## 3 Results and Discussion
### Mathematical Empirical Models  
Mathematical, empirical models developed for draught force, \(F_{x}\), and vertical force, \(F_{y}\) (Equation (2)) were based on analysing the influence of key independent variables and their interactions. These models extended previous approaches, often only considering basic operating parameters such as working depth, \(d\), tool width, \(w\), and movement speed, \(v\). The developed models were more complex, as they additionally considered the effects of time stiffness, \(k\), and soil moisture content, \(MC\). Entering stiffness \(k\) as a variable was essential to research on modelling soil-tool interactions. As shown earlier, tool stiffness can significantly affect draught forces, which is confirmed in the literature [40]. Stiffness affects the susceptibility of the tool to adapt to soil conditions, especially when the soil is more compact or when there is significant working resistance resulting from the operation of the tools at specific technical parameters. Including this parameter in the mathematical model enabled a more accurate prediction of the values of the forces acting on the tool. The empirical models developed included both linear and non-linear effects of variables and interactions between variables. These interactions were crucial for correctly mapping the impact of several variables simultaneously, which is particularly important in studying soil-tool interactions. The literature notes that considering variable interactions, especially double and triple interactions, improves the precision of predictive models, which was also reflected in the present study [42]. A general regression equation included all relevant factors affecting these forces in the models developed for draught force \(F_{x}\) (Equation (3)) and vertical force \(F_{y}\) (Equation (4)). These equations were more complex than traditional models, reflecting the complexity of the soil-tool interaction process. These models considered the linear relationships between variables and their squares and interactions between variables, allowing for more precise prediction results. \[ \begin{split} F_{x}=486-36.0k-502\frac{d}{w}-368v-42.2 MC-542\Big{(} \frac{d}{w}\Big{)}^{2}+17.8v^{2}+95.3k\frac{d}{w}++48.2 kv+4.18 kMC\\ +1073\frac{d}{w}v+104\frac{d}{w}MC+24.9 vMC-150k\frac{d}{w}v-10.1k \frac{d}{w}MC--3.63 kvMC-85.6\frac{d}{w}vMC\\ +13.0k\frac{d}{w}vMC\\ F_{y}=-267+1705\frac{d}{w}-176v-776\Big{(}\frac{d}{w}\Big{)}^{2 }-90.6k\frac{d}{w}+29.2 kv+0.847 kMC-56.3\frac{d}{w}MC+18.7 vMC\\ +5.65k\frac{d}{w}MC-2.89 kvMC-7.64\frac{d}{w}vMC+1.48k\frac{d}{w} vMC\end{split}  \tag{3}\]  
One of the critical elements of the models was the inclusion of the constant, \(\beta_{0}\), which served as an equalisation of the prediction level. The lack of a constant in the model would artificially inflate the multiple correlation coefficient, R, values, which could lead to erroneous conclusions about the model's validity; including a constant allowed the model to fit the data more realistically, which is especially important for non-linear relationshipsbetween variables [44]. The values of constants in both models were statistically significant, confirmed by the results of the Student's \(t\)-tests, and were of great importance in improving the prediction quality (Table 4). The analysis of variance and evaluation of model fit confirms their statistical significance, \(p\) < 0.0001 (Table 5). The values of multiple correlation coefficients, R, for the developed models were 0.4996 for \(F_{x}\) and 0.6227 for \(F_{y}\), indicating a moderate fit of the models to the experimental data. This may be due to random variability of the compacted soil, too low implement movement speed, instability of the working depth related to tine stiffness, and the associated change in the duckfoot rake angle. Still, they were statistically significant and showed a satisfactory reflection of the real-world phenomena occurring in the soil-tool interaction.