## 5 Point Cloud Uncertainty Evaluation: Gaussian Process Error Models
### Gaussian Process Models for Rotation Errors (Er)  
The GP models in Section 5.3 used, perhaps, with a simple parametric error model can simulate a wide range of plausible CMM behaviour, but it relates only to one probing configuration and does not, without modification, allow us to evaluate the uncertainties associated with different probe configurations. An extension of the model is to use GP models to model both the location and rotation errors:  
\[\mathbf{x}_{i}=\mathbf{x}_{i}^{*}+\mathbf{e}_{i}+R(\mathbf{\alpha}_{i})\mathbf{p}+\mathbf{e}_{i}, \tag{22}\]  
where \(\mathbf{\alpha}_{i}=(\alpha_{i,x},\alpha_{i,y},\alpha_{i,z})^{\top}\) represents three spatially correlated rotation errors acting on the probe offset vector \(\mathbf{p}\) through the rotation matrix \[R(\mathbf{\alpha}_{i})=R_{z}(\alpha_{i,z})R_{y}(\alpha_{i,y})R_{x}(\alpha_{i,x}), \tag{23}\]  
the product of rotations about each of the three coordinate axes:  
\[R_{x}(\alpha_{x})=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\alpha_{x}&-\sin\alpha_{x}\\ 0&\sin\alpha_{x}&\cos\alpha_{x}\end{array} \right],\quad R_{y}(\alpha_{y})= \left[ \begin{array}{ccc}\cos\alpha_{y}&0&\sin\alpha_{y}\\ 0&1&0\\ -\sin\alpha_{y}&0&\cos\alpha_{y}\end{array} \right],\]  
and  
\[R_{z}(\alpha_{z})=\left[ \begin{array}{ccc}\cos\alpha_{z}&-\sin\alpha_{z}&0 \\ \sin\alpha_{z}&\cos\alpha_{z}&0\\ 0&0&1\end{array} \right].\]  
We assume that the rotational effects about one axis are independent from the rotational effects about the other two axes, but other more general approaches are possible. For measurements involving multiple probes, the degree of spatial correlation associated with \(\mathbf{\alpha}_{i}\) and \(\mathbf{\alpha}_{j}\) depends on  
\[\|\mathbf{x}_{i}^{*}-\mathbf{x}_{j}^{*}\|\doteq\|(\mathbf{x}_{i}-\mathbf{p}_{k(i)})-(\mathbf{x}_ {j}-\mathbf{p}_{k(j)})\|.\]  
If the rotational errors are similar along each axis, then the GP model for the rotational errors is specified by the two statistical hyper-parameters \(\sigma_{ER}\), and \(\lambda_{ER}\). The variance associated with location errors for any coordinate is \(\sigma_{ET}^{2}\). If the maximum probe length is \(P\), then the maximum variance associated with rotational effects for a coordinate is given by \(P^{2}\sigma_{ER}^{2}\). Hence, the maximum variance associated with location and rotation errors for a coordinate is given by \(\sigma_{ET}^{2}+P^{2}\sigma_{ER}^{2}\). This maximum can be compared with statements of maximum permissible error. We note that if the variance matrix associated with \(\mathbf{\alpha}=(\alpha_{x},\alpha_{y},\alpha_{z})^{\top}\) with \(\mathbf{\alpha}=\mathbf{0}\) is \(V_{\mathbf{\alpha}}\), then the variance matrix \(V_{\mathbf{p}}\) associated with \(R(\mathbf{\alpha})\mathbf{p}\), with \(R(\mathbf{\alpha})\) as in Equation (23), is given by \(GV_{\mathbf{\alpha}}G^{\top}\), where  
\[G=\left[ \begin{array}{ccc}0&p_{z}&-p_{y}\\ -p_{z}&0&p_{x}\\ p_{y}&-p_{x}&0\end{array} \right],\quad\mathbf{p}=(p_{x},p_{y},p_{z})^{\top}. \tag{24}\]  
As for the case of the kinematic error model, the explicit dependence on the probe offset allows for the uncertainty contributions for different probe configurations to be taken into account. If \(V_{AR}\) is the \(3m\times 3m\) variance matrix associated with \(\mathbf{\alpha}_{1:m}\) determined from the correlation kernel (or otherwise), then the variance contribution \(V_{XR}\) to the measurements \(\mathbf{x}_{1:m}\) is given by  
\[V_{XR}=G_{XR}V_{AR}G_{XR}^{\top},\]  
where \(G_{XR}\) is a \(3m\times 3m\) block-diagonal matrix. If the \(i\)th measurement is associated with the \(k\)th probe, then the \(3\times 3\)\(i\)th diagonal is equal to \(G_{k}\), where \(G_{k}\) is constructed from \(\mathbf{p}_{k}\) as in Equation (24). Although the model for the rotation angles about the three axes are mutually independent, the sensitivity matrices \(G_{XR}\) in general will introduce correlation between the effects applied to the \(x\)-, \(y\)- and \(z\)-coordinates.