# Dynamic Analysis and Trajectory Tracking Control for a Parallel Manipulator with Joint Friction
## 5 Simulation and Experiment Analysis
### Dynamics Simulation  
Due to the large number of joints in the parallel Manipulator with two moving platforms, the control performance of parallel manipulator was significantly affected by the friction of the motion joints. Especially when the robot was used for the artificial hip joint friction and wear test, the 300 N-3000 N dynamic load was loaded directly on the joint, and the friction at the artificial joint and thrust ball bearing would have a more significant effect. During the dynamic simulation of the hip joint simulator, only the effect of the frictional force of the intermediate branch chain with higher loading force on the dynamic characteristics of the manipulator is discussed. Subsequently, the Coulomb friction model of the artificial joints and thrust ball bearings is established. \[\mathbf{\tau}_{m\!f\!f}=\frac{\mathbf{\omega}_{j}}{\|\mathbf{\omega}_{j}\|}r_{h}\mu_{h}\| \mathbf{F}_{e\!j}\|+\frac{\mathbf{\omega}_{j}}{\|\mathbf{\omega}_{j}\|}r_{t}\mu_{t}\|\mathbf{F}_ {e\!j}+m_{m}g\| \tag{16}\]  
where \(r_{h}\) and \(\mu_{h}\) denote the radii and friction factors of the hip joint, and \(r_{t}\) and \(\mu_{t}\) denote the radii and friction factors of the thrust ball bearing, respectively. The MATLAB software was used to validate the dynamic model of the parallel manipulator. In the manuscript, parts of the virtual prototype are established in the Solidworks software, and the mass and inertia parameters can be obtained, as shown in Table 2.  
From the aforementioned analysis, the contributions of each part to the input torques actuated by the motors were obtained, as shown in Figure 8.  
As shown in Figure 8, in the working process of the parallel manipulator, the maximum driving torques of the three linear modules related to the inertia of the moving platform were 300.18 N-mm, 289.79 N-mm, and 267.69 N-mm, respectively. The maximum driving torques of three linear modules related to the gravity and inertia of the joint lever were 26.22 N-mm, 30.97 N-mm, and 26.38 N-mm, respectively. The maximum driving torques of the three linear modules related to the gravity and inertia of the slider were 118.41 N-mm, 119.30 N-mm, and 115.11 N-mm, respectively. The maximum driving torques of the three  
Figure 7: Errors of the posture angles of the moving platform. (**a**) AA error; (**b**) IER error; (**c**) EE Error. linear modules related to the inertia of the lead screw were 74.261 N\(\cdot\)mm, 72.69 N\(\cdot\)mm, and 44.12 N\(\cdot\)mm, respectively. The driving torques of the three linear modules related to the moving platform were greater than those associated with the other parts. The driving torque can be reduced to a certain degree by reducing the masses of the moving platform and slider. The block diagram of the designed augmented PD controller can be seen in Figure 9.  
The dynamics experiment platform consists of an industrial control computer (IPC), a drive system, a hydraulic loading system, and a data acquisition system. The control and compensation program are programmed by LabVIEW and installed in the IPC. The servo motor is controlled by a motion control card (GTS-400-PV from Goodco) and a servo driver. The hydraulic cylinder is controlled by PLC and the hydraulic system applies the loading force by controlling the opening size of the proportional valve. The data acquisition system consists of an encoder, attitude heading reference system, and pull wire displacement sensor. The structure of the control and data acquisition system is shown in Figure 10.