## 3 Numerical Investigation of the Actuator
This means that when high voltage is applied, it will be complicated to achieve the planar and rotational motion of the payload independently. The same studies were performed for contact points driven by Planar\({}_{2}\) and Planar\({}_{3}\) segments in order to fully investigate displacement amplitudes and indicate vibrations coupling between all actuator segments while 200V\({}_{\text{P-P}}\) is applied. A summary of the investigation is given in Figure 7.  
As can be found in Figure 7a, displacement amplitudes of contact points are similar, and differences of displacement amplitudes do not exceed 9.5%. In addition, it can be found that coupling ratios between planar segments are similar in all cases. On the other hand, displacement amplitudes of contact points located on rotary segments (Figure 7b), are low and ensure high coupling ratios i.e., up to 24.6 times. Therefore, it can be stated that the vibrations of passive planar segments will have a minor influence on motion direction when excitation of planar motion is used, and rotary segments will be almost at rest at the same conditions. Investigation of displacement amplitudes of all contacting points and vibration coupling ratio while single segment Rotary\({}_{1}\) is affected by excitation signal was performed as well. Two studies were performed at different frequency ranges while the amplitude of excitations signal was set to 50 V\({}_{\text{P-P}}\) and 200 V\({}_{\text{P-P}}\). Results of calculations are given in Figure 8.  
The maximum displacement amplitudes were obtained at a frequency of 46.18 kHz. It confirms the results of impedance and phase-frequency characteristics calculations. The maximum displacements of 5.92 \(\upmu\)m and 23.15 \(\upmu\)m were obtained when excitation voltage of 50 V\({}_{\text{P-P}}\) and 200 V\({}_{\text{P-P}}\) were amplitudes to segment Rotary\({}_{1}\), respectively. The ratio between displacement amplitude and applied voltage is 118.4 nm/V\({}_{\text{P-P}}\) and 115.7 nm/V\({}_{\text{P-P}}\), respectively. The coupling ratio between vibrations of rotary segments is up to 4.19 times when 50 V\({}_{\text{P-P}}\) was applied and 4.1 times when 200 V\({}_{\text{P-P}}\) was applied. It can be seen that the coupling ratio of vibrations between rotary segments is similar at different excitation voltages. Moreover, the coupling ratio of vibrations between rotary and planar segments is up 17.4 times while 50 V\({}_{\text{P-P}}\) was applied and up to 17.14 times while 200 V\({}_{\text{P-P}}\) was applied. In order to fully investigate the coupling ratio, the numerical investigation was performed when rotary segments Rotary\({}_{2}\) and Rotary\({}_{3}\) were affected by 200 V\({}_{\text{P-P}}\). A summary of the investigations is given in Figure 9.  
Figure 6: Displacement—frequency characteristics of six spherical contacts when segment Planar\({}_{1}\) is excited by the voltage of 50 V\({}_{\text{P-p}}\) (**a**) and 200 V\({}_{\text{P-p}}\) (**b**). As can be found in Figure 9a, displacement amplitudes of all contact points located on rotary segments are similar. The differences do not exceed 6.16%, while vibrations coupling ratios are also almost the same. The difference in the coupling ratio does not exceed 11%. Therefore, it can be stated that the influence of coupling vibrations between rotary segments will have a minor influence on angular motion direction. In addition, considering to results represented in Figure 9b, it can be found that the coupling ratio between rotary and planar segments is up to 17.39 times, while fluctuations between coupling ratio values do not exceed 6.15%. Therefore, it can be assumed that during the excitation of angular motion, the actuator will not generate planar motion and will stay in a constant position. Motion trajectories of contact spheres at the resonant frequencies of 18.017 kHz and 46.199 kHz were studied as well. Time-domain studies were set for both vibrations modes. Time ranges were set to one period of vibrations (T).