## 2 The Loop Equation of the Single-DOF Spherical Kinematic Chain with Only Revolute Pairs
### The Loop Equation  
A single-DOF two-loop Stephenson-III spherical mechanism with a spherical slider that contains a spherical prismatic pair, which contains two closed kinematic chains, \(\mathrm{A}_{0}\mathrm{ABB}_{0}\) and \(\mathrm{B}_{0}\mathrm{BDCC}_{0}\), is shown in Figure 2.  
Figure 2: The schematic diagram of the Stephenson-III spherical mechanism with a spherical prismatic pair. In Figure 2, the yellow solid spherical line is the fixed spherical link; the blue solid spherical lines are the unfixed spherical links. C represents both the spherical prismatic pair C\({}_{\text{P}}\) and the revolute pair C\({}_{\text{R}}\); CC\({}_{0}\) is the blue dotted spherical line that represents the arc distance the spherical slider C with prismatic pair C\({}_{\text{P}}\) reaches on the fixed track that coincides with CC\({}_{0}\). Therefore, \(\alpha_{56}\) is a variable that represents the arc distance that the spherical slider C reaches; \(\theta_{5}\) is a constant, since arc distance CC\({}_{0}\) coincides with the fixed track. Input and output joints are spherical revolute joints B\({}_{0}\) and B, respectively; the input angle is \(\theta_{4}\), and the output angle is \(\theta_{3}\). The fixed yellow spherical link A\({}_{0}\)B\({}_{0}\)C\({}_{0}\) consists of two separate spherical sub-links that are at an angle, \(\beta\), which is a constant. The triangular spherical link ABD consists of three spherical sub-links, two of which, AB and BD, are at an angle, \(\lambda\), which is also a constant. The loop equation of the spherical four-bar kinematic chain A\({}_{0}\)ABB\({}_{0}\) is shown in Equation (8). \[\mathbf{Z}_{2}\mathbf{S}_{23}\mathbf{Z}_{3}\mathbf{S}_{34}\mathbf{Z}_{4} \mathbf{S}_{41}\mathbf{Z}_{1}\mathbf{S}_{12}=\mathbf{I} \tag{8}\]  
The loop equation of the spherical five-bar kinematic chain B\({}_{0}\)BDCC\({}_{0}\) is shown in (9). \[\mathbf{Z}_{6}\mathbf{S}_{56}\mathbf{Z}_{5}\mathbf{S}_{45}\mathbf{Z}_{4} \mathbf{S}_{34}\mathbf{Z}_{3}\mathbf{S}_{73}\mathbf{Z}_{7}\mathbf{S}_{67}= \mathbf{I} \tag{9}\]  
In Figure 2, the rotation angles \(\theta_{i}\) of \(\mathbf{Z}_{3}\), \(\mathbf{Z}_{4}\), and the rotation angles \(\theta_{i}\) of \(\mathbf{Z}_{3^{\prime}}\) and \(\mathbf{Z}_{4^{\prime}}\) in Equations (8) and (9), are different; \(\mathbf{Z}_{3}\) and \(\mathbf{Z}_{4}\) correspond to \(\theta_{3}\) and \(\theta_{4}\), respectively; however, \(\mathbf{Z}_{3^{\prime}}\) and \(\mathbf{Z}_{4^{\prime}}\) correspond to \((\lambda-\theta_{3})\) and \((\theta_{4}-\beta)\), respectively. According to Equations (3), (6) and (7), Equations (8) and (9) can be simplified into Equations (10) and (11), respectively, by eliminating the variables \(\theta_{1}\), \(\theta_{2}\), \(\theta_{6}\), and \(\theta_{7}\). \[\mathbf{Z}^{\text{T}}\mathbf{S}_{23}\mathbf{Z}_{3}\mathbf{S}_{34}\mathbf{Z}_{4 }\mathbf{S}_{41}\mathbf{z}=\mathbf{z}^{\text{T}}\mathbf{S}_{12}{}^{\text{T}} \mathbf{z} \tag{10}\]  
\[\mathbf{z}^{\text{T}}\mathbf{S}_{56}\mathbf{Z}_{5}\mathbf{S}_{45}\mathbf{Z}_{4 ^{\prime}}\mathbf{S}_{34}\mathbf{Z}_{3^{\prime}}\mathbf{S}_{73}\mathbf{z}= \mathbf{z}^{\text{T}}\mathbf{S}_{6^{\prime}}{}^{\text{T}}\mathbf{z} \tag{11}\]  
Equations (10) and (11) have three unknown variables \(\alpha_{56}\), \(\theta_{3}\), and \(\theta_{4}\); \(\alpha_{56}\) is the arc distance of the spherical slider. The tangent half-angle formulas and discriminant method are utilized to eliminate \(\theta_{4}\) and \(\alpha_{56}\). Firstly, the equations \(\cos\theta_{3}=\frac{1-l^{2}}{1+l^{2}}\) and \(\sin\theta_{3}=\frac{2l}{1+l^{2}}\), where \(l\) equals \(\tan\frac{\theta_{3}}{2}\), are employed in Equation (10). Secondly, the equations \(\cos\alpha_{56}=\frac{1-m^{2}}{1+m^{2}}\) and \(\sin\alpha_{56}=\frac{2m}{1+m^{2}}\), where \(m\) equals \(\tan\frac{\alpha_{56}}{2}\), are employed in Equation (11). Hence, Equations (10) and (11) can be, respectively, simplified into Equations (12) and (13).