_applied sciences_  
_Article_  
The Kinematic Investigation of the Stephenson-III Spherical Mechanism with a Spherical Slider Containing a Spherical Prismatic Pair  
[PERSON] \({}^{\copyright}\), [PERSON] \({}^{\copyright}\) and [PERSON] \({}^{\ast}\)  
Institute of Additive Manufacturing and Industrial Robotics, School of Mechanical Engineering,  
Hubei University of Technology, Wuhan 430068, China; [EMAIL_ADDRESS]  
\({}^{\ast}\) Correspondence:[PERSON]S] (M.Y.); [EMAIL_ADDRESS] (Y.H)  
**Featureued Application: The methodology proposed in this paper studies the mobility of a multi-loop spherical mechanism with a spherical slider containing a spherical prismatic pair. By analyzing the motion of each closed kinematic chain that the mechanism consists of, not only can the proposed methodology thoroughly and accurately identify the mechanism's singularity configurations, but the method can also demonstrate the mechanism's range of motion. Therefore, the proposed method ultimately identifies the mechanism's motion continuity and smoothness, which are two of the most essential characteristics a multi-loop spherical mechanism possesses. Hence, the proposed methodology can help mechanism designers create multi-loop mechanisms with spherical sliders without motion deficiencies by avoiding the mechanisms' singularity configurations.**  
**Abstract: Multi-loop spherical mechanisms are extremely beneficial for creating versatile mechanical devices, including robotic joints and surgical tools, since multi-loop spherical mechanisms possess unique capabilities to operate in spatial situations with relatively simple movement. Nevertheless, the research on multi-loop spherical mechanisms with spherical sliders containing spherical prismatic pairs is lacking. Therefore, the main innovation of this paper is to propose the Stephenson-III two-loop spherical mechanism that possesses a spherical slider containing a spherical prismatic pair and to analyze the proposed spherical mechanism's motion characteristics. An algebraic approach was employed to obtain the branch graphs of the proposed spherical mechanism with a spherical slider. The branch graphs were categorized into two types, according to whether branch points existed. With the algebraic approach, loop equations of the two spherical kinematic chains inside the proposed spherical mechanism were established to identify the input-output curves and singularity curves, with which the branch graphs were obtained. With the branch graphs, the joint rotation spaces (JRSs) of the proposed mechanism were recognized and so were the dead center positions, branches, sub-branches, and branch points. The results from the mathematical analysis were simulated and verified by three-dimensional (3D) models of the proposed spherical mechanism. The analytical results demonstrate that the spherical prismatic pair diversifies the motion of the proposed spherical mechanism by producing rotational sliding movement, which can cover the entire circumference of a specific greater circle on the proposed mechanism's sphere.**  
**Keywords: multi-loop spherical mechanism; spherical prismatic pair; branch graph; singularity configuration; three-dimensional simulation**