  Through assembling the navigation parameters as 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
is 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 error in continuous time are given
in matrix Lie group SE_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.
