The core part of any study of rolling stock behavior is the wheel-track interaction patch because the forces produced at the wheel-track interface govern the dynamic behavior of the whole railway vehicle. It is significant to know the nature of the contact force to design more effective vehicle dynamics control systems and condition monitoring systems. However, it is hard to find the status of this adhesion force due to its complexity, highly non-linear nature, and also affected with an unpredictable operation environment. The purpose of this paper is to develop a model-based estimation technique using the Extended Kalman Filter (EKF) with inertial sensors to estimate non-linear wheelset dynamics in variable adhesion conditions. The proposed model results show the robust performance of the EKF algorithm in dry, wet/rain, greasy, and fully contaminated track conditions in traction and braking modes of a railway vehicle. The proposed model is related to the other works in the area of wheel-rail systems and a tradeoff exists in all conditions. This model is very useful in condition monitoring systems for railway asset management to avoid accidents and derailment of a train.

The rapid development in railway traffic across the world demands better acceleration and braking performance. The railway wheelset and wheel-rail contact patch play a vital role in the acceleration and braking performance of railway operation [

The estimation of adhesion coefficient, slip ratio, and lateral dynamics of railway vehicle successively in traction and braking modes is essential for both trip safety and passenger ease. But estimation of wheelset dynamics is a complicated process because the wheel-rail interface is an open system with changing external conditions. Many scholars have proposed some wheel-rail contact estimation techniques most of which are model-based, which have been summarized in Refs. [

Several methods have been proposed in the literature to accurately estimate the adhesion condition in railway transport. Most of these techniques are designed to work during the normal operation (steady-state) of a railway vehicle. Accurate adhesion information is not only important during the normal running conditions, it is also important during the traction and braking modes to avoid wheel-slip during traction and wheel-slide during braking. But due to the highly nonlinear behavior of the wheelset dynamics during the traction and braking modes, it is very difficult to accurately estimate the adhesion condition. In addition to the presence of nonlinearities during the traction and braking modes, the unpredictable environmental conditions present a serious challenge for researchers to accurately estimate adhesion at the wheel-rail interface. In this paper, we extend the works reported in Refs. [

This research work focuses on the extension of the railway wheelset dynamics as reported in Refs. [

In the above figure torque (T_{m}) of traction motor attached on one side of the wheelset, traction force (F_{t}), the normal force (F_{N}), and linear velocity (V_{x}) are labeled being important parameters involved in modeling.

The Adhesion coefficient is one of the most important parameters of wheelset because the dynamic response of the wheelset depends on it. The adhesion coefficient _{a} that is generated between the wheel-rail contact area to normal force N.

In normal condition with small slip ratio, the adhesion force is linear with slip ratio but for the large slip ratio the adhesion force becomes nonlinear and can be expressed as:

where j = longitudinal and lateral directions.

The total tangential force F_{a} of longitudinal and lateral directions can be calculated using the Polach formula [

where k_{A} is the reduction factor around adhesion, K_{S} is the reduction factor in a slip,

where

Each curve can be divided into three parts to describe the stable and unstable behavior of the wheelset. The initial portion is almost linear, the middle portion is nonlinear and is called the high slip ratio region and the last portion having a negative slope is the unstable region of the curve [

The values of Polach parameters used for tuning of the creep curves given in

Parameter | Dry condition | Wet condition | Greasy condition | Extremely slippery condition |
---|---|---|---|---|

k_{A} |
1 | 1 | 1 | 1 |

k_{S} |
1 | 1 | 1 | 1 |

u_{0} |
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 |

The slip ratio γ is the relative speed of the wheel to rail, the slip ratios of both wheels of wheelset in the longitudinal and lateral direction, and total slip ratio are presented in

The total slip ratio will be:

A complete wheelset model includes all relevant motions related to wheel-rail contact forces that help to study the wheelset dynamics. The equations of motion of the railway wheelset for longitudinal, lateral, rotational, torsional, and yaw dynamics are given below [

where

The centripetal force component

The description of the model parameters of the railway wheelset is given in

No. | Symbol | Parameter | Value and/or Unit |
---|---|---|---|

1 | γ_{xR,} γ_{xL} |
Right and left wheel slip ratios in longitudinal direction | |

