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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 Sua 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 signicant 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 aected 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 prole 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 dicult 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 dierent 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 coecient 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 eectively 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
identication 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 coecient 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 aected 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 simplied longitudinal creep equations
because only yaw and lateral dynamics are sucient for detecting these changes. Further
, so the simplied 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 coecient 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 coecient with respect to creepage for all track
conditions i.e. dry, wet, poor and worst conditions.
Figure 2. Creep v/s Adhesion Coefcient for all conditions of wheel-rail interface.
Following equations illustrate creep forces and adhesion coecient.
(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 coecient, 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 coecient at zero creep velocity, A is ratio of friction coecient
at innity creep velocity to u
0
and B is coecient of exponential friction decrease.
=
(10)
While a and b are half-axes of contact ellipse and c is coecient of contact shear stiness
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 coefcient 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
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No. Symbol Parameter Value Unit
33 T
L
Traction torque on left wheel calculated Nm
34 I
R
Right wheel inertia 134 (constant) Kgm
2
35 I
L
Left wheel inertia 64 (constant) Kgm
2
36 K
s
Torsional stiffness 6063260 (constant) N/m
37 θ
s
Twist angle calculated rad
3. DESIGNING OF EXTENDED KALMAN FILTER FOR ESTIMATING
NON-LINEAR WHEELSET MODEL
Being non-linear nature of railway wheelset model, it is dicult to estimate the wheelset
dynamics with ordinary estimation techniques. Therefore Extended Kalman lter is used
to estimate wheelset dynamics and contact force in all adhesion conditions. Kalman lter
utilizes measurements associated to the state and error covariance matrices to produce a
gain known as Kalman gain. Figure 3 shows the block diagram of Kalman lter with
generic scheme.
Figure 3. Block diagram of the Kalman lter with generic scheme.
Extended Kalman lter (the extension form of Kalman lter) linearizes the current mean
and covariance by assessing Jacobian matrices and their partial derivatives (Ngigi et al.,
2012)
From non-linear model of railway wheelset, single equation (18) in matrix form is furnished
after putting the values of F
xR
, F
xL
, F
yR
and F
yL
in equations (13) and (14).
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(18)
If left and right wheel creep are same (
and
) then
(19)
Here are state variables of wheelset model i.e y’(lateral velocity), Ψ’ (Yaw rate) γ (Creep
or slip), U (friction coecient) and F (Creep force) taken for EKF algorithm. Lateral
acceleration (y’’) and yaw rate (Ψ ’’ ) can be measured along noise with accelerometer and
gyroscope. From equation (19):
(20)
(21)
(22)
(23)
(24)
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Now it is required to discretize equations (20)-(24) by using Forward Euler (FE) method in
order to design Extended Kalman lter for estimation.
(25)
(26)
(27)
(28)
(29)
As the Extended Kalman lter uses a 2 step predictor-corrector algorithm (Welch & Bishop,
2001). The predictor step is given by
(30)
(31)
And the equations of corrector step are,
(32)
(33)
(34)
Where f and h are non-linear functions relating to process and measurement states, while:
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and
Nomenclature of EKF algorithm is given in below table.
Table 2. Nomenclature of EKF algorithm.
Symbol Description
x ̂-k discretized a-priori estimated process
x ̂k discretized a-postriori estimated process
Pk- a-priori estimate of the covariance of process error
Pk estimate of the covariance of measurement error
Fk Jacobian matrix of process
Hk Jacobian matrix of measurement
Qk process noise covariance
Rk measurement noise covariance
Kk Kalman gain
y ̃k measured output
The Jacobean matrix of process matrix:
is
(35)
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And the Jacobian matrix of measurement matrix is
(36)
4. SIMULATION RESULTS
The simulation models of non-linear railway wheelset and EKF are developed in Simulink/
MATLAB and are simulated 50 microseconds step size. As the vehicle is kept on constant
velocity i.e. motor torque is applied zero, only random track disturbance of ±7mm
magnitude in lateral direction is applied as input to the model for exciting lateral dynamics.
Curves of Figure 2 are tuned with Polach parameters k
A
, k
S
, u
0
, A and B. Table 3 contains
the values which are used to tune these creep curves. Along with Kalman gain and Jacobian
matrices, the other EKF tuning parameters are measurement noise covariance of inertial
sensors and process noise covariance for entire range of track conditions which are set in
equation (37)-(40). The measurement noise covariance matrix in equation (37) is calculated
by adding noise power for accelerometer and gyro sensor, while the process noise matrices
of equations (38)-(40) are calculated based on ne tuning of results.
R = [1x10
-7
1x10
-13
]
(37)
Q1= [5x10
-14
1x10
-14
1x10
-14
1x10
-14
1x10
-14
] for dry condition (38)
Q2= [0.5x10
-12
9x10
-17
1x10
-12
1x10
-12
1x10
-12
] for wet condition (39)
Q3= Q4= [1x10
-13
9x10
-17
1x10
-12
1x10
-12
1x10
-12
] for poor and worst condition (40)
Table 3. Polach parameters.
