arxiv: 2605.13266 · v1 · submitted 2026-05-13 · 💻 cs.RO

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Galilean State Estimation for Inertial Navigation Systems with Unknown Time Delay

Giulio Delama , Martin Scheiber , Yixiao Ge , Tarek Hamel , Stephan Weiss , Robert Mahony

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:07 UTC · model grok-4.3

classification 💻 cs.RO

keywords inertial navigation systemstime delay estimationequivariant filterGalilean symmetryGNSS measurementsstate estimationUAV navigation

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The pith

An equivariant filter using Galilean symmetry estimates both navigation states and unknown time delays consistently.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to model unknown time delays in inertial navigation by drawing on the symmetries of Galilean space-time transformations. This allows derivation of an equivariant filter that estimates the navigation state and the delay parameter together. The filter stays consistent and accurate even when delays increase, which is crucial for real-world systems where GNSS receivers introduce lags of 50 to 300 milliseconds. A sympathetic reader would see this as a way to avoid the error growth that plagues standard filters without needing extra tuning.

Core claim

The central claim is that leveraging Galilean symmetry provides a joint representation of space and time for consistent state estimation in time-delayed inertial navigation systems, enabling derivation of an Equivariant Filter for coupled estimation of navigation states and time delay that preserves accuracy and consistency where the Extended Kalman Filter does not.

What carries the argument

The Equivariant Filter derived from Galilean symmetry for joint space-time representation in delayed INS measurements.

If this is right

The EqF preserves accuracy and consistency on UAV flights with 90ms and 120ms GNSS lags.
Simulations confirm no significant performance degradation for delays up to 500ms.
The approach provides consistent estimation without post-hoc parameter tuning or data exclusions.
Validation on fixed-wing UAVs lasting two to three minutes supports real-time applicability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar symmetry-based methods could address time offsets in other robotic sensor fusion tasks.
The framework may improve robustness in autonomous navigation under uncertain timing.
Testing on longer flights or different platforms would further validate scalability.

Load-bearing premise

Galilean symmetry can be leveraged to provide a joint representation of space and time that remains consistent for time-delayed INS measurements.

What would settle it

An experiment measuring whether the EqF's error covariance matches the actual estimation errors across a range of increasing time delays, in contrast to the EKF.

Figures

Figures reproduced from arXiv: 2605.13266 by Giulio Delama, Martin Scheiber, Robert Mahony, Stephan Weiss, Tarek Hamel, Yixiao Ge.

**Figure 1.** Figure 1: Overview of the real-world flight experiments. The top row shows the two fixed-wing UAVs used for data collection, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Galilean transformations for a rigid body [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Galilean transformations between two rigid bodies [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Representation of the time-varying Galilean frames between bodies [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Real-world comparison of the proposed EqF (solid blue) against three EKF baselines: a standard EKF without delay [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Results of a Monte Carlo simulation with 50 different realizations of randomized initial states, process noise, [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The Galilean EqF folds time delay into the navigation state via symmetry and keeps covariance consistent where the EKF degrades, but only on short 2-3 minute flights.

read the letter

The main takeaway is that the authors treat the unknown GNSS delay as part of a joint space-time state using Galilean symmetry, then derive an equivariant filter that estimates both navigation variables and the delay together. This avoids the usual EKF state augmentation and preintegration adjustments. In simulation the EqF maintains statistical consistency up to 500 ms delays while the EKF loses it. On two real fixed-wing UAV flights with 90 ms and 120 ms measured lags the same pattern holds for the short duration tested.

Referee Report

2 major / 2 minor

Summary. The paper claims that Galilean symmetry provides a joint space-time representation for inertial navigation systems (INS) subject to unknown GNSS measurement delays (50-300 ms). From this, an Equivariant Filter (EqF) is derived for simultaneous estimation of navigation states and time delay. On two 2-3 minute fixed-wing UAV flights (90 ms and 120 ms lags) and simulations with delays up to 500 ms, the EqF is reported to preserve both accuracy and statistical consistency, whereas the EKF loses consistency and degrades as delay increases.

