## 4 Application of the Monte Carlo Method to Assessing the Probability of Singular Points Occurrence  
Input data for the identification of proximity of a singular point is the endpoint position with respect to the extended axes of the robot's joints. The Monte Carlo method makes it possible to generate random positions of the robot arms' configuration. Individual positions are generated based on an even distribution of the probability of occurrence in the given interval of movement. In such a case, each position has an equal chance of materializing. Each position can then be statistically evaluated in terms of attaining proximity to a singular state.  
Surroundings of the singular point may be considered a critical area. The size of these surroundings is given by precision based on which a decision is made whether the Jacobian determinant will be considered to be quasi-zero. The agreed precision of calculation is, at the same time, a reference value for each mechanism type. If the critical value is exceeded, the point in the space this value refers to will be considered a singular point.  
Figure 4: Arms singularity projected onto a plane.  
Figure 5: Singularity of the wrist projected onto a plane.  
Figure 6 shows the size of the area critical for singularity with respect to the workspace that originated through point generation for the telescopic mechanism (Figure 1). In this case, there is only one singular point, and its size represents the calculation precision of setting the Jacobian determinant with respect to the mechanism's workspace. At the threshold precision value for calculating the Jacobian determinant of 0.01, the size of the tolerance field corresponds approximately to one-twentieth of the workspace span, which meets the simulation requirements.  
This precision has been maintained in all mechanisms. The mechanism with the revolute joints in Figure 2 has several singular points. The Monte Carlo analysis has yielded a map of singularities (Figure 7). Only outer singularities can be seen on the picture, as the second arm is shorter than the first one, so the kinematics of the mechanism does not allow the endpoint to pass through the central axis.  
The generation of a large number of attempts will show the ratio and the distribution of the singular states versus the non-singular states. This is enabled by the visual rendering of the given workspace section. The distribution of singular points depends on the chosen size of tolerance of the Jacobian determinant and on the number of points generated. Figure 8 shows singular points in space, singular points in the planar section of the workspace, and pairs of singular and non-singular points. These pairs represent the mutual configuration of robotic arms that can be attained in several ways.  
Figure 6: Workspace of a planar mechanism with a rotational and prismatic joint RP: (**a**) Singular and nonsingular points of workspace, (**b**) singular positions only.  
Figure 7: Workspace of a planar mechanism with two rotational joints RR: (**a**) Singular (red) and nonsingular (blue) workspace points; (**b**) in singular positions only.  
As the analysis has shown, a non-singular configuration can be attained in immediate proximity to singular points in all robot types. However, it needs to be decided here to what degree the established precision of calculating the Jacobian determinant is going to be critical for the given type of mechanism.  
To compare the robots, two known designs (Stanford, Kuka) and one special design (URM-unlimited rotational module) have been selected (Table 1).