## 6 Simulation Results  
We investigate the workspace of the parallel manipulator without considering the rotation of the rotary part of the mobile platform. Consider the change in the workspace and the intersection of the links with restrictions on the angles of rotation of the movable platform \(E_{1}E_{2}E_{3}\). Simulation is performed for \(l_{1,2,3}\in\left[300\;\;\text{mm}\;,\;600\;\;\text{mm}\;\right]\), \(\text{R}=400\;mm\), \(\text{r}=100\;mm\), \(h=100\;\text{mm}\). Conversion of the obtained sets into an STL file for three-dimensional visualization of the workspace has been implemented. Also, the export for the formation of a 3D model of the manipulator in STL format for the positions at which the intersections of the links occur has been given. The results showed that for the ranges of angles \(\theta\in[-90^{\circ};90^{\circ}]\) and \(\psi\in[-90^{\circ};90^{\circ}]\) (Figure 6a), there are no intersections of the links. The volume of the workspace was \(6.48\times 10^{-6}\;\text{m}^{3}\). Increasing the angle ranges will result in overlaps. At the same time, the volume of the workspace increases slightly. Figure 6b shows the workspace for \(\theta\in[-95^{\circ};95^{\circ}]\) and \(\psi\in[-95^{\circ};95^{\circ}]\), and Figure 7 shows some cases of intersections. The volume of the workspace increased insignificantly and amounted to \(6.59\times 10^{-6}\;\text{m}^{3}\).  
Figure 6: Workspace of the parallel manipulator—(a)—for the range of angles \(\theta\) and \(\psi\pm 90^{\circ}\), (b)—for the range of angles \(\theta\) and \(\psi\pm 95^{\circ}\).  
Simulations for other ranges of rod lengths have similarly shown the occurrence of intersections when the range of angles \(\theta\) and \(\psi\) is expanded over \(\pm 90^{\circ}\). Based on the results for further research, the above restriction on the angles of rotation of the movable platform of the parallel manipulator was adopted.  
To select the sign in the condition of constancy of the determinant of the Jacobi matrix to exclude singularity zones, the workspace was determined both with the condition of positiveness (Figure 8a) of the determinant, and with the condition of negativity (Figure 8b).  
Figure 8 shows that the inner part of the workspace corresponds to a negative value of the determinant. Based on this, we add the condition for the negative determinant of the Jacobi matrix to exclude singularities. The volume of the workspace, taking into account singularity zones, was \(5.71\times 10^{-6}\) m\({}^{3}\).  
Figure 8: Workspace of the parallel manipulator—(**a**)—with a negative sign of the determinant of the Jacobi matrix, (**b**)—with a positive sign.  
Let us define the workspace of the parallel manipulator, taking into account the rotation of the platform. The range of the angle of the rotary part of the platform \(\varphi\in[0^{\circ};90^{\circ}]\) was selected with a step of \(2^{\circ}\) for simulation. The results are shown in Figure 9a. The volume of the workspace due to the rotation of the platform increased from \(5.71\times 10^{-6}\) to \(62.04\times 10^{-6}\) m\({}^{3}\). The workspace shows the many achievable end effector positions. However, to avoid collisions of the manipulator with external objects, it is necessary to determine the technological area, that is, the area of possible positions of all intermediate links. So, in addition to the coordinates of the end effector, an iterative discrete determination of the coordinates of points lying on the axes of all links and the moving platform has been added. The discrete step value does not exceed the approximation accuracy. The results in an approximation accuracy of \(\delta=5\) mm, as shown in Figure 9b.