## 2 System Overview and Vehicle Model
### Vehicle Model  
The nomenclature used in this paper is presented in Table 1. In this section, we present a 7-DOF dynamic model of an electric vehicle for the design of the adaptive-layer controller to enhance both the lateral stability and battery energy savings, as shown in Figure 2. The longitudinal, lateral, and yaw motions of the vehicle body are considered with the pitch, roll, and vertical motions being ignored. The rotational motion of the front left, front right,  
Figure 1: Adaptive stability and energy-saving control system for the 4 WID-EV. rear left, and rear right wheels are indexed by \(\beta\), \(f\), \(f\), \(r\), \(l\), and \(rr\), respectively. Let \(O\) denote the origins of the vehicle body frame. Note that the origin \(O\) is tied to the center of gravity (CG) of the vehicle body. In the vehicle body frame, \(l_{f}\) and \(l_{r}\) represent the distances from the origin \(O\) to the front and rear axles, resulting in a wheelbase \(L=l_{f}+l_{r}\). Let \(b_{f}\) and \(b_{r}\) represent the front and rear track width, and \(\delta\) and \(i_{t}\) represent the steering angles and the steering ratio, respectively. The Ackermann steering characteristics are employed to model the kinematics of the vehicle, which means that all wheels possess an instantaneous center of rotation. Based on geometric correlations, the following relation holds:  
\[\delta_{f}=\frac{\delta}{i_{t}} \tag{1}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Definition** & **Symbol** & **Definition** & **Symbol** \\ \hline vehicle mass & \(m\) & vertical loads at each wheel & \(F_{z,i}\) \\ steering angle & \(\delta\) & gravitational acceleration & \(g\) \\ front wheel steering angle & \(\delta_{f}\) & steering ratio & \(i_{t}\) \\ front right wheel steering angle & \(\delta_{fr}\) & rolling friction coefficient & \(f\) \\ front left wheel steering angle & \(\delta_{fl}\) & wheel rolling inertia & \(l_{w}\) \\ front to rear axle distance & \(L\) & yaw moment of inertia & \(l_{z}\) \\ distance from CG to front axle & \(l_{f}\) & motor torque on each wheel & \(T_{i}\) \\ distance from CG to rear axle & \(l_{r}\) & motor rotational speed & \(n_{i}\) \\ front track width & \(b_{f}\) & motor efficiency & \(\eta_{i}\) \\ rear track width & \(b_{r}\) & angel velocity of each wheel & \(\omega_{i}\) \\ vehicle longitudinal velocity & \(v_{x}\) & wheel rolling radius & \(r\) \\ vehicle lateral velocity & \(v_{y}\) & road friction coefficient & \(\mu\) \\ yaw rate in vehicle body & \(\gamma\) & cornering stiffness of total tires & \(K\) \\ steady-state yaw rate & \(\gamma_{\rm ss}\) & cornering stiffness of the front tires & \(K_{f}\) \\ side slip angle in vehicle body & \(\beta\) & cornering stiffness of the rear tires & \(K_{r}\) \\ steady-state side slip angle & \(\beta_{\rm ss}\) & external yaw moment & \(M_{z}\) \\ tire longitudinal forces & \(F_{x,i}\) & modified external yaw moment & \(M_{z}^{\prime}\) \\ tire lateral forces & \(F_{y,j}\) & total longitudinal torque & \(T_{\rm req}\) \\ total lateral forces of front tires & \(F_{y,f}\) & maximum torque on each wheel & \(T_{\rm max}\) \\ total lateral forces of rear tires & \(F_{y,r}\) & state of charge & SOC \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nomenclature. Figure 2: Schematic of the vehicle dynamics model and the tire longitudinal dynamics model.