## 3 Theoretical Modeling
### Milling Force Model  
A cutting force model for cylindrical helical end milling is incorporated into the equation of motion of the robot to obtain the force vector \(\mathbf{F}_{E}\), without considering the regenerative effect and the  
Figure 2: The kinematic scheme. effect of the dynamic response of the robot on the cutting force [40; 41]. In our system, up milling is chosen due to the concern with a backlash of the system. The readers are referred to Reference [42] for more detail about the cutting force modeling of down milling, which is similar in principle to that of up milling. In this cutting force model, the instantaneous cutting forces are assumed to be proportional to the area of contact between the flute and the workpiece [43]. Then, a mathematical model of the instantaneous milling force can be constructed by calculating the uncut chip thickness in a certain time. Then, the change of the cutter edge length during milling is analyzed to obtain the continuous milling force in the whole period. Figure 3 shows the geometric size and force condition of a single cutting edge of the cylindrical helical end mill. The tool coordinate system \(\{o_{l}\_x_{t}y_{t}z_{t}\}\) (also see Figure 2) is established as follows. The origin is at the center of the milling cutter tool tip, the \(z_{t}\) axis is along the centerline of the milling cutter, the \(x_{t}\) axis is along the direction of feed rate, and the \(y_{t}\) axis is determined from the right-hand rule. \(N\) represents the number of milling teeth; \(\beta\) and \(R\) represent the helical angle and the radius of the cutter, respectively; and \(a_{p}\) and \(a_{e}\) represent the axial and radial depth of cut, respectively. Due to the existence of the helix angle of the cutter tool, the axial height of cutter teeth varies with the cutting thickness. Therefore, a microelement method is used to model a single cutter tooth. First, the cutter edge is discretized into numerous cutting-edge elements along the \(z_{t}\) axis direction, and the height of each axial element is denoted as \(d_{z}\). \(\varphi\) represents the projection angle of the entire cutting edge in the \(x_{t}o_{t}y_{t}\) plane, and this projection angle is denoted as \(\psi\) when the axial height reaches \(z^{\prime}\). Then we have the following equations [42; 43; 44]:  
\[\varphi=\frac{2\pi n_{c}}{60}t\text{ and }\psi=\frac{z^{\prime}tan\beta}{R}, \tag{3}\]  
where \(n_{c}\) represents the spindle speed with a unit of rpm and \(t\) represents time. The cutting-edge angle \(\theta\) between the microelement with an axial height of \(z^{\prime}\) and the \(y_{t}\) axis can be calculated as  
\[\theta=\varphi-\psi. \tag{4}\]  
The determination of the cutting forces requires the area of contact between the end mill and the workpiece. For an axial element of thickness \(dz\), this is equivalent to requiring the uncut chip thickness \(h\). The uncut chip thickness \(h\) with an axial height of \(z^{\prime}\) can be approximately expressed as  
Figure 3: The instantaneous milling force modeling. \[h(\theta)=f_{t}\sin(\theta), \tag{5}\]  
where \(f_{t}=v_{f}/(nN)\) represents the feed per tooth and \(v_{f}\) represents the feed rate. Because dz is very small, the area of contact with a height of dz between the flute and the workpiece at time \(t\) can be approximately equal to \(h(\theta)\)dz. Then, we can calculate the elemental cutting force at this moment. Next, the force generated by this cutting tooth can be obtained by adding up all the elemental cutting forces. This is a definite integral problem, and we can get the result by integrating the axial height numerically. The upper and lower limits of integration can be obtained from the height of the cutting tooth involved during milling. In addition, there can be multiple cutting teeth involved in the cutting at time \(t\). Then, we need to add up the forces from all the cutting teeth involved in the cutting to obtain the overall cutting force. The milling force is periodic because of the intermittence of the cutter teeth.