## 2 The Loop Equation of the Single-DOF Spherical Kinematic Chain with Only Revolute Pairs
### The Branch Graph of the Proposed Spherical Mechanism
#### 3.2.1 The Type-One Branch Graph  
In the first category, branch points do not exist. Table 1 enumerates the parameters for the proposed mechanism that has the type-one branch graph. With the given parameters, the branch graph is shown in Figure 3a. The three-dimensional model of the proposed mechanism with the type-one branch is shown in Figure 3b. In Figure 3a, the red input-output curves of A\({}_{0}\)ABB\({}_{0}\) and the blue singularity curves of B\({}_{0}\)BDCC\({}_{0}\) do not intersect; there are no common solutions for Equations (12) and (17). Hence, no branch point exists. The two red curves are identical and repeated with an interval of \(2\pi\). The two blue curves are different. Each of the two blue curves is also repeated with an interval of \(2\pi\). Since the branch graph shown in Figure 3a emphasizes both the two red curves and the two blue curves, the repeated appearance of the two blue curves is also clearly presented in Figure 3a. JRS-2, which is the light gray area surrounded by blue curves, is the JRS of B\({}_{0}\)BDCC\({}_{0}\); JRS-2 is determined by \(\Delta_{2}\geq 0\). The red input-output curves are completely covered by JRS-2. Therefore, the two branches are the two red input-output curves. Each point on the two branches satisfies the equation, \(\Delta_{2}>0\). When \(\Delta_{1}\) equals 0, A\({}_{0}\)ABB\({}_{0}\) encounters singularity configurations. The set of input and output \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline parameters & \(\alpha_{41}\) & \(\alpha_{12}\) & \(\alpha_{23}\) & \(\alpha_{34}\) & \(\alpha_{45}\) & \(\alpha_{67}\) & \(\alpha_{73}\) & \(\theta_{5}\) & \(\beta\) & \(\lambda\) \\ \hline values & 54.1\({}^{\circ}\) & 22.5\({}^{\circ}\) & 25\({}^{\circ}\) & 100\({}^{\circ}\) & 30\({}^{\circ}\) & 140\({}^{\circ}\) & 50\({}^{\circ}\) & 0\({}^{\circ}\) & 55\({}^{\circ}\) & 95\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the proposed mechanism that has the type-one branch graph. values, where A\({}_{0}\)ABB\({}_{0}\) encounters a singularity configuration, is a dead center position. In Figure 3a, the four dead center positions are indicated by numbers 1 to 4, which are enumerated in Table 2.  
The values of \(\theta_{4}\) can be obtained from \(\Delta_{1}\) = 0; Equations (15) and (16) are utilized to obtain the values of \(\theta_{3}\). Dead center positions 1 and 2 divide branch 1 into two sub-branches, (1-2) and (2-1). On (1-2), which starts from dead center positions 1 to 2 clockwise, Equation (15) can be utilized to obtain the values of \(\theta_{3}\); on (2-1), which starts from dead center positions 2 to 1 clockwise, Equation (16) can be utilized to obtain the values of \(\theta_{3}\). Dead center positions 3 and 4 divide branch 2 into two sub-branches, (3-4) and (4-3). On (3-4), which starts from dead center positions 3 to 4 clockwise, Equation (15) can be utilized to obtain the values of \(\theta_{3}\); on (4-3), which starts from dead center positions 4 to 3 clockwise, Equation (16) can be utilized to obtain the values of \(\theta_{3}\). With the values of the variables \(\theta_{3}\) and \(\theta_{4}\) acquired, the value of \(\pi_{56}\) can be obtained by Equations (18) and (19). Inside a sub-branch, the mechanism's motion is continuous and smooth. Equation (10) can be rewritten as \(\mathbf{Z_{3}S_{3}Z_{4}S_{4}I_{2}S_{1}S_{2}Z_{2}S_{23}}\) = \(\mathbf{I}\), which can be simplified into \(\mathbf{z^{T}S_{3}Z_{4}S_{4}I_{2}S_{1}Z_{5}I_{2}z}\) = \(\mathbf{z^{T}S_{23}}^{T}\mathbf{z}\), where \(\theta_{1}\) is the only unknown variable. Therefore, the value of \(\theta_{1}\) can be obtained. Equation (10) can also be rewritten as \(\mathbf{Z_{4}S_{4}I_{2}S_{1}\mathbf{Z_{5}I_{2}Z_{5}S_{2}Z_{3}S_{34}}}\) = \(\mathbf{I}\), which can be simplified into \(\mathbf{z^{T}S_{4}I_{2}S_{1}\mathbf{Z_{5}I_{2}Z_{5}Z_{3}z}}\) = \(\mathbf{z^{T}S_{34}}^{T}\mathbf{z}\), where \(\theta_{2}\) is the only unknown variable. Therefore, the value of \(\theta_{2}\) can be obtained.