## 3 Theoretical Background  
Clarifying the essence of singularity states is the basis for their modeling. The following specific conditions hold for kinematic systems:  
Horizontal movement corresponds to the \(x\)-axis or the \(y\)-axis. Thus, all gravitational forces can be easily described in the cartesian system. Robots with predominantly articulated arms are more suitable for work in constrained quarters than are the robots with cartesian configuration.  
For example, the mounting capability of a jointed-arm mechanism in space is significantly better than that of a similar translational mechanism with a smaller number of building elements. This is a substantial advantage. A disadvantage lies in the fact that in the given design, the precision is reduced due to angular errors. Most often, the movement of the robot's arms is rotational, which is why calculating the trajectory of the endpoint necessitates conversion into the polar system. In terms of mathematics, this is an isomorphic transformation from the real space described by cartesian coordinates into the real space described by spherical coordinates. The Jacobian determinant is a transformational element enabling conversion between the two systems.  
In the case of a planar telescopic arm anchored at the outset, the Jacobian determinant equals zero if the arm is reduced to zero (Figure 1). This state of inner singularity may materialize only if the required endpoint trajectory passes directly through the joint. Figure 1 shows a gradual worsening of the state dependent on the trajectory, which gradually approaches the joint. This example enables an illustrative observation of change in differential values in individual systems. It follows from the analysis of the movement that a small differential change of a parameter in the cartesian system corresponds to a large change in the polar system (Figure 1b). The inadequate step change of the angle occurs around the joint, where the movement portion under consideration, is the one always lying in the plane perpendicular to the joint's axis.  
The same case will materialize if the mechanism in Figure 2 is used instead of a telescopic arm (Figure 2). Angle \(\varphi\) is replaced with a general parameter \(q\) in line with the marking used in robotics.  
Drawing on the examples above, singularity may be avoided in two ways. The first way is that the trajectory of the endpoint mechanism will not pass in the vicinity of a singular state. The second way is that the robot's redundant mechanism will be used. In the redundant mechanism case, the angular movements can be evenly distributed among all joints so that no singularity occurs in the movement across the joint's axis (Figure 3).  
This principle of the inner singularity origin also holds for other types of known singular state designs. For example, a known arm singularity may be evaluated in the plane perpendicular to the axis of rotation (Figure 4). In principle, this state corresponds with the examples in Figures 1 and 2.  
Figure 1: Movement of the endpoint mechanism (**a**) along a distant trajectory AB, (**b**) along a close trajectory AB.  
Figure 3: Redundant mechanism with three joints.  
By analogy, the wrist singularity, too, may be recognized on a similar principle (Figure 5).  
The above examples make it clear that the principle of the origin of internal singularities is always the same:  
_\"Inner singularity occurs in the case in which the nonredundant manipulator's endpoint trajectory in a plane perpendicular to the axis of rotation passes through that axis\"._  
In reality, it is sufficient for the singularity to occur also when the movement is in close proximity to such a point. This fact may be beneficially taken advantage of in the analysis and detection of singular points in the robot's workspace through the Monte Carlo method.