## 6 Experimental Scheme Design and Optimization Results Discussion
### Factor Effect Analysis  
In GRA, the average GRG of each factor level is a significant indicator to determine the best combination of design parameters. The GRG is classified according to the identical level of each column of design parameters in the Taguchi array, and the average GRG is calculated for each factor at the same level.  
Taking the average GRG of each level of the \(\alpha\) factor as an example, the third column in Table 5 shows that the factor \(\alpha\) level in the 1 st, 6 th, 11 th, and 16 th experiments performed is level 1, and the corresponding value of GRG is the result listed in Table 7. For level 1 of factor \(\alpha\), the GRG of the 1 st, 6 th, 11 th, and 16 th experiments are 0.5343, 0.3994, 0.5379, and 0.6203, respectively. Therefore, the level 1 (\(\alpha_{1}\)) average GRG of factor \(\alpha\) is  
\[\alpha_{1}=(0.5343+0.3994+0.5379+0.6203)/4=0.5229 \tag{17}\]  
Similarly, the average GRG of \(\alpha_{2}\), \(\alpha_{3}\), and \(\alpha_{4}\) are 0.5466, 0.4968, and 0.4529, respectively. The same method is used to calculate all average GRG for each factor level, and the calculation results are shown in Table 8.  
As can be seen from Table 8, the maximum values of the average GRG of the factors \(H\), \(\beta\), \(\alpha\), \(R\), and \(r\) are \(H_{4}\), \(\beta_{3}\), \(\alpha_{2}\), \(R_{1}\), and \(r_{1}\), respectively. Therefore, the performance of the  
\begin{table}
\begin{tabular}{c c c} \hline
**Number** & **GRG** & **Ranking** \\ \hline
1 & 0.5343 & 8 \\
2 & 0.4000 & 12 \\
3 & 0.3624 & 15 \\
4 & 0.3387 & 16 \\
5 & 0.3713 & 14 \\
6 & 0.3994 & 13 \\
7 & 0.4390 & 10 \\
8 & 0.6133 & 3 \\
9 & 0.4229 & 11 \\
10 & 0.5358 & 7 \\
11 & 0.5379 & 6 \\
12 & 0.5887 & 4 \\
13 & 0.4982 & 9 \\
14 & 0.5886 & 5 \\
15 & 0.8264 & 1 \\
16 & 0.6203 & 2 \\ \hline \end{tabular}
\end{table}
Table 7: The GRG and ranking of experimental results.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Factor** & **Level 1** & **Level 2** & **Level 3** & **Level 4** & **Max-Min** \\ \hline \(H\) & 0.4089 & 0.4558 & 0.6411 & 0.7981 & 0.3892 \\ \(\beta\) & 0.4567 & 0.4810 & 0.5414 & 0.5403 & 0.0847 \\ \(\alpha\) & 0.5229 & 0.5466 & 0.4968 & 0.4529 & 0.0937 \\ \(R\) & 0.5377 & 0.5124 & 0.4725 & 0.4969 & 0.0652 \\ \(r\) & 0.6275 & 0.4706 & 0.4622 & 0.4591 & 0.1684 \\ \hline \end{tabular}
\end{table}
Table 8: Average GRG.  
mechanism is optimal when the structural parameters of the 6-UPS parallel mechanism \(H\) = 1100 mm, \(\beta\) = 30\({}^{\circ}\), \(\alpha\) = 85\({}^{\circ}\), \(R\) = 800 mm, and \(r\) = 500 mm. In addition, in Table 8, the absolute deviation between the maximum and minimum value of the average GRG of factor \(H\) is the largest. Therefore, factor \(H\) has the most excellent effect on the load-bearing capacity and motion range of the 6-UPS parallel mechanism, followed by factors \(r\), \(\alpha\), \(\beta\), and \(R\).  
The effects of different levels of each factor on the multi-objective performance are shown in Figure 11. From the slopes of each factor curve in Figure 11, it can be seen that the initial height \(H\) between the mobile and fixed platform has a positive effect on the performance of the 6-UPS parallel mechanism; the distribution circle radius \(r\) of the geometric center of the spherical hinge of the mobile platform has a negative effect on the performance of the 6-UPS parallel mechanism, while other factors have an uncertain effect on the performance of the 6-UPS parallel mechanism.