# 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
\right)-T_{s}\delta\varphi_{ \text{Ts}}-f_{s}\delta x_{s}, \tag{27}\]  
\[\overset{\rightarrow}{f}_{1}=(-f_{1}\cos\alpha,f_{1}\sin\alpha), \tag{28}\]  
\[\overset{\rightarrow}{f}_{2}=(-f_{2}\cos(\alpha+\beta),f_{2}\sin(\alpha+\beta )), \tag{29}\]  
\[\overset{\rightarrow}{G}_{1}=(h_{1}\sin\alpha,h_{1}\cos\alpha), \tag{30}\]  
\[\overset{\rightarrow}{G}_{2}=(l_{1}\sin\alpha+h_{2}\sin(\alpha+\beta),l_{1} \cos\alpha+h_{2}\cos(\alpha+\beta)), \tag{31}\]  
\[\varphi_{\text{Ts}}=\frac{\theta_{0}}{i}-\alpha, \tag{32}\]  
\[T_{s}=k_{2}\varphi_{\text{Ts}}=k_{2}(\frac{\theta_{0}}{i}-\alpha), \tag{33}\]  
\[x_{s}=r(\alpha+\beta), \tag{34}\]  
\[f_{s}=k_{1}x_{s}=k_{1}r(\alpha+\beta), \tag{35}\]  
Furthermore, the derivatives of the two sides of the Equations (30)-(32) and (34) are obtained:  
\[\overset{\rightarrow}{\delta\!G_{1}}=(h_{1}\cos\alpha\cdot\delta\alpha,-h_{1 }\sin\alpha\cdot\delta\alpha), \tag{36}\]  
\[\overset{\rightarrow}{\delta\!G_{2}}=(l_{1}\cos\alpha\cdot\delta\alpha+h_{2} \cos(\alpha+\beta)\cdot(\delta\alpha+\delta\beta),\]  
\[\overset{\rightarrow}{f}_{2}=(-f_{2}\cos(\alpha+\beta),f_{2}\sin(\alpha+\beta )), \tag{29}\]0  
\[\overset{\rightarrow}{f}_{2}=(-f_{2}\cos(\alpha+\beta),f_{2}\sin(\alpha+\beta )), \tag{29}\]1  
\[\overset{\rightarrow}{f}_{2}=(-f_{2}\cos(\alpha+\beta),f_{2}\sin(\alpha+\beta )), \tag{29}\]2  
Combining the above equations, one can obtain:  
\[ \begin{array}{c}T_{0}\cdot\delta\theta_{0}\!=\{f_{1}h_{1}+f_{2}h_{2}+f_{2}l _{1}[\cos(\alpha+\beta)\cos\alpha+\sin(\alpha+\beta)\sin\alpha]\}\delta\alpha\\ +f_{2}h_{2}\delta\beta+k_{2}(\frac{\theta_{0}}{i}-\alpha)(\frac{\delta\theta_{0 }}{i}-\delta\alpha)+k_{1}r^{2}(\alpha+\beta)(\delta\alpha+\delta\beta)\end{array} \, \tag{40}\]  
According to the mechanization of the partial effective transmission mechanism, the rotational angle of the reduced motor, stroke angle of the partial effective transmission mechanism, and the joint angles of the proximal joint axis and distal axis have the following relationship:  
\[\beta=\frac{\theta_{0}}{i}-\varphi_{0}-\alpha, \tag{41}\]  
Thus,  
\[ \begin{array}{c}\{f_{1}h_{1}+f_{2}h_{2}+f_{2}l_{1}[\cos(\alpha+\beta)\cos \alpha+\sin(\alpha+\beta)\sin\alpha]+k_{1}r^{2}(\alpha+\beta)-iT_{0}\}\delta \alpha\\ -\left[-k_{2}(\beta+\varphi_{0})+iT_{0}-f_{2}h_{2}-k_{1}r^{2}(\alpha+\beta) \right]\cdot\delta\beta=0,\\ \left(\begin{matrix}f_{1}h_{1}+f_{2}h_{2}+f_{2}l_{1}\cos\beta+k_{1}r^{2}( \alpha+\beta)-iT_{0}\\ k_{2}(\beta+\varphi_{0})-iT_{0}+f_{2}h_{2}+k_{1}r^{2}(\alpha+\beta)\end{matrix} \right)^{T}\cdot\left(\begin{matrix}\delta\alpha\\ \delta\beta\end{matrix}\right)=0,\end{array}  \tag{42}\]Then, one can obtain the grasping forces of both phalanges under the self-adaptive grasping mode:  
\[ \begin{pmatrix}f_{1}\\ f_{2}\end{pmatrix} = \begin{pmatrix}\frac{k_{2}}{h_{1}}(\beta+\varphi_{0})-\frac{l_{1 }\left[iT_{0}-k_{2}(\beta+\varphi_{0})-k_{1}r^{2}(\alpha+\beta)\right]\cos\beta }{h_{1}h_{2}}\\ \frac{iT_{0}-k_{2}(\beta+\varphi_{0})-k_{1}r^{2}(\alpha+\beta)}{h_{2}}\end{pmatrix} , \tag{43}\]  
Similarly, after the parameters are included, the grasping forces distribution can be obtained, as shown in Figure 13 below. When the ESMGM hand enters the self-adaptive grasping mode from the parallel grasping mode, both the proximal and distal phalanges provide the grasping forces towards the object, changing with various joint angles. The grasping force of distal phalange \(f_{2}\) is relatively smaller than that under the parallel grasping mode. With increased joint angles of both proximal phalange \(\alpha\) and distal phalange \(\beta_{s}f_{2}\) decreases slightly. Overall, \(f_{2}\) maintains a stable level of 10 N to 20 N. The grasping force of proximal phalange \(f_{1}\) is slightly affected by the joint angle of proximal phalange \(\alpha\), but increases rapidly with the joint angle of distal phalange \(\beta\). When \(\beta\) increases to a certain degree, \(f_{2}\) could reach about 70 N, which is sufficient for the grasping of most things that a human hand can manage in daily life and industrial production.