## 1 State-of-the-Art in Studying Spherical Mechanisms  
Multi-loop spherical mechanisms are the backbone of many mechanical structures, such as surgical devices [1] and rehabilitation devices [2]. Compared with planar mechanisms, multi-loop spherical mechanisms are able to operate in three-dimensional spaceswhile maintaining relatively simple movement. Similar to a planar link of a multi-loop planar mechanism, a spherical link of a multi-loop spherical mechanism has at most three degrees of freedom (DOF); similar to a planar kinematic pair of a multi-loop planar mechanism, a spherical kinematic pair of a multi-loop spherical mechanism has at most two degrees of constraint. Therefore, the algebraic method to analyze the motion of a multi-loop planar mechanism's closed kinematic chains is similar to the algebraic method to study the motion characteristics of a multi-loop spherical mechanism's closed kinematic chains. A spherical mechanism, which consists of one closed loop which is also called a closed kinematic chain, or multiple closed loops, has multiple spherical links and kinematic pairs. For instance, a single-DOF single-loop four-bar spherical mechanism with only revolute pairs has four spherical links and four revolute pairs, which together form a closed four-bar kinematic chain. On the basis of the N-bar rotatability laws [3; 4], the concept of joint rotation space (JRS) [5], and the discriminant method [6], input-output curves, singularity curves, dead center positions, branches, sub-branches, and branch points are crucial elements to understanding a multi-loop spherical mechanism's motion. The input-output curve of a single-DOF multi-loop spherical mechanism describes the relationship between the input and output joints. Dead center positions, where the single-DOF multi-loop spherical mechanism's motion is impeded, sometimes appear on input-output curves. A singularity curve is a collection of all the singularity configurations of a closed spherical kinematic chain; a branch is the range of motion where a multi-loop spherical mechanism operates continuously without impediments; branch points are the intersection points of the input-output curves and singularity curves or the intersection points of singularity curves. Singularity analysis [7; 8] and branch identification [9; 10] are some of the most efficient approaches to identify input-output curves, singularity curves, dead center positions, branches, sub-branches, and branch points of multi-loop spherical mechanisms with closed kinematic chains. Since multi-loop spherical mechanisms can generate spatial motion that multi-loop planar mechanisms cannot create, many scholars around the world have designed novel multi-loop spherical mechanisms of closed kinematic chains with only revolute pairs and introduced different methods to study the mechanisms' motion characteristics. Some of them have also created various practical mechanical structures with those spherical mechanisms. With the concept of JRS and the discriminant method, [PERSON] et al. [11] identified and analyzed motion characteristics, branches, and singularity configurations of the two-DOF two-loop spherical seven-bar mechanism with only spherical revolute pairs that contain two closed five-bar spherical kinematic chains. [PERSON] et al. [12] put forward a modular approach for analyzing kinematic and mechanical errors of serial spherical mechanisms that can be decomposed into predefined modules. [PERSON] et al. [13] presented a new spherical decoupled mechanism, which is a new five-bar spherical parallel linkage that generates a remote center of motion of two angular decoupled degrees of freedom, and they conducted kinematic analysis of the proposed mechanism via kinematic and velocity models. A robotic manipulator [14] based on the spherical decoupled mechanism in [13] was invented to perform neuro-endoscopy for brain exploration. [PERSON] et al.