## 5 Algorithm That Determines Singular Points Generation
### Analysis of the Configuration alongside Trajectory of the Endpoint Movement  
The smoothness of the robotic arms' movement was determined from a special analysis of the configuration along a straight line delimited by endpoints in the workspace. Individual A\({}_{k}\) samples alongside the examined trajectory were ordered under the k index ranking. The essence of the analysis lies in the fact that a straight line or a line with a tolerance array was selected in the workspace. Only the non-singular points that fall within the straight line's tolerance array in the Monte Carlo simulation are evaluated. The configuration of the angle parameters was saved for each of these points. When the points were ordered along the straight line, angle parameters can be rendered graphically. Angular parameters are selected with a certain restriction on the movement of joints so as to reduce the number of possible configurations in a single trajectory point. This restriction may be adjusted to correspond to the existing real robot constraints. The constraints may be changed at will according to the analysis result. In this way, unambiguous solutions for the robot's movement can be obtained and the computational time reduced. The shape of the angle parameters determines the manner in which robotic arms may change from the initial point A to endpoint B in the course of the endpoint's movement along the straight line. If the parameters change smoothly, such movement along the straight line is well-programmable. A smooth change means the angular changes between two adjacent points on the trajectory at the same robot configuration are small. Figure 11 shows configurations with angle parameters along a straight line for a robotic mechanism featuring three joints (3 degrees of freedom) RRR in space (Figure 4). The analysis follows a point in space. In the case shown below, the system is non-redundant. A difference in the robot's configuration is best seen in the change of the joint \(q_{2}\). Figure 11: Change in four possible configurations along the straight line in the workspace composed of points A\({}_{k}\) (angles \(q_{1}\), \(q_{2}\), \(q_{3}\) have a constraint set from 0 to \(2\pi\), repeated simulation \(5\times 25,000\) samples, the straight line does not intersect the \(z\) axis). Figure 12 shows spatial layout of the straight line and the mechanism in the coordinate system. Only one configuration of the robotic arm layout is shown out of four possible arrangements. The analysis will change significantly if the straight line passes through a singular point in this mechanism, i.e., intersects the \(z\)-axis (Figures 13 and 14). Figure 14: Change in four possible configurations along the straight line in the workspace featuring singularity (\(q_{1},\ q_{2},\ q_{3}\) in the absolute range from 0 to 2\(\pi\) ). Figure 12: Straight line’s position with respect to the RRR mechanism. Figure 13: Position of the straight line with respect to the robot featuring a singular point. Analysis of the Kuka and URM robots' workspace requires a certain configuration modification. A body in space has six degrees of freedom, to which the kinematics of non-redundant robots is adjusted. If only a single point is followed in space, three degrees are superfluous, and the system becomes redundant. Due to this, three angle parameters in the robots will be constrained, thus yielding a non-redundant system and the configuration with the remaining three parameters will be focused on. Practically, this is achieved by fixating the \(q_{4},\ q_{5},\ q_{6}\) joints. This case concerns the end effector the orientation of which, with respect to the coordinate system, is the one mentioned above. The position of the straight line with respect to the Kuka robot workspace is shown in Figure 15. Conversely, Figure 16 shows a configuration of a robot composed of the URM modules. As the analysis has shown, more singular points were found along the straight line of the URM than of the Kuka robot.