## 2 Materials and Methods
### Optical Coordinate Measurement Machine Model  
The vision subsystem model of the optical coordinate measurement machine used in this paper is the so-called affine camera, in which the optical centre is located at an infinity point [17]. This model is used to model systems with telecentric lenses/lens systems [18], allowing for the transformation of the coordinates of a point in space (3D) called \"_coordinates of the world system_\" into the coordinates of a point of an image (2D) called \"_coordinates of the image system digitized_\". This inverse transformation is represented in the matrix form by:  
\[ \begin{array}{l}\mathbf{X}_{lm}^{t}=\mathbf{Q}^{t}\cdot\mathbf{U}_{psi}\\ \mathbf{X}_{lm}^{t}=\left[\mathbf{R}^{t}\mid\mathbf{t}\right]^{-1}\cdot\mathbf{ R}^{-1}\cdot\mathbf{U}_{psi}\end{array} \Bigg{\}}\quad\text{with}\quad\mathbf{K}^{-1}= \left( \begin{array}{ccc}1/\alpha_{x}&-s/\left(\alpha_{x}\cdot\alpha_{y}\right) &-\left(\alpha_{y}\cdot x_{0}+s\cdot y_{0}\right)/\left(\alpha_{x}\cdot\alpha_ {y}\right)\\ 0&1/\alpha_{y}&-y0/\alpha_{y}\\ 0&0&1\end{array} \right), \tag{1}\]where \(\mathbf{X}_{hm}^{\prime}=\left(x_{m},y_{m},1\right)^{T}\) are the coordinates of a point in the world system, and \(\mathbf{]}\) are the pixel coordinates of the digitized image system. \(\mathbf{R}^{\prime}\) is an orthogonal rotation matrix \(3\times 3\), and \(\mathbf{t}\) is a translation matrix \(3\times 1\). With these two matrixes, the position of the camera with respect to the world system is defined. \(s\) characterizes the possibility of the lack of perpendicularity between the axes, \(x_{0}\), \(y_{0}\) represent the image centre coordinates, \(\alpha_{x},\alpha_{y}\) model the camera pixels size along the \(X\)- and \(Y\)-axes, and \(\delta_{u^{\prime}},\delta_{v^{\prime}}\) represent the geometric distortion of the coordinates \(u^{\prime}\) and \(v^{\prime}\):  
\[ \begin{array}{l}\delta_{u^{\prime}}=\overbrace{k_{1}\Delta u^{\prime}\left( \Delta u^{\prime}{}^{2}+\Delta v^{\prime}{}^{2}\right)+k_{2}\Delta u^{\prime} \left(\Delta u^{\prime}{}^{2}+\Delta v^{\prime}{}^{2}\right)^{2}}+\overbrace{p_ {1}\left(3\Delta u^{\prime}{}^{2}+\Delta v^{\prime}{}^{2}\right)+2p_{2} \Delta u^{\prime}\Delta v^{\prime}{}}^{Decantizing}+\overbrace{s_{1}\left( \Delta u^{\prime}{}^{2}+\Delta v^{\prime}{}^{2}\right)}^{Min\ prism}\,\\ \delta_{v^{\prime}}=k_{1}\Delta v^{\prime}\left(\Delta u^{\prime}{}^{2}+\Delta v ^{\prime}{}^{2}\right)+k_{2}\Delta v^{\prime}\left(\Delta u^{\prime}{}^{2}+ \Delta v^{\prime}{}^{2}\right)^{2}+2p_{1}\Delta u^{\prime}\Delta v^{\prime}{} +p_{2}\left(\Delta u^{\prime}{}^{2}+3\Delta v^{\prime}{}^{2}\right)+s_{2}\left( \Delta u^{\prime}{}^{2}+\Delta v^{\prime}{}^{2}\right)\end{array}  \tag{2}\]  
where \(\Delta u^{\prime}=u^{\prime}-x_{0}\), \(\Delta v^{\prime}=v^{\prime}-y_{0}\) and \(k_{1},k_{2}\), \(p_{1}\), \(p_{2},s_{1},s_{2}\) represent the geometric distortion coefficients. By employing a fixed frequency grid distortion target, it is possible to calibrate the instrument vision subsystem [15] so that the traceability of the measures subsequently made by the equipment is ensured. The results of the calibration of a MITUTOY equipment, model Ultra-QV350 (measuring range (\(X\times Y\times Z\)): 350 mm \(\times\) 350 mm \(\times\) 150 mm, resolution: 0.01 \(\upmu\)m) available in the Centro Espanol de Metrologia (CEM) (Tres Cantos, Spain) are shown below. The distortion target images (dot diameter: 65 \(\upmu\)m and step: 125 \(\upmu\)m) have been obtained by using diascopic illumination at 20% of its nominal value, with a magnification of 10\(\times\), which determines a nominal pixel size of 0.89 \(\upmu\)m. The temperature during the measurement time is kept in the range 20 \(\pm\) 0.2 \({}^{\circ}\)C. Table 1 shows the results of the simulations performed for a number of replications \(M=10^{4}\) (the Monte Carlo method is used for determining the uncertainty associated with the model calibration parameters). Since the above variables have common input variables, there are correlated [15].