## 5 Experiments
### Experiment 1-Star Loop Motion Trajectory (a Two-Dimensional Motion)  
In Experiment 1, the PH-LDR tracks the plane trajectory of a star loop motion trajectory. The tracking objective is a star loop, which is composed of five oblique straight lines. The profile and the direction of movement for the trajectory are shown in Figure 9. The formulation is:  
\[\left\{\begin{array}{ll}1\text{-segment}:&x_{i}=-S\cdot\sin 72+x_{i-1},\quad y _{i}=S\cdot\sin 18+y_{i-1}\\ 2\text{-segment}:&x_{i}=S\cdot\sin 36+x_{i-1},\quad y_{i}=-S\cdot\sin 54+y_{i-1} \\ 3\text{-segment}:&x_{i}=x_{i-1},\quad y_{i}=S+y_{i-1}\\ 4\text{-segment}:&x_{i}=-S\cdot\sin 36+x_{i-1},\quad y_{i}=-S\cdot\sin 54+y_{i-1} \\ 5\text{-segment}:&x_{i}=S\cdot\sin 72+x_{i-1},\quad y_{i}=S\cdot\sin 18+y_{i-1} \\ \end{array}\right.. \tag{35}\]  
In Equation (35), \(S\) is the position increment, \(x_{i}\) is the \(x\)-axis trajectory, and \(y_{i}\) is the \(y\)-axis trajectory of the moving platform. The actuated joint trajectories for the PH-LDR are computed by the inverse kinematics. To prevent the sliders and the shock absorber from touching at the ends of the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs stroke during the trajectory tracking, the center position of the star motion trajectory is shifted by a bias that is larger than the amplitude of the trajectory. Initially, the moving platform moves along the fifth-order path in 3 s and the moving platform moves from the zero position to the starting point \(P_{1}=(10,0,-60\text{cm})\) for the star motion trajectory. For segment 1 to segment 5, the platform moves along a star loop path with edges of length \(S=19\text{cm}\) through five positioning points at the vertices of the star loop. The trajectory comprises six segments, which define five positioning points at the vertices of the stat: \(P_{1}=(10,0,-60\text{cm})\), \(P_{2}=(-8.1,5.9,-60\text{cm})\), \(P_{3}=(3.1-9.5,-60\text{cm})\), \(P_{4}=(3.1,9.5,-60\text{cm})\) and \(P_{5}=(-8.1,-5.9,-60\text{cm})\). The time that is required to travel along each of the segments 1 to 5 is 2 s.  
Figure 10 shows the desired star loop motion trajectory and the movement of the moving platform and the overall error \(e_{3D}(t)\). Figure 11 shows the desired trajectories and the system responses for the three rod-less PAs. Figure 12 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 spool 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 13 shows the tracking errors of the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs. The tracking error for the 1 st-axis is within \(\pm 0.6\) cm; the tracking error for the 2 nd-axis is within \(\pm 0.58\) cm; the tracking error for the 3 rd-axis is within \(\pm 0.6\) cm; and the maximum of \(e_{3D}(t)\) for the star loop motion is less than 0.65 cm. There is a large tracking error in the start loop trajectory when the command position is suddenly changed; otherwise, the tracking error is very small. The results show good contour tracking for the star loop motion trajectory.  
Figure 9: The star loop trajectory.  
Figure 11: Position tracking for the star trajectory for the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs.  
Figure 10: Star trajectory tracking and the overall error, as defined in Equation (34).  
Figure 12: Control voltages for the star trajectory for the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs.  
Figure 13: Tracking errors \(e_{i}(i=1,2,3)\) for the star trajectory for the 1 st-axis, the 2 nd-axis and the 3 rd-axis rod-less PAs.