## Appendix C IMU Faults Propagation in the Full-State EKF
\begin{array}{l}\overset{\rightarrow}{\omega}_{ib}^{b^{\prime}}= \overset{\rightarrow}{\omega}_{ib}^{b}+\Delta\overset{\rightarrow}{\omega} _{ib}^{b}\\ \overset{\rightarrow}{\omega}_{ib}^{b^{\prime}}=\overset{\rightarrow}{f}_{ib} +\Delta\overset{\rightarrow}{f}_{ib}^{b^{\prime}}\\ \overset{\cdot}{C}_{b}^{e^{\prime}}=C_{b}^{e}+\Delta C_{b}^{e}\\ \overset{\rightarrow}{v}_{cb}^{e^{\prime}}=\overset{\rightarrow}{v}_{cb}^{e }+\Delta\overset{\rightarrow}{v}_{cb}^{e}\end{array}  \tag{13}\]  
where \(\Delta\overset{\rightarrow}{\omega}_{ib}^{b}\) is the gyroscope fault vector (rad/s), \(\Delta f_{ib}\) is the accelerometer fault vector (m/s\({}^{2}\)), \(\Delta C_{b}^{e}\) is the difference between the true transformation matrix and the faulty transformation matrix, and \(\Delta\overset{\rightarrow}{v}_{cb}^{e^{\prime}}\) is the difference between the true velocity vector and the faulty velocity vector (m/s). Substituting Equation (A24) into Equation (A23), the inertial navigation equation under the IMU faults condition becomes  
\[ \begin{array}{l}\overset{\cdot\cdot}{\overrightarrow{\sigma}}_{b}^{e^{\prime}}=C _{b}^{e}\bigg{[}\bigg{(}\overset{\rightarrow}{\omega}_{ib}^{b}+\Delta \overset{\rightarrow}{\omega}_{ib}^{b}\bigg{)}\times\bigg{]}-\bigg{[}\overset{ \rightarrow}{\omega}_{ie}^{e}\times\bigg{]}\overset{\rightarrow}{\sigma}_{b}^{e} \\ \overset{\cdot\cdot}{\overrightarrow{\sigma}}_{cb}^{e^{\prime}}=\big{(}C_{b}^{e}+ \Delta C_{b}^{e}\bigg{)}\bigg{(}\overset{\rightarrow}{f}_{ib}^{b}+\Delta \overset{\rightarrow}{f}_{ib}^{b}\bigg{)}-2\bigg{[}\overset{\rightarrow}{ \omega}_{ie}^{e}\times\bigg{]}\overset{\rightarrow}{\sigma}_{eb}^{e}+ \overset{\rightarrow}{g}^{e}\\ \overset{\cdot\cdot}{\overrightarrow{\sigma}}_{cb}^{e^{\prime}}=\overset{ \rightarrow}{\overrightarrow{\sigma}}_{cb}^{e}+\Delta\overset{\rightarrow}{ \overrightarrow{\sigma}}_{eb}^{e}\end{array} \] (A25)  
Rearrange Equation (A25),  
\[ \begin{array}{l}\overset{\cdot}{\overrightarrow{\sigma}}_{b}^{e^{\prime}}=C _{b}^{e}\bigg{[}\overset{\rightarrow}{\omega}_{ib,meas}^{b}-\overset{ \rightarrow}{b}_{g}-G_{g}\overset{\rightarrow}{f}_{ib}^{b}-\overset{ \rightarrow}{\overrightarrow{\sigma}}_{g}\bigg{)}\times\bigg{]}-\bigg{[} \overset{\rightarrow}{\omega}_{ie}^{e}\times\bigg{]}\overset{\rightarrow}{ \sigma}_{b}^{e^{\prime}}+\overset{\rightarrow}{f}_{\overset{\cdot}{ \overrightarrow{\sigma}}_{b}^{e^{\prime}}}\\ \overset{\cdot}{\overrightarrow{\sigma}}_{cb}^{e^{\prime}}=C_{b}^{e}\overset{ \rightarrow}{f}_{ib,meas}^{e}-b_{a}-\overset{\rightarrow}{\overrightarrow{ \sigma}}_{a}^{e}\bigg{)}-2\bigg{[}\overset{\rightarrow}{\omega}_{ie}^{e} \times\bigg{]}\overset{\rightarrow}{\overrightarrow{\sigma}}_{cb}^{e^{\prime}} +\overset{\rightarrow}{g}+\overset{\rightarrow}{f}_{\overset{\cdot}{ \overrightarrow{\sigma}}_{eb}^{e^{\prime}}}\\ \overset{\cdot}{\overrightarrow{\sigma}}_{cb}^{e^{\prime}}=\overset{ \rightarrow}{\overrightarrow{\sigma}}_{cb}^{e}+\overset{\rightarrow}{f}_{ \overset{\cdot}{\overrightarrow{\sigma}}_{eb}^{e^{\prime}}}\end{array} \] (A26)  
where \(\overset{\rightarrow}{f}_{\overset{\cdot}{\overrightarrow{\sigma}}_{b}^{e^{ \prime}}}\), \(\overset{\rightarrow}{f}_{\overset{\cdot}{\overrightarrow{\sigma}}_{eb}^{e^{ \prime}}}\), and \(\overset{\rightarrow}{f}_{\overset{\cdot}{\overrightarrow{\sigma}}_{ab}^{e^{ \prime}}}\) are the resultant bias vectors for the attitude, velocity, and position states due to the IMU faults, respectively. In the EKF, the accelerometer measurement of the specific force \(\overset{\rightarrow}{f}_{ib,meas}^{b}\) (m/s\({}^{2}\)) is represented as Equation (A27), which includes the true specific force, \(\overset{\rightarrow}{f}_{ib}^{b}\) (m/s\({}^{2}\)); bias in the accelerometer, \(\overset{\rightarrow}{b}_{a}\) (m/s\({}^{2}\)); and process noise, \(\overset{\rightarrow}{\overrightarrow{w}}_{a}\).