## 2 Dimensional Synthesis Equations for Six-Bar Slider-Crank Mechanisms
### Synthesis Equations of Stephenson III Six-Bar Slider-Crank Function Generator  
A Stephenson III slider-crank linkage is shown in Figure 2a. The input link \(r_{1}\) rotates about the fixed pivot \(O\), and the output slider translates along a line parallel to \(y\)-axis. The angle between links \(r_{2}\) and \(r_{4}\) is \(\beta\). The angular displacements of links \(r_{2}\), \(r_{3}\), \(r_{4}\), and \(r_{5}\) are denoted by \(\phi\), \(\alpha\), \(\phi+\beta\), and \(\delta\), respectively. As shown in Figure 2b, \(\Delta\theta_{j}\), \(\Delta\phi_{j}\), \(\Delta\alpha_{j}\), and \(\Delta\delta_{j}\) represent the rotation angles of links \(r_{1}\) to \(r_{5}\), measured from their initial positions to \(j\)th positions, respectively. The prescribed precision points are given as Equation (1). The dimensional synthesis equations of the Stephenson III six-bar slider-crank function generator can be derived using the closure equations of the two loops, \(OABC\) and \(OADE\), shown in Figure 2 as follows. Loop \(OABC\):  
For the initial and \(j\)th positions, the loop closure equations can be written as  
\[\mathbf{r}_{1}+\mathbf{r}_{2}=\mathbf{r}_{0}+\mathbf{r}_{3}, \tag{20}\]  
and  
\[\mathbf{r}_{1}e^{i\Delta\theta_{j}}+\mathbf{r}_{2}e^{i\Delta\phi_{j}}=\mathbf{ r}_{0}+\mathbf{r}_{3}e^{i\Delta\alpha_{j}},\,j=2,\ldots,n, \tag{21}\]  
Figure 2: Stephenson III six-bar slider-crank mechanism: (**a**) Design parameters; (**b**) The link vectors drawn in its first and \(j\)th positions. respectively. Subtracting Equation (20) from Equation (21) gives  
\[\mathbf{r}_{1}(e^{i\Delta\theta_{j}}-1)+\mathbf{r}_{2}(e^{i\Delta\phi_{j}}-1)= \mathbf{r}_{3}(e^{i\Delta\alpha_{j}}-1),\,j=2,\ldots,n. \tag{22}\]  
Loop _OADE_:  
For the initial and \(j\)th positions, the loop closure equations can be written as  
\[\mathbf{r}_{1}+\mathbf{r}_{4}=h+ip_{1}+\mathbf{r}_{5}, \tag{23}\]  
and  
\[\mathbf{r}_{1}e^{i\Delta\theta_{j}}+\mathbf{r}_{4}e^{i\Delta\alpha_{j}}=h+ip_ {j}+\mathbf{r}_{5}e^{i\Delta\delta_{j}},j=2,\ldots,n, \tag{24}\]  
respectively. Subtracting Equation (23) from Equation (24) gives  
\[\mathbf{r}_{1}(e^{i\Delta\theta_{j}}-1)+\mathbf{r}_{4}(e^{i\Delta\alpha_{j}}-1 )=i\Delta p_{j}+\mathbf{r}_{5}(e^{i\Delta\delta_{j}}-1),\,j=2,\ldots,n. \tag{25}\]  
Separating the real and imaginary parts of Equations (22) and (25) yields  
\[r_{1}\big{\{}\cos(\theta_{1}+\Delta\theta_{j})-\cos\theta_{1}\big{\}}+r_{2} \big{\{}\cos(\phi_{1}+\Delta\phi_{j})-\cos\phi_{1}\big{\}}=r_{3}\big{\{}\cos( \alpha_{1}+\Delta\alpha_{j})-\cos\alpha_{1}\big{\}},\,j=2,\ldots,n, \tag{26}\]  
\[r_{1}\big{\{}\sin(\theta_{1}+\Delta\theta_{j})-\sin\theta_{1}\big{\}}+r_{2} \big{\{}\sin(\phi_{1}+\Delta\phi_{j})-\sin\phi_{1}\big{\}}=r_{3}\big{\{}\sin( \alpha_{1}+\Delta\alpha_{j})-\sin\alpha_{1}\big{\}},\,j=2,\ldots,n, \tag{27}\]  
and  
\[r_{1}\big{\{}\cos(\theta_{1}+\Delta\theta_{j})-\cos\theta_{1}\big{\}}+r_{4} \big{\{}\cos(\phi_{1}+\beta+\Delta\alpha_{j})-\cos(\phi_{1}+\beta)\big{\}}=r_{ 5}\big{\{}\cos(\delta_{1}+\Delta\delta_{j})-\cos\delta_{1}\big{\}},\,j=2, \ldots,n, \tag{28}\]  
\[\mathbf{r}_{1}e^{i\Delta\theta_{j}}+\mathbf{r}_{2}e^{i\Delta\phi_{j}}=\mathbf{ r}_{0}+\mathbf{r}_{3}e^{i\Delta\alpha_{j}},\,j=2,\ldots,n, \tag{21}\]0  
respectively. Among the unknowns in Equations (26) through (29), \(\Delta\phi_{j}\), \(\Delta\alpha_{j}\), and \(\Delta\delta_{j}\) should satisfy trigonometric identities  
\[ \begin{array}{l}\cos^{2}\Delta\phi_{j}+\sin^{2}\Delta\phi_{j}=1,\\ \cos^{2}\Delta\alpha_{j}+\sin^{2}\Delta\alpha_{j}=1,\\ \cos^{2}\Delta\delta_{j}+\sin^{2}\Delta\delta_{j}=1.\end{array}  \tag{30}\]  
where \(j=2,\,\ldots,\,n\).