## 3 Mechanical Model and Kinematic Analysis of the Low-Power Holding Mechanism  
### Mechanical Model of the Low-Power Holding Mechanism  
The low-power holding mechanism adopts four locking tabs to share the maximum load of the closing spring. Figure 3 shows the force analysis of the spring support cylinder and single locking tab in the holding process, in which \(F_{\text{ml}}/4\) is the maximum load of the closing spring shared by the single locking tab, \(R_{16}\) and \(R_{61}\) are the interactive force between the spring support cylinder and the single locking tab, \(R\) is the radial component force acting on the spring support cylinder by the single locking tab in the symmetrical direction, \(R_{63}\) and \(R_{36}\) are the interactive force between the single locking tab and the actuator working cavity, \(F_{\text{r}}\) and \(F^{\prime}{}_{\text{r}}\) are the interactive radial thrust between the single locking tab and the locking ring. Figure 3: The force analysis of the spring support cylinder and single locking tab in the holding process. 1—Spring support cylinder, 2—Closing spring, 3—Actuator working cavity, 4—Electromagnet, 5—Locking ring sleeve, 6—Locking tab, 7—Locking ring. In the holding process, the spring support cylinder and the single locking tab are in a state of mechanical equilibrium. Figure 4 shows the force diagram of the spring support cylinder and the single locking table as shown in Figure 4a, the relation between \(R_{61}\) and \(F_{\mathrm{ml}}\) can be expressed as:  
\[R_{61}=\frac{F_{\mathrm{ml}}}{4\cos(\lambda-\varphi)}, \tag{1}\]  
where \(\lambda\) is the angle of the slope of the single locking tab; \(\varphi\) is the friction angle; \(F_{\mathrm{ml}}\) is the maximum load of the closing spring. As shown in Figure 4b, the relation between \(R_{61}\) and \(F_{\mathrm{r}}\) can be given as:  
\[\frac{R_{16}}{\sin(90^{\circ}+\varphi)}=\frac{R_{61}}{\sin(90^{\circ}+\varphi )}=\frac{F_{r}^{\prime}}{\sin(\lambda-2\varphi)}=\frac{F_{r}}{\sin(\lambda-2 \varphi)}, \tag{2}\]  
Combining Equations (1) and (2), the relation between \(F_{\mathrm{r}}\) and \(F_{\mathrm{ml}}\) can be expressed as:  
\[F_{\mathrm{r}}=\frac{R_{16}\sin(\lambda-2\varphi)}{\sin(90^{\circ}+\varphi)}= \frac{F_{\mathrm{ml}}\sin(\lambda-2\varphi)}{4\cos(\lambda-\varphi)\cos\varphi}, \tag{3}\]  
Figure 5 shows the force diagram of the contact between the single locking tab and the locking ring in the holding process, in which \(R_{67}\) and \(R_{76}\) are the interactive forces between the single locking tab and the locking ring. To facilitate the analysis, the force diagram in Figure 5 is simplified, as shown in Figure 6. This assumes that O\({}_{1}\) is the center of the actuator working cavity; O\({}_{2}\) is the center of the notch of the locking ring; O\({}_{3}\) is the center of the curved surface of the locking tab; C is the contact point between the locking tab and locking ring; the straight line, MN, is the normal direction of the contact point C; S is the starting point of the notch of the locking ring, which is the contact point between the notch and the actuator working cavity; E is the ending point of the notch of the locking ring, which is the intersection of the notch and the line O\({}_{1}\)O\({}_{2}\); \(\delta\) is the angle of the notch of the locking ring. The angle between \(R_{76}\) and MN is \(\varphi\) since \(R_{76}\) is the resultant force of normal pressure and friction; the direction of \(F_{\mathrm{r}}\) is parallel to O\({}_{1}\)O\({}_{3}\). Because the notch of the locking ring and the actuator working cavity are not concentric, a circumferential force \(F_{\mathrm{t}}\) is introduced at the contact point C, which is not only the resistance to overcome by the low-power retaining mechanism in the holding process, but also the driving force of the backward rotation of the locking ring in the unlocking process.