Galilean State Estimation for Inertial Navigation Systems with Unknown Time Delay

TL;DR

This paper presents a novel geometric framework using Galilean symmetry and an Equivariant Filter for joint state and delay estimation in INS, outperforming traditional EKF methods especially with unknown GNSS delays.

Contribution

It introduces a new Galilean symmetry-based geometric model and an Equivariant Filter for accurate, consistent joint estimation of navigation states and unknown time delays in INS.

Findings

The EqF maintains accuracy and consistency across delays up to 500 ms.

The EKF's performance degrades significantly with increasing delays.

Validation on UAVs shows the proposed method outperforms state-of-the-art EKF.

Abstract

Many Inertial Navigation Systems (INS) use Global Navigation Satellite System (GNSS) position as the primary measurement to drive filter performance and bound error growth. However, commercial-grade GNSS receivers introduce unknown measurement delays ranging from 50 ms to 300 ms depending on sensor quality and operating mode. Such time delays can significantly degrade INS performance unless they are explicitly compensated for. Existing algorithms commonly estimate this delay offline, run the filter concurrently with GNSS measurements using buffered Inertial Measurement Unit (IMU) data, and predict the current state by forward-integrating buffered inertial measurements via IMU preintegration. The state-of-the-art online method is an Extended Kalman Filter (EKF) that explicitly models the time delay as a state parameter, which defines the preintegration duration. This paper introduces a novel geometric framework for modeling time-delayed INS, in which Galilean symmetry is leveraged to provide a joint representation of space and time for consistent state estimation. An Equivariant Filter (EqF) is derived for the coupled estimation of navigation states and time delay. Validation is performed on two fixed-wing Uncrewed Aerial Vehicles (UAV) with GNSS time lags of 90 ms and 120 ms. The test flights last two to three minutes. Simulations further investigate delays up to 500 ms and provide a statistical comparison against the state-of-the-art EKF. Results show that the EqF preserves accuracy and consistency, while the EKF lacks consistency and its performance degrades significantly with increasing measurement delays.