## 3 Titanium Alloy Turning Process Morphology Correction Method
### Corrective Modeling of Surface Morphology for Turning of Titanium Alloys  
The topographical parameters of titanium alloy machined surfaces exhibit stochastic variations across distinct sampling locations. These spatial-dependent characteristics were mathematically modeled through Equations (24)-(29), which establish functional relationships between surface parameters and sampling coordinates. \[x_{k}^{i}\Big{(}z_{k}^{i},\theta_{k}^{i}\Big{)}=\Big{\{}x_{k}^{1}\Big{(}z_{k}^{1}, \theta_{k}^{1}\Big{)},x_{k}^{2}\Big{(}z_{k}^{2},\theta_{k}^{2}\Big{)},x_{k}^{3} \Big{(}z_{k}^{3},\theta_{k}^{3}\Big{)},\ldots,x_{k}^{0}\Big{(}z_{k}^{0},\theta _{k}^{0}\Big{)}\Big{\}} \tag{24}\]  
\[x_{m}^{i}\Big{(}z_{m}^{i},\theta_{m}^{i}\Big{)}=\Big{\{}x_{m}^{1}\Big{(}z_{m}^{ 1},\theta_{m}^{1}\Big{)},x_{m}^{2}\Big{(}z_{m}^{2},\theta_{m}^{2}\Big{)},x_{m} ^{3}\Big{(}z_{m}^{3},\theta_{m}^{3}\Big{)},\ldots,x_{m}^{0}\Big{(}z_{m}^{0}, \theta_{m}^{0}\Big{)}\Big{\}} \tag{25}\]  
\[\Delta x_{k}^{i}\Big{(}z_{k}^{i},\theta_{k}^{i}\Big{)}=\Big{\{}\Delta x_{k}^{1 }\Big{(}z_{k}^{1},\theta_{k}^{1}\Big{)},\Delta x_{k}^{2}\Big{(}z_{k}^{2},\theta _{k}^{2}\Big{)},\Delta x_{k}^{3}\Big{(}z_{k}^{3},\theta_{k}^{3}\Big{)},\ldots, \Delta x_{k}^{0}\Big{(}z_{k}^{0},\theta_{k}^{0}\Big{)}\Big{\}} \tag{26}\]  
\[\Delta x_{m}^{i}\Big{(}z_{m}^{i},\theta_{m}^{i}\Big{)}=\Big{\{}\Delta x_{m}^{1} \Big{(}z_{m}^{1},\theta_{m}^{1}\Big{)},\Delta x_{m}^{2}\Big{(}z_{m}^{2},\theta _{m}^{2}\Big{)},\Delta x_{m}^{3}\Big{(}z_{m}^{3},\theta_{m}^{3}\Big{)},\ldots, \Delta x_{m}^{0}\Big{(}z_{m}^{0},\theta_{m}^{0}\Big{)}\Big{\}} \tag{27}\]  
\[R_{m}^{i}\Big{(}z_{m}^{i},\theta_{m}^{i}\Big{)}=\Big{\{}R_{m}^{1}\Big{(}z_{m}^ {1},\theta_{m}^{1}\Big{)},R_{m}^{2}\Big{(}z_{m}^{2},\theta_{m}^{2}\Big{)},R_{m }^{3}\Big{(}z_{m}^{3},\theta_{m}^{3}\Big{)},\ldots,R_{m}^{0}\Big{(}z_{m}^{0}, \theta_{m}^{0}\Big{)}\Big{\}} \tag{28}\]  
\[Ra^{i}\Big{(}z_{m}^{i},\theta_{m}^{i}\Big{)}=\Big{\{}Ra^{1}\Big{(}z_{m}^{1}, \theta_{m}^{1}\Big{)},Ra^{2}\Big{(}z_{m}^{2},\theta_{m}^{2}\Big{)},Ra^{3}\Big{(} z_{m}^{3},\theta_{m}^{3}\Big{)},\ldots,Ra^{0}\Big{(}z_{m}^{0},\theta_{m}^{0} \Big{)}\Big{\}} \tag{29}\]  
Variations in machining parameters necessitate the identification of deterministic surface morphology variables in turning operations. Multivariate regression models through Equations (24)-(29) were developed to establish correlations between process parameters, sampling locations, and topographical descriptors of machined surfaces. \[S_{a}=\{x_{k}(z_{k},\theta_{k}),x_{m}(z_{m},\theta_{m}),\Delta x_{k}(z_{k}, \theta_{k}),\Delta x_{m}(z_{k},\theta_{m}),R_{m}(z_{m},\theta_{m}),Ra\} \tag{30}\] \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Plan**} & \multicolumn{3}{c}{**Turning Process Parameters**} \\ \cline{2-4}  & _ap_ **(mm)** & \(n\) **(r/min)** & \(f\) **(mm)** \\ \hline
1 & 0.1 & 590 & 0.1 \\
2 & 0.15 & 590 & 0.1 \\
3 & 0.2 & 590 & 0.1 \\
4 & 0.25 & 590 & 0.1 \\
5 & 0.2 & 491 & 0.1 \\
6 & 0.2 & 786 & 0.1 \\
7 & 0.2 & 1180 & 0.1 \\
8 & 0.2 & 590 & 0.06 \\
9 & 0.2 & 590 & 0.14 \\
10 & 0.2 & 590 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 3: Turning experiment plan. Figure 5: Ultra-depth of field microscope measurement site. Let's denote the characteristic covariates within the parameter set. The corresponding regression model is expressed by Equation (31), with its coefficients systematically summarized in Table 4.  
\[s_{a}=\beta\cdot{a_{p}}^{\alpha_{1}}\cdot{n^{\alpha_{2}}}\cdot{f^{\alpha_{3}}} \cdot{z^{\alpha_{4}}}\cdot{\theta^{\alpha_{5}}}\cdot{s_{0}} \tag{31}\]  
where \(\beta\), \(\alpha_{1}\)-\(\alpha_{5}\) are the coefficients of the fitted regression equation, and \(s_{0}\) represents the value of each characteristic parameter corresponding to the ideal uniform surface. When the machined surface is ideal and uniform, \(\beta\) = 1, \(\alpha_{1}\)-\(\alpha_{5}\) are all 0, meaning that \(s_{a}=s_{0}\). \(S_{0}\) is related to tool parameters, process parameters, and other factors, with the set of variables shown in Equation(32).