# Solution to the Inverse Problem of the Angular Manipulator Kinematics with Six Degrees of Freedom
## 1 Introduction  
Solving inverse kinematics problems is a key step in the development of control systems for robots and manipulators [1, 2, 3, 4, 5, 6, 7]. Additionally, such solutions play an important role in other fields, such as motion design in capture systems and character animation in computer graphics [8]. Today, inverse kinematics problems for robots and manipulators are addressed using both analytical and numerical methods, and their combination is also a viable approach. Recently, methods have been developed--and, in some cases, have demonstrated their effectiveness--that combine traditional approaches with artificial intelligence techniques [9, 10, 11]. Despite these advancements, classical analytical and numerical methods remain the primary means of solving such problems. Geometric methods are widely used [12, 13, 14]. One of the approaches that has experienced widespread adoption in recent years is the implementation of quaternion algebra [8, 15, 16, 17]. Its use, in some cases, reduces computational complexity and eliminates singularities [15]. In many cases, numerical methods are employed [7, 8, 18, 19, 20, 21, 22]. Although, in the most general cases, obtaining solutions without their use is impossible, one of their main shortcomings is the lack of numerical stability and the need for a large number of computationaloperations to achieve solution convergence. Starting with the seminal paper by [PERSON] (D-H) [20], which established the standard formalism for describing mechanisms with kinematic pairs of the fifth class, most subsequent studies have adopted this approach (see recent publications, e.g., [6]). However, the Denavit-Hartenberg (D-H) notation is not always optimal due to its constraints, such as the mandatory designation of the axes of kinematic pairs as the applicate axes of the corresponding link coordinate systems. These constraints often result in unnecessary coordinate system rotations of \(90^{\circ}\) or \(180^{\circ}\). The algorithms proposed in this article eliminate these limitations by directly describing displacements and rotations of the coordinate system of one link relative to another, by explicitly specifying the axes of motion and rotation [10; 23; 24; 25]. To solve the inverse problem of the kinematics of an angular manipulator with six degrees of freedom, a new analytical solution algorithm is proposed in this article. It is based on the well-known approach of [PERSON] [1; 26], which uses the fact of the intersection of the axes of rotation of the last three links. This allows us to consider the orientation problem and the transfer problem separately with a reduction in the number of degrees of freedom from six to three (in each problem). For one of the three considered models, in which the axis of rotation of the fourth link does not pass through the point of intersection of the fifth and sixth links, a modification of the solution was obtained. It is found in the preliminary solution of the transcendental equation for the rotation angle of the fourth link. Moreover, the solution of the inverse problem does not depend on whether the \"miss\" of the axis of the fourth link is large from the point of intersection of the fifth and sixth links. Two systems of \"geographical\" angles (\"longitude\", \"latitude\" and \"self-rotation angle\") are connected to the output link. One system is rotated relative to the other by some constant angle. This makes it possible to solve the problem of the degeneracy of such systems since when one degenerates, the other, non-degenerate one is used. The specified angle systems are built based on the given absolute coordinates of the three points of the link, the motion of which is specified by the operational process. For models of manipulators that do not coincide with the considered ones, but in some sense are close to them, the approach of constructing a solution using [PERSON]'s method, the initial approximation for which is taken as the solution of the considered models, is presented in this article.