# A General Euler Angle Error Model of Strapdown Inertial Navigation Systems
## 3 Simulation Experiments
### Simulation and Analysis of Convected Euler Angle Errors  
In order to validate the proposed convected Euler angle error models (Equations (41) and (42)) and their conclusions, four simulation experiments are carried out, where the initial attitude of SINS is supposed by \(\left[ \begin{array}{cc}\theta_{0}&\gamma_{0}&\varphi_{0}\\ \end{array} \right]=\left[ \begin{array}{cc}-45^{\circ}&-45^{\circ}&-45^{\circ }\\ \end{array} \right]\). In the first group simulation, there is the coupling of initial Euler angle errors in three axes \(\left[ \begin{array}{cc}\delta\theta_{0}&\delta\gamma_{0}&\delta\varphi_{0} \\ \end{array} \right]=\left[ \begin{array}{cc}0.1^{\circ}&0.1^{\circ}&0.1^{\circ }\\ \end{array} \right]\) and the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame along the \(Y\) axis \(\omega_{nby}^{b}=1\ ^{\circ}/s\). The second group simulation presents the coupling of initial head errors \(\delta\varphi_{0}=0.1^{\circ}\), the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame \(\mathbf{\omega_{nb}^{b}}=\left[ \begin{array}{cc}1\ ^{\circ}/s&1\ ^{\circ}/s&1\ ^{\circ}/s\\ \end{array} \right]\). The coupling of initial Euler angle errors in three axes \(\left[ \begin{array}{cc}\delta\theta_{0}&\delta\gamma_{0}&\delta\varphi_{0} \\ \end{array} \right]=\left[ \begin{array}{cc}0.1^{\circ}&0.1^{\circ}&0.1^{\circ }\\ \end{array} \right]\) and the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame along the \(X\) axis \(\omega_{nbx}^{b}=1\ ^{\circ}/s\) are given in the third group simulation. In the fourth group simulation, there is the coupling of initial Euler angle errors in three axes \(\left[ \begin{array}{cc}\delta\theta_{0}&\delta\gamma_{0}&\delta\varphi_{0} \\ \end{array} \right]=\left[ \begin{array}{cc}0.1^{\circ}&0.1^{\circ}&0.1^{\circ }\\ \end{array} \right]\) and the rotation angular velocity of the body frame with respect to the navigation frame denoted in the body frame, \(\mathbf{\omega_{nb}^{b}}=\left[ \begin{array}{cc}1\ ^{\circ}/s&1\ ^{\circ}/s&1\ ^{\circ}/s\\ \end{array} \right]\). The results of the four group simulation experiments are given in Figures 4-7, respectively. The convected Euler angle error model can precisely describe the attitude error propagation rules of SINS. When the SINS' attitude errors are regarded as error references, the root mean square error (RMSE) of the attitude angle error calculated by the convected Euler angle error model is superior to \(0.4441^{\prime\prime}\), as shown in Figures 4-7 and Table 1. The angular velocity in the \(Y\) axis \(\omega_{nby}^{b}\) and the initial head error \(\delta\varphi_{0}\) does not affect the convected Euler angle errors, which are greatly affected by the pitch magnitude. The conclusions above have been validated. Figure 4: The first group I simulation with rotation in the \(Y\) axis \(\omega_{nby}^{b}\). RMSE: root mean square error. Figure 5: The second group II simulation with the initial head error \(\delta\varphi_{0}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Simulations** & **Head Error \(\delta\varphi^{*}\) (\({}^{\prime}\))** & **Pitch Error \(\delta\theta^{*}\) (\({}^{\prime}\))** & **Roll Error \(\delta\gamma^{*}\) (\({}^{\prime}\))** & **Maximum Error (\({}^{\prime}\))** \\ \hline First group I & 2.168 \(\times\)\(10^{-9}\) & 2.067 \(\times\)\(10^{-9}\) & 1.471 \(\times\)\(10^{-9}\) & 2.168 \(\times\)\(10^{-9}\) \\ Second group II & 8.484 \(\times\)\(10^{-10}\) & 4.571 \(\times\)\(10^{-10}\) & 8.967 \(\times\)\(10^{-10}\) & 8.967 \(\times\)\(10^{-10}\) \\ Third group III & 0.4441 & 0.1350 & 0.2573 & 0.4441 \\ Fourth group IV & 0.2437 & 0.0405 & 0.2847 & 0.2847 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Attitude angle errors (RMSE (root mean square error)) solved by the convected Euler angle error model. Figure 6: The third group III simulation with rotation in the \(X\) axis \(\omega_{nbx}^{b}\).