# An End-of-Line Test System for Pneumatic ABS Controllers in Commercial Vehicles
## 3 Wheel Speed Simulation  
### Wheel Speed Simulation Model  
The wheel speed simulation model is the basis for generating wheel speed signals and ABS valve theoretical drive signals under specific test environments. In order to derive the relationship among the test conditions, the wheel speed, and ABS valve operation logic, a wheel speed simulation model was built based on vehicle dynamics and the working principles of the pneumatic ABS system. The test conditions were used as inputs, while the wheel linear velocity and ABS valve operation logic acted as outputs. The wheel speed simulation model is shown in Figure 4.  
As the test conditions proposed in this paper all belong to linear braking conditions, the lateral motion, the yaw motion, and the roll motion of the vehicle were not considered when establishing the dynamic equations. The aerodynamic force was also ignored. The vehicle's motion equation can be obtained by force analysis, which is expressed as follows:  
\[-m\dot{v}=-(F_{x1}+F_{x2}+F_{x3}+F_{x4}) \tag{1}\]  
where \(m\) (kg) is the mass of the whole vehicle, \(\dot{v}\) (m\(\cdot\)s\({}^{-2}\)) is the acceleration of the vehicle, and \(F_{xi}\) (\(i\) = 1, 2, 3, 4) (N) is the longitudinal force of each tire. The moment balance of the wheel rotation is as follows:  
\[J\dot{\omega}_{i}=-T_{bi}+F_{xi}\cdot R \tag{2}\]  
where \(J\) (N\(\cdot\)rad\({}^{-1}\)) is the moment of inertia, \(T_{bi}\) (N\(\cdot\)m) is the braking torque of each wheel, \(\omega_{i}\) (rad/s) is the angular velocity of each wheel, and \(R\) (m) is the effective radius of the wheel. During the braking process, the vertical load of each wheel changes due to acceleration. Considering the influence of vertical load transfer, the vertical load of each wheel can be defined as follows:  
\[\left\{ \begin{array}{l}F_{Zi}=\frac{mg_{i}}{2^{I}}+\Delta F_{Zb}\left(i=1,2 \right)\\ F_{Zi}=\frac{mg_{i}}{2^{I}}-\Delta F_{Zb}\left(j=3,4\right)\end{array} \right. \tag{3}\]  
where \(F_{Zi}\) (N) and \(F_{Zj}\) (N) are the vertical loads of each wheel, \(\Delta F_{Zb}\) (N) is the change in the vertical reaction, and \(l_{f}\) (m) and \(l_{r}\) (m) are the distance from front and rear axle to centroid. Figure 4: Structure of wheel speed simulation model. It is assumed that the road surface is ideal, i.e., there is no horizontal deformation or normal ground deformation. According to the magic formula [14], the relationship between the slip rate and tire longitudinal force can be expressed as follows:  
\[F_{x}(S,F_{z},\mu)=f(\mu)D_{x}\sin\{C_{x}\arctan\left[B_{x}S-E_{x}(BS-\arctan B_{ x}S)\right]\} \tag{4}\]  
where \(\mu\) is the road adhesion coefficient, \(S\) is the slip rate, and \(F_{z}\) is the vertical load. The tire force data can be obtained by IPG/TruckMaker. The coefficients in the mathematical model of the tire force are obtained by fitting them as seen below:  
\[\left\{ \begin{array}{l}D_{x}=T_{1}F_{z}{}^{2}+T_{2}F_{z}+T_{3}\\ B_{x}C_{x}D_{x}=T_{4}F_{z}{}^{2}+T_{5}F_{z}\\ C_{x}=T_{6}\{1\pm\left[1-2\arcsin(T_{7})/\pi\right]\}\\ E_{x}=T_{8}F_{z}+T_{9}\end{array} \right. \tag{5}\]  
The coefficients in Equation (5) are shown in Table 2.  
Since the braking torque lag has little effect on the EOL test system, the brake model can be obtained by ignoring it. The model can be expressed as:  
\[T_{b}=K_{b1}\cdot p\cdot A_{b}\cdot\mu_{b}\cdot r_{b}=K_{b1}\cdot K_{b2}\cdot p \tag{6}\]  
where \(K_{b1}\) is the efficiency coefficient from brake chamber to brake, \(p\) (Pa) is the pressure of chamber, \(A_{b}\) (m\({}^{2}\)) is the brake area, \(r_{b}\) (m) is the effective brake radius, \(\mu_{b}\) is the friction coefficient, and \(K_{b2}\) is the brake efficiency coefficient. System identification [15] is a method for establishing the mathematical mode according to the input-output relationship of a dynamic system. The most commonly used identification methods include the step response method and frequency response method.