# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 2 Error Separation Models for In Situ Measurement of Cylindricity
### Parallel Scan EST
Referencing the sequential three-point straightness EST [5; 15], the iterative formulas used to extract \(\mathbf{R}\big{(}z_{J},1\big{)}\)\((J=1,2,\cdots,M)\), representing the LSC vectors of each cross-sectional profile, ware derived in in Appendix of Reference [14]. When the carriage locates at the first position, let \(\mathbf{\Theta}_{cx}(1,1)\) and \(\mathbf{R}(z_{1},1)\) be equal to \"zero\". The first harmonic vector \(\mathbf{R}\big{(}z_{J},1\big{)}\) of the \(J\)th \((J=2,3,\cdots,M)\) cross-sectional profile is extracted by:  
\[\mathbf{R}\big{(}z_{J},1\big{)}=\mathbf{R}\big{(}z_{J-1},1\big{)}+\mathbf{T}_{ 4}(J-1,1)-\hat{\mathbf{E}}_{x}\big{(}z_{J-1},1\big{)}-\mathbf{\Theta}_{cx}(J-1,1) \tag{6a}\] \[\mathbf{\Theta}_{cx}(J,1)=\mathbf{\Theta}_{cx}(J-1,1)-\big{[}\mathbf{T}_{5}(J- 1,1)-\mathbf{T}_{4}(J-1,1)\big{]}+\big{[}\mathbf{T}_{4}(J,1)-\hat{\mathbf{E}}_ {x}\big{(}z_{J},1\big{)}\big{]}. \tag{6b}\]  
Figure 3: Schematic of parallel scan EST for measurement of cylindrical profile. The 1–5 are the displacement probes. The LSC vector of the \(J\)th \((J=1,2,\cdots,M)\) cross-sectional profile is \(2\mathbf{R}(z_{J},1)/N\) or \(2\mathbf{R}\big{(}z_{J},N-1\big{)}/N\), with which a curved median line of the cylinder can be accurately fitted, here, \(\mathbf{R}\big{(}z_{J},N-1\big{)}=\text{conjugation}\big{[}\mathbf{R}\big{(}z_{ J},1\big{)}\big{]}\). It is noteworthy that the curved median line fitted through the LSCs of each cross-sectional profile, as a whole, undergoes little movement and inclination because \(\mathbf{R}(z_{1},1)\) and \(\mathbf{\Theta}_{\text{cv}}(1,1)\) are all set to \"zero\", rather than their truth-values. The correctness of the curved median line profile will not be affected. Now, by means of the three-point roundness EST and Equation (6), \(\mathbf{R}\big{(}z_{J},p\big{)}\)\((p=0,2,3,\cdots,N-2)\) as well as \(\mathbf{R}\big{(}z_{J},1\big{)}\) and \(\mathbf{R}\big{(}z_{J},N-1\big{)}\), indicating the \(p\)th \((p=0,1,2,\cdots,N-1)\) harmonic vectors of the \(J\)th \((J=1,2,\cdots,M)\) cross-sectional profiles, can be detected accurately. A reconstructed cylindrical profile, denoted by \(r\big{(}z_{J},i\big{)}\)\((J=1,2,\cdots,M)\)\((i=0,1,\cdots,N-1)\), can be achieved by;  
\[r\big{(}z_{J},i\big{)}=\text{IDFT}\big{[}\mathbf{R}\big{(}z_{J},p\big{)}\big{]} \text{ }(i,p=0,1,2,\cdots,N-1). \tag{7}\]  
Where, IDFT is the inverse discrete Fourier transform operator. Theoretical analysis and numerical validation about this EST have been performed in Reference [8]. The results verify that error motions of the cylinder and carriage are all removed, i.e., full harmonic error separation is realized. The LSCs of each cross-sectional profile of the cylinder are accurately extracted even if the error motions of the cylinder are not repeatable in each rotation. The spatial curved median line of the cylinder is accurately determined by fitting through the LSCs of the sequential cross-sectional profiles. The cylindrical profile conforming to the standard ISO 12180 [6], can then be reconstructed by Equation (7), as shown in Equation (3). The cylindrical form errors, not only cylindricity but also out-of-roundness, generatrix straightness, taper angle, and radius deviation, can be evaluated simultaneously by using the reconstructed cylindrical profile.