## 4 Point Cloud Uncertainty Evaluation: Explicit Models
### Scale and Squareness Effects (S)  
From practical experience, it is well known that scale and squareness effects are a major component of CMM behaviour. Scale and squareness effects are special cases of a class of models in which the systematic effects in Equation (3) are taken to be functions \(\mathbf{e}_{i}=\mathbf{e}(\mathbf{x}_{i},\mathbf{b})\) of location \(\mathbf{x}\) and additional parameters \(\mathbf{b}=(b_{1},\dots,b_{p})^{\top}\) that model some aspect of CMM behaviour. The model below incorporates scale effects and three squareness effects through  
\[\mathbf{x}_{i}=B(\mathbf{b})\mathbf{x}_{i}^{*}, \tag{10}\]  
where  
\[B(\mathbf{b})=\left[ \begin{array}{ccc}(1+b_{aa}+b_{xx})&b_{xy}&b_{xz}\\ 0&(1+b_{aa}+b_{yy})&b_{yz}\\ 0&0&(1+b_{aa}+b_{zz})\end{array} \right]\]  
depends on effects \(\mathbf{b}=(b_{aa},b_{xx},b_{yy},b_{zz},b_{xy},b_{xz},b_{yz})^{\top}\). (The model as it stands assumes the slightly unrealistic case of a zero probe offset). The term \(b_{aa}\) models a global scale effect, while \(b_{xx}\), \(b_{yy}\) and \(b_{zz}\) model scale effects for each axis and \(b_{xy}\), \(b_{xz}\) and \(b_{yz}\) model the squareness effects. The \(3m\times 7\) sensitivity matrix \(G_{X|S}\) for this model is assembled from \(3\times 7\) matrices of the form  
\[G(\mathbf{x}_{i}^{*})=\left[ \begin{array}{cccccc}x_{i}^{*}&x_{i}^{*}&0&0&y_{i}^{ *}&z_{i}^{*}&0\\ y_{i}^{*}&0&y_{i}^{*}&0&0&0&z_{i}^{*}\\ z_{i}^{*}&0&0&z_{i}^{*}&0&0&0\end{array} \right]. \tag{11}\]  
In practice, \(\mathbf{x}_{i}^{*}\) is unknown but can be approximated accurately by the measured coordinate \(\mathbf{x}_{i}\), and the sensitivity matrix is approximated by  
\[G_{i}=G(\mathbf{x}_{i})=\left[ \begin{array}{cccccc}x_{i}&x_{i}&0&0&y_{i}&z_{i}& 0\\ y_{i}&0&y_{i}&0&0&0&z_{i}\\ z_{i}&0&0&z_{i}&0&0&0\end{array} \right]. \tag{12}\]  
The model is completed by specifying the variance matrix \(V_{B}\) associated with the scale and squareness effects, e.g.,  
\[V_{B}=\left[ \begin{array}{cccccc}\sigma_{S}^{2}&0&0&0&0&0&0\\ 0&\sigma_{S,x}^{2}&0&0&0&0&0\\ 0&0&\sigma_{S,y}^{2}&0&0&0&0\\ 0&0&0&\sigma_{S,z}^{2}&0&0&0\\ 0&0&0&0&\sigma_{Q}^{2}&0&0\\ 0&0&0&0&0&\sigma_{Q}^{2}&0\\ 0&0&0&0&0&\sigma_{Q}^{2}\end{array} \right]. \tag{13}\]  
If we assume that the individual axis scale effects are associated with the same variance so that \(\sigma_{S,x}^{2}=\sigma_{S,y}^{2}=\sigma_{S,z}^{2}=\sigma_{S,\sigma}^{2}\), the model is associated with four statistical hyperparameters \(\mathbf{\sigma}=(\sigma_{S},\sigma_{S,a},\sigma_{Q},\sigma_{R})^{\top}\). Given a data point \(\mathbf{x}_{i}\), the variance matrix \(V_{\mathbf{x}_{i}}=V_{\mathbf{x}_{i}}(\mathbf{\sigma})\), due to scale and squareness effects, is given by  
\[V_{\mathbf{x}_{i}}=\sigma_{S}^{2}\mathbf{x}_{i}\mathbf{x}_{i}^{\top}+\left[ \begin{array}[] {cccc}x_{i}^{2}\sigma_{S,a}^{2}+(y_{i}^{2}+z_{i}^{2})\sigma_{Q}^{2}&0&0\\ 0&y_{i}^{2}\sigma_{S,a}^{2}+z_{i}^{2}\sigma_{Q}^{2}&0\\ 0&0&z_{i}^{2}\sigma_{S,a}^{2}\end{array} \right].\]For a working volume of \([-L,L]^{3}\), the variance associated with \(x\)-coordinate point \(\mathbf{x}_{L}=(L,L,L)^{\top}\) is \(L^{2}(\sigma_{S}^{2}+\sigma_{S,a}^{2}+2\sigma_{Q}^{2})\), the maximum variance associated with any coordinate in the working volume. This maximum can be compared with statements of maximum permissible error. Over modest working volumes for this model, the variance matrix \(V_{X|S}\) associated with a set of coordinates is given by  
\[V_{X|S}=G_{X|S}V_{S}G_{X|S}^{\top}, \tag{14}\]  
where \(G_{X|S}\) is the \(3m\times 7\) sensitivity matrix constructed from \(G_{i}\), defined as in Equation (12), and \(V_{S}\) typically has the form in Equation (13).