## 1 Introduction
[19] provided the first two and last three joint angles given by analytical expression, and the other joint angles solved by considering the optimization of elbow locations. [PERSON] et al. [20] determined the position of elbow of the 7-DOF manipulator by minimizing the force imposed on the elbow and the joint angles are obtained in analytical expressions by using geometric approach. [PERSON] et al. [21] proposed a method for a 7-DOF manipulator where four joint angles are computed analytically and the remaining three are computed using the pseudoinverse of Jacobian matrix. Due to the addition of the closed-form solution for some joint variables, the hybrid method can efficiently improve the computational efficiency [13; 21]. In this work, the hyper-redundant robot is a kind of \(2n\)-DOF serial manipulator with \(n\) identical universal joints. These universal joints have the same mechanical mechanism, the same angle limits, and the same kinematic characteristics. Research about the inverse kinematics problem for this type of manipulator has been carried out. However, these methods involve, more or less, numerical approaches. [PERSON] et al. [22] studied the inverse kinematics of the manipulators with three universal joints and one rotary joint. The last universal joint and the rotary joint are regarded as a spherical wrist. Based on this, the closed-form solution of the last six joint angles is given by functions about the first joint angle. The first joint angle was determined by minimizing total energy. Therefore, this inverse kinematics solution is not a rigorous closed-form solution. [PERSON] et al. [10] computed the Cartesian position of each universal joint by modifying the backbone curve method in which a spatial backbone is defined using a mode function to represent the geometry of the manipulator [11]. All of the universal joint angles are solved analytically based on the above computed universal joint positions. However, the numerical optimization still exists during the process of determining the Cartesian positions of the universal joints. Thus, this approach is not a rigorous closed-form solution. [PERSON] et al. [23] provided the inverse kinematics of the single-section manipulator. However, for the inverse kinematics of the whole manipulator, the proposed approach fell into the Jacobian analysis. Compared to the above inverse kinematics solutions for the \(2n\)-DOF hyper-redundant serial robot with \(n\) identical universal joints, the inverse kinematics solution proposed in this paper is rigorous closed-form. In this inverse kinematics model, the universal joint is modeled as a general spherical joint through introducing a virtual revolution between two adjacent universal joints. It is surprising that all of the joints of the \(2n\)-DOF manipulator are general spherical joints. Simultaneously, these general spherical joints can realize the decoupling of the positions and orientations. A novel method for determining the positions of these universal joints is proposed. In this method, the Cartesian coordinates of a universal joint are obtained according to three distances between the universal joint to be solved and another three different known ones. According to inverse kinematics of a single general spherical joint section, the closed-form solution for the whole manipulator is solved. How to choose the distances for solving the positions of universal joints is important in the proposed closed-form solution. The main contribution of this paper is as follows. Firstly, a novel rule for link frames attached to the links is proposed during the process of forward kinematics, which is more intuitive and briefer than the classical (Denavit-Hartenberg) D-H convention. Secondly, a novel concept named \"general spherical joint\", which exists between any two adjacent link frames, is introduced. The general spherical joint is similar to spherical wrist and can realize the decoupling of the positions and orientations. Thirdly, a completely closed-form solution for the \(2n\)-DOF manipulator is proposed.