## 3 Magnetic Device Analysis and Results  
### Validation of Multi-Link Magnet Device for Wireless Training  
To verify the proposed mechanism, we performed various magnetic simulations using COMSOL software. We also performed multiple experiments. In the simulations, the finger position changes and the M1 driving currents were the same as in the actual experiments, as shown in Figure 5. Figure 5a-d show the finger flexion process up to a 25 A driving current. The generated magnetic field is directed upward as a result of the positive current. In the case shown in Figure 5a, M1 and M3 generate an attractive force and their flux direction is opposite to the field generated by the driving coil (16 A). In particular, \(F_{M1,3}\) is stronger than that between M1 and the electromagnet (\(F_{M1,E}\)) at maximum extension. Figure 5: Magnetic simulation results according to changes in finger positions due to driving current variation. (**a–d**) Finger flexion: actual positions for driving currents of 16, 20, 23 and 25 A, and their magnetic couplings in magnetic simulations. (**e–h**) Extension motion due to negative driving currents from \(-\)16 to \(-\)24 A and their magnetic couplings in magnetic simulations. When the driving current is 20 A, the generated magnetic field from the coil pushes M1. Under that condition, we can observe that the magnetic fluxes from M1 and the coil repel each other. In that case, \(F_{M1,E}\) is stronger than \(F_{M1,3}\). The increased driving currents further increase the bending of the finger via the stronger \(F_{M1,E}\). At this time, M1 is close to M2, which is on the palm, and an attraction force \(F_{M1,2}\) is generated between M1 and M2, yielding the maximum grasping motion, as shown in Figure 5c,d. In that case, the 25 A driving current generates a maximum flexion angle of 180\({}^{\circ}\). For finger extension, we changed the driving current from 25 to \(-\)25 A. In Figure 5e, the field generated by the coil exhibits initial extension due to strong \(F_{M1,2}\) between M1 and M2 at a \(-\)16 A driving current. In that case, the driving magnetic force \(F_{M1,E}\) is stronger than \(F_{M1,2}\). Because of downward field from the negative current, M1 moves in the field direction (finger extension), as shown in Figure 5f-h. Moreover, when M1 is close to M3, \(F_{M1,3}\) can enhance the finger stretching motion, as shown in Figure 5h. Therefore, through simulation, we successfully confirmed the magnetic coupling condition according to changes in the M1 position due to variations in the driving current for flexion and extension. In particular, the magnetic axial coupling between M1 and M2 or between M1 and M3 enhances the finger movement induced by \(F_{M1,2}\) and \(F_{M1,3}\). In addition, the magnetic device generates magnetic torque in the applied magnetic field. Figure 6 shows the calculated maximum magnetic torque at the positive and negative driving currents of \(\pm\) 25 A. For torque calculation, we used the measured magnetic field flux density using the device in Figure 3b. There are a total of eight cylindrical permanent magnets built into the multi-link magnet device. When wearing a multi-link magnet device, all fingers are restricted to moving together, so the torque generated by each magnet was calculated and all the results were summed. A magnetic torque can be expressed as follows:  
\[\tau=m\times B \tag{1}\]  
where \(m\) is the magnetic moment and \(B\) is the external magnetic field. The magnetic moment value of the permanent magnet can be calculated from the residual magnetic flux density \(B_{r}\) by the following equation:  
\[m=\frac{1}{\mu_{0}}B_{r}V \tag{2}\]  
where \(\mu_{0}\) is the permeability of vacuum (\(H/m\)) and \(V\) is the volume of the magnet. During the flexion and extension movement, the position of the device produces maximum torque at a height of 66 to 96 mm from the coil surface. In this height range, it can be confirmed that the angle between the magnetic field and the magnetic moment is 90 degrees.