## 2 Materials and Methods
### Upper-Stability Controller Design
#### 2.4.2 Design of Direct Yaw Moment Controller  
When the sprayer turns, the direct yaw moment controller identifies the sprayer's driving intention based on the steering angle signal. Utilizing the previously established chassis dynamics model, the required yaw moment for the sprayer is calculated through the corresponding algorithm. This ensures the sprayer maintains neutral steering to the greatest extent possible, enhancing its driving stability under extreme conditions. The specific control system structure is shown in Figure 9.  
By incorporating an additional yaw moment \(\Delta\)T into the two-degree-of-freedom sprayer dynamics model, the new dynamic equation is obtained as follows:  
\[ \begin{cases}\hat{\beta}_{\text{centroid}}=\frac{k_{\text{f}}+k_{\text{r}}}{M \alpha_{\text{x}}}\beta_{\text{centroid}}+\left(\frac{k_{\text{f}}k_{\text{f}} -b_{\text{f}}k_{\text{r}}}{M\alpha_{\text{x}}^{\text{ex}}}-1\right)\omega_{ \text{z}}-\frac{k_{\text{f}}}{M\alpha_{\text{x}}}\delta_{\text{f}}\\ \dot{\omega}_{\text{z}}=\frac{l_{\text{z}}k_{\text{f}}-l_{\text{z}}k_{\text{r} }}{l_{\text{z}}}\beta_{\text{centroid}}+\frac{l_{\text{z}}^{2}k_{\text{f}}^{ \text{ex}}}{L_{\text{z}}\alpha_{\text{x}}}\omega_{\text{z}}-\frac{l_{\text{z} }k_{\text{f}}}{l_{\text{z}}}\delta_{\text{f}}+\frac{\Delta\text{T}}{l_{\text{z }}}\end{cases}  \tag{8}\]  
Figure 8: Slip rate control module structure. Figure 7: Single-wheel slip controller. Design of Yaw Rate Sliding Mode Control Algorithm  
In the study of chassis control for self-propelled sprayers, the sliding mode control algorithm is widely considered due to its excellent robustness [30]. A sliding surface was designed using the actual and ideal yaw rates to further calculate the additional yaw rate moment \(T_{\omega}\). The yaw rate tracking error and its derivative are defined using Equation (9) as follows:  
\[\left\{ \begin{array}{l}e_{\omega}=\omega_{\mathrm{z}}-\omega_{\mathrm{z}_{ \mathrm{e}}\exp}\\ \dot{e}_{\omega}=\dot{\omega}_{\mathrm{z}}-\dot{\omega}_{\mathrm{z}_{\mathrm{e }}\exp}\end{array} \right. \tag{9}\]  
The sliding mode surface \(s_{\omega}\) is defined as follows:  
\[s_{\omega}=c_{\omega}e_{\omega}+\dot{e}_{\omega} \tag{10}\]  
where \(c_{\omega}\) is the relative weighting coefficient between the error and the error rate of change; the sliding mode approach employs a constant rate reaching law; then, \(\dot{s}_{\omega}=c_{\omega}\dot{e}_{\omega}+\dddot{e_{\omega}}=-\epsilon_{ \omega}sign(s_{\omega})\); \(\epsilon_{\omega}\) is a positive constant that determines the reaching speed; \(sign(s_{\omega})\) is the sign function. By combining Equations (8)-(10), we obtain the following:  
\[\dot{s}_{\omega} =c_{\omega}\dot{e}_{\omega}+\frac{l_{\mathrm{z}}k_{\mathrm{f}}-l _{\mathrm{b}}k_{\mathrm{r}}}{l_{\mathrm{z}}}\dot{g}_{\mathrm{ centroid}}+\frac{l_{\mathrm{z}}^{2}k_{\mathrm{f}}+l_{\mathrm{b}}^{2}k_{ \mathrm{r}}}{l_{\mathrm{z}}v_{\mathrm{x}}}\dot{\omega}_{\mathrm{z}}-\frac{l_ {\mathrm{z}}k_{\mathrm{f}}}{l_{\mathrm{z}}}\dot{\delta}_{\mathrm{f}}+\frac{ \Delta T}{l_{\mathrm{z}}}-\tilde{\omega}_{\mathrm{z}_{\mathrm{e}}\exp} \tag{11}\]  
The additional yaw rate torque \(T_{\omega}\) is finally obtained as follows:  
\[T_{\omega}=\int\Biggl{\{}-\mathrm{I}_{\mathrm{z}}\Biggl{(}c_{\omega}\dot{e}_ {\omega}+\frac{l_{\mathrm{z}}k_{\mathrm{f}}-l_{\mathrm{b}}k_{\mathrm{r}}}{l_ {\mathrm{z}}}\dot{\beta}_{\mathrm{centroid}}+\frac{l_{\mathrm{z}}^{2}k_{ \mathrm{f}}+l_{\mathrm{b}}^{2}k_{\mathrm{r}}}{l_{\mathrm{z}}v_{\mathrm{x}}} \dot{\omega}_{\mathrm{z}}-\frac{l_{\mathrm{z}}k_{\mathrm{f}}}{l_{\mathrm{z}}} \dot{\delta}_{\mathrm{f}}-\ddot{\omega}_{\mathrm{z}_{\mathrm{e}}\exp}+ \epsilon_{\omega}\cdot sign(s_{\omega})\Biggr{)}\Biggr{\}}dt \tag{12}\]  
Design of Centroid Slip-Angle Sliding Mode Control Algorithm  
The sliding mode surface was designed using the actual and desired values of the centroid sideslip angle to further solve for the additional centroid sideslip angle torque \(T_{\beta}\).