# Dimensional Synthesis for Multi-Linkage Robots Based on a Niched Pareto Genetic Algorithm
## 5 Engineering Example of the Method Application and Analysis
### Method Application in Chemical Industry  
2,2'-azobis(2,4-dimethylvaleronitrile)(shorted ADVN) (molecular formula: C14H24N4) is widely used in industrial production for polyvinyl chloride(PVC), polyacrylamide(PAN), polymethyl methacrylate(PMMA), anticorrosive paint, etc. Due to its strong toxicity, flammability, and explosive risk, the workers should have little or no contact with it as much as possible. In this, the use of linkage robots is very important to ensure human health. In the process of barreled solid ADVN scattering, there is some surplus in the barrel. In order not to damage the material barrel, we designed the 3R linkage robot for the secondary scattering of remaining materials, as shown in Figure 11.  
In this application, the height of the barrel is 59 cm, the maximum diameter is 40 cm, and bunghole diameter is 26 cm. The dimensional synthesis data of three-link robot obtained by the KPCA is shown in Table 4.  
In Table 4, dimensional parameters, we take \(l_{1}=15.0000\), \(l_{2}=23.0000\), \(l_{3}=25.0000\), \(\theta_{1}\in[0,67^{\circ}]\), \(\theta_{2}\in[0,97^{\circ}]\), \(\theta_{3}\in[0,119^{\circ}]\) after radian convert to angles. Design solution is carried out based on the NPGA method. According to the mathematical model constructed and the application example of two-link robot in salt field, population size is 150, chromosome length in binary is 32 bits, crossover operator is 0.07, mutation operator is 0.70, termination generation is 150. The output results are indicated in Figure 12. It is observed that data convergence is good. We gained the global Pareto sets, some data are listed in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Links** & \(l_{1}(\mathbf{cm})\) & \(l_{2}(\mathbf{cm})\) & \(\theta_{1}(\mathbf{rad})\) & \(\theta_{2}(\mathbf{rad})\) \\ \hline Team1 & 11.0361 & 17.0021 & 0.5756 & 2.0311 \\ Team2 & 10.9512 & 16.8802 & 0.5608 & 2.1051 \\ Team3 & 11.1004 & 16.7560 & 0.5732 & 1.9601 \\ Team4 & 10.9913 & 17.0103 & 0.5714 & 1.9985 \\ Team5 & 11.0032 & 17.1022 & 0.5746 & 2.5543 \\ Team6 & 10.8239 & 16.9823 & 0.5611 & 1.9723 \\ Team7 & 10.9025 & 17.1005 & 0.5802 & 1.9028 \\ Team8 & 11.0344 & 16.9562 & 0.5913 & 2.0048 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Some dimensional synthesis data of 2R linkage robot. Figure 10: Output results of NPGA in salt industry. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Design Method** & **Joint Angle** & **Elapsed Time** \\ \hline KPCA & 36\({}^{\circ}\), 120\({}^{\circ}\) & 6’12” \\ NPGA & 33\({}^{\circ}\), 117\({}^{\circ}\) & 5’26” \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Application data of the two methods in the salt field. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Links** & \(I_{1}(\mathbf{cm})\) & \(I_{2}(\mathbf{cm})\) & \(I_{3}(\mathbf{cm})\) & \(\theta_{1}(\mathbf{rad})\) & \(\theta_{2}(\mathbf{rad})\) & \(\theta_{3}(\mathbf{rad})\) \\ \hline Team1 & 15.1023 & 23.1212 & 25.1223 & 1.1684 & 1.6863 & 2.0615 \\ Team2 & 14.8623 & 23.0258 & 25.0782 & 1.1697 & 1.6912 & 2.1109 \\ Team3 & 14.7956 & 23.1203 & 24.9036 & 1.1716 & 1.7065 & 2.0738 \\ Team4 & 15.0023 & 22.7250 & 24.9956 & 1.1702 & 1.6544 & 2.0752 \\ Team5 & 15.0320 & 22.8155 & 24.7152 & 1.1686 & 1.6764 & 2.0813 \\ Team6 & 15.1780 & 23.2013 & 24.8574 & 1.1581 & 1.7106 & 2.1024 \\ Team7 & 15.2311 & 23.0035 & 25.1129 & 1.1573 & 1.6872 & 2.0623 \\ Team8 & 14.9055 & 23.1322 & 25.3001 & 1.1592 & 1.6928 & 2.0668 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Partial data table of three-link dimensional synthesis (NPGA). Figure 11: Scattering of remaining ADVN. Figure 12: Output results of NPGA in chemical industry. In Table 5, synthesizing various parameters, we take \(l_{1}=16.0000\), \(l_{2}=22.0000\), \(l_{3}=26.0000\), \(\theta_{1}\in[0,65^{\circ}]\), \(\theta_{2}\in[0,96^{\circ}]\), \(\theta_{3}\in[0,117^{\circ}]\) after radian convert to angles.