# Kinematic Modeling of a Trepanation Surgical Robot System
## 4 Conclusions  
The technical solution adopted in this paper is a prototype version of a robot with a series-parallel structure. The main task, in this case, was to verify the hypotheses and test the validity of the technical solution adopted. Further development of the project will focus on changing the construction materials and the way in which they are joined to those that will ensure adequate rigidity of the structure and compatibility with medical applications. Another area of research will be the selection of accurate, safety-compliant medical equipment in direct contact with the patient, linear electric drives with higher accuracy, and absolute encoders made for use in operating-room conditions. The method of secure and stable attachment of the robot to the operating table and floor in the prototype robot solution has not been finally resolved and will be the subject of further development of the proposed design. The current stage of development of the prototype of a trepanning robot makes it possible to test the engineering assumptions adopted and to lay a solid foundation for the development of control and safety systems based on the solution of a forward and inverse kinematics task. The study of the dynamics of this type of robot is usually neglected due to the very low dynamics of the displacements (low speed of the drives) of the parts of the robot that support the surgical procedures. Basically, the robot remains in a static position once it has reached a certain position. This is to provide support and guidance for the trephe drill. An important direction for future research into the design of the robot and its use in the operating theatre will be the development of a human-machine interface (HMI) that allows medical staff to easily use the robotic tool for specific medical procedures while integrating medical imaging from MRI and CT sources.  
Figure 8: Analysis of the manipulability index of the tripod end-effector. (**a**) Slice of the manipulability index \(w(q_{2},\,q_{3})\) for \(q_{1}=0.14686\). (**b**) Slice of the manipulability index \(w(q_{1},\,q_{3})\) for \(q_{2}=0.14686\). (**c**) Slice of the manipulability index \(w(q_{1},\,q_{2})\) for \(q_{3}=0.14686\).  
**Author Contributions:** Conceptualization, A.W., R.T., V.M. and K.M.; methodology, A.W., R.T. and V.M.; validation, A.W., R.T., V.M. and K.M.; formal analysis, A.W., R.T., V.M. and K.M.; investigation, A.W., R.T., V.M. and K.M.; resources, R.T.; writing--original draft preparation, A.W., R.T. and V.M.; writing--review and editing, A.W., R.T., V.M. and K.M.; visualization, A.W. and R.T.; supervision, A.W. and R.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was performed within the framework of statutory research of the Department of Automatic Control and Robotics and the Department of Robotics and Mechatronics at the Bialystok University of Technology and financed with funds from the Ministry of Science and Higher Education, Poland.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the funding agency policy.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Symbols and Abbreviations**  
The following symbols and abbreviations are used in this manuscript:  
\begin{tabular}{l l} MRI & magnetic resonance imaging \\ CT & computer tomography \\ IPA & The Fraunhofer-Institut fur Produktionstechnik und Automatisierung \\ RCM & temote centre of motion \\ FK & forward kinematics \\ IK & inverse kinematics \\ EAA & equivalent axis-angle \\ D-H & Denavit-Hartenberg (parameters) \\ MC & Monte Carlo \\ \(q_{1}\), \(q_{2}\), \(q_{3}\), \(q_{4}\) & configuration variables for joints 1 to 4 \\ \(q_{21}\), \(q_{22}\), \(q_{23}\), \(q_{31}\), \(q_{32}\), \(q_{33}\) & additional configuration variables for the redundant joints of the \\ \(\{