# Path Tracking of Agricultural Vehicles Based on 4 WIS-4 WID Structure and Fuzzy Control
## 2 Materials and Methods
### Path-Tracking Model Based on 4 WIS-4 WID Structure and Fuzzy Control
#### 2.2.2 Influence Analysis of Steering Center Position on Vehicle Position Correction  
As shown in Figure 4, provided that point \(O^{\prime}\) is the steering center of the 4 WIS-4 WID prototype, point \(O\) is the gravity center, \(OO^{\prime}\) is the steering radius \(R\), and the angle between \(OO^{\prime}\) and the vehicle axial direction (forward vehicle direction) is denoted as \(\alpha\). According to the Ackermann steering geometry, the following equation determines the steering angle of each wheel, namely \(\delta_{1}\)-\(\delta_{4}\).  
\[\left\{\begin{array}{l}\delta_{1}=\arcsin(\frac{R\cos\alpha+1/2}{R_{1}})= \arctan(\frac{R\cos\alpha+1/2}{R\sin\alpha+W/2})\\ \delta_{2}=\arcsin(\frac{R\cos\alpha+1/2}{R_{2}})=\arctan(\frac{R\cos\alpha+1 /2}{R\sin\alpha-W/2})\\ \delta_{3}=\arcsin(\frac{R\cos\alpha-1/2}{R_{3}})=\arctan(\frac{R\cos\alpha-1 /2}{R\sin\alpha-W/2})\\ \delta_{4}=\arcsin(\frac{R\cos\alpha-1/2}{R_{4}})=\arctan(\frac{R\cos\alpha-1 /2}{R\sin\alpha+W/2})\end{array}\right. \tag{1}\]  
where \(W\) is the axle track in meters, \(l\) is the wheelbase in meters. \(R\) is the steering radius at the gravity center of the prototype vehicle in meters, and the turning radius of each wheel is \(R_{1}\)-\(R_{4}\) in meters. Then, the following equation can determine the linear velocities of each wheel.  
\[\left\{\begin{array}{l}v_{1}=v\frac{R_{1}}{R}\\ v_{2}=v\frac{R_{3}}{R}\\ v_{3}=v\frac{R_{3}}{R}\\ v_{4}=v\frac{R_{4}}{R}\end{array}\right. \tag{2}\]  
Figure 4: Steering model when the center of steering is at a random position.  
In Equation (2), \(v\) is the linear velocity of the gravity center \(O\) of the test prototype and \(v_{1}\)\(\sim\)\(v_{4}\) are the linear velocities of four wheels in ms\({}^{-1}\). The analysis of Equations (1) and (2) and Figure 4 leads to the following two conclusions.  
1. When \(\alpha\) is closer to \(0^{\circ}\) or \(180^{\circ}\), steering radius \(R\) is closer to the longitudinal axis line OB, and the movement trend of the prototype vehicle tends to be the lateral crab movement. The prototype performs a quick response of lateral correction. Conversely, when \(\alpha\) is closer to \(90^{\circ}\), steering radius \(R\) is perpendicular to the prototype's axle, and the instantaneous movement direction is closer to the longitudinal axis line \(OB\). The prototype performs a weak response of lateral correction.
2. The smaller the steering radius \(R\) is, the closer the steering center \(O^{\prime}\) is to the gravity center \(O\), the motion of the prototype tends to be in situ rotation, and the prototype performs more vital heading control ability. Conversely, the larger the steering radius \(R\) is, the farther the steering center \(O^{\prime}\) is from the prototype gravity center \(O\), the movement of the prototype vehicle tends to be linear motion, and the prototype performs a weak ability to control heading direction.  
As discussed earlier, two variables determine the steering center \(O^{\prime}\), namely, \(\alpha\) and \(R\). These two parameters determine the location of the steering center \(O^{\prime}\) and also the control capability of the steering.