## 4 Simulation Results and Discussions
The 2-DOF PIDM was driven at a frequency of 10 Hz with a voltage of 100 V and a duty ratio of 98% in both the \(x\)-axis and \(y\)-axis directions. At the same time, the two corresponding independent 1-DOF PIDMs were driven at the same conditions. The results of the mentioned eight curves are shown in Figure 6. The output of slider was important and was analyzed. The results show that the average stepping displacements of the slider for the 2-DOF PIDM in the two axes (curve-d\({}_{\text{xx}}\) and curve-d\({}_{\text{sy}}\)) are both about 19.5 \(\upmu\)m, but the average stepping displacement of the two 1-DOF PIDMs (curve-d\({}_{\text{s1}}\) and curve-d\({}_{\text{s2}}\)) are both about 6.0 \(\upmu\)m. In the parallel 2-DOF PIDM, the backlash displacement of the slider was decomposed, and the stepping displacement in the two axes was improved in a circle. The two directions of the 2-DOF PIDM were driven at the same frequency of 10 Hz with a duty ratio of 98%, but the driving voltage was 100 V in the direction of the \(x\)-axis and 50 V in the direction of the \(y\)-axis. At the same time, one of the two corresponding independent 1-DOF PIDMs was excited by a voltage of 100 V at a frequency of 10 Hz with a duty ratio of 98%, and the other was excited by a  
Figure 5: Model comparison: (**a**) 2-DOF PIDM with single input; (**b**) The driving voltage of the 1-DOF PIDM is \(\surd\) times than that of the 2-DOF PIDM in the direction of the \(x\)-axis (\(y\)-axis). Figure 6: Results of two directions in the 2-DOF PIDM and the two 1-DOF PIDMs driven at the same driving signal. voltage of 50 V at a frequency of 10 Hz with a duty ratio of 98%. The results of the eight curves are shown in Figure 7. The average stepping displacements of the slider for the 2-DOF PIDM in the two axes (curve-d\({}_{\text{xx}}\) and curve-d\({}_{\text{sy}}\)) are about 10.8 and 5.8 \(\upmu\)m, while the average stepping displacements of the two 1-DOF PIDMs (curve-d\({}_{\text{s1}}\) and curve-d\({}_{\text{s2}}\)) are about 6.0 and \(-\)3.3 \(\upmu\)m. The 2-DOF motions are quite different from the corresponding 1-DOF motions. The two directions of the 2-DOF PIDM were excited by the same driving voltage of 100 V with a duty ratio of 98%, but the driving frequency was 10 Hz in the direction of the \(x\)-axis and 20 Hz in the direction of the \(y\)-axis. At the same time, one of the two corresponding independent 1-DOF PIDMs was excited by a driving voltage of 100 V at a frequency of 10 Hz with a duty ratio of 98%, and the other was excited by a voltage of 100 V at a frequency of 20 Hz with a duty ratio of 98%. The results of the eight curves are shown in Figure 8. The average stepping displacements of the slider for the 2-DOF PIDM in the two axes (curve-d\({}_{\text{xx}}\) and curve-d\({}_{\text{sy}}\)) are about 23.8 and 43.1 \(\upmu\)m, while the average stepping displacements of the two 1-DOF PIDMs (curve-d\({}_{\text{s1}}\) and curve-d\({}_{\text{s2}}\)) are 6.0 and 44.3 \(\upmu\)m. It is obvious that the high frequency motion achieves a slight decrease and the low frequency motion obtains a large increase. The results from Figures 6-8 indicate that the coupled motion interactions of the 2-DOF parallel linear PIDM are complex and they are not a simple superposition of double 1-DOF motions.