## 2 Modeling of Flute-grinding Processes
### Kinematic of CNC Flute-Grinding  
To demonstrate the flute-grinding operations, another framework \(\mathbf{O_{T}}\) is established in Figure 2, which is denoted as a tool coordinate system. In the modeling, \(\mathbf{O_{T}}\) is static while \(\mathbf{O_{G}}\) is moving with the grinding wheel. The flute-grinding operations consist of two steps: (1) wheel set-up and (2) wheel moving with a helix trajectory. Figure 1: Modeling of conical grinding wheel. In the wheel set-up operation, the wheel is configured in \(\mathbf{O_{T}}\) with a specified location and orientation shown in Figure 2. The wheel location is defined by the wheel center \(\mathbf{O_{G}}\), which is denoted by the coordinate value \([dx\;dy\;dz]\). The wheel orientation is defined as an angle, which can be viewed as rotating the grinding wheel about the \(\mathbf{Y_{T}}\) axis in a counter clockwise direction through the angle \(\beta\). To this end, the set-up operation in the tool coordinate system can be expressed using the homogeneous transformation matrix in Equation (3). \[\mathbf{T_{M_{1}}}=\left[ \begin{array}{cccc}\cos\beta&0&\sin\beta&dx\\ 0&1&0&dy\\ -\sin\beta&0&\cos\beta&dz\\ 0&0&0&1\end{array} \right]. \tag{3}\]  
In the wheel helix motion, the wheel moves along the \(\mathbf{Z_{T}}\) axis with a translation velocity \(v\) while the cutter rotates about the \(\mathbf{Z_{T}}\) axis with a velocity \(\omega\). In the tool coordinate system, the kinematics matrix of the helix motion is represented as follows:  
\[\mathbf{T_{M_{1}}}=\left[ \begin{array}{cccc}\cos(\omega\cdot t)&0&-\sin( \omega\cdot t)&0\\ \sin(\omega\cdot t)&1&\cos(\omega\cdot t)&0\\ 0&0&1&v\cdot t\\ 0&0&0&1\end{array} \right], \tag{4}\]  
where \(t\) represents the grinding time. To guarantee the helix angle \(\lambda\), the translation \(v\) and the rotation \(\omega\) is supposed to satisfy the following condition:  
\[\text{cot}\lambda=\frac{v}{r_{T}\cdot\omega}. \tag{5}\]  
Based on the above operations, the kinematic of grinding wheels can be obtained with regarding machining time \(t\) in the tool coordinate system by integrating Equations (1)-(5), listed in Equation (6). In addition, to simplify the calculation, the rotation speed \(\omega\) is generally set as \(1\) in the following equations. \[ \begin{array}{c}{}^{T}W(h,\theta,t)={}^{T}\!M_{2}\cdot{}^{T}\!M_{1}\cdot{}^{ G}W(h,\theta)=\\ \left[\begin{array}{c}dx\cdot\cos-dy\cdot\sin t+h\cdot\sin\beta\cdot\cos t- f\cdot\sin\theta\cdot\sin t+f\cdot\cos\beta\cdot\cos\theta\cdot\cos t\\ dy\cdot\cos t+dx\cdot\sin t+h\cdot\sin\beta\cdot\sin t+f\cdot\sin\theta\cdot \cos t+f\cdot\cos\beta\cdot\cos\theta\cdot\sin t\\ h\cdot\cos\beta+v\cdot t-f\cdot\sin\beta\cdot\cos\theta+dz\end{array}\right].\end{array}  \tag{6}\]  
Figure 2: Flute-grinding processes: (**a**) \(\text{X}_{\text{T}}\text{Y}_{\text{T}}\) view; (**b**) \(\text{X}_{\text{G}}\text{Y}_{\text{G}}\) view. Geometrically, the flutes are generated by the envelope of grinding wheels. The envelope surface consists of a group of curves, which are called the contact curve [19]. As shown in Figure 3, the contact curve is composed of two parts. One is generated by the wheel surface, which can be deduced using envelope theory. The other is formed by part of the wheel edges. For the first part of the contact curve, it can be obtained using the conjugate theory as shown in Equation (7):  
\[{}^{T}N\cdot{}^{T}V=0, \tag{7}\]  
where \({}^{T}N={}^{T}M_{2}\cdot{}^{T}M_{1}\cdot{}^{G}N\) and \({}^{T}V(h,\theta,t)=\frac{d\left({}^{T}W(h,\theta,t)\right)}{dt}\). By solving Equation (7), the equation of this contact curve can be deduced as Equation (8). \[ \begin{array}{c}v\cdot\left(dz-\sin\beta\cdot\cos\theta+\cot\alpha\cdot\cos \beta\right)-\left(dy+\sin\theta\cdot f\right)\cdot\left(dx+\cot\alpha\cdot \sin\beta+\cos\beta\cdot\cos\theta\right)\\ \qquad\qquad\qquad\qquad\qquad+\left(dy+\sin\theta\right)\cdot\left(dx+h\cdot \sin\beta+\cos\beta\cdot\cos\theta\cdot f\right)=0.\end{array}