## 3 Results and Discussion
### Optimal Parameters of Bar Thickness  
The simulation calculation is carried out according to the 20 test schemes in Table 2. The results show that the amplitude displacement value in the Y direction is the largest, and the maximum harmonic response displacement value of the controllable mechanism welding robot's execution end occurs at different frequencies: 16 Hz, 18 Hz, 24 Hz, 25 Hz,  
Figure 12: Interaction diagrams of the maximum amplitude displacement at the end of a series-parallel controllable mechanism welding robot (The sign * in the figure means the interaction of two factors).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Response** & **Fitting Value** & **Fitting Value Error** & **95\% Confidence Interval** & **95\% Prediction Interval** \\ \hline Deformation (16 Hz) & 0.2880 & 0.0191 & (0.2459, 0.3301) & (0.2266, 0.3495) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Response optimizer optimization results of excitation design.  
Figure 13: Response optimizer optimization diagram of excitation design.  
Considering Figures 15 and 16, it can be seen that the thickness of the rod at the end g has the greatest influence on the amplitude and displacement value of the execution end of the welding robot, and the thickness of the rod at the connecting rod 2 and the swing arm g have some influence on the amplitude and displacement of the execution end.  
Taking the execution end amplitude displacement value of 0 mm as the optimization target, the optimization is carried out through the response optimizer, and the optimization result is shown in Figure 17.  
The optimization results in Figure 17 are shown in Table 6.  
Figure 16: Interaction diagrams for the amplitude displacement at the end of the welding robot: (**a**) 16 Hz; (**b**) 18 Hz; (**c**) 24 Hz; (**d**) 25 Hz; (**e**) 45 Hz. (The sign * in the figure means the interaction of two factors).  
As depicted in Figure 17 and Table 6, the target of the amplitude displacement at the end of welding execution is 0 mm at 16 Hz, 18 Hz, 24 Hz, 25 Hz and 45 Hz. The optimal result is achieved when the thicknesses of the connecting rod 2, swing arm g and end g are 16.0 mm, 9.9951 mm and 10 mm, respectively, and the end displacement value of the controllable mechanism welding robot is the target value. The displacement values at 16 Hz, 18 Hz, 24 Hz, 25 Hz and 45 Hz are 0.0003 mm, 0.0155 mm, 0.0013 mm, 0.0876 mm, and 0.0102 mm, respectively. Compared with the initial amplitude response displacement, the displacement is 0.0150 mm, 0.0041 mm, 0.0336 mm, 0.0302 mm, and 0.0942 mm, respectively. After optimization, the amplitude response displacement decreases by 98%, 74%, 96%, 66%, and 89%, respectively.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Response** & **Fitting Value** & **Fitting Value Error** & **95\% Confidence Interval** & **95\% Prediction Interval** \\ \hline Deformation (16 Hz) & 0.0003 & 0.1220 & (\(-\)0.2760, 0.2770) & (\(-\)0.4350, 0.4360) \\ Deformation (18 Hz) & 0.0155 & 0.0147 & (\(-\)0.0177, 0.0487) & (\(-\)0.0367, 0.0677) \\ Deformation (24 Hz) & 0.0013 & 0.0139 & (\(-\)0.0303, 0.0328) & (\(-\)0.0483, 0.0509) \\ Deformation (25 Hz) & 0.0876 & 0.1450 & (\(-\)0.2410, 0.4160) & (\(-\)0.4290, 0.6040) \\ Deformation (45 Hz) & 0.0102 & 0.0133 & (\(-\)0.0200, 0.0404) & (\(-\)0.0373, 0.0578) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Response optimizer optimization results of bar thickness design.  
Figure 17: Optimization diagram of bar thickness design.  
The amplitude response curve of the execution end obtained after optimization is shown in Figure 18.  
The results in Figure 18 illustrate that when the excitation frequency is 25 Hz, the amplitude value of the execution end is the largest at 0.0922 mm. In Table 5, the response amplitude at 25 Hz based on the response optimization solver is 0.0876 mm. The optimized simulation calculation results are within the confidence interval of the optimization solver's solution results, so the optimization results are reasonable and accurate.