## 7 Geometric Parameters Optimization  
Next, we will optimize the geometric parameters of the mechanism. The volume of the workspace, taking into account singularity zones and the interference of links, is an optimization criterion. Determination of the optimal sizes of links a, c, d, and e was carried out in several stages. At each stage, the volume of the workspace was calculated taking into account the interference of the links for various combinations of sizes. The ranges of resizing and the iteration step are reduced with each stage to reduce the computational complexity. Numerical values are given in Table 4. The last column contains the volume of the workspace, which is a criterion for excluding or including in the next stage a certain part of the size range. Constant dimensions for modeling: \(h=100\) mm, D\({}_{link}=30\) mm. The sum of the dimensions \(a\), \(c\), \(d\), and \(e\) was taken to be constant and equal to 1850 mm. The amount was selected in accordance with the dimensions of the industrial robot ABB IRB 360: \(a\approx 600\) mm, \(c\approx 100\) mm, \(d\approx 350\) mm, \(e\approx 100\) mm. It should be noted that at the first stage, the computations were carried out for all configurations; however, the condition for the required volume of the workspace was fulfilled only for configurations 1 and 2/3/5. At the second stage, the volume condition was met only for the first configuration. In this regard, modeling was carried out only for the first configuration starting from the third stage. Figure 14: Workspace of the mechanism after excluding the areas of interference: (**a**) for configuration 1, (**b**) 2, (**c**) 3. Figure 13: Examples of identified link interference: (**a**) links B\({}_{31}\)C\({}_{31}\) and B\({}_{32}\)C\({}_{32}\) with a fixed platform and link A\({}_{3}\)B\({}_{3}\) with link B\({}_{32}\)C\({}_{32}\), (**b**) links B\({}_{31}\)C\({}_{31}\) and B\({}_{32}\)C\({}_{32}\) with a fixed platform, (**c**) link A\({}_{2}\)B\({}_{2}\) with link B\({}_{22}\)C\({}_{22}\). The maximum workspace is reached at \(a=61\) mm, \(c=86\) mm, \(d=513\) mm, and \(e=1190\) mm. For such a ratio of dimensions, the total volume of the workspace, taking into account singularity zones, is 4.597 m\({}^{3}\), of which 2.714 m\({}^{3}\) is an area without interference of links. Thus, assuming the conditional size of the side of the base \(a=1\), the ratios of the sizes \(c\), \(d\), and \(e\) were 1.41, 8.41, and 19.51, respectively. The dependence of the volume of the workspace for various configurations on the change in the parameters \(d\) for \(a=61\) mm and \(c=86\) mm, \(e=(1850-a-c-d)\) is shown in Figure 15. The numerical values of the workspace volume for the plot are obtained by iterative execution of the developed program for various parameters. Figure 16 shows the dependence of the proportion of areas in which interference occurs in the total volume of the workspace depending on the parameters \(d\) and \(e\). The graphs show that for the first configuration, the fraction of interference decreases with increasing size \(d\). For the rest of the configurations, the percentage of interference for almost all sizes \(d\) is more than 90%. Let us perform the second stage of optimization. It takes into account the requirements of the aliquoting process. The required workspace height \(h_{c}\) is based on the requirements of tube and pipette sizes (350 mm). We can conclude that the manipulator workspace is round based on the previous analysis. Based on this, an optimization problem can be posed, which maximizes the radius \(r_{c}\) of a cylinder of a given height \(h_{c}\), which can be inscribed into the manipulator's workspace. In this case, the sum of the dimensions \(a,\ c,\ d,\) and \(e\) of the manipulator should also be minimized. Therefore, the general criterion k can be written as:  
\[k=\frac{a+c+d+e}{r_{c}} \tag{26}\]