## 3 Propagation and Update of the ICEKF
### Propagation  
For the linearized continuous-time errors of an ICEKF state, the following equation exists [24]:  
\[\widetilde{\dot{\mathbf{X}}}=\mathbf{F}_{c}\widetilde{\dot{\mathbf{X}}}+\mathbf{G}_{c}\mathbf{n}, \tag{18}\]  
where \(\mathbf{n}=\left[\mathbf{n}_{\omega c}^{T}\ \mathbf{n}_{a_{mi}}^{T}\ \mathbf{n}_{\omega c}^{T}\ \mathbf{n}_{\omega_{mi}}^{T}\ \mathbf{n}_{b_{ai}}^{T}\ \mathbf{n}_{b_{ai}}^{T}\right]^{T}\) is the propagation noise vector for the ICEKF following the Gaussian distribution. \(\mathbf{F}_{c}\) and \(\mathbf{G}_{c}\) are considered constant during every iteration.  
To discretize Equation (18) during the period \(\Delta t\), we write the description of the discrete state transition matrix \(\mathbf{F}_{d}\) and the discrete noise covariance matrix \(\mathbf{Q}_{d}\) as follows [24]:  
\[\left\{\begin{array}{l}\mathbf{F}_{d}=\exp(\mathbf{F}_{c}\Delta t)=\mathbf{I}_{d}+\mathbf{F}_ {c}\Delta t+\frac{1}{2\pi}\mathbf{F}_{c}^{2}\Delta t^{2}+\dots\\ \mathbf{Q}_{d}=\int_{t}^{t+\Delta t}\mathbf{F}_{d}(\tau)\mathbf{G}_{c}\mathbf{Q}_{c}\mathbf{G}_{c}^ {T}\mathbf{F}_{d}(\tau)^{T}d\tau\end{array}\right., \tag{19}\]  
where \(\mathbf{Q}_{c}=diag(\sigma_{\mathbf{n}_{c}c}^{2},\sigma_{\mathbf{n}_{a_{i}}}^{2},\sigma_{ \mathbf{n}_{\omega c}^{2}}^{2},\sigma_{\mathbf{n}_{\omega i}}^{2},\sigma_{\mathbf{n}_{b_{ai} }}^{2},\sigma_{\mathbf{n}_{b_{ai}}}^{2})\) is the diagonal matrix converted from Gaussian noise.  
Section 2.5 presents the specific expression of \(\widetilde{\hat{\mathbf{X}}}\). Considering that the algorithm operates at a high frequency, only the first-order expansion in Equation (19) is considered. The complete expression of \(\mathbf{F}_{d}\) is derived as follows:  
\[\mathbf{F}_{d}=\left[\begin{array}{ccccc}\mathbf{I}_{3}&\mathbf{A}_{1}&\mathbf{A}_{2}&\ldots& \mathbf{0}_{3\times 19}\\ \mathbf{0}_{3}&\mathbf{I}_{3}&\ldots&\mathbf{0}_{3\times 22}\\ \mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{I}_{3}&\mathbf{0}_{3}&\mathbf{A}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}& \mathbf{A}_{4}&\mathbf{0}_{3}&\mathbf{0}_{3\times 1}\\ \mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{A}_{5}&\mathbf{0}_{3}&\mathbf{A}_{6}&\mathbf{A}_{7}& \mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3\times 1}\\ \mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{A}_{8}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}& \mathbf{A}_{9}&\mathbf{0}_{3\times 1}\\ \mathbf{0}_{13\times 15}&\ldots&\mathbf{I}_{13\times 13}\end{array}\right]_{\mathbf{28 }\ast 28}, \tag{20}\]  
where \(A_{1}=\lambda(1-\mu_{v})\mathbf{R}_{c}^{i}\Delta t\), \(A_{2}=\mu_{v}\mathbf{I}_{3}\Delta t\), \(A_{3}=-\hat{\mathbf{R}}_{i}\lfloor\hat{\mathbf{a}}_{i}\times\rfloor\Delta t\), \(A_{4}=-\hat{\mathbf{R}}_{i}\Delta t\), \(A_{5}=\mathbf{I}_{3}-\left(\mu_{w}\lfloor\left(\mathbf{R}_{w_{w}c}\hat{\mathbf{\alpha}}_{ i}\right)\times\rfloor+(1-\mu_{w})\lfloor\hat{\mathbf{\alpha}}_{c}\times\rfloor \right)\Delta t\), \(A_{6}=(1-\mu_{w})\mathbf{I}_{3}\Delta t\), \(A_{7}=\mu_{w}\mathbf{R}_{w_{w}c}\hat{\mathbf{\alpha}}\), \(A_{8}=\mathbf{I}_{3}-\lfloor\hat{\mathbf{\alpha}}_{I}\times\rfloor\Delta t\), and \(A_{9}=-\mathbf{I}_{3}\Delta t\).  
\(\mathbf{Q}_{d}\) can be further derived by combining Equation (20) and \(\mathbf{G}_{c}\), and the explicit form of \(\mathbf{G}_{c}\) is recovered according to Equation (18) as follows:  
\[\mathbf{G}_{c}=\left[\begin{array}{ccccc}\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{ 0}_{3}&\mathbf{0}_{3}\\ \mathbf{I}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}\\ \mathbf{0}_{3}&-\hat{\mathbf{R}}_{i}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}\\ \mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}\\ \mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&-\mathbf{I}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}\\ \mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}&\mathbf{0}_{3}\\ \mathbf{0}_{7\times 18}\end{array}\right]_{\mathbf{28}\ast 18}. \tag{21}\]