## 3 Determination of Wheel Location and Orientations with a 2D Projection
### Calculation Procedure with the Improved GA and PSO
In addition, two key points in \(\mathrm{P_{S}}\) and \(\mathrm{P^{\prime}_{S}}\) in the projection mode were deduced in the following:

\[\mathrm{P_{S}}=\left[ \begin{array}{c}dx+R\cdot\cos\beta\cdot\cos\theta_{2} \\ dy+R\cdot\sin\theta_{2}\end{array} \right], \tag{12}\]

\[\mathrm{P^{\prime}_{S}}=\left[ \begin{array}{c}dx+R\cdot\cos\beta\cdot\cos \theta_{2}\\ dy+R\cdot\sin\theta_{2}\end{array} \right]. \tag{13}\]

The open angle between \(\mathrm{P_{S}}\) and \(\mathrm{P^{\prime}_{S}}\) can be represented as: \(\left|\mathrm{O_{T}P_{S}}\right|=\left|\mathrm{O_{T}P^{\prime}_{S}}\right|=r_{T}\). ## Appendix B The Geometrical Condition for Interference-Free

Improper wheel setting would result in the interference in the flute-grinding operations. As shown in Figure 11, the interference is generally caused by the wheel edge grinding in the rake face. Geometrically, the interference would happen while the point \(\mathrm{P^{\prime}_{S}}\) crosses \(\mathrm{P_{S}}\) in the counterclockwise direction. To simplify this problem, the angle \(\theta_{ss}\) is introduced in Figure 12 and 13, which is defined as follows:

\[\theta_{ss} \begin{pmatrix}>0&No\ inter\ \mathit{ference}\\ =0&Critical\\ <0&Inter\ \mathit{ference}\end{pmatrix} . \tag{14}\]

In the flute-grinding operations, the angle \(\theta_{ss}\) can be obtained by mapping the angle \(\theta_{s}\) with a phase difference in the projection model: \(\theta_{ss}=\theta_{s}-\Delta\Omega\). As mentioned, in the flute-grinding process, the tool-stock rotates with speed \(\omega\), while the wheel translates in speed \(v\) along the \(\mathbf{Z_{T}}\) direction. Therefore, the points \(\mathrm{P_{S}}\) and \(\mathrm{P^{\prime}_{S}}\) would be located in the cross-section with a phase difference, which can be calculated by the following equation:

\[\Delta\Omega=\frac{f\cdot\sin\beta\cdot(\cos\theta_{2}-\cos\theta_{1})}{v} \cdot\omega. \tag{15}\]

Therefore, to avoid interference, the following constraint can be got for the open-angle \(\theta_{s}\) in the projection mode:

\[\theta_{s}\geq\frac{f\cdot\sin\beta\cdot(\cos\theta_{2}-\cos\theta_{1})}{v} \cdot\omega. \tag{16}\]

## References

* [PERSON] et al. (2016) [PERSON]; [PERSON]; [PERSON] Analytical model of milling forces based on time-variant sculptured shear surface. _Int. J. Mech. Sci._**2016**, _115-116_, 190-201. [CrossRef]
* [PERSON] et al. (2011) [PERSON]; [PERSON]; [PERSON]; [PERSON] Three-dimension surface characterization of grinding wheel using white light interferometer. _Int. J. Adv. Manuf. Tech._**2011**, _55_, 133-141. [CrossRef]
* [PERSON] et al. (2019) [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] Effect of the Relative Position of the Face Milling Tool towards the Workpiece on Machined Surface Roughness and Milling Dynamics. _Appl. Sci._**2019**, \(9\), 842. [CrossRef]
* [PERSON] et al. (2020) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Stability Analysis of Milling Process with Multiple Delays. _Appl. Sci._**2020**, _10_, 3646. [CrossRef][PERSON]; [PERSON]; [PERSON]; [PERSON] An accurate method for five-axis flute grinding in cylindrical end-mills using standard 1V1/1A1 grinding wheels. _Precis. Eng._**2016**, 43, 387-394. [CrossRef]
* _[PERSON] et al. (2013)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] A New and Accurate Mathematical Model for Computer Numerically Controlled Programming of 4Y1 Wheels in 21/2-Axis Flute Grinding of Cylindrical End-Mills. _J. Manuf. Sci. Eng._**2013**, _135_, 04100801-04100811. [CrossRef]
* [PERSON] and [PERSON] (2010) [PERSON]; [PERSON] A manufacturing model of an end mill using a five-axis CNC grinding machine. _Int. J. Adv. Manuf. Tech._**2010**, _48_, 461-472. [CrossRef]
* [PERSON] and [PERSON] (2014) [PERSON]; [PERSON]; [PERSON] Process modeling of end mill groove machining based on Boolean method. _Int. J. Adv. Manuf. Tech._**2014**, _75_, 959-966. [CrossRef]
* [PERSON] et al.