## 3 Derivation of Theoretical Force Model  
### Theoretical Force Analysis of the EPM  
Figure 3 shows a full schematic of the EPM device for holding the rotating shaft of the wobble motor. Alnico 5 and NdFeB magnets are described as square magnets in the schematic, but the two magnets used in the theoretical analysis and the experiment were replaced with cylindrical magnets. To predict the attractive force, first, the structure is assumed to be a simple structure with a constant gap, and a core that contains two permanent magnets. Additionally, we assume that the rotating shaft is replaced with the plate. Second, in order to simplify the calculations for magnetic flux density, including the leakage flux, we simplified the path through which the leakage flux flows. Additionally, the fringing flux was assumed to be the leakage flux, and the path of flux was consisted as a combination of arcs and straight line, as shown in Figure 4a [18]. The fringing fluxes were considered a kind of leakage flux. It was also assumed that the paths detouring on the top outside of the surrounding air gap were connected in the quadrant with a straight line and the length of the air gap [19]. Figure 4b shows a consideration of the mean leakage flux line. The specifications of the electromagnetic elements in use and their parameters are summarized in Table 1.  
For the leakage flux paths of \(P_{dx}\), as shown in Figure 4a, the length \(l_{x}\) of the magnetic flux path is equal to \(g+\pi x\), which is the sum of the lengths of two quadrant arcs of radius \(x\) and the length of the gap. The permeance in the path is as follows:  
\[P_{dx}=\frac{1}{\Re_{dx}}=\frac{\mu_{0}dA_{x}}{l_{x}}=\frac{\mu_{0}bdx}{g+\pi x} \tag{1}\]  
The permeance of the total leakage flux can be obtained by integrating Equation (1). Figure 2: Basic principle driving an electropermanent magnet: (**a**) In the no current state, the two magnets are oppositely polarized; (**b**) In the switching current state, the two magnetic magnets are polarized in the same direction. Figure 4: Paths model for leakage flux: (**a**) Paths for the leaked flux; (**b**) Mean length of flux path. Figure 3: Schematic of the switchable eletropermanent magnet construction. If the core and the plate have a constant air gap, as shown in Figure 5a, it is assumed that the leakage flux paths diverting to the outside of the gap are connected to two quadrants of the same magnetic flux in a straight line along the length of the air gap. Each leakage flux path indicated by a line has a reluctance, and all of them in the magnetic circuit are connected in parallel, expressing it as shown in Figure 5b. Therefore, it is possible to use a permeance that has an inverse relationship with a reluctance. In Figure 5b, the reluctance \(\Re_{c}\) and \(\Re_{o}\) in the iron are ignored. Therefore, when the permeances of the leakage flux path are \(P_{go}\), \(P_{gl}\), \(P_{mo}\), \(P_{mi}\) and \(P_{g}\), respectively, they can be expressed as reciprocal reluctances \(\Re_{go}\), \(\Re_{gl}\), \(\Re_{mo}\), \(\Re_{mi}\) and \(\Re_{g}\), \(\Re_{m}\), \(\Phi_{r}\) and \(\Phi_{m}\) are the source of the magnet, the magnetic flux of the magnetic flux passing through the air gap. To more accurately consider the leakage flux generated in the gap, it is regarded as a geometry as shown in Figure 6. The permeance of each of the flux paths through the gap can be expressed as permeance \(P_{go}=P_{1}+2(P_{2}+P_{3})+4P_{4}\) and \(P_{gl}=2(P_{5}+P_{6})+4P_{7}\). Permeance \(P_{1}\) is \(\mu_{0}ab/g\). The mean length of the flux path of permeance \(P_{2}\) can be considered to be equal to the length of a line drawn midway between the radius and a quarter of circumference as shown in Figure 6b. Permeance \(P_{2}\) and \(P_{3}\) with a quadrant of the cylindrical volume are \(0.528\mu_{0}b\) and \(0.528\mu_{0}a\), respectively.