## 2 Materials and Methods  
### Parallel Robot for Laparoscopic Pancreatic Surgery  
The concept of the parallel robot designed for laparoscopic pancreatic surgery is presented in Figure 1[21]. The kinematic model of the robot has been previously presented in [22]; thus, only the final equations of the inverse geometric model are presented here in Equation (1). \[ \begin{cases}q_{1}=&Y_{p}-a_{02}-\sqrt{-\left(X_{p}+a_{01}\right)^{2}-\left(Z _{p}+a_{03}\right)^{2}+a_{1}^{2}-a_{4}^{2}+2\sqrt{\left(X_{p}a_{4}-a_{01}a_{4} \right)^{2}+\left(Z_{p}a_{4}-a_{03}a_{4}\right)^{2}}}\\ q_{2}=&Y_{p}-a_{02}+\sqrt{-\left([PERSON]_{p}+a_{01}\right)^{2}-\left([PERSON]_{p}+a_{03} \right)^{2}+a_{1}^{2}-a_{4}^{2}+2\sqrt{\left([PERSON]_{p}a_{4}-a_{01}a_{4}\right)^{2 }+\left([PERSON]_{p}a_{4}-a_{03}a_{4}\right)^{2}}}\\ q_{3}=&-a_{2\text{min}}-a_{5}+1/2(-4a_{4}\cos(\lambda)\sqrt{4{a_{3}}^{2}-\left( q_{1}+q_{2}\right)^{2}}-8X_{p}a_{4}\cos(\lambda)+8a_{01}a_{4}\cos(\lambda)\\ &-8\sin(\lambda)Z_{p}a_{4}+8\sin(\lambda)a_{03}a_{4}+4X_{p}\sqrt{4{a_{3}}^{2}- \left(q1+q2\right)^{2}}-4a_{01}\sqrt{4{a_{3}}^{2}-\left(q1+q_{2}\right)^{2}} \\ &+4{X_{p}}^{2}-8X_{p}a_{01}+4{Z_{p}}^{2}-8Z_{p}a_{03}+4{a_{01}}^{2}+4{a_{03}}^{ 2}+4{a_{4}}^{2}-(q1+q2)^{2})^{1/2}\end{cases}  \tag{1}\]The parallel robot consists of two main modules that work together to support and manipulate the surgical instrument. The first module is a 3-DOF active parallel mechanism, while the second is a passive spherical mechanism that constrains the Remote Center of Motion (RCM) of the instrument. The active module is designed with two passive universal joints (U1P and U2P), three actively controlled prismatic joints (q1, q2, and q3), and a set of nine passive revolute joints (R1P through R9P). A global coordinate system is defined at the RCM, which serves as the pivot for the instrument, and an additional coordinate system (OXYZ) is fixed directly to the robot body. The critical geometric parameters of the system include the link lengths (a1 through a5), the minimum stroke (a2 min) of the prismatic joint q3, as well as the instrument's total length (a) and the inserted portion length (l_ins) relative to the RCM. Complementing the active module, the spherical mechanism is designed with five passive revolute joints (R1S through R5S) and one passive cylindrical joint (C1S). This arrangement enables not only the rotational movements needed to adjust the instrument but also facilitates its insertion during surgery. The spherical mechanism is connected to the active parallel robot via a linkage that features two additional spherical joints (S1 and S2). These joints are crucial for fine-tuning the positioning and orientation of the passive module, ensuring the mechanism can be appropriately aligned with respect to the active robot. Together, the coordination of these modules enables enhanced accuracy and dynamic control in surgical applications. The robot uses the pivot point of the passive mechanism as the RCM (the trocar inserted into the patient); from this point the instrument can only perform three motions: insertion of the instrument along its longitudinal axis, as well as rotations around X and Y axes. The passive spherical mechanism supplements the active system by enforcing a strict RCM constraint, thereby decoupling the translational and rotational motions. This dual-module approach allows for enhanced dexterity and precision. Figure 1: The kinematic scheme of the parallel robot for laparoscopic pancreatic surgery.