## 2 System Modeling  
### Vehicle System Modeling  
The picture of 4 WIS/4 WID electric vehicle is shown in Figure 1a. All the wheels of the vehicle can be independently steering and driving. In Figure 1b, the 3-degree of freedom (DOF) nonlinear vehicle dynamics model is applied to present vehicle dynamics features. In this model, the longitudinal degree of freedom is added to the normal two-degree-of-freedom model. Since this research mainly takes into account lateral, longitudinal, and yaw dynamics, other vehicle motions here, such as pitch, roll, and vertical motions are neglected in the presented vehicle model [25].  
In Figure 1b, CG is the vehicle's center of gravity; \(B_{1}\) s wheelbase; \(\beta\) is the vehicle sideslip angle; and \(a\) and \(b\) are the distance from CG to front axle and rear axle, respectively. The vehicle lateral dynamics including the longitudinal, lateral, and yaw motions can be given as follows:  
\[\left\{\begin{array}{l}\dot{v}_{x}=v_{y}\cdot\dot{\varphi}-\frac{1}{2m}C_{D}{ \it{Av}_{x}}^{2}-\mu g+\frac{1}{m}\sum F_{x}+\tau_{x}\\ \dot{v}_{y}=-v_{x}\cdot\dot{\varphi}+\frac{1}{m}\sum F_{y}+\tau_{y}\\ \ddot{\varphi}=\frac{1}{l_{z}}\sum M_{z}+\tau_{x}\end{array}\right. \tag{1}\]  
where \(m\) denotes the mass of the vehicle; \(\dot{v_{x}}\) and \(\dot{v_{y}}\) are the longitudinal and lateral acceleration, respectively; \(C_{D}\) denotes the aerodynamic resistance coefficient; \(A\) denotes windward area; \(l_{z}\) denotes the rotational inertia; \(\mu\) is the tire-road friction coefficient; \(g\) is the gravitational acceleration; and \(v_{x}\) and \(v_{y}\) represent the longitudinal and lateral velocity, respectively. \(\dot{\varphi}\) denotes the yaw rate and \(\ddot{\varphi}\) denotes the angular acceleration. \(\Sigma F_{x}\), \(\Sigma F_{y}\), and \(\Sigma M_{z}\) are the longitudinal tire resultant force, lateral tire resultant force, and yawing moment. \(\tau_{x}\), \(\tau_{y}\), and \(\tau_{z}\) represent the external disturbance input.  
Figure 1: 4 WIS/4 WID electric vehicle (**a**) Picture of 4 WIS/4 WID electric vehicle. (**b**) Schematic of 4 WIS/4 WID vehicle model.  
\(\Sigma F_{x}\), \(\Sigma F_{y}\), and \(\Sigma M_{z}\) can be written as:  
\[\sum F_{x}=F_{x1}+F_{x2}+F_{x3}+F_{x4} \tag{2}\]  
\[\sum F_{y}=F_{y1}+F_{y2}+F_{y3}+F_{y4} \tag{3}\]  
\[\sum M_{z}=\frac{B}{2}(F_{x2}-F_{x1}+F_{x3}-F_{x4})+a\big{(}F_{y1}+F_{y2}\big{)} -b\big{(}F_{y3}+F_{y4}\big{)} \tag{4}\]  
where \(F_{xi}\) and \(F_{yi}\) (\(i\) = 1, 2, 3, 4) are the tire forces of each wheel in the \(x\) direction and \(y\) direction (the subscript \(i\) denotes the front left, front right, rear left, and rear right wheels).  
As shown in Figure 2a,b, the tire forces of each wheel in the \(x\) direction and \(y\) direction are related with the wheel steering angle \(\delta_{i}\) (\(i\) = 1, 2, 3, 4) and longitudinal and lateral tire force \(F_{mi}\), \(F_{ni}\) (\(i\) = 1, 2, 3, 4), respectively, which can be expressed by matrix form:  
\[\left[\begin{array}{c}F_{x1}\\ F_{x2}\\ F_{x3}\\ F_{x4}\end{array}\right]=M\left[\begin{array}{c}F_{a1}\\ F_{a2}\\ F_{a3}\\ F_{x4}\end{array}\right]+N\left[\begin{array}{c}F_{b1}\\ F_{b2}\\ F_{b3}\\ F_{b4}\end{array}\right] \tag{5}\]  
\[\left[\begin{array}{c}F_{y1}\\ F_{y2}\\ F_{y3}\\ F_{y4}\end{array}\right]=P\left[\begin{array}{c}F_{a1}\\ F_{a2}\\ F_{a3}\\ F_{a4}\end{array}\right]+M\left[\begin{array}{c}F_{b1}\\ F_{b2}\\ F_{b3}\\ F_{b4}\end{array}\right] \tag{6}\]  
where matrices \(M\), \(N\), and \(P\) are given as follows:  
\[M=\left[\begin{array}{cccc}\cos\delta_{1}&&&\\ &\cos\delta_{2}&&\\ &&\cos\delta_{3}&\\ &&&\cos\delta_{4}\end{array}\right] \tag{7}\]  
\[N=\left[\begin{array}{cccc}-\sin\delta_{1}&&&\\ &-\sin\delta_{2}&&\\ &&&\sin\delta_{3}&\\ &&&\sin\delta_{4}\end{array}\right] \tag{8}\]  
\[P=-N \tag{9}\]