## 4 Highly Dynamic Simulations and Results
### Simulation of Inertial Calculation
#### 4.2.2 Stability Simulation under Highly Dynamic Circumstances  
With the help of the simulation above, the velocity error caused by the inertial calculation has been quantized, and the stability of the tracking loop is related to Equation (27) directly. So, in this part, the correctness of Equation (27) will be proved. The parameters of both the INS and GNSS systems are set as the same as before; it should be mentioned that the Kalman Filter (KF) parameters should be configured according to the system and measurement parameters, and this article will not discuss the setting of KF parameters. The structure of the carrier loop is determined as the third-order phase-locked loop aided by the second-order frequency-locked loop, so \(n\) in Equation (27) is 3. The phase detection results of one picked tracking channel aided with three different precision IMUs are shown in Figure 4, and the phase errors calculated by the left-hand side of Equation (27) are shown in Table 4.  
Figure 3: Histogram of root-mean-square errors (RMSEs) of velocity errors. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**RMSE of Velocity Errors** & **High-Precision IMU** & **Mid-Precision IMU** & **Low-Precision IMU** \\ \hline East & 0.0150 & 0.0206 & 0.0805 \\ North & 0.0222 & 0.1241 & 0.5268 \\ Up & 0.0456 & 0.2267 & 0.6751 \\ Calculated by Equation (28) & 0.0601 & 0.1574 & 0.5724 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMSEs of the velocity errors in one cycle. Figure 4 shows that the high-precision IMU helps maintain tracking through almost all the process. Only at the times of 265 s and 270 s does there exist a phase detecting error saltation, because the acceleration changed from 19 m/s/s to 300 m/s/s suddenly and the tracking loop cannot adjust to the great jerk instantly. During the 5 s acceleration of X-43, the high-precision IMU can help maintain tracking, and the middle-precision IMU performs worse than the high-precision IMU, but the low-precision IMU cannot guarantee the stability of the tracking loop even when the vehicle is under a low dynamic environment. Further, according to Table 4, the calculated phase errors show that both high-precision and middle-precision IMUs could satisfy the stability condition of Equation (27) but the low-precision IMU could not; this result agrees well with the phenomenon observed in Figure 4.  
The navigation results aided by the three kinds of IMUs are further compared in Figures 5 and 6. The detailed statistical analysis of Figures 5 and 6 is shown in Table 5. \begin{table}
\begin{tabular}{c c} \hline
**IMUs** & **Calculated Phase Error (\({}^{\circ}\))** \\ \hline High-precision & 5.08280 (\(<\)15) \\ Mid-precision & 11.7625 (\(<\)15) \\ Low-precision & 34.1919 (\(>\)15) \\ \hline \end{tabular}
\end{table} 
Table 4: The phase errors calculated by the left-hand side of Equation (27) for the three kinds of IMUs. Figure 4: Comparison diagram of phase detecting error aided by three kinds of IMUs. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**RMSE of System Errors** & **High-Precision** & **Mid-Precision** & **Low-Precision** \\ \hline East-position (m) & 1.9686 & 7.0726 & 407.51 \\ North-position (m) & 2.8281 & 6.9240 & 279.73 \\ Up-position (m) & 5.5569 & 11.482 & 210.98 \\ East-velocity (m/s) & 0.0740 & 0.4287 & 6.5085 \\ North-velocity (m/s) & 0.1307 & 0.4583 & 30.695 \\ Up-velocity (m/s) & 0.2115 & 0.8553 & 13.535 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: RMSEs of the deeply coupled system errors aided by the three kinds of IMUs. Figure 5: Comparison diagram of position error aided by the three kinds of IMUs. Figures 5 and 6 and Table 5 show that with the help of the high-precision IMU, the deeply coupled navigation system performs very well in highly dynamic environments. The middle-precision IMU can guarantee the navigation precision of the system but performs worse than the high-precision IMU. However, the low-precision IMU causes the system to lose positioning ability throughout the whole process.