditions i.e. dry, wet, poor and worst conditions.
Figure 2. Creep v/s Adhesion Coefcient for all conditions of wheel-rail interface.
Following equations illustrate creep forces and adhesion coecient.
(7)
i = Right and left wheels, j = longitudinal and lateral directions
F
i
is the total creep force and can be calculated by Polach formula (Polach, 2005).
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(8)
Where U is friction coecient, is gradient of the tangential stress in area of adhesion, k
A
is reduction factor in the area of adhesion and is the reduction factor in slip. Both U and
are illustrated as:
(9)
Where u
0
is maximum friction coecient at zero creep velocity, A is ratio of friction coecient
at innity creep velocity to u
0
and B is coecient of exponential friction decrease.
=
(10)
While a and b are half-axes of contact ellipse and c is coecient of contact shear stiness
in N/m
3
.
(11)
The equations of motion of railway wheelset at any point of creep curve of Figure 2 are
expressed as (Hussain and Mei, 2009):
(12)
(13)
(14)
(15)
(16)
(17)
Where
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F
C
is centripetal force component and can be neglected when vehicle does not run in curves
and C
S
is material damping of shaft which is normally very small. Hence both terms are
not considered in this research.
In Table 1 detailed information of all parameters used in simulated wheelset model is given.
Table 1. Parameters used in modeling on non-linear wheelset.
No. Symbol Parameter Value Unit
1 γ
xR
Right wheel creep in longitudinal direction calculated ratio
2 γ
xL
Left wheel creep in longitudinal direction calculated ratio
3 γ
yR
Right wheel creep in lateral direction calculated ratio
4 γ
yL
Left wheel creep in lateral direction calculated ratio
5 γ
R
Total creep of right wheel calculated ratio
6 γ
L
Total creep of left wheel calculated ratio
7 r
0
Wheel radius 0.5 (constant) m
8 L
g
Half gauge of track 0.75 (constant) m
9 λ
w
Wheel conicity 0.15 (constant) rad
10 ɷ
L
Angular velocity of left wheel calculated rad/sec
11 ɷ
R
Angular velocity of right wheel calculated rad/sec
12 v Vehicle’s forward velocity calculated m/sec
13 y Lateral displacement Output m
14 y
t
Track disturbance in lateral direction input m
15 Ψ Yaw angle output rad
16 F
xR
Right wheel creep force in longitudinal direction calculated Newton
17 F
xL
Left wheel creep force in longitudinal direction calculated Newton
18 F
yR
Right wheel creep force in lateral direction calculated Newton
19 F
yL
Left wheel creep force in lateral direction calculated Newton
20 F
R
Total creep force of right wheel calculated Newton
21 F
L
Total creep force of left wheel calculated Newton
22 µ Adhesion coefcient between track and wheel calculated ratio
23 N Normal load on wheel constant Newton
24 M
v
Vehicle mass 15000 (constant) Kg
25 I
w
Yaw moment of inertia of wheelset 700 (constant) Kgm
2
26 K
w
Yaw stiffness 5x10
6
(constant) N//rad
27 m
w
Wheel weight with induction motor 1250 (constant) Kg
28 v
0
Vehicle’s forward velocity at initial input m/sec
29 ɷ
0
Angular velocity of wheelset at initial input Rad/sec
30 T
m
Torque of traction motor input Nm
31 T
s
Torsional torque calculated Nm
32 T
R
Traction torque on right wheel calculated Nm