## 3 Modeling and Simulations of the Assembly Error Effect on the Stability Accuracy
### Modeling and Simulation of the Assembly Error Effects on Stability Accuracy
#### 3.5.3 Modeling and Simulation of Assembly Errors of Gyroscope Locations  
The gyroscopes are the core components for realizing the stabilizing function, and the assembly of them will directly affect the stability accuracy. The coordinates of the gyroscopes \([ga]\) and \([gp]\) are ideally parallel to \([p]\). Due to bad assembly, there are angular deviations between the real and the ideal locations. The deviations are denoted as \(\overrightarrow{\epsilon}_{a}=\left[\epsilon_{ax},\epsilon_{ay},0\right]^{T}\) for the azimuth gyroscope and \(\overrightarrow{\epsilon}_{p}=\left[\epsilon_{px},0,\epsilon_{pz}\right]^{T}\) for the pitch one. Based on the earlier analysis, it is easy to infer the rotation transformation matrixes caused by assembly errors of gyroscope locations \(R_{ag}\) and \(R_{pg}\), which are expressed as Equation (21).  
\[\begin{split} R_{ag}=\left[\begin{array}{ccc}c\epsilon_{ay}&0& -s\epsilon_{ay}\\ 0&1&0\\ s\epsilon_{ay}&0&c\epsilon_{ay}\end{array}\right]\left[\begin{array}{ccc}1&0& 0\\ 0&c\epsilon_{ax}&s\epsilon_{ax}\\ 0&-s\epsilon_{ax}&c\epsilon_{ax}\end{array}\right],\\ R_{pg}=\left[\begin{array}{ccc}c\epsilon_{pz}&s\epsilon_{pz}&0\\ -s\epsilon_{pz}&c\epsilon_{pz}&0\\ 0&0&1\end{array}\right]\left[\begin{array}{ccc}1&0&0\\ 0&c\epsilon_{px}&s\epsilon_{px}\\ 0&-s\epsilon_{px}&c\epsilon_{px}\end{array}\right],\end{split} \tag{21}\]  
Based on Equation (8) and Equation (21), the sensitive velocities of the two gyroscopes in \(\{i\}\) affected by \(R_{ag}\) and \(R_{pg}\) are denoted as \(\omega^{\prime}_{gyro\_z}\) and \(\omega^{\prime}_{gyro\_y}\), and they could be calculated by using Equation (22).  
\[\begin{split}\omega^{\prime}_{gyro\_z}=[0,0,1]R_{ag} \overrightarrow{\omega}_{ib}^{p},\\ \omega^{\prime}_{gyro\_y}=[0,1,0]R_{pg}\overrightarrow{\omega}_{ib}^{p}, \end{split} \tag{22}\]  
Figure 12: The signals affected by errors of run-outs.  
By using a typical second-order model, the transformation function between the input and output signals of gyroscopes \(Trans_{g}(s)\) could be expressed as Equation (23), where \(K_{g}\), \(\omega_{g}=2\pi f_{g}\) and \(\zeta_{g}\) represent the amplification coefficient, natural frequency, and damping coefficient of the gyroscopes.  
\[Trans_{g}(s)=\frac{K_{g}\omega_{g}^{2}}{s^{2}+2\zeta_{g}\omega_{g}s+\omega_{g}^ {2}}, \tag{23}\]  
Based on Equation (23) and analysis above, the signals and assembly errors of the gyroscopes are simulated in Simulink as shown in Figure 13. Based on Equation (22), \(\overset{\rightarrow}{\omega}_{ib}^{p}\) is set as the input. Furthermore, the value of \(\overset{\rightarrow}{\omega}_{ib}^{p}\) can be considered equal to the real-time value of \(\overset{\rightarrow}{\omega}_{ip}^{p}\), which can be obtained from the kinematics module in Section 3.3. The assembly error vectors of \(\overset{\rightarrow}{\epsilon}_{a}\) and \(\overset{\rightarrow}{\epsilon}_{p}^{p}\), the transformation matrixes \(R_{ag}\) and \(R_{pg}\), and \(Trans_{g}(s)\) are imbedded in the middle. The final sensitive velocities of the two gyroscopes \(\omega_{gyro\_z}^{\prime}\) and \(\omega_{gyro\_y}^{\prime}\) are set as the output, and they will be used in other modules. The value of \(\overset{\rightarrow}{\epsilon}_{a}\) and \(\overset{\rightarrow}{\epsilon}_{p}^{p}\) can be set as needed in the future simulation experiments, and also can be measured by using high-precision instruments and relevant methods [23; 24].