## 2 Materials and Methods
### Path-Tracking Method Design
#### 2.2.2 Kinematics Equation of the Vehicle  
Different from traditional rear wheel-driven vehicles, to control a 4 WIS-4 WID vehicle, steering angle \(\delta_{1}\)-\(\delta_{4}\) and speed \(v_{1}\)-\(v_{4}\) for each of the four wheels need to be considered, thus leading to an even more complicated kinematics model as Figure 5.  
According to the Ackermann steering principle of Equation (2),  
\[\cot\delta_{1}-\cot\delta_{2}=\frac{2W}{l} \tag{2}\]  
where:  
\(l=\) wheelbase, m;  
\(W=\) left and right wheel tread, m;  
\(\delta_{1}=\) steering angle of the front outer wheel, \({}^{\circ}\);  
\(\delta_{2}=\) steering angle of the front inner wheel, \({}^{\circ}\).  
Since \(2W/l>0\), the value of \(\delta_{1}\) is less than \(\delta_{2}\). In the simplified steering model, the front and rear wheels on the same side have unique steering angles but opposite steering directions, i.e., \(\delta_{1}=-\delta_{3}\), \(\delta_{2}=-\delta_{4}\), as in Equation (3).  
\[\left\{\begin{array}{l}\delta_{1}=-\delta_{3}=\arctan\frac{l}{2R+W}\\ \delta_{2}=-\delta_{4}=\arctan\frac{l}{2R-W}\end{array}\right. \tag{3}\]  
where:  
\(l=\) wheelbase, m;  
\(W=\) left and right wheel tread, m.  
Figure 4: Pure pursuit model.  
Figure 5: Kinematics model of the vehicle.  
The wheel is required to turn clockwise in a positive direction and counter-clockwise in a negative direction.  
Thus, rotation radii for the four wheels are determined by Equation (4).  
\[\left\{\begin{array}{l}R_{1}=R_{3}=\frac{2R+W}{2\cos 6}\\ R_{2}=R_{4}=\frac{2R-W}{2\cos 6}\end{array}\right. \tag{4}\]  
To ensure that the four wheels are always pure rolling with the ground during steering, the rotation speeds of the four wheels should be in accordance, and the speeds of the four wheels \(v_{1}\)-\(v_{4}\) are listed as Equation (5).  
\[\left\{\begin{array}{l}v_{1}=v_{3}=\frac{vR_{1}}{R}\\ v_{2}=v_{4}=\frac{vR_{2}}{R}\end{array}\right. \tag{5}\]  
where: \(v\) = driving speed of the prototype vehicle, m/s.  
Thus, the kinematic control model of the prototype vehicle could be constructed. In the model, deviation (\(d\), \(\theta\)) can be transferred into the steering angle \(\delta_{1}\)-\(\delta_{4}\) and wheel speed \(v_{1}\)-\(v_{4}\).  
In the pure pursuit model, the forward distance \(LD\) has a significant impact on the path-tracking quality. A smaller forward distance quickly converges the prototype to the desired path (desired path) but also reduces the stability of the path tracking; a larger forward distance enables the prototype to smoothly converge to the desired path, but it takes a longer time and passes over a longer distance. The experience shows that when the deviation of the prototype vehicle is large, a small look-ahead distance should be used to quickly adjust the vehicle position and converge to the desired path; when the deviation of the prototype vehicle is small, a large look-ahead distance is used to avoid the oscillation caused by over-sensitive system adjustment. Currently, there is no mature mathematical function used to determine the look-ahead distance for vehicle with a 4 WIS-4 WID structure. A fuzzy controller was designed to dynamically determine the look-ahead distance \(LD\) using the driver's driving experience as the control rule.