# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 2 Error Separation Models for In Situ Measurement of Cylindricity
### V-Block Scan EST  
Ref. [1; 12; 13; 17] presented the V-block scan EST for measurement of cylindrical profile as shown in Figure 5. An element connecting the two V-blocks functions as guideway, along which a carriage moves. A displacement probe is mounted onto the carriage. When the cylinder driven by the spindle rotates, probe keeps collecting data. At the moment, two V-blocks should contact reliably with the surface of the cylinder, and the measuring system should avoid rotating around and moving in the \(Z\)-axis. Because five points are contact with the measured surface during measuring, this system may be regarded as an analogous five-point EST measuring system. Figure 4: Schematic of spiral scan EST [10]. The 1-5 are the displacement probes. As shown in Figure 5, the OXYZ system was assumed in such a way that the \(Z\)-axis coincides with the axis of the nominal cylinder, and the plane determined by the \(X\)- and \(Y\)-axes coincide with the plane of the left V-block. The \(Y\)-axis is the angular bisector of the V-block. Let the total number of scanning points per revolution be \(N\), and the scanning angular interval should be \(\delta=2\pi/N\). Let the angular position of the probe be \(\varphi=k\delta\) with respect to the \(X\)-axis, and the half angle of the V-block be \(\alpha=u\delta\). Because the system rides on the cylinder during measuring, the radial and tilt error motions of the cylinder will have nothing to do with the output of the probe. However, as those two cross-sectional profiles that are contacted by two V-blocks have roundness errors, it would make the measuring system generate the radial and tilt error motions. At this moment, the probe situated in measured cross-section will collect the radial mixed errors, including the cross-sectional profile \(r(z,i)\)\((i=0,1,\cdots,N-1)\), the measuring system's radial error motions \(e_{x}(z,i)\), \(e_{y}(z,i)\)\((i=0,1,\cdots,N-1)\) in the cross-section where the probe is, as well as the carriage straightness error motions \(e_{sx}(z)\), \(e_{sy}(z)\) in the location where the probe is. According to Equation (4), the output of the probe \(t(z,i)\)\((i=0,1,\cdots,N-1)\) should be:  
\[t(z,i)=r(z,i+k)+[e_{x}(z,i)-e_{sx}(z)]\cos(\varphi)+\left[e_{y}(z,i)-e_{sy}(z) \right]\sin(\varphi). \tag{12}\]  
where, the initial zero-setting value of the probe is omitted due to without impact on accurate measurement. It is worth noting that \(e_{x}(z,i)\), \(e_{y}(z,i)\)\((i=0,1,\cdots,N-1)\), indicating the measuring system's radial error motions in measured cross-section, are caused by the cross-sectional profiles in contacted with the two V-blocks. There are obviously four points on the V-blocks in contact with the cross-sections defined by the plane \(z=0,L\) of the cylinder. If the contacted points are regarded as the virtual probes, output of the virtual probes in the \(z=0,L\) cross-sections should be \"zero\". It is obvious that the \(z=0,L\) cross-sectional profiles \(r(z,i)|_{z=0,L}\)\((i=0,1,\cdots,N-1)\) control the radial error motions of the measuring system in the \(z=0,L\) cross-sections, denoted by \(e_{x}(z,i)\), \(e_{y}(z,i)|_{z=0,L}\)\((i=0,1,\cdots,N-1)\). At this moment, if the carriage moves, and the probe locates in the \(z=0,L\) cross-sections, respectively, the probe collects the radial mixed errors during cylinder rotating. The outputs of the probe should be \(t(z,i)|_{z=0,L}\)\((i=0,1,\cdots,N-1)\), as shown in Equation (12). Taking DFT to \(t(z,i)|_{z=0,L}\)\((i=0,1,\cdots,N-1)\), we will achieve the \(p\)th\((p=0,1,\cdots,N-1)\) harmonic vector of the \(z=0,L\) cross-sectional profiles \(\mathbf{R}(z,p)|_{z=0,L}\), as follow.