## 4 Simulation Results of Workspace Taking into Account Singularity Zones  
The results of modeling the workspace were obtained for the following parameters of the DeLi manipulator: \(a\) = 450 mm, \(c\) = 200 mm, \(d\) = 150 mm, \(e\) = 230 mm. The visualization of the results of modeling the workspace is carried out by converting the covering set describing the workspace into a universal format of 3D models--an STL file. The constructed workspace without taking into account singularity zones is shown in Figure 7. Figure 7a shows the workspace in full, 7b--in section by the XOZ plane. In order to increase the performance, the processor multithreading is applied using the OpenMP package. The computation time for the approximation accuracy \(\delta\) = 2 mm and the grid dimension for performing computations of the 32 \(\times\) 32 \(\times\) 32 functions was 67 s for each inverse kinematics case. The results of modeling the workspace taking into account singularity zones are shown in Figure 8. Figure 8a shows the regions corresponding to the singularity zones when using the \"+\" sign in inverse kinematics (2), Figure 8b--when using the \"-\" sign, Figure 8c--with all eight possible combinations of the \"+\" and \"-\" signs in the inverse kinematics, Figure 8d--for all combinations within the workspace. The average computation time for an approximation accuracy \(\delta\) = 2 mm and a grid dimension of 64 \(\times\) 64 \(\times\) 64 on a personal computer was 32 s.  
The simulation results show the following conclusions. The inner part of the workspace has a positive sign of the determinant of the Jacobi matrix if two or three kinematic chains have a \"-\" sign in the inverse kinematics. The negative sign of the determinant is obtained  
Figure 8: Areas corresponding to singularity zones: (**a**) for “+” in inverse kinematics, (**b**) for “-”, (**c**) for all possible combinations, (**d**) inside the workspace. Figure 7: (**a**) Volumetric workspace, (**b**) in section in the XOZ plane. in the opposite cases, that is, if the \"+\" sign is selected for at least two chains. With this in mind, when determining the workspace, taking into account singularity zones for four possible solutions, we add the condition of the positive determinant, i.e., det (\(J_{A}\)) \(>\) 0, and for the other four, the condition of negativity, i.e., det (\(J_{A}\)) \(<\) 0. Table 1 shows the simulation results for these cases. As can be seen from the table, out of eight options for the inverse kinematics, four groups of solutions can be distinguished: 1--all kinematic chains with \"+\", 2--all with \"-\", 3--one chain with \"+\" and two with \"-\", and also 4--two with \"+\" and one with \"-\". The first two groups correspond to one option, the second two to three options each. Considering the symmetry of the robot structure, the volume of the resulting workspace is practically the same for all variants of the group. This is confirmed by the calculated values of the volume of the workspace. The table also shows that options 2, 3, and 5, as well as 4, 6, and 7 correspond to the same volume of the workspace. Figure 9 shows the workspaces corresponding to each of the groups. In Figure 10, it can be seen that taking into account the singularity zones led to a decrease in the size of the workspace by 4.58-14.88%, depending on the combination of possible solutions to the inverse kinematics. In addition to singularity zones, the shape and volume of the workspace are directly influenced by the interferences of the robot links with the platforms and with each other.