## 1 State-of-the-Art in Studying Spherical Mechanisms
[15] created a spherical mechanism for a robotic manipulator to perform craniotomy, which is a novel 3-RRR (R represents the revolute pair) spherical parallel mechanism whose workspace and kinematic performance were systematically studied. A screw theory approach [16] was created for the computation of the instantaneous rotation axes of indeterminate spherical linkages. [PERSON] et al. [17] put forward a wavelet feature parameters method for the non-periodic design requirements in the path synthesis of a spherical four-bar linkage mechanism. [PERSON] et al. [18] established the dynamic modeling and base inertial parameter determination of a general 5R two-DOF spherical parallel manipulator via a new geometric approach where inverse and forward kinematic problems were transformed to the problem of determining the intersection of two cones with a common vertex. [PERSON] et al. [19] redesigned the 3-RRR parallel manipulator with reconfigurability. The new design can be used directly for applications, such as flight simulation; the workspace, dexterity, and singularity of the new design were investigated. [PERSON] et al. [20] developed and validated an error kinematic model of a mini spherical robot, which is a spatial five-bar linkage with two rotational degrees of freedom. [PERSON] et al. [21] developed an analytical model to solve the dynamic problem of a 3-RRR spherical parallel manipulator. As per the abovementioned mechanisms, multi-loop spherical mechanisms' versatility is demonstrated by their various industrial and medical applications. Multi-loop spherical mechanisms' versatility is generated by the fact that the mechanism has a special structure where rotation axes of all its links intersect at the center of the mechanism's sphere, thus making the mechanisms' workspace wider and increasing the mechanisms' stability. Other scholars have invented other types of spherical mechanisms and other spatial mechanisms that either include spherical kinematic chains or generate spherical motion without spherical links, and they analyzed the mechanisms' motion characteristics. Some of the mechanisms have valuable industrial usage. [PERSON] et al. [22] analyzed the dynamic property of the parallel spherical manipulators with three degrees of freedom and presented determination of acceleration input links, oscillations, and the problem of control. [PERSON] et al. [23] presented two novel metamorphic linkages as the spherical-planar 6R metamorphic linkage and the Bennett-spherical 6R metamorphic linkage, both of which have three various distinguished motion branches, and they revealed the conditions of various motion branches and a set of transformations for switching motion branches after establishing the close-loop equations of the spherical-planar 6R metamorphic linkages. [PERSON] et al. [24] presented a geometric method to establish static balancing of mechanisms constructed using spherical chain units. The method was proven with a Bennett-Plano-Spherical hybrid linkage and a 3-RRS (S represents the spherical pair) parallel mechanism. [PERSON] et al. [25] presented the crease pattern of a Kirigami model, which is kinematically equivalent to a Bennett Plano-spherical linkage, and introduced a new reconfigurable parallel manipulator with three hybrid kinematic limbs, each of which consists of a closed-loop sub-chain, a Bennett Plano-spherical linkage, and a R(RR) serial chain. [PERSON] et al. [26] designed a novel spherical parallel manipulator, which consists of three identical limbs, each of which is formed by a planar parallelogram linkage, a universal joint, and a revolute one successively, and they analyzed the manipulator's mobility via the reciprocal screws. [PERSON] et al. [27] proposed a novel 6-DOF parallel manipulator, which is in the form of a five-bar spatial linkage with spherical-universal-universal-spherical architecture, where actuation comes from two spherical joints. The manipulator's active spherical joint is a spherical 3-RRR parallel mechanism.