Parameter Dry condition Wet condition Poor condition Worst condition
kA 1 1 1 1
kS 1 1 1 1
u0 0.46 0.3 0.2 0.1
A 0.4 0.4 0.1 0.1
B 0.6 0.2 0.2 0.2
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Following tests are performed on wheelset with EKF algorithm.
(i) Dry condition, (ii) Wet condition, (iii) Poor condition, (iv) Worst condition and (v)
Transition from dry condition to worst condition.
4.1. DRY CONDITION TEST
The lateral velocity and yaw rate of wheelset as well as creep and creep force are computed
along with error on dry condition curve (Dry curve of Figure 2) and shown in Figure 4 to 7.
Figure 4. lateral velocity comparison (top) and Error (bottom) for dry condition of wheel-rail interface.
Figure 5. Yaw rate comparison (top) and Error (bottom) for dry condition of wheel-rail interface.
Figure 6. Creep comparison (top) and Error (bottom) for dry condition of wheel-rail interface.
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Figure 7. Creep force comparison (top) and Error (bottom) for dry condition of wheel-rail interface.
4.2. WET CONDITION TEST
The lateral velocity and yaw rate of wheelset as well as creep and creep force are computed
along with error on wet track condition curve (Wet curve of Figure 2) and shown in Figure
8 to 11.
Figure 8. Lateral velocity comparison (top) and Error (bottom) for wet condition of wheel-rail interface.
Figure 9. Yaw rate comparison (top) and Error (bottom) for wet condition of wheel-rail interface.
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Figure 10. Creep comparison (top) and Error (bottom) for wet condition of wheel-rail interface.
Figure 11. Creep force comparison (top) and Error (bottom) for wet condition of wheel-rail interface.
4.3. POOR CONDITION TEST
The lateral velocity and yaw rate of wheelset as well as creep and creep force are computed
along with error on poor track condition curve (Poor curve of Figure 2) and shown in
Figure 12 to 15.
Figure 12. Lateral velocity comparison (top) and Error (bottom) for poor condition of wheel-rail interface.
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Figure 13. Yaw rate comparison (top) and Error (bottom) for poor condition of wheel-rail interface.
Figure 14. Creep comparison (top) and Error (bottom) for poor condition of wheel-rail interface.
Figure 15. Creep force comparison (top) and Error (bottom) for poor condition of wheel-rail interface.
4.4. WORST CONDITION TEST
The lateral velocity and yaw rate of wheelset as well as creep and creep force are computed
along with error on worst track condition curve (Worst curve of Figure 2) and shown in
Figure 16 to 19.
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Figure 16. Lateral velocity comparison (top) and Error (bottom) for worst condition of wheel-rail interface.
Figure 17. Yaw rate comparison (top) and Error (bottom) for worst condition of wheel-rail interface.
Figure 18. Creep comparison (top) and Error (bottom) for worst condition of wheel-rail interface.
Figure 19. Creep force comparison (top) and Error (bottom) for worst condition of wheel-rail interface.
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4.5. TRANSITION TEST FROM DRY CONDITION TO WORST CONDITION
The lateral velocity and yaw rate of wheelset as well as creep and creep force are computed
along with error on all adhesion condition curves (Dry to worst curves of Figure 2) and
shown in Figure 20 to 23. During simulation adhesion condition changed at every 2
seconds from dry to worst adhesion conditions in 8 seconds of simulation time and then the
condition again changed from worst to wet. The graphs show the changings of adhesion
conditions and match estimated results with actual results.
Figure 20. Lateral velocity comparison (top) and Error (bottom) for all track conditions of wheel-rail interface.
Figure 21. Yaw rate comparison (top) and Error (bottom) for all adhesion condition of wheel-rail interface.
Figure 22. Creep comparison (top) and Error (bottom) for all adhesion condition of wheel-rail interface.
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Figure 23. Creep force comparison (top) and Error (bottom) for all adhesion condition of wheel-rail interface.
4.5. ERROR ANALYSIS
It is shown from Figure 4 to 23 that the Extended Kalman lter is a valid estimation technique
to estimate wheelset dynamics with authenticity. However, estimated creep force error in
Figure 23 became high for few moments during simulation (a spike seen at 2 seconds) due
to sudden change of adhesion condition from dry to wet.
Overall, EKF estimates the wheelset dynamics perfectly for dry, wet, poor and worst
adhesion conditions and can be used for condition monitoring of rolling stock.
5. CONCLUSION
As wheel-rail contact force is complex and non-linear function of slip and aected with
other vehicle parameters, therefore it is dicult to estimate by simple estimating techniques.
In this paper, the Extended Kalman lter is used to estimated lateral velocity and yaw rate
of railway wheelset as well as creep and creep force of wheel-rail interface and validated
through Simulink/MATLAB. EKF estimates not only wheelset dynamics for dry, wet, poor
and worst adhesion conditions but perfectly estimates for transition of all track conditions
during simulation.
Further, research is going to estimate wheel-rail dynamics in traction and braking modes,
also work is going on to implement the simulation work on FPGA platform.
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ACKNOWLEDGEMENT
The authors would like to acknowledge “Condition Monitoring System Lab at Mehran
University of Engineering and Technology, Jamshoro, part of NCRA project of Higher
Education Commission Pakistan, for supporting their work.
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