## 1 Introduction
### Background and Related Work  
The configuration space is an important concept used in motion planning algorithms. The main idea is to represent the robot as a point in a multidimensional space and to map the obstacles [24; 25; 26; 27; 28]. The concept is applicable to both mobile [24] and stationary robots [25; 28]. A configuration of a stationary robot uniquely defines the mutual arrangement of its links. For this reason, this configuration is determined by the position of the controllable motors. The set of all configurations that a robot can occupy defines its configuration space. Each configuration of the robot in this space represents an n-dimensional point. Although tasks are formulated in real Cartesian space, in order for the robot to understand and execute them, they are translated in its configuration space. The movements of the robot are represented by the movement of a point in this space. A map is created with free regions and regions that are inadmissible due to collision with obstacles. The concept transforms the problem of planning the motion of a robot in an environment with obstacles (where the shape and dimensions of the robot and the obstacles must be considered) into a problem of planning the motion of a point in the robot's free configuration space. Configuration space has played a crucial role in the development of many motion-planning algorithms for environments with obstacles [26]. Many applications in robotics, computer-aided design, and related fields can be reduced to computational problems with configuration spaces. In [24], two important challenges related to configuration spaces are addressed: how to efficiently compute an approximate representation of high-dimensional configuration spaces; and how to efficiently perform geometric proximity queries and motion planning in high-dimensional configuration spaces. Algorithms for building a configuration space, ones based on machine learning and geometric approximation techniques, are presented. The motion of a robotic arm in a rectangular tunnel and the corresponding configuration space are studied in [25]. Techniques from geometric group theory are used to find the optimal way to move the arm from one position to another. The authors of [26] propose a decomposition of the configuration space in order to check for collisions of the robot with an obstacle. The algorithm is implemented for a robot with seven degrees of freedom. The research in [27] presents a simple algorithm for approximating the free configuration space of robots with a small number of DOFs. Various approaches are applied to planning motions in the configuration space: network methods, Voronoi decomposition and artificial potential fields. In [28], a configuration space warping method and the classic Rapidly-exploring Random Tree (RRT) search method are compared in terms of their computation time and path length. The methods for finding a solution to IK presented in [2; 3; 4; 5; 6; 7; 8; 9; 10; 11] can be defined as analytical or analytically geometric. Works [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23] present numerical methods for finding a solution to IK. Some of the presented works combine numerical, analytical and geometric approaches. Inverse kinematics is the basis for determining the configuration space of a robot. The investigations in [24; 25; 26; 27; 28] consider algorithms for generating a configuration space in the presence of obstacles, which is important for the proper implementation of robot control algorithms. The analysis of redundant robots [9; 10; 11; 12; 13] shows that infinitely many IK solutions are possible for a single position and orientation of the end effector. For classic industrial robots such as [2; 3; 4; 5; 6; 7; 8], depending on their joint constraints, it is possible to have more than one solution to the inverse problem, as shown in [2].