## 3 Mechanical Error Analysis  
The mechanical errors, specifically the kinematic error equations, of the output link (i.e., link 9 shown in Figure 4) caused by dimensional errors (or tolerances) of link members can be derived analytically through the concept of tolerance sensitivity analysis [26; 27; 28]. To this end, the nine-link type double-toggle mold/die clamping mechanism is structurally divided into two sub-mechanisms as shown in Figure 5. As can be seen, the lower and upper sub-mechanisms are both Watt II six-link sub-mechanisms, and whose linear positions of the output link, \(L_{L}(x)\) and \(L_{U}(x)\), are defined by the horizontal positions of joints F and C relative to the fixed pivots, respectively. It must be emphasized that an error means the difference between the actual (measured) and nominal (theoretical) dimensions of a link member that has been machined, while a tolerance means the allowable deviation range (defined by allocated upper and lower deviations) of the nominal dimension of a link member. When performing the mechanical error analysis, the upper or lower deviations of the allocated dimensional tolerances are considered as the dimensional errors for simulating a worst situation of machined link members. The actually measured dimensional errors can also be used in the mechanical error analysis for evaluating a real condition.  
Figure 5: Two sub-mechanisms divided from the nine-link type double-toggle mold/die clamping mechanism.  
If only \(\Delta\tau_{7b}\) is considered to induce the output error \(\Delta L_{U}\), the output error termed \(\Delta L_{7b}\) can be derived and expressed as  
\[\begin{array}{ll}\Delta L_{7b}&=-\frac{\partial F_{U}/\partial\tau_{Tb}}{ \partial F_{U}/\partial L_{U}}\Delta\tau_{7b}\\ &=\Big{[}-\frac{(\tau_{7b}-\tau_{7b}\cos\alpha_{7})(\cos\phi_{7}\tan\theta_{8 }-\sin\theta_{7})}{\tau_{7a}\sin\alpha_{7}}+\cos\theta_{7}+\sin\theta_{7}\tan \theta_{8}\Big{]}\Delta\tau_{7b}\end{array} \tag{34}\]  
If only \(\Delta\tau_{7c}\) is considered to induce the output error \(\Delta L_{U}\), the output error termed \(\Delta L_{7c}\) can be derived and expressed as  
\[\begin{array}{ll}\Delta L_{7c}&=\ -\frac{\partial F_{U}/\partial\tau_{Tc}}{ \partial F_{U}/\partial L_{U}}\Delta\tau_{7c}\ =\ \Big{[}\frac{\tau_{7c}(\cos\theta_{7}\tan\theta_{8}-\sin\theta_{7})}{\tau_{7a} \sin\alpha_{7}}\Big{]}\Delta\tau_{7c}\end{array} \tag{35}\]  
If only \(\Delta\tau_{8}\) is considered to induce the output error \(\Delta L_{U}\), the output error termed \(\Delta L_{8}\) can be derived and expressed as  
\[\begin{array}{ll}\Delta L_{8}&=\ -\frac{\partial F_{U}/\partial\tau_{8}}{ \partial F_{U}/\partial L_{U}}\Delta\tau_{8}\ =\ \Big{(}\frac{1}{\cos\theta_{8}}\Big{)}\Delta\tau_{8}\ =\ \Delta\tau_{8}\sec\theta_{8}\end{array} \tag{36}\]  
If only \(\Delta e_{U}\) is considered to induce the output error \(\Delta L_{U}\), the output error termed \(\Delta L_{eU}\) can be derived and expressed as  
\[\begin{array}{ll}\Delta L_{eU}&=\ -\frac{\partial F_{U}/\partial e_{U}}{ \partial F_{U}/\partial L_{U}}\Delta e_{U}\ =\ (\tan\theta_{8})\Delta e_{U}\end{array} \tag{37}\]  
As shown in Equations (30)-(37), the kinematic error equations of the output link for the upper sub-mechanism shown in Figure 5 are derived and can be calculated analytically. The analytical expressions of the partial derivatives appearing in Equations (29)-(37) are detailed in Appendix B.