## 3 Magnetic Device Analysis and Results
### Motion Analysis of Multi-Link Magnet Device  
Using the designed multi-link magnet device, we performed motion analysis based on video tracking according to changes in the driving current, considering two cases. That is, finger movements with and without the M2 auxiliary magnet on the palm were observed for comparison for a 0.4 Hz driving frequency, as shown in Figure 7. First, we observed finger flexion and extension without M2, as shown in Figure 7a. In that case, the maximum movement range was found to be \(-15\) to \(150^{\circ}\) at a driving range of \(-25\) to \(25\) A. Figure 7b shows the motion trajectory for flexion and extension without M2, while Figure 7b1-b12 show the finger positions according to the driving current variations. The black and red dotted lines in Figure 7b indicate finger flexion and extension, respectively. For finger flexion, the driving current varied from 0 A to 25 A. With no current, the fingers have initial and maximum angles of \(38^{\circ}\) and \(155^{\circ}\), respectively, with a maximum current of 25 A, corresponding to the position shown in Figure 7b1-b3. Figure 7b4-b8 corresponds to the extension trajectories for driving-current variation from 25 A to \(-25\) A. The finger angle decreases from \(155^{\circ}\) to \(60^{\circ}\) while the driving current reduction from 25 to 0 A (Figure 7b4-b7). For \(-8.3\) A driving current, a maximum stretched finger angle of approximately \(-7^{\circ}\) appears; this is maintained up to \(-25\) A because of \(F_{M1,3}\). Although the driving current is positive to approximately 16.5 A, the attractive force \(F_{M1,3}\) maintains the extended position. At 17.2 A, the finger angle increases rapidly to \(56.6^{\circ}\) (Figure 7b10). Above 17 A, the finger movement changes rapidly from extension to flexion (16-25 A), as shown in Figure 7b10-b12. In the second experiment, flexion and extension motions were observed for a device containing M1, M2, and M3, as shown in Figure 7c. In this configuration, \(F_{M1,E}\) between M1 and the driving coil generates basic finger flexion and extension, while \(F_{M1,2}\) and \(F_{M1,3}\) enhance the flexion and extension and maintain the maximum flexion and extension, improving the training effect. Because of the effect of \(F_{M1,2}\), a maximum flexion angle of approximately \(170^{\circ}\) appears, which is maintained at \(146.8^{\circ}\) until a \(-16.5\) A driving current, as shown in Figure 7d1-d6. Next, M1 on the finger accelerates under the influence of \(F_{M1,E}\) and \(F_{M1,3}\), and the two forces produce a rapid stretching motion for a small \(-8\) A current variation, as shown in Figure 7d6-d8. In particular, \(F_{M1,3}\) maintains stretching motion, which can improve the training effect up to a driving current of approximately 18 A, as shown in Figure 7d8-d10. In this case, although the driving current is positive, the generated \(F_{M1,E}\) is weaker than \(F_{M1,3}\) such that the stretching motion is maintained. A bending motion occurs when the driving current exceeds 18 A, and the motion changes occur quickly in response to the stronger \(F_{M1,E}\) with the assisting force \(F_{M1,2}\), as shown in Figure 7d10-d12. A comparison of the two methods examined in the above experiments reveals that the use of auxiliary magnets M2 and M3 yields greater finger movement. Note that the experimenter's hand muscle strength was  
Figure 6: Magnetic torque calculation according to height, with the negative driving currents represented by black bars and the positive by red bars. not used, and the hand was actively moved by magnetic forces only. Furthermore, both \(\textbf{F}_{\textbf{M},\textbf{2}}\) and \(\textbf{F}_{\textbf{M},\textbf{3}}\) can manually improve hand muscle strength without an external magnetic field because the two forces interfere with finger movement. Thus, through magnetic simulation and motion analysis, we confirmed active extension and flexion for wireless training using magnetic coupling.