EXTENDED KALMAN FILTER FOR ESTIMATION OF CONTACT FORCES AT WHEEL-RAIL INTERFACE

The wheel-track interface is the most significant 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 affected 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.


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 profile by using conicity and wheel-rail contact forces.
A method using Kalman filter 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 filters are proposed for (i) wheel slip detection and, (ii) real time contact condition and adhesion estimation in Hussain andMei (2010, 2011). In Hussain, Mei and Ritchings (2013) and Ward, Goodall and Dixon (2011), the development of techniques based on Kalman-Bucy filter 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 filter is difficult 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 filter 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 different processing methods, i.e., model-based approach using Kalman-Bucy filter and non-model based using direct data analysis, is presented for onboard indirect detection of low adhesion condition in Hubbard et al. (2013aHubbard et al. ( , 2013b. 3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143 Edición Especial Special Issue Abril 2020 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 coefficient 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 effectively 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.  (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 filter is designed for non-linear railway wheelset model to estimate lateral velocity and yaw rate of wheelset as well as creep and creep force.

MODELING OF NON-LINEAR WHEELSET
The motion of a railway vehicle is directed by interaction forces produced at wheeltrack contact, which change non linearly with respect to creepage and are affected 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. 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) 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 simplified longitudinal creep equations because only yaw and lateral dynamics are sufficient for detecting these changes. Further , so the simplified creep equations used in above model become as: (3) (4) 3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 -4143 Edición Especial Special Issue Abril 2020 The creepages in lateral direction are expressed as: While in equations (6) total creep of the wheels is depicted.
As the wheel-rail contact forces govern railway vehicle's dynamics are creep forces and are the function of creeps. The adhesion coefficient is the ratio of tangential force that is creep force to normal force and hence is also a function of creep. Figure   Following equations illustrate creep forces and adhesion coefficient.
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).
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 -4143 Edición Especial Special Issue Abril 2020 (8) Where U is friction coefficient, 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: Where u 0 is maximum friction coefficient at zero creep velocity, A is ratio of friction coefficient at infinity creep velocity to u 0 and B is coefficient of exponential friction decrease.
Where 3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 -4143 Edición Especial Special Issue Abril 2020 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.

DESIGNING OF EXTENDED KALMAN FILTER FOR ESTIMATING NON-LINEAR WHEELSET MODEL
Being non-linear nature of railway wheelset model, it is difficult to estimate the wheelset dynamics with ordinary estimation techniques. Therefore Extended Kalman filter is used to estimate wheelset dynamics and contact force in all adhesion conditions. Kalman filter 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 filter with generic scheme.

2012)
From non-linear model of railway wheelset, single equation (18) As the Extended Kalman filter uses a 2 step predictor-corrector algorithm (Welch & Bishop, 2001). The predictor step is given by And the equations of corrector step are, Where f and h are non-linear functions relating to process and measurement states, while: and Nomenclature of EKF algorithm is given in below table.

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

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    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.

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.

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.

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

ERROR ANALYSIS
It is shown from Figure 4 to 23 that the Extended Kalman filter 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.

CONCLUSION
As wheel-rail contact force is complex and non-linear function of slip and affected with other vehicle parameters, therefore it is difficult to estimate by simple estimating techniques.
In this paper, the Extended Kalman filter 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.