# An Improved Self-Born Weighted Least Square Method for Cylindicity Error Evaluation
## 4 Cylindicity Error Uncertainty Evaluation  
### GUM Assessment Method  
Currently, the GUM method is the most commonly used method in uncertainty evaluation. According to the GUM method, the uncertainty propagation rate model is determined, and the uncertainty source of each parameter is determined at the same time. The composite standard uncertainty and expanded uncertainty of the cylindricity error uncertainty evaluation are calculated. The sources of uncertainty are analyzed as follows:  
Figure 2: ISWLS cylindricity error algorithm evaluation process.  
(1) Repeat the measurement at a single point 10 times, and the introduced uncertainty component \(u_{1}\) is calculated according to the Bessel formula [3]:  
\[u_{1}=\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}{n-1}} \tag{27}\]  
(2) The uncertainty component caused by the indicated value error can be derived from the measurement experience of the instrument. The maximum allowable indication error \(\sigma\) of the CMM is 0.5 \(\upmu\)m. The coverage factor \(k\) is \(\sqrt{3}\). The indication error is expressed as \(u_{2}\)[3]:  
\[u_{2}=\frac{\sigma}{k} \tag{28}\]  
(3) During the test, the room temperature is kept constant at 20 degrees Celsius, so the uncertainty \(u_{3}\) caused by the temperature is approximately 0 \(\upmu\)m [3].  
(4) In practical measurements, the uncertainty component \(u_{4}\) is approximated as 0 \(\upmu\)m due to the deformation caused by the measuring force [3].  
Then the single-point uncertainty \(u_{0}\) is calculated as [3]:  
\[u_{0}=\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{4}^{2}} \tag{29}\]  
According to the actual situation, the conditions of a single measurement tend to be the same, and the single-point uncertainty is:  
\[u_{xmax}=u_{ymax}=u_{zmax}=u_{xmin}=u_{ymin}=u_{zmin}=u_{0} \tag{30}\]  
The key to calculating the uncertainty is to find the uncertainty u and \(\rho\) of each parameter as the correlation coefficient of each parameter. Uncertainties of \(a,b,p,q\) are obtained from actual data. Suppose the vector \(D=[a,b,p,q]^{T}\), then  
\[cov(D)=\left[\begin{array}{cccc}var(a)&cov(a,b)&cov(a,p)&cov(a,q)\\ cov(b,a)&var(b)&cov(b,p)&cov(b,q)\\ cov(p,a)&cov(q,a)&cov(p,b)&cov(q,b)\\ var(p)&cov(q,p)&cov(p,q)&var(q)\end{array}\right]\]  
The correlation coefficient can be calculated according to Equation (31) as follows:  
\[\rho_{ab}=\frac{u_{ab}}{u_{a}u_{b}} \tag{31}\]  
Finally, the synthetic standard uncertainty \(u_{\delta}\) is calculated according to the GUM method:  
\[\begin{array}{l}u_{\delta}^{2}=\left(\frac{\partial f}{\partial x_{max}}u_ {xmax}\right)^{2}+\left(\frac{\partial f}{\partial y_{max}}u_{ymax}\right)^ {2}+\left(\frac{\partial f}{\partial z_{max}}u_{zmax}\right)^{2}\\ +\left(\frac{\partial f}{\partial x_{min}}u_{xmin}\right)^{2}+\left(\frac{ \partial f}{\partial y_{min}}u_{ymin}\right)^{2}+\left(\frac{\partial f}{ \partial z_{min}}u_{zmin}\right)^{2}\\ +\left(\frac{\partial f}{\partial y}u_{p}\right)^{2}+\left(\frac{\partial f} {\partial y}u_{q}\right)^{2}+\left(\frac{\partial f}{\partial u}u_{q}\right) ^{2}+\left(\frac{\partial f}{\partial b}u_{p}\right)^{2}\\ +2\frac{\partial f}{\partial d}\frac{\partial f}{\partial b}\rho_{ab}u_{ab}+2 \frac{\partial f}{\partial d}\frac{\partial f}{\partial p}\rho_{ap}u_{ap}+2 \frac{\partial f}{\partial a}\frac{\partial f}{\partial b}\rho_{aq}u_{aq}\\ +2\frac{\partial f}{\partial b}\frac{\partial f}{\partial b}\rho_{bp}u_{bp}+2 \frac{\partial f}{\partial b}\frac{\partial f}{\partial q}\rho_{bq}u_{bb}+2 \frac{\partial f}{\partial q}\frac{\partial f}{\partial b}\rho_{q}u_{pq} \end{array} \tag{32}\]