## 3 Realization of the Specific Ultra-tight GPS/INS Integration  
The carrier tracking loop employing pre-filter 1 is in essence equivalent to a third-order loop. However, a _N-th_ order phase locked loop cannot correctly track the signal whose phase varies by time to the _N_th or higher-order power (the estimates of signal parameters can still have low variance, but not low bias) [18]. That indicates, even improved by baseband signal processing, an independent GPS receiver is vulnerable to extreme high dynamics, and may give inaccurate tracking and navigation results. On the contrary, INS is robust to vehicle dynamics. Based on this fact, INS can be used to assist GPS, which usually means baseband signal processing is connected closely with the estimation of navigation parameters. In the ultra-tight GPS/INS integration shown in Figure 2, the integrated navigation filter is responsible for obtaining corrected inertial navigation information, and delivering them back to GPS tracking channels. Because the feedback from the corrected inertial navigation information has the advantage of high accuracy in a short time, GPS tracking channel can perform better under highly dynamic conditions. The detailed work procedure of this ultra-tight GPS/INS integration is as follows: in each GPS tracking channel, in-phase and quadrature signals are processed by phase discriminators and pre-filters to estimate tracking parameters, such as carrier frequency and code phase; the observations of the integrated navigation filter can be converted from tracking parameters as well as deduced from inertial navigation information and satellite ephemeris, and then the corrected inertial navigation information can be obtained; finally, auxiliary tracking parameters, carrier frequency, for example, can be deduced from these external navigation information and satellite ephemeris, and then be applied to assist signal tracking. Through the above analysis, it can be seen that identifying conversion relations between tracking parameters and navigation parameters as well as finding a rational way to assist tracking are the keys to designing a good ultra-tight GPS/INS integration. Figure 2: The structure of the ultra-tight GPS/INS integration. In the ideal situation, pseudorange rate and variation of pseudorange output by GPS tracking channel can be respectively derived from:  
\[\dot{\rho}_{G}=-f_{d}\frac{c}{f_{L1}} \tag{27}\]  
\[\Delta\rho_{G}=\Delta\theta^{code}\frac{c}{f_{CA}} \tag{28}\]  
where \(c\) is the speed of light. But in fact, under highly dynamic conditions, the carrier phase of satellite signal may be proportional to time to the power of \(N\) or higher value, and hence the estimated carrier frequency may contain a large error \(\delta f^{carrier}\). Since the incorrect GPS outputs could destroy the estimation of INS errors, it is necessary to analyze this error in detail. Therefore Equation (27) is rewritten as:  
\[\dot{\rho}_{G} =-\left(\hat{f}^{carrier}-f_{IF}\right)\frac{c}{f_{L1}} \tag{29}\] \[=-\left(f_{d}+\delta f^{carrier}\right)\frac{c}{f_{L1}}\]  
On the other hand, Equation (28) can remain unchanged due to the fact that the code loop dynamic stress error is usually small. \(\rho_{G}\), which represents the current pseudorange output by the GPS tracking channel, can be calculated from \(\Delta\rho_{G}\) and the previous pseudorange. Then, in Earth Centered Earth Fixed (ECEF) coordinate system, the observation equation of integrated navigation filter can be expressed as:  
\[\left\{ \begin{array}{l}\delta\rho=\rho_{I}-\rho_{G}=e_{1}\delta x+e_{2} \delta y+e_{3}\delta z-\delta t_{u}+v_{\rho}\\ \delta\dot{\rho}=\dot{\rho}_{I}-\dot{\rho}_{G}=e_{1}\delta\dot{x}+e_{2}\delta \dot{y}+e_{3}\delta z-\delta t_{ru}+\delta v_{G}+v_{\dot{\rho}}\end{array} \right.