## 2 Existing Designs
### Multiple-Joint Mechanism  
Figure 1d shows a multiple-joint mechanism from patent [1], while Figure 2 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 2a.  
As shown in Figure 2c, the multiple-joint mechanism contains two independent vector loops (Loop I and Loop II), such as Equations (1) and (2).  
\[\overset{\rightarrow}{r}_{2}+\overset{\rightarrow}{r}_{3}-\overset{ \rightarrow}{r}_{4}=0 \tag{1}\]  
\[\overset{\rightarrow}{r}_{4}+\overset{\rightarrow}{r}_{5}-\overset{ \rightarrow}{r}_{6}=0 \tag{2}\]  
Here, Loop I is represented by Equation (1), and Loop II is represented by Equation (2). The vectors \(\overset{\rightarrow}{r}_{2}\), \(\overset{\rightarrow}{r}_{3}\), \(\overset{\rightarrow}{r}_{4}\), \(\overset{\rightarrow}{r}_{5}\) and \(\overset{\rightarrow}{r}_{6}\) represent the links of the mechanism.  
The horizontal component X and vertical component Y equations are  
\[r_{2x}+r_{3}\cos\theta_{3}-r_{4}\cos\theta_{4}=0 \tag{3}\]  
\[r_{2y}+r_{3}\sin\theta_{3}-r_{4}\sin\theta_{4}=0 \tag{4}\]  
\[r_{4}\cos\theta_{4}+r_{5}\cos\theta_{5}-r_{6}=0 \tag{5}\]  
\[r_{4}\sin\theta_{4}+r_{5}\sin\theta_{5}=0 \tag{6}\]  
Equations (3) and (4) represent the horizontal and vertical components of Loop I, respectively. Equations (5) and (6) represent the horizontal and vertical components of Loop II, respectively. Additionally, \(\theta_{3}\), \(\theta_{4}\), and \(\theta_{5}\) correspond to the angles between each link and the positive \(X\)-axis.  
Figure 1: Feasible mechanisms for mold clamping using generalized chain configurations (6, 7) [14].  
Assuming mechanical efficiency of 100%, input power equals output power, i.e.,  
\[F_{i}V_{i}=F_{o}V_{o} \tag{7}\]  
Here, the \(F_{i}\) and \(V_{i}\) represent input force and input velocity. Similarly, \(F_{o}\) and \(V_{o}\) represent output force and output velocity.  
According to research [14; 15; 16] and based on Equations (3)-(6), if the design parameters are as shown in Table 1 and the input (\(S_{in}\) = 242.25 mm) is driven in 1 s with an MS curve, the output (\(S_{out}\)) displacement is 250 mm, as shown in Figure 2(a). Then, the velocity and acceleration can be calculated using the finite difference method, as shown in Figure 2(b),c. According to Equation (7) and Figure 2(b), the force ratio is calculated and shown in Figure 2(d). The peak values for the multi-joint mechanism are summarized in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(\mathbf{r_{2y}}\) & \(\mathbf{r_{3}}\) & \(\mathbf{r_{4}}\) & \(\mathbf{r_{5}}\) & \(\mathbf{r_{6}}\) (\(\mathbf{S_{out}}\)) & \(\mathbf{r_{2}}\) (\(\mathbf{S_{in}}\)) \\ \hline unit & & & mm & & \\ value & 140 & 140 & 250 & 250 & 250 & 242.25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters of the multiple-joint mechanism.  
Figure 2: Kinematic skeletons of a multiple-joint mechanism, constructed based on the descriptions in Patent No. TW 542097 U [1]. The original patent does not provide this schematic; the diagram was interpreted and redrawn by the authors for analysis purposes.