## 2 Materials and Methods
### General Formula of the Empirical Model
For example, the product \(k\times v\) represents the interaction between the time stiffness and the speed of movement, which is essential because, with higher stiffness and higher velocity, there is greater soil inertia, which can increase draught forces. On the other hand, the term \((d/w)\times MC\) describes the influence of soil moisture on the ratio of depth to width of the tool, which is important for the efficiency of soil cutting and the forces occurring during the operation. For the estimation of regression coefficients and the regression model, the Levenberg-Marquardt method was chosen as the most effective and fastest converging method, the algorithm of which is described in detail in the literature [38]. The method of estimating regression coefficients was based on the Levenberg-Marquardt algorithm, which provides stable convergence even for correlated input variables. A procedure was performed to eliminate statistically insignificant components to ensure that the model is not overly fitted to the experimental data, leaving only those significantly affecting the results. The final formula of the models included statically significant regression coefficients, which were left behind after the backward stepwise modelling procedure. Discrimination of regression coefficients was performed using the Student's \(t\)-test. The fit of mathematical models was assessed based on the Fisher-Snedecor test and the \(p\)-value for the model. Statistical conclusions were formulated at the probability level of 0.05. The statistical analysis was conducted using the standard statistical package Statistica v.13.3 (StatSoft Poland Ltd., Cracow, Poland). The final model allows the estimation of the draught force \(F_{x}\) and vertical force \(F_{y}\) in a given range of experimental parameters, and its accuracy was verified by comparing the predictive values with the experimental results. Despite some limitations, the model identifies key relationships and provides tools for optimising the operating parameters of tillage tools. For the developed empirical models, partial derivatives concerning independent variables \(k\), \(d/w\), \(v\), and \(MC\) were determined, and an optimisation process was carried out to find the maximum and minimum values of \(F_{x}\) and \(F_{y}\). The system of four equations from partial derivatives was solved numerically using the SLSQP method, available in the open-source Python software's v1.15.2 minimise function from the SciPy library. The numerical analysis took into account the initial conditions of the decision variables \(k_{0}\) = 5.3 kN\(\cdot\)m\({}^{-1}\), \(d_{0}/w_{0}\) = 0.15, \(v_{0}\) = 0.84 m\(\cdot\)s\({}^{-1}\) and \(MC_{0}\) = 10%, and the constraints on the range of variables used: stiffness \(k\) = [5.3, 8.3] in kN\(\cdot\)m\({}^{-1}\), depth-width ratio \(d/w\) = [0.15, 0.67] (dimensionless quantity), movement speed \(v\) = [0.84, 2.31] in m\(\cdot\)s\({}^{-1}\), and soil moisture \(MC\) = [10, 14] in %. For the values of the optimal parameters, the values of the objective functions \(F_{x}\) and \(F_{y}\) were calculated, and relative errors were determined concerning the values tested for the same operating parameters.