## 3 Mathematical Model of the Proposed MEMS Gyroscope
### Electrostatic Tuning for Mode-Matching  
The configuration scheme of electrostatic tuning combs implemented in the MEMS gyroscope design to achieve the resonant frequency match between the drive and sense axis is shown in Figure 1. The interdigitated combs are varying-gap structures with an equal air gap and attached to both sides of the sense frame. Since the movable combs can move only in the sense direction, the comb-drive-based tuning structures only affect the sense-mode dynamics. Considering a single tuning-comb structure, the sense frame displacement in the positive \(y\)-direction leads to an increase in the upper capacitance \(C_{1t}\) and a decrease in the bottom capacitance \(C_{2t}\) created between fixed and moving comb fingers. This capacitance variation can be expressed as:  
\[\begin{array}{l}C_{1t}(y)=\frac{\epsilon_{0}A_{t}}{(y_{0t}-y)}\\ C_{2t}(y)=\frac{\epsilon_{0}A_{t}}{(y_{0t}+y)}\end{array} \tag{5}\]  
where \(\epsilon_{0}\) is the free space permittivity, \(A_{t}\) is the overlap area and \(y_{0t}\) is the nominal gap between the fingers. When a DC tuning tension \(V_{t}\) is given to the anchored comb, the moving comb finger is subjected to two electrostatic forces, namely \(F_{1t}\) and \(F_{2t}\) and expressed as nonlinear functions of the \(y\)-displacement:  
\[\begin{array}{l}F_{1t}=\frac{1}{2}\frac{\partial C_{1t}}{\partial y}V_{t}^{2 }=\frac{1}{2}\frac{\epsilon_{0}A_{t}}{(y_{0t}-y)^{2}}V_{t}^{2}\\ F_{2t}=\frac{1}{2}\frac{\partial C_{2t}}{\partial y}V_{t}^{2}=-\frac{1}{2} \frac{\epsilon_{0}A_{t}}{(y_{0t}+y)}V_{t}^{2}\end{array} \tag{6}\]  
Considering \(N_{t}\) moving fingers, the comb-drive-based electrostatic tuning structures can be modeled as a series of capacitors. Accordingly, the total electrostatic tuning force \(F_{t}\) can be obtained as:  
\[F_{t}=N_{t}(F_{1t}+F_{2t})=\frac{N_{t}V_{t}^{2}}{2}\left(\frac{\epsilon_{0}A _{t}}{(y_{0t}-y)^{2}}-\frac{\epsilon_{0}A_{t}}{(y_{0t}+y)^{2}}\right) \tag{7}\]  
The electrostatic spring constant \(k_{ty}\) due to the force nonlinearity can be calculated by taking the derivative of the total electrostatic force \(F_{t}\) with respect to the \(y\)-axis displacement:  
\[k_{ty}=\frac{\partial F_{t}}{\partial y}=-N_{t}V_{t}^{2}\left(\frac{\epsilon_ {0}A_{t}}{(y_{0t}-y)^{3}}+\frac{\epsilon_{0}A_{t}}{(y_{0t}+y)^{3}}\right) \tag{8}\]Considering \(y\ll y_{0t}\), Equation (8) can be simplified as follows:  
\[k_{ty}=\frac{\partial F_{t}}{\partial y}=-2N_{t}V_{t}^{2}\bigg{(}\frac{e_{0}A_{t }}{y_{0t}}\bigg{)} \tag{9}\]  
The electrostatic tuning force acts only in the \(y\)-direction and does not cause a shift in the drive resonant frequency \(\Omega_{d}\), while the sense resonant frequency \(\Omega_{s}\) always decreases with increasing DC tension across the tuning electrodes, according to the following expression:  
\[\Omega_{s}=\sqrt{\frac{k_{y}+k_{ty}}{(m_{d}+m_{s})}}=\sqrt{\frac{k_{y}-2N_{t} V_{t}^{2}\bigg{(}\frac{e_{0}A_{t}}{y_{0t}^{3}}\bigg{)}}{(m_{d}+m_{s})}} \tag{10}\]  
According to Equation (10), the sense resonant frequency can be tuned by adjusting the tuning voltage \(V_{t}\) applied to tuning electrodes in order to achieve the mode-matching condition, i.e., \(\Omega_{s}=\Omega_{d}\) at a certain voltage.