## 4 Simulation Results and Discussions  
The output displacements of the PA and slider in the direction of the \(x\)-axis and \(y\)-axis of the 2-DOF PIDM were compared to the output displacements of the PA and slider of the two corresponding independent 1-DOF PIDMs in order to analyze the coupled motion interactions. The PAs were actuated by sawtooth driving voltage signals. The curves of output displacements of the PA in the direction of the \(x\)-axis and \(y\)-axis in the 2-DOF PIDM are shown as curve-d\({}_{\rm px}\) and curve-d\({}_{\rm py}\), respectively. The output displacement curves of the PA are the periodic sawtooth curves (dotted curves), which are similar to the driving signals. The output displacement curves of the slider in the direction of the \(x\)-axis and \(y\)-axis in the 2-DOF PIDM are shown as curve-d\({}_{\rm xx}\) and curve-d\({}_{\rm sy}\), and the output curves are usually the stepping curves (solid curves). The output displacements of the PAs and sliders in the two corresponding independent 1-DOF PIDMs are curve-d\({}_{\rm p1}\), curve-d\({}_{\rm p2}\), curve-d\({}_{\rm s1}\), and curve-d\({}_{\rm s2}\), respectively. The curve-d\({}_{\rm p1}\) and curve-d\({}_{\rm s1}\) correspond to the curves of the 2-DOF PIDM (curve-d\({}_{\rm px}\) and curve-d\({}_{\rm s3}\)), which was driven by the \(x\)-axis alone. The curve-d\({}_{\rm p2}\) and curve-d\({}_{\rm s2}\) correspond to the curves of the 2-DOF PIDM (curve-d\({}_{\rm py}\) and curve-d\({}_{\rm sy}\)), which was driven by the \(y\)-axis alone. According to the above principles, curve-d\({}_{\rm px}\) and curve-d\({}_{\rm p1}\) are the same dotted curves, and curve-d\({}_{\rm py}\) and curve-d\({}_{\rm p2}\) are the same dotted curves. In order to verify the correctness of the friction distribution in the two orthogonal coupled motions, the 2-DOF PIDM was excited by a driving voltage of 100 V at a frequency of 10 Hz with a duty ratio of 98% in the direction of the \(x\)-axis and a driving voltage of 0 V at a frequency of 10 Hz with a duty ratio of 98% in the direction of the \(y\)-axis. In addition, 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 driving voltage of 0 V. The results of the mentioned eight curves are shown in Figure 5a. The 2-DOF PIDM was driven by a voltage of 100/\(\surd\)2 V at the frequency of 10 Hz with duty ratio of 98% in both the direction of the \(x\)-axis and \(y\)-axis. Simultaneously, one of the two  
Figure 4: Simulation block: (**a**) 2-DOF whole block diagram; (**b**) 2-DOF subsystem block diagram; (**c**) 1-DOF whole block diagram; (**d**) 1-DOF subsystem block diagram. corresponding independent 1-DOF PIDMs was driven by a voltage of 100 V at a frequency of 10 Hz with duty ratio of 98%, and the other was driven by a voltage of 0 V. The results of the mentioned eight curves are shown in Figure 5b. The results in Figure 5a show that curve-d\({}_{\text{px}}\) and curve-d\({}_{\text{p1}}\) are the same sawtooth curves, and the results of curve-d\({}_{\text{xx}}\) and curve-d\({}_{\text{s1}}\) are the same output stepping displacement curves. All curves in the \(y\) direction and the corresponding 1-DOF PIDM are horizontal lines with a magnitude of 0. The results in Figure 5b show that curve-d\({}_{\text{px}}\) and curve-d\({}_{\text{py}}\) are the same sawtooth curves (dotted curves), and curve-d\({}_{\text{xx}}\) and curve-d\({}_{\text{sy}}\) are also the same sawtooth curves (solid curves). Figure 5b shows that the maximum value of curve-d\({}_{\text{p1}}\) is about \(\surd\) times than that of curve-d\({}_{\text{px}}\) (in one cycle), and the maximum value of curve-d\({}_{\text{s1}}\) is about \(\surd\) times than that of curve-d\({}_{\text{xx}}\) (in one cycle). The results indicate that the two-dimensional extended Karnopp friction model in the 2-DOF parallel linear PIDM is correct.