# Solution to the Inverse Problem of the Angular Manipulator Kinematics with Six Degrees of Freedom
## 5 Conclusions  
An analytical solution has been obtained for three models of robots. Two of them are structurally identical to PUMA 560 and one is fundamentally different, due to the design options of the manipulator being developed. In this model, the rotation axis of the fourth link does not pass through the intersection point of the rotation axes of the fifth and sixth, which is the case in the first two models and in the overwhelming majority of operated manipulators. This takes the third model out of the class of the so-called Pieper's models, so solving the inverse kinematics problem for it is new. It is shown that in this case, an analytical solution can be constructed determining the rotation angle of the fourth link from the solution of the transcendental equation obtained here (the second equation in System (15) for the scheme with the \"upper elbow\" or (16) for the scheme with the \"lower elbow\"). New here also is the construction of an algorithm for solving the inverse problem of kinematics of the models under consideration without using four-dimensional Denavit-Hartenberg matrices to determine the rotation angles of the last three links and direct geometric expressions for the angles of the first three links. The latter makes the algorithm clear and simple. It should be noted that all expressions are obtained for two main configurations of the manipulator (with an upper and lower \"elbows\"). For more complex models relative to those considered in the article, a scheme of a numerical iterative algorithm for solving inverse problems of kinematics based on the Newton method is proposed. In this case, analytical solutions for the considered models can be used as an initial approximation if they are close to those considered. The practically important issue of setting the orientation of the output link of the mechanism for performing the required operations is also considered. Here, three \"geographical\" angles are determined by the given coordinates of three points of the output link, which are included in the expressions of the elements of the rotation matrices. Since such a system of three angles, like any other, can degenerate, the use of two such systems associated with the characteristic points of the output link is proposed. The peculiarity of such systems is that if one of them degenerates, the other will remain non-degenerate, and vice versa. Therefore, the algorithm uses a parameter that regulates the use of one or another system of angles. Since the manipulator under consideration is only being developed, it was not possible to conduct any experiments with it, so the testing of the proposed algorithms was carried out by calculating the direct kinematics of the models after obtaining the laws of varying in the angles in the joints from the inverse kinematics problem. As an example of such a simulation, a calculation of a specially designed motion of the output link is given. It represented a spatial motion in which the initial position corresponded to the degeneration of one system of \"geographical\" angles, and the final position corresponded to the degeneration of another.  
Conceptualization, Y.A. and D.B.; methodology Y.A. and K.N.; software, Y.A. and H.S.; validation, Y.A. and K.N.; formal analysis, Y.A. and H.A.; investigation, Y.A. and H.S.; writing--original draft preparation, Y.A. and D.B.; writing--review and editing, D.B., K.N. and H.A. All authors have read and agreed to the published version of the manuscript.  
**Funding:** [PERSON] acknowledges the support by the Volkswagen Foundation within the programme \"Visiting research program for refugee Ukrainian scientists\" (Az. 9D257) and EURIZON project #3053 Investigation of thermal influence on errors in orientation systems used in aerospace engineering.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.