## 2 Position Analysis  
In order to analyze the mechanical errors in a nine-link type double-toggle mold/die clamping mechanism, the position analysis must be performed first. Figure 4 shows the structural diagram of a nine-link type double-toggle mold/die clamping mechanism, in which, \(r_{1L}\), \(r_{1U}\), \(r_{2L}\), \(r_{2L}\), \(r_{3}\), \(r_{4L}\), \(r_{4b}\), \(r_{4c}\), \(\alpha_{4}\), \(r_{5}\), \(r_{6}\), \(r_{7a}\), \(r_{7b}\), \(r_{7c}\), \(\alpha_{7}\), and \(r_{8}\) are the geometric parameters (i.e., the dimensions) of respective links, \(e_{L}\) and \(e_{U}\) are the vertical offsets of the revolute joints between the output link (link 9) and the frame (link 1), \(x\) is the positional parameter of the input link (link 2) (i.e., the input variable that increases positively when link 2 moves rightward), \(L(x)\) is the linear position of the output link (link 9), and \(\theta_{3}\), \(\theta_{4}\), \(\theta_{5}\), \(\theta_{6}\), \(\theta_{7}\), and \(\theta_{8}\) are the angular positions of respective links increasing positively when the links rotate counterclockwise from the horizontal. Since the nine-link type mold/die clamping mechanism must satisfy the geometric relation of a symmetric configuration, thus the correlations of \(r_{1L}=r_{1U}\), \(r_{2L}=r_{2U}\), \(r_{3}=r_{6}\), \(r_{4a}=r_{7a}\), \(r_{4b}=r_{7b}\), \(r_{4c}=r_{7c}\), \(\alpha_{4}=\alpha_{7}\), \(r_{5}=r_{8}\), and \(e_{L}=e_{U}\) exist. The linear position of the output link (link 9), \(L(x)\), can be expressed as  
\[L(x)\ =\ r_{4b}\cos\theta_{4}+\sqrt{r_{5}^{2}-\left(r_{4b}\sin\theta_{4}-e_{L} \right)^{2}} \tag{2}\]  
or  
\[L(x)\ =\ r_{7b}\cos\theta_{7}+\sqrt{r_{8}^{2}-\left(r_{7b}\sin\theta_{7}+e_{U} \right)^{2}} \tag{3}\]  
in which, Equations (2) and (3) are derived from the lower and upper slider-crank chains of the mechanism shown in Figure 4, respectively, and \(\theta_{4}\) and \(\theta_{7}\) can be expressed as  
\[\theta_{4}\ =\ 90^{\circ}-\left(\alpha_{4}+\gamma_{1L}+\gamma_{2L}\right) \tag{4}\]  
and  
\[\theta_{7}\ =\ 270^{\circ}+\left(\alpha_{7}+\gamma_{1U}+\gamma_{2U}\right) \tag{5}\]  
in which,  
\[\alpha_{4}\ =\ \cos^{-1}\left(\frac{r_{4a}^{2}+r_{4b}^{2}-r_{4c}^{2}}{2r_{4a}r _{4b}}\right) \tag{6}\]  
\[\gamma_{1L}\ =\ \sin^{-1}\left(\frac{x}{l_{L}}\right) \tag{7}\]\[\gamma_{2L}\ =\ \cos^{-1}\left(\frac{l_{L}^{2}+r_{4a}^{2}-r_{3}^{2}}{2l_{L}r_{4a}}\right) \tag{8}\]  
\[\alpha_{7}\ =\ \cos^{-1}\left(\frac{r_{7a}^{2}+r_{7b}^{2}-r_{7c}^{2}}{2r_{7a}r_{ 7b}}\right) \tag{9}\]  
\[\gamma_{1U}\ =\ \sin^{-1}\left(\frac{x}{l_{1U}}\right) \tag{10}\]  
\[\gamma_{2U}\ =\ \cos^{-1}\left(\frac{l_{L}^{2}+r_{7a}^{2}-r_{6}^{2}}{2l_{U}r_{ 7a}}\right) \tag{11}\]  
and  
\[L(x)\ =\ r_{7b}\cos\theta_{7}+\sqrt{r_{8}^{2}-\left(r_{7b}\sin\theta_{7}+e_{U} \right)^{2}} \tag{3}\]0  
\[L(x)\ =\ r_{7b}\cos\theta_{7}+\sqrt{r_{8}^{2}-\left(r_{7b}\sin\theta_{7}+e_{U} \right)^{2}} \tag{3}\]1  
Thus, as \(\theta_{4}\) and \(\theta_{7}\) are solved by using Equations (4) and (5), respectively, the position of the output link, \(L(x)\), can be further solved by substituting them into Equations (2) and (3), respectively. Since the nine-link type mechanism essentially has a symmetric configuration, the use of Equations (2) or (3) will yield the same calculating result. For the remaining angular positions, they can be expressed as  
\[L(x)\ =\ r_{7b}\cos\theta_{7}+\sqrt{r_{8}^{2}-\left(r_{7b}\sin\theta_{7}+e_{U} \right)^{2}} \tag{3}\]2  
\[L(x)\ =\ r_{7b}\cos\theta_{7}+\sqrt{r_{8}^{2}-\left(r_{7b}\sin\theta_{7}+e_{U} \right)^{2}} \tag{3}\]3  
\[L(x)\ =\ r_{7b}\cos\theta_{7}+\sqrt{r_{8}^{2}-\left(r_{7b}\sin\theta_{7}+e_{U} \right)^{2}} \tag{3}\]4  
\[L(x)\ =\ r_{7b}\cos\theta_{7}+\sqrt{r_{8}^{2}-\left(r_{7b}\sin\theta_{7}+e_{U} \right)^{2}} \tag{3}\]5 Structural diagram of a nine-link type double-toggle mold/die clamping mechanism.