## 2 Materials and Methods
### Calibration of Contact Parameters Between Sesbania Seeds  
With the contact characteristic parameters of Sesbania seed as experimental factors, the simulated stacking angle as the experimental variable, and the actual stacking angle as the target, a three-factor and three-level Box-Behnken response surface methodology (RSM) experiment was designed. Drawing on the measured contact characteristic parameters and relevant references [27; 28], the experimental factors and levels were determined, as presented in Table 2. The range of all factors includes the measured values and refers to the parameters of legume seeds in the references. The experimental scheme and outcomes are summarized in Table 3.  
The quadratic polynomial regression model used to determine the relative disparity between the measured and simulated stacking angles was established by analyzing the experimental data in Table 3 using the Design-Expert 10 software.  
\[Y=41.74-0.59A+10.8B+7.98C-0.24 AB-0.54 AC+3.65 BC-0.53A^{2}-7.38B^{2}-2.62C^{2} \tag{7}\]  
where \(Y\) is the simulated stacking angle (in \({}^{\circ}\)).  
The variance analysis of the test data in Table 3 is summarized in Table 4. In light of the results of the analysis, the simulated stacking angle \(Y\) exhibits a highly significant  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Levels**} & \multicolumn{3}{c}{**Factors**} \\ \cline{2-4}  & **Coefficient of** & **Coefficient of** & **Coefficient of** \\  & **Recovery A** & **Static Friction B** & **Rolling Friction C** \\ \hline \(-1\) & 0.10 & 0.10 & 0.01 \\
0 & 0.30 & 0.42 & 0.15 \\
1 & 0.50 & 0.74 & 0.29 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Factors and levels for the parameter calibration experiment.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test Group**} & \multirow{2}{*}{**A**} & \multirow{2}{*}{**B**} & \multirow{2}{*}{**C**} & **Simulated** \\  & & & & **Sracking Angle (\({}^{\circ}\))** \\ \hline
1 & 0 & \(-1\) & \(-1\) & 17.41 \\
2 & 1 & 1 & 0 & 44.19 \\
3 & \(-1\) & 1 & 0 & 45.00 \\
4 & 0 & 0 & 0 & 43.03 \\
5 & 0 & 0 & 0 & 42.65 \\
6 & 0 & 0 & 0 & 41.91 \\
7 & \(-1\) & 0 & \(-1\) & 30.26 \\
8 & \(-1\) & \(-1\) & 0 & 22.98 \\
9 & 0 & 1 & \(-1\) & 31.74 \\
10 & 0 & 0 & 0 & 40.49 \\
11 & 1 & \(-1\) & 0 & 23.14 \\
12 & 1 & 0 & 1 & 45.83 \\
13 & 1 & 0 & \(-1\) & 29.33 \\
14 & 0 & 0 & 0 & 40.60 \\
15 & 0 & 1 & 1 & 53.39 \\
16 & \(-1\) & 0 & 1 & 48.93 \\
17 & 0 & \(-1\) & 1 & 24.44 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Scheme and outcomes of the parameter calibration experiment.  
correlation with the coefficient of static friction \(B\), the coefficient of rolling friction \(C\), the interaction term \(BC\), and \(B^{2}\) and \(C^{2}\). Among these factors, the coefficient of static friction \(B\) exerts the most substantial influence on the research outcomes. The \(p\)-value for the lack-of-fit term is 0.312, which exceeds 0.05, indicating that the lack-of-fit of the regression model is not significant. This suggests that the regression model demonstrates high fit and the quadratic polynomial simulation is reliable. The coefficient of determination (R\({}^{2}\)) of the model is 0.9932, and the adjusted coefficient of determination (Adj R\({}^{2}\)) is 0.9845, both of which are nearly identical to 1. Additionally, the coefficient of variation (C.V.) is 3.57%, and the precision value is 37.296, which exceeds 4, further confirming that the regression model has high credibility and accuracy.  
With the physical stacking angle as the target value, the Design-Expert software optimizes and solves the regression model. The objective function and constraint conditions are presented in Formula (8).  
\[\left\{\begin{array}{l}Y(A,\ B,C)=34.5\\ 0.10\leq A\leq 0.50\\ 0.10\leq B\leq 0.74\\ 0.01\leq C\leq 0.29\end{array}\right. \tag{8}\]  
The calibrated values of the coefficient of restitution, coefficient of static friction, and coefficient of rolling friction between the Sesbania seeds were determined to be 0.168, 0.339, and 0.083, respectively.