## 3 Robot Dynamic Modeling  
### Differential Equations of Motion of the Robot  
The Lagrange equation in matrix form describing the motion of the robot has form (8).  
\[\left[\mathrm{M(q)}\right]_{\left(8\times 8\right)}\ddot{\bar{\mathrm{q}}}_{ \left(8\times 1\right)}+\left[\mathrm{C(q,\dot{\mathrm{q}})\dot{\mathrm{q}} }\right]_{\left(8\times 1\right)}+\left[\mathrm{G(q)}\right]_{\left(8\times 1 \right)}+\left[\mathrm{Q}\right]_{\left(8\times 1\right)}=\left[\mathrm{U}\right]_{ \left(8\times 1\right)} \tag{8}\]  
In which \(\left[\mathrm{M(q)}\right]_{8\times 8}\) is the mass matrix of the robot system, calculated through mass \(\mathrm{m_{i}}\), translation Jacobian matrix \(\mathrm{J_{Ti}}\), rotation Jacobian matrix \(\mathrm{{}^{cI}}_{\mathrm{J_{Ri}}}\) and inertia tensor \(\mathrm{{}^{C}}\mathrm{\Theta_{Ci}}\) of link i (9).  
\[\left[\mathrm{M(q)}\right]_{\left(8\times 8\right)}=\left[\sum_{i=1}^{8} \left(\mathrm{J_{Ti}^{T}m_{i}J_{Ti}+^{i}J_{Ri}^{T}\dot{\mathrm{\mathrm{d}}} \mathrm{\Theta_{ci}}^{cI}j_{Ri}}\right)\right]_{\left(8\times 8\right)} \tag{9}\]  
\(\left[\mathrm{C(q,\dot{\mathrm{q}})\dot{\mathrm{q}}}\right]_{8\times 1}\) is the vector of the generalized forces of Coriolis and centrifugal inertial forces, calculated through the partial derivative of the elements of matrix \(\left[\mathrm{M(q)}\right]\) with respect to the generalized coordinates \(\mathrm{q}\), and the generalized velocity \(\dot{\mathrm{q}}\) (10).  
\[\left[\mathrm{C(q,\dot{\mathrm{q}})\dot{\mathrm{q}}}\right]_{\left(8\times 1 \right)}=\left[\mathrm{c}_{1},\mathrm{c}_{2},\ldots,\mathrm{c}_{8}\right]^{ \mathrm{T}};\ \mathrm{c}_{j}=\sum_{k,l=1}^{8}\left(k,l;j\right)\dot{\mathrm{q}}_{k}\dot{ \mathrm{q}};\ \ \left(k,l;j\right)=\frac{1}{2}\left(\frac{\partial\mathrm{m_{kj}}}{ \partial\mathrm{q_{li}}}+\frac{\partial\mathrm{m_{lj}}}{\partial\mathrm{q_{k}}} -\frac{\partial\mathrm{m_{kl}}}{\partial\mathrm{q_{j}}}\right) \tag{10}\]  
(k,l,j) is Christoffel notation with 3 indexes of the first kind; \(\mathrm{m_{kl}}\) (k,l = 1,..., 8) are the elements of matrix \(\left[\mathrm{M(q)}\right]\).  
\(\left[\mathrm{G(q)}\right]_{8\times 1}\) is the vector of the generalized forces of the conservative forces (11).  
\[\left[\mathrm{G(q)}\right]_{\left(8\times 1\right)}=\left[\mathrm{G}_{1}, \mathrm{G}_{2},\ldots,\mathrm{G}_{8}\right]^{\mathrm{T}};\mathrm{G}_{\mathrm{ j}}=\frac{\partial\Pi}{\partial\mathrm{q_{j}}} \tag{11}\]  
\(\Pi\) is the potential energy of the system.  
\(\left[\mathrm{U}\right]_{8\times 1}\) is the vector of the generalized forces of the driving forces/torques (12).  
\[\left[\mathrm{U}\right]_{\left(8\times 1\right)}=\left[\mathrm{U}_{1}, \mathrm{U}_{2},\ldots,\mathrm{U}_{8}\right]^{\mathrm{T}};\mathrm{U}_{\mathrm{ i}}=\tau_{\mathrm{i}} \tag{12}\]  
\(\tau_{\mathrm{i}}\) is the driving force at joint i. \(\tau_{\mathrm{i}}\) is the force or torque depending on the prismatic joint or revolute joint.  
\(\left[\mathrm{Q(q)}\right]_{8\times 1}\) is the vector of the generalized forces of the non-conservative forces, including the friction forces at the contact point between the tool and the workpiece, the forces that separate the chip from the workpiece; the acting forces between the tool and the workpiece are called the cutting forces, collectively. In addition, there are also the friction forces at the joints, the abnormal perturbation from the environment, etc., of which the value of cutting forces and the effect of cutting forces on motion are great. Therefore, this paper limits the investigation of cutting forces and other types of force are ignored. The next subsection introduces the investigation of cutting forces in milling process with the robot.