## 2 Evaluation Method
### Uncertainty Evaluation Model
(3) Uncertainty components caused by stability and reproducibility  
The stability index usually has a significant influence on the electronic measuring instruments but less on the CMM geometric measurement; the stability is equivalent to the reproducibility caused by time variation, so this uncertainty component can be ignored. The uncertainty component \(u_{\text{R}}\) caused by CMM task-oriented reproducibility indicates the consistency between the measurement results of the same measuring object when the measurement conditions are changed, that is, different measurement conditions such as personnel changes and different measurement strategies have led to the variations of the measurement mean value for the same measurement task. In the actual uncertainty component quantization, the repeatability experiment is used to determine the influence of the component. For CMM, multiple groups of repeatability quantization experiments are completed by different surveyors according to the sampling strategy with the measuring points determining themselves based on the habits for measurement. Therein, the uncertainty component caused by sampling strategy is affected by the number and distribution of the sampling points. Sampling point distribution determines the probability of extracting the extreme error point of the errors for measured elements when the number of sampling points is the same. It has become a consensus in CMM applications that minimal measurement uncertainty occurs when the sampling points are evenly distributed. The number of sampling points reflects the CMM ability to extract the form information of the measured elements. When there is a small number of sampling points, the measuring points will have a higher probability of excluding the extreme point of the form tolerance. From the point of view of information extraction only, the larger the number of sampling points, the better. However, the measurement time similar to contact triggered CMM will increase sharply as the number of sampling points increases, which has violated the CMM characteristics of measurement efficiency. At the same time, the excessive increase of the number of measuring points will multiply the impact of the CMM residual mechanism errors on the measurement results. Generally, for contact triggered CMM, the suitable number and preferable distribution of sampling points can be confirmed according to BS7172 [12]. As the evaluation software requires, the number of measurement points just needs to be greater than the minimum number of the points required by the mathematical requirements of the geometric elements to be measured. Therefore, the primary model is as follows for evaluating the CMM task-oriented uncertainty based on the method for statistical analysis of measurement characteristics:  
\[u_{\mathrm{c}}=f(u_{\mathrm{E}},u_{\mathrm{r}},u_{\mathrm{R}}) \tag{1}\]  
Formula (1) shows three inputs, namely, the maximum permissible error \(\delta_{\mathrm{E}}\), measurement repeatability \(\delta_{\mathrm{r}}\) and measurement reproducibility \(\delta_{\mathrm{R}}\) of the instrument. The expected values are 0 for all inputs and all are the measurement characteristics of the measurement task output \(Y\). \(y\) is the measurement estimated value of \(Y\). Therefore, the analysis model is as follows for measurement uncertainty:  
\[Y=y+\delta_{\mathrm{E}}+\delta_{\mathrm{r}}+\delta_{\mathrm{R}} \tag{2}\]  
Based on the Guide to the expression of uncertainty in measurement (GUM) [32], Formula (1) can be written as:  
\[uu_{\mathrm{c}}\sqrt{u_{\mathrm{E}}^{2}+u_{\mathrm{r}}^{2}+u_{\mathrm{R}}^{2}} \tag{3}\]