## 3 Error Overboundings against IMU Faults
### IMU Faults Propagation in the Error-State EKF  
In this section, the IMU faults propagation process in the state vector for the error-state EKF is explained. While following a similar approach, it is worth mentioning that the method derived here is a different approach from the work published by [PERSON] et al. [18] where error states instead of full navigation states are considered. The differences between the output measurement value and the true value of the IMU are expressed in Equation (3):  
\[\left\{ \begin{array}{l}\delta\varphi_{\beta}^{\alpha}=\widetilde{\varphi}_{ \beta}^{\alpha}-\varphi_{\beta}^{\alpha}\\ \delta\varphi_{\beta\alpha}^{r}=\widetilde{\varphi}_{\beta\alpha}^{r}-\varphi_ {\beta\alpha}^{r}\\ \delta r_{\beta\alpha}^{r}=\widetilde{\varphi}_{\beta\alpha}^{r}-r_{\beta \alpha}^{r}\end{array} \right. \tag{3}\]  
where \"\(\sim\)\" represents the measurement value, \(\varphi_{\beta}^{\alpha}\) is the attitude vector (rad), \(\varphi_{\beta\alpha}^{r}\) is the velocity vector (m/s), and \(r_{\beta\alpha}^{r}\) is the position vector (m). The inertial navigation error equations in the Earth-centered, Earth-fixed (ECEF) coordinate frame can be seen in Equation (4), including the attitude error, velocity error, and position error update equations:  
\[ \begin{array}{l}\dot{\delta\widetilde{\varphi}}_{\alpha b}^{e}=C_{b}^{e} \left(\left(\overrightarrow{\omega_{b}^{b}}\times\right)-\left(\overrightarrow {\omega_{b}^{b}}\times\right)\right)-\left(\overrightarrow{\omega_{b}^{e}} \times\right)\left(\overrightarrow{\varphi_{eb}^{e}}-\overrightarrow{ \varphi_{eb}^{e}}\right)\\ \dot{\delta\widetilde{v}}_{\alpha b}^{e}=C_{b}^{e}\left(\overrightarrow{ \overrightarrow{r}_{eb}^{b}}-\overrightarrow{\overrightarrow{r}_{eb}^{b}} \right)-2\left(\overrightarrow{\omega_{b}^{e}}\times\right)\left( \overrightarrow{\overrightarrow{v}_{eb}}-\overrightarrow{\overrightarrow{v}_ {eb}^{e}}\right)+\overrightarrow{\overrightarrow{g}}^{e}-\overrightarrow{g }\\ \dot{\delta\widetilde{r}}_{\alpha b}^{e}=\overrightarrow{v}_{eb}^{e}- \overrightarrow{\overrightarrow{v}_{eb}^{e}}\end{array}  \tag{4}\]  
where \(C_{b}^{e}\) is the coordinate transformation matrix from the body frame to the Earth frame (a detailed expression can be found in Appendix B.2), \(\overrightarrow{\omega_{ie}^{e}}\) is the Earth rotation rate (rad/s), \(\overrightarrow{g}^{e}\) is the gravity vector in the Earth-centered, Earth-fixed frame (m/s\({}^{2}\)), and \(\left[\star\times\right]\) represents the calculation of a skew-symmetric matrix. Neglecting the effects of gravity errors, Equation (4) can be simplified as  
\[ \begin{array}{l}\dot{\delta\overline{\varphi}}_{eb}^{e}=C_{b}^{e}\left[ \overrightarrow{\omega_{b}^{b}}\times\right]-\left[\overrightarrow{\omega_{b}^ {e}}\times\right]\left(\overrightarrow{\omega_{eb}^{e}}\times\right.\\ \dot{\delta\widetilde{v}}_{eb}^{e}=C_{b}^{e}\delta\overrightarrow{\overrightarrow {r}_{eb}^{b}}-2\left(\overrightarrow{\omega_{i}^{e}}\times\right)\delta \widetilde{v}_{eb}^{e}\\ \dot{\delta\widetilde{r}}_{eb}^{e}=\delta\overrightarrow{v}_{eb}^{e}\end{array} . \tag{5}\]  
Figure 2 shows the updating process of the attitude, velocity, and position in the ECEF coordinate frame for the IMU. 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 error equation in Equation (5) becomes Equation (6), where the apostrophe (\({}^{\prime}\)) indicates the faulty parameter.