# Development of an Augmented Reality Surgical Trainer for Minimally Invasive Pancreatic Surgery
## 2 Materials and Methods
### ATHENA Parallel Robot for Minimally Invasive Pancreatic Surgery  
Designed as an assistant for the surgeon, the ATHENA robotic system provides precise control for accurate simulation of surgical procedures (Figure 2). It can act as an organ retractor or handle any other active surgical instrument used as a third surgical tool (e.g., the endoscopic camera), adjusting its orientation and position according to the requirements of the primary surgeon. The ATHENA robot [14] has a modular architecture consisting of two parallel modules (Figure 3). The first module is a parallel passive Spherical Mechanism (SM) with an architecturally constrained Remote Center of Motion (RCM), ensuring that the surgical instrument always passes through the RCM. It has four degrees of freedom (DOFs): three orientations of the surgical instrument relative to the fixed coordinate system OXYZ (identical with the RCM) and one translation along the axis of the surgical instrument for its insertion and retraction, which is possible since one of the revolute joints of the SM was replaced with a cylindrical one, namely \(C_{18}\). The other revolute joints of the SM are \(R_{ss}\), \(i=1.4\). Figure 2: ATHENA parallel robot for minimally invasive surgery in the medical environment. The second module is an active 4-DOF Parallel Mechanism (PM) with three kinematic chains. The first one consists of the \(q_{1}\) and \(q_{2}\) active prismatic joints connected to the links \(l_{1}\) through the \(U_{1P}\) and \(U_{2P}\) universal joints. The \(l_{1}\) links are connected through the \(R_{9P}\) passive revolute joint. The second parallel kinematic chain consists of the same \(q_{1}\) and \(q_{2}\) active joints connected to the links \(l_{3}\) through the \(R_{9P}\) and \(R_{7P}\) revolute prismatic joints. The \(l_{3}\) links are connected through the \(R_{4P}\) passive revolute joint. This kinematic chain is connected to the third kinematic chain through the revolute joint \(R_{5P}\). It is a serial kinematic chain constating of the following joints: the active revolute joint \(q_{3}\) and the passive revolute joints \(R_{3P}\) and \(R_{8P}\) connected through the \(l_{2}\) links. The surgical instrument is connected to the 4-DOF parallel module through a universal joint consisting of two passive revolute joints (\(R_{1P}\) and \(R_{2P}\)) intersecting in point P, as the center of the revolute joint \(R_{1P}\). The parallel robot controls the surgical instrument, which has 4 DOFs, namely the orientations \(\psi\), \(\theta\), and \(\varphi\) and the insertion length \(l_{ins}\) or the coordinates of point \(E(X_{E},Y_{E},Z_{E})\) and the orientation angle \(\varphi\). These sets of coordinates are constrained by the following equation:  
\[\left\{ \begin{array}{l}\psi=\text{atan2}(Y_{E},X_{E})\\ \theta=\arccos\left(Z_{E}/\sqrt{{X_{E}}^{2}+{Y_{E}}^{2}+{Z_{E}}^{2}}\right)\\ l_{ins}=\sqrt{{X_{E}}^{2}+{Y_{E}}^{2}+{Z_{E}}^{2}}\end{array} \right. \tag{1}\]  
Besides the geometrical parameters of the robot (\(l_{1}\)-\(l_{4}\) and the surgical instrument's length \(l\)), the coordinates of the mobile coordinate system \(\alpha\)_xyz_ in the fixed coordinate system \(OXYZ\) (\(l_{01}\), \(l_{02}\), and \(l_{03}\)) are required. The coordinates of point \(P\) are as follows:  
\[\left\{ \begin{array}{l}X_{P}=\cos(\psi)\sin(\theta)(l-l_{ins})\\ Y_{P}=\sin(\psi)\sin(\theta)(l-l_{ins})\\ Z_{P}=\cos(\theta)(l-l_{ins})\end{array} \right. \tag{2}\]  
The input-output equations of the ATHENA robot are presented in Equation (3), from where the inverse kinematics (the active joints \(q_{1}\), \(q_{2}\), \(q_{3}\), and \(q_{4}\)) is derived:  
Figure 3: ATHENA parallel robot kinematic scheme. \[\left\{ \begin{array}{l}f_{1}:Y_{P}+l_{5}-(q_{2}+q_{2})/2=0\\ f_{2}:(q_{2}-q_{1}/2)^{2}+t_{1}-l_{1}^{2}=0\\ f_{3}:((g_{1}-l_{4})\cdot\sin(g_{3})+g_{2}+(g_{1}-l_{4})\cdot\cos(g_{3})-l_{2} \cdot sin(q_{3}))^{2}-l_{2}^{2}=0\\ f_{4}:q_{4}-\varphi=0\end{array} \right. \tag{3}\]  
where  
\[