## 3 Error Overboundings against IMU Faults
### IMU Faults Propagation in the Error-State EKF
The accelerometer bias and gyroscope bias are usually modeled as constant errors [23]:  
\[\delta\overset{\cdot\cdot\epsilon}{\partial\tau}_{eb}^{\epsilon} =\bigg{(}C_{b}^{\epsilon}\bigg{)}^{\prime}\bigg{(}\delta\overset{ \rightarrow}{f}_{ib}^{b}\bigg{)}-2\bigg{[}\overset{\omega_{ie}^{e}}{\times} \bigg{]}\delta\overset{\omega_{cb}^{\epsilon}}{\partial\tau}_{eb}^{\epsilon}\]0  
\[\delta\overset{\cdot\cdot\epsilon}{\partial\tau}_{eb}^{\epsilon} =\bigg{(}C_{b}^{\epsilon}\bigg{)}^{\prime}\bigg{(}\delta\overset{ \rightarrow}{f}_{ib}^{b}\bigg{)}-2\bigg{[}\overset{\omega_{ie}^{e}}{\times} \bigg{]}\delta\overset{\omega_{cb}^{\epsilon}}{\partial\tau}_{eb}^{\epsilon}\]1  
According to Equations (10) and (11), the accelerometer measurement of the specific force error and the gyroscope measurement of the angular rate error are derived from the accelerometer measurement of the specific force, \(\overrightarrow{f}_{ib,meas}^{b}\), and the gyroscope measurement of the angular rate, \(\overrightarrow{\omega}_{ib,meas}^{b}\), as shown in Equation (12):  
\[ \begin{array}{l}\delta\overrightarrow{\omega}_{ib}^{b}&=\overrightarrow{ \omega}_{ib,meas}^{b}-\overrightarrow{\omega}_{ib}^{b}\\ &=\overrightarrow{b}_{g}+G_{g}\overrightarrow{f}_{ib}^{b}+\overrightarrow{w} _{g}\\ \delta\overrightarrow{f}_{ib}^{b}&=\overrightarrow{f}_{ib,meas}^{b}- \overrightarrow{f}_{ib}\\ &=\overrightarrow{b}_{a}+\overleftarrow{w}_{a}\end{array} . \tag{12}\]  
Substituting Equation (12) into Equation (9), the inertial navigation error equation under the IMU faults condition is  
\[ \begin{array}{l}\delta\overrightarrow{\omega}_{eb}^{e}=C_{b}^{e}\bigg{[} \bigg{(}\overrightarrow{b}_{g}+G_{g}\overrightarrow{f}_{ib}^{b}+ \overrightarrow{w}_{g}\bigg{)}\bigg{]}-\bigg{[}\overrightarrow{\omega}_{ib}^ {e}\times\bigg{]}\delta\overrightarrow{\omega}_{eb}^{e}+\overrightarrow{f}_{ \delta\overrightarrow{\omega}_{eb}^{e}}\\ \delta\overrightarrow{\omega}_{eb}^{e}=C_{b}^{e}\bigg{[}\bigg{(} \overrightarrow{b}_{a}+\overrightarrow{w}_{a}\bigg{)}\bigg{]}-2\bigg{[} \overrightarrow{\omega}_{ib}^{e}\times\bigg{]}\delta\overrightarrow{\omega}_ {eb}^{e}+\overrightarrow{f}_{\delta\overrightarrow{\omega}_{eb}^{e}}\\ \delta\overrightarrow{\tau}_{eb}^{e}=\delta\overrightarrow{\omega}_{eb}^{e}+ \overrightarrow{f}_{\delta\overrightarrow{\tau}_{eb}^{e}}\end{array} . \tag{13}\]  
Rearranging Equation (13), the inertial navigation error equation under the IMU faults condition (Equation (8)) becomes a continuous form of an EKF error-state equation as a function of EKF error states, process noise, and the state fault vector, as shown in Equation (14):  
\[ \begin{array}{l}\delta\widehat{\phi}_{ab}^{e}=\underbrace{-\big{[} \widehat{\omega}_{e}^{e}\times\big{]}\delta\widehat{\phi}_{ab}^{e}+C_{b}^{e} \big{[}\widehat{b}_{g}\times\big{]}}+C_{b}^{e}\big{[}\big{(}G_{g}\overrightarrow {f}_{a}^{b}+\overrightarrow{w}_{g}\big{)}\times\big{]}+\widetilde{f}_{\delta \overrightarrow{\omega}_{eb}}\\ \delta\widehat{\psi}_{eb}^{e}=\underbrace{-\big{[}\widehat{\omega}_{e}^{e} \times\big{]}\delta\widehat{\psi}_{eb}^{e}+C_{b}^{e}\big{[}\big{]}}+C_{b}^{e} \big{[}\big{(}\overleftarrow{w}_{a}^{e}\big{)}+\widetilde{f}_{\delta \overrightarrow{\omega}_{eb}^{e}}\\ \delta\widehat{\tau}_{eb}^{e}=\delta\widehat{\psi}_{eb}^{e}+\widehat{f}_{ \delta\overrightarrow{\omega}_{eb}^{e}}\end{array} . \tag{14}\]  
Based on the integral relationship between the velocity and the position in the error-state inertial navigation equations, Equation (14) can be rewritten as  
\[