## Appendix C IMU Faults Propagation in the Full-State EKF  
For either the full-state or error-state in the extended Kalman filter for a linear system, the essence is to estimate the error. For nonlinear systems, the second or higher order errors are neglected. This is also true for other estimation methods (e.g., least squares). Therefore, the state error equations for the error-state EKF are the same as for the full-state EKF. However, there is a difference between the error-state EKF and the full-state EKF in the system propagation step--the control input is added to the full-state EKF. Thus, when faults occur, the faults propagation process makes a difference, which needs to be considered. The velocity update equation is used as an example to explain the IMU faults propagation in the full-state EKF in [18]. In this appendix, refer to the methods proposed in [18], the detailed derivation of error propagation using full-state EKF in position, velocity and attitude equations are given. The inertial navigation equations in the Earth-centered, Earth-fixed (ECEF) coordinate frame can be seen in Equation (10), including the attitude, velocity, and position update equations:  
\[ \begin{array}{l}\overset{\cdot\cdot}{\varphi}_{b}^{e}=\frac{\partial_{ \varphi b}^{\omega c}}{\partial t}=C_{b}^{e}\!\!\left[\overset{\rightarrow}{ \omega}_{ib}^{b}\!\!\times\!\right]-\left[\overset{\rightarrow}{\omega}_{ib} ^{e}\!\!\times\!\right]\overset{\rightarrow}{\varphi}_{b}^{e}\\ \overset{\cdot}{\overline{v}}_{cb}^{e}=\frac{\partial_{\varphi b}^{\omega e }}{\partial t}=C_{b}^{e}\!\!\left[\overset{\rightarrow}{f}_{ib}-2\!\left[ \overset{\rightarrow}{\omega}_{ib}^{e}\!\!\times\!\right]\overset{ \rightarrow}{v}_{cb}^{e}+\overset{\rightarrow}{g}\\ \overset{\cdot}{\overline{r}}_{cb}^{e}=\frac{\partial_{\varphi b}^{\omega e }}{\partial t}=\overset{\rightarrow}{v}_{cb}^{e}\end{array}  \tag{11}\]  
where \(\overset{\rightarrow}{\varphi}_{b}^{e}\) is the attitude vector (rad), \(\overset{\rightarrow}{v}_{cb}^{e}\) is the velocity vector (m/s), \(\overset{\rightarrow}{r}_{cb}^{e}\) is the position vector (m), \(C_{b}^{e}\) is the coordinate transformation matrix from the body frame to the Earth frame, \(\overset{\rightarrow}{\omega}_{ib}^{e}\) is the Earth rotation rate (rad/s), \(\overset{\rightarrow}{g}^{e}\) gravity vector in the Earth-centered, Earth-fixed frame (m/s\({}^{2}\)) and \([\bullet\times]\) represents the calculation of a skew-symmetric matrix. When IMU faults occur in the gyroscope or accelerometer, the computed attitude, velocity, and position are sequentially affected, finally resulting in position errors. Under the condition of a faulty IMU, the inertial navigation equation (Equation (10)) becomes Equation (10), where the apostrophe (\({}^{\prime}\)) indicates the faulty parameters. The parameters affected by the fault are expressed as the sum of a normal term and a faulty additional term, as shown in Equation (11):  
\[ \begin{array}{l}\overset{\cdot\cdot}{\varphi}_{b}^{e^{\prime}}=C_{b}^{e} \!\!\left[\overset{\rightarrow}{\omega}_{ib}^{b^{\prime}}\!\!\times\! \right]-\left[\overset{\rightarrow}{\omega}_{ib}^{e^{\prime}}\!\!\times\! \right]\overset{\rightarrow}{\varphi}_{b}^{e}\\ \overset{\cdot}{\overline{v}}_{cb}^{e^{\prime}}=C_{b}^{e^{\prime}}\!\! \overset{\rightarrow}{\overline{r}}_{cb}^{b^{\prime}}-2\!\left[\overset{ \rightarrow}{\omega}_{ib}^{e^{\prime}}\!\!\times\!\right]\overset{ \rightarrow}{v}_{cb}^{e}+\overset{\rightarrow}{g}^{e}\\ \overset{\cdot}{\overline{r}}_{cb}^{e^{\prime}}=\overset{\rightarrow}{ \overline{v}}_{cb}^{e^{\prime}}\end{array} , \tag{12}\]  
\[