## 1 Introduction  
To increase the manipulator's kinematic dexterity, more degrees of freedom (DOFs) beyond required for accomplishing a given task are introduced. These redundant DOFs can be used to accomplish additional various tasks, including obstacle avoidance, singularity avoidance, and physical limitations [1]. However, the inverse kinematics of the redundant manipulators would be more challenging and complicated than the non-redundant one because of the existence of infinite number of solutions for a given task [2]. To solve the inverse kinematics of such redundant manipulators, many algorithms have been proposed. According to the solution type of the inverse kinematics problem, these algorithms can be categorized into three groups: analytical method, numerical method, and the hybrid method [2, 3, 4, 5]. Upon the analytical method, the joint variables can be explicitly interpreted by a function of the given pose of end-effector. Hence, the analytical method is characterized by computational efficiency and can provide all solutions for a given pose within the workspace of the manipulator [5]. However, the closed-form solution is strongly dependent on the configuration of the manipulator. It is generally known that the common 6-DOF non-redundant manipulator has a closed-form inverse kinematics solution if three consecutive revolute joint axes of the manipulator intersect at a common point or three consecutive revolute joint axes of the manipulator are parallel [6], especially the former. Actually, for the redundant manipulators, adding reasonable constraints is an effective way to pursuit the closed-form inverse kinematics solution. That is because that the number of DOF is bigger than the given conditions provided by the given position and orientation. Meanwhile, the existence of the above two criterions for the non-redundant 6-DOF manipulator will greatly increase the possibility of getting the closed-form solution [7; 8; 9; 10]. For example, [PERSON] et al. [7] modeled the redundant manipulator as a multi-section robot composed of spherical joints and proposed an approach, which computed per-section endpoint locations as the constraints. Based on these endpoint locations, the closed-form inverse kinematics of the multi-section robot is obtained by applying the inverse kinematics of single-section robot. [PERSON] et al. [9] directly enumerated the different combinations of the segment positions as the constraints. [PERSON] et al. [10] modeled the manipulator as a multi-group manipulator and determined the Cartesian coordinates of each node by modifying the well-known backbone curve method [11]. How to add right constraints to obtain the closed-form solution requires the algebraic and geometric intuition. In contrast to the analytical method, the solution of the joint variables solved by the numerical method is implicit in the nonlinear equations relating the joint variables and the given pose. Thus, various algorithms solving nonlinear equations can be applied to solve the inverse kinematics of the redundant manipulators, for example, Newton-Raphson algorithm [12], Levenberg-Marquardt algorithm [13], and Cyclic Coordinate Descent (CCD) method [14]. Moreover, intelligent algorithms, such as neural network [15; 16], fuzzy logic [17], and generic algorithm [18] are also used for the inverse kinematics problem of the redundant manipulators. These algorithms inevitably bear sensibility to the choice of initial values, high computational burden, and numerical errors. Upon the hybrid method combining analytical and numerical method, part of the joint variables can be given by the closed-form algorithm and the rest are solved by the numerical method. Alternatively, the solution may be in closed-form, while the closed-form solution is based on the conditions provided by some numerical method like numerical optimization. For example, to solve the inverse kinematics of the proposed \((2n+1)\)-DOF hyper-redundant manipulator, [PERSON] et al.