# Dynamic Analysis and Trajectory Tracking Control for a Parallel Manipulator with Joint Friction
## 5 Simulation and Experiment Analysis  
### Kinematics Simulation  
To investigate the control behavior of hip joint prosthesis motion, experiments were carried out on ISO14242-1:2014(E), in which the spin axes of the \(X\)-axis, \(Y\)-axis, and \(Z\)-axis were 18\({}^{\circ}\)-25\({}^{\circ}\), \(-\)4\({}^{\circ}\)-7\({}^{\circ}\), and \(-\)10\({}^{\circ}\)-2\({}^{\circ}\), respectively. To illustrate the role of the wear test machines, the control input is shown in Figure 4.  
Based on a preliminary parameter optimization considering the mounting dimension and maximal rotary angle of the spherical joint, maximum stroke and driving force of the linear module, and orientation working space of the moving platform, the main structural parameters of the parallel manipulator are obtained, as listed in Table 1.  
The control strategy of the PD controller based on the workspace is shown in Figure 5, and the driving force matrix of the active branch chains can be expressed as follows:  
\[F_{j}=J_{j}^{-1}[K_{d}\dot{\omega}_{j}+K_{p}\omega] \tag{15}\]  
where \(\Delta\omega_{j}=\omega_{j}^{d}-\omega_{j}\) is the attitude angle error of the moving platform.  
To avoid singular positions in the operation process of the parallel manipulator, the initial position is obtained from the motion curves in Figure 6b at zero after a 30\({}^{\circ}\) rotation  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline Length of joint lever (\(l_{l}\)) /mm & 328 & Height (\(\theta_{1}\))/mm & 899 \\ Radius (\(e\))/mm & 144 & Height (\(\theta_{2}\))/mm & 490 \\ Radius (\(E\))/mm & 265 & Lead of the lead screw (\(p\))/mm & 10 \\ \hline \end{tabular}
\end{table}
Table 1: Main structure parameters of the parallel manipulator.  
Figure 4: Loading and displacement parameters for wear-testing machines. (**a**) Motion curves; (**b**) Loading force.  
Figure 5: Block diagram of the PD controller.  
around the \(Z\)-axis. The displacements, velocities, and accelerations of the sliders were simulated as shown in Figure 6.  
As shown in Figure 6, the curves of the displacements, velocities, and accelerations of the sliders were smooth, and there were no rush points. The maximum displacements of sliders S\({}_{11}\), S\({}_{12}\), and S\({}_{13}\), corresponding to the moving platform m\({}_{1}\) were 665.58 mm, 660.47 mm, and 624.42 mm, respectively. The maximum displacements of sliders S\({}_{21}\), S\({}_{22}\) and S\({}_{23}\) corresponding to moving platform m\({}_{2}\) were 256.58 mm, 251.47 mm, and 215.42 mm, respectively. Sliders S\({}_{11}\) and S\({}_{21}\), S\({}_{12}\), and S\({}_{22}\), and S\({}_{13}\) and S\({}_{23}\) were installed on the same lead screws and the corresponding maximum velocities and accelerations are 275.15 mm/s, 267.80 mm/s, 143.97 mm/s, and 2152.40 mm/s\({}^{2}\), 2100.80 mm/s\({}^{2}\), 1270.90 mm/s\({}^{2}\), respectively.  
With moving platform m\({}_{2}\) as the control object and moving platform m\({}_{1}\) as the follower, the posture angles were detected by the attitude heading reference system installed on the moving platform. The errors in the output posture angles of the moving platform are shown in Figure 7.  
From Figure 7, it can be observed that the error of the posture angle around the \(Z\)-axis was larger than that around the \(X\)-axis and \(Y\)-axis. In the intermediate branch chain, the position of the rotation center of the moving platform m\({}_{1}\) in the absolute reference frames along the \(Z\)-axis direction was influenced by the position of the mobile platform m\({}_{2}\), and the attitude error of the moving platform m\({}_{1}\) was greater than that of the moving platform m\({}_{2}\) because of the cumulative effect of the error.  
Figure 6: Motion curves of the slider. (**a**) Displacement; (**b**) Velocity; (**c**) Acceleration.