## 2 Problem Formulation and Modeling
### Kinematics Model for Car-like Vehicles  
Car-like vehicles are a class of vehicles that are capable of independently controlling their forward speed and steering, with the most commonly used model being the equivalent kinematics bicycle model [1; 22; 23]. The bicycle model as illustrated in Figure 1 consists of two wheels connected by a rigid link and is restricted to movement in a plane, where the front wheel is allowed to rotate about the axis vertical to the plane and the rear wheel fixed to the body provides forward momentum. More generally, vehicles with these constraints on maneuverability are referred to as nonholonomic vehicles. The parameters and notations of the vehicle model used in Figure 1 are presented in Tables 1 and 2, respectively. The motion in the lateral and yaw directions of the vehicle's center of gravity (C.G.) is given by the following equation [22; 24; 25]. \[mV(\hat{\beta}+\lambda) =F_{f}+F_{r} \tag{9}\] \[I_{z}\lambda =l_{f}F_{f}-l_{r}F_{r}\]  
where cornering forces generated in the front and rear tires, denoted as \(F_{f}\) and \(F_{r}\), respectively, are expressed as follows. \[F_{f} =-K_{f}(\beta+\frac{l_{f}\lambda}{V}-\delta) \tag{10}\] \[F_{r} =-K_{r}(\beta+\frac{l_{r}\lambda}{V})\]  
The motion equation for the vehicle model is derived by substituting Equation (10) into Equation (9), resulting in the following expression. \[mV\dot{\beta}+(K_{f}+K_{r})\beta+\{mV+\frac{l_{f}K_{f}-l_{r}K_{r}}{V}\} \lambda=K_{f}\delta \tag{11}\] \[(l_{f}K_{f}-l_{r}K_{r})\beta+l_{z}\dot{\lambda}+\frac{l_{f}^{2}K_{f }+l_{r}^{2}K_{r}}{V}\lambda=l_{f}K_{f}\delta\]  
where \(\delta\) is an additional freedom degree of the front tire that rotates to steer [26]. Due to vehicle mechanics, the tire angle \(\delta\) is usually limited to a range, where \(\delta\in[\delta_{min},\delta_{max}]\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Value** \\ \hline \(m\) & Vehicle mass & 1188 kg \\ \(V\) & Vehicle velocity & 28 km/h \\ \(l_{z}\) & Vehicle yawing inertia & 2243.1 kgm\({}^{2}\) \\ \(l_{f}\) & Front Axle-C.G. distance & 1.1281 m \\ \(l_{r}\) & Rear Axle-C.G. distance & 1.4719 m \\ \(K_{f}\) & Front cornering power & 76,744 N/rad \\ \(K_{r}\) & Rear cornering power & 119,320 N/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vehicle parameters for simulation. Figure 1: Bicycle model of a ground car-like vehicle. This model is a two-wheeled vehicle model, assuming that the left and right wheels of the front and rear of the vehicle are concentrated at the center position of the axle. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Unit** \\ \hline \(\delta\) & Front tire angle & rad \\ \(\beta\) & Side slip angle & rad \\ \(\lambda\) & Yaw rate & rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Vehicle variables for simulation. Let \(\mathbf{x}=[x,y,\phi]^{T}\in\mathbb{R}^{3}\) denote the posture of the vehicle's C.G. in the fixed ground frame \(\{O\}\). Here, the heading angle \(\phi\) corresponds to the orientation of the vehicle, measured as the angle between the x-axis of frame \(\{O\}\) and the direction in which the vehicle is moving. Positive values of the heading angle indicate a counterclockwise rotation. The kinematics model that describes the differential constraints of the vehicle can be given by Equation (12). \[\dot{x} =V\cos(\phi+\beta) \tag{12}\] \[\dot{y} =V\sin(\phi+\beta)\] \[\dot{\phi} =\lambda\]  
The path-following problem based on this model is depicted in Figure 2. In the figure, \(e_{p}\) denotes the cross-track error, which represents the deviation between the tangent at the nearest point on the path and the C.G. of the vehicle. When the vehicle is on the left side of the path, \(e_{p}\) is greater than zero (\(e_{p}>0\)) and conversely, when the vehicle is on the right side of the path, \(e_{p}\) is less than zero (\(e_{p}<0\)). \(\psi\) signifies the relative orientation between the C.G. of the vehicle and the path.