# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 2 Error Separation Models for In Situ Measurement of Cylindricity
### V-Block Scan EST
\[\mathbf{R}(z,p)|_{z=0,L}=\frac{\mathbf{T}(z,p)+\mathbf{M}(z,p)}{\mathbf{ \Omega}(p)}|_{z=0,L};\ \mathbf{\Omega}(p)=\exp(jp\varphi)-\mathbf{\Gamma}(p) \tag{13}\]  
\[\mathbf{\Gamma}(p)=\left\{\exp(jpa)\left[\frac{\cos(\varphi)}{2\cos(\alpha)}+ \frac{\sin(\varphi)}{2\sin(\alpha)}\right]+\exp[jp(\pi-\alpha)]\left[-\frac{ \cos(\varphi)}{2\cos(\alpha)}+\frac{\sin(\varphi)}{2\sin(\alpha)}\right]\right\}\]  
\[\mathbf{M}(z,p)|_{z=0,L}=\mathrm{DFT}\left[e_{sx}(z)\cos(\varphi)+e_{sy}(z) \sin(\varphi)\right]|_{z=0,L}\]  
Figure 5: Schematic of V-block scan EST [1, 12, 13]. The left shows the measured cylinder in the OXYZ system; the right shows the cross-section of the cylinder in the plane of the V-block. where, \(\mathbf{\Omega}(p)\) is the transfer function, and \(\mathbf{T}(z,p)|_{z=0,L}=\mathrm{DFT}[t(z,i)]|_{z=0,L}\) are the \(p\mathrm{th}(p=0,1,\cdots,N-1)\) harmonic vectors of the output of the probe when it locates in the \(z=0,L\) cross-sections. \(\mathbf{M}(z,p)|_{z=0,L}\) are the \(p\mathrm{th}(p=0,1,\cdots,N-1)\) harmonic vectors of the disturbances from the carriage straightness error motions when the probe locates in the \(z=0,L\) cross-sections. If stiffness of the measuring system is sufficient, the radial error motions of the measuring system in measured cross-section, denoted by \(e_{x}(z,i)\), \(e_{y}(z,i)(i=0,1,\cdots,N-1)\), should be the linear combination of \(e_{x}(z,i)\big{|}_{z=0,L}\), \(e_{y}(z,i)\big{|}_{z=0,L}\). Here, their \(p\mathrm{th}\)\((p=0,1,\cdots,N-1)\) harmonic vectors, denoted by \(\mathbf{E}_{x}(z,p)=\mathrm{DFT}[e_{x}(z,i)]\) and \(\mathbf{E}_{y}(z,p)=\mathrm{DFT}[e_{y}(z,i)]\), can be achieved. Now, taking DFT to Equation (12), we will obtain the \(p\mathrm{th}\)\((p=0,1,\cdots,N-1)\) harmonic vector of the output of the probe when it locates in measured cross-section:  
\[\mathbf{T}(z,p)=\mathrm{DFT}[t(z,i)]=\exp(jp\varphi)\mathbf{R}(z,p)+\mathbf{E }_{x}(z,p)\cos(\varphi)+\mathbf{E}_{y}(z,p)\sin(\varphi)-\mathbf{M}(z,p). \tag{14}\]  
Bringing \(\mathbf{E}_{x}(z,p)\), \(\mathbf{E}_{y}(z,p)\) and \(\mathbf{R}(z,p)|_{z=0,L}\)\((p=0,1,\cdots,N-1)\) into Equation (14), we can determine \(\mathbf{R}(z,p)\), indicating the \(p\mathrm{th}\)\((p=0,1,\cdots,N-1)\) harmonic vectors of the measured cross-sectional profile of the cylinder. \[\mathbf{R}(z,p)=\left\{\mathbf{T}(z,p)+\frac{(L-z)\mathbf{T}(0,p)+z\mathbf{T }(L,p)}{L\mathbf{\Omega}(p)}\mathbf{\Gamma}(p)\right\}\exp(-jp\varphi)+ \mathbf{\Delta}(z,p). \tag{15}\]  
\[\mathbf{\Delta}(z,p)=\left\{\mathbf{M}(z,p)+\frac{(L-z)\mathbf{M}(0,p)+z \mathbf{M}(L,p)}{L\mathbf{\Omega}(p)}\mathbf{\Gamma}(p)\right\}\exp(-jp\varphi)\]  
where, \(\mathbf{T}(z,p)\) is the \(p\mathrm{th}(p=0,1,\cdots,N-1)\) harmonic vector of the output of probe in measured cross-section. \(\mathbf{M}(z,p)\) is the \(p\mathrm{th}(p=0,1,\cdots,N-1)\) harmonic vector of the disturbances from the carriage straightness error motions in measured cross-section. A reconstructed cylindrical profile \(r(z,i)(i=0,1,\cdots,N-1)\) can be achieved by taking IDFT to \(\mathbf{R}(z,p)(p=0,1,\cdots,N-1)\), similarly to Equation (6). However, it is worth noting that Equations (13)-(15) are true if the transfer function \(\mathbf{\Omega}(p)\
eq 0\), in addition, the disturbances from the carriage straightness error motions will impact the accuracy of the V-block scan EST.