## 2 Definitions and Conditions for the Solution Types  
Many of the stationary robots can be represented as planar [1; 13]. For others, the spatial motions are obtained after rotation or translation of a planar mechanism [29], which means that the study of planar robots is important. Still, the preponderance of robots mainly use rotational and translational joints, and a planar mechanism with a serial topology can be schematically represented as in Figure 2a. Here, the axes of the rotational joints are marked with green letters \(O,\ P_{1},\ P_{2},\ \dots\ P_{n}\) the center of the gripper with P, and the lengths of the links with \(a_{i}\). The length of the gripper with \(a_{g}\). The axis of a translational joint can pass through the axes of two adjacent rotational joints, such as between points \(P_{1}\) and \(P_{2}\), or be arbitrarily loc  
Figure 1: Problematic configurations in solving inverse kinematics for a planar robot with 3 DOF. Incorrectly defined solutions with respect to (**a**) existence, (**b**) uniqueness, and (**c**) stability. Figure 2: (**a**) Scheme of a planar serial robot with n rotational and translational degrees of freedom—and (**b**) its analogous representation with only rotational joints. rotations, such as the translation located between \(P_{k-1}\) and \(P_{k}\). However, for both cases it can be said that the main effect of the translations is to change the distance \(a_{k}\) between their neighboring rotations. Similar considerations can be made even for cases where there are several translations located between two rotational joints. For these reasons, in our further reasoning, it is used a simplified equivalent model of a planar serial robot, one which uses only rotational joints, and where there are translations, the effect is reflected with a variable length of the link- _var._\(a_{k}\) Thus, three definitions are introduced for an n-link robot, which can be represented as shown in Figure 2b. **Definition 1**. : _Right-hand relative configuration of two adjacent links (for example \(a_{k}\) and \(a_{k+1}\)) of the robot is called such a configuration, for which the relative angle \(\theta_{kR}\), between the two links, is within the range:_  
\[0\pm 2i\pi<\theta_{kR}<\pi\pm 2i\pi;\ i=0,1,2,3,\ldots,n. \tag{1}\]  
_Such configuration is denoted with \(R\) (Figure 3a)._  
For example: Usually the angles of rotation of the links, relative to each other, are less than a full revolution. Then, if the joint constraints of joint k are in the interval \(0<\theta_{kR}<\pi\), the configuration is right-hand, and it can be said that the angle \(\theta_{kR}\) is positive for a right-hand relative configuration of the two adjacent links. If a person bends their right arm, their shoulder and forearm will assume a right-hand configuration. **Definition 2**. : _Left-hand relative configuration of two adjacent links (for example \(a_{k}\) and \(a_{k+1}\)) of a robot is called such a configuration, for which the relative angle \(\theta_{kL}\) between the two links is within the range:_  
\[-\pi\pm 2i\pi<\theta_{kL}<0\pm 2i\pi;\ i=0,1,2,3,\ldots,n. \tag{2}\]  
_Such configuration is denoted with \(L\) (Figure 4a)._  
Figure 4: (a) Left-hand relative configuration L between two links of a serial robot. (b) Intervals in which the angle \(\theta_{k}\) changes in the left-hand configuration represented on a numerical axis. Figure 3: (a) Right-hand relative configuration R between two links of a serial robot. (b) Intervals in which the angle \(\theta_{k}\) changes in the right-hand configuration represented on a numerical axis. **Definition 3**. : _Relative singularity configuration \(\theta_{ks}\), between two adjacent links of a robot, is called such a configuration, which cannot be presented, neither as right-hand, nor as left-hand.