## 2 Materials and Methods
### ETS Alignment  
Since CT0 and verification scans take place on different days, it is worth underlining that the main aim of this study was to avoid performing step #2 (ETS alignment) within the CT bunker, taking advantage of a more flexible optical-tracking configuration in a dedicated non-clinical area (Figure 2c,d). A schematic representation of the new workflow proposed in this study is depicted in Figure 6.  
Based on the optimal gaze direction provided by the treatment planning software, the ETS reference configurations were generated in Matlab (The Mathworks, Inc, Natick, MA, USA), as described by [PERSON] [18]. The experimental setup was meticulously prepared to ensure optimal reproducibility and precision to align the ETS. As described in Figure 6, this involved: (1) Proper orientation of three near-infrared cameras to optimize visibility of the ETS handling area. (2) Optimization of camera parameters (focus, zoom, diaphragm aperture) to enhance calibration outcome and minimize residuals. Figure 4: The ETS position used for CT0 planning image acquisition is shown. The fixation point is projected along the patient’s straight-forward gaze direction. Figure 5: Optimization of ETS positioning for the verification CT scan using the software application available in the clinic. In (**a**), the green dotted line represents the gaze direction; the blue dots are a virtualization of the markers; the blue area on the ocular region of the patient depicts the FoV projection of the ETS on the mask. In (**b**), the patient gaze is directed towards the fixation light reflected on the mirror. An example is shown with polar 28\({}^{\circ}\) and azimuth 90\({}^{\circ}\) (**b**). OTS features the 3D localization of passive, reflective markers through real-time processing and triangulation of the image data coming from the cameras [22; 23]. OTS optical calibration was performed using a wand-based method to estimate both intrinsic and extrinsic camera parameters. The calibration process was divided into two stages: static and dynamic. The static phase is performed by means of an orthogonal set of wands to establish a spatial coordinate system. In the dynamic stage, a single wand with three markers is moved within the calibration volume, to refine camera parameters and ensure comprehensive 3D motion-tracking coverage [24; 25; 26]. The required mapping among the native OTS reference system, with respect to which the 3D coordinates of the markers are reconstructed, and the CT/patient reference system were obtained by acquiring the ETS-mounted markers' positions after driving the robotic manipulator in the same configuration used for CT0 acquisition. As a result, the corrections estimated through optical tracking are executed by the robotic systems in a geometrically consistent way. Corrections of the ETS position are estimated from a point-based rigid registration between the current ETS fiducial configuration measured in real time by OTS and a pre-defined geometrical reference. As already described (step 2 of the clinical treatment simulation workflow), the ETS alignment is performed by driving the robotic manipulator until the misalignment between the current ETS-mounted markers and the reference position detected by the OTS is mini  
Figure 6: Workflow scheme—setup preparation and ETS position alignment with proposed method. Differences (highlighted in red box) between the current method in setup preparation and the use of a different OTS. To verify the accuracy between the two results, CT acquisition was performed to extract the coordinates of the center of radiopaque markers on the ETS surface for both configurations (standard method and proposed method). mized (Figure 7). The OTS detects the position of the ETS markers in real time and provides motion correction parameters (6 degrees of freedom (DOF)) for accurate alignment of the actual configuration against a predefined reference and indirectly of the fixation point to minimize misalignment between the ETS-mounted markers and the reference position detected by the OTS.