## 3 Error Overboundings against IMU Faults
### IMU Faults Propagation in the Error-State EKF
\begin{array}{l}\delta\overrightarrow{\omega}_{eb}^{e}=F_{\delta \overrightarrow{\omega}_{eb}^{e}}\delta\overrightarrow{\omega}_{eb}^{e}+G_{g} \overrightarrow{w}_{g}+\overrightarrow{f}_{\delta\overrightarrow{\omega}_{eb }^{e}}\\ \delta\overrightarrow{\tau}_{eb}^{e}=F_{\delta\overrightarrow{\omega}_{eb}^{e} }\delta\overrightarrow{\omega}_{eb}^{e}+G_{a}\overrightarrow{w}_{a}+ \overrightarrow{f}_{\delta\overrightarrow{\omega}_{eb}^{e}}\\ \delta\overrightarrow{\tau}_{eb}^{e}=F_{\delta\overrightarrow{\tau}_{eb}^{e} }\delta\overrightarrow{\omega}_{eb}^{e}+G_{r}\overrightarrow{w}_{r}+ \overrightarrow{f}_{\delta\overrightarrow{\tau}_{eb}^{e}}\end{array} . \tag{15}\]Hence, an EKF state equation under the IMU faults condition generalized as Equation (16) in a continuous form. The error-state vector contains the IMU position error, the IMU velocity error, the IMU attitude error, accelerometer biases, gyroscope biases, the GNSS receiver offset, and the GNSS receiver drift:  
\[\overrightarrow{X}^{\prime}=F\overrightarrow{X}+G_{w}\overrightarrow{w}+ \overrightarrow{f}_{\overrightarrow{X}} \tag{16}\]  
where  
\[\overrightarrow{X}=\left[ \begin{array}{c}\delta\overrightarrow{\sigma}^{e} _{eb}\\ \delta\overrightarrow{\sigma}^{e}_{eb}\\ \delta\overrightarrow{\tau}^{b}_{eb}\\ \overrightarrow{b}_{a}\\ \overrightarrow{\tau}_{b}\\ \overrightarrow{b}_{g}\\ \delta\rho_{c}\\ \delta\dot{\rho}_{c}\end{array} \right]\overrightarrow{w}=\left[ \begin{array} []{c}\overrightarrow{w}_{a}\\ \overrightarrow{w}_{g}\\ \overrightarrow{w}_{g}\\ \end{array} \right]f_{\overrightarrow{X}}=\left[ \begin{array}{c} \overrightarrow{f}_{\overrightarrow{\delta\dot{\dot{\rho}}}^{e}_{eb}}\\ \overrightarrow{f}_{\overrightarrow{\delta\dot{\dot{\tau}}}^{e}_{eb}}\\ \overrightarrow{f}_{\overrightarrow{\delta\dot{\tau}}^{e}_{eb}}\\ 0_{3}\\ 0_{3}\\ 0_{3}\\ 0\\ 0\end{array} \right]G_{w}=\left[ \begin{array}{c}0_{3}&-C_{b}^{e}\\ -C_{b}^{e}&0_{3}\\ 0_{3}&0_{3}\\ 0_{3}&0_{3}\\ 0_{3}&0_{3}\\ 0&0\\ 0&0\end{array} \right]\]  
\[F=\left[ \begin{array}{cccccc}\left[\overrightarrow{\omega}^{e}_{ie}\times \right]&0_{3}&0_{3}&0_{3}&C_{b}^{e}&0&0\\ -\left[\left(\dot{C}_{b}\overrightarrow{\rho}^{b}_{ib}\right)\times\right]&- 2\left[\overrightarrow{\omega}^{e}_{ie}\times\right]&0&-C_{b}^{e}&0_{3}&0&0\\ 0_{3}&I_{3}&0&0_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&0_{3}&0_{3}&0&0\\ 0&0&0&0&0&0&I\\ 0&0&0&0&0&0&0\end{array} \right].\]  
The discrete form of Equation (16) is shown in Equation (17) [18]:  
\[\overrightarrow{X}_{k}^{\prime}=\varPhi_{k}\overrightarrow{X}_{k-1}+ \overrightarrow{\omega}_{k}+\overrightarrow{f}_{\overrightarrow{X}_{k}}. \tag{17}\]  
The discrete form of \(F\) is shown as  
\[\varPhi\approx\left[ \begin{array}{cccccc}I_{3}-\left[\overrightarrow{ \omega}^{e}_{ie}\times\right]\tau_{s}&0_{3}&0_{3}&0_{3}&C_{b}^{e}\tau_{s}&0&0\\ -\left[\left(\dot{C}_{b}^{e}\overrightarrow{\rho}^{b}_{ib}\right)\times \right]\tau_{s}&I_{3}-2\left[\overrightarrow{\omega}^{e}_{ie}\times\right] \tau_{s}&\left[\overrightarrow{\omega}^{e}_{ie}\times\right]^{2}\tau_{s}&-C_{b }^{e}\tau_{s}&0_{3}&0&0\\ 0_{3}&I_{3}\tau_{s}-\left[\overrightarrow{\omega}^{e}_{ie}\times\right]^{2}\tau _{s}^{2}&I_{3}\tau_{s}+\left[\overrightarrow{\omega}^{e}_{ie}\times\right] \tau_{s}z^{2}/2&0_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&I_{3}&0_{3}&0&0\\ 0_{3}&0_{3}&0_{3}&0_{3}&I_{3}&0&0\\ 0&0&0&0&0&I_{3}&\tau_{s}\\ 0&0&0&0&0&0&I_{3}\end{array} \right] \tag{18}\]  
where \(\tau_{s}\) is the time interval of the adjacent propagation. Thus, the predicted state under the IMU faults condition at epoch k is  
\[\hat{X}_{k}^{-\prime}=\varPhi_{k}\hat{X}_{k-1}+\overrightarrow{f}_{ \overrightarrow{X}_{k}}. \tag{19}\]  
Substitute Equation (21) into Equation (26):  
\[