# Multi-Objective Optimal Control Method for the 6-DOF Robotic Crusher
## 5 Numerical Example
### Equivalent Wear Calculation  
According to previous research, when the swing frequency is 4 \(\leq\) f \(\leq\) 5.57, the crushing chamber is divided into 10 crushing layers. Under the reference conditions, the hourly wear of each crushing layer is 0.0505, 0.0797, 0.1233, 0.1578, 0.1974, 0.2642, 0.2412, 0.2367, 0.2390  
Figure 3: Coordinate transformation of suspension point. and 0.2472 mm, respectively. The wear of each crushing layer under different conditions is defined as the relative wear compared with the wear amount under reference conditions. Taking the second, fifth, and ninth crushing layers as an example, the specific wear frequency histogram of all feasible solutions is shown in Figure 4:  
The variation range of the relative wear of each crushing layer is different, and the difference between the maximum value and the minimum value is large, and it presents the characteristics of the long-tail distribution. The satisfaction boundary of each crushing layer needs to be determined separately. Among all feasible schemes, the minimum specific wear of the broken layer is the most satisfactory wear \(\Delta h_{e,\min}\). The satisfaction boundaries calculated by exponential interpolation are \(Bou_{1}=\Delta h_{e,\min}^{\frac{3}{3}}\), \(Bou_{2}=\Delta h_{e,\min}^{\frac{1}{4}}\), \(Bou_{3}=\Delta h_{e,\min}^{-\frac{1}{5}}\), \(Bou_{4}=\Delta h_{e,\min}^{-\frac{3}{5}}\), respectively. The satisfaction boundary of each crushing layer is shown in Table 3:  
The range of wear from the upper part to the lower part of the crushing chamber gradually increases. Therefore, the boundary gap of satisfaction also increases. When choosing different satisfaction schemes for each decision maker, the weight results calculated are different. Since the theory within the feasible region can select an infinite number of schemes, it is necessary to select the scheme by sampling. The scheme selection is as follows:  
1. Select the most satisfactory scheme of each decision maker (each crushing layer), respectively;
2. Select the scheme that is less than and closest to \(Bou_{1}\) among all decision makers;
3. Select the scheme that is less than and closest to \(Bou_{2}\) among all decision makers. Taking method \(1\) as an example, after removing the duplicate scheme, the selected scheme is shown in Table 4: \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Crushing Layer** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \(R_{1}\) & 0.3202 & 0.2299 & 0.1444 & 0.1059 & 0.0762 & 0.0525 & 0.0543 & 0.0519 & 0.0451 & 0.0484 \\ \(R_{2}\) & 0.3109 & 0.2277 & 0.1449 & 0.1063 & 0.0759 & 0.0509 & 0.0528 & 0.0506 & 0.0439 & 0.0473 \\ \(R_{3}\) & 0.2866 & 0.2205 & 0.1462 & 0.1090 & 0.0774 & 0.0497 & 0.0517 & 0.0497 & 0.0432 & 0.0465 \\ \(R_{4}\) & 0.2997 & 0.2246 & 0.1455 & 0.1074 & 0.0762 & 0.0498 & 0.051 & 0.0495 & 0.0430 & 0.0464 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Satisfaction of decision makers in the scheme selected according to method 1. Figure 4: (**a**) Relative wear of the 2 nd crushing layers, (**b**) Relative wear of the 5 th crushing layers (**c**) Relative wear of the 9 th crushing layers. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Crushing Layer** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \(Bou_{1}\) & 0.8571 & 0.8482 & 0.8294 & 0.8198 & 0.7846 & 0.7527 & 0.7162 & 0.6641 & 0.6013 & 0.6  
According to Formula (8), the objective function of wear weight optimization is  
\[f(\lambda_{1},\lambda_{2},...,\lambda_{t})=\min\sum\limits_{k=1}^{t}d(\widetilde{ R},\widetilde{R}_{k}) \tag{10}\]  
\[s.t.\left\{ \begin{array}{l}0\leq\lambda_{k}\leq 1\\ \lambda_{1}\leq\lambda_{2}\leq...\leq\lambda_{10}\\ \sum\limits_{k=1}^{10}\lambda_{k}=1\end{array}