## 7 Geometric Parameters Optimization
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Stage** & **a\({}_{\text{min}}\)** & **a\({}_{\text{max}}\)** & **c\({}_{\text{min}}\)** & **c\({}_{\text{max}}\)** & **d\({}_{\text{min}}\)** & **d\({}_{\text{max}}\)** & **e\({}_{\text{min}}\)** & **e\({}_{\text{max}}\)** & **Step** & **Required** \\  & & & & & & & & & & & \\
1 & 50 & 1700 & 50 & 1700 & 50 & 1700 & 50 & 1700 & 100 & 1.5 \\
2 & 50 & 320 & 50 & 320 & 200 & 800 & 750 & 1320 & 30 & 2.4 \\
3 & 51 & 139 & 81 & 169 & 346 & 634 & 1020 & 1316 & 8 & 2708 \\
4 & 60 & 72 & 82 & 94 & 469 & 520 & 1173 & 1227 & 3 & 27,105 \\
5 & 61 & 65 & 83 & 87 & 512 & 519 & 1183 & 1190 & 1 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Stages of determining the optimal link lengths. Figure 15: Dependence of the volume of the workspace on the change in the length of the links \(d\) and \(e\). The optimization was performed in several stages, similar to the optimization performed above to maximize the workspace. Initial data for optimization: \(a\in[30;1500],\ c\in[30;1500],\ d\in[30;1500],\ e\in[30;1500],\ h_{c}=350\) mm. As a result, the optimal value of the criterion \(k=1.778\) was obtained for the following dimensions: \(a=61\) mm, \(c=87\) mm, \(d=478\) mm, \(e=1485\) mm, \(r_{c}=1187,29\) mm. The simulation of the workspace was performed for the obtained dimensions. Figure 17 shows that the workspace completely includes the cylinder with the calculated radius. An additional simulation was performed to compare the results of the two optimizations. The volume of the workspace and the maximum cylinder diameter were calculated for both optimal results. The parameters obtained during the first optimization were changed while maintaining the ratio 1:1.41:8.41:19.51. The change is necessary due to the incorrectness of comparing results that have a different sum of parameters. The sum of the parameters for the second optimization was 2111 mm. Therefore, the dimensions of the first optimization were changed so that their sum was 2111 mm. The obtained initial data and simulation results are shown in Table 5.  
Figure 16: Dependence of the ratio of areas of interference to the workspace on the length of the links. Figure 17: Workspace for optimal parameters. As can be seen from the table, the workspace volume was larger as expected for the parameters obtained as a result of the first optimization and the maximum radius of the inscribed cylinder for the second optimization. However, it should be noted that the workspace volume in the first case turned out to be only 0.47% larger, while the radius for the second case was larger by 9.34%. Therefore, in this case, the radius of the inscribed cylinder is the main criterion for the problem of aliquoting. So, we can conclude that second optimization results should be currently accepted as the final results. However, a formal optimization algorithm will be planned in the future for a further optimization of the obtained results. The aliquoting process assumes that Uni and DeLi should have a common part of workspaces. The Uni kinematic scheme is shown in Figure 18.  
Workspace of Uni is calculated using D-H parameters and transformation matrices, where D-H parameters are equal to \(a_{2}=240\) mm, \(a_{3}=210\) mm, \(d_{1}=150\) mm, \(d_{4}=110\) mm, \(d_{5}=85\) mm, and \(d_{6}=81\) mm. The end-effector positions were calculated in discrete steps using the forward kinematics. The calculated positions were entered into the workspace array. The resulting array was exported to the STL model. Likewise, a program was written in the C++ programming language to implement the developed algorithm. The Uni workspace is shown in Figure 19. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Optimization** & & & & & & & \\
**No.** & & & & & & & \\ \hline
1 & 69.606 & 98.133 & 585.375 & 1357.886 & 2111 & 4.024 & 1085.91 \\
2 & 61 & 87 & 478 & 1485 & 2111 & 4.005 & 1187.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Stages of determining the optimal link lengths. Figure 18: The kinematic scheme of Uni robot.