## 4 Conclusions and Discussions  
According to the literature review in the first section, past research on multi-loop spherical mechanisms rarely included spherical sliders. However, prismatic pairs are frequently used in spatial mechanisms to diversify spatial mechanisms' motion. Therefore, it is necessary to investigate whether or not spherical sliders containing spherical prismatic pairs can diversify the motion of multi-loop spherical mechanisms. This paper proposed the single-DOF two-loop Stephenson-III spherical mechanism with a spherical slider that possesses a spherical prismatic pair. The approach to study the proposed mechanism's motion is algebraic. The loop equations of the two spherical kinematic chains of the proposed spherical mechanism were established to identify the input-output curves of the closed four-bar kinematic chain A\({}_{0}\)ABB\({}_{0}\) and the singularity curves of the closed five-bar kinematic chain B\({}_{0}\)BDCC\({}_{0}\) before the branch graphs of the mechanism could be obtained. The branch graphs of the proposed spherical mechanism were categorized into two types according to if branch points existed or not. With the branch graph, the dead center positions, branches, and branch points could be identified. With the obtained dead center positions and branch points, the precise range of the arc distance reached by the spherical slider was obtained. Therefore, whether or not the spherical slider can travel the entire circumference of the fixed track could be determined. The mathematical investigation of the proposed spherical mechanism was paired with three-dimensional simulation, where the three-dimensional models of the proposed spherical mechanism simulated the mechanism's motion and were utilized to verify the results of the mathematical investigation. As per the three-dimensional simulation of the proposed mechanism's motion with different parameters, spherical prismatic pairs are capable of producing rotational sliding motion that a spherical link with only revolute pairs cannot not achieve, thus diversifying the motion of the two-loop spherical mechanism proposed in this paper. The spherical slider containing the prismatic pair with specific parameters for the proposed mechanism is able to move along the entirety of the fixed track's circumference, which is a specific greater circle on the surface of the mechanism's sphere. In this paper, much focus is on incorporating the spherical prismatic pair into multi-loop spherical mechanisms, as well as the algebraic approach and three-dimensional simulation to study the kinematics of the proposed spherical mechanism. Less focus is on how the proposed mechanism can be adopted in real-life situations, which is one of the limitations of this paper. In future research, how the incorporated prismatic pairs advance  
Figure 17: The displacement range of \(\alpha_{56}\) in sub-branch 2 of branch 2; (**a**) maximal negative displacement; (**b**) minimal negative displacement. the performance of devices based on spherical mechanisms should be thoroughly studied. The next limitation is that the proposed mechanism only has one degree of freedom and incorporates only one spherical prismatic pair. If a spherical mechanism has two degrees of freedom and possesses one or more spherical prismatic pairs, the mechanism's motion can be further diversified. Therefore, in future research, the focus should also be on the two-DOF two-loop seven-bar spherical mechanism with one and two spherical prismatic pairs. The final limitation is that the proposed mechanism only has two closed kinematic chains. Whether or not the algebraic approach mentioned in this paper can be extended to study the kinematics of a spherical mechanism with more than three closed kinematic chains should be the topic of future research, as well. Conceptualization, [PERSON] and M.Y. ; methodology, [PERSON] and M.Y. ; software, [PERSON], M.Y. and Y.H. ; validation, [PERSON], M.Y. and Y.H. ; formal analysis, [PERSON], M.Y. and Y.H. ; investigation, [PERSON], M.Y. and Y.H.