## 4 Results
### Example 2  
Figure 14 shows a 3D printed model of a redundant robot with 4 parallel axes of rotation. The main movements take place in the plane. The rotation \(\theta_{4}\) is used to change the orientation of the gripper, and \(\theta_{1},\ \theta_{2},\ \theta_{3}\) to define its position. Here, only the gripper position is of interest and the configuration space is considered as three-dimensional. The dimensions and joint constraints of the robot are given in Figure 15a. Figure 13: Solutions of the inverse kinematics problem and determination of zones with the proposed algorithm for a two-link mechanism from Figure 11. Transformation of trajectories. Octants and Types of Solutions in the Configuration Space  
The joint angles \(\theta_{1},\ \theta_{2},\ \theta_{3}\) are independently controllable, defining three mutually perpendicular axes and planes that define a three-dimensional configuration space. The three coordinate planes divide space into eight parts, and each part is known as an octant. In Figure 14c, the octants are marked with Roman numerals. The sign of the coordinates of a point in this space depends on the octant in which it is located. If the rotational joints are rotated by an angle \(-\pi<\theta_{i}<\pi\), each different type of solution will fall into a precisely defined octant. The angle \(\theta_{1}\) does not affect the solution type, therefore one solution type can fall into two adjacent octants. Table 1 presents the signs of the coordinates of the joint angles and the solution types according to the octants. Relative singularity configuration types are easier to define in the configuration space. The parameter \(\theta_{2}=0\) means that the singularity configurations \(S_{0,2}\) fall in the coordinate plane determined by the axes \(\theta_{1}\) and \(\theta_{3}\), and \(\theta_{3}=0\) sets \(S_{0,3}\) in the plane \(O,\theta_{1},\theta_{2}\) (see Figure 14c). Using the proposed algorithm, a Visual LISP for AutoCAD program is created to determine IK solutions and divide them into types. The program runs in AutoCAD and uses dialog boxes to set the input parameters. The results are presented in the AutoCAD model space. Figure 14: (**a**) Model and (**b**) prototype of a 3D printed educational robot. (**c**) Three-dimensional configuration space divided by types. Figure 15: (**a**) Dialog window for setting the basic geometrical parameters of the robot. (**b**) Dialog window for generating points in polar coordinates. (**c**) Generated workspace divided by type of available solutions. In the \"Geometric Parameters\" dialog box, Figure 15a, the lengths of the robot's links are set, as well as the joint constraints. This is where the scope of the solution is determined, either to find a single point, to find multiple points from a segment or arc, or to define the robot's workspace. If the \"Include obstacles\" option is selected, the program will check for collisions with obstacles. To define the workspace, multiple points are generated in polar coordinates. They are entered with the dialog window \"Positions of end effector\" (Figure 15b). The last line \"Number of radial points 2\" sets the number of points at which link \(a_{3}\) will be rotated according to the algorithm. The result, based on the parameters set by the dialog windows, is given in Figure 15c. The types of solutions are represented with small circles placed in different layers with different diameter and color. For clarity, Figure 16 shows the individual types of solutions separately. If the cross-section of these four main types of solutions (singularity configurations are not taken into account for now) is plotted, 10 regions numbered from 1 to 10 and given in Figure 17 are obtained. Since the resulting type map is symmetrical, the areas 1 through 6 are examined in more detail. The \"Point\" option from the \"Geometric parameters\" dialog box (Figure 15a is used to find solutions for point \(P\) from the corresponding regions.