本研究旨在改善皮穿刺腫瘤消融手術中手臂空間與影像轉換的關係，以提高手術過程中針對手臂特定姿態下的操作準確性。在手術過程中，我們使用術中電腦斷層掃描所得的手臂末端點和影像標記點，來計算手臂空間與影像空間的轉換。然而，由於影像中的點是人工標記的，可能導致轉換關係產生誤差。因此，本研究採用蒙地卡羅演算法來尋求更小的誤差，以優化轉換關係。演算法的目標即為尋找最佳的轉換參數，使得誤差最小化。
在論文中，我們首先在影像軟體中標註三個點和手臂空間中的三個點，以符合卡布希演算法(Kabsch Algorithm)最少資料輸入的條件，進而計算出兩座標間的轉換關係，並評估轉換前後的誤差。接著，我們使用蒙特卡羅演算法(Monte Carlo Algorithm)為基礎，將誤差視為演算法的參數。利用機器學習的概念，調整學習率大小和迭代次數，以獲得最準確的轉換關係。然後，我們建立了導引套管和機械手臂的模擬模型。根據影像引導或操作者需求，使用羅里格斯旋轉公式(Rodrigues Rotation Formula)計算適合操作的旋轉角度，並進一步計算移動至病灶點所需的距離，從而獲得最終的針對操作姿態。最後，我們將這些姿態應用於遠端運動中心軌跡(Remote Center motion)或五次插值多項式軌跡規劃(Quintic Interpolation Polynomial Trajectory Planning)中，並計算不同姿態下的誤差。
最終，我們進行了MATLAB模擬並前往新竹竹北醫院驗證演算法的可行性。同時，我們詳細描述了整合系統的流程，並進一步比較了這套系統與傳統使用氣動臂進行消融手術之之間的差異。

The aim of this research is to improve the relationship between arm space and image transformation in percutaneous tumor ablation surgery, with the goal of enhancing the accuracy of operations in specific arm postures during the procedure. In the surgical process, we utilize the arm's endpoint obtained from intraoperative computed tomography scans and image-marked points to calculate the transformation between arm space and image space. However, as the points in the image are manually marked, it may introduce errors in the transformation relationship. Therefore, this study employs the Monte Carlo algorithm to minimize the errors and optimize the transformation relationship. The algorithm's objective is to find the optimal transformation parameters to minimize the errors.
In the paper, we first mark three points in the image software and three points in arm space to satisfy the minimal data input requirements of the Kabsch algorithm. Subsequently, we calculate the transformation relationship between the two coordinates and evaluate the errors before and after the transformation. Next, we use the Monte Carlo algorithm as the basis, treating the errors as parameters. Leveraging concepts from machine learning, we adjust the learning rate and iteration times to obtain the most accurate transformation relationship. Then, we establish simulation models for guiding sheaths and robotic arms. Depending on image guidance or operator requirements, we compute suitable rotational angles for manipulation using the Rodrigues Rotation Formula and further calculate the distance required to move to the lesion point, thereby obtaining the final operating posture. Finally, we apply these postures to remote center motion or quintic interpolation polynomial trajectory planning and calculate the errors under different postures.
In conclusion, we conducted MATLAB simulations and verified the feasibility of the algorithm at Zhubei Hospital in Hsinchu. Additionally, we provided a detailed description of the system integration process and further compared this system with the traditional use of pneumatic arms for ablation surgery.

[1] T. C. Huber, T. Bochnakova, Y. Koethe, B. Park, and K. Farsad, "Percutaneous Therapies for Hepatocellular Carcinoma： Evolution of Liver Directed Therapies," Journal of Hepatocellular Carcinoma, vol. Volume 8, pp. 1181-1193, 2021, doi： 10.2147/jhc.s268300.
[2] A. Gupta, B. Musaddaq, C. von Stempel, and S. Ilyas, "Percutaneous Renal Ablation," in Seminars in Ultrasound, CT and MRI, 2020, vol. 41, no. 4： Elsevier, pp. 351-356.
