# Stability Analysis of Lane-Keeping Assistance System for Trucks under Crosswind Conditions
## 2 Materials and Methods
### Simulation Model
#### 2.1.3 Linear Two-Degrees-of-Freedom Vehicle Dynamics Model  
Since the LKAS primarily focuses on the lateral motion of the vehicle, an accurate vehicle lateral dynamics model is required. In this study, a linear two-degrees-of-freedom vehicle dynamics model was adopted, which includes the vehicle's lateral motion along the \(y\)-axis and yaw motion along the \(z\)-axis. The linear two-degrees-of-freedom vehicle model is illustrated in Figure 3.  
First, the forces acting in the \(y\)-direction and the torques along the \(z\)-axis of the vehicle were analyzed to establish the equilibrium equations:  
\[\left\{\begin{array}{l}F_{y}=m\big{(}\ddot{y}+v_{x}\ddot{\varphi}\big{)}\\ M_{z}=I\ddot{\varphi}\end{array}\right., \tag{1}\]  
where \(m\), \(y\), \(v_{x}\), and \(\varphi\) are the mass, lateral displacement, longitudinal velocity, and yaw angle of the vehicle, respectively, and \(I\) is the moment of inertia of the vehicle along the \(z\)-axis.  
Based on Figure 3, the resultant force acting on the vehicle in the \(y\)-direction and the moment around the center of mass can be obtained as:  
\[\left\{\begin{array}{l}F_{y}=F_{yf}\cos\delta_{f}+F_{yr}\\ I\ddot{\varphi}=aF_{yf}\cos\delta_{f}-bF_{yr}\end{array}\right., \tag{2}\]  
Figure 3: Linear two-degrees-of-freedom vehicle model.  
Figure 2: Road model drawing.  
where, \(F_{yf}\) and \(F_{yr}\) are the lateral reaction forces from the ground on the front and rear wheels, respectively, \(\delta_{f}\) is the steering angle of the front wheels, and \(a\) and \(b\) are the distances from the vehicle's center of mass to the front and rear axes, respectively.  
Considering that the lateral tire forces are approximately linearly related to the tire slip angles when the slip angles are small, the lateral force can be calculated using the following equation:  
\[\left\{\begin{aligned} & F_{yf}=C_{f}\alpha_{f}\\ & F_{yr}=C_{r}\alpha_{r}\end{aligned}\right., \tag{3}\]  
where \(C_{f}\) and \(C_{r}\) are the front and rear tire stiffnesses, respectively, and \(a_{f}\) and \(a_{r}\) are the front and rear tire slip angles, respectively.  
Assuming that the steering angle is very small and approaches \(1^{\circ}\), Equations (1)-(3) yield:  
\[\left\{\begin{aligned} & m\big{(}\ddot{y}+v_{x}\ddot{\varphi}\big{)}=C_{f} \alpha_{f}+C_{r}\alpha_{r}\\ & I\ddot{\varphi}=aC_{f}\alpha_{f}-bC_{r}\alpha_{r}\end{aligned}\right.. \tag{4}\]  
Based on the linear two-degrees-of-freedom vehicle model and assuming small sideslip angles, the following equation can be obtained:  
\[\left\{\begin{aligned} &\alpha_{f}=\frac{\dot{\varphi}\alpha+v_{y}}{v_{x}}- \delta_{f}\\ &\alpha_{r}=\frac{v_{y}-\dot{\varphi}b}{v_{x}}\end{aligned}\right.\, \tag{5}\]  
where \(v_{y}\) is the lateral velocity of the vehicle.  
By combining Equations (1)-(5), the linear two-degrees-of-freedom dynamic model can be derived as:  
\[\left[\begin{aligned} &\ddot{y}\\ &\ddot{\varphi}\end{aligned}\right]=\left[\begin{aligned} &\frac{C_{f}+C_{r}}{m\
u_{x}}& \frac{aC_{f}-bC_{r}}{m\
u_{x}}-v_{x}\\ &\frac{aC_{f}-bC_{r}}{Ic_{x}}&\frac{a^{2}C_{f}+b^{2}C_{r}}{Ic_{ x}}\end{aligned}\right]\left[\begin{aligned} &\dot{y}\\ &\dot{\varphi}\end{aligned}\right]+\left[\begin{aligned} &-\frac{C_{f}}{\mu}\\ &-\frac{aC_{f}}{I}\end{aligned}\right]\delta_{f}. \tag{6}\]