## 2 Fault Identification Method for Deep Integration  
### The Cause of Fault Propagation  
In the deeply coupled GNSS/INS integrated system, the vector tracking loop is a feedback loop. GNSS IF (Intermediate Frequency) data are processed by multiple receiving channels and then put into the navigation filter. After fusing all the received data in a centralized manner, the navigation filter outputs the positioning errors to correct INS. Next, the carrier frequency and code phase of each receiving channel that are predicted by making use of both the INS data and ephemeris are used to configure the corresponding NCO (Numerically Controlled Oscillator). The structure of deep GNSS/INS integration is shown in Figure 1.  
The navigation filter is based on the Extended Kalman Filtering algorithm, and the system state is defined as  
\[\Delta\textbf{X}=\left[\Delta\textbf{R}^{e},\Delta\textbf{V}^{e},\Delta\textbf{q} ^{e},\Delta\textbf{W}^{b},\Delta\textbf{A}^{b},\Delta\textbf{C}\right]^{\text{T}}, \tag{1}\]  
where \(\Delta\textbf{R}\) represents position error vector, \(\Delta\textbf{V}\) represents velocity error vector, \(\Delta\textbf{q}\) represents attitude error vector, \(\Delta\textbf{W}\) represents the bias error of gyroscope, and \(\Delta\textbf{A}\) represents the bias error of accelerometer. \(\Delta\textbf{C}=\left[\Delta t_{b},\Delta t_{d}\right]\), \(\Delta t_{b}\) and \(\Delta t_{d}\) represent the clock bias error and clock drift error of GNSS receiver, respectively. The superscripts \(e\) and \(b\) represent the ECEF (Earth-Centered Earth-Fixed) coordinate system and the body coordinate system, respectively. Pseudo-range error and pseudo-range rate error are adopted as the measurements of navigation filter, and then the measurement vector at time \(k\) is expressed as  
\[\textbf{Z}_{k}^{\text{Nav}}=\left[\Delta\rho_{1},\Delta\dot{\rho}_{1},\Delta \rho_{2},\Delta\dot{\rho}_{2},\ldots\ldots,\Delta\rho_{N},\Delta\dot{\rho}_{N} \right]^{\text{T}}, \tag{2}\]  
where \(\Delta\rho_{i}\) and \(\Delta\dot{\rho}_{i}\) represent pseudo-range error and pseudo-range rate error of the \(i\)th receiving channel, respectively. \(N\) is the number of receiving channels. \(\Delta\rho_{i}\) is obtained from code phase error \(\Delta\tau_{i}\), and \(\Delta\dot{\rho}_{i}\) is obtained from carrier frequency error \(\Delta f_{i}\). Their expressions are shown in Equations (3) and (4), respectively:  
\[\Delta\rho_{i}=\frac{\text{c}}{f_{i,code}}\Delta\tau_{i}, \tag{3}\]  
\[\Delta\dot{\rho}_{i}=\frac{\text{c}}{f_{i,carr}}\Delta f_{i}, \tag{4}\]  
where \(f_{i,code}\) denotes code frequency, \(f_{i,carr}\) denotes carrier frequency, and c denotes the speed of light. Extended Kalman Filter (EKF) adopts an iterative method to estimate the system state, including two processes of prediction and correction. Prediction:  
\[\Delta\textbf{X}_{ikk-1}=F\Delta\textbf{X}_{k-1}, \tag{5}\]  
\[\textbf{P}_{ikk-1}=\textbf{FP}_{k-1}\textbf{F}+\textbf{Q}, \tag{6}\]  
Correction:  
\[\textbf{W}_{k}=\textbf{P}_{ikk-1}\textbf{H}_{k}^{\text{T}}\big{(}\textbf{H}_ {k}\textbf{P}_{ikk-1}\textbf{H}_{k}^{\text{T}}+\textbf{R}\big{)}^{-1}, \tag{7}\]  
\[\Delta\textbf{X}_{k}=\Delta\textbf{X}_{ikk-1}+\textbf{W}_{k}\big{(}\textbf{Z} _{k}^{\text{Nav}}-\textbf{H}_{k}\Delta\textbf{X}_{ikk-1}\big{)}, \tag{8}\]  
Figure 1: The structure of deep GNSS (Global Navigation Satellite System)/INS (Inertial Navigation System) integration. \[\Delta\textbf{X}=\left[\Delta\textbf{R}^{e},\Delta\textbf{V}^{e},\Delta\textbf{q} ^{e},\Delta\textbf{W}^{b},\Delta\textbf{A}^{b},\Delta\textbf{C}\right]^{\text{T}}, \tag{1}\]0  
where \(P_{k}\) is the covariance matrix of system state error, \(W_{k}\) is the gain matrix, \(Q\) is the covariance matrix of process noise, and \(R\) is the covariance matrix of measurement noise. \(H_{k}\) is the measurement matrix whose coefficients are unit vectors in the line-of-sight direction from the receiver antenna to satellites. \(F\) is the state transition matrix, whose coefficients are available in [32].