## 8 Determination of Links Interference of a Multi-Robotic System  
In the process of collaborative execution of tasks by a multi-robotic system, two problems arise within the framework of determining a possible intersection of links. The first task is to determine the presence of intersections for the given positions of each of the  
Figure 11: Serial manipulator areas: (a) workspace, (b) technological space. manipulators. Each of the positions can be obtained by discretizing the trajectory while performing the operation, taking into account the time. The second task is to determine the presence of intersections, taking into account all achievable positions of the manipulators for a given relative position of the mobile platforms. To solve the first problem, we use the approach to determining the intersections of links, described in Section 3. In this case, the intersection of all combinations of manipulator links with each other is checked. The coordinates specify the mutual position of the platforms \(x_{s}\) and \(y_{s}\) by the angle \(\lambda\) of rotation of the coordinate system of the mobile platform of the serial manipulator relative to the coordinate system of the parallel manipulator. Checking the intersection of the links of the serial manipulator with the rotating platform of the parallel manipulator includes turning relative to the X axis by an angle at which the plane of the turntable coincides with the XOY plane. In this case, checking is reduced to determine the intersections of the links of the serial manipulator with rectangles parallel to the XOY, XOZ and YOZ planes. Simulation was performed for \(x_{s}=-100\) mm, \(y_{s}=-100\) mm, \(\theta=\psi=0^{\circ}\), \(z_{0}=500\) mm, \(\varphi=78^{\circ}\). In the first case, for \(\lambda=110^{\circ}\), \(\theta^{(1)}=30^{\circ}\), \(\theta^{(2)}=35^{\circ}\), \(\theta^{(3)}=-20^{\circ}\), an intersection was found between the links (Figure 12a), in the second for \(\lambda=110^{\circ}\), \(\theta^{(1)}=40^{\circ}\), \(\theta^{(2)}=40^{\circ}\), \(\theta^{(3)}=10^{\circ}\) there is no intersection (Figure 12b), in the third, for \(\lambda=60^{\circ}\), \(\theta^{(1)}=30^{\circ}\), \(\theta^{(2)}=35^{\circ}\), \(\theta^{(3)}=-20^{\circ}\), an intersection occurs between the link of the serial manipulator and the turntable parallel manipulator (Figure 12c). To verify the results, visualization was implemented through automatic generation of a 3D model of a multi-robotic system for given positions (Figure 12). The results of checking the intersections of the manipulators links coincide with the results of checking using the generated 3D model of the multi-robotic system. The algorithm is fast enough to check about \(10^{6}\) different positions per second, which makes it possible to check intersections in real time, discretely checking the trajectories of the manipulators. To solve the second problem, let us determine the minimum distance between the regions. The distance \(\Delta\) between the centers of the boxes included in the technological areas of the manipulators is determined iteratively. The dimensions of the boxes for each of the measurements are equal to the approximation accuracy \(\delta\). Therefore, we take into account the distance from the center of the box to the top of the box, which is \(\frac{\sqrt{2\delta}}{2\pi}\) for each  
Figure 12: Relative position of the multi-robotic system—(**a**) the presence of an intersection between the links of the manipulators, (**b**) the absence of intersections, (**c**) the intersection of the serial manipulator with the rotating platform of the parallel manipulator. of the two boxes, between the centers of which we determine the distance. Based on this, we write down the condition for the absence of a possible intersection of the links:  
\[\Delta>D_{link}+\sqrt{2}\delta \tag{56}\]  
If the condition is not met for any of the combinations of boxes, the iterative check stops.