## 2 Dynamics of MEMS Gyroscopes  
The schematic structure of a gyroscope model is described in Figure 1. As shown in Figure 1, the vibrating gyroscope contains a proof mass, spring beams, and dampers. The proof mass is driven by electrostatic actuators and it is constrained only to move in the X-O-Y plane where states including position and velocity information of the proof mass can be acquired detecting capacity combs. Some fixed angular velocity is applied to the gyroscope along the Z axis. Gyroscope dynamics [28; 29] can be described in the following form:  
\[\left\{ \begin{array}{l}m\ddot{x}+d_{xx}\dot{x}+d_{xy}\dot{y}+k_{xx}x+k_{xy}y =u_{x}+2m\Omega_{2}\dot{y}\\ m\ddot{y}+d_{xy}\dot{x}+d_{yy}\dot{y}+k_{xy}x+k_{yy}y=u_{y}-2m\Omega_{2}\dot{x }\end{array} \right. \tag{1}\]  
\(m\) is the proof mass weight;  
\(x\) and \(y\) are the displacements of the mass; and  
\(k_{xx}k_{yy}\) and \(d_{xx}d_{yy}\) are the coefficients for springs and dampers in the X and Y directions. \(k_{xy}\) and \(d_{xy}\) are coefficients of the coupling terms which are mainly caused by the asymmetric structure due to limits of manufacture technology. Figure 1: Schematic model for a gyroscope. \(\Omega_{z}\) is the angular velocity and \(u_{x}\) and \(u_{y}\) represent the control forces in the X and Y direction. We can get the motion dynamics in non-dimensional form through dividing both sides of (1) by the resonance frequency \(\omega_{0}^{2}\), reference length \(q_{0}\) and reference mass \(m\). \[\left\{ \begin{array}{l}\ddot{\mathbf{q}}_{x}+D_{xx}\dot{\mathbf{q}}_{x}+D_{ xy}\dot{\mathbf{q}}_{y}+K_{x}{}^{2}\mathbf{q}_{x}+K_{xy}\mathbf{q}_{y}=U_{x}+2 \Theta\dot{\mathbf{q}}_{y}\\ \ddot{\mathbf{q}}_{y}+D_{xy}\dot{\mathbf{q}}_{x}+D_{yy}\dot{\mathbf{q}}_{y}+K_ {xy}\mathbf{q}_{x}+K_{y}{}^{2}\mathbf{q}_{y}=U_{y}-2\Theta\dot{\mathbf{q}}_{x} \end{array} \right. \tag{2}\]  
where \(q_{x}=\frac{x}{q_{0}}\), \(q_{y}=\frac{y}{q_{0}}\), \(\dot{q}_{x}=\frac{\dot{x}}{q_{0}\omega_{0}}\), \(\dot{q}_{y}=\frac{\dot{y}}{q_{0}\omega_{0}}\), \(\ddot{q}_{x}=\frac{\ddot{x}}{q_{0}\omega_{0}^{2}}\), \(\ddot{q}_{y}=\frac{\ddot{y}}{q_{0}\omega_{0}^{2}}\), \(D_{xx}=\frac{d_{xy}}{m\omega_{0}}\), \(D_{xy}=\frac{d_{yy}}{m\omega_{0}}\), \(\Theta=\frac{\Omega_{z}}{\omega_{0}}\), \(K_{x}=\sqrt{\frac{k_{xx}}{m\omega_{0}^{2}}}\), \(K_{y}=\sqrt{\frac{k_{yy}}{m\omega_{0}^{2}}}\), \(K_{xy}=\frac{k_{yy}}{m\omega_{0}^{2}}\), \(U_{x}=\frac{u_{x}}{m\omega_{0}\omega_{0}^{2}}\), and \(U_{y}=\frac{u_{y}}{m\omega_{0}\omega_{0}^{2}}\). (2) is rewritten in vector form as:  
\[\ddot{\mathbf{q}}+\mathbf{D}\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}=\mathbf{U}- 2\Omega\dot{\mathbf{q}} \tag{3}\]  
where \(\mathbf{q}=\left[ \begin{array}{c}q_{x}\\ q_{y}\end{array} \right]\mathbf{U}=\left[ \begin{array}{c}U_{x}\\ U_{y}\end{array} \right]\mathbf{D}=\left[ \begin{array}{cc}D_{xx}&D_{xy}\\ D_{xy}&D_{yy}\end{array} \right]\mathbf{K}=\left[ \begin{array}{cc}K_{x}{}^{2 }&K_{xy}\\ K_{xy}&K_{y}{}^{2}\end{array} \right]\), and \(\mathbf{\Omega}=\left[ \begin{array}{cc}0&-\Theta\\ \Theta&0\end{array} \right]\)  
Due to manufacture errors and environment changes, actual values for springs, dampers, and their nominal values are not exactly the same. Considering these effects, we modified system dynamics by adding extra terms concerning parameter deviation and external disturbance and the new motion equation in the described in the following form:  
\[\ddot{\mathbf{q}}=\mathbf{U}-(\mathbf{D}+2\mathbf{\Omega})\dot{\mathbf{q}}- \mathbf{K}\mathbf{q}+\mathbf{F} \tag{4}\]  
where \(\mathbf{F}\) is the lumped unknown disturbances vector including unknown system dynamics (caused by parametric deviation) and external disturbance.