# Research on Active Anti-Slip Control of High-Speed Trains Based on High-Order Sliding Mode
## 2 Problem Formulation  
### Train Model  
The primary focus of this paper is on anti-slip control of trains considering wheel-rail adhesion. Therefore, a single-point mass model of the train body is employed in this study. The dynamic analysis shown in Figure 1 leads to the formulation of the dynamic equations governing the wheels and the vehicle body as follows:  
\[\frac{dx}{dt}=v \tag{1}\]  
\[M\frac{dv}{dt}=\sum_{i=1}^{32}F_{\mu i}-F_{d} \tag{2}\]  
\[J\frac{dv_{wi}}{dt}=rT_{i}-F_{\mu i}r^{2}-B_{i}w_{wi}r \tag{3}\]  
where \(x\) represents the displacement, \(M\) is the train's mass, and \(F_{d}\) denotes the basic resistance force acting on the train. The term \(F_{\mu i}\) corresponds to the adhesive force for the \(i\)th wheel. The variables \(w_{wi}\) and \(v_{wi}\) represent the angular velocity and linear velocity of the \(i\)th wheel, respectively. The torque applied by the motor or air brake on the \(i\)th wheel is denoted by \(T_{i}\). for the motor or air brake applied to the \(i\)th wheel moment. In Equation (2), the summation is performed over 32 axes, reflecting the total number of axes on the train. Furthermore, the parameter \(r\) represents the wheel radius, while \(B_{t}\) denotes the damping coefficient for the \(i\)th wheel. For a specific vehicle model, the expression for the basic resistance force \(F_{d}\) can be fitted from wind tunnel tests. However, the parameters of this expression are generally difficult to obtain accurately and may vary with different operating environments. It can be modeled as  
\[F_{d}=(6.796+0.062v+0.00143v^{2})M/1000 \tag{4}\]  
where \(v\) represents the train's velocity, and \(M\) denotes the train's mass. According to (2), the adhesive force \(F_{\mu i}\) can be expressed as the product of the coefficient of adhesion \(\mu\) and the axial weight \(Wg\):  
\[F_{\mu i}=\mu Wg \tag{5}\]  
where \(\mu\) is the coefficient of adhesion, \(W\) represents the axial load, and \(g\) is the gravitational acceleration. Experiments have shown that the coefficient of adhesion \(\mu\) can be expressed as a function of creep speed \(v_{s}\) and vehicle speed, as described in Equation (6). The creep speed \(v_{s}\) is defined as the difference between the wheel speed and the vehicle speed:  
\[\left\{ \begin{array}{l}\mu=\frac{2\eta_{\sigma f}}{\pi}(\frac{k_{x}\epsilon} {1+(k_{x}\epsilon)^{2}}+\arctan(k_{s}\epsilon))\\ \epsilon=\frac{1}{4}\frac{C\pi abG}{Q_{\mu\sigma}}s_{x}\end{array} \right. \tag{6}\]  
\[\mu_{\sigma f}=\mu_{\sigma f0}\left[(1-A)\mathrm{e}^{-Bv_{s}}+A\right] \tag{7}\]  
In Equations (6) and (7), \(k_{A}\), \(k_{s}\), \(A\), \(B\) and \(\mu_{\sigma f0}\) are parameters that need to be set based on different operating conditions. \(C\) is the Kalker creep coefficient, \(G\) represents the shear modulus, \(a\) and \(b\) are the long and short semi-axes of the elliptical contact patch, and \(Q\) is the axial load. The longitudinal creep rate \(s_{x}\) is defined by the following equation:  
\[s_{x}=v_{s}/v_{t} \tag{8}\] \[v_{s}=v-v_{w}\]  
As shown in Figure 1, the traction wheel circumference braking force of the train is provided by the motor, while the braking force of the brake wheels is assumed to be supplied solely by the air brake system. The motor control method used is vector control, although the principles of motor vector control are not discussed in this paper. The air brake is modeled as follows:  
\[F_{b}=\sum K\varphi_{k}\frac{R}{r} \tag{9}\]  
\[K=\frac{\pi}{4}d_{z}^{2}p\gamma_{z}\eta_{z} \tag{10}\]  
\[G(s)=\frac{1}{1+T_{p}s} \tag{11}\]  
where \(K\) represents the pressure on each brake pad, \(\varphi_{k}\) is the friction coefficient of the brake pad, and \(R\) denotes the average friction radius of the brake disc. The parameter \(d_{z}\) refers to the working diameter of the brake cylinder, \(p\) represents the brake cylinder pressure, \(\gamma_{z}\) is the brake clamp multiplier, and \(\eta_{z}\) is the basic brake transmission efficiency. \(T_{p}\) is the time constant of the brake cylinder for flushing and exhausting.