## 4 Experiments and Discussions  
We set up the experimental apparatus to measure the force and the displacement by the solenoid actuator to examine the performance of this solenoid actuator. As shown in Figure 6a, we construct an experimental set-up to measure the force generated by the solenoid using a load cell (CASTM PW4 MC3, HBM, Darmstadt, Germany) the fixture to hold the solenoid, and the power supply. We use an infrared sensor (2Y0A02, Sharp Corporation, Osaka, Japan) as shown in Figure 6b to measure the displacement of the segmented robot. The distance measuring range of the infrared sensor system is from 20 to 150 cm. Additionally, the tolerance of this sensor system is approximately 0.3 mm. Table 1 shows the theoretical and experimental results of electromagnetic forces acting on the PM plunger by the solenoid at the center of the solenoid (\(z_{p}=0\)), using the optimal design parameters: the solenoid length (\(L=20\) mm), the number of coil turns (\(N=1250\) turns), and the inner diameter (\(d=5\) mm). The experiments are performed 10 times and the averaged values are compared to the theoretical ones. The theoretical forces are calculated using Equation (7) for various numbers of coil  
Figure 6: Experimental apparatus: (**a**) the measurement of the output force by the solenoid actuator; (**b**) the displacement measurement of the segmented robot using an infrared sensor. turns, from 1100 to 1600 turns, at the constant input current (\(\pm\)0.6 A). The resistance of the proposed solenoid actuator is about 20 ohms. The experimental pushing and pulling forces are measured while changing the turns of wounded coils and the direction of the constant current. The electromagnetic force by the solenoid depends on the positions of the solenoid and the plunger. The theoretical results show that the pushing and pulling forces are the same at the constant input current. However, the experimental results indicate that the pushing and pulling forces are slightly different when the input current is reversed. The proportional dependence of the solenoid-induced force increasing on the coil turns is similar in both theory and experiments. The theoretical electromagnetic forces from Equation (9) and the experimental data are plotted in Figure 7 for various coil turns when the plunger positions are changed from \(z_{p}=-5\) to 5 mm at the center of the solenoid. Figure 7a-f represent the pushing and pulling forces for six cases of coil turns: 1100, 1200, 1300, 1400, 1500, and 1600 turns. Other variables are fixed for the experiments such as the current applied to the coil (\(i=0.6\) A), the diameter of coil (0.2 mm), the solenoid length (\(L=20\) mm), and the remanence of PM (4000 Gauss). The electromagnetic force from the solenoid also depends on the positions of the solenoid and the plunger. The dashed circle and dashed star lines in the graphs indicate the experimental pushing and pulling electromagnetic forces, respectively. As mentioned in Table 1, the theoretical pushing and pulling forces are the same. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No. of Turns of Coils**} & \multicolumn{2}{c}{**Theoretical Force (N)**} & \multicolumn{2}{c}{**Experimental Force (N)**} \\ \cline{2-5}  & **Pushing (\(\widehat{\mathcal{Z}}\))** & **Pulling (\(-\widehat{\mathcal{Z}}\))** & **Pushing (\(\widehat{\mathcal{Z}}\))** & **Pulling (\(-\widehat{\mathcal{Z}}\))** \\ \hline
1100 & 0.5644 & 0.5664 & 0.5335 & 0.5077 \\
1200 & 0.6104 & 0.6104 & 0.6036 & 0.5881 \\
1300 & 0.6554 & 0.6554 & 0.6392 & 0.6580 \\
1400 & 0.6996 & 0.6996 & 0.6557 & 0.6926 \\
1500 & 0.7429 & 0.7429 & 0.7082 & 0.7906 \\
1600 & 0.7854 & 0.7854 & 0.7659 & 0.8370 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The pulling and pushing electromagnetic forces from the solenoid actuator and the comparison of theoretical forces with experimental results for various coil turns.