## 3 Results  
Following the two-step multi-objective optimization algorithm, several geometric parameters of the parallel robot for pancreatic surgery were optimized and further implemented into the constructive design of the robot. The instrument trajectories for the input values of the points, recorded by manipulating manually the surgical instrument with respect to an un-optimized RCM point given as input data for the optimization algorithm, are presented in Figure 5a,b. The figure presents the trajectories of the instrument using the same input points but constrain the instrument trajectory to the optimized RCM position. The points below the RCM position inside the patient are the same, but as can be observed, the RCM position is more central than the initial one. The initial position of the RCM was [100, 0, 300] and the optimized position of the RCM is [119.003337, \(-\)146.610801, 269.07376]. The result of the second step of optimization implied the determination of the optimal length for links a\({}_{1}\), a\({}_{3}\), and a\({}_{2}\)min. The optimal length for these links given by the genetic algorithm was 213.5455 mm for link a\({}_{1}\), 248.48918 for link a\({}_{3}\), and 48.5886 mm for link a\({}_{2}\)min. For constructive reasons, the values of these links have been rounded to 215 mm, 250 mm, and 50 mm, respectively, and a graphical simulation using Siemens NX was performed to test the validity of the design with the rounded length of the links. The result of the workspace simulation using optimized link lengths is presented in Figure 6. The input points from the genetic algorithm were integrated into a cone shape of yellow color in Figure 6 while the workspace of the instrument holder is represented with grey color. As can be observed, the tip of the instrument is not limited only to the input points, as it can go deeper in some cases. Using the values of the geometrical parameters optimized in the previous section, the constructive design of the parallel robot for laparoscopic pancreatic surgery is presented in Figure 7. The parallel robot is integrated within the operating room, and the patient is placed on the operating table, ready for the medical procedure. The robot has been equipped with a supplementary independent DOF to allow vertical sliding of the entire robot, thus providing the possibility to vertically displace the RCM of the passive mechanism. Each geometric parameter analyzed has been implemented into the constructive design of the robot. The integration method into the mechanisms of the robot is presented in Figure 8,  
Figure 5: Position of the instrument holder: (**a**) before the optimization process; (**b**) after the optimization. where the lengths of the components resulting from the genetic algorithm optimization were used. To provide better access of the instrument to the operating area, the sliding axis for q1 and q2 was placed as front-facing as possible to the passive RCM mechanism. To further validate the optimization of the design, a virtual simulation using the constructive design of the parallel robot for pancreatic surgery. The graphical simulation was performed using Siemens NX 2406. The starting point of the simulation was P1 (107, \(-\)102, \(-\)140) and the target point was P2 (158, \(-\)160, 190). The results of the simulation are  
Figure 8: Integration of the optimized parameters into the design of the robot. Figure 6: Parallel robot workspace after the optimization. Figure 7: The parallel robot integrated into the operating room. presented in Figure 9 for the coordinates at the tip of the instrument and in Figure 10 for the active joints of the parallel robot. In Figure 10, the first graphic represented with cyan color represents the displacement of the tip of the instrument on X, Y, and Z, the second graph represents the velocity (blue color) of the tip of the instrument on X, Y, and Z, and the third graph represents the acceleration (red color) of the tip of the instrument on the X, Y, and Z.