## 1 Introduction
### Literature Review  
Serial robots are the most common industrial manipulators. They are designed as a series of links connected by motor-actuated joints that extend from a base to an end-effector. Robot kinematics deals with two types of problems--forward and inverse kinematics problems. For serial robots the forward kinematics problem is straightforward, and a single solution is easily found; the most popular method for the solution is proposed by [PERSON] and [PERSON] [1]. Inverse kinematics (IK) is a much more difficult problem than forward kinematics due to the presence of singularities and nonlinearities. Completely analytical solutions exist only for a small class of kinematically simple manipulators [2]. For many industrial robot designs, specific inverse kinematic solutions have been found using various algorithmic approaches that do not involve direct analytical solutions. Deriving kinematic models for robots with a serial topology is essential for analyzing, researching and programming industrial robots. In [2], an analytical solution for a two-link robot is described, as well as for the basic structure of the Stanford Manipulator. The research in [3] investigates popular serial-structure mechanisms with two and three degrees of freedom, whose inverse kinematics permit closed-form analytical solutions. A Python program was developed for the serial-structure mechanisms. Geometric solutions of the inverse kinematics problem for robots with three and six rotational degrees of freedom are given in [4] and [5], respectively. The geometric approach is usually more intuitive and significantly reduces the complexity and time for calculations of the IK. However, it requires an individual approach to each different type of manipulator. In [6], a hybrid algorithm is proposed that combines the geometric method with an analytical one and is applied to an industrial robot PUMA 560. In [7], the inverse kinematics problem is solved with a reduced set of equations, an appropriate choice of the coordinate system associated with the end effector and an application of the principle of orthogonality in rotation. The approach does not require the calculation of the forward kinematics problem and can be used for manipulators with different geometries. For redundant robots, determining analytically all solutions of the inverse kinematics problem is a very difficult task [8; 9; 10; 11]. Despite the complexity, [8; 9; 10; 11] contain several examples of finding solutions analytically. The Pioneer 2 robotic arm is a redundant robot with five degrees of freedom. The analytical equations for its inverse kinematics are derived in [8]. Article [9] presents solutions to the forward and inverse problems of a Power Cube 4-DOF redundant robot, using the Denavit and Hartenberg parameters, the law of cosines and the method of vector coordinates. It is shown that the different position and the change of an additional input variable have a significant influence on the robot configuration. The human hand is modeled as a redundant serial manipulator in [10]. The null space of the Jacobian matrix for the proposed model is determined. The results in [10] suggest that the redundancy in the human hand is used to reduce errors in trajectory and anisotropy arising from external disturbances. The authors of [11] propose a method for speeding up calculations in analytical and numerical methods for solving the inverse kinematics problem for a robot with seven degrees of freedom. Numerical algorithms usually do not give all the solutions. In general, the choice of initial value and the searching algorithm have a big influence on the accuracy of their solution [6]. In [12], a numerical method for IK of serial robots is proposed which uses a dual quaternion formulation of kinematics in a converging paths algorithm. According to [12], the traditional algebraic, geometric and iterative methods are inadequate if the manipulator's structure is more complex.