# Simulation-Based Reliability Design Optimization Method for Industrial Robot Structural Design
## 3 Demonstrative Example
### Reliability Design Optimization Model  
In the structure optimization design of the big arm and the small arm, the impact of uncertainties was considered. With the lightest weight of the robotic arm as the goal, the yield stress and deformation meeting the allowable requirements as the performance constraints, an initial model of the reliability design optimization of the manipulator was constructed. The initial reliability design optimization model of the big arm was:  
\[ \begin{array}{ll}\min&f(\mu_{x_{1}},\mu_{x_{2}},\mu_{x_{3}})\\ &=4.16(70+\mu_{x_{1}})^{2}\times 10^{-4}+0.02\mu_{x_{2}}\\ &-2.1(72.5-2\mu_{x_{2}})^{2}\times 10^{-4}\\ &+0.83\times(105+2\mu_{x_{3}})^{2}\times 10^{-4}+6.69\\ s.t.&C_{1}=\text{Prob}[(\sigma_{s}-\hat{g}_{1}(x_{1},x_{2},x_{3}))>0]\geq 0.9987 \\ &C_{2}=\text{Prob}[(\varepsilon_{1}-\hat{g}_{2}(x_{1},x_{2},x_{3}))>0]\geq 0.9987 \\ &2\leq\mu_{x_{1}}\leq 12\ \ 3\leq\mu_{x_{2}}\leq 15\ \ 2\leq\mu_{x_{3}}\leq 12 \end{array}  \tag{7}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Uncertainty**} & \multirow{2}{*}{**Distribution**} & \multicolumn{2}{c}{**Mean**} & \multirow{2}{*}{**Standard**} \\ \cline{3-3} \cline{5-5}  & & **Low** & & **Up** \\ \hline \(x_{1}\) & Normal & 2 mm & 12 mm & 0.1\(\mu_{x_{1}}\) \\ \(x_{2}\) & Normal & 3 mm & 15 mm & 0.1\(\mu_{x_{2}}\) \\ \(x_{3}\) & Normal & 2 mm & 12 mm & 0.1\(\mu_{x_{3}}\) \\ \(x_{4}\) & Normal & 6 mm & 18 mm & 0.1\(\mu_{x_{4}}\) \\ \(x_{5}\) & Normal & 2 mm & 15 mm & 0.1\(\mu_{x_{5}}\) \\ \(x_{6}\) & Normal & 8 mm & 20 mm & 0.1\(\mu_{x_{5}}\) \\ \(\sigma_{s}\) & Normal & 210 MPa & 210 MPa & 18 MPa \\ \(\varepsilon_{1}\) & Normal & 0.08 mm & 0.08 mm & 0.008 mm \\ \(\varepsilon_{2}\) & Normal & 0.1 mm & 0.1 mm & 0.01 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Distribution of the uncertainties. Figure 11: Geometrical structure of the big arm. Figure 12: Geometrical structure of the small arm. The initial model for the reliability design optimization of the boom was:  
\[ \begin{array}{ll}\min f(\mu_{x_{4}},\mu_{x_{5}},\mu_{x_{6}})=4.06\times 10^{-2}+4.16 \mu_{x_{4}}^{2}\times 10^{-4}\\ \hskip 14.226378 pt+2.08\mu_{x_{4}}\times 10^{-2}+1.91\mu_{x_{5}}^{2}\times 10^{-3}\\ \hskip 14.226378 pt+9.57\mu_{x_{5}}\times 10^{-2}+6.66\mu_{x_{6}}^{2}\times 10^{-4} \\ \hskip 14.226378 pt+3.33\mu_{x_{6}}\times 10^{-2}\\ \hskip 14.226378 pts.t.\hskip 5.690551 ptC_{3}=\mathrm{Prob}[(\sigma_{s}-\hat{g}_{3 }(x_{4},x_{5},x_{6}))>0]\geq 0.9987\\ \hskip 14.226378 ptC_{4}=\mathrm{Prob}[(\varepsilon_{2}-\hat{g}_{4}(x_{4},x_{5}, x_{6}))>0]\geq 0.9987\\ \hskip 14.226378 ptC_{5}=\mu_{x_{4}}-\mu_{x_{5}}\geq 1.5;C_{6}=\mu_{x_{6}}-\mu_{x_{ 4}}\geq 1.5\\ \hskip 14.226378 pt6\leq\mu_{x_{4}}\leq 18,\;2\leq\mu_{x_{5}}\leq 15,\;8\leq\mu_{x_{6}} \leq 20\end{array}  \tag{8}\]  
In Equations (7) and (8), \(C_{1}\) and \(C_{2}\) are the constraints of the strength reliability and stiffness reliability of the big arm, \(C_{3}\) and \(C_{4}\) are the constraints of the strength reliability and stiffness reliability of the small arm, \(C_{5}\) and \(C_{6}\) are the geometric structure constraints of the small arm. The strength reliability and the stiffness reliability were modeled by the stress and the deformation of the robot arms, respectively. The minimum allowable reliability was \(R_{t}=0.9987\). The function \(\hat{g}_{i}(i=1,2,3,4)\) of the reliability constraints is the implicit function. Through adding the reliability constraints in the design optimization, a lightweight design of the robot could be obtained under the condition of meeting certain reliability performance requirements. It is noted that the models of Equations (7) and (8) only consider the reliability requirements under static performance and do not consider the reliability requirements under dynamic performance. Additionally, the establishment of the model also did not consider the influence of the jump resonance. However, the jump resonance will inevitably occur due to the decay of the device mechanical properties [29].