## 3 Robot Dynamic Modeling
### Cutting Forces in Milling
The cutting force components are not constants because the materials are heterogeneous, and the depth of cut, the feed rate, etc., may be changed during the machining process. The cutting forces are complex distribution forces on thecontact area between the surface of the cutting edge and the workpiece, with the resultant force placed at the center of the contact area (placed at the upper point of the cutting edge between the cutter entry and exit angles). According to the authors of [25; 26], the approximate method of calculating the milling forces in the x, y, z directions of the Z cutting teeth is represented by (21). \[\mathrm{F_{x}}=\sum_{i=1}^{Z}\mathrm{F_{x,i}};\quad\mathrm{F_{y}}=\sum_{i=1}^{ Z}\mathrm{F_{y,i}};\quad\mathrm{F_{z}}=\sum_{i=1}^{Z}\mathrm{F_{z,i}} \tag{21}\]  
where \(\mathrm{F_{x,i}}\), \(\mathrm{F_{y,i}}\), \(\mathrm{F_{z,i}}\) are the cutting forces acting on the \(i\)th cutting edge in the x, y, z directions, respectively, calculated according to (22). \[\mathrm{F_{f,i}}(\theta_{i}(z))=\int\limits_{z_{i,1}}^{z_{i,2}}\mathrm{d} \mathrm{F_{f,i}}(\theta_{i}(z))\mathrm{d}z;\;\mathrm{f}\;=\;\mathrm{x,y,z} \tag{22}\]  
In which \(\mathrm{z_{i,1}}(\theta_{i}(z))\), \(\mathrm{z_{i,2}}(\theta_{i}(z))\) are upper and lower limitations of the cutting area of the \(i\)th cutting edge, in axial direction of the cutter shaft. \(\mathrm{d}\mathrm{F_{x}^{i}},\mathrm{d}\mathrm{F_{y}^{i}},\mathrm{d}\mathrm{F _{z}^{i}}\) are the differential cutting forces acting on the infinitesimal cutting edge segment of the \(i\)th cutting edge, in x, y, z directions, respectively (23). \[\left[ \begin{array}{c}\mathrm{d}\mathrm{F_{xi}}\\ \mathrm{d}\mathrm{F_{yi}}\\ \mathrm{d}\mathrm{F_{zi}}\end{array} \right]=\left[ \begin{array}{ccc}-\cos \theta_{i}&-\sin\theta_{i}\sin\kappa&-\sin\theta_{i}\cos\kappa\\ \sin\theta_{i}&-\cos\theta_{i}\sin\kappa&-\cos\theta_{i}\cos\kappa\\ 0&-\cos\kappa&-\sin\kappa\end{array} \right]\left[ \begin{array}{c}\mathrm{d} \mathrm{F_{ti}}\\ \mathrm{d}\mathrm{F_{ri}}\\ \mathrm{d}\mathrm{F_{ai}}\end{array} \right] \tag{23}\]  
\(\theta_{i}(z)\) is the immersion angle for the cutting edge i at axial depth of cut z (24):  
\[\theta_{i}(z)=\theta(t)+\theta_{\mathrm{pi}}-\psi(z);\mathrm{i}\;=\;1,\;\ldots, \;Z;\theta(t)=\frac{2\pi n}{60}t;\theta_{\mathrm{pi}}=(1-i)\frac{2\pi}{Z} \tag{24}\]  
\(\kappa\) is the angle between \(\mathrm{d}\mathrm{F_{ri}}\) and the cutter axial; \(Z\) is number of cutting edge/flute/tooth of the cutter; \(\Psi(z)\) is the radial lag angle; n is spindle speed (rpm). At instant t, at any point P of the cutting area, a segment of the \(i\)th cutting edge performs cutting, the differential cutting forces \(\mathrm{d}\mathrm{F_{ii}}\), \(\mathrm{d}\mathrm{F_{ri}}\), \(\mathrm{d}\mathrm{F_{ai}}\) in tangential, radial and axial directions, respectively, acting on point P, represented in Figure 2 and determined by (25). Figure 2: Representation of the cutting force model of the cutting edge i of a disk dz in milling. \[\left\{ \begin{array}{l}\mathrm{dF_{ii}}=\mathrm{K_{tc}a_{i}(\theta_{i},\kappa) db+K_{te}dS}\\ \mathrm{dF_{ri}}=\mathrm{K_{rc}a_{i}(\theta_{i},\kappa)db+K_{re}dS}\\ \mathrm{dF_{ai}}=\mathrm{K_{ac}a_{i}(\theta_{i},\kappa)dB+K_{ae}dS}\end{array} \right. \tag{25}\]  
Of which:  
Ktc, Krc, Kac are tangential, radial and axial cutting force coefficients in milling. Kte, Kre, Kae are tangential, radial and axial edge force coefficients in milling. dS is thickness of an element of the ith cutting edge. db is length of cut in the direction of the cutting velocity vector. a\({}_{i}(\theta_{i},\kappa)\) is chip thickness that cut by the segment of the ith cutting edge (26). \[\mathrm{a}(\theta_{i},\kappa)=\mathrm{S_{z}\sin\theta_{i}\sin\kappa} \tag{26}\]  
The force components according to (21)-(23) can be conveniently represented on the x, y, z axes of the tool frame O\({}_{\mathrm{E}}\)x\({}_{\mathrm{E}}\)y\({}_{\mathrm{E}}\)z\({}_{\mathrm{E}}\). Summing the cutting force system with respect to the origin of the tool frame, we obtain the torques of the cutting forces.