2 | γ_{yR,} γ_{yL} |
Right and left wheel slip ratios in lateral direction | |

3 | γ_{R,} γ_{L} |
Total slip ratios of right and left wheel | |

4 | r_{0} |
Wheel radius | 0.5 meter |

5 | L_{g} |
Half gauge of track | 0.75 meter |

6 | λ_{w} |
Wheel conicity | 0.15 rad |

7 | ɷ_{R,} ɷ_{L} |
Angular velocities of right and left wheel | |

8 | V | Vehicle’s forward velocity | |

9 | Y | Lateral displacement | Output in meter |

10 | y_{t} |
Track disturbance in lateral direction | Track disturbance in meter |

11 | Ψ | Yaw angle | Radians |

12 | F_{xR,} F_{xL} |
Right and left wheel creep forces in longitudinal direction | |

13 | F_{yR,} F_{yL} |
Right and left wheel creep forces in lateral direction | |

14 | F_{R,} F_{L} |
Total creep forces of right and left wheel | |

15 | M_{v} |
Vehicle mass | 15000 Kg |

16 | I_{w} |
Yaw moment of inertia of wheelset | 700 Kgm^{2} |

17 | K_{w} |
Yaw stiffness | 5x10^{6} N//rad |

18 | m_{w} |
Wheel weight with induction motor | 1250 Kg |

19 | T_{m} |
Torque of traction motor | Input in Nm |

20 | T_{s} |
Torsional torque | |

21 | T_{R,} T_{L} |
Tractive torques on right and left wheel | |

22 | I_{R} |
Right wheel inertia | 134 Kgm^{2} |

23 | I_{L} |
Left wheel inertia | 64 Kgm^{2} |

24 | K_{s} |
Torsional stiffness | 6063260 N/m |

25 | θ_{s} |
Twist angle |

In the study of wheelset dynamics, it is important to develop and use a complete model that comprises all related motions associated with the contact forces because of powerful interactions among various motions of the wheelset play in both the longitudinal and lateral directions. Wheelsets are the element of railway vehicles that interact directly with the rail path and subsequently, the wheelset dynamics are directly affected by changing contact conditions.

A Simulink model of complete and non-linear railway wheelset, based on _{t} of 5 mm step is generated to simulate wheelset dynamics to show the existence of track disturbances. The dry and wet condition creep curves of

The main objective of this study is to develop a novel model-based technique to estimate the wheelset dynamics in different contact conditions. The model-based estimation schemes using discrete Kalman filter and Bucy Kalman filter have been successfully used by many researchers for estimation of wheelset parameters [

The nonlinear wheelset model discussed in Section 2 is used to develop the EKF algorithm.

The main objective of this study is to develop a state-of-the-art technique to detect the changes in wheel-rail contact conditions and only yaw and lateral dynamics are sufficient for detecting these changes. Therefore, longitudinal dynamics are not taken in

As the Extended Kalman filter like simple KF is a 2 step predictor-corrector algorithm [

Equations of predictor step:

Equations of corrector step:

where

Symbol | Description |
---|---|

discretized a-priori estimated process | |

discretized a-postriori estimated process | |

P_{k}^{-} |
a-priori estimate of the covariance of process error |

P_{k} |
an estimate of the covariance of measurement error |

F_{k} |
Jacobian matrix of process |

H_{k} |
Jacobian matrix of measurement |

Q_{k} |
process noise covariance |

R_{k} |
measurement noise covariance |

K_{k} |
Kalman gain |

m ̃_{k} |
measured output |

The five variables given in

The process variables are reproduced from

As the chosen process variables are extracted from the wheelset model and are continuous but the EKF algorithm is a discrete one, therefore equations from

Now the Jacobean matrix of process matrix

And the Jacobian of the measurement matrix

where:

The performance of EKF not only depends on Jacobian matrices but also the selection of Kalman gain and noise covariance contribute significantly. Kalman gain is calculated by using

A simulation model of the proposed estimation technique shown in

The measurement noise covariance matrix

Simulations are carried out in traction and braking modes of vehicle in five different conditions i.e., (i) Dry condition, (ii) Wet condition, (iii) Greasy condition, (iv) Extremely slippery condition, and (v) Transition from dry condition to extremely slippery condition.

The simulation in dry track conditions is carried out for 50 seconds in both accelerating and decelerating modes of a vehicle to calculate adhesion coefficient, slip ratio, and yaw rate.

In traction mode of a vehicle for 25 seconds of simulation time, tractive torque is applied to increase the linear velocity up to 30 m/sec (108 km/h), and then in braking mode of vehicle tractive torque applied in the reverse direction to reduce the velocity up to initial velocity i.e., 5 m/sec (18 km/h). In only 25 seconds, linear velocity increased from 18 km/h up to 108 km/h and in 25 seconds linear velocity decreased from 108 km/h back to the initial velocity. Both wheelset and estimator remain stable during the whole simulation time.

