## 3 Experimental Results
### In Vivo Experiment
The \(\mathbf{P}_{iso}\) was located at a point 696.0 mm away from the source of the C-arm, and the actual length of 1 pixel was 0.262 mm when the SID was at a maximum of 1250 mm. The rotation angle of the C-arm was displayed through the control program at a resolution of 1\({}^{\circ}\). During the procedure, we set the SID to the maximum configuration (1250 mm), and we selected \(V_{n\!rf}\) as the AP view and \(V_{corr}\) as the RAO 15\({}^{\circ}\) view (\(\mathbb{A}_{15^{\circ},0^{\circ}}\)) for the right SFA and the LAO 15\({}^{\circ}\) view (\(\mathbb{A}_{-15^{\circ},0^{\circ}}\)) for the left SFA to avoid collision between the C-arm and the RAMAN system. The MMR utilized in this experiment was a separable and recombinable magnetic robot (SRMR) [24] that can be separated and recombined with a delivery catheter, and the mini pig was a Yucatan miniature. Before the main procedure, the SRMR attached to the delivery catheter was inserted near the target lesion by the feeding device. Then, the C-arm was rotated by RAO 15\({}^{\circ}\) (\(\mathbb{A}_{15^{\circ},0^{\circ}}\)) or LAO 15\({}^{\circ}\) (\(\mathbb{A}_{-15^{\circ},0^{\circ}}\)), and a contrast agent was injected into the vessels to capture \(I_{corr}\). The C-arm was then rotated to the AP view, and a contrast agent was injected again to capture \(I_{n\!rf}\). The pixel coordinates of the four corners were detected using the vision assistant module of LabVIEW from the captured \(I_{n\!rf}\) and \(I_{corr}\). Then, the relative rotational angles between the two views (\(\mathbb{A}_{\theta,\,\,\varphi}^{opt}\)) were calculated utilizing the proposed method in Section 2.4,  
Figure 6: Experimental environments for the in vivo experiment. (**a**) Operating room; and (**b**) control room. Figure 7: Corners of the upper electromagnets of the RAMAN system utilized as corresponding points: (**a**) side view; and (**b**) top view. and were \(\mathbb{A}_{15^{\circ},0^{\circ}}^{opt}=(16.3^{\circ},-0.7^{\circ})\) and \(\mathbb{A}_{-15^{\circ},0^{\circ}}^{opt}=(-16.5^{\circ},0^{\circ})\), and the average distance errors between the four corresponding points were 0.341 mm and 0.324 mm, respectively. Also, 3D coordinates of the corresponding points were calculated in \(C_{\textit{ref}}\). Subsequently, the contour of the vessel where an SRMR will be operated was manually selected from \(I_{\textit{ref}}\) and \(I_{\textit{corr}}\), and the centerlines of the blood vessel were enhanced by skeletonization in \(I_{\textit{ref}}\) and \(I_{\textit{corr}}\). Finally, the 3D coordinates of a blood vessel's centerline were calculated in the \(C_{\textit{ref}}\) using the calculated relative rotation angles and the method proposed in Section 2.5 and were then transformed into the 3D coordinate system of the MNS workspace by considering the calculated coordinates of the four corresponding points. Once two 2D X-ray images are ready, it takes 18.7 s in a computer with an Intel Core i7-11700 KF CPU (Intel, Santa Clara, CA, USA), 32 GB DDR4 RAM, and an NVIDIA GeForce GTX 760 GPU (NVIDIA, Santa Clara, CA, USA) to compute the 3D coordinates of 596 points. Figure 8 shows the X-ray images of right SFA of the mini pig and the 3D plot of corresponding points and the centerline of the SFA calculated using the method proposed in Section 2.5. The C-arm was maintained in the AP view during the rest of the procedure. To control the SRMR inserted into the vessel, the current \(\mathbf{I}_{\text{opt}}\) needs to be calculated and supplied to the electromagnets of the RAMAN system to generate the required magnetic field (\(\mathbf{B}\)) and magnetic force (\(\mathbf{F}\)). The RAMAN system generates a nonuniform magnetic field within the workspace and controls the magnetic field depending on the position \(\mathbf{P}\) in the 3D coordinate system of MNS workspace. When the SRMR is positioned at a specific location \(\mathbf{P}\), the magnetic field and force can be calculated using the following equation [25]:  
\[ \begin{bmatrix}\mathbf{B}(\mathbf{P})\\ \mathbf{F}(\mathbf{m},\mathbf{P})\end{bmatrix}