## 1 State-of-the-Art in Studying Spherical Mechanisms
This new design has only two sub-chains, which bring about kinematic advantages, such as large workspace and small footprint ratio. [PERSON] et al. [28] proposed a construction approach from Origami to create multiple spherical integrated mechanisms and built a novel quadruped robot with the kinematotropic metamorphic 8R mechanism as the reconfigurable trunk. [PERSON] et al. [29] invented a novel mechanism active ankle that consists of three degrees of freedom and operates in an almost spherical work modality. [PERSON], [PERSON] et al. [30] reviewed the state-of-the-art of kinematics, dynamics, and design optimization of spherical mechanisms and novel spherical mechanisms with emerging applications. Some of the aforementioned spherical and spatial mechanisms only contain revolute pairs, while the rest only possess revolute and spherical pairs. None of them contains prismatic pairs. Many researchers invented non-spherical spatial mechanisms with prismatic pairs. For instance, [PERSON], [PERSON] et al. [31] designed a four-universal joint-prismatic joint-spherical joint/universal joint-prismatic joint-universal joint (4 UPS-UPU) spatial parallel mechanism to validate the proposed dynamic modeling method for a spatial parallel mechanism with multi-spherical joint clearances based on the Lagrange multiplier method. [PERSON] et al. [32] designed a four-UPS/1-RPS parallel mechanism, which has five prismatic pairs, and they turned it into a grinding robot by equipping it with a constant force actuator. Hence, prismatic pairs are an indispensable element in creating spatial mechanisms, as they have diversified spatial mechanisms' motion. Nevertheless, very few researchers have incorporated spherical sliders, which contain spherical prismatic pairs, in designing spherical mechanisms. [PERSON], [PERSON] et al. [33] created a novel three-DOF spherical parallel mechanism with arc prismatic pairs whose singularity analysis was conducted. [PERSON], [PERSON] et al. [34] built a two-DOF curved biaxial swing spherical mechanism with two active arc sliders and solved the forward and inverse kinematics for posture control of the mechanism's end effector. The spherical mechanism has rotational degrees of freedom in two directions around a fixed center. [PERSON], [PERSON] et al. [35] theoretically confirmed the influence that the arc sliders' friction had on the workspace of the mechanism proposed in [34] and used roller guides to reduce friction created by the arc sliders. The two-DOF curved biaxial swing spherical mechanism with two active arc sliders can be utilized in automobiles such as stabilization platforms and acceleration impact reduction mechanisms. [PERSON], [PERSON] et al. [36] presented a three-DOF spherical parallel mechanism for the shoulder complex of the humanoid robot and chose 3-PSS/S as the original configuration for the shoulder complex. With the genetic algorithm, the optimization of the 3-PSS/S spherical parallel mechanism was performed, and the orientation workspace of the prototype mechanism was enlarged. Some scholars focused on designing the single-DOF single-loop spherical four-bar crank-slider mechanism [37; 38] and studied its output motion. However, the abovementioned spherical mechanisms with prismatic pairs are not multi-loop spherical mechanisms that only consist of closed spherical kinematic chains. To investigate whether or not spherical prismatic pairs are capable of diversifying multi-loop spherical mechanisms' motion, the Stephenson-III spherical mechanism with a spherical slider containing a spherical prismatic pair, which has two closed spherical kinematic chains, is proposed. An algebraic approach will be employed to establish the loop equations of the two closed spherical kinematic chains. Afterwards, the mechanism's input-output curves and singularity curves can be obtained via the discriminant method mentioned in [6; 11]; JRS, branches, dead center positions, sub-branches, and branch points of the two spherical kinematic chains can be identified via the branch graphs of the proposed mechanism.