# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 2 Error Separation Models for In Situ Measurement of Cylindricity
### Spiral Scan EST  
Reference [10] presented the spiral five-probe arrangement for measurement of cylindrical profile, as shown in Figure 4. Five displacement probes arranged in a spiral scanning-path are mounted onto the carriage, which moves along the \(Z\) direction. This system makes it possible to transform the cylindrical profile measurement to the one-dimensional straightness measurement of the spiral profile on the cylinder given that the rotation of the cylinder keeps pace with the movement of the carriage. Let the total number of scanning points per revolution be \(N\), and the scanning angular interval should be \(\delta=2\pi/N\). When probes move along the spiral line, the variable \(z\) of the carriage moving along the \(Z\) direction is proportional to the variable \(\theta=i\times\delta\) of the cylinder rotating, i.e., \(z=\theta\times T/(2\pi)=i\times T/N\), where, constant \(T\) is the pitch of the spiral line. The angular positions of Probes 1-5 are \(\varphi_{n}\)\((n=1,2,\cdots,5)\) with respect to the \(X\) direction, and axial positions are \(z_{\text{sH}}=\tau\times\varphi_{n}/(2\pi)\)\((n=1,2,\cdots 5)\), respectively. When the cylinder rotates and the carriage moves synchronously, several error motions come up as shown in Figure 2, including the cylinder's radial error motions \(e_{\text{cx}}(i),\ e_{\text{cy}}(i)\) and tilt error motions \(\tau_{\text{cx}}(i),\ \tau_{\text{cy}}(i)\), as well as the carriage straightness error motions \(e_{\text{ex}}(i),\ e_{\text{ey}}(i)\) and pitching \(\tau_{\text{ex}}(i)\), yawing \(\tau_{\text{sy}}(i)\)\((i=0,1,\cdots,N\times M-1)\). Here, \(M\) is number of total rotating circle of the spiral line on the cylinder. Let \(r(i)\)\((i=0,1,\cdots,N\times M-1)\) be the spiral line profile on the cylinder. According to Equation (4), the outputs of the five probes, denoted by \(t_{n}(i)\)\((n=1,2,\cdots,5)\)\((i=0,1,\cdots,N\times M-1)\), can be written as follow:  
\[ \begin{array}{c}t_{n}(i)=r(i+k_{n})-D_{n}+\big{\{}\big{[}e_{\text{cx}}(i)-e_ {\text{ex}}(i)+z\tau_{\text{cy}}(i)\big{]}\!+\!z_{\text{sn}}[\tau_{\text{cy}} (i)-\tau_{\text{sy}}(i)]\big{\}}\cos(\varphi_{n})\\ \qquad\qquad+\{[e_{\text{cy}}(i)-e_{\text{sy}}(i)+z\tau_{\text{cx}}(i)]\!+\!z _{\text{sn}}[\tau_{\text{cx}}(i)-\tau_{\text{ex}}(i)]\}\sin(\varphi_{n}). \end{array}  \tag{8}\]  
where, \(k_{n}=\varphi_{n}/\delta\)\((n=1,2,\cdots,5)\); \(D_{n}\)\((n=1,2,\cdots,5)\) are the initial zero-setting values of the probes. Similarly to the three-point straightness EST based on DFT [5; 16], the weighted output of Probes 1-5 should be \(t(i)=\sum_{n=1}^{5}c_{n}t_{n}(i)\). Let the weighted coefficients \(c_{n}\)\((n=1,2,\cdots,5)\) be \(\sum_{n=1}^{5}c_{n}\cos(\varphi_{n})=0\) and \(\sum_{n=1}^{5}c_{n}\sin(\varphi_{n})=0\), and \(\sum_{n=1}^{5}c_{n}z_{\text{sn}}\cos(\varphi_{n})=0\), \(\sum_{n=1}^{5}c_{n}z_{\text{sn}}\sin(\varphi_{n})=0\). The weighted output \(t(i)\)\((i=0,1,\cdots,N\times M-1)\) can be rewritten as: \(t(i)=\sum_{n=1}^{5}c_{n}r(i+k_{n})-D\)Where, \(D=\sum_{n=1}^{5}c_{n}D_{n}\) is the weighted initial zero-setting value. Obviously, all error motions of the cylinder and carriage are removed from \(t(i)\). Moreover, applying the DFT to \(t(i)\) and introducing the time-shift theorem, we can obtain the \(p\)th\((p=0,1,\cdots,N\times M-1)\) harmonic vectors of the weighted output \(t(i)\)\((i=0,1,\cdots,N\times M-1)\) as follows:  
\[\mathbf{T}(p)=\mathrm{DFT}[t(i)]=\mathbf{R}(p)\times\mathbf{\Omega}(p)- \mathbf{D}(p);\mathbf{\Omega}(p)=\sum_{n=1}^{5}c_{n}\exp\left(\frac{ip\varphi_{ n}}{M}\right). \tag{9}\]  
where, \(\mathbf{D}(p)=\mathrm{DFT}[D]\). If the transfer function \(\mathbf{\Omega}(p)\
eq 0\)\((p=0,1,\cdots,N\times M-1)\), the \(p\)th \((p=0,1,\cdots,N\times M-1)\) harmonic vectors of the spiral line profile on the cylinder are:  
\[\mathbf{R}(p)=\frac{\mathbf{T}(p)+\mathbf{D}(p)}{\mathbf{\Omega}(p)}. \tag{10}\]  
According to the properties of the DFT, \(\mathbf{D}(p)|_{p=0}\
eq 0\) and \(\mathbf{D}(p)|_{p\
eq 0}=0\).