Log-linear Error State Model Derivation without Approximation for INS

TL;DR

This paper derives exact log-linear error models for inertial navigation systems using matrix Lie groups, eliminating the need for approximation and enhancing initial alignment robustness.

Contribution

It presents the first derivation of exact, non-approximated log-linear error models for INS on matrix Lie groups, validating their use with large initial errors.

Findings

Exact log-linear models derived without approximation

Models applicable to large initial errors in INS

Supports improved initial alignment in INS systems

Abstract

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.