## 5 Experiments
### Experiment 2--Helical Circular Motion Trajectory  
In Experiment 2, a helical circular motion trajectory is used to conduct space trajectory tracking control for the PH-LDR. The tracking objective is to force the moving platform along the helical circular motion trajectory. To prevent the slider and the shock from touching at the ends of the stroke of the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs during trajectory tracking, the center position of the specified helical circular trajectory is shifted by a bias that is larger than the amplitude of the trajectory. The overall trajectory has two segments: a fifth-order path that takes the slider from the zero position to the starting point of the helical circular motion trajectory, and a segment wherein the platform moves along a helical circular motion trajectory. The helical circular motion trajectory is shown in Figure 14. The trajectory is formulated as [25]:  
\[\left\{\begin{array}{l}x(t)=\gamma\sin(\theta_{i}(t))\\ y(t)=\gamma\cos(\theta_{i}(t))\\ z(t)=z_{i-1}(t)+\Delta z\end{array}\right. \tag{36}\]  
where \(\theta_{i}(t)=\theta_{i}(t-1)+\Delta\theta\), and \(\Delta\theta\), \(\Delta z\), \(\gamma\), \(x(t)\), \(y(t)\), \(z(t)\) are the increment of the angle, the increment of the z-axis, the radius, the x-axis position, the y-axis position, and the z-axis position of the moving platform, respectively. The helical circular tracking motion has a trajectory that is described by Equation (36), where the radius \(\gamma=10\) cm, \(\theta_{i}(t)\) are within the interval of \([0~{}10\pi]\), \(\Delta\theta=\pi/300\), \(\Delta z=1/50\) mm and \(z(t)\) are from \(-65\) cm to \(-60\) cm. The desired path and the system response for the moving platform for the helical circular trajectory are shown in Figure 15. The maximum of \(e_{3D}(t)\) for the overall helical circular motion is less than 0.61 cm. Figure 16 shows the desired trajectories and the system responses for the three rod-less PAs. Figure 17 shows the control voltages of the proportional servo valve for the 1 st-axis, the 2 nd-axis, and the 3 rd-axis rod-less PA when the end-effector moves along the star loop motion trajectory. The proportional servo valve can adjust a pool according to input-voltage to change compressed air that flows into the chambers of a rod-less PA. Initially, the input-voltage is set to 5 V to remain the piston at the center of the rod-less PA in the experiments. Figure 18 shows the tracking errors of the 1 st-axis, 2 nd-axis and 3 rd-axis for the rod-less PAs when the platform moves along the helical circular trajectory, where the tracking error for the 1 st-axis is within \(\pm 0.6\) cm, the tracking error for the 2 nd-axis is within \(\pm 0.7\) cm, and the tracking error for the 3 rd-axis is within \(\pm 0.7\) cm. The tracking error in the 3-DOF motion contour tracking for the overall helical circular motion is small. Figures 15-18 show that the proposed rod-less PAs based PH-LDR is effective and accurate.  
Figure 14: Helical circular trajectory.  
Figure 16: Position tracking for the helical circular trajectory for the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs.  
Figure 15: Helical circular trajectory tracking and the overall error, as defined in Equation (34).  
Figure 17: Control voltages for the helical circular for the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs.  
Figure 18: Tracking errors for the helical circular trajectory for the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs.