# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 1 Introduction
### Mathematical Modeling of the Measuring System  
Cylindricity errors are three-dimensional form deviations, and accurate measurement of cylindricity is relatively complex, should include measuring and reconstructing cylindrical profile, and evaluating out-of-cylindricity. Any in situ measuring system must employ several probes mounted onto a carriage to sense the radial deviations of the cylinder while the cylinder is rotating and the carriage is sliding along the longitudinal direction. It is known that rotating of the cylinder or sliding of the carriage have inevitable five degree-of-freedom geometric error motions due to their manufacturing and assembly errors. Therefore, the method of measuring cylindrical profile must be able to eliminate error motions of the cylinder and carriage in data analysis. Error separation techniques (ESTs) have been used in measurement of the form errors, such as straightness, flatness, roundness, and so on. Error separation techniques ensure measurement accuracy by separating form errors and error motions with specific designed setup and special mathematical modeling. In general, the three-point roundness EST [3; 4; 9] can remove the radial error motions in the cross-section of the cylinder and accurately determine radius deviations and out-of-roundness of the cross-sectional profile of the cylinder [3; 5; 9]. However, the first harmonic of the cross-sectional profile are inseparable from that of the radial error motions in the cross-section of the cylinder [3; 4; 5; 8; 9]. The first harmonic of the cross-sectional profile denotes the vector from rotating center to the LSC of the cross-sectional profile. Median line of the cylinder is a spatial curve that is fitted through the LSCs of sequential cross-sectional profiles [6; 8]. Therefore, three-point roundness EST is insufficient to accurately determine a cylindrical profile. Modeling of a probe sensing the cylindrical profile is particularly foundational for cylindrical profile measurement. Reference [10] built such a measurement model, as shown in Figure 2. \(OXYZ\) is the global coordinate system, in which the cylinder coordinate system \(O_{c}X_{c}Y_{c}Z_{c}\) and the carriage coordinate system \(O_{s}X_{s}Y_{s}Z_{s}\) are located. Theoretically, there are ten terms of error motions, including the cylinder's translational error motions \(e_{cx}\), \(e_{cy}\), \(e_{cz}\) and tilt error motions \(\tau_{cx}\), \(\tau_{cy}\), as well as the carriage's straightness error motions \(e_{sx}\), \(e_{sy}\) and three angular error motions of pitch \(\tau_{sx}\), yaw \(\tau_{sy}\), and roll \(\tau_{sz}\). The vectors of the \(O_{c}\) and \(O_{s}\) in the \(OXYZ\) system are \(\mathbf{e}_{c}=\left[e_{cx},\ e_{cy},\ 0\right]\) and \(\mathbf{e}_{s}=\left[e_{sx},\ e_{sy},\ z\right]\), respectively. Let the rotation angle of the cylinder be \(\theta\). In the \(O_{s}X_{s}Y_{s}Z_{s}\) system, the measuring point \(\mathbf{P}\) locates at \(\mathbf{r}^{s}=\left[r_{0}\cos(\varphi),\ r_{0}\sin(\varphi),\ z_{s}\right]\), and point \(\mathbf{P}\) in the \(OXYZ\) system would be offset due to the carriage error motions. If \(e_{cz}\), \(\tau_{sz}\), and quadratic terms of the errors are omitted, the actual position of point \(\mathbf{P}\) in the \(OXYZ\) system is:  
\[\zeta_{x}=r_{0}\cos(\varphi)+e_{sx}+z_{s}\tau_{sy};\ \zeta_{y}=r_{0}\sin( \varphi)+e_{sy}+z_{s}\tau_{sx}.\]  
In a similar way, in the \(O_{c}X_{c}Y_{c}Z_{c}\) system, because of point \(\mathbf{P}\)'s offsets caused by the error motions of the cylinder, the actual position of point \(\mathbf{P}\) in the \(OXYZ\) system is:  
\[ \begin{array}{l}\xi_{x}=r(z+z_{s},\theta+\varphi)\cos(\varphi)+e_{cx}+(z+z_ {s})\tau_{cy};\\ \zeta_{y}=r(z+z_{s},\theta+\varphi)\sin(\varphi)+e_{cy}+(z+z_{s})\tau_{cx}. \end{array} \]  
Obviously, the output of the probe indicates their difference in \(OXYZ\) system:  
\[