## 1 Motivation  
The authors' scientific research aim is to develop numerical and experimental methods to design a surgical robot with a remote center of motion (RCM) mechanism that is free of mechanical vibrations and has the necessary strength stability for soft tissue surgery. A key consideration is the robot's functionality, which refers to its ability to operate on cardiovascular tissue in a manner acceptable to cardiac surgeons, including performing procedures, like coronary aortic anastomosis, that are difficult to perform with classical techniques. For soft tissue surgery, the robot must also ensure the required speed of tools, the smoothness of their movement, positioning accuracy, and repeatability, which are crucial for the robot's usefulness in the minimally invasive robotic surgery technique (MIRS) [1; 2; 3]. Achieving positive results on these technical issues would enable the submission of a surgical robot prototype for animal testing and eventually human testing. Currently, teleoperators (controller-responder structures) are used for soft tissue surgery. The surgeon's movement instructions are transmitted through the controller and converted into the movement of the responder, an endoscopic surgical instrument in direct contact with the cardiovascular tissue, after making small incisions in the chest. In this study, the authors analyzed the structure of a surgical robot with a RCM that moves in a spherical manner after the endoscopic tool passes through the surgical port, or trocar, which is placed in the outer shells of the human body during (MIRS) surgery. The authors' goal is to determine the optimization model, taking into account relevant physical criteria and specifying the constraints for decision variables. The decision variables are defined as geometric dimensions of abstracted links because they affect the criteria adopted. The discussed mechanical system can be observed in the clinically used da Vinci or Robin Heart surgical robots, which allows the operation to be performed in a minimally invasive manner. This structure is not based on the open kinematic chain found in the mechanisms of industrial robots. It consists of rigid links connected in joints forming parallel systems, enabling the retrieval of constant point kinematics. The structure is illustrated in Figure 1, together with the resulting shape of the working space. The authors validated the correctness of Formula (1) using a computer aid design (CAD) model. Due to the spherical shape of the working space, as shown in Figure 1, it is convenient to describe it using a triple integral in spherical coordinates, which allows for the calculation of the volume of the working space using Formula (1). The integral depends on the connector variables of the RCM mechanism: \(\psi_{1},\psi_{2},\delta\). \[|V|=4\int\limits_{0}^{\frac{\psi_{1}}{2}}d\psi_{1}\int\limits_{\psi_{2}}^{\frac {\pi}{2}}sin(\psi_{2})d\psi_{2}\int\limits_{0}^{\delta}\delta^{2}d\delta \tag{1}\]  
where \(\psi_{1},\psi_{2},\delta\)--RCM link variables. Before manufacturing the prototype of the surgical robot with the mechanism (RCM) shown in Figure 1, it is necessary to determine the optimal values for the relevant strength criteria (such as natural frequency, buckling coefficient, static and dynamic safety coefficient, and effector displacement) using an established method with a known mathematical basis that applies to the mechanical structure, assuming that forces and displacements maintain physical relationships for the selected construction material.