## 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
Equation (11) can be rewritten as \(\mathbf{Z_{5}S_{4}Z_{4}S_{3}Z_{5}S_{3}Z_{5}S_{7}Z_{5}S_{6}Z_{6}S_{6}}\) = \(\mathbf{I}\), which can be simplified into \(\mathbf{z^{T}S_{4}Z_{4}S_{3}Z_{5}S_{7}S_{6}Z_{6}S_{6}}\) = \(\mathbf{z^{T}S_{56}}^{T}\mathbf{z}\), where \(\theta_{7}\) is the only unknown variable. Therefore, the value of \(\theta_{7}\) can be obtained. Equation (11) can also be rewritten as \(\mathbf{Z_{4}S_{3}Z_{5}S_{7}Z_{5}S_{6}Z_{6}S_{6}Z_{6}S_{6}Z_{6}S_{6}Z_{6}S_{6} }\) = \(\mathbf{I}\), which can be simplified into \(\mathbf{z^{T}S_{34}Z_{5}S_{7}S_{7}Z_{5}S_{6}Z_{6}S_{6}Z_{6}S_{6}}\) = \(\mathbf{I}\), which can be simplified into \(\mathbf{z^{T}S_{34}Z_{5}S_{7}S_{7}Z_{5}S_{6}Z_{6}S_{6}}\) = \(\mathbf{z^{T}S_{4}}^{T}\mathbf{z}\), where \(\theta_{6}\) is the only unknown \begin{table}
\begin{tabular}{c c c} \hline \hline
**Joint** & \(\theta_{3}\) & \(\theta_{4}\) \\ \hline Dead center position 1 & \(165.735^{\circ}\) & \(167.039^{\circ}\) \\ \hline Dead center position 2 & \(194.265^{\circ}\) & \(192.961^{\circ}\) \\ \hline Dead center position 3 & \(165.735^{\circ}\) & \(-192.961^{\circ}\) \\ \hline Dead center position 3 & \(194.265^{\circ}\) & \(-167.039^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dead center positions. Figure 3: The branch graph and the three-dimensional model. (**a**) The branch graph. (**b**) The three-dimensional model. variable. Therefore, the value of \(\theta_{6}\) can be obtained. Figure 4a-d are the configurations of the proposed mechanism at dead center positions 1, 2, 3, and 4, respectively. In Figure 4, the dotted lines with arrows OA\({}_{0}\), OA, and OB are coplanar; the four values of \(\alpha_{56}\), each of which is represented by a blue dotted spherical line, are all highlighted in blue. The three-dimensional model of the mechanism shown in Figure 3b simulates the mechanism's motion and verifies each joint's displacement and the configurations in Figure 4. Since curves in the branch graph shown in Figure 3a do not intersect, Figure 3a does not have branch points, and the proposed mechanism with the type-one branch has decoupled motion, which is created by the situation where a two-loop spherical mechanism's motion is entirely decided by one of the two loops. For the two-loop Stephenson-III spherical mechanism shown in Figure 3b with the dimensions given in Table 1, the mechanism's decoupled motion is completely decided by the spherical kinematic chain \(\Lambda_{0}\)AB\({}_{0}\), which is one of the two loops (or spherical kinematic chains) in the mechanism. Branch 1 has the input range for \(\theta_{4}\), (167.039\({}^{\circ}\), 192.961\({}^{\circ}\)) and the output range for \(\theta_{3}\), (160.428\({}^{\circ}\), 200.700\({}^{\circ}\)). Branch 2 has the input range for \(\theta_{4}\), (\(-\)192.961\({}^{\circ}\), \(-\)167.039\({}^{\circ}\)) and the output range for \(\theta_{3}\), (160.428\({}^{\circ}\), 200.700\({}^{\circ}\)). The minimal and maximal arc distances reached by the spherical slider in branch 1 is shown in Figure 5a,b; \(\alpha_{56}\)(\(\in\)(71.620\({}^{\circ}\), 105.310\({}^{\circ}\)). Figure 4: Configurations at dead center positions. (**a**) Dead center position 1; (**b**) Dead center position 2; (**c**) Dead center position 3; (**d**) Dead center position 4. The minimal and maximal arc distances reached by the spherical slider in branch 2 is shown in Figure 6a,b; \(\alpha_{56}\)\(\in\)(\(-3.736^{\circ}\), \(25.382^{\circ}\)). In Figures 5 and 6, the values of \(\alpha_{56}\), each of which is represented by a blue dotted spherical line, are all highlighted in blue. The three-dimensional model of the mechanism shown in Figure 3b simulates the mechanism's motion and verifies each joint's displacement and the configurations in Figures 5 and 6. The three-dimensional simulation conducted by the three-dimensional model verified the decoupled motion of the proposed spherical mechanism.