[3] A. Páez-Carpio et al., "Image-guided percutaneous ablation for the treatment of lung malignancies： current state of the art," Insights into Imaging, vol. 12, no. 1, 2021, doi： 10.1186/s13244-021-00997-5.
[4] C.-H. Wu et al., "Iodized oil computed tomography versus ultrasound-guided radiofrequency ablation for early hepatocellular carcinoma," Hepatology International, vol. 15, no. 5, pp. 1247-1257, 2021, doi： 10.1007/s12072-021-10236-0.
[5] Y. S. Kwoh, J. Hou, E. A. Jonckheere, and S. Hayati, "A robot with improved absolute positioning accuracy for CT guided stereotactic brain surgery," IEEE transactions on biomedical engineering, vol. 35, no. 2, pp. 153-160, 1988.
[6] P. Hlivák, H. Mlčochová, P. Peichl, R. ČIHÁK, D. Wichterle, and J. Kautzner, "Robotic navigation in catheter ablation for paroxysmal atrial fibrillation： midterm efficacy and predictors of postablation arrhythmia recurrences," Journal of cardiovascular electrophysiology, vol. 22, no. 5, pp. 534-540, 2011.
[7] D. Stoianovici et al., "Acubot： a robot for radiological interventions," IEEE Transactions on Robotics and Automation, vol. 19, no. 5, pp. 927-930, 2003, doi： 10.1109/tra.2003.817072.
[8] A. Melzer et al., "INNOMOTION for Percutaneous Image-Guided Interventions," IEEE Engineering in Medicine and Biology Magazine, vol. 27, no. 3, pp. 66-73, 2008, doi： 10.1109/emb.2007.910274.
[9] S. Groetz, K. Wilhelm, W. Willinek, C. Pieper, H. Schild, and D. Thomas, "A new robotic assistance system for percutaneous CT-guided punctures： Initial experience," Minimally Invasive Therapy & Allied Technologies, vol. 25, no. 2, pp. 79-85, 2016, doi： 10.3109/13645706.2015.1110825.
[10] J. J. Craig, Introduction to robotics： mechanics and control. Pearson Educacion, 2005.
[11] Rasmus Skovgaard Andersen： Kinematics of a UR5. Aalborg University, 2018.5.31.
[12] M. W. Spong and M. Vidyasagar, Robot dynamics and control. John Wiley & Sons, 2008.
[13] Z. Yun-ping and Z. Fan, "A Novel Remote Center-of Motion Parallel manipulator for Minimally Invasive Celiac Surgery," International Journal of Research in Engineering and Science (IJRES) ISSN (Online), pp. 2320-9364.
[14] N. Aghakhani, M. Geravand, N. Shahriari, M. Vendittelli, and G. Oriolo, "Task control with remote center of motion constraint for minimally invasive robotic surgery," in 2013 IEEE international conference on robotics and automation, 2013: IEEE, pp. 5807-5812.
[15] M. M. Marinho, K. Harada, and M. Mitsuishi, "Comparison of remote center-of-motion generation algorithms," in 2017 IEEE/SICE International Symposium on System Integration (SII), 2017: IEEE, pp. 668-673.
[16] T. R. Inc, Software-Mannual-TMflow. Taoyuan,Taiwan: TM Robot., 2020.
[17] D. S. Community, 3D Slicer Documentation. America: National Institutes of Health, 2022.
[18] W. Kabsch, "A solution for the best rotation to relate two sets of vectors," Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography, vol. 32, no. 5, pp. 922-923, 1976.
[19] K. S. a. H. Arun, T. S. and Blostein, S. D, "Least-Squares Fitting of Two 3-D Point Sets," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. Volume 9 Issue 5, May 1987.
[20] R. RASALA, "The Rodrigues Formula andPolynomial Differential Operators," JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, pp. 443-482, 1981.