## 1 Introduction  
Nowadays, the use of manipulator robots is widely extended in the industry because these robots carry out tasks that pose risks to humans, improve the quality of the products manufactured, and increase productivity. Nevertheless, robotics is not limited to the industry, but has also expanded to other fields such as medicine (robotic implants or remote surgical intervention systems), agriculture (sheep shearing or pork cutting), space (planetary surface exploration), and geological and military research, among others [1; 2; 3]. Manipulator robots can be classified into serial and parallel based on their structure. The links of a serial manipulator are normally assembled together by means of rotational or prismatic joints that form an open kinematic chain whose structure is similar to a human arm. This configuration allows for assembling a clamp or any other tools that can be moved to any position within the robot's workspace [4]. The industrial robots available in the global markets are not redundant, but they do have the degrees of freedom necessary for performing the tasks for which they were designed [5]. However, if a failure occurs in the actuator of an industrial robot, without a safety backup, its respective joint blocks, and the robot losses the mobility of that joint. This may lead to situations that imply risks to the operators, difficulties for the users, interruption of the mining or industrial process, and economic losses, among other consequences [6]. Therefore, interest has arisen in the study of the behavior of redundant manipulator robots, owing to a design that gives them more degrees of freedom than necessary for performing a specific task [7]. In general, the kinematic and dynamic study of robotic systems can be conducted in a didactic way, using graphic simulators. The kinematic problem can be solved through the Geometric Method, but, in practice, this method is not widely used because of the complexity of applying it to redundant manipulator robots, which has led to the development of several alternative methods. Kinematics is divided into direct and inverse kinematics (IK) that can be derived based on the standard Denavit-Hartenberg (D-H) model, which describes the geometry of the robot. Besides the almost universally used the D-H parameters [8; 9; 10; 11], other methods such Algebra and Lie Groups are also useful to obtain the kinematics of the system [12]. The inverse kinematics chain of redundant manipulator robots is much more difficult to obtain. Its solution is fundamental to deal with problems such as route planning, visually guided movement, grabbing objects, etc. [13]. Currently, the conventional methods for solving inverse kinematics are divided into two categories: analytical (geometric-algebraic) and numerical (iterative algorithms) [14]. Some recent articles show that analytical solutions have only been found for some specific structures of redundant manipulator robots. For instance, [PERSON] et al. [15] present a close loop intuitive solution for a space manipulator robot with 7-DoF and a PYR-P-PYR (Pitch-Yaw-Roll) structured based on the blockage of the first joint P. [PERSON] and [PERSON] [16] present the Inverse Coordinates Method for a special S-RRR (Spherical-Rotary) configuration, which is based on the inversion of the base and the end effector of the manipulator robot. Additionally, [PERSON] et al. [17] employ analytical, geometric and algebraic methods for a 6R industrial robot, which are combined with the Paden-Kahan subproblem and the matrix theory. Furthermore, since there are no possible analytical solutions for the inverse kinematics of some robots, some methods are now under discussion, for example, the Iterative Method, which is based on numerical methods [18]. [PERSON] and [PERSON] [19] present a numerical solution, which is achieved through the application of the Sequential Recovery Method to a robot with and RPY (Roll-Pitch-Yaw) configuration based on the modification of the Newton-Raphson Method.