## 2 Solenoid Design and Analysis
### Theoretical Analysis  
The theoretical analysis on the electromagnetic force generated by the solenoid is implemented to design a solenoid actuator with a permanent magnet plunger. We develop a new type of solenoid actuator for bi-directional motion in the miniaturized robot. This solenoid actuator has a simple structure composed of coils and a permanent magnet plunger. The resultant force by the solenoid depends on the length, air-gap, number of coil turns, applied current, and the type of permanent magnet, _etc._ We induce the equation of electromagnetic force actuated by the solenoid as the functions of these design variables to develop a suitable solenoid for segmented robots. The magnetic field along the axis of a circular current loop of radius \(r\) and steady current \(i\) can be expressed by the Biot-Savart law in Figure 1a:  
\[\mathbf{B_{sol}}=\frac{\mu_{0}i}{4\pi}\int\frac{d\mathbf{l}\times\mathbf{r}^{\prime}}{|\bm {r}^{\prime}|^{3}} \tag{1}\]  
Figure 1: The design of a solenoid with a permanent magnet (PM) plunger: (**a**) the design concept of multi-segment robots with linear and turning motions using a pair of solenoids; (**b**) the cross-sectional view of the proposed solenoid actuator with PM plunger. Here, \(d\mathbf{l}\) and \(\mathbf{r}^{\prime}\) are the line segment and distance vector from the viewpoint to the source charge; \(\mu_{0}=4\pi\ \times 10^{-7}\ Wb/A\cdot m\) represents the permeability of vacuum. By the use of \(d\mathbf{l}=R\ d\psi\ \widehat{\mathbf{\Psi}}\), \(\mathbf{r}^{\prime}=z\widehat{\mathbf{z}}\ -r\widehat{\mathbf{r}}\) and \(|\mathbf{r}^{\prime}|=\sqrt{2^{2}+r^{2}}\), the magnetic field \(\mathbf{B_{sol\mathbf{z}}}\) at an arbitrary point P on the axis of the current loop becomes  
\[\mathbf{B_{sol\mathbf{z}}}=\ \frac{\mu_{0}ir^{2}}{4\pi}\int_{0}^{2\pi}\frac{d\psi}{\left(r^ {2}+\ z^{2}\right)^{3}\!/_{2}}\ \widehat{\mathbf{z}}=\frac{\mu_{0}}{2}\frac{ir^{2}}{\left(r^{2}+\ z^{2}\right)^{3 }\!/_{2}}\ \widehat{\mathbf{z}} \tag{2}\]  
Figure 2b shows the cross-sectional view of the solenoid actuator. In particular, for currents with special symmetry, [PERSON]'s law in integral form offers an extraordinarily efficient means for finding the electromagnetic field. The axial electromagnetic field of the solenoid is obtained by Equation (2). Using the design parameters of the solenoid actuator such as inner diameter \(d\), outer diameter \(D\), number of coil turns \(N\), and length \(L\), Equation (2) becomes  
\[\mathbf{B_{sol\mathbf{z}}}=\ \int_{0}^{L}\int_{d/_{2}}^{D/_{2}}\frac{\mu_{0}nir^{2}}{2 \left[r^{2}+\ \left(z_{p}+\ \left(z_{p}+\ \left/_{2}-z\right)^{2}\right)^{3}\!/_{2} \right.}\ \widehat{\mathbf{z}}\ dr\ dz \tag{3}\]  
Here, \(n=\ N/L\) and \(z_{p}\) represent the wounded coils per unit length and the distance of point \(P\) from the center of the solenoid; \(z=\ L/2+z_{p}\) is the moving distance of the PM plunger inside the solenoid or the length of the air-gap. By the integration of Equation (3), the electromagnetic field at point \(P\) becomes  
\[ \begin{split}&\mathbf{B_{sol\mathbf{z}}}=\ \frac{\mu_{0}Ni}{2L(D\ -\ d)}\ \left(L\ +\ 2z_{p}\right)ln\left[\frac{D+\ \sqrt{D^{2}\ +\ \left(L\ +\ 2z_{p}\right)^{2}}}{d\ +\ \sqrt{d^{2}\ +\ \left(L\ +\ 2z_{p}\right)^{2}}} \right]\\ &+\ \frac{\mu_{0}Ni}{2L(D\ -\ d)}\ \left(L\ -\ 2z_{p}\right)ln\left[\frac{D+\ \sqrt{D^{2}\ +\ \left(L\ -\ 2z_{p}\right)^{2}}}{d\ +\ \sqrt{d^{2}\ +\ \left(L\ -\ 2z_{p}\right)^{2}}} \right]\ \widehat{\mathbf{z}}\end{split}  \tag{4}\]  
Figure 2: The solenoid dimensions for the electromagnetic analysis: (**a**) one turn of coil flowing with current \(i\); (**b**) the cross-sectional view of a solenoid with PM plunger and design parameters.