## 1 Introduction  
Nine-link type double-toggle mold/die clamping mechanisms are widely used in modern injection molding machines and die casting machines in order to provide sufficient mold/die clamping force for counteracting the pressure occurred inside molds/dies [1, 2]. The perspective view of a nine-link type mold/die clamping mechanism is illustrated in Figure 1, and its perspective sectional view is also depicted in Figure 2. Such a mechanism, also known as a five-point double-toggle mold/die clamping mechanism, is part of an injection molding machine or of a die casting machine. The mechanism shown in the figures basically consists of a fundamental structure A1, a moving platen A2, and a linkage assembly A3. The fundamental structure A1, as an immovable frame of the mechanism, is mainly comprised of a base A1-1, a stationary platen A1-2, a tail stock platen A1-3, four tie bars A1-4, and a hydraulic cylinder A1-5. The moving platen A2, which is constrained by the four parallel tie bars A1-4, can only translate along the axial direction of the tie bars. In order to drive the moving platen, the linkage assembly A3 is set up between the tail stock platen A1-3 and the moving platen A2 and is actuated by a piston of the hydraulic cylinder A1-5. Two half-molds/dies (not shown in the figures) are firmly installed on the stationary platen A1-2 and the moving platen A2, respectively, and can be closed or opened during the injection molding or die casting process when the moving platen A2 is driven by the linkage assembly A3 to translate forward or backward. Three configurations of the nine-link type mold/die clamping mechanism, for illustrating its initial position, intermediate position, and toggle position, respectively, are shown in Figure 3. When the mechanism is in its initial position, as shown in Figure 3a, both the piston of the hydraulic cylinder (i.e., the input link member) and the moving platen (i.e., the output link member) are at their initial positions, and the two half-molds/dies (not shown) are entirely opened. When the mechanism is in its intermediate position, as  
Figure 1: Perspective view of a nine-link type double-toggle mold/die clamping mechanism. Figure 2: Perspective sectional view of a nine-link type double-toggle mold/die clamping mechanism. shown in Figure 3b, both the input and output link members translate forward (rightward) from their initial positions, and the two half-molds/dies (not shown) are closing. When the mechanism is in its toggle position, as shown in Figure 3c, both the input and output link members are at their limit positions, and the two half-molds/dies (not shown) are entirely closed. It must be noted in Figure 3c that, the three revolute joints A, B, and C are collinear at that instant, so are the three revolute joints D, E, and F; such geometric relations form the well-known double-toggle configuration. Such a double-toggle configuration can theoretically achieve an infinite mechanical advantage (MA) [3] for providing sufficient mold/die clamping force to counteract large pressure occurred inside molds/dies. Referring to Figures 1-3, the nine-link type double-toggle mold/die clamping mechanism topologically consists of nine links, ten revolute joints, and two prismatic joints. According to Grubler-Kutzbach criteria [3], the number of degree-of-freedom (DOF, i.e., the mobility) of such a planar mechanism should be  
\[F_{p}\ =\ 3(N-1)-2(J_{R}+J_{P})\ =\ 3(9-1)-2(10+2)\ =\ 0 \tag{1}\]  
in which, \(N\), \(J_{R}\), and \(J_{P}\) are the numbers of links, revolute joints, and prismatic joints, respectively. When the number of DOF of a mechanism is less than one, it is theoretically  
Figure 3: Configurations of a nine-link type double-toggle mold/die clamping mechanism: (**a**) initial position, (**b**) intermediate position, and (**c**) toggle position. a structure or an overconstrained mechanism, i.e., the mechanism should be immovable. However, this mold/die clamping mechanism is actually movable and does not agree with the Grubler-Kutzbach criteria.