## Methods
### Design of Unmanned Agricultural Machine Operation under Improved Fuzzy Adaptive Algorithm
#### 3.1.1 Kinematic Model and Improved Adaptive Algorithm Design
\[\dot{\theta}=(v\tan\delta)/L \tag{1}\]  
In Equation (1), \(v\) represents the vehicle speed, \(\delta\) represents the front wheel angle of the control variable, and \(L\) represents the wheelbase of the farm machine. The front wheel steering angle is usually measured relative to the longitudinal centreline of the vehicle. The front wheel steering angle represents the angle between the front wheel and the longitudinal centreline of the vehicle body, or the steering angle of the front wheel. In the process of constructing the kinematic model, it is assumed that both the vehicle body and wheels are rigid, without considering factors such as elastic deformation. The speed of the vehicle remains relatively stable, while only the change in steering angle is considered during the turning process, and the position of the centre of mass of the vehicle does not change with the action of external forces during the motion process. The kinematic model of the farm machine can be expressed as Equation (2). Figure 1: Kinematic model of unmanned agricultural machinery. \[\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\\ \dot{\delta}\end{array} \right]=\left[ \begin{array}{c}v\sin\theta\\ v\cos\theta\\ (v\tan\theta)/L\\ 0\end{array} \right]+\left[ \begin{array}{c}0\\ 0\\ 0\\ 1\end{array} \right]\omega \tag{2}\]  
In Equation (2), \(\dot{x}\) denotes the component of velocity \(v\) at the rear axle of the vehicle due east, \(\dot{y}\) is the component of velocity due north, \(\dot{\delta}\) is the front wheel angle vector, and \(\omega\) is the steering speed. The lateral deviation of this farm machine can be expressed as Equation (3). \[|d|=\sqrt{\left(x_{m}-x_{t}\right)^{2}+\left(y_{m}-y_{t}\right)^{2}} \tag{3}\]  
In Equation (3), \(x_{t}\) denotes the horizontal coordinates of the point \(t\) on the desired path, and \(y_{t}\) denotes the vertical coordinates of the point \(t\) on the desired path. \(x_{m}\) is the horizontal coordinates of the position \(m\) where the vehicle is located, and \(y_{m}\) is the vertical coordinates of the position \(m\) where the vehicle is located. Subsequently, based on the kinematic model, this study builds a path tracking steering simulation model based on MATLAB, see Figure 2. MATLAB models can quickly develop prototypes and test them, and their built-in mathematical and engineering functions are suitable for testing agricultural machinery models proposed in research. Compared to other simulation software such as Python, LabVIEW, etc., it provides a more efficient, intuitive, and systematic solution, which is particularly effective for developing complex control systems and conducting detailed data analysis. Moreover, MATLAB's interactive features and powerful visualisation capabilities make it particularly outstanding for monitoring and evaluating the performance of control systems, making the verification and display of simulation results intuitive and easy to understand, such as path tracking effects, error curves, etc. In this model, the lateral deviation and heading angle deviation are taken as the physical quantity feedback inputs, and the output result is the desired front wheel angle, and the deviation in the vehicle trajectory is displayed according to the tracking effect on the desired path. This model has flexibility, intuitiveness, and strong scalability in design and simulation and can be optimised and debugged using rich tools and resources. The key to path tracking steering is to calculate the motion and attitude information from  
Figure 2: Simulation model of path tracking and turning for autonomous driving of unmanned agricultural machinery. the sensors. Then, the steering wheel control system executes the corresponding motion based on the calculated cornering results, and the accuracy is evaluated by the lateral deviation results. However, the vehicle steering needs to take into account the influence of the lateral deviation and heading angle deviation, but the deviation results are more easily affected by subjective and objective factors.