## 3 Error Overboundings against IMU Faults
### IMU Faults Propagation in the Error-State EKF
The parameters affected by thefault are expressed as the sum of a normal error term and a faulty additional error term, as shown in Equation (7):  
\[\delta\overset{\cdot\cdot\epsilon}{\partial\varphi}_{eb}^{\epsilon} =C_{b}^{\epsilon}\bigg{(}\delta\bigg{[}\overset{\omega_{ib}^{b}}{ \times}\bigg{]}\bigg{)}^{\prime}-\bigg{[}\overset{\omega_{ie}^{e}}{\times} \bigg{]}\delta\overset{\omega_{cb}^{\epsilon}}{\partial\varphi}_{eb}^{\epsilon} \tag{6}\] \[\delta\overset{\cdot\cdot\epsilon}{\partial\tau}_{eb}^{\epsilon} =\bigg{(}C_{b}^{\epsilon}\bigg{)}^{\prime}\bigg{(}\delta\overset{ \rightarrow}{f}_{ib}^{b}\bigg{)}-2\bigg{[}\overset{\omega_{ie}^{e}}{\times} \bigg{]}\delta\overset{\omega_{cb}^{\epsilon}}{\partial\tau}_{eb}^{\epsilon}\] (7) \[\delta\overset{\cdot\cdot\epsilon}{\partial\tau}_{eb}^{\epsilon} =\bigg{(}\delta\overset{\rightarrow}{\overline{\upsilon}}_{eb}^{ \epsilon}\bigg{)}^{\prime}\] \[\delta\overset{\omega_{ib}^{b}}{\omega_{ib}^{b}} =\delta\overset{\omega_{ib}^{b}}{\omega_{ib}^{b}}+\Delta\overset{ \rightarrow}{\overline{\omega}}_{ib}^{b}\] \[\delta\overset{\rightarrow}{\overline{\upsilon}}_{eb}^{\epsilon} =\delta\overset{\rightarrow}{\overline{\upsilon}}_{eb}^{\epsilon}+ \Delta\overset{\rightarrow}{\overline{\upsilon}}_{eb}^{\epsilon}\]  
Substituting Equation (7) into Equation (6), the inertial navigation error equation under the IMU faults condition can be shown as  
\[\delta\overset{\cdot\cdot\epsilon}{\partial\varphi}_{eb}^{\epsilon} =C_{b}^{\epsilon}\bigg{(}\delta\bigg{[}\overset{\omega_{ib}^{b}}{ \times}\bigg{]}+\Delta\overset{\rightarrow}{\omega_{ib}^{b}}\bigg{)}-\bigg{[} \overset{\omega_{ie}^{e}}{\times}\bigg{]}\delta\overset{\omega_{cb}^{\epsilon} }{\partial\tau}_{eb}^{\epsilon} \tag{8}\] \[\delta\overset{\cdot\cdot\epsilon}{\partial\tau}_{eb}^{\epsilon} =\big{(}C_{b}^{\epsilon}+\Delta C_{b}^{\epsilon}\bigg{)}\bigg{(} \delta\overset{\rightarrow}{f}_{ib}^{b}+\Delta\overset{\rightarrow}{f}_{ib} ^{b}\bigg{)}-2\bigg{[}\overset{\omega_{ie}^{e}}{\times}\bigg{]}\delta\overset{ \omega_{cb}^{\epsilon}}{\partial\tau}_{eb}^{\epsilon}\.\]  
Rearrange Equation (8),  
\[\delta\overset{\omega_{ie}^{e}}{\varphi}_{eb}^{\epsilon} =C_{s}^{\epsilon}\delta\big{[}\overset{\omega_{b}^{e}}{\times} \big{]}-\bigg{[}\overset{\omega_{e}^{e}}{\times}\bigg{]}\delta\overset{ \omega_{a}^{e}}{\times}+C_{s}^{\epsilon}\Delta\overset{\omega_{b}^{e}}{\omega_ {a}^{e}} \tag{9}\] \[\delta\overset{\omega_{ie}^{e}}{\varphi}_{eb}^{\epsilon} =\delta\overset{\omega_{a}^{e}}{\times}+\Delta\overset{\omega_{ab}^{ e}}{\underset{f_{is}}{\varphi}}\]  
where \(\overset{\rightarrow}{f}_{ib,\omega_{b}^{e}}\), \(\overset{\rightarrow}{f}_{j,\omega_{b}^{e}}\), and \(\overset{\rightarrow}{f}_{j,\omega_{b}^{e}}^{\epsilon}\) represent the resultant bias vectors for the attitude, velocity, and position error states due to the IMU faults. In the EKF, the accelerometer measurement of the specific force \(\overset{\rightarrow}{f}_{ib,\text{meas}}^{b}\) is represented as Equation (10), which includes the true specific force, the accelerometer  
Figure 2: Navigation parameters update process for the IMU. bias, and the process noise. In addition, the gyroscope measurement of the angular rate, \(\overrightarrow{\omega}_{ib,meas}^{b}\), is represented as Equation (11), which includes the true angular rate, the gyroscope bias, a g-dependent bias related to the specific force, and process noise.