## 2 Dimensional Synthesis Equations for Six-Bar Slider-Crank Mechanisms
### Synthesis Equations of Watt II Six-Bar Slider-Crank Function Generator  
A Watt II six-bar slider-crank linkage is shown in Figure 1a, in which the input link \(r_{1}\) rotates about the fixed pivot \(O\), and the output slider translates vertically. A fixed frame is positioned so that its origin is at \(O\) and oriented so that its \(x\)-axis is perpendicular to the direction of slide. The horizontal distance from \(O\) to the line of path of the moving pivot \(E\) is the offset denoted by \(h\). In the fixed frame, the input angle \(\theta\) is measured from the positive \(x\)-axis and the output displacement of the slider \(p\) is the vertical distance from the \(x\)-axis to \(E\), the center of the revolute joint attached to the slider. The angle of link \(r_{0}\) is \(\eta\), and the angle between links \(r_{3}\) and \(r_{4}\) is \(\beta\). The angular displacements of links \(r_{2}\), \(r_{3}\), \(r_{4}\), and \(r_{5}\) are denoted by \(\phi\), \(\alpha\), \(\alpha+\beta\), and \(\delta\), respectively. Figure 1b shows the mechanism in its first and \(j\)th positions. In the Figure, \(\Delta\theta_{j}\), \(\Delta\phi_{j}\), \(\Delta\alpha_{j}\), and \(\Delta\delta_{j}\) denote the rotation angles of links \(r_{1}\) to \(r_{5}\), respectively, and \(\Delta p_{j}\) represents the displacement of the slider, all measured from the initial position to \(j\)th position of the mechanism. Let \(\theta_{1}\) and \(p_{1}\) denote the angle of input link \(r_{1}\) and the position of output slider in the initial position, respectively. Then, the prescribed positions of the function generator to be synthesized are given pairwise as  
\[\Delta\theta_{j}=\theta_{j}-\theta_{1},\Delta p_{j}=p_{j}-p_{1},j=2,\ldots,n, \tag{1}\]  
where \(\theta_{j}\) and \(p_{j}\) are the input angle and the position of the slider in \(j\)th position, respectively, and \(n\) is the number of precision points to be satisfied. The dimensional synthesis equations of the Watt II six-bar slider-crank function generator can be constructed by the closure equations of the two loops \(OABC\) and \(OCDE\) shown in Figure 1 as follows. Loop \(OABC\):  
Figure 1: Watt II six-bar slider-crank mechanism: (**a**) Design parameters; (**b**) The link vectors drawn in its first and \(j\)th positions. For the initial and \(j\)th positions, the loop closure equations can be written as  
\[\mathbf{r}_{1}+\mathbf{r}_{2}=\mathbf{r}_{0}+\mathbf{r}_{3}, \tag{2}\]  
and  
\[\mathbf{r}_{1}e^{i\Delta\theta_{j}}+\mathbf{r}_{2}e^{i\Delta\phi_{j}}=\mathbf{r }_{0}+\mathbf{r}_{3}e^{i\Delta\alpha_{j}},j=2,\ldots,n, \tag{3}\]  
respectively. Subtracting Equation (2) from Equation (3) gives  
\[\mathbf{r}_{1}(e^{i\Delta\theta_{j}}-1)+\mathbf{r}_{2}(e^{i\Delta\phi_{j}}-1)= \mathbf{r}_{3}(e^{i\Delta\alpha_{j}}-1),j=2,\ldots,n. \tag{4}\]  
Loop OCDE:  
For the initial and \(j\)th positions, the loop closure equations can be written as  
\[\mathbf{r}_{0}+\mathbf{r}_{4}=h+ip_{1}+\mathbf{r}_{5}, \tag{5}\]  
and  
\[\mathbf{r}_{0}+\mathbf{r}_{4}e^{i\Delta\alpha_{j}}=h+ip_{j}+\mathbf{r}_{5}e^{ i\Delta\delta_{j}},j=2,\ldots,n, \tag{6}\]  
respectively. Subtracting Equation (5) from Equation (6) gives  
\[\mathbf{r}_{4}(e^{i\Delta\alpha_{j}}-1)=i\Delta p_{j}+\mathbf{r}_{5}(e^{i \Delta\delta_{j}}-1),j=2,\ldots,n. \tag{7}\]  
Separating the real and imaginary parts of Equations (4) and (7) leads to  
\[r_{1}\big{\{}\cos\big{(}\theta_{1}+\Delta\theta_{j}\big{)}-\cos\theta_{1} \big{\}}+r_{2}\big{\{}\cos(\phi_{1}+\Delta\phi_{j})-\cos\phi_{1}\big{\}}=r_{3 }\big{\{}\cos(\alpha_{1}+\Delta\alpha_{j})-\cos\alpha_{1}\big{\}},j=2,\ldots,n, \tag{8}\]  
\[r_{1}\big{\{}\sin(\theta_{1}+\Delta\theta_{j})-\sin\theta_{1}\big{\}}+r_{2} \big{\{}\sin(\phi_{1}+\Delta\phi_{j})-\sin\phi_{1}\big{\}}=r_{3}\big{\{}\sin( \alpha_{1}+\Delta\alpha_{j})-\sin\alpha_{1}\big{\}},j=2,\ldots,n, \tag{9}\]  
and  
Figure 1:0  
Figure 1:1  
respectively. Note that \(\theta_{1}\), \(\phi_{1}\), \(\alpha_{1}\), and \(\delta_{1}\) denote the angles of the corresponding links shown in Figure 1a in the initial position.