## 6 Uncertainties Associated with Distances Derived from Point Clouds
### Plausible Values for Statistical Parameters Based on an MPE Statement  
The analysis above shows how the uncertainty \(u(d)\) associated with length measurement depends on the various influence factors considered. The analysis provides an estimate of \(u(d)\) in terms of the statistical hyper-parameters assigned to the influence factors. A statement of maximum permissible error can be used to derive plausible values on these statistical hyper-parameters or, at a minimum, provide upper bounds for them. Suppose the MPE statement is \(|d-d^{*}|\leq A+d^{*}/B\). We can interpret this statement statistically as  
\[Ku(d)\leq A+d/B,\]  
where \(K\) is, say, 2 or 3. From the summary information given in Section 6.7 above and specifically Equation (33), for \(d_{ij}\approx 0\), the uncertainty \(u(d_{ij})\) is such that \(u^{2}(d_{ij})=\sigma_{A}^{2}\) which implies  
\[\sigma_{A}^{2}=2\Big{(}\sigma_{R}^{2}+\sigma_{PQ}^{2}+\sigma_{P}^{2}+2\sigma_ {P_{0}}^{2}\Big{)}\leq A^{2}/K^{2} \tag{35}\]  
in order to conform to the MPE statement. This inequality relates \(\sigma_{A}\) directly to \(A\) and puts constraints on the size of \(\sigma_{A}\) defined above and, therefore, on the statistical parameters \(\sigma_{R}\), \(\sigma_{PQ}\), \(\sigma_{P_{0}}\) and \(\sigma_{P}\). Similarly, for larger distances, the MPE constraint implies  
\[\sigma_{S}^{2}+\sigma_{S,a}^{2}+\sigma_{Q}^{2}\leq\frac{u^{2}(d)}{d^{2}}\leq \frac{1}{d^{2}K^{2}}(A+d/B)^{2}\approx\frac{1}{K^{2}B^{2}},\]  
so that the scale and squareness hyper-parameters are effectively bounded by \(1/(KB)\). The relationship \[2\sigma_{ET}^{2}\Big{(}1-e^{-d^{2}/\lambda_{ET}^{2}}\Big{)}+2\sigma_{ER}^{2}P^{2} \Big{(}1-e^{-d^{2}/\lambda_{ER}^{2}}\Big{)}\leq u^{2}(d)\leq\frac{1}{K^{2}}(A+d/B )^{2},\]  
constrains the combined effect of the kinematic errors as specified by \(\sigma_{ET}\) and \(\sigma_{ER}\) along the the spatial correlation lengths \(\lambda_{ET}\) and \(\lambda_{ER}\). In general, given a summary estimate \(u(d)\) of the uncertainty in distance due to the combined effects, we can evaluate  
\[C(d)=\frac{Ku(d)}{A+d/B}. \tag{36}\]  
If \(C(d)\leq 1\) over the working volume, then the values of the statistical parameters do not violate the MPE statement. If \(C_{\text{max}}=\max_{d\leq t_{\text{max}}}C(d)\) is the maximum value of \(C(d)\) over the working volume, then the simple procedure of dividing all the statistical parameters representing standard deviations, \(\sigma_{R}\), etc., by \(C_{\text{max}}\) will lead to conformance with the MPE statement. The MPE statement can be used, along with any other information, expert judgement, etc., to guide the assignment of the statistical hyper-parameters. The assignment cannot be unique, as demonstrated in Section 3, but the MPE can be used to determine a range of plausible values for the parameters. Making an assignment involves setting a balance between the random effects, the scale and squareness effects and the translational and rotational effects. Table 3 gives two examples of MPE statements, labelled MPE1 and MPE2, along with assignments of values for the statistical hyper-parameters that have been derived to be consistent with the MPE statements. MPE1 corresponds to a more accurate CMM relative to MPE2. Graphs of the MPE and uncertainty components associated with distance as a function of \(d\) derived for statistical parameters in Table 3 and expansion factor \(K=2\) are shown in Figures 4 and 5. The figures shows that the statistical parameters conform to the MPE statements. The uncertainty contribution from scale and squareness effects is linear in \(d\), while the spatially correlated location and rotation effects start off as linear in \(d\) but begin to level off for larger values of \(d\). For both characterisations, the spatial correlation length scale parameter \(\lambda_{ER}\) associated with the rotation effects is larger than that \(\lambda_{ET}\) for the location effects and the levelling off occurs later for the rotation effects.