## 4 Results and Discussion
### Semi-Physical Simulation Based on Telemetered Missile Trajectory  
Highly dynamic conditions, which are needed for validating the effectiveness of the ultra-tight GPS/INS integration, are extremely difficult to achieve in practice. To simulate this condition, in this paper, a missile trajectory lasting 180.23 s is derived from Fiber Optic Gyroscope (FOG) and accelerometer data, which are telemetered during a missile test, and is treated as the reference trajectory. The position of the missile relative to the launch site in geodetic coordinate system, and the velocity of the missile in ECEF coordinate system are shown in Figure 9. Figure 10 shows the (strapdown) INS used in the missile test. The statistical characteristics of the FOG and accelerometer are listed in Table 4. If the same errors in turn are added to the original FOG and accelerometer data, a set of new data can be generated. The trajectory corresponding to these new inertial measurements can be used as the INS navigation result, which needs to be calibrated by the ultra-tight GPS/INS integration. In addition, based on the reference trajectory as well as the satellite configuration at the missile launch site shown in Figure 11, the IF signal can be simulated. Figure 8: Frequency difference between local and input carriers in field tests. Figure 9: The reference trajectory for simulation. Figure 12 provides a close look at the absolute acceleration of the missile following the reference trajectory. It is observed that this value does not remain unchanged during the missile ascent phase. Indeed, this value has abrupt changes at about 4 s and 9 s. That usually means that even baseband signal preprocessing cannot ensure the carrier frequency estimation accuracy. Taking PRN5 as example, the Doppler shift tracking error obtained from the independent GPS receiver equipped with pre-filter-D is shown in Figure 13. Obviously, the rapid variation of acceleration introduces a significant error into the estimated Doppler shift after 4 s, and the sudden reversal of jerk at 9 s causes the sign of this error to change. As a comparison, the same dynamic simulation is also performed with the ultra-tight GPS/INS integration. The local carrier generation can be directly controlled by the frequency derived from the external navigation information (GPS/INS-f), but even so, the carrier tracking loop is still equivalent to a third-order phase locked loop. Hence, in the integrated system, the Doppler shift tracking error follows the same trend as the absolute acceleration. However, the absolute maximum \begin{table}
\begin{tabular}{c c c} \hline  & **Gyros** & **Accelerometers** \\ \hline In-Run Bias Error & 0.5\({}^{\circ}\)/h & 2 \(\times 10^{-4}\) g \\ Noise (1\(\sigma\), smoothing window: 10 ms) & 15\({}^{\circ}\)/h & 2 \(\times 10^{-2}\) g \\ \hline \end{tabular}
\end{table} 
Table 4: Gyro and accelerometer specifications. Figure 11: Satellites sky plot of the simulation. Figure 10: The (strapdown) INS used in the missile test. value of this error drops to 5 Hz from 15 Hz, and the whole tracking process goes more smoothly. According to Equation (36), the external navigation information can also be used to correct the state feedback value. In this case, local carrier generation is controlled by state feedback (GPS/INS-u), and the absolute maximum value of the Doppler shift estimation error is 3 Hz. The following discussions focus on GPS/INS-u. It can be seen from Equation (29) that \(\delta v_{G}\) is proportional to the estimation error of input carrier frequency, so Figure 13 also shows that the estimation of \(\delta v_{G,1}\) (PRN5 is locked by channel #1) is accurate. In this sense, the proposed navigation filter can ensure INS is properly calibrated. If the traditional 17-dimensional filter is applied, in fact, the significant Doppler shift tracking error caused by missile ascent will lead the navigation results to gradually deviate from the reference trajectory until the whole integrated system cannot work.