# Development of a Path Tracker Based on a 4 WS Vehicle for Low-Speed Automated Driving Systems
## 3 Path-Tracker Design
### Proposed Path Tracker for 4 WS Vehicle
#### 3.2.1 Front Wheel Controller Based on the Curvature of the Reference Path Tracker  
The calculation of the front steering angle for path-tracking control utilizes the geometric relationship between the vehicle and the reference path. As shown in Figure 5, the vehicle selects the nearest waypoint from the reference path to determine the target point for path tracking. At this point, the reference point for the vehicle's current position is selected as the center of the longitudinal axis of the vehicle, taking into account the center of gravity of the vehicle, which was selected as the 4 WS vehicle platform for the driving experiments in this study.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Gain Parameter** & & **Description** \\ \hline \multirow{2}{*}{\(k_{e}\)} & - & Controls the influence of the lateral position error, \(e\), on the steering command \\  & - & A higher \(k_{e}\) value improves tracking accuracy but may cause aggressive steering corrections \\ \hline \multirow{2}{*}{\(k_{h}\)} & - & Controls the influence of the heading error, \(\psi_{e}\), on the steering command \\  & - & A higher \(k_{h}\) value results in a faster correction but may introduce oscillations \\  & - & A lower \(k_{h}\) value results in a smoother response but may lead to understeering \\ \hline \multirow{2}{*}{\(k_{p}\)} & - & Serves as a weight factor for the feedforward term, ensuring smoother transitions in curved \\  & - & Peru when the heading angle error, \(\psi_{e}\), and lateral deviation, \(\psi_{e}\), are both zero, \(C_{p}\) acts as a feedforward term to improve tracking performance \\ \hline \multirow{2}{*}{\(k_{r}\)} & - & Limits the rear wheel steering angle but also determines the steering mode \\  & - & Adjusts the rear steering angle, \(\delta_{r}\), relative to the front steering angle, \(\delta_{f}\), based on curvature \\  & \(C_{t.}\) mode \\ \hline \multirow{2}{*}{\(k_{t}\)} & - & Ensures the lowest layer of the style \\  & - & Ensures smooth trajectory adjustments as the feedforward term, particularly in confined \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description of the gain parameters of the proposed path tracker.  
Figure 4: Architecture of the proposed path tracker.  
Based on the selected nearest waypoint, the lateral distance error, \(e\), between the vehicle and the reference path, as well as the heading angle error, \(\psi_{e}\), is calculated [11,26]. Additionally, the curvature, \(C_{p}\), of the reference path is incorporated as a feedforward term to improve the response performance of the path tracker, with the aim of enhancing the vehicle's path-tracking performance [26]. \(C_{p}\) is the inverse of the road radius, \(R_{p}\). Equation (8) represents the heading angle error between the vehicle and the path, and Equation (9) calculates the curvature, \(C_{p}\), of the waypoint on the reference path after the nearest waypoint [27,28]. Finally, as shown in Equation (10), the proposed path tracker generates the front steering angle, \(\delta_{f}\).  
\[\psi_{e}=\psi-\psi_{p} \tag{8}\]  
\[C_{p}=\frac{1}{R_{p}} \tag{9}\]  
\[\delta_{f}=k_{h}\psi_{e}(t)+\tan^{-1}\frac{k_{e}e(t)}{V_{x}(t)}+k_{p}\tan^{-1} \bigl{(}C_{p}(t)L\bigr{)} \tag{10}\]  
As the heading error, \(\psi_{e}\), in the first term and the lateral deviation, \(e\), in the second term of Equation (8) increase, a greater control command is applied to follow the path. Additionally, even when the error between the vehicle and the reference path is zero--meaning that the lateral distance error, \(e\), and the heading error, \(\psi_{e}\), are both zero--the curvature, \(C_{p}\), of the reference path is incorporated as a feedforward term. This allows the vehicle to account for upcoming path information in advance, improving the vehicle's response performance and leading to better path-tracking accuracy [29].