## 2 Existing Designs
### Optimal Watt-Chain Mechanism  
Figure 1a shows a Watt-chain mechanism from patent [2], while Figure 4 shows the corresponding mechanism and kinematic skeleton in both the closed and open positions. The mold closing position features double-toggle effects (\(L_{1}\) and \(L_{2}\)), as shown in Figure 4a.  
As shown in Figure 4c, the Watt-chain mechanism contains two independent vector loops (Loop I and Loop II), such as Equations (8) and (9).  
\[\overset{\rightarrow}{r}_{2}+\overset{\rightarrow}{r}_{3}-\overset{ \rightarrow}{r}_{4a}-\overset{\rightarrow}{r}_{4b}=0 \tag{8}\]  
\[\overset{\rightarrow}{r}_{4a}+\overset{\rightarrow}{r}_{5}-\overset{ \rightarrow}{r}_{6}=0 \tag{9}\]  
Here, Loop I is represented by Equation (8), and Loop II is represented by Equation (9), where the vectors \(\overset{\rightarrow}{r}_{2}\), \(\overset{\rightarrow}{r}_{3}\), \(\overset{\rightarrow}{r}_{4a}\), \(\overset{\rightarrow}{r}_{4b}\), \(\overset{\rightarrow}{r}_{5}\), and \(\overset{\rightarrow}{r}_{6}\) represent the links of the mechanism.  
The horizontal component X and vertical component Y equations are  
\[r_{2x}+r_{3}\cos\theta_{3}-r_{4a}\cos\theta_{4a}-r_{4b}\cos\theta_{4b}=0 \tag{10}\]  
\[r_{2y}+r_{3}\sin\theta_{3}-r_{4a}\sin\theta_{4a}-r_{4b}\sin\theta_{4b}=0 \tag{11}\]  
\[r_{4a}\cos\theta_{4a}+r_{5}\cos\theta_{5}-r_{6}=0 \tag{12}\]  
\[r_{4a}\sin\theta_{4a}+r_{5}\sin\theta_{5}=0 \tag{13}\]  
Figure 3: The kinematic characteristics and force ratio of the multiple—joint mechanism [1].  
Equations (10) and (11) represent the horizontal and vertical components of Loop I, respectively. Equations (12) and (13) represent the horizontal and vertical components of Loop II, respectively. Additionally, \(\theta_{3}\), \(\theta_{4a}\), \(\theta_{4b}\) and \(\theta_{5}\), correspond to the angles between each link and the positive \(X\)-axis.  
According to research [16], Table 3 shows the design parameters of the optimal Watt-chain mechanism. If the input (\(S_{in}\) = 276.29 mm) is driven in 1 s with an MS curve, Figure 4(a)-c show the displacement, velocity, and acceleration of the output (\(S_{out}\)), and Figure 4(d) shows the corresponding force ratio. The peak values for the optimal Watt-chain mechanism are summarized in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters** & \(\mathbf{r_{2y}}\) & \(\mathbf{r_{3}}\) & \(\mathbf{r_{4b}}\) & \(\mathbf{r_{4c}}\) & \(\mathbf{r_{5}}\) & \(\mathbf{r_{6}}\) (\(\mathbf{S_{out}}\)) & \(\mathbf{r_{2}}\) (\(\mathbf{S_{in}}\)) & \(\mathbf{\alpha}\) \\ \hline unit & & & mm & & & & degree \\ value & 170 & 144 & 250 & 30 & 250 & 250 & 276.29 & 60 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The peak values of the optimal Watt–chain mechanism.  
Figure 4: Kinematic skeletons of the Watt–chain mechanism, constructed based on the descriptions in Patent No. TW M257273 [2]. The original patent does not provide this schematic; the diagram was interpreted and redrawn by the authors for analysis purposes.  
Tables 2 and 4 show that the optimal Watt\(-\)chain mechanism [2] performs better than the multi\(-\)joint mechanism [1]. The maximum acceleration \(\mid A_{max}\mid\) decreased by 16.2%, and the maximum force ratio (\(F_{in}\)/\(F_{out}\))\({}_{max}\) decreased by 26.6%.