## 2 Mathematical Model of a Deeply Coupled Navigation System  
The core concept of the deeply coupled system aided by an IMU is to extract the doppler frequency and code phase deviation based on the inertial information and satellite ephemeris, and this can help to control the code/carrier NCO (Numerically Controlled Oscillator) in the tracking loop. This structure can ensure the system maintains high performance of the tracking loop in narrow bandwidth conditions. The overall architecture of the system is shown in Figure 1.  
The performance of the inertial devices is obviously affected by highly dynamic circumstances, so the IMUs' error model should be established first. Due to the smaller and lighter tendency of the deeply coupled system, this article mainly discusses the MEMS device. The error model of gyroscope and accelerometer are given as follows:  
\[\widetilde{\omega}_{ib}^{b}=\omega_{ib}^{b}+b_{g}+g_{s}f_{ib}^{b}+\eta_{g}, \tag{1}\]  
Figure 1: Structure diagram of the GNSS/INS deeply coupled system. \[\widehat{f}_{ib}^{b}=f_{ib}^{b}+b_{a}, \tag{2}\]  
where \(\widetilde{\omega}_{ib}^{b}\) is the angular velocity measurement of the gyroscope, \(i\) denotes the inertial coordinate system, \(b\) denotes the body coordinate system, \(b_{g}\) is the gyroscope bias, \(g_{s}\) is the matrix of g-sensitivity coefficients, \(f_{ib}^{b}\) is the specific force, \(\eta_{g}\) is the measurement error of gyroscope, \(\widehat{f}_{ib}^{b}\) is the specific force measurement of the accelerometer, and \(b_{a}\) is the accelerometer bias [8; 9]. In order to analyze the relationship between the IMU errors and the INS navigation errors in the deeply coupled system, the state variables related to the inertial system in the integrated navigation filter could be expressed as follows:  
\[X=\left[ \begin{array}{cccc}\delta R^{e}&\delta V^{e}&\delta\psi^{e}&\delta \omega^{b}&\delta A^{b}&\delta T\end{array} \right]^{T}, \tag{3}\]  
where \(\delta R^{e}\) are the error vectors of the 3D position, \(\delta V^{e}\) are the error vectors of 3D velocity, and \(\delta\psi^{e}\) are the error vectors of 3D attitude; e denotes the ECEF coordinate system; \(\delta\omega^{b}\) are the error vectors of the gyroscopes; and \(\delta A^{b}\) are the error vectors of the accelerometers in the body coordinate system. \(\delta T\) are the clock error vectors including clock bias error \(\delta T_{bias}\) and clock drift error \(\delta T_{drift}\)[10; 11]. The equation of state is given as follows:  
\[\left[ \begin{array}{c}\delta\hat{R}^{e}\\ \delta\hat{V}^{e}\\ \delta\hat{\psi}^{e}\\ \delta\hat{\omega}^{b}\\ \delta\hat{A}^{b}\\ \delta\hat{T}\end{array} \right]=\left[ \begin{array}{cccccc}0&I&0&0&0\\ G-\left(\widetilde{\Omega}^{e}\right)^{2}&-2\widetilde{\Omega}^{e}&\widetilde{ F}^{e}&0&C_{b}^{e}&0\\ 0&0&-\widetilde{\Omega}^{e}&C_{b}^{e}&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&\Lambda\end{array} \right]\left[ \begin{array}{c}\delta R^{e}\\ \delta V^{e}\\ \delta\psi^{e}\\ \delta\omega^{b}\\ \delta A^{b}\\ \delta T\end{array} \right]+\left[ \begin{array}{c}0\\ 0\\ 0\\ \eta_{g}\\ b_{a}\\ \omega_{T}\end{array} \right], \tag{4}\]  
\[G=\frac{\mu}{r^{3}}\left[ \begin{array}{cccc}-1+\frac{3r^{2}}{r^{2}}&\frac{ 3 xy}{r^{2}}&\frac{3 xz}{r^{2}}\\ \frac{3 xy}{r^{2}}&-1+\frac{3y^{2}}{r^{2}}&\frac{3 yz}{r^{2}}\\ \frac{3 xz}{r^{2}}&\frac{3 yz}{r^{2}}&-1+\frac{3z^{2}}{r^{2}}\end{array} \right], \tag{5}\]  
\[\widetilde{\Omega}^{e}=\left[ \begin{array}{cccc}0&\omega_{e}&0\\ -\omega_{e}&0&0\\ 0&0&0\end{array} \right]\widetilde{F}^{e}=\left[ \begin{array}{cccc}0&f_{2}&- f_{y}\\ -f_{z}&0&f_{x}\\ f_{y}&-f_{x}&0\end{array} \right]\Lambda=\left[ \begin{array}{cccc}0&1\\ 0&0\end{array} \right] \tag{6}\]  
where \(G\) denotes the parameter of gravity, \(\mu=3.986005\times 10^{14}(m^{3}/s^{2})\), which represents the product of the gravitation constant and the mass of earth; \(\left[ \begin{array}{cccc}x&y&z\end{array}