## 3 Experiments
The measurement equation is as follows:  
\[\mathrm{d}=1-\mathrm{l_{s}}(1-\alpha\Delta\mathrm{T}) \tag{1}\]  
where d is deviation value, 1 is the measured length of the CMM under measurement at 20 \({}^{\circ}\)C, \(\alpha\) is thermal expansion coefficient of the CMM under measurement, \(\Delta\mathrm{T}\) is the temperature difference between the temperature of the CMM under measurement at 20 \({}^{\circ}\)C, and \(\mathrm{l_{s}}\) is the standard length of the hole plate at 20 \({}^{\circ}\)C.  
The measurement uncertainty of the hole plate method originates from four influence sources to contribute to the measured value \(\varepsilon\), including the length of the CMM 1, the standard length of the hole plate \(\mathrm{l_{s}}\), the thermal expansion coefficient \(\alpha\) and the temperature difference \(\Delta\mathrm{T}\). The measured length contains repeatability of CMM measurement, resolution of the CMM, and setup errors of the hole plate. \[\varepsilon=\mathrm{f}(\mathrm{l},\ \mathrm{l_{s}},\ \alpha,\ \Delta\mathrm{T}) \tag{2}\]  
The combined standard uncertainty is expressed as follows:  
\[ \begin{array}{l}\mathrm{u_{c}^{2}(\varepsilon)=\left(\frac{\partial\varepsilon }{\partial\mathrm{f}}\right)^{2}u(\mathrm{l})^{2}+\left(\frac{\partial \varepsilon}{\partial\mathrm{l_{s}}}\right)^{2}u(\mathrm{l_{s}})^{2}+\left( \frac{\partial\varepsilon}{\partial\mathrm{\alpha}}\right)^{2}u(\alpha)^{2}+ \left(\frac{\partial\varepsilon}{\partial\mathrm{\Delta T}}\right)^{2}u( \mathrm{\Delta T})^{2}\\ =u^{2}(\mathrm{l})+u^{2}(\mathrm{l_{s}})+(\mathrm{l_{s}}\Delta\mathrm{T})^{2}u ^{2}(\alpha)+(\mathrm{\alpha_{s}})^{2}u^{2}(\mathrm{\Delta T})\end{array}  \tag{3}\]  
\(\mathrm{u(l)}\), \(\mathrm{u(l_{s})}\), \(\mathrm{u(\alpha)}\) and \(\mathrm{u(\Delta T)}\) are called standard uncertainties. The uncertainty \(\mathrm{u_{c}}\) is evaluated using six uncertainty sources. (1) Repeatability of CMM measurement  
Repeatability is a source of uncertainty that arises from multiple measurements. The measured results of the CMM are calculated as the average of the measurement results and their corresponding standard deviations. The standard uncertainty is determined as the maximum standard deviation of 0.39 \(\mathrm{\upmu m}\) divided by the square root of six, which is referred to as type A.  
\[\mathrm{u(l_{2})=0.39\ \mathrm{\upmu m}/\sqrt{6}=0.16\ \mathrm{\upmu m} \tag{4}\]  
Figure 9: The DCC implementation of administrative and measurement information of CMM in PDF. The degree of freedom is 5. According to the specification, the resolution of the CMM is 0.1 \(\upmu\)m. The standard uncertainty distribution is assumed to have a rectangular probability by a square root of twelve, called type B.  
\[\text{u}(1_{1})=0.1\ \upmu\text{m}/\sqrt{12}=0.03\ \upmu\text{m} \tag{5}\]  
Traceability of the hole plate  
The calibration certificates of the hole plate declare that the same expanded uncertainty is 0.68 \(\upmu\)m with a coverage factor k of 2 (95% confidence level). The standard uncertainty is obtained by dividing 0.68 \(\upmu\)m by 2, called type B.  
\[\text{u}(1_{\text{s1}})=0.68\ \upmu\text{m}/2=0.34\ \upmu\text{m} \tag{6}\]  
The relative standard uncertainty is estimated as 5%, the degree of freedom is 200. Setup errors of the hole plate  
The hole plate needs to be adjusted within 50 \(\upmu\)m, and the collimation error is 0.002 \(\upmu\)m. The standard uncertainty distribution assumes a rectangular probability by a square root of twelve, called type B.  
\[\text{u}(1_{\text{s2}})=0.002\ \upmu\text{m}/\sqrt{12}=0.01\ \upmu\text{m} \tag{7}\]  
Thermal expansion coefficient of CMM  
The maximum axial thermal expansion coefficient of the coordinate measuring instrument is \(11\times 10^{-6}\). The measurement range 550 mm.