Table 1 Two-sample inertial navigation computation in local-level frame and Earth frame

Attitude update

\(\begin{aligned} & {\varvec{\upomega}}_{ie}^{n} = \left[ {\Omega \cos L\;\;\Omega \sin L\;\;0} \right]^{T} \\ & {\varvec{\upomega}}_{en}^{n} = \left[ {{{v_{E}^{n} } \mathord{\left/ {\vphantom {{v_{E}^{n} } {\left( {R_{E} + h} \right)}}} \right. \kern-0pt} {\left( {R_{E} + h} \right)}}\;\;{{v_{E}^{n} \tan L} \mathord{\left/ {\vphantom {{v_{E}^{n} \tan L} {\left( {R_{E} + h} \right)}}} \right. \kern-0pt} {\left( {R_{E} + h} \right)}}\;\;{{ - v_{N}^{n} } \mathord{\left/ {\vphantom {{ - v_{N}^{n} } {\left( {R_{N} + h} \right)}}} \right. \kern-0pt} {\left( {R_{N} + h} \right)}}} \right]^{T} \\ & {\varvec{\upsigma}}_{n} = T\left( {{\varvec{\upomega}}_{ie}^{n} + {\varvec{\upomega}}_{en}^{n} } \right) \\ & {\mathbf{q}}_{{n_{k} }}^{{n_{k + 1} }} = \cos \frac{{\left| {{\varvec{\upsigma}}_{n} } \right|}}{2} + \frac{{{\varvec{\upsigma}}_{n} }}{{\left| {{\varvec{\upsigma}}_{n} } \right|}}\sin \frac{{\left| {{\varvec{\upsigma}}_{n} } \right|}}{2} \\ \end{aligned}\)

\(\begin{aligned} & {\varvec{\upomega}}_{ie}^{e} = \left[ {\begin{array}{*{20}c} 0 & 0 & \Omega \\ \end{array} } \right]^{T} \\ & {\varvec{\upsigma}}_{e} = T{\varvec{\upomega}}_{ie}^{e} \\ & {\mathbf{q}}_{{e_{k} }}^{{e_{k + 1} }} = \cos \frac{{\left| {{\varvec{\upsigma}}_{e} } \right|}}{2} + \frac{{{\varvec{\upsigma}}_{e} }}{{\left| {{\varvec{\upsigma}}_{e} } \right|}}\sin \frac{{\left| {{\varvec{\upsigma}}_{e} } \right|}}{2} \\ \end{aligned}\)

\({\varvec{\upsigma}}_{b} = \Delta {\varvec{\uptheta}}_{1} + \Delta {\varvec{\uptheta}}_{2} + \frac{2}{3}\Delta {\varvec{\uptheta}}_{1} \times \Delta {\varvec{\uptheta}}_{2} ,\quad {\mathbf{q}}_{{b_{k} }}^{{b_{k + 1} }} = \cos \frac{{\left| {{\varvec{\upsigma}}_{b} } \right|}}{2} + \frac{{{\varvec{\upsigma}}_{b} }}{{\left| {{\varvec{\upsigma}}_{b} } \right|}}\sin \frac{{\left| {{\varvec{\upsigma}}_{b} } \right|}}{2}\)

\({\mathbf{C}}_{b}^{n} \left( {k + 1} \right) = {\mathbf{C}}_{{n_{k} }}^{{n_{k + 1} }} {\mathbf{C}}_{b}^{n} \left( k \right){\mathbf{C}}_{{b_{k + 1} }}^{{b_{k} }}\)

\({\mathbf{C}}_{b}^{e} \left( {k + 1} \right) = {\mathbf{C}}_{{e_{k} }}^{{e_{k + 1} }} {\mathbf{C}}_{b}^{e} \left( k \right){\mathbf{C}}_{{b_{k + 1} }}^{{b_{k} }}\)

Velocity update

\({\mathbf{u}}^{b} = \Delta {\mathbf{v}}_{1} + \Delta {\mathbf{v}}_{2} + \frac{1}{2}\left( {\Delta {\varvec{\uptheta}}_{1} + \Delta {\varvec{\uptheta}}_{2} } \right) \times \left( {\Delta {\mathbf{v}}_{1} + \Delta {\mathbf{v}}_{2} } \right) + \frac{2}{3}\left( {\Delta {\varvec{\uptheta}}_{1} \times \Delta {\mathbf{v}}_{2} + \Delta {\mathbf{v}}_{1} \times \Delta {\varvec{\uptheta}}_{2} } \right)\)

\(\begin{aligned} {\mathbf{v}}^{n} \left( {k + 1} \right) & = {\mathbf{v}}^{n} \left( k \right) + {\mathbf{C}}_{b}^{n} \left( k \right){\mathbf{u}}^{b} \\ & \quad - T\left( {2{\varvec{\upomega}}_{ie}^{n} + {\varvec{\upomega}}_{en}^{n} } \right) \times {\mathbf{v}}^{n} \left( k \right) + T{\mathbf{g}}^{n} \left( k \right) \\ \end{aligned}\)

\(\begin{aligned} {\mathbf{v}}^{e} \left( {k + 1} \right) & = {\mathbf{v}}^{e} \left( k \right) + {\mathbf{C}}_{b}^{e} \left( k \right){\mathbf{u}}^{b} \\ & \quad - 2T{\varvec{\upomega}}_{ie}^{e} \times {\mathbf{v}}^{e} \left( k \right) + T{\mathbf{g}}^{e} \left( k \right) \\ \end{aligned}\)

Position update

\(\begin{aligned} & {\mathbf{r}} = {{T\left( {{\mathbf{v}}^{n} \left( k \right) + {\mathbf{v}}^{n} \left( {k + 1} \right)} \right)} \mathord{\left/ {\vphantom {{T\left( {{\mathbf{v}}^{n} \left( k \right) + {\mathbf{v}}^{n} \left( {k + 1} \right)} \right)} 2}} \right. \kern-0pt} 2} \\ & {\mathbf{p}}^{n} \left( {k + 1} \right) = {\mathbf{p}}^{n} \left( k \right) + {\mathbf{R}}_{c} \left( k \right){\mathbf{r}} \\ \end{aligned}\)

\({\mathbf{p}}^{e} \left( {k + 1} \right) = {\mathbf{p}}^{e} \left( k \right) + {{T\left( {{\mathbf{v}}^{e} \left( k \right) + {\mathbf{v}}^{e} \left( {k + 1} \right)} \right)} \mathord{\left/ {\vphantom {{T\left( {{\mathbf{v}}^{e} \left( k \right) + {\mathbf{v}}^{e} \left( {k + 1} \right)} \right)} 2}} \right. \kern-0pt} 2}\)