# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 3 Applicative and Comparative Analysis for In Situ Measurement of Cylindricity
### Applicative Analysis of V-Block Scan EST  
First, we should evaluate the validity and accuracy of the V-block scan EST based on the standard ISO 12180 [6]. From Equation (15), we seem to obtain the measured cross-sectional profiles \(\mathbf{R}(z,p)\ (p=0,1,2,\cdots,N-1)\). However, from Equation (13), it can be verified that the transfer function \(\mathbf{\Omega}(p)|_{p=1,N-1}\equiv 0\), as shown below. \[\mathbf{\Omega}(1)=\exp(j\varphi)-\Big{\{}\exp(j\alpha)\Big{[} \frac{\cos(\varphi)}{2\cos(\alpha)}+\frac{\sin(\varphi)}{2\sin(\alpha)}\Big{]} -\exp[j(\pi-\alpha)]\Big{[}\frac{\cos(\varphi)}{2\cos(\alpha)}-\frac{\sin( \varphi)}{2\sin(\alpha)}\Big{]}\Big{\}}\] \[\qquad\qquad\qquad\qquad=\exp(j\varphi)-\{\cos(\varphi)+j\sin( \varphi)\}=0\]  
Therefore, Equations (13) and (15) are not true in this case, the first harmonic suppression occurs in the V-block scan EST. It can be seen that \(\mathbf{R}(0,p)|_{p=1,N-1}\) and \(\mathbf{R}(L,p)|_{p=1,N-1}\), which indicate the first and \((N-1)\)th harmonic vectors of the \(z=0,L\) cross-sectional profiles, cannot be accurately extracted. In other word, the LSCs vectors of the \(z=0,L\) cross-sectional profiles are unknowable. On the basis of routine, \(\mathbf{R}(0,p)|_{p=1,N-1}\) and \(\mathbf{R}(L,p)|_{p=1,N-1}\) will be set as \"zero\". Then, \(\mathbf{E}_{x}(z,p)|_{p=1,N-1}\) and \(\mathbf{E}_{y}(z,p)|_{p=1,N-1}\) could be also regarded as \"zero\" because the radial and tilt error motions of the measuring system are controlled by the \(z=0,L\) cross-sectional profiles. In view of the properties of DFT, there are \(\mathbf{M}(z,0)\
eq 0\) and \(\mathbf{M}(z,p)|_{p\
eq 0}=0\) in Equation (13). From Equation'(15), \(\mathbf{R}(z,p)|_{p=1,N-1}\), which indicate the first and \((N-1)\)th harmonic vectors of the measured cross-sectional profiles of the cylinder, are:  
\[\mathbf{R}(z,1)=\mathbf{T}(z,1)\exp(-j\varphi);\ \mathbf{R}(z,N-1)=\text{ conjugation}[ \mathbf{R}(z,1)]. \tag{16}\]  
However, \(\mathbf{R}(0,p)|_{p=1,N-1}\) and \(\mathbf{R}(L,p)|_{p=1,N-1}\) are actually existent and are typical of the LSC vectors of the \(z=0,L\) cross-sectional profiles in contact with two V-blocks. They induce \(\mathbf{E}_{x}(z,p)|_{p=1,N-1}\) and \(\mathbf{E}_{y}(z,p)|_{p=1,N-1}\), indicating the first harmonic vectors of the radial error motions of the measuring system. Substituting \(\mathbf{E}_{x}(z,1)\) and \(\mathbf{E}_{y}(z,1)\) into Equation (14), \(\mathbf{T}(z,p)|_{p=1,N-1}\), which indicate the first and \((N-1)\)th harmonic vectors of the output of the probe, can be rewritten as follow:  
\[\mathbf{T}(z,1)=\Big{[}\mathbf{R}(z,1)-\frac{(L-z)\mathbf{R}(0,1)+z\mathbf{R }(L,1)}{L}\Big{]}\exp(j\varphi);\ \mathbf{T}(z,N-1)=\text{ conjugation}[\mathbf{T}(z,1)].\]  
Equation above shows that \(\mathbf{T}(z,1)\) comprises \(\mathbf{R}(z,1)\) and linear combination of \(\mathbf{R}(0,1)\) and \(\mathbf{R}(L,1)\). However, from Equation (16), the LSC vectors in the measured cross-sections \(2\mathbf{R}(z,1)/N\) are just based on the line connecting the actual LSCs of the \(z=0,L\) cross-section, i.e., \(2\mathbf{R}(0,1)/N\) and \(2\mathbf{R}(L,1)/N\). Hence, the spatial curved median line fitted by \(2\mathbf{R}(z,1)/N\) or \(2\mathbf{R}(z,N-1)/N\) has a little movement and inclination, based on which the cylindrical profile can be reconstructed accurately even if the suppression of the first harmonic exists. The non-first harmonic suppression [3; 4; 5; 8; 14], which denotes the transfer function \(\mathbf{\Omega}(p)|_{p>1}=0\), will impact the measurement accuracy of the out-of-roundness of each cross-sectional profile of the cylinder. Whether suppression of the non-first harmonic exists in this EST will depend on the values of angular parameters of \(\varphi\) and \(\alpha\) as shown in Figure 5. Digital simulations were performed to verify how to configure parameters of \(\varphi\) and \(\alpha\) to avoid the non-first harmonic suppression. Figure 13 demonstrates the relationships between the angular parameters and the \(p\)th harmonic suppression.