## 4 Modeling of Low-Speed Steering Resistance Torque Considering Tire/Pavement Friction
### Aligning Torques Caused by the Kingpin Caster and Pneumatic Trail  
Due to the centrifugal effect, the lateral force \(F_{y}\) exists at the center of steered wheel when the vehicle steers. Accordingly, the lateral reaction force \(F_{Y}\), which is also known as the cornering force, is generated by the pavement on the steered wheel. The cornering force produces the aligning torques which is related to the kingpin caster trail and pneumatic trail. The detailed solution procedure of the aligning torques caused by the kingpin caster and pneumatic trail is shown in Figure 6. The 2-degree-of-freedom (2-dof) vehicle dynamic model is employed here to calculate the ideal side-slip angle and yaw rate. It can also be used to identify and estimate the actual side-slip angle. Under the 2-dof vehicle model, the lateral force and the aligning torque are considered acting on the front axle as a whole, which is consistent with the processing ideas in the calculation of \(M_{f}\) and \(M_{\theta}\) in Sections 4.1 and 4.2. The corresponding vehicle motion differential equations of the 2-dof vehicle model in Figure 6 are expressed as Equation (12) [31]:  
\[\left\{ \begin{array}{c}(k_{1}+k_{2})\beta+\frac{1}{u}(ak_{1}-bk_{2})\omega_{r}- k_{1}\delta=m\big{(}\dot{v}+u\omega_{r}\big{)}\\ (ak_{1}-bk_{2})\beta+\frac{1}{u}\big{(}a^{2}k_{1}+b^{2}k_{2}\big{)}\omega_{r}- ak_{1}\delta=I_{Z}\dot{\omega}_{r}\end{array} \right. \tag{12}\]  
where \(a\) and \(b\) are respectively the distance from the front and rear axle to the center of mass, \(k_{1}\) and \(k_{2}\) are respectively the cornering stiffness of the front and rear wheel, \(u\) and \(v\) are respectively the longitudinal velocity along the \(x\)-axis and the lateral velocity along the \(y\)-axis of the vehicle, \(I_{Z}\) is the rotational inertia around the \(z\)-axis, \(\beta\) is the side-slip angle of the mass center, \(\omega_{r}\) is the yaw rate and \(\delta\) is the steered wheel angle. According to the 2-dof vehicle motion differential equations in Equation (12), the simulation model is established in the MATLAB/Simulink environment, which takes the forward velocity \(u\) and steered wheel angle \(\delta\) as input, and takes the side-slip angle of mass center \(\beta\) and yaw rate \(\omega_{r}\) as output. The desired side-slip angle of the front wheel \(\alpha_{1}\) can be obtained by one more step calculation, which is:  
\[\alpha_{1}=-(\delta-\xi)=\beta+\frac{a\omega_{r}}{u}-\delta \tag{13}\]  
When the vehicle is steering, the tire easily gets into a nonlinear state. Combined with the side-slip angle of the tire and vertical load, the 'Magic Formula' tire model is used to calculate the lateral force \(F_{y}\) and aligning torque caused by the pneumatic trail \(M_{Z}\), so as to accurately describe the mechanical characteristics of the tire. The expressions of the lateral force \(F_{y}\) and aligning torque \(M_{Z}\) are as follows [32]:  
\[\left\{ \begin{array}{c}F_{y}=D_{1}\sin\{C_{1}\text{arctan}[B_{1}(1-E_{1}) \alpha+E_{1}\text{arctan}(B_{1}\alpha)]\}\\ M_{Z}=D_{2}\sin\{C_{2}\text{arctan}[B_{2}(1-E_{2})\alpha+E_{2}\text{arctan}(B _{2}\alpha)]\}\end{array} \right. \tag{14}\]  
where \(D_{1}\), \(B_{1}\), \(C_{1}\) and \(E_{1}\) are respectively the peak factor, stiffness factor, shape factor and curvature factor for the solution of lateral force. \(D_{2}\), \(B_{2}\), \(C_{2}\) and \(E_{2}\) are respectively the peak factor, stiffness factor, shape factor and curvature factor for the solution of the aligning torque and \(\alpha\) is the side-slip angle of the tire. Under a single steering condition, the coefficients of the 'Magic Formula' can be referred to in the literature [33]. The curves of the lateral force \(F_{y}\) and the aligning torque caused by the pneumatic trail \(M_{Z}\) with the side-slip angle of the tire \(\alpha\) are drawn according to Equation (14) in the MATLAB, which are shown in Figure 7a,b. Figure 6: Solution procedure of the aligning torques caused by the kingpin caster and pneumatic trail.