## 2 Materials and Methods
### General Formula of the Empirical Model  
For the draught force \(F_{x}\) and the vertical force component \(F_{y}\), mathematical, empirical models were determined to bind the independent variables, tine stiffness, \(k\), depth-to-width ratio, \(d/w\), movement speed, \(v\), and soil moisture, \(MC\), using a general regression equation, to analyse the effect of the studied parameters on the dependent variables \(F_{x}\) and \(F_{y}\), Equation (1). \begin{table}
\begin{tabular}{c c c c c} \hline
**Duckfoot, Width** & **Stiffness \(k\),** & **Working Depth \(d\),** & **Movement Speed \(v\),** & **Soil Moisture** \\ \(w\), mm & **kN\(\cdot\)m\({}^{-1}\)** & **mm** & **m\(\cdot\)s\({}^{-1}\)** & **Content DM, \%** \\ \hline \multirow{8}{*}{A105, 105} & & 30 & 0.84, 1.67, 2.31 & \multirow{8}{*}{10} \\  & 5.3 & 50 & 0.84, 1.67, 2.31 & \multirow{8}{*}{14} \\  & & 70 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & & 30 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & & 70 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & & 30 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & 5.3 & 50 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & & 70 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & & 30 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & 8.3 & 50 & 0.84, 1.67, 2.31 & \\ \cline{1-1} \cline{2-5}  & & 70 & 0.84, 1.67, 2.31 & \\ \hline B135, 133 & & & repeat the combination as for A105 & \\ C200, 202 & & & repeat the combination as for A105 & \\ \hline \end{tabular}
\end{table} 
Table 1: Combinations of variables used in the experiment. \[F_{x,y}=\beta_{0}+\sum_{i=1}^{4}\beta_{i}x_{i}+\sum_{i=1}^{4}\beta_{ii}x_{i}^{2}+ \sum_{i=1}^{3}\sum_{j=i+1}^{4}\beta_{ii}x_{i}x_{j}+\sum_{i=1}^{2}\sum_{j=i+1}^{3} \sum_{k=j+1}^{4}\beta_{ijk}x_{i}x_{j}x_{k}+\beta_{1234}x_{1}x_{2}x_{3}x_{4} \tag{1}\]  
where \(F_{x,y}\)--draught force (horizontal) \(F_{x}\), vertical force \(F_{y;\ x_{i,j}}\)--\(i\)-\(j\)-\(th\) independent variable (factor); \(x_{1}=k\)--stiffness; \(x_{2}=d/w\)--ratio of depth to width; \(x_{3}=v\)--movement speed; \(x_{4}=MC\)--soil moisture content; and \(\beta_{0}\), \(\beta_{i}\), \(\beta_{ii}\), \(\beta_{ij}\), \(\beta_{ijk}\), \(\beta_{1234}\)--regression coefficients. The higher-order interactions in Equation (1) account for the cumulative effects of the operating variables on the forces \(F_{x}\) and \(F_{y}\), allowing for a better fit of the model to the actual operating conditions of the tool. Equation (1) contains the constant component \(\beta_{0}\), which represents the underlying value of the force in case all independent variables are zero. The \(\beta_{i}\) coefficients describe the effect of individual independent variables \(x_{i}\), while the \(\beta_{ii}\) coefficients determine the nonlinear effect of a given variable in the form of its square. In turn, \(\beta_{ij}\) coefficients are responsible for dual interactions between variables, which means that the impact of one variable depends on the value of another. Taking into account the triple \(\beta_{ijk}\) and quadruple \(\beta_{1234}\) interactions allows for the modelling of more complex relationships between variables, which may result from synergistic or compensatory effects. The developed formula of the empirical model is in the form of Equation (2). \[F_{x,y}=\beta_{0}+\beta_{1}k+\beta_{2}\frac{d}{w}+\beta_{3}v+\beta_{4}MC+\beta _{5}k^{2}+\beta_{6}\left(\frac{d}{w}\right)^{2}+\beta_{7}v^{2}+\beta_{8}MC^{2 }+\beta_{9}k\frac{d}{w}++\beta_{10}kv+\beta_{11}kMC\]  
\[+\beta_{12}\frac{d}{w}v+\beta_{13}\frac{d}{w}MC+\beta_{14}vMC+\beta_{15}k \frac{d}{w}v+\beta_{16}k\frac{d}{w}MC+\beta_{17}kvMC+\beta_{18}\frac{d}{w}vMC\]  
\[+\beta_{19}k\frac{d}{w}vMC\]  
The physical meaning of individual terms in Equation (2) can be considered in the context of the influence of individual parameters on the draught force \(F_{x}\) and the vertical force \(F_{y}\).