# Shape Design Optimization of a Robot Arm Using a Surrogate-Based Evolutionary Approach
## 4 Simulation Results
### The Relationship between Objectives and Design Variables
#### 4.5.1 Main Effect Analysis  
A main effect is frequently used in the context of factorial designs and regression models to distinguish main effects from interaction effects. A main effect occurs when the mean response changes across the levels of a factor. According to the ranges of design variables in Table 1, the lower bound is denoted as level (-1), the upper bound is denoted as level (+1), and the average of the two is denoted as level (0) when running the DOE. The main effect plots for the objectives of weight (W), deformation (D), and moment of inertia (J) are shown in Figure 11. Several results observed in the main effect plots are stated below. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**2 nd Robot Arm**}} & \multicolumn{3}{c}{**Objectives (Predicted by Model/ANSYS)**} \\ \cline{2-4}  & **Weight W (kg)** & **Moment of Inertia J (kg. m\({}^{2}\))** & **Deformation D (mm)** \\ \hline (0) Baseline Design & –/40.26 & –/8.404 & –/0.0644 \\ (1) First Design Model-1 & 40.75/41.28 & 7.8078/8.0081 & 0.0311/0.0333 \\ (2) Renewal Design Model-2 & 32.69/33.89 & 6.6873/6.5079 & 0.0409/0.0501 \\ (3) Renewal Design Model-3 & 33.02/33.80 & 6.3328/6.4906 & 0.0491/0.0513 \\ Improvement \%: ((3)/(0))\(-\)(1) & \(-\)18.0\%/\(-\)16.1\% & \(-\)24.7\%/\(-\)22.8\% & \(-\)23.8\%/\(-\)20.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of objectives with the baseline design. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Population** & **Time (sec) per** & **Converge at** & **Total Time (sec)** & **All Process Time (sec)** \\  & **Size** & **Generation** & **Generation** & **to Convergence** & **(Total Time + ANSYS)** \\ \hline ANSYS & 1035 & 31.050 & 3 & 93.150 & 93.150 \\ Model-1 & 1035 & 0.249 & 25 & 6.225 & 6.225+30 \(\times\) 25 = 756.225 \\ Model-2 & 1035 & 0.325 & 22 & 7.150 & 7.15+30 \(\times\) 22 = 667.15 \\ Model-3 & 1035 & 0.299 & 15 & 4.485 & 4.485+30 \(\times\) 15 = 454.485 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Computational efficiency comparison between the model and ANSYS. On the objective D:  
1. All factors, except \(R_{h}\), can cause the response to increase when they move from low level (\(-1\)) to high level (\(+1\)). While \(H_{i}\) and \(H_{m}\) can only cause the response to increase slightly as the level changes, \(V_{m}\) and \(T_{a}\) can cause the response to increase significantly, especially \(T_{a}\). 2. Factor \(R_{h}\) can cause the response to decrease when its level increases, but the slope of the response only changes slightly. 3. Consequently, all five factors appear to have a main effect on the response W. Among them, \(T_{a}\) is the most important factor because of its high significance. On the objective J:  
1. Factors \(H_{i}\) and \(R_{h}\) present no significant main effects because the response remains nearly the same when they move from low level to high level. 2. Factors \(V_{m}\) and \(T_{a}\) can cause the response to increase significantly when they move from low level to high level, and the slope of the increased response induced by \(T_{a}\) is the sharpest. 3. Factor \(H_{m}\) can cause the response to increase sharply when it moves from low level to intermediate level, and after the saddle point, the response remains almost constant until high level. 4. Consequently, factors \(H_{m}\), \(V_{m}\), and \(T_{a}\) appear to cause a main effect on the response J. Among them, \(T_{a}\) is the most important factor because of its high significance. On the objective D:  
1. When the factors move from low level to high level, only \(R_{h}\) causes response D to decrease, while the others cause response D to increase. 2. Factors \(H_{i}\), \(V_{m}\), and \(T_{a}\) cause the response D to decrease sharply when the factors move from low level to high level, and the slopes of the responses remain almost the same. 3. Consequently, all five factors appear to have main effects on the response.