## 3 Simulation Model
### Modelling Road  
Road conditions highly determine vehicle stability, which connects to its interaction with the wheel [56]. In terms of modeling the road, it mainly comprises a radius of circular curve, superelevation, and rutting. A set of horizontal curves with different values of radius were established, in order to test vehicle stability on curve segments, with a total length of 1200 m. Superelevation was set in the form of a lateral pavement gradient. Concerning modeling pavement rutting, the selection of rutting parameters is of particular importance. When a vehicle drives parallel to the rutting direction, minor effects on driving safety may occur. However, in the case of changing lanes or directions, a vehicle directly drives across a rut, thereby wandering behavior occurs [57], in which the vehicle may show unstable, lateral, or yaw motions, as if the vehicle went through a hole. As presented in Figure 5, \(h\) denotes the rutting depth, and \(w\) is the rutting width. The rutting shape is commonly described with trigonometric functions [58], as given by Equation (16). The inclination angle of the tire is expressed as Equation (17). \[z=\frac{h}{2}\bigg{(}1-\cos\bigg{(}\frac{2\pi y}{w}\bigg{)}\bigg{)} \tag{16}\]  
\[\psi=\frac{dz}{dy}=\frac{\pi h}{w}\sin\bigg{(}\frac{2\pi y}{w}\bigg{)} \tag{17}\]  
Though conjoint force analysis with a rut, the equation of vehicle motion is given by:  
\[m\ddot{y}+\frac{C_{Fa}}{V_{x}}\dot{y}+\frac{2\pi^{2}h}{w^{2}}\big{(}F_{zi}-C_{ F\dot{\psi}}\big{)}y=0 \tag{18}\]  
where \(C_{Fa}\) denotes the cornering stiffness of the tire, \(C_{F\dot{\psi}}\) denotes the camber stiffness of the tire, \(F_{z}\) is the vertical force exerted at tire, \(m\) equals tire mass, and \(V_{x}\) is the driving speed along \(x\) axis. Equation (18) presents free vibration equation with damping when the tire negotiates a rut. It can be found that the damping coefficient is proportional to \(C_{Fa}\), but inversely proportional to \(V_{x}\). Additionally, stiffness \(\frac{2\pi^{2}h}{w^{2}}\big{(}F_{z}-C_{F\dot{\psi}}\big{)}\) is obviously decided by rutting depth and rutting width. So, a higher driving speed will lead to a large vibration frequency while crossing a rut. Meanwhile, the rutting dimension has an impact on the vibration properties of the vehicle. Figure 5: Diagram of the wheel driving across a rut. The lateral vehicle dynamics when instantaneously crossing a rut are rather complex, not only for a single tire or a single axle, but also for the whole vehicle. So, evaluating the dynamic condition of the vehicle considering rutting is more crucial than roughness, cracking, and other pavement distresses. Further, from the equation of vehicle motion (18), it implies that the rutting depth and rutting width show strong connection with the dynamic vibration properties of the vehicle. The three main parameters of rutting are generally used to characterize driving safety. Normally, the maximum rutting depth (RD) is used to evaluate the rutting conditions, which affects the driving safety by the pattern of height difference and fluctuation. The rutting width (RW) is also an important parameter to characterize the rutting shapes. When the ruts are narrow, the wheels experience large changes in elevation in a very short period of time, and the vehicle's recovery adaptations are unable to quickly follow the changes, making it inclined to lose stability. A smaller RW will present a greater impact on the driving stability. The width-height ratio (WHR) of the rutting sidewall denotes the dip angle of rutting sidewall, which refers to the ratio of the width-to-height of the rut sidewall. A higher WHR indicates a greater resistance to be overcome when changing the trajectory of the vehicle, and the greater the resistance to steering that the vehicle will experience. Therefore, in order to excellently characterize the effects of height fluctuation and lateral crossing when driving across rutting, WHR is selected as a factor that influences driving stability in different simulation cases.