# Assessment of Active Wheelset Steering System Using Computer Simulations and Roller Rig Tests
## 3 Computer Simulations
### Track Radius Estimation  
In general, two main approaches for the track radius estimation exist:  
1. Utilization of the track map and detecting the position of the vehicle on the track. This method assumes the knowledge of the curvature along the track, which must be available to the controller, for example in the form of a look up tables. The position of the vehicle on the track could be detected by GPS navigation or by measurement of wheelset revolutions. Integration wheelset revolutions is rather inaccurate due to slips in wheel-rail contacts and unknown value of exact wheel radius. However, it can be refined by track marks at certain known positions. Most of the railway tracks are already equipped with such marks, for example balise transmission modules of European Train Control System could be utilized for this purpose. 2. Estimation of the track radius using on board sensors. Although vehicle positioning systems based on the GPS navigation exist and achieve continuously improving parameters in terms of accuracy and reliability, estimation of track radius by of onboard sensors was finally chosen. The main advantages of an onboard system are:  
* independence of GPS signal which can be difficult to reach in urban areas with high buildings around the track, deep valleys, or in tunnel sections;
* independence of wheel radius measurement or estimation;
* independence of the track map. This gives the possibility to operate the vehicle on any track without the need to provide the controller by track data, which can be very important in the event of unexpected obstacles on the track, lockouts, etc. Due to the above reasons the estimation of the track radius by onboard sensors is proposed in this study. As an input for the tack radius estimation the rotation angle around vertical axis of the bogie towards the car body can be used. Assuming that the wheelsets follow the track centreline, the track curvature can be expressed by:  
\[\rho=\frac{1}{\bar{R}}=\frac{2\sin\alpha}{u}\cong\frac{2\alpha}{u} \tag{1}\]where \(R\) is track radius, \(\rho\) is track curvature, \(u\) is bogie distance, and \(\alpha\) is angle of rotation of bogies towards the car body, see Figure 6.  
Equation (1) is fully valid only when the whole vehicle is in a curved track of a constant curvature. Figure 7 shows simulation results of a vehicle negotiation of a \(R=250\) m track curve. It can be seen that in the transition track section, where the track curvature changes, the track curvature estimated by Equation (1) lags behind the actual track curvature on the leading bogie, whereas track curvature estimated on trailing bogie is ahead. This phenomenon could be considerably eliminated by the method proposed in [18]. The method utilizes not only the angle of bogie rotation towards the car body \(\alpha\), but also its derivative:  
\[\rho=\frac{1}{R}=\frac{2\alpha}{u}+\frac{\dot{\alpha}}{\
u} \tag{2}\]  
where \(\dot{\alpha}\) is time derivative of \(\alpha\) and \(\
u\) is vehicle forward velocity. Figure 8 shows track curvature estimation by Equation (2) in \(R=250\) m curve. Figure 6: Planar view of a vehicle in the constant curvature track. Figure 7: Track curvature estimated by bogie to car body rotation. The curvature estimation leg is virtually eliminated, although small overshoots occur around points where the second derivative of track curvature changes. The error of the track curvature estimation for both calculation algorithms is plotted on Figure 9. The algorithm according Equation (2) reduces the track estimation error in the transition track by more than 50% compared to the Equation (1). The track estimation error of Equation (2) in the transition track is less than 7%. One problem is that the assumption of wheelsets that are perfectly aligned with the track centreline is not fully satisfied. A wheelset can move towards the track centreline in the lateral direction within the gauge clearance.