## 2 Vehicle Model Used in Agricultural Vehicle Path Tracking Control
### Kinematic Vehicle Model  
The path tracking of agricultural vehicles based on the kinematic model determines the front wheel rotating angle according to the position and heading parameters of the agricultural machinery, and then controls the vehicle to work along the predetermined path [29; 30]. The existing research shows that a path-tracking controller based on a kinematic model can achieve better tracking accuracy for agricultural machinery under low-speed conditions. [PERSON] [31; 32] developed an automatic control system based on a simple kinematic model and tested it with a large agricultural tractor. The standard deviation of the lateral position in the linear tracking test is less than 2.5 cm, which proves that only the GPS controller can guide the tractor along a straight line very accurately. [PERSON] [33] designed a path-tracking algorithm suitable for different curves using the kinematic model of agricultural machinery. Assuming that agricultural machinery operates on flat ground, without considering the sideslip, offset, pitching, etc., various four-wheel agricultural types of machinery, such as tractors, harvesters, and rice transplants, can be simplified into bicycle model, which has the characteristics of simple structure and convenient parameter acquisition. As shown in Figure 2, a commonly used bicycle model in the automatic navigation of four-wheeled agricultural machinery is presented. The coordinates of the center point of the rear axle of the vehicle are (\(x_{1}\), \(y_{1}\)), and (\(x_{2}\), \(y_{2}\)) are the coordinates of the center point of the front axle of the vehicle. \(v_{1}\) and \(v_{2}\) represent centerline velocities of the rear and front axles, m/s, \(L\) is the wheelbase of the agricultural machinery, m, \(\varphi\) is the heading angle, rad, \(\delta\) is the front wheel angle, rad. The bicycle model can be determined using Equation (1), where \(\dot{x}\) denotes vehicle speed in the \(x\)-axis direction, m/s, \(\dot{y}\) denotes vehicle speed in the \(y\)-axis direction, m/s, and \(\dot{\theta}\) denotes heading angular velocity of the vehicle, rad/s. \[\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{array} \right]=\left[ \begin{array}{c}\cos\varphi\\ \sin\varphi\\ \frac{\tan\delta}{L}\end{array} \right]v_{1} \tag{1}\]  
According to the motion characteristics of tracked walking mechanism, its kinematic model is a typical differential speed model, and the motion form depends on the speed of  
Figure 2: Bicycle kinematic model. both sides of the tracks. Assuming that the centroid of the whole machine coincides with the geometric center and that the tracks on both sides do not slip and deflect, the kinematic model can be established. As shown in Figure 3, \(C\) represents the centroid of the whole machine, (\(x\), \(y\), \(\theta\)) is the position and orientation coordinates of the tracked vehicle, \(W\) is the track gauge, m, \(v_{L}\) and \(v_{R}\) are the linear speeds of the left and right tracks, m/s. The agricultural tracked vehicle kinematic model can be determined using Equation (2)  
\[\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{array} \right]=\left[ \begin{array}{cc}\frac{\cos\theta}{ \sin^{2}\theta}&\frac{\cos\theta}{2}\\ \frac{\sin\gamma}{12}&\frac{\sin\gamma}{2}\\ \frac{1}{w}&-\frac{1}{w}\end{array} \right]\left[ \begin{array}{c}v_{R}\\ v_{L}\end{array} \right] \tag{2}\]  
The agricultural wheeled articulated vehicle is divided into two parts: the front and rear body. Different from the traditional front axle steering structure, it adds complexity to the path tracking control through the hinge point bending steering method. Based on the motion characteristics of articulated vehicles, the kinematic model is established, which is helpful in determining the path-tracking control variables of the agricultural wheeled articulated vehicle.