# Electromagnetic Switching Multiple Grasping Modes Robot Hand
## 4 Mechanical Analysis of ESMM Hand
### Kinematic Analysis of the ESMGM Hand  
#### 4.1.1 Parallel and Self-Adaptive Grasping Mode (PASA Mode)  
Based on the motion trajectory of the tip end of the distal phalange, the kinematic analysis of the ESMGM hand was performed. We assumed that the motor provides a constant speed of n0 revolutions per minute. The mechanical analysis of ESMGM hand under PASA mode is shown in Figure 8.  
If the ESMGM hand is under parallel grasping mode, as shown in Figure 8, taking the proximal joint axis as the origin, the coordinate of the end of the distal phalange can be represented as:  
\[x_{d}=l_{1}\sin\alpha=l_{1}\sin\frac{6n_{0}t}{i}, \tag{5}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Interpretation** & **Unit** & **Default** \\ \hline \(f_{1}\) & Force exerted on object by proximal phalange & N & / \\ \(f_{2}\) & Force exerted on object by distal phalange & N & / \\ \(h_{1}\) & Arm of force \(f_{1}\) with respect to the proximal joint axis & mm & 20 \\ \(h_{2}\) & Arm of force \(f_{2}\) with respect to the distal joint axis & mm & 20 \\ \(f_{s}\) & Tension of tension spring & N & / \\ \(T_{s}\) & Torque of torsion spring & mN.m & / \\ \(k_{1}\) & Stiffness coefficient of tension spring & N/mm & 1 \\ \(k_{2}\) & Stiffness coefficient of torsion spring & N.mm/rad & 385 \\ \(T_{0}\) & Torque of the reduced motor & mN.m & 200 \\ \(n_{0}\) & Rotation speed of motor & round/min & 50 \\ \(\theta_{0}\) & Rotational angle of reduced motor & rad & / \\ \(\varphi_{0}\) & Stroke angle of partial effective transmission mechanism & rad & 90 \\ \(\psi_{0}\) & Return angle of partial effective transmission mechanism & rad & 60 \\ \(i\) & Reduction ratio of worm gear transmission mechanism & / & 10 \\ \(r\) & Distance of tension force point from gear center & mm & 12 \\ \(\eta\) & Overall transfer efficiency & / & 80\% \\ \(t_{a}\) & Time of parallel/coupled grasping period & s & / \\ \(t_{\varphi}\) & Time of idle-stroke phase & s & / \\ \(t_{\beta}\) & Time of self-adaptive grasping period & s & / \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._  
Figure 8: Mechanical analysis of the ESMGM hand (PASA mode): (**a**) Parallel grasping; (**b**) self-adaptive grasping.  
\[y_{d}=l_{2}+l_{1}\cos\alpha=l_{2}+l_{1}\cos\frac{6n_{0}t}{i}, \tag{6}\]  
Due to the limitation of the stroke angle of the partial effective transmission mechanism, the joint angle of proximal phalange \(\alpha\) and time of parallel grasping period \(t_{\alpha}\) have the following ranges:  
\[0\leq\alpha\leq\varphi_{0}, \tag{7}\]  
\[0\leq t_{\alpha}\leq\frac{i\varphi_{0}}{6n_{0}}, \tag{8}\]  
After the parallel grasping mode, the ESMGM hand will execute a period of the idle-stroke phase, and the idle-stroke phase time \(t_{\varphi}\) can be calculated as follows:  
\[t_{\varphi}=\frac{i\varphi_{0}}{6n_{0}}-t_{\alpha}, \tag{9}\]  
Then, the ESMGM hand will enter the self-adaptive grasping mode, and the coordinate of the end of the distal phalange can be represented as:  
\[x_{d}=l_{1}\sin\alpha+l_{2}\sin(\alpha+\beta)=l_{1}\sin\frac{6n_{0}t_{\alpha}} {i}+l_{2}\sin(\frac{6n_{0}t}{i}-\varphi_{0}), \tag{10}\]  
\[y_{d}=l_{1}\cos\alpha+l_{2}\cos(\alpha+\beta)=l_{1}\cos\frac{6n_{0}t_{\alpha} }{i}+l_{2}\cos(\frac{6n_{0}t}{i}-\varphi_{0}), \tag{11}\]  
\[t_{\beta}\geq\frac{i\varphi_{0}}{6n_{0}}, \tag{12}\]  
Thus, one can draw the trajectory of the distal phalange end during the PASA mode, as shown in Figure 9.