# 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  
#### 2.2.1 Relationship between Traditional Pure Tracking Model and Vehicle Structure  
The traditional pure pursuit (TPP) algorithm is a pure geometric path-tracking model based on look-ahead aiming points [19]. As shown in Figure 3a, line \(P_{1}P_{2}\) is the desired path of the farm machinery, and \(S\) is the look-ahead point on line \(P_{1}P_{2}\). For the ordinary two-wheel steering and two-wheel drive (2 WS-2 WD) vehicle structure, \(W\) is the axle track, \(l\) is the wheelbase, Point O is the center of the rear axle, Point A is the projection of \(O\) on \(P_{1}P_{2}\), \(AS\) is the look-ahead distance (\(LD\)), Point \(G\) is the midpoint of the look-ahead straight-line OS, and O'G is the vertical bisector of \(OS\). The length of line OA is the lateral  
Figure 1: Test prototype and test environment. 1. Motor driver, 2. Microcontroller, 3. Lithium battery, 4. Wheel motor, 5. UWB base station, 6. Mobile tag, 7. Servo, 8. Electronic gyroscope. deviation \(d\), and the angle between axle \(OB\) and \(P_{1}P_{2}\) is the heading deviation \(\theta\). From the TPP model, the location of the steering center \(O^{\prime}\) is determined by the look-ahead straight-line \(OS\) and the rear axle line \(OO^{\prime}\) during the path tracking of the farm robots. According to the Ackermann steering geometry, the rear vehicle axis is perpendicular to the vehicle body. Therefore the location of point \(O^{\prime}\) is determined by the look-ahead straight-line \(OS\), that is, it has one degree of freedom (DOF). By repeatedly tracking the look-ahead points on the desired path, the prototype's location and orientation are gradually adjusted so that the lateral deviation \(d\) and the heading deviation \(\theta\) converge to 0 gradually. From Figure 3a, we can also obtain that the farther the steering center \(O^{\prime}\) is from the prototype vehicle (i.e., the larger the steering radius \(R\)), the smaller the front wheel steering angle is needed, and the vehicle's tendency to move forward is stronger; conversely, the closer the point \(O^{\prime}\) is from the prototype, the larger the steering angle of the front wheel, and the stronger the vehicle's ability to adjust its heading behavior. We should note that when point \(O^{\prime}\) is located inside the prototype vehicle, the steering angle of the front wheels will be significant, and the prototype will make a point-turn motion if the steering mechanism limitation has not been considered. A similar control method is widely used to control the conventional 2 WS-2 WD structure for the test prototype with the 4 WIS-4 WID design (Figure 3b) [18]. The steering radius  
Figure 3: Traditional pure pursuit models on different structures of prototype. (**a**) 2 WS–2 WD, (**b**) 4 WIS–4 WID. Figure 2: Schematic diagram of control system. Note: \((x,y)\) are coordinates of mobile tag; \(\varphi\) is heading angle, \((^{\circ})\); \(\delta\) is steering angle, \((^{\circ})\); \(v\) is vehicle speed, \(\text{ms}^{-1}\); \(U\) is voltage, \(V\); PCM means pulse code modulation; PWM means pulse code modulation; USART is universal synchronous/asynchronous receiver/transmitter; TIM is general timer; \(ds\) is distance, mm. \(O^{\prime}O\) consisting of the steering center \(O^{\prime}\) and the gravity center \(O\) is always perpendicular to the vehicle body. The vertical foot is point \(O\). The trajectories of the inner front and rear wheels coincide, and the same for the outer front and rear wheels. The prototype could avoid lateral movement and obtain high vehicle stability. However, the steering tends to lack flexibility. For agricultural machinery used in large farm fields or greenhouses, its transverse acceleration and lateral force can be ignored due to low speed. Therefore, the following section will mainly focus on the control rule when the steering center is randomly positioned in the prototype vehicle. The steering wheel's steering corrects the prototype's position and orientation.