## 3 Determination of Wheel Location and Orientations with a 2D Projection
### Grinding Operation Projection  
Geometrically, the flutes are formed by the intersection between the wheel and the bar-stock. To represent the intersection, the grinding wheel and the bar-stock were projected in the cross-section, which is shown in Figure 5. The bar-stock is simplified as a circle area and the wheel edge can be expressed with an ellipse area. The ellipse could be represented with regard to the wheel's location and orientation denoted as \(r_{\mathrm{c}}|_{(dx,dy,\beta)}\). In order to assure the accuracy of the core radius, the ellipse will intersect with the circle \(r_{\mathrm{T}}\) and tangent with the circle \(r_{c}\) at a tangent point. The tangent point is defined by the parameter \(\theta_{c}\), denoted by \(\mathrm{P_{c}}|_{(\theta_{c})}\). It can be seen that the point \(\mathrm{P_{c}}|_{(\theta_{c})}\) can be used to locate the ellipse. The algebra relation between \(\mathrm{P_{c}}|_{(\theta_{c})}\) and \(r_{\mathrm{c}}|_{(dx,dy,\beta)}\) is deduced in Appendix A. In light of the above geometrical relation, the wheel path determination equation can be re-organized concerning the parameters \(\theta_{c}\) and \(\beta\), shown in Equation (13).  
\[\left\{\begin{array}{l}f_{\mathit{take}}\left(\theta_{C},\;\beta\right)= \gamma_{0}\\ f\;_{\mathit{flute}}\left(\theta_{C},\;\beta\right)=\phi_{0}\\ f\;_{\mathit{core}}\left(\theta_{C},\;\beta\right)=r_{c0}\end{array}\right., \tag{13}\]  
To solve this equation, an optimization model is defined as following:  
\[\mathit{min}\xi(\theta_{C},\;\beta)=s.t.\xi=max\left\{\left|\frac{f_{\mathit{ make}}(\theta_{C},\;\beta)-\gamma_{0}}{\gamma_{0}}\right|\left|\frac{f_{ \mathit{flute}}(\theta_{C},\;\beta)-\phi_{0}}{\phi_{0}}\right|\right|\left| \frac{f_{\mathit{core}}(\theta_{C},\;\beta)-r_{c0}}{r_{c0}}\right|\right\}, \tag{14}\]  
where \(\xi\) is the grinding error in the following description.  
In addition, several constraints were defined as follows:  
* **Constraint 1:** the tangent point \(\mathrm{P_{c}}|_{(\theta_{c})}\) should always locate in the first or the second quadrant, \[\theta_{C}\in[0,\pi].\] (15)  
Figure 5: Illustration of the projection of the wheel edge and cutter profile within the cross-section.  
* **Constraint 2:** the wheel edge cannot be separated with the bar-stock and overcut the core radius, \[r_{c}(\theta_{C},\ \beta)=|\mathrm{O}_{T}\mathrm{P}_{C}|\in[r_{c0},r_{T}],\] (16)
* **Constraint 3:** to avoid interference, the open-angle \(\theta_{S}\) should satisfy the following condition (see Appendix B), \[\theta_{S}\geq\Omega,\] (17) To sum up, the flute-grinding operations can be re-formulated by the following optimization problem: \[\min\xi(\theta_{C},\ \beta),\]  
subjected to: Equations (15)-(17).  
Compared with current flute-grinding optimization models, this project method has two advantages: (1) instead of three decision parameters \((dx,dy,\beta)\), only two parameters \((\theta_{C},\beta)\) were required to be considered, which will simplify the calculation during the iterations; (2) three constraints could be used to confine the feasible area and generate the proper initial points, which would improve the robustness and efficiency of the optimization.