## 3 Realization of the Specific Ultra-tight GPS/INS Integration
\tag{30}\]  
where \(\rho_{I}\), \(\dot{\rho}_{I}\) respectively represent the pseudorange and pseudorange rate deduced from inertial navigation information and satellite ephemeris, \(\delta x\), \(\delta y\), \(\delta z\), \(\delta\dot{x}\), \(\delta\dot{y}\), \(\delta\dot{z}\) are vehicle position errors and velocity errors in ECEF coordinate system, \(e_{1}\), \(e_{2}\), \(e_{3}\) are unit vectors in the \(x\), \(y\), and \(z\) direction, \(\delta t_{u}\), \(\delta t_{ru}\) represent clock bias and drift respectively, \(\delta v_{G}\) is the error of pseudorange rate output by the GPS tracking channel and equals to \(\delta f^{carrier}c/f_{L1}\), \(\
u_{\rho}\), \(v_{\dot{\rho}}\) are observation noises. It is noted that Equation (30) should usually be transformed into the geodetic coordinate system to facilitate INS error correction. Then, taking all the available tracking channels into consideration, the observation matrix \(\widetilde{H}\) can be obtained naturally, where the superscript \"\(\sim\)\" denotes the navigation filter designed in this paper. If there are \(m\) satellites being locked by tracking channels, the state vector of the integrated navigation filter can be expressed as:  
\[\widetilde{X}=\left[ \begin{array}{cccc}X&\delta v_{G,1}&\delta v_{G,2}& \cdots&\delta v_{G,m}\end{array} \right]^{T} \tag{31}\]  
where \(X\) is the transpose of the state vector adopted by a traditional integrated navigation filter, namely  
\[X=\left[ \begin{array}{cccc}\varphi_{E}&\varphi_{N}&\varphi_{U}&\delta v_{E }&\delta v_{U}&\delta L&\delta\lambda&\delta h&\varepsilon_{bx}&\varepsilon_{ by}&\varepsilon_{bz}&\
abla_{bx}&\
abla_{by}&\
abla_{bz}&\delta t_{u}&\delta t_{ru} \end{array} \right] \tag{32}\]  
and the first 15 variables of \(X\) represent three INS error states each in attitude, velocity, position, gyro bias, accelerometer bias. Accordingly, the state transition matrix \(\Phi\) and the noise matrix \(G\) of the traditional integrated navigation filter can be expanded to:  
\[\widetilde{\Phi}=\left[ \begin{array}{cc}\Phi_{17\times 17}&0_{17\times m}\\ 0_{m\times 17}&I_{m\times m}\end{array} \right] \tag{33}\]  
\[\widetilde{G}=\left[ \begin{array}{cc}G_{17\times 8}&0_{17\times m}\\ 0_{m\times 8}&I_{m\times m}\end{array} \right] \tag{34}\]In addition, \(\widetilde{H}\) can also be written as:  
\[\widetilde{H}=\left[ \begin{array}{cc}H_{2m\times 17}&\left[\begin{array}{c}0_{m \times m}\\ I_{m\times m}\end{array}\right]\end{array} \right] \tag{35}\]  
where \(H\) is the traditional observation matrix. The specific information of \(\Phi\), \(G\) and \(H\) can be easily found in many references [19; 20; 21; 22] and will not be discussed in detail here. According to the conversion relations indicated by Equations (27) and (28), the corrected inertial navigation information can in turn feed back into GPS tracking channels. The corresponding carrier frequency estimate is usually accurate enough, and can be used to directly control local carrier generation. However, in the new baseband signal preprocessing model, it is hoped that the phase error can return to 0 in the shortest amount of time, so the external information assists in local carrier generation through correcting the state feedback value, and the input carrier frequency estimated output by pre-filter 1 can be expected to remain relatively stable and accurate. In other words, instead of being given by Equation (17), the local carrier frequency \(u_{k}^{curr}\) is calculated from:  
\[u_{k}^{curr}=\frac{\Delta\hat{\delta}_{k}^{curr}}{2\pi T_{coh}}+\left\{-\frac{ f_{L1}}{c}\left[e_{1}\left(v_{x}^{s}-v_{x}\right)+e_{2}\left(v_{y}^{s}-v_{y}\right)+e_{3} \left(v_{z}^{s}-v_{z}\right)\right]+f_{IF}\right\}\right|_{k}+\frac{1}{2} \hat{\tilde{f}}_{k}^{curr}T_{coh} \tag{36}\]  
where \(v_{x}\), \(v_{y}\), \(v_{z}\) are the corrected vehicle velocity components in ECEF coordinate system, \(v_{x}^{s}\), \(v_{y}^{s}\), \(v_{z}^{s}\) are the satellite velocity components.