Significance. If the central consistency claim holds, the work supplies a symmetry-derived, parameter-free alternative to ad-hoc EKF delay augmentation or offline calibration for time-delayed INS. The Galilean construction is a genuine strength: it grounds the filter in an external physical principle rather than fitted parameters, and the reported preservation of consistency under increasing delay is a non-trivial result for real-time navigation. The short experimental horizon, however, leaves open whether the advantage persists when delay-estimation residuals integrate over longer periods.

major comments (2)

[Experimental Results] Experimental Results (UAV flights): The two flights last only 2-3 minutes each. For INS consistency—defined as the filter covariance correctly bounding actual error growth—this duration is too short to stress accumulation of small delay-estimation biases into position drift. The central claim that the EqF “preserves consistency” while the EKF does not therefore rests on evidence that cannot yet rule out a short-horizon artifact.
[Simulation Results] Simulation section: No Monte-Carlo run count, NEES statistics, or error-bar details are supplied for the consistency comparison up to 500 ms delay. Without these, it is impossible to assess whether the reported superiority of the EqF over the EKF is statistically reliable or merely qualitative.

minor comments (2)

[§3] The abstract states that the EqF is “derived” from Galilean symmetry but does not indicate whether the resulting filter equations are presented in closed form or require numerical integration; a brief statement in §3 would improve clarity.
[§2] Notation for the augmented state (navigation + delay) is introduced without an explicit table; adding one would help readers track the Galilean group action.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and have revised the paper to strengthen the presentation of results and acknowledge limitations.

read point-by-point responses

Referee: [Experimental Results] Experimental Results (UAV flights): The two flights last only 2-3 minutes each. For INS consistency—defined as the filter covariance correctly bounding actual error growth—this duration is too short to stress accumulation of small delay-estimation biases into position drift. The central claim that the EqF “preserves consistency” while the EKF does not therefore rests on evidence that cannot yet rule out a short-horizon artifact.

Authors: We agree that the 2–3 minute UAV flights provide limited opportunity to observe long-term integration of small delay-estimation residuals into position drift. The consistency claim for these flights rests on NEES remaining within the 3-sigma bounds for the EqF while exceeding them for the EKF. In the revised manuscript we have added an explicit limitations paragraph in the discussion section that acknowledges the short experimental horizon and notes that the simulation results (which extend to 500 ms delays over longer effective horizons) provide supporting evidence. We also outline the need for future extended-duration flight tests. revision: partial
Referee: [Simulation Results] Simulation section: No Monte-Carlo run count, NEES statistics, or error-bar details are supplied for the consistency comparison up to 500 ms delay. Without these, it is impossible to assess whether the reported superiority of the EqF over the EKF is statistically reliable or merely qualitative.

Authors: We thank the referee for identifying this omission. The revised manuscript now reports that the simulation results are based on 100 independent Monte-Carlo runs, includes NEES time-series plots with 1-sigma error bars, and provides tabulated statistics showing that the EqF NEES remains within the expected consistency envelope for all tested delays (0–500 ms) while the EKF NEES grows beyond the envelope as delay increases. These quantitative details make the statistical comparison explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation grounded in external Galilean symmetry

full rationale

The paper constructs an Equivariant Filter by leveraging the Galilean group to jointly represent space and time for delayed INS measurements. This symmetry principle is an external mathematical structure independent of the target estimation result. No equations reduce by construction to fitted parameters, no load-bearing self-citations close the derivation loop, and no ansatz is smuggled via prior author work. The central claim (EqF preserves consistency where EKF does not) follows from the symmetry-induced filter equations rather than tautological renaming or input-output equivalence. Short flight durations affect empirical validation but do not create circularity in the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that Galilean symmetry extends consistently to time-delayed measurement models; no free parameters or invented entities are identified from the abstract.

axioms (1)

domain assumption Galilean symmetry provides a joint representation of space and time for consistent state estimation in time-delayed INS
Invoked to derive the Equivariant Filter for coupled navigation-state and delay estimation.

pith-pipeline@v0.9.0 · 5581 in / 1300 out tokens · 123432 ms · 2026-05-14T18:07:07.242736+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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