## 2 Design of the State Vector
### Error of the State Vector
By subjecting Equation (12) to Equation (11) and disregarding the high-order terms, the following is obtained:  
\[\delta\ddot{\overline{\mathbf{q}}}_{ic}=\left\{\begin{array}{l}\frac{\mu_{c}}{2} \delta\overline{\mathbf{q}}_{ic}\otimes\left[\begin{array}{l}0\\ \mathbf{R}_{\mu_{\omega}c}^{\,Then, with the small angle assumption \(\delta\overline{\mathbf{q}}_{ic}=\left[q_{0}.\delta\mathbf{q}_{ic}\right]^{T}\approx \left[1,\frac{1}{2}\delta\theta_{ic}\right]^{T}\), the error of the coupled equivalent Euler angle is simplified as follows:  
\[\delta\dot{\mathbf{q}}_{ic}=-\mu_{\omega}\Big{[}\Big{[}\left(\mathbf{R}_{\mu_{\omega}c }{}^{T}\dot{\mathbf{\omega}}_{i}\right)\times\Big{]}\delta\mathbf{\theta}_{ic}-\mathbf{R}_ {\mu_{\omega}c}{}^{T}\Delta\omega_{i}\Big{]}-(1-\mu_{\omega})\big{[}\lfloor \dot{\mathbf{\omega}}_{c}\times\rfloor\delta\theta_{ic}-\Delta\omega_{c}\big{]}, \tag{14}\]  
The expectation of the body angular velocity of the IMU is \(\dot{\mathbf{\omega}}_{i}=\mathbf{\omega}_{mi}-\dot{\mathbf{b}}_{\omega i}\), and \(\Delta\mathbf{b}_{\omega i}=\mathbf{b}_{\omega i}-\dot{\mathbf{b}}_{\omega i}\). Resembling the process above, the error of the rotation of the IMU is written as follows:  
\[ \begin{array}{l}\delta\dot{\overline{\mathbf{q}}}_{i}=\dot{\overline{\mathbf{q}}}_{i }^{*}\otimes(\dot{\overline{\mathbf{q}}}_{i}-\dot{\overline{\mathbf{q}}}_{i}\otimes \delta\overline{\mathbf{q}}_{i})=\frac{1}{2}\delta\overline{\mathbf{q}}_{i}\otimes( \dot{\overline{\mathbf{\omega}}}_{i}+\Delta\overline{\mathbf{\omega}}_{i})-\frac{1}{2 }\dot{\overline{\mathbf{\omega}}}_{i}\otimes\delta\overline{\mathbf{q}}_{i}\\ =\frac{1}{2}\left[\begin{array}{cc}0&-\dot{\mathbf{\omega}}_{i}\\ \dot{\mathbf{\omega}}_{i}&-\lfloor\dot{\mathbf{\omega}}_{i}\times\rfloor\end{array} \right]\left[\begin{array}{cc}q_{0}\\ \delta\dot{\mathbf{q}}_{i}\end{array}\right]-\frac{1}{2}\left[\begin{array}{cc}0& -\dot{\mathbf{\omega}}_{i}\\ \dot{\mathbf{\omega}}_{i}&\lfloor\dot{\mathbf{\omega}}_{i}\times\rfloor\end{array} \right]\left[\begin{array}{cc}q_{i0}\\ \Delta\mathbf{\omega}_{i}&-\lfloor\Delta\mathbf{\omega}_{i}\times\rfloor\end{array} \right]\left[\begin{array}{cc}q_{i0}\\ \delta\dot{\mathbf{q}}_{i}\end{array}\right]\\ \approx\left[\begin{array}{cc}0&\\ -\lfloor\dot{\mathbf{\omega}}_{i}\times\rfloor\delta\dot{\mathbf{q}}_{i}\end{array} \right]-\left[\begin{array}{cc}0&\\ \frac{1}{2}\delta\dot{\mathbf{q}}_{i0}\Delta\omega_{i}\end{array}\right]\end{array}  \tag{15}\]  
Equation (15) can be simplified with the small angle assumption as follows:  
\[\dot{\delta\dot{\mathbf{\theta}}}_{i}=-\lfloor\dot{\mathbf{\omega}}_{i}\times\rfloor \delta\dot{\mathbf{\theta}}_{i}-\Delta\mathbf{\omega}_{i}=-\lfloor\dot{\mathbf{\omega}}_{i }\times\rfloor\delta\dot{\mathbf{\theta}}_{i}-\Delta\mathbf{b}_{\omega i}-\mathbf{n}_{ \omega_{mi}}, \tag{16}\]  
The derivatives of the remaining terms in \(\widetilde{\mathbf{X}}\) are as follows:  
\[\dot{\Delta\mathbf{v}}_{c}=\mathbf{n}_{v_{c}},\Delta\dot{\mathbf{\omega}}_{c}=\mathbf{n}_{ \omega_{c}},\Delta\dot{\mathbf{\omega}}_{i}=\frac{d}{dt}(\Delta\mathbf{b}_{\omega i}+ \mathbf{n}_{\omega_{mi}})=\mathbf{n}_{\omega i},\Delta\dot{\mathbf{b}}_{ai}=\mathbf{n}_{\mathbf{b} _{ai}},\Delta\dot{\mathbf{b}}_{\omega i}=\mathbf{n}_{\mathbf{b}_{ai}},\Delta\dot{\Delta}=n_ {\lambda}, \tag{17}\]