# Development of a Path Tracker Based on a 4 WS Vehicle for Low-Speed Automated Driving Systems
## 2 4 WS Vehicle Model
### 4 WS-Based Kinematic Vehicle Model  
There are two commonly used methods for modeling the dynamic characteristics of a vehicle. One approach models the vehicle as a bicycle using only two wheels, and the other models it using all four wheels. In this study, simulations were conducted using a simplified bicycle model that considers only two wheels, based on a 4 WS-based kinematic vehicle model [13; 14; 15; 16; 17; 18]. A common simplification of an Ackerman-steered vehicle used for geometric path tracking is the bicycle model. For the purpose of geometric path tracking, it is only necessary to state that the bicycle model simplifies the 4 WS-based car by combining the two front wheels and the two rear wheels to form a two-wheeled model, like a bicycle [11]. Most studies on path trackers based on geometric, kinematic, and dynamic models adopt the bicycle model [4; 5; 8; 12].  
Figure 1 illustrates the kinematic bicycle model of a four-wheel steering vehicle, where the steering angles of the front and rear wheels differ. The primary assumption in developing the kinematic model is that the velocity vectors at each wheel align with the respective steering angles. Specifically, the velocity vector at the front wheel forms an angle, \(\delta_{f}\), with the vehicle's longitudinal axis, while the velocity vector at the rear wheel forms an angle \(\delta_{r}\). This assumption is equivalent to considering the slip angles at both wheels as zero, which is reasonable for low-speed (\(<\)5 [m/s]) vehicle motion, where the lateral forces generated by the tires remain minimal [19]. Also, the bicycle model is assumed to perform rigid body motion on a fixed X-Y coordinate plane, with no wheel slip. Point C represents the vehicle's center of gravity, and this point indicates the position of the vehicle in the fixed X-Y coordinate system.  
The vector, \(\mathbf{V}\), represents the linear velocity vector of the vehicle at its center of gravity. \(\delta_{f}\) and \(\delta_{r}\) represent the front-wheel steering angle and the rear-wheel steering angle of the vehicle, respectively. \(\psi\) denotes the vehicle's heading angle, and \(\beta\) represents the side-slip angle, which is the angle between the vehicle's longitudinal axis and the linear velocity vector at the vehicle's center of gravity. \(\mathbf{v}_{f}\) and \(\mathbf{v}_{r}\) represent the linear velocity vectors of  
Figure 1: 4 WS-based on kinematic vehicle model.  
the front and rear wheels of the bicycle model, respectively. \(l_{f}\) is the distance from the vehicle's center of gravity to the front axle, and \(l_{r}\) is the distance from the center of gravity to the rear axle. The kinematic equations representing these relationships are shown in Equations (1)-(5). The equations below represent the velocity in the X direction in the global coordinate system, the velocity in the Y direction in the global coordinate system, the rate of change of the heading angle in the vehicle coordinate system, the side-slip angle in the vehicle coordinate system, and the velocity in the vehicle coordinate system, respectively.  
\[\dot{X}=\mathbf{V}cos(\psi+\beta) \tag{1}\]  
\[\dot{Y}=\mathbf{V}\text{sin}(\psi+\beta) \tag{2}\]  
\[\dot{\mathbf{\Psi}}=\frac{\mathbf{V}\text{cos}\,\beta\Big{(}\tan\delta_{f}\ -\tan\delta_{r}\Big{)}}{l_{f}+l_{r}} \tag{3}\]  
\[\beta=\tan^{-1}\Bigg{\{}\frac{l_{r}\tan\delta_{f}\ +l_{f}\tan\delta_{r}}{l_{f}+l_{ r}}\Bigg{\}} \tag{4}\]  
\[\mathbf{V}=\frac{\mathbf{v}_{r}\text{cos}\,\delta_{f}\ +\mathbf{v}_{r}\text{cos}\,\delta_{r}}{2 \text{cos}\,\beta} \tag{5}\]