## 3 Robot Dynamic Modeling
### Cutting Forces in Milling  
The cutting forces are generated at the cutting point of the tool and the workpiece and this is a complex process force system. However, in general, the process force system can be represented as the force system from the workpiece acting on the robot including the force and the torque (force couple) with notation (\(\mathrm{F_{c}M_{c}}\)). The notation (\(\mathrm{F_{r}M_{r}}\)) represents the force and torque vectors from a tool acting on the workpiece (13). \[\mathrm{F_{c}}=\left[\mathrm{F_{x}},\mathrm{F_{y}},\mathrm{F_{z}}\right]^{ \mathrm{T}};\mathrm{M_{c}}=\left[\mathrm{M_{x}},\mathrm{M_{y}},\mathrm{M_{z}} \right]^{\mathrm{T}};\mathrm{F_{r}}=-\mathrm{F_{c}};\ \mathrm{M_{r}}=-\mathrm{M_{c}} \tag{13}\]  
The generalized force components of the force \(\mathrm{F_{c}}\) and the torque \(\mathrm{M_{c}}\) are determined by the robot kinematic chain (14). \[\mathrm{Q_{c}}=\mathrm{J_{TE}^{T}F_{C}}+\mathrm{J_{RE}^{T}M_{c}}=\left[ \mathrm{J_{TE}^{T}J_{RE}^{T}}\right]\left[\mathrm{F_{c}},\mathrm{M_{c}}\right] ^{\mathrm{T}}=\mathrm{J_{E}^{T}R_{c}} \tag{14}\]  
\(\mathrm{r_{E}}\) is the position vector of the acting point of the force \(\mathrm{F_{c}}\) with respect to the base frame, \(\mathrm{\omega_{E}}\) is the angular velocity of the end effector, determined by the robot kinematic chain. The Jacobian matrices in (14) are calculated according to (15). \[\mathrm{J_{TE}}=\frac{\partial\mathrm{r_{E}}}{\partial\mathrm{q}};\mathrm{J_{ RE}}=\frac{\partial\mathrm{\omega_{E}}}{\partial\mathrm{\dot{q}}} \tag{15}\]  
The generalized force components of the force \(\mathrm{F_{r}}\) and the torque \(\mathrm{M_{r}}\) are determined according to the platform kinematic chain (16). \[\mathrm{Q_{r}}=\mathrm{J_{T}^{T}F_{r}}+\mathrm{J_{R}^{T}M_{r}}=\left[ \mathrm{J_{T}^{T}J_{RF}^{T}}\right]\left[\mathrm{F_{r}},\mathrm{M_{r}}\right] ^{\mathrm{T}}=\mathrm{J_{r}^{T}R_{r}} \tag{16}\]  
\(\mathrm{r_{k}}\) is the position vector of acting point of the force \(\mathrm{F_{r}}\) with respect to the base frame and \(\mathrm{\omega_{k}}\) is the angular velocity of the workpiece determined by the platform kinematic chain. The Jacobian matrices in (16) are calculated according to (17). \[\mathrm{J_{Tr}}=\frac{\partial\mathrm{r_{k}}}{\partial\mathrm{q}};\mathrm{J_{ RF}}=\frac{\partial\mathrm{\omega_{k}}}{\partial\mathrm{\dot{q}}} \tag{17}\]  
Note that Equation (17) is used for the general case; with the robot system investigated in this paper, the angular velocity \(\mathrm{\omega_{k}}\) is zero because the platform carrying the workpiece has only two prismatic joints, and only performs translational motion. It is known that on the robot kinematic chain, there are only joint coordinates \(\mathrm{q_{1}}\),\(\mathrm{q_{2}}\),..., \(\mathrm{q_{6}}\); while on the platform kinematic chain, there are only joint coordinates \(\mathrm{q_{7}}\), \(\mathrm{q_{8}}\), therefore we have (18) and (19). \[\mathrm{Q_{c}}=\left[\mathrm{Q_{c1}},\mathrm{Q_{c2}},\ldots,\mathrm{Q_{c6}},0, 0\right]^{\mathrm{T}} \tag{18}\]  
\[\mathrm{Q_{r}}=\left[0,0,\ldots,0,\mathrm{Q_{r7}},\mathrm{Q_{r8}}\right]^{ \mathrm{T}} \tag{19}\]  
So, the sum of the generalized force components of the non-conservative forces is expressed as (20). \[\left[\mathrm{Q(q)}\right]_{(8\times 1)}=\mathrm{Q_{c}}+\mathrm{Q_{r}}=\left[ \mathrm{Q_{c1}},\mathrm{Q_{c2}},\ldots,\mathrm{Q_{c6}},\mathrm{Q_{r7}}, \mathrm{Q_{r8}}\right]^{\mathrm{T}} \tag{20}\]  
Recognizing that in order to calculate the generalized forces of cutting forces, the Jacobian matrices calculated according to (15) are quite complex, because the expressions for determining the position of cutting points and the angular velocity of the end effector are often cumbersome and complex. The cutting forces expressing as (13) appear in milling process depending on the cutting parameters, the geometric parameters of the chip geometry, cutting conditions, etc.