## 3 Mechanical Model and Kinematic Analysis of the Low-Power Holding Mechanism
### Kinematic Analysis of the Low-Power Holding Mechanism  
In the locking or unlocking process, the locking tab moves linearly in the cam-like notch of the locking ring, and the locking ring rotates around the actuator working cavity. Figure 8 shows the relative position between the locking tab and the locking ring in the locking state and unlocking state, respectively. Thus, the constraint condition for the locking tab to complete the locking and unlocking actions can be expressed as:  
\[L\leq[L], \tag{17}\]  
where \([L]\) is the permissible stroke of the locking tab; \(L\) is the working stroke of the locking tab.  
The permissible stroke of the locking tab depends on the size of the low-power holding mechanism. To facilitate the analysis, the movement of the locking ring and the locking tab is equivalent so that the locking ring remains fixed, while the locking tab not only moves linearly in the notch but also rotates around the actuator working cavity during the locking and unlocking processes. Figure 9 shows the kinematic analysis of the unlocking process of the low-power holding mechanism. Assuming that O\({}_{32}\) is the center of the curved surface of the locking tab in an unlocking state; O\({}_{31}\) is the intersection of the segment O\({}_{1}\)O\({}_{32}\) and the arc formed by the rotation of O\({}_{3}\) around O\({}_{1}\), \(m\) is the length of the segment  
Figure 7: The force analysis of the locking ring sleeve and the locking ring in the holding process. 3—Actuator working cavity, 4—Electromagnet, 5—Locking ring sleeve, 7—Locking ring, 8—Housing.  
\(\mathrm{O_{1}O_{32}O_{2}}\). The other symbols have the same values as depicted in Figure 6. The permissible stroke of the locking tab, \([L]\), can be expressed as:  
\[[L]=m-h. \tag{18}\]  
As shown in Figure 9, the rotation angle of the locking ring in the locking or unlocking process \(\theta\) can be given as:  
\[\theta=\alpha-\angle\mathrm{O_{32}O_{1}O_{2}}=\alpha-\arcsin\left(\frac{r_{3} }{r_{1}}\right). \tag{19}\]  
So \(\angle\mathrm{O_{1}O_{32}O_{2}}\) and \(m\) can be expressed as follows, respectively:  
\[\angle\mathrm{O_{1}O_{32}O_{2}}=\arcsin\left(\frac{l\sin\angle\mathrm{O_{32}O_ {1}O_{2}}}{r_{2}-r_{3}}\right)=\arcsin\left(\frac{lr_{3}}{r_{1}(r_{2}-r_{3})} \right), \tag{20}\]  
\[m^{2}=(r_{2}-r_{3})^{2}+l^{2}+2l(r_{2}-r_{3})\cos\left[\arcsin\left(\frac{r_{3 }}{r_{1}}\right)+\arcsin\left(\frac{lr_{3}}{r_{1}(r_{2}-r_{3})}\right)\right]. \tag{21}\]  
By substituting Equations (15) and (21) into Equation (18), the permissible stroke of the locking tab \([L]\) can be rewritten as:  
Figure 8: The relative position between the locking tab and the locking ring in (**a**) the locking state, (**b**) the unlocking state. 3—Actuator working cavity, 6—Locking tab, 7—Locking ring.  
Figure 9: The kinematic analysis of the unlocking process of the low-power holding mechanism.  
\[\begin{split}[L]&=m-h\\ =&\left\{(r_{2}-r_{3})^{2}+l^{2}+2l(r_{2}-r_{3})\cos \left[\arcsin\left(\frac{r_{2}}{r_{1}}\right)+\arcsin\left(\frac{lr_{3}}{r_{1}( r_{2}-r_{3})}\right)\right]\right\}^{\frac{1}{2}}-\\ &\left\{(r_{2}-r_{3})^{2}+l^{2}+2l(r_{2}-r_{3})\cos\left[\alpha+ \arcsin\left(\frac{l\sin\alpha}{r_{2}-r_{3}}\right)\right]\right\}^{\frac{1}{2 }}\end{split} \tag{22}\]