## 3 Determination of Wheel Location and Orientations with a 2D Projection
### Calculation Procedure with the Improved GA and PSO
What is more, to further test the efficiency of the proposed method, the computing time was recorded as shown in Figure 7b. The integrated method also showed better efficiency in convergence, which could save about 15%-40% computing time. In addition, the initial setting parameters of the grinding wheel solved by IGA are provided in Table 4. In light of the IGA solution, three instances were selected and simulated in the software CATIA. The simulated results were obtained and measured, as shown in Figure 8, which also shows that they are highly consistent with the designed flute parameters. In summary, according to the simulation tests for various optimization methods, it is demonstrated that the proposed IGA and IPSO based on the projection model is effective, efficient, and robust solving the flute-grinding problem. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wheel Parameters** & **Wheel 1** & **Wheel 2** & **Wheel 3** \\ \hline wheel width H (mm) & 5 & 20 & 40 \\ wheel radius R (mm) & 30 & 75 & 75 \\ wheel angle \(\alpha\) (deg.) & 75 & 75 & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification of grinding wheel. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameters Setting** \\ \hline GA or IGA & \
\begin{tabular}{c} initial population size: 100 \\ range of crossover probability: 0.2 \\ range of mutation probability: 0.1 \\ stopping condition: iterations \(>\) 100 or \(\xi<1\times 10^{-4}\) \\ \end{tabular} \\ \hline PSO or IPSO &
\begin{tabular}{c} initial population size: 100 \\ inertia weight: 1 \\ inertia weight damping Ratio: 0.99 \\ personal learning coefficient: 1.5 \\ global learning coefficient: 2.0 \\ stopping condition: iterations \(>\)1 00 or \(\xi<1\times 10^{-4}\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of optimization algorithms. Figure 7: Comparisons with various algorithms: **(a)** the grinding errors; **(b)** the computing time. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Flute Size** & **Case No. ** & \begin{tabular}{c} **Cutter** \\ **Radius** \\ \end{tabular}  & \begin{tabular}{c} **Desired Flute** \\ **Parameters** \\ \end{tabular}  & \begin{tabular}{c} **Flute Parameters** \\ **Solved by ICA** \\ \end{tabular}  & \begin{tabular}{c} **Flute Parameters** \\ **Solved by ICA** \\ \end{tabular}  & \ \begin{tabular}{c} **Flute Parameters** \\ **Solved by PSO** \\ \end{tabular}  & \begin{tabular}{c} **Flute Parameters** \\ **Solved by PSO** \\ \end{tabular}  \\ \hline \multirow{3}{*}{small size flutes} & F1 & 0.3 & \((0.2,6.75)\) & \((0.200,5.995,74.946)\) & / & / & / \\  & F2 & 0.5 & \((0.3,6.75)\) & \((0.300,5.996,74.976)\) & / & \((0.300,5.919,74.796)\) & / \\  & F3 & 1 & \((0.6,6.75)\) & \((0.600,5.998,75.962)\) & \((0.602,5.991,7.386)\) & \((0.600,6.000,75.027)\) & / \\ \hline \multirow{3}{*}{medium size flutes} & F4 & 7 & \((5,9.75)\) & \((5.000,9.057,0.510)\) & \((4.989,8.894,75.077)\) & \((5.000,9.000,74.997)\) & \((5.21,8.895,74.781)\) \\  & F5 & 9 & \((5,9.75)\) & \((5.000,8.996,75.009)\) & \((5.149,8.894,75.410)\) & \((5.000,9.000,75.000)\) & \((4.973,8.986,74.926)\) \\  & F6 & 11 & \((6,25,110)\) & \((6.002,5.016,109.981)\) & \((5.955,2.507,10.930)\) & \((6.000,2.500,110.020)\) & \((5.946,2.994,109.951)\) \\  & F7 & 17 & \((10,9.75)\) & \((10.000,9.002,75.005)\) & \((9.997,9.006,74.916)\) & \((10.000,8.998,75.003)\) & \((9.995,8.984,74.989)\) \\ \hline \multirow{3}{*}{large size flutes} & F8 & 20 & \((15,9.78)\) & \((1000,8.995,75.942)\) & \((14.944,9.098,75.345)\) & \((15.000,9.000,75.00)\) & \((14.979,8.995,75.139)\) \\  & F9 & 25 & \((17.25,110)\) & \((17.000,24.995,110.143)\) & \((17.681,25.571

## 5 Conclusions

In the CNC flute-grinding processes, the accuracy of generated flute parameters is determined by setting the wheel's location and orientation. The existing methods for the solution of wheel path optimization were time-consuming and cannot handle grinding the extreme-size cutters.