# Solution to the Inverse Problem of the Angular Manipulator Kinematics with Six Degrees of Freedom
## 4 Discussion  
This article is devoted to the development of algorithms for solving inverse problems of robotic kinematics using six-degree angular manipulators as an example. An analytical solution is obtained for three models of PUMA 560 robots. The first two are models considered in the literature as \"Pieper models\" [26], the peculiarity of which is the intersection of the rotation axes of the last three links at one point. The third model is outside the class of Pieper models, so the analytical solution obtained in the article is new, expanding the approach based on [PERSON]'s ideas to a wider class of 6R manipulator models. In all the algorithms given, angles of the \"geographic\" system are used to specify the orientation of the output link. This is convenient in the case of welding manipulators. Such a system of angles degenerates at latitude values of \(\pm 90\)\". In order to eliminate the degeneracy of equations for such angles, two systems of \"geographic\" angles are used. Both are associated with the points of the working tool and form a constant angle between themselves. At a specific moment in time, a non-degenerate system is used. It seems that this is a relatively new approach. The main idea of the algorithm for solving the inverse kinematics problem and its significant limitation are the use of the fact that the axes of the last three (or two) links intersect at one point. This fact is the basis for many algorithms for solving the inverse kinematics problem for robots of the PUMA 560 type. In this article, based on the assignment of the position of the output link, an original solution to the transition problem is given, based on the consideration of triangles constructed on the axes of the first three links. This gives clear expressions for determining the rotation angles of these links without using three-dimensional rotation matrices or four-dimensional coordinate transformation matrices, in particular, the Denavit-Hartenberg matrices. The expressions become simpler without considering the additional angles specially introduced by [PERSON] [1] to resolve transcendental equations. This approach also shows some novelty. After solving the transition problem, the orientation of the third link becomes known. Considering the given orientation of the working tool, the orientation problem is completed by determining the angles of the last three links. Here, Denavit-Hartenberg matrices can be used, as in the [PERSON] method [1]. The article shows that this can be performed using only link rotation matrices. This approach is traditional. The issue of developing analytical algorithms for models that do not have the Pieper's property must be resolved in each specific case. Here, the proposed algorithm is extended to the case of intersection of the rotation axes of the fifth and sixth links and the absence of intersection with this point of the axis of the fourth. The formulation and solution of such a problem seems new to us. Consideration of other cases may be the task of subsequent studies. The algorithms are derived and used in the article for calculations of the welding manipulator being developed in Kharkiv, the work on which has been temporarily stopped. Therefore, it is not possible to conduct any experiments. The only way to check the validity of the proposed algorithms is to calculate the forward kinematics and compare the results with the original ones for the inverse kinematics or vice versa. To illustrate this method and, at the same time, to check the operability of the algorithms with the degeneration of \"geographical\" angles, the results of modeling a special motion of the working element are given. In this case, analytical expressions are specified in the form of (21) of the spiral laws of motion of points K and E relative to a fixed point D (see Figure 9c).