## 3 Results and Discussion
### Optimisation of Parameters and Values of Objective Functions \(F_{x}\) and \(F_{y}\)  
Partial derivatives concerning decision variables are presented as a system of four equations, separately for the model \(F_{x}\), Equation (5) and \(F_{y}\), Equation (6). \[\left. \begin{array}{l}\frac{\partial F_{x}}{\partial k}=-36.0+95.3\left( \frac{d}{w}\right)+48.2v+4.18 MC-150\left(\frac{d}{w}\right)v-10.1\left(\frac{d }{w}\right)MC-3.63 vMC+13.0\left(\frac{d}{w}\right)vMC=0\\ \frac{\partial F_{x}}{\partial\left(\frac{d}{w}\right)}=-502-1084\left(\frac{ d}{w}\right)+95.3k+1073v+104 MC-150 kv-10.1 kMC-85.6 vMC+13.0 kvMC=0\\ \frac{\partial F_{x}}{\partial v}=-368+35.6v+48.2k+1073\left(\frac{d}{w} \right)+24.9 MC-150k\left(\frac{d}{w}\right)-3.63 kMC-85.6\left(\frac{d}{w} \right)MC+13.0k\left(\frac{d}{w}\right)MC=0\\ \frac{\partial F_{x}}{\partial MC}=-42.2+4.18k+104\left(\frac{d}{w}\right)+24.9v-10.1k\left(\frac{d}{w}\right)-3.63 kv-85.6\left(\frac{d}{w}\right)v+13.0k \left(\frac{d}{w}\right)v=0\end{array} \right\} \tag{5}\]  
\[\left. \begin{array}{l}\frac{\partial F_{y}}{\partial k}=-90.6\left(\frac{d }{w}\right)+29.2v+0.847 MC+5.65\left(\frac{d}{w}\right)MC-2.89 vMC+1.48\left( \frac{d}{w}\right)vMC=0\\ \frac{\partial F_{y}}{\partial\left(\frac{d}{w}\right)}=1705-1552\left(\frac{ d}{w}\right)-90.6k-56.3 MC+5.65 kMC-7.64 vMC+1.48 kvMC=0\\ \frac{\partial F_{y}}{\partial v}=-176+29.2k+18.7 MC-2.89 kMC-7.64\left(\frac{ d}{w}\right)MC+1.48k\left(\frac{d}{w}\right)MC=0\\ \frac{\partial F_{y}}{\partial\text{MC}}=0.847k+56.3\left(\frac{d}{w}\right)+1 8.7v+5.65k\left(\frac{d}{w}\right)-2.89 kv-7.64\left(\frac{d}{w}\right)v+1.48k \left(\frac{d}{w}\right)v=0\end{array} \right\} \tag{6}\]  
The maximum value of the draught force of 438.55 N (Table 6) was achieved with higher values of the variables \(k\), \(d/w\), and \(v\). This confirms that tools with higher rigidity working at greater depths at higher speeds require more pulling power. A graphical interpretation of the objective function \(F_{x}\) and \(F_{y}\) is presented in graphs (Figure 2). In the literature [30], the tool's higher stiffness and speed significantly increase its pulling resistance, especially in soils with higher moisture. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variables** & \(F_{\mathbf{xmin}}\) & \(F_{\mathbf{xmax}}\) & \(F_{\mathbf{ymin}}\) & \(F_{\mathbf{ymax}}\) \\ \hline \hline Tine stiffness, \(k\), kN\(\cdot\)m\({}^{-1}\) & 5.3 & 8.3 & 5.3 & 8.3 \\ \hline Tool depth-to-width ratio, \(d/w\) & 0.15 & 0.67 & 0.15 & 0.62 \\ \hline Tool speed, \(v\), m\(\cdot\)s\({}^{-1}\) & 0.84 & 2.31 & 0.84 & 2.31 \\ \hline Soil moisture, \(MC\), & 14 & 14 & 10 & 14 \\ \hline Optimum force value, N & 98.98 & 438.55 & \(-\)84.55 & 135.25 \\ \hline Value of measured force, N & 102.88 & 431.15 & \(-\)80.67 & 132.74 \\ \hline Model error, \% & 3.79 & \(-\)1.72 & \(-\)4.81 & \(-\)1.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values of optimal parameters and target functions for \(F_{x}\) and \(F_{y}\) models. Figure 2: Draught force \(F_{x}\) (**a,c,e,g**) and vertical force \(F_{y}\) (**b,d,f,h**) for the movement speed \(v\) and depth-width ratio \(d/w\) for soil moisture \(MC\) 10% (**a–d**) and 14% (**e–h**) and times with a stiffness \(k\) of 5.3 kN\(\cdot\)m\({}^{-1}\) (**a,b,e,f**) and 8.3 kN\(\cdot\)m\({}^{-1}\) (**c,d,g,h**). On the other hand, minimising the draught force of 98.98 N required working at lower depths at lower speeds, significantly reducing soil resistance. This aligns with research [46] showing that shallow tools generate less draught force, promoting energy savings in agricultural processes. Vertical force optimisation, \(F_{y}\), showed that, as in the case of pulling power, the maximum and minimum values were reached at the ends of the ranges of the technical variables. A maximum vertical force of 135.25 N was achieved at higher values of \(k,d/w\), and \(v\), which means that higher tool stiffness, a larger depth-to-width ratio, and higher working speed led to a higher vertical force.