# MPC and PSO Based Control Methodology for Path Tracking of 4 WS4 WD Vehicles
## 2 Vehicle System Modelling
### Tire Model  
The actual generation of wheel forces are from the contact between tires and the ground. The wheel forces depend on the surface friction, load, and the intrinsic properties of the tire. To analyze the wheel forces \(\mathbf{F}_{wi}\) in Equation (9), the tire model is built as shown in Figure 4.  
Considering the generation of wheel forces and external disturbances, \(\mathbf{F}_{wi}\) can be expressed as  
\[\mathbf{F}_{wi}=\begin{bmatrix}F_{wli}&F_{wLi}\end{bmatrix}^{T}=\begin{bmatrix} F_{li}-F_{gli}&F_{Li}-F_{gli}\end{bmatrix}^{T}, \tag{10}\]  
where \(F_{li}\) and \(F_{Li}\) (\(i=1,\ldots,4\)) are longitudinal and lateral forces caused by wheel slip, respectively. \(F_{gli}\) and \(F_{gli}\) denote the terrain disturbances corresponding to \(F_{li}\) and \(F_{Li}\).  
According to [20], the longitudinal force can be considered proportional to the slip ratio in small range, which is expressed by  
\[F_{li}=\begin{cases}\begin{array}{c}k_{li}F_{Ni}\frac{\zeta_{i}}{\Omega,1}, \end{array}|\begin{array}{c}\xi_{i}|\leq 0.1,\\ k_{li}F_{Ni},\end{array}|\begin{array}{c}\xi_{i}|>0.1,\end{array}\end{cases} \tag{11}\]  
Figure 3: Driving unit dynamic model.  
where \(F_{Ni}\) is the weight force on wheel \(i\). In the simulation, the load transfer due to the longitudinal and lateral accelerations and roll angle are taken into consideration. \(F_{Ni}\) can be calculated by considering longitudinal and lateral load transfer according to [21]. The longitudinal slip stiffness \(k_{li}\) is determined by the tire type and terrain condition. \(\varsigma_{i}\) is the longitudinal slip ratio of wheel \(i\) presented in [22]:  
\[\varsigma_{i}=\begin{cases}\dfrac{R_{w}\omega_{i}-v_{wi}}{R_{w}\omega_{i}},&R _{w}\omega_{i}>v_{wi}>0,\\ \dfrac{R_{w}\omega_{i}-v_{wi}}{v_{wi}},&R_{w}v_{wi}>\omega_{i}>0,\\ 0,&R_{w}v_{wi}=\omega_{i}\geq 0,\end{cases} \tag{12}\]  
where \(R_{w}\) and \(\omega_{i}\) denote the tire radius and angular velocity of wheel \(i\) as shown in Figure 4b. \(v_{wi}\) represents the actual velocity of the wheel \(i\), which can be calculated by the vehicle geometry as well as velocities \(V_{l}\), \(V_{r}\) and \(\Omega\) shown in Figure 2. Note that the model is proposed only considering the vehicle is moving forward. Thus, \(v_{wi}\) and \(\omega_{i}\) are set to be nonnegative. As is discussed in Equation (12), the slip ratio is zero when the vehicle is resting.  
Finally, as per Figure 4b, the wheel dynamic equation indicating the relationship between \(T_{i}\) and \(F_{li}\) can be written as  
\[T_{i}=(F_{li}+F_{ri})R_{w}+\dot{\omega}_{i}J_{w}, \tag{13}\]  
where \(J_{w}\) represents the wheel inertia, and \(F_{ri}\) is the rolling resistance calculated by  
\[F_{ri}=F_{Ni}(K_{r0}+K_{r1}V_{l}^{2}), \tag{14}\]  
where \(K_{r0}=0.015\) and \(K_{r1}=7\times 10^{-6}\) s\({}^{2}\)/m\({}^{2}\) are the parameters for common car tires [23].  
The steering motion of wheels result in the lateral slip velocities \(v_{si}\), which causes the actual direction of wheel velocity \(v_{wi}\) to differ from the wheel center plane by the slip angle \(\alpha_{i}\). According to the tire lateral characteristic curve [24], a linear model for \(F_{Li}\) is given in  
Figure 4: Tire model. (**a**) Top view of the tire; (**b**) lateral view of the tire.  
\[F_{Li}=\begin{cases}-k_{Li}F_{Ni}\frac{\alpha_{i}}{5},&|\alpha_{i}|\leq 5^{ \circ},\\ -k_{Li}F_{Ni},&\alpha_{i}>5^{\circ},\\ k_{Li}F_{Ni},&\alpha_{i}<-5^{\circ},\end{cases} \tag{15}\]  
where \(k_{Li}\) is the lateral slip stiffness of the wheel \(i\). Note that, due to the sign conventions in \(X_{Wi}-O_{Wi}-Y_{Wi}\) and \(X_{L}-O_{L}-Y_{L}\), a negative slip angle causes a positive lateral force and vice versa.  
As shown in Figure 4a, the slip angle \(\alpha_{i}\) can be calculated by steering angle \(\delta_{i}\) and side slip angle \(\beta_{i}\),  
\[\alpha_{i}=\beta_{i}-\delta_{i}, \tag{16}\]  
where the expression of \(\beta_{i}\) is available in our previous work presented in [25].