## 2 Materials and Methods
### Data Analysis and Statistics  
Spatial coordinates of the maximal excursion were manually identified on-screen by the operator recording the session, by marking the point, which coordinates are stored in the program and used for the further steps of the analysis. This operation leads to variability in the results of the session (Intra-operator reliability). An automatic detection method for the maximum excursion can be developed in future following examples available in the literature [13]. The position of the inter-incisal edge point (IEP) was derived from the position of the markers in space by knowing the relationship between the Mandibular Tracking Support (MTS) and the teeth, measured by the 3D-scan of the dental resin models with mounted external support.  
Figure 3: Acquisition set-up. The volunteer is sitting straight facing the cameras with the head in a neutral position and he is wearing the cranial and mandibular marker supports.  
A dedicated custom-made algorithm automatically detected on-line the measurements (mm) between the identified spatial coordinates of the lower central incisors inter-incisal edge point (IEP) at the reference limit points defined as:  
1. The mandibular opening difference between IEP coordinates including vertical over-bite during the active opening.
2. The maximum extent of the lateral mandibular movement difference between IEP coordinates during active lateral excursions (right and left).
3. The maximal retrusion difference between IEP during active retrusion.
4. The maximal protrusion difference between IEP during active retrusion corrected by including the horizontal overbite [22].  
All measurements were relative to the cranial reference-the volunteer's head and neck movements were subtracted from the raw mandibular movement using three cranial markers and the reference zero point using an application of the original mathematical algorithms by [PERSON] et al. [25].  
For the calculation of the mandibular motion relative to the head, a series of mathematical transformations were applied to the collected data. The raw data were recorded in the fixed coordinate system \(W\), represented by the camera set-up. In this coordinate system, the head-related and mandible-related systems were defined by the spatial position of the Cranial Tracking Support CTS and the spatial position of the Mandibular Tracking Support MTS. The absolute movement of the head in the coordinate system of the cameras W was described by the transformation matrix \({}^{W}\)H\({}_{CTS}\) and the absolute movement of the mandible was represented by the transformation matrix \({}^{W}\)H\({}_{MTS}\). To obtain the relative motion of the mandible with respect to the head, the transformation (1) was applied to all the coordinates of the mandibular points.  
\[{}^{CTS}\mathrm{H}_{MTS}=(^{W}\mathrm{H}_{CTS})^{-1}\cdot{}^{W}\mathrm{H}_{ MTS} \tag{1}\]  
The mandibular range of motion at the interincisor point was calculated starting from the maximum y value for the mouth opening, x values (right and left) for the maximum lateral movement, and z values for the retromandibular and protusive movements. Calculations were performed after 0.10 mm of movement as the precision of the instrument was set at 0.05 mm.