## 3 Results and Discussion
### Mathematical Empirical Models
#### 3.2.1 Detailed Model Description for Draught Force, \(F_{x}\)
The conjunction of stiffness and speed \(k\times v\), the ratio of depth to width and soil moisture (\(d/w\times MC\)), the movement speed and soil moisture (\(v\times MC\)), the stiffness and soil moisture (\(k\times MC\)), and the stiffness and the ratio of depth to width had a significant influence on the draught force (\(k\times d/w\)). In addition, the quadruple interaction of all independent variables (\(k\times d/w\times v\times MC\)) was significant. This indicates the complexity of the soil-tool interaction process, in which many factors act simultaneously and affect the final values of draught forces. Previous research shows that the draught force was greater for rigid tools than for flexible tools, and that increasing the speed of the flexible tool increased the draught force. The \(d/w\) ratio also affects the effectiveness of soil loosening, directly affecting soil compaction and crop yields [30]. The analysis also showed that certain variables had a negative effect on the pulling power, indicating their mitigating effect on the tool's operating resistance. The most significant negative regression coefficient (\(-542\)) was assigned to the square of the tool depth-to-width ratio (\(d/w\))\({}^{2}\). Such quadratic relationships are often observed in research on agricultural tools, where an increase in technical parameters above a specific value ceases to bring benefits in the form of increased tool efficiency [46]. The value of the draught force was negatively affected by regression coefficients with linear forms of all four independent variables \(k\), \(d/w\), \(v\), and \(MC\). Similarly, all triple conjunctions of the analysed parameters had a negative effect on the draught force: (\(k\times d/w\times v\)), (\(k\times d/w\times MC\)), (\(k\times v\times MC\)), and (\(d/w\times v\times MC\)). However, this negative effect of the variables should be considered in combination with other variables. Still, these negative values of regression coefficients with the variables highlighted may suggest that, under certain conditions, an increase in these variables did not necessarily lead to an increase in draught force. For example, increasing the moisture content of specific soil types may have led to a decrease in pulling resistance, which is consistent with the literature, which indicates that moist soil with one particular granulometric structure has less adhesion to implements, which reduces resistance to tool movement [41]. The models include the square of the speed, and the analyses above of empirical models show that the draught force is also affected by linear speed. Therefore, it can be concluded that further modification of the models should consider both forms of speed:quadratic and linear. It was indicated that the [PERSON], [PERSON], and Perumpral models should be modified for comprehensive tools, as the models have a more significant agreement between the predictive and experimental values for a narrow tool [50]. Overall, the model for the draught force, \(F_{x}\), was highly complex, considering both linear and nonlinear relationships between independent variables, as well as interactions between these variables. The regression analysis showed that the tool movement speed and depth-to-width ratio significantly impacted the draught force. In contrast, stiffness and soil moisture had a more minor but significant effect. This model is essential to developing predictive methods for agricultural implements, enabling more accurate prediction of draught forces under varying operating conditions.