## 3 Determination of Wheel Location and Orientations with a 2D Projection
### Calculation Procedure with the Improved GA and PSO
In addition, the current model ignored the definition of various constraints in machining, which would strongly affect the machining accuracy and calculation stability. In the present work, a novel projection model for flute-grinding operations was developed to generate the grinding wheels' configuration. Based on the projection model, the wheel's location was re-formulated with the projection parameters, which simplified the following calculation of machined flute parameters. To minimize the flute-grinding errors regarding the wheel configuration, the projection model was used to generate the proper initial points and was integrated with the GA and PSO as a heuristic regulation. In the numerical simulation, the improved GA and PSO are more accurate, efficient, and robust, with a wide range of applications for various flute sizes. It is noted that the proposed flute-grinding algorithm was verified with a simulation-based method. For the actual grinding, a great deal of topics, such as the dynamics of the grinding machine, the material of the grinding wheel and work-piece, the post-processing, the grinding speed, etc., should be considered in the future experiments. Conceptualization, Y.F., L.W. and J.Y. ; formal analysis, Y.F., L.W. and J.Y. ; funding acquisition, L.W. ; Methodology, Y.F., and L.W. ; supervision, L.W. and J.L. ; Validation, Y.F. and L.W. ; writing--original draft, Y.F., L.W. and J.Y. ; writing--review and editing, Y.F., L.W. and J.Y. All authors have read and agreed to the published version of the manuscript. This work was partially supported by the Funds of the Key Research and Development Plan of Shandong Province (2019 CSF108005), the Shandong Provincial Natural Science Foundation, China (ZR2017 BEE018) and China Postdoctoral Science Foundation (2016M592182). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case No.** & \(\beta\) & \(\theta_{\mathbf{c}}\) & \(dx\) & \(dy\) \\ \hline F1 & 52.9353 & 87.7783 & 0.4304 & 30.1917 \\ F2 & 50.3040 & 82.2274 & 1.7030 & 30.1838 \\ F3 & 49.6645 & 82.7015 & 1.6804 & 30.4926 \\ F4 & 53.2624 & 95.5550 & \(-\)3.0894 & 80.4462 \\ F5 & 48.4607 & 80.5059 & 6.3067 & 79.4745 \\ F6 & 55.3177 & 100.8935 & \(-\)5.7797 & 80.4662 \\ F7 & 48.4607 & 77.1924 & 9.5815 & 83.9187 \\ F8 & 54.5717 & 85.3600 & 3.2566 & 89.8680 \\ F9 & 57.7373 & 99.4303 & \(-\)6.3210 & 91.4772 \\ F10 & 52.8445 & 79.5509 & 8.6418 & 94.2074 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial setting parameters of grinding wheels calculated by IGA. Figure 8: Simulated cutting tool flutes by CATIA. **Acknowledgments:** Our deepest gratitude goes to the editors and the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature**

R

## Appendix A Representation of the Wheel's Location in the Projection Model

In the projection model, the wheel edge can be represented as:

\[r_{e}=\left[ \begin{array}{c}dx+R\cdot\cos\beta\cdot\cos\theta\\ dy+R\cdot\sin\theta\end{array} \right], \tag{1}\]The wheel edge is tangent with the core at the point \(\mathrm{P_{c}}\), which satisfies the following condition:

\[\left\{ \begin{array}{c}r_{c}=r_{e}\\ r_{c}\cdot r^{\prime}_{e}=0\end{array} \right., \tag{10}\]

where \(r_{c}=\left[ \begin{array}{c}r_{c0}\cdot\cos\theta_{c}\\ r_{c0}\cdot\sin\theta_{c}\end{array} \right]\) and \(r^{\prime}_{e}\) is the derivative of \(r_{e}\). Solving Equation (10), we get:

\[\left[ \begin{array}{c}dx\\ dy\end{array} \right]=\left[ \begin{array}{c}r_{c0}\cdot\cos\theta_{c}-\mathrm{ R}\cdot\cos\beta\cdot\cos\theta_{e}\\ r_{c0}\cdot\sin\theta_{c}-R\cdot\sin\theta_{e}\end{array} \right], \tag{11}\]

where \(\theta_{c}=\arctan\left(\frac{\tan\theta_{c}}{\cos\beta}\right)\). In summary, the wheel's location \([dx\ dy]\) can be expressed regarding the parameter \(\theta_{c}\).