# Dynamic Analysis and Trajectory Tracking Control for a Parallel Manipulator with Joint Friction
## 5 Simulation and Experiment Analysis
### Dynamics Simulation
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline Radius of the articular head & & Radius of the thrust ball & \\ (\(r_{h}\))/mm & 15 & bearing (\(r_{t}\)) / mm & 64 \\ Friction factor of the hip joint (\(\mu_{h}\)) & 0.095 & Friction factor of the thrust ball bearing (\(\mu_{t}\)) & 0.025 \\ Mass of the joint lever (\(m_{l}\))/kg & 1.43 & Acceleration of gravity (g)/ & 9.8 \\ Rotational inertia of the lead & & & \\ screw (\(l_{l}\))/kg\(\cdot\)mm\({}^{2}\) & 54.91 & Mass of the slider (\(m_{L}\))/kg & 3.00 \\ Inertia tensor of the joint lever & diag(26,395.81, 26,339.29, & Inertia tensor of the moving & diag(248,030.50, 249,940.25, \\ (\(l_{l}\))/kg\(\cdot\)mm\({}^{2}\) & 480.07) & platform (\(l_{m}\))/kg\(\cdot\)mm\({}^{2}\) & 386,502.35) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mass and inertia parameters of the parallel manipulator. Figure 8: Contributions of each part to the input torques. (**a**) Moving platform; (**b**) Joint lever; (**c**) Slider; (**d**) Lead screw. The rotation angle, angular velocity, and angular acceleration of the servo motor are measured directly by the encoder. The attitude of the moving platform is measured by an attitude heading reference system with a single experiment time less than five minutes caused by the zero point drift of the device. Owing to the measurement delay, the pull-wire displacement sensor cannot be used for feedback of the control system, but for calibration of the zero point of the slider. As observed in Figure 11, the results from the dynamic model and the experiment showed the same trends in their variation. The drive torque of the three linear modules was similar in one cycle, which indicates that the load of the linear modules were balanced. The driving torque was larger at 0.4-0.6 s, with maximum values reached 850 N\(\cdot\)mm, 806 N\(\cdot\)mm, and 743 N\(\cdot\)mm, respectively. Further, there was a large difference in the driving torque at 0 s\(\cdot\)0.6 s and 0.6 s\(\cdot\)1 s. Combined with Figure 4b, it can be seen that the friction torque of the moving platform was larger at 0 s\(\cdot\)0.6 s because of the larger pressure output by the loading system, and the friction torque of the moving platform was smaller at 0.6-1 s because of the smaller and stable pressure output by the loading system. Figure 10: Control and data acquisition system of the parallel manipulator. Figure 9: Block diagram of the augmented PD controller. The difference between the theoretical and actual parameters of the parallel manipulator resulted in errors in the driving torque. The errors of the three linear module drive torque were larger around 0.55 s, and the maximum values reached 158 N\(\cdot\)mm, 220 N\(\cdot\)mm, and 186 N\(\cdot\)mm, respectively, which may be caused by the rotation direction variation of the moving platform. At this point, the hydraulic loading system outputs a larger loading force, and the nonlinear dynamics become significant. Experiments were conducted to verify the correctness of the dynamic model, which demonstrated that the established dynamic model can accurately describe the performance of the parallel manipulator.