## Appendix C IMU Faults Propagation in the Full-State EKF
In addition, the gyroscope measurement of the angular rate, \(\overset{\rightarrow}{\omega}_{ib,meas}^{b}\) (rad/s), is represented as Equation (A28), which includes the true angular rate, \(\overset{\rightarrow}{\omega}_{ib}^{b}\) (rad/s); bias in the gyroscope, \(\overset{\rightarrow}{b}_{g}\) (rad/s); a g-dependent bias related to the specific force, \(G_{g}\) (rad\(\cdot\)s/m); and process noise, \(\overset{\rightarrow}{\overrightarrow{w}}_{g}\):  
\[\overset{\rightarrow}{f}_{ib,meas}^{b}=\overset{\rightarrow}{f}_{ib}^{b}+ \overset{\rightarrow}{b}_{a}+\overset{\rightarrow}{w}_{a}\] (A27)  
\[\overset{\rightarrow}{\omega}_{ib,meas}^{b}=\overset{\rightarrow}{\omega} _{ib}^{b}+\overset{\rightarrow}{b}_{g}+G_{g}\overset{\rightarrow}{f}_{ib} ^{b}+\overset{\rightarrow}{w}_{g}\] (A28)  
Rearranging Equation (A27) and Equation (A28), the true specific force and angular rate can be expressed as follows:  
\[\overset{\rightarrow}{f}_{ib}^{b}=\overset{\rightarrow}{f}_{ib,meas}^{b}- \overset{\rightarrow}{b}_{a}-\overset{\rightarrow}{w}_{a}\] (A29)  
\[\overset{\rightarrow}{\omega}_{ib}^{b}=\overset{\rightarrow}{\omega}_{ib,meas }^{b}-\overset{\rightarrow}{b}_{g}-G_{g}\overset{\rightarrow}{f}_{ib}- \overset{\rightarrow}{w}_{g}.\] (A30)  
Substituting Equations (A29) and (A30) into Equation (A26) is expressed as  
\[ \begin{array}{l}\overset{\cdot}{\overrightarrow{\sigma}}_{b}^{e^{\prime}}=C _{b}^{e}\bigg{[}\bigg{(}\overset{\rightarrow}{\omega}_{ib,meas}^{b}- \overset{\rightarrow}{b}_{g}-G_{g}\overset{\rightarrow}{f}_{ib}^{b}- \overset{\rightarrow}{w}_{g}\bigg{)}\times\bigg{]}-\bigg{[}\overset{ \rightarrow}{\omega}_{ie}^{e}\times\bigg{]}\overset{\rightarrow}{\sigma}_{b}^{e}+ \overset{\rightarrow}{f}_{\overset{\cdot}{\overrightarrow{\sigma}}_{b}^{e^{ \prime}}}\\ \overset{\cdot}{\overrightarrow{\sigma}}_{cb}^{e^{\prime}}=C_{b}^{e}\bigg{(} \overset{\rightarrow}{f}_{ib,meas}^{b}-\overset{\rightarrow}{b}_{a}- \overset{\rightarrow}{w}_{a}\bigg{)}-2\bigg{[}\overset{\rightarrow}{\omega} _{ie}^{e}\times\bigg{]}\overset{\rightarrow}{\overrightarrow{\sigma}}_{cb}^{e^{ \prime}}+\overset{\rightarrow}{g}+\overset{\rightarrow}{f}_{\overset{\cdot}{ \overrightarrow{\sigma}}_{eb}^{e^{\prime}}}\\ \overset{\cdot}{\overrightarrow{\sigma}}_{cb}^{e^{\prime}}=\overset{ \rightarrow}{\overrightarrow{\sigma}}_{cb}^{e}+\overset{\rightarrow}{f}_{ \overset{\cdot}{\overrightarrow{\sigma}}_{eb}^{e^{\prime}}}\end{array} .\] (A31)  
Contrasting Equation (A46) with Equation (27), it can be found that the IMU faults present the same propagation law in both the error-state and full-state EKFs, resulting in the same state error. According to Section 3.3, for the full-state EKF, the innovation at epoch k, written as \(\overrightarrow{\gamma}_{k}\), can be expressed as shown in Equation (A47):  
\[ \begin{split}\vec{\gamma}_{k}=&\vec{Z}_{k}-H_{k} \hat{X}_{k}^{\cdot\cdot}\\ =& H_{k}\vec{X}_{k}+\vec{\gamma}_{k}-H_{k}\hat{X}_{k} ^{\cdot\cdot}\\ =& H_{k}\left(\[PERSON]_{k}\vec{X}_{k\cdot+}+\underline{ \Gamma_{uk}\vec{u}_{k}}+\underline{\bar{\alpha}}_{k}\right)+\vec{\
u}_{k}-H_{k }\left(\[PERSON]_{k}\hat{X}_{k\cdot+}+\underline{\Gamma_{uk}\vec{u}_{k}}+\underline {\tilde{f}_{\vec{x}_{k}}}\right)\cdot\\ =& H_{k}\Phi_{k}\left(\underline{\left[\vec{X}_{k \cdot+}-\vec{X}_{k\cdot+}\right]}+H_{k}\vec{\alpha}_{k}+\vec{\
u}_{k}-H_{k} \tilde{f}_{\vec{x}_{k}}\right.\\ =&-H_{k}\Phi_{k}\delta X_{k\cdot+}+H_{k}\vec{\alpha} _{k}+\vec{\
u}_{k}-H_{k}\tilde{f}_{\vec{x}_{k}}\end{split} \] (A47)  
As seen in Equations (A47) and (28), the relational expressions between the innovation and the state error for the full-state and error-state EKFs are the same. The reason is that the control items for the full-state EKF are offset, as seen in Equation (A47). Therefore, the error overboundings for the error-state EKF and full-state EKF should also be consistent, and the derivation is exactly the same.