# Kinematic Modeling of a Trepanation Surgical Robot System
Important engineering issues to be considered in the practical implementation and use of parallel robots, in the various applications of parallel mechanisms, are above all the problems of determining the singular positions of the mechanism, as well as the selection of optimal design parameters of the links forming the kinematic chain, so as to obtain the largest possible working space [22; 23; 24]. However, most of these technical solutions are currently at the stage of laboratory prototypes or test or development platforms when it comes to medical applications. An example of such a solution is the hexapod functional prototype robot developed by The Fraunhofer-Institut fur Produktionstechnik und Automatisierung (IPA), Stuttgart, Germany, dedicated to microsurgery [25]. Another interesting solution is a novel skull-mounted robot with a compact and lightweight parallel mechanism for positioning surgical tools in minimally invasive neurosurgery [26]. An analysis of technical solutions based on a parallel mechanism for the RCM (remote centre of motion) of surgical robots, with possible application to neurological procedures, is also presented in reference [27]. An overview of the different solutions of neurosurgical robots that have been used and developed in the last 30 years has been described in references [28; 29]. A conceptual solution for a parallel robot to assist in cranial trepenation surgery is presented in this paper. The main motivation for the topic presented in this paper was the need to create a controllable, accurate robot to support the surgeon's activities in the operating theatre, with a kinematic structure as light and rigid as possible, with the possibility of attaching it to the operating table, in order to increase the accuracy of trepenation openings performed during neurological surgical procedures. The robotic structure developed should allow the future integration of the developed robotic mechanism into medical imaging systems based on MRI and CT so that the surgical tool can be precisely positioned and thus have the smallest possible size after the procedure to close the holes through the bone wax. The steps taken to create a new parallel robotic solution to support selected neurosurgical procedures, in particular cranial trepenation, are mainly due to the fact that the source analysis carried out on existing technical solutions on the market and prototype service robotic solutions in the respective area of interest showed that this area of application is practically \"undeveloped\" scientifically, except for the solution presented in reference [25]. The organization of the contents of this paper is as follows: (1) the first section (Section 1) is an introductory chapter on the application of the trepenning robot technology discussed; (2) the next section (Section 2.1) describes the design of a parallel manipulator for positioning a trepenation tool in the cranial region of the patient; this section also describes in detail the components of the manipulator and the advantages of the use of closed-loop manipulators for accurate tool positioning; (3) the next section (Section 2.2) presents the mathematical analytical equations describing the inverse kinematics task of a parallel manipulator; (4) a forward kinematics task for a manipulator is solved analytically in the fourth section (Section 2.4); (5) in the next section (Section 3.1), the workspace analysis of the robot system is performed, and (6) the last section (Section 4) is a summary of the results obtained, defining, among other things, the areas for further development and future research of a parallel robot dedicated to assisting in cranial trepenation procedures.