## 4 Example: Kinematic and Dynamic Modelling of a Real Robot  
In this section, the kinematic and dynamic modelling of a newly designed hybrid robot Class III are presented. The robot was designed and implemented with the purpose of handling material for a hot forging press shop floor. The 3D design of the robot and the robot prototype are shown in Figure 5a,b, respectively. The functional tests of the robot prototype and the numerical simulation of the kinematic and dynamic responses shows clearly the effectiveness and advantages of the method proposed in this study. The kinematic diagram of the robot is presented in Figure 6, which is designed with 4 serial links \(\left\{l_{i}\right\}_{i=1+4}\), and 4 local closed loops \(\left\{\Theta_{k}\right\}_{k=1+4}\) ( Figure 7). The reference frame \(O_{0}x_{0}y_{0}z_{0}\) is located on the ground. The coordinate systems \(O_{1}x_{1}y_{1}z_{1}\), \(O_{2}x_{2}y_{2}z_{2}\), \(O_{3}x_{3}y_{3}z_{3}\) and \(O_{4}x_{4}y_{4}z_{4}\) are located on the serial links. Figure 5: The forging robot: (**a**) The 3D design, and (**b**): The robot prototype. Figure 6: The kinematic chain of the forging robot. The joint variables of the main serial chain are \(\mathbf{q}=\left[ \begin{array}{cccccc}q_{1}&q_{2}&q_{3}&q_{4}\end{array} \right]^{T}\) (Figure 6). Each local closed loop presented in Figure 7 is a four-bar linkage which is made up of two serial links (in red colour) and other two additional links (in green colour). Note that \(\Theta_{1}\) and \(\Theta_{2}\) are the loops of Type A; \(\Theta_{3}\) and \(\Theta_{4}\) are the loops of Type B. The loop \(\Theta_{1}\) includes the link \(l_{1}\), \(l_{2}\), \(l_{11}\) and \(l_{12}\). The joint variables of the loop are \(q_{2}\), \(q_{11}\) and \(q_{12}\). The loop \(\Theta_{2}\) consists of \(l_{2}\), \(l_{3}\), \(l_{21}\) and \(l_{22}\). The joint variables of the loop are \(q_{3}\), \(q_{21}\) and \(q_{22}\). The loop \(\Theta_{3}\) consists of \(l_{1}\), \(l_{2}\), \(l_{31}\) and \(l_{32}\). The joint variables of the loop are \(q_{2}\), \(q_{31}\) and \(q_{32}\). The loop \(\Theta_{4}\) consists of \(l_{3}\), \(l_{32}\), \(l_{41}\) and \(l_{42}\). The joint variables of the loop are \(q_{3}\), \(q_{41}\) and \(q_{42}\). Table 1 presents the Denavit-Hartenberg notations of the four loops. Table 2 presents the dynamics parameters of the robot. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & \multicolumn{3}{c}{**Loop \(\Theta_{1}\)**} & \multicolumn{3}{c}{**Loop \(\Theta_{2}\)**} & \multicolumn{3}{c}{**Loop \(\Theta_{3}\)**} & \multicolumn{3}{c}{**Loop \(\Theta_{4}\)**} \\ \hline Link & \(l_{2}\) & \(l_{11}\) & \(l_{12}\) & \(l_{3}\) & \(l_{21}\) & \(l_{22}\) & \(l_{2}\) & \(l_{31}\) & \(l_{32}\) & \(l_{3}\) & \(l_{41}\) & \(l_{42}\) \\ \hline \(\theta\) & \(q_{2}\) & \(q_{11}\) & \(0\) & \(q_{3}\) & \(q_{21}\) & \(0\) & \(q_{2}\) & \(q_{31}\) & \(q_{32}\) & \(q_{3}^{\prime}\) & \(q_{41}\) & \(q_{42}\) \\ \hline \(d\) & \(0\) & \(0\) & \(q_{12}\) & \(0\) & \(0\) & \(q_{22}\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) \\ \hline \(a\) & \(L_{2}\) & \(-l_{1}\cos q_{11}\) & \(0\) & \(L_{3}-b_{3}\) & \(-l_{2}\cos q_{21}\) & \(0\) & \(L_{2}\) & \(l_{2}\) & \(0\) & \(L_{3}-b_{3}\) & \(L_{3}-b_{3}\) & \(0\) \\ \hline \(a\) & \(0\) & \(\frac{\pi}{2}\) & \(\frac{\pi-2}{2}\) & \(0\) & \(\frac{\pi-2}{2}\) & \(\frac{\pi}{2}\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(\frac{\pi-2}{2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The D-H notations of the four local closed loops. Figure 7: The local closed-loop linkages of the forging robot: (**a**) Loop \(\Theta_{1}\),(**b**) Loop \(\Theta_{2}\), (**c**) Loop \(\Theta_{3}\), and (**d**) Loop \(\Theta_{4}\). The length of the links are given as follows: \(L_{1}=0.7\); \(L_{2}=0.7\); \(L_{3}=0.8\); \(L_{4}=0.2\); \(L_{11}=0.4\); \(L_{21}=0.4\); \(L_{42}=0.2\); \(b_{1}=0.3\); \(b_{2}=0.3\); \(b_{3}=0.2\); \(c_{1}=0.2\); \(c_{2}=0.2\).