## 2 Dimensional Synthesis Equations for Six-Bar Slider-Crank Mechanisms
### Procedure for Simplifying the Synthesis Equations of Stephenson III Six-Bar Slider-Crank  
The synthesis equations, Equations (26)-(30), can be reduced from \(7(n-1)\) to \(n-1\) by eliminating \(\Delta\alpha_{j}\), \(\Delta\delta_{j}\), and \(\Delta\phi_{j}\); \(j=2,\,\ldots,\,n\). Applying the similar procedure used in Section 2.2, the unknown \(\Delta\alpha_{j}\) can be eliminated from Equations (26) and (27) as  
\[ \begin{array}{l}r_{3}^{2}\ \ =\big{[}r_{1}\big{\{}\cos(\theta_{1}+\Delta\theta_{j})- \cos\theta_{1}\big{\}}+r_{2}\big{\{}\cos(\phi_{1}+\Delta\phi_{j})-\cos\phi_{1} \big{\}}+r_{3}\cos\alpha_{1}\big{]}^{2}\\ \ \ +\big{[}r_{1}\big{\{}\sin(\theta_{1}+\Delta\theta_{j})-\sin\theta_{1}\big{\}}+ r_{2}\big{\{}\sin(\phi_{1}+\Delta\phi_{j})-\sin\phi_{1}\big{\}}+r_{3}\sin \alpha_{1}\big{]}^{2},\end{array} \,\,\,\,\,j=2,\ldots,n, \tag{31}\]  
and the unknown \(\Delta\delta_{j}\) can be eliminated from Equations (28) and (29) as  
\[ \begin{array}{l}r_{3}^{2}\ \ =\big{[}r_{1}\big{\{}\cos(\theta_{1}+\Delta\theta_{j})- \cos\theta_{1}\big{\}}+r_{4}\big{\{}\cos(\phi_{1}+\beta+\Delta\phi_{j})-\cos( \phi_{1}+\beta)\big{\}}+r_{3}\cos\delta_{1}\big{]}^{2}\\ \ \ +\big{[}r_{1}\big{\{}\sin(\theta_{1}+\Delta\theta_{j})-\sin\theta_{1}\big{\}}+ r_{4}\big{\{}\sin(\phi_{1}+\beta+\Delta\phi_{j})-\sin(\phi_{1}+\beta)\big{\}}+r_{5}\sin \delta_{1}\big{\}-\Delta p_{j}\big{\}}^{2},\end{array} \,\,\,\,\,j=2,\ldots,n. \tag{32}\]Let the \(x\)- and \(y\)-components of the initial positions of the links \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\), and \(r_{5}\) be  
\[ \begin{array}{l}r_{1}\cos\theta_{1}=r_{1x},r_{2}\cos\phi_{1}=r_{2x},r_{3}\cos \alpha_{1}=r_{3x},r_{4}\cos(\phi_{1}+\beta)=r_{4x},r_{5}\cos\delta_{1}=r_{5x}, \\ r_{1}\sin\theta_{1}=r_{1y},r_{2}\sin\phi_{1}=r_{2y},r_{3}\sin\alpha_{1}=r_{3y}, r_{4}\sin(\phi_{1}+\beta)=r_{4y},r_{5}\sin\delta_{1}=r_{5y}.\end{array}  \tag{33}\]  
Substituting Equation (33) into Equations (31) and (32), and gathering the coefficients of \(\cos\Delta\phi_{j}\) and \(\sin\Delta\phi_{j}\) gives  
\[L_{1j}+L_{2j}\cos\Delta\phi_{j}+L_{3j}\sin\Delta\phi_{j}=0,\,j=2,\ldots,n, \tag{34}\]  
where  
\[ \begin{array}{ll}L_{1j}=&-2\cos\Delta\theta_{j}\Big{(}r_{1x}+r_{1y}^{2}+r_ {1y}^{2}+r_{1y}r_{2x}-r_{1y}r_{2y}+r_{1y}r_{2y}-r_{1y}r_{y}\Big{)}+2\sin\Delta \theta_{j}\big{(}-r_{1x}r_{2y}+r_{1y}r_{2x}+r_{1x}r_{3y}-r_{1y}r_{3x}\big{)} \\ &+2\big{(}r_{2x}^{2}+r_{2y}^{2}-r_{2y}r_{3x}-r_{2y}r_{3y}\big{)}+2\big{(}r_{1 x}^{2}+r_{1y}^{2}+r_{1x}r_{2x}-r_{1x}r_{3x}+r_{1y}r_{2y}-r_{1y}r_{3y}\big{)}, \end{array}  \tag{34a}\]  
\[L_{2j}=2\cos\Delta\theta_{j}\big{(}r_{1x}r_{2x}+r_{1y}r_{2y}\big{)}+2\sin \Delta\theta_{j}\big{(}r_{1x}r_{2y}-r_{1y}r_{2x}\big{)}-2\big{(}r_{1x}^{2}+r_{ 1y}^{2}r_{2y}-r_{2y}r_{3x}-r_{2y}r_{3y}\big{)}, \tag{34b}\]  
\[L_{3j}=2\cos\Delta\theta_{j}\big{(}-r_{1x}r_{2y}+r_{1y}r_{2x}\big{)}+2\sin \Delta\theta_{j}\big{(}r_{1x}r_{2x}+r_{1y}r_{2y}\big{)}+2\big{(}r_{1x}r_{2y}- r_{1y}r_{2x}\big{)}+2\big{(}r_{2x}r_{3y}-r_{1y}r_{3x}\big{)}, \tag{34c}\]  
and  
\[Q_{1j}+Q_{2j}\cos\Delta\phi_{j}+Q_{3j}\sin\Delta\phi_{j}=0,\,j=2,\ldots,n, \tag{35}\]  
where  
\[ \begin{array}{ll}Q_{1j}=&-2\cos\Delta\theta_{j}\big{(}r_{1x}^{2}+r_{1y}^{2} +r_{1x}r_{4x}-r_{1y}r_{5x}+r_{1y}r_{4y}-r_{1y}r_{5y}+r_{1y}\Delta\eta_{j}\big{)} \\ &+2\sin\Delta\theta_{j}\big{(}-r_{1x}r_{4y}+r_{1y}r_{4x}+r_{1y}r_{5y}-r_{1y}r_ {5x}-r_{1x}\Delta\eta_{j}\big{)}+2\big{(}r_{1x}^{2}+r_{1y}^{2}+r_{1y}^{2}+r_{1 x}r_{4x}-r_{1x}r_{5x}+r_{1y}r_{4y}-r_{1y}r_{5y}\big{)}\\ &+2\big{(}r_{4x}^{2}+r_{4y}^{2}-r_{4x}r_{5x}-r_{4y}r_{5y}\big{)}+2\Delta\theta _{j}\big{(}r_{1y}+r_{4y}-r_{5y}\big{)}+2\theta_{j}^{2},\end{array}  \tag{35a}\]  
\[ \begin{array}{ll}Q_{2j}=&2\cos\Delta\theta_{j}\big{(}r_{1x}r_{4x}+r_{1y}r_{ 4y}\big{)}+2\sin\Delta\theta_{j}\big{(}r_{1x}r_{4y}-r_{1y}r_{4x}\big{)}-2\big{(} r_{1x}r_{4x}+r_{1y}r_{4y}\big{)}\\ &-2\big{(}r_{4x}^{2}+r_{4y}^{2}-r_{4x}r_{5x}-r_{4y}r_{5y}\big{)}-2r_{4y}\Delta \eta_{j},\end{array}  \tag{35b}\]  
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