## 4 Parameters Design and Performance Analysis
### The Design of \(Q_{2}\) and the ADA Improvement with DOB  
The FOG-based DOB method depends on the model of the platform. The fine tracking model of the MOTS is as follows, which contains a differential element, a mechanical resonance element with natural frequency \(\omega_{n}\), and an inertial element with electrical time constant \(T_{e}\). Then, with the spectrum fitting method in system identification, the detailed parameters could be determined. Commonly, \(\omega_{n}\) is several Hz and \(T_{e}<<1\). \[G_{p}=\frac{Ks}{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2\overline{s}}{\omega_{n}^{ 2}}s+1}\cdot\frac{1}{T_{e}s+1} \tag{19}\]  
Before adding the DOB structure into the inner loop, we should complete the design of the velocity closed loop. To get a high-bandwidth inner loop with a sufficient PM and GM, the controller can be set as follows, which would make the open-loop transfer function as an approximate integration element. The anti-resonant link can be used to compensate the resonance in \(G_{p}\), \(T_{e}s+1\) to promote the phase lag, and the inertial element can be used to filter the high-frequency noise. \[C_{p}=\frac{k_{p}\Big{(}\frac{s^{2}}{\omega_{n}^{2}}+\frac{2\overline{s}}{ \omega_{n}}s+1\Big{)}}{s^{2}}\cdot\frac{(T_{e}s+1)}{T_{p}s+1} \tag{20}\]  
After establishing the velocity closed loop, the position open-loop transfer function could still be an approximate integration element and the position controller \(C_{p}\) should be set as a proportional link according to Section 4.1. After completing the VPDC design, the DOB could be introduced without affecting the stability of the system. According to Figure 6 and Equation (12), the ideal DOB controller \(Q_{2}\) should be the inverse of \(G_{p}\), as shown below,  
\[Q_{2}=(G_{v})^{-1}=\frac{\Big{(}\frac{s^{2}}{\omega_{n}^{2}}+\frac{2\overline{ s}}{\omega_{n}}s+1\Big{)}(T_{e}s+1)}{Ks} \tag{21}\]  
Figure 7: The simulation of the performance promotion with EBO. In Equation (21), the order of the numerator order is higher than that of the denominator, which cannot be accomplished in physics. In addition, as the FOG's signal is commonly accompanied with drift at the low frequency, the integration in \(Q_{2}\) would exacerbate this situation and result in the saturation of the driver, which would affect the stability. To solve the problem, the controller \(Q_{2}\) should make a compromise, although this would sacrifice some performances at the very low and high frequency. The practical controller is shown as follows. \[Q_{2}=\frac{\frac{s^{2}}{\omega_{n}^{2}}+\frac{2\sigma}{\omega_{n}}s+1}{K(s+b )(T_{1}s+1)} \tag{22}\]  
In \(Q_{2}\), the integral is changed to be \(1/(s+b)\), making the integral effect disappear at the low frequency. What's more, an inertial element, in which \(T_{1}<<1\), is used to filter the high-frequency signals and abandon the compensation for high frequency disturbances because the model of the platform is commonly not accurate at the very high frequency. From Equation (12), the ADA promotion from DOB is due to \(1-Q_{2}\widetilde{G}_{v}\), and now attention is attached to its value. \[ \begin{array}{ll}1-Q_{2}\widetilde{G}_{v}&=1-\frac{\frac{s^{2}}{\omega_{n}^ {2}}+\frac{2\sigma}{\omega_{n}}s+1}{K(s+b)(T_{1}s+1)}\cdot\frac{Ks}{\frac{s^{2 }}{\omega_{n}^{2}}+\frac{2\sigma}{\omega_{n}}s+1}\frac{1}{T_{s}s+1}\\ &=1-\frac{s}{(s+b)(T_{1}s+1)(T_{s}s+1)}\cdot\frac{1}{K(s+1)}\\ &=\frac{T_{1}T_{s}s^{3}+(T_{1}+T_{e}+bT_{1}T_{e})s^{2}+(bT_{1}+bT_{1})s+b}{T_ {1}Ts^{3}+(T_{1}+T_{e}+bT_{1}T_{e})s^{2}+(1+bT_{1}+bT_{e})s+b}\end{array}  \tag{23}\]  
At the very low frequency, \(1-Q_{2}\widetilde{G}_{v}\approx b/b=1\), there is no improvement for ADA. At the medium frequency, \(1-Q_{2}\widetilde{G}_{v}\approx b/(s+b)\), there is a maximum \(-20\) dB promotion and as \(\omega\) grows, the promotion is smaller. At the very high frequency, \(1-Q_{2}\widetilde{G}_{v}\approx(T_{1}T_{e}s^{3}/T_{1}T_{e}s^{3})\approx 1\), the lift disappears again.