# Simulation-Based Reliability Design Optimization Method for Industrial Robot Structural Design
## 3 Demonstrative Example
### Optimization Result Analysis  
The reliability design optimization model by Equations (7) and (8) was solved by the SORA method. The results for the big arm and the small arm are shown in Tables 4 and 5, respectively. In order to better analyze the design results by different design methods, the results by the deterministic optimization and original design are also shown in Tables 4 and 5. It can be seen in Tables 4 and 5 that the weight and the stiffness reliability by the proposed reliability design optimization were between those obtained with the deterministic design optimization and the original design. All three designs had the same strength reliability. Compared with the deterministic design optimization, the results of the proposed method had a higher weight (the big arm weighed 9.43 Kg, and the small arm 2.11 Kg, while the weights by deterministic design optimization were 8.96 Kg and 1.67 Kg, respectively). However, the stiffness reliability of the deterministic design optimization (for the big arm, it was only 0.63407, and for the small arm, it was 0.5) was much lower than that of the proposed method (for the big arm, it was 0.999989, and for the small arm, it was 0.99926). This shows that although the deterministic optimization could obtain the lightest weight, its design results could not satisfy the design requirements of the stiffness reliability. This indicates that the design of the proposed method is more reliable than that of the deterministic optimization. For the original design, both the weight and the reliability were the highest. Compared with the original design, the proposed method could reduce the weights of the big arm and the small arm by 7.73% and 25.70%, respectively, and meanwhile satisfy the 0.9987 reliability requirements. This indicates that the design optimization of the proposed method can reduce the weight as much as possible while satisfying the reliability design requirements. All the above descriptions demonstrate that the method proposed in this paper can effectively complete the structural reliability design optimization of industrial robots. Modal analysis is an important means to understand the vibration of a designed industrial robot; therefore, a modal analysis of the big arm and the small arm was performed. The ANSYS results under the first-order modality are shown in Figures 13 and 14. The frequencies under the original design and the reliability optimization design for the different modality shapes are presented in Table 6. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Item** & **Original Design** & \
\begin{tabular}{l} **Deterministic** \\ **Optimization** \\ \end{tabular} &
\begin{tabular}{l} **Proposed Reliability** \\ **Optimization** \\ \end{tabular} \\ \hline (\(x_{1},x_{2},x_{3}\)) & (11, 9.5, 8) mm & (2, 3, 2) mm & (8.79, 3.54, 2) mm \\ Weight & 10.22 Kg & 8.96 Kg & 9.43 Kg \\ Strength reliability & 1 & 1 & 1 \\ Stiffness reliability & 0.999989 & 0.63407 & 0.99884 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Reliability design optimization results for the big arm. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Item** & **Original Design** & \
\begin{tabular}{l} **Deterministic** \\ **Optimization** \\ \end{tabular} &
\begin{tabular}{l} **Proposed Reliability** \\ **Optimization** \\ \end{tabular} \\ \hline (\(x_{4},x_{5},x_{6}\)) & (16.5, 13, 18) mm & (9.8, 8.3, 11.3) mm & (12.08, 10.58, 13.58) mm \\ Weight & 2.84 Kg & 1.67 Kg & 2.11 Kg \\ Strength reliability & 1 & 1 & 1 \\ Stiffness reliability & 0.99926 & 0.5 & 0.99872 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Reliability design optimization results for the small arm. The modal analysis results showed that the frequencies of the big arm and the small arm under the first-order modality in the design of the reliability optimization were 164.37 Hz and 503.84 Hz, respectively. In Table 6, the first six natural frequencies of the big arm range from 164.37 Hz to 2048.3 Hz for the reliability optimization design and from 150.48 Hz to 2003.7 Hz for the original design.