# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 2 Error Separation Models for In Situ Measurement of Cylindricity  
### Parallel Scan EST  
Reference [8] presented a parallel scan EST for measuring cylindrical profile, as shown in Figure 3. Let the cylinder be divided by \(M\) cross-sections, and the axial distance between each two cross-sections be \(d\). Five displacement probes are mounted onto the carriage, which moves along the \(Z\) direction. When the carriage is located at the \(J\)th \((J=1,2,\cdots,M)\) position and the cylinder rotates one revolution, Probes 1-3 sense the mixed radial errors in the \(J\)th cross-section, meanwhile, Probes 4 and 5 sense the \(X\) directional mixed errors in the \((J+1)\)th and \((J+2)\)th cross-sections. When data collection is completed, the carriage moves a distance \(d\) along the \(Z\) direction and enters the \((J+1)\)th position. At this moment, Probes 1-3 locates in the \((J+1)\)th cross-section and Probes 4 and 5 locate in the \((J+2)\)th and \((J+3)\)th cross-sections, respectively. Data collection of new revolution will be carried on. Figure 2: Probe senses cylindrical profile and error motions [10]. When the carriage locates at the \(J\)th \((J=1,2,\cdots,M)\) position, the cylinder is rotating and Probes 1-5 collects data. It was verified that the \(p\)th \((p=0,2,3,\cdots,N-2)\) harmonic vector \(\mathbf{R}(z_{J},p)\) of the \(J\)th cross-sectional profile \(r(z_{J},i)\)\((i=0,1,\cdots,N-1)\) of the cylinder could be detected with the three-point roundness EST [3; 4; 5; 8; 9] by the outputs of Probes 1-3, denoted by \(t_{n}(z_{J},i)\)\((n=1,2,3)\)\((i=0,1,\cdots,N-1)\). Therefore, \(r_{0}\big{(}z_{J}\big{)}\) and \(r_{2}\big{(}z_{J},\ i\big{)}(i=0,1,\cdots,N-1)\), indicating the radius deviation and out-of-roundness of the \(J\)th cross-sectional profile, can be accurately determined. In addition, \(\hat{e}_{x}\big{(}z_{J},i\big{)}\)\((i=0,1,\cdots,N-1)\), indicating the separated result of the \(X\) directional error motions in the \(J\)th cross-section of the cylinder, can be obtained by the outputs of Probes 1 and 2, as shown in Ref. [8; 14]. The first harmonic vector should be \(\hat{\mathbf{E}}_{x}\big{(}z_{J},1\big{)}=\text{DFT}\big{[}\hat{e}_{x}\big{(}z _{J},i\big{)}\big{]}|_{p=1}\), here, DFT is the discrete Fourier transform operator. Simultaneously, Probes 4 and 5 sense the X directional mixed errors, including the \((J+1)\)th and \((J+2)\)th cross-sectional profiles, denoted by \(r\big{(}z_{J+1},\ i\big{)}\) and \(r\big{(}z_{J+2},\ i\big{)}\)\((i=0,1,\cdots,N-1)\), the \(X\) directional error motions in the \(J\)th cross-section \(e_{cx}\big{(}z_{J},\ i\big{)}\) and the tilt error motion of the cylinder around the \(Y\) direction \(\tau_{cy}(J,i)\), as well as the \(X\) directional straightness deviations of the carriage \(e_{sx}\big{(}z_{J}\big{)}\) and yawing \(\tau_{cy}(J)\), as shown in Figures 2 and 3. From Equation (4), the outputs of Probes 4 and 5, denoted by \(t_{n}(J,i)\)\((n=4,5)\)\((i=0,1,\cdots,N-1)\), are given by  
\[t_{4}(J,i)=r\big{(}z_{J+1},i\big{)}-\mathbf{D}_{4}+\big{\{}[e_{cx}\big{(}z_{J},i\big{)}+d\times\tau_{cy}(J,i)]-\big{[}e_{sx}\big{(}z_{J}\big{)}+d\times\tau_ {sy}(J)\big{]}\big{\}} \tag{5a}\] \[t_{5}(J,i)=r\big{(}z_{J+2},i\big{)}-\mathbf{D}_{5}+\big{\{}[e_{cx}\big{(}z_{J},i\big{)}+2d\times\tau_{cy}(J,i)]-\big{[}e_{sx}\big{(}z_{J}\big{)}+2d\times \tau_{sy}(J)\big{]}\big{\}}. \tag{5b}\]  
Applying Discrete Fourier transform (DFT) to \(t_{4}(J,i)\) and \(t_{5}(J,i)\), we may obtain the \(p\)th \((p=0,1,\cdots,N-1)\) harmonic vectors of the outputs of Probes 4 and 5, denoted by \(\mathbf{T}_{4}(J,p)=\text{DFT}[t_{4}(J,i)]\) and \(\mathbf{T}_{5}(J,p)=\text{DFT}[t_{5}(J,i)]\), in which the first harmonic vectors are \(\mathbf{T}_{4}(J,1)\) and \(\mathbf{T}_{5}(J,1)\), respectively. \(\mathbf{\Theta}_{cx}(J,1)\)\((J=1,2,\cdots,M)\) is designated as the first harmonic vector of the \(X\) directional additional error motions induced by the tilt error motions of the cylinder around the \(Y\)-axis.