## 4 Conclusions  
The main research problem was the development of mathematical, empirical models for the prediction of the fraught force \(F_{x}\) and the vertical force component \(F_{y}\), which take into account the critical parameters of duckfoots, such as tine stiffness, \(k\), tool depth-to-width ratio, \(d/w\), tool movement speed, \(v\), and soil moisture, \(MC\), which have not been thoroughly analysed in the literature so far. The developed empirical models aimed to accurately represent the values of the forces acting on the tool, considering the non-linear relationships between the independent variables and their interactions. These models allowed for a more comprehensive understanding of implement-soil interactions, which was crucial for optimising the operation of agricultural implements under different operating conditions. Previous studies, although focused on selected parameters such as working depth or movement speed, rarely considered the stiffness of the duckfoot fixing tine and soil moisture. The results of statistical analyses and correlations showed that the values of forces \(F_{x}\) and \(F_{y}\) were most affected by the ratio \(d/w\) and the tool movement speed \(v\). The \(d/w\) variable had the most significant effect on both forces (\(F_{x}\) and \(F_{y}\)), which was confirmed by the relatively high values of the Pearson's correlation coefficients. The values of the multiple correlation coefficients for the models were not very large and were 0.4996 for \(F_{x}\) and 0.6227 for \(F_{y}\). Still, all regression coefficients were statistically significant, so the type of study can be considered acceptable. The operating parameters were optimised using partial derivatives concerning \(k\), \(d/w\), \(v\), and \(MC\). Then, the equation system was solved using the SLSQP method. The optimisation showed that the maximum values of the forces \(F_{x}\) and \(F_{y}\) were achieved at the extremes of the considered intervals of the technical variables. The maximum fraught force was 438.55 N, achieved with higher variable values such as stiffness, the ratio of depth to width of the tool, and the speed of the tool movement. The minimum value of the fraught force (98.98 N) was reached at lower values of the same variables. Similarly, optimising the vertical force \(F_{y}\) indicated that the maximum value (135.25 N) was associated with higher values of the same parameters. In comparison, the minimum value of the vertical force (\(-84.55\) N) was achieved with lower values of the variables. The optimal values were at the ends of the ranges, suggesting the possibility of further research into the complex soil-duckfoot relationships. The optimisation should also be extended to soil loosening efficiency, which is directly related to the state of soil compaction. The analysis of the results indicates that relatively small relative errors for the optimal values were obtained for the developed mathematical empirical models, which confirms the model's agreement with the experimental results. The relative errors ranged from \(-1.72\%\) to \(-4.81\%\), which is evidence of good prediction accuracy, and the negative values of the errors indicate a slight overestimation of the models compared to the actual values. Based on the justifications carried out, there were no grounds to reject the hypothesis that the ratio of the depth to the width of the tool, \(d/w\), the movement speed, \(v\), the time stiffness, \(k\), and the soil moisture, \(MC\) have a significant effect on the draught \(F_{x}\) and the vertical \(F_{y}\) forces, as the experimental results showed agreement with the predictive values of empirical models. The developed empirical model differs from traditional theoretical models such as the [PERSON], Sohne, or Perumpral models [50], which are based on simplified geometrical and mechanical assumptions of soil-tool interactions. These models often treat soil as a homogeneous medium based on strength criteria such as the Mohr-Coulomb theory [45].