## 2 Sensing and Actuation Principle
### Electrostatic Actuation  
According to the electrode division, corresponding differential capacitance pairs are formed in three axes between the TM and the EH. If a non-conservative force is imposed on the TM and compels it to deviate from the center position, the capacitance sensor will generate a voltage signal caused by the change in the differential capacitance. The voltage signal will be amplified by a signal amplifier and sent to the PID controller for closed-loop feedback control. A feedback voltage from the controller is loaded on the corresponding electrode plate to generate a feedback electrostatic force between the TM and the electrode to balance the non-conservative force. Therefore, the magnitude and direction of the output feedback voltage can characterize the magnitude and direction of the non-conservative force applied to the TM. Figure 4 exhibits the basic flow diagram of the position measurement and control of the TM. Figure 3: Schematic diagram of the electrode configuration. (**a**) Electrode arrangement of each axis under the average degree of freedom; (**b**) Electrode arrangement of each axis under the non-average degree of freedom. In Figure 4, \(\rm V_{P}\) is the preload driving voltage of the electrostatic force, a is the linear acceleration of the TM relative to the EH, \(\rm\alpha\) is the angular acceleration of the TM relative to the EH, and \(\rm V_{f}\) is the feedback control voltage. Figure 5a shows the initial state of electrostatic loading, in order to explain the motion state more clearly, some local details have been enlarged. The distance between the TM and the electrode is usually much smaller than the side length of the TM. Therefore, the edge effect can be ignored. The electrostatic force of a parallel plate capacitor can be expressed as [39]:  
\[\rm F=\frac{1}{2}\epsilon_{0}\epsilon_{r}A\left(\frac{V_{P}}{d_{0}}\right)^{2} \tag{4}\]  
For a translational control process, the load control method (single channel) of IS is shown in Figure 5b. When the TM has a displacement x relative to the center position, the electrostatic forces from the upper electrode plate \(\rm F_{1}\) and the lower electrode plate \(\rm F_{2}\) are applied to the TM. The resultant of the electrostatic forces can be written as:  
\[\rm F=F_{1}-F_{2}=ma=2\epsilon_{0}\epsilon_{r}A\frac{V_{P}V_{f}\kappa^{2}-(V_{ P}^{2}+V_{f}^{2})d_{0}x+V_{P}V_{f}d_{0}^{2}}{\left(d_{0}^{2}-x^{2}\right)^{2}} \tag{5}\]  
where the higher order terms are ignored, \(\rm m\) is the mass of the TM, and a is the acceleration. In the closed-loop state, the resultant of electrostatic forces and the external force on the TM are balanced. Figure 4: Flow diagram of the electrostatic actuation. Figure 5: Types of electrostatic force voltage configuration. (**a**) Initial state of the TM; (**b**) Feedback voltage configuration under the TM translation state; (**c**) Feedback voltage configuration under the TM rotation state. When x is close to zero, the acceleration can be expressed as:  
\[\text{a}=\frac{2\varepsilon_{0}\varepsilon_{\text{r}}\text{AV}_{\text{P}}}{ \text{md}_{0}^{2}}\text{V}_{\text{f}} \tag{6}\]  
Equation (6) shows that the input acceleration is proportional to the feedback voltage. The coefficient of V\({}_{\text{f}}\) is the system acceleration gain. Similarly, for a rotation control process, when the TM rotates an angle of \(\theta\) relative to the EH (Figure 5c), according to the principle of torque balance, we can get:  
\[\alpha=\frac{2\varepsilon_{0}\varepsilon_{\text{r}}\text{AV}_{\text{P}}}{ \text{Jd}_{0}^{2}}\text{V}_{\text{f}} \tag{7}\]  
where l is the horizontal distance from the electrode center to the center of the TM, and J is the moment of inertia of the TM relative to the axis of rotation. The measurement and control of the six-degrees-of-freedom motion position of the TM are usually implemented through the channel control.