# Electromagnetic Switching Multiple Grasping Modes Robot Hand
## 4 Mechanical Analysis of ESMM Hand
### Contact Forces Analysis of the ESMGM Hand
#### 4.2.1 PASA Mode  
1. Parallel grasping mode  
When the ESMGM hand grasps the object with parallel grasping mode, the grasping force will be provided by a distal phalange, as shown in Figure 8a. The tension spring and torsion spring will not deform; thus, in the case of neglecting friction, the power provided by the motor will all be converted into mechanical energy when the robot hand grasps. According to the principle of virtual work, one can obtain the following equations:  
\[-T_{0}\cdot\delta\theta_{0}=\overset{\rightarrow}{f}_{2}\cdot\overset{ \rightarrow}{\delta}\overset{\rightarrow}{G}_{2}^{t}, \tag{21}\]  
\[\overset{\rightarrow}{f}_{2}=(-f_{2},0), \tag{22}\]  
\[\overset{\rightarrow}{G}_{2}=(l_{1}\sin\alpha,l_{1}\cos\alpha+h_{2}), \tag{23}\]  
Figure 11: Kinematic analysis of the ESMGM hand (COSA mode): (**a**) Trajectory along the \(x\)-axis; (**b**) trajectory along the \(y\)-axis. After the speed reduction of the worm gear mechanism and gear transmission mechanism, the relationship between the joint angle of the proximal phalange and the rotation angle of the motor is as follows:  
\[\alpha=\frac{\theta_{0}}{i}, \tag{24}\]  
Therefore, the grasping force under the parallel grasping mode can be calculated as:  
\[-T_{0}\cdot\delta\theta_{0}=-f_{2}l_{1}\cos\alpha\frac{\delta\theta_{0}}{i}, \tag{25}\]  
\[f_{2}=\frac{iT_{0}}{l_{1}\cos\alpha}, \tag{26}\]  
From the equation, when the size parameters of the ESMGM robot hand and the components such as the motor, worm, gear, spring, and torsion spring are determined, the grasping force of each phalange is a function related to the joint angles of the proximal joint axis and distal joint axis. Since the actual specifications and dimensions of the ESMGM robot hand were determined, we could quantitatively analyze the grasping forces of the robot hand according to the equations derived above. Thus, during the parallel grasping mode, the grasping force of the proximal phalange is zero, and the grasping force of the distal phalange changes with the joint angles. From Figure 12, one can conclude that, under the parallel grasping mode, the grasping force of the distal phalange increases with the increase of the joint angle of the proximal joint axis. When the joint angle is less than 60 degrees, the grasping force is stable. When the joint angle increases, the grasping force increases rapidly with a larger growth rate. Overall, the ESMGM hand can provide a relatively stable and large grasping force, which is larger than 40 N, in the parallel grasping mode. This characteristic perfectly meets the needs of the parallel grasping mode, which is often applied to the clamping of parts in industrial production and requires a large grasping force and good stability. 2 Self-adaptive grasping mode  
After the ESMGM hand enters the self-adaptive grasping mode from the parallel grasping mode, the torsion spring will deform because the proximal phalange is obstructed by the object. The tension spring will be elongated because the partial effective transmission mechanism enters the effective-stroke stage. Therefore, part of the energy provided by the  
Figure 12: Grasping force under the parallel grasping mode. motor will be converted into the elastic potential energy of the torsion spring. As shown in Figure 8b, one can obtain the following equations according to the principle of virtual work:  
\[-T_{0}\cdot\delta\theta_{0}=[\overset{\rightarrow}{f}_{1},\overset{ \rightarrow}{f}_{2}]\left( \begin{array}{c}\overset{\rightarrow}{\delta\!G_{1 }}^{t}\\ \overset{\rightarrow}{\delta\!G_{2}}^{t}\end{array}