Applied torque and varying linear velocity are shown in

The simulation in wet track conditions is carried out for 50 seconds in both traction and braking modes of the vehicle to calculate the adhesion coefficient, slip ratio, and yaw rate.

In traction mode of the vehicle for 25 seconds of simulation time, tractive torque is applied to increase the linear velocity up to 90 km/h, and then in braking mode of vehicle tractive torque applied in the reverse direction to reduce the velocity up to the initial velocity i.e., 18 km/h. Applied torque and varying forward velocity are shown in

The simulation in greasy track condition is carried out for 50 seconds in both traction and braking modes of the vehicle to calculate adhesion coefficient, slip ratio, and yaw rate.

In the traction mode of the vehicle for 25 seconds of simulation time, tractive torque is applied to increase the linear velocity up to 63 km/h and then in the braking mode of the vehicle tractive torque is applied in the reverse direction to reduce the velocity up to initial velocity.

The simulation in extremely slippery track conditions is carried out for 50 seconds in both traction and braking modes of the vehicle to calculate adhesion coefficient, slip ratio, and yaw rate.

In the traction mode of the vehicle for 25 seconds of simulation time, tractive torque is applied to increase the linear velocity up to about 40 km/h, and then in braking mode of the vehicle tractive torque is applied in the reverse direction to reduce the velocity back up to 18 km/h. Applied torque and varying forward velocity are shown in

In this sub-section, the wheel-rail interface condition is changed during simulation from normal to extremely slippery adhesion condition in 25 seconds of simulation time and reversely adhesion condition changed from extremely slippery to dry track condition in the remaining time of the simulation. In the traction mode of the vehicle for 25 seconds of simulation time, tractive torque is applied to increase the linear velocity maximally 63 km/h, and then in the braking mode of the vehicle tractive torque is applied in the reverse direction to reduce the velocity up to initial velocity.

Further applied torque and varying linear velocity are shown in

As shown in _{a}) given in

The absolute accuracy indices for the estimation for all adhesion conditions are shown in

Track condition | Adhesion coefficient | Slip ratio | Yaw rate | |||
---|---|---|---|---|---|---|

MV | ||||||

0.42 | 0.00102 | 0.011 | 0.000006 | 0.083 | 0.00004 | |

0.27 | 0.00006 | 0.032 | 0.000002 | 0.013 | 0.00001 | |

0.177 | 0.00003 | 0.033 | 0.000002 | 0.0071 | 0.00001 | |

0.088 | 0.000008 | 0.042 | 0.000001 | 0.0032 | 0.000009 | |

0.42 | 0.00098 | 0.185 | 0.000006 | 0.035 | 0.00004 |

MV: Maximum Value; AAI: Absolute Accuracy Index.

It can be seen that the values of absolute accuracy indices confirm the effectiveness of the estimator. To test the performance of the proposed estimation setup, some related and recent works are chosen as the comparison techniques with equivalent system and setup. The work reported in Refs. [

The proposed EKF-based estimator shows outperformance in wheelset dynamics for dry, wet, greasy, and extremely slippery track conditions in both traction and braking modes of railway vehicles. Therefore, the proposed model can very well suit for condition monitoring of rolling stock.

The performance of railway operation mainly is affected by wheel-rail contact forces but it is not possible to measure these contact forces and interrelated dynamics directly, therefore it is necessary to estimate these wheelset dynamics through state of art technique. In this research paper, a railway wheelset model and a novel observer-based estimator are developed in Simulink/MATLAB to calculate and estimate nonlinear wheelset dynamics. The estimator based on the extended Kalman filter is used to estimate adhesion coefficient, slip ratio, and yaw rate effectively in dry, wet, greasy and extremely slippery track conditions. The functioning of the EKF algorithm is assessed by using absolute accuracy indices and compared with other relevant and recent research work. The estimator not only verified excellent performance in the normal operation of a railway vehicle on a normal track but equally depicted robustness in traction and braking modes of the vehicle in wet, oily, and extremely slippery track conditions. The validity of the estimator is also checked in the transition of adhesion conditions from dry to extremely slippery and vice-versa during the simulation. In the future, this approach will be implemented on Field Programmable Gate Arrays (FPGA) platform for real-time condition monitoring of wheelset dynamics to avoid the accidents and derailment of railway vehicle.

The authors would like to acknowledge the “NCRA Condition Monitoring Systems Lab” at Mehran University of Engineering and Technology, Jamshoro, part of the NCRA project of Higher Education Commission Pakistan, for supporting this work.