Log-Linear Error State Model Derivation Without Approximation for INS

Through assembling the navigation parameters as the matrix Lie group state, the corresponding inertial navigation system (INS) kinematic model possesses a group-affine property. The Lie logarithm of the navigation state estimation error satisfies a log-linear autonomous differential equation. These log-linear models are still applicable even with arbitrarily large initial errors, which are very attractive for INS initial alignment. However, in existing works, the log-linear models are all derived based on first-order linearization approximation, which seemingly goes against their successful applications in INS initial alignment with large misalignments. In this work, it is shown that the log-linear models can also be derived without any approximation; the error dynamics for both left- and right-invariant errors in continuous time are given in the matrix Lie group
$\mathrm{S}{\mathrm{E}}_2( 3 )$
for the first time. This work provides another evidence for the validity of the log-linear model in situations with arbitrarily large initial errors.