# Crawling Magnetic Robot to Perform a Biopsy in Tubular Environments by Controlling a Magnetic Field
## 4 Design and Fabrication of the CMR
### Design of the Biopsy Part  
To design the biopsy part, we considered the torque and force constraints between the CM, FM, RM, and external magnetic field during the uncovering motion, biopsy, and covering motion. To keep the cover closed during the crawling motion, as shown in Figure 5a, the magnetic torque exerted on RM should satisfy the following equation:  
\[\mathbf{T_{e}}+\mathbf{T_{23}}-\mathbf{T_{21}}<0 \tag{7}\]  
where \(\mathbf{T_{e}}\), \(\mathbf{T_{21}}\), and \(\mathbf{T_{23}}\) are the magnetic torque exerted on the RM by the external magnetic field, the torque exerted on the RM by the FM, and the torque exerted on the RM by the CM (magnet 3). In this situation, an attractive force between the RM and FM and a repulsive force between the RM and CM are generated to keep the cover closed. To perform an uncovering motion, we increase the external magnetic field, as shown in Figure 5b, in such a way, that the magnetic torque exerted on the RM satisfies the following equation:  
\[\mathbf{T_{e}}+\mathbf{T_{23}}-\mathbf{T_{21}}>0 \tag{8}\]  
Once Equation (8) is satisfied, the RM rotates 180\({}^{\circ}\) in the same direction as the external magnetic field, which generates the repulsive force between the RM and the FM. This repulsive force opens the cover, and the biopsy needle is exposed. Then, we apply the external precessional magnetic field, as shown in Figure 5c, to generate a crawling motion to puncture the tissue. After the needle is inserted into the tissue, we rotate the uniform magnetic field with respect to the y-axis by 180\({}^{\circ}\) to rotate the CMs. This rotates the direction of movement of the crawling body and flexible legs to the opposite direction; at the same time, the polarity of the RM changes, as shown in Figure 5d, due to the attractive forces between the CM and RM and between the RM and FM. However, the attractive force between the RM and FM should be greater than that between the CM and RM to close the cover as follows:  
\[\mathbf{F_{21}}-\mathbf{F_{23}}>0 \tag{9}\]  
where \(\mathbf{F_{21}}\) and \(\mathbf{F_{23}}\) are the magnetic attractive force exerted on the RM by the FM and the magnetic attractive force exerted on the RM by the CM (magnet 3). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Variable** & **Value** \\ \hline \multirow{4}{*}{Distance between magnets} & d\({}_{12}\) & 8.0 mm \\  & d\({}_{23}\) & 18.5 mm \\  & d\({}_{34}\) & 13.0 mm \\  & d\({}_{45}\) & 13.0 mm \\ \hline \multirow{4}{*}{Fixed cylindrical magnet} & Diameter & 3.0 mm \\  & Length & 3.0 mm \\  & Grade & N52 \\  & Magnetic moment & 0.0245 A-m\({}^{2}\) \\ \hline \multirow{4}{*}{Rotating ring magnet} & Outer diameter & 2.0 mm \\  & Inner diameter & 1.0 mm \\ \cline{1-1}  & Length & 0.8 mm \\ \cline{1-1}  & Grade & N52 \\ \cline{1-1}  & Magnetic moment & 0.0022 A-m\({}^{2}\) \\ \hline \multirow{4}{*}{Crawling magnet} & Diameter & 3.0 mm \\ \cline{1-1}  & Length & 6.0 mm \\ \cline{1-1}  & Grade & N52 \\ \cline{1-1}  & Magnetic moment & 0.0490 A-m\({}^{2}\) \\ \hline Flexible leg & Length & 25 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design variables of the CMR. After we designed the CMs, the spacing between CMs, the length of legs, and the minimum magnetic field for the crawling motion described in Section 4.1, we determined \(\mathrm{d}_{23}\) to be 18.5 mm, which includes the rotational space for the flexible leg to change the direction of movement (12.5 mm), the opening and closing space for the cover to perform the uncovering and covering motions (4 mm), and the thickness of the biopsy structure required for three-dimensional (3D) printing (2 mm). We designed the RM, the FM, and \(\mathrm{d}_{12}\) in such a way that Equations (7)-(9) are satisfied for the crawling motion with the safely covered needle, the uncovering motion to expose the needle, and the motion to cover the needle and collected sample, respectively.