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arxiv: 2606.29123 · v2 · pith:RBC3FIHNnew · submitted 2026-06-28 · 💻 cs.RO

On the Identifiability of Aided Inertial Navigation Under Measurement Delays: A Geometric Approach

Jonathan Kelly This is my paper

Pith reviewed 2026-07-01 07:16 UTC · model grok-4.3

classification 💻 cs.RO
keywords aided inertial navigationmeasurement time delaysidentifiabilitygeometric analysiscontinuous symmetrydegenerate trajectoriessensor fusion
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The pith

A continuous symmetry in the delayed measurement model prevents unique recovery of both the delay and initial navigation state for a larger class of trajectories.

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

The paper examines the conditions under which an unknown constant delay in aiding sensor measurements can be recovered along with the initial navigation state that defines the trajectory. Geometric analysis reveals that identifiability fails for more degenerate trajectories than earlier results indicated, because a continuous symmetry maps different delay-state pairs to identical measurements. A reader would care about this result because navigation systems must know when unique recovery is impossible in order to avoid relying on unrecoverable parameters. The work focuses on a single aiding sensor and shows that both the timing of measurements and the shape of the trajectory determine whether the symmetry appears.

Core claim

Our geometric analysis shows that, for a larger class of uninformative (i.e., degenerate) trajectories than has previously been reported, the delayed measurement model admits a continuous symmetry that prevents unique delay-and-state recovery.

What carries the argument

The continuous symmetry admitted by the delayed measurement model for degenerate trajectories.

Load-bearing premise

The analysis assumes a single aiding sensor whose measurements have an unknown but constant delay relative to the inertial-measurement data stream, with the trajectory parameterized by the initial navigation state.

What would settle it

Construct a specific degenerate trajectory, vary the delay while adjusting the initial state to preserve the symmetry, and check whether the resulting measurement sequences are identical; distinct sequences would falsify the symmetry claim.

read the original abstract

In aided inertial navigation, measurements from different sensors are often subject to unknown relative time delays. Consider a single aiding sensor whose measurements have an unknown but constant delay relative to the inertial-measurement data stream. We study the identifiability of the delay and the initial navigation state that parameterizes the trajectory. Identifiability depends on both the temporal structure of the aiding measurements and the form of the trajectory itself. Our geometric analysis shows that, for a larger class of uninformative (i.e., degenerate) trajectories than has previously been reported, the delayed measurement model admits a continuous symmetry that prevents unique delay-and-state recovery.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that, for aided inertial navigation with a single aiding sensor subject to an unknown constant delay, identifiability of the delay together with the initial navigation state (which parameterizes the trajectory) depends on both the temporal structure of the aiding measurements and the trajectory itself. A geometric analysis is used to show that a larger class of uninformative (degenerate) trajectories than previously reported admits a continuous symmetry in the delayed measurement model, preventing unique delay-and-state recovery.

Significance. If the geometric construction holds, the result meaningfully extends prior identifiability analyses in inertial navigation by enlarging the set of trajectories for which delay estimation is impossible. The geometric symmetry argument supplies a parameter-free derivation of the degeneracy condition and yields falsifiable predictions about which trajectories are uninformative; these are clear strengths.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'geometric analysis' is used without indicating the concrete geometric objects (e.g., Lie-group actions, orbits, or invariants) employed; a single sentence clarifying the construction would improve accessibility.
  2. The manuscript would benefit from an explicit statement (perhaps in the introduction or a dedicated section) of how the newly identified class of degenerate trajectories strictly contains the classes reported in the cited prior literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee summary accurately captures the core contribution regarding the enlarged class of degenerate trajectories under delayed aiding measurements.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claim rests on a geometric analysis deriving continuous symmetries in the delayed measurement model for certain degenerate trajectories. This is presented as a direct consequence of the model equations under the stated assumptions (constant unknown delay, trajectory parameterized by initial state). No load-bearing steps reduce to self-definition, fitted parameters renamed as predictions, or chains of self-citations; the derivation is self-contained against the model itself without external fitted values or ansatzes imported circularly.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides insufficient detail to exhaustively list free parameters or invented entities; the constant-delay assumption is the primary domain assumption visible.

axioms (1)
  • domain assumption Aiding measurements have an unknown but constant delay relative to the inertial data stream.
    Directly stated in the abstract as the problem setup.

pith-pipeline@v0.9.1-grok · 5625 in / 1050 out tokens · 31526 ms · 2026-07-01T07:16:34.391366+00:00 · methodology

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discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

  1. [1]

    Determining the Time Delay Between Inertial and Visual Sensor Measurements,

    J. Kelly, N. Roy, and G. S. Sukhatme, “Determining the Time Delay Between Inertial and Visual Sensor Measurements,”IEEE Transactions on Robotics, vol. 30, no. 6, pp. 1514–1523, Dec. 2014.DOI:10. 1109/TRO.2014.2343073

    work page arXiv 2014

  2. [2]

    Online temporal calibration for camera-IMU systems: Theory and algo- rithms,

    M. Li and A. I. Mourikis, “Online temporal calibration for camera-IMU systems: Theory and algo- rithms,”The International Journal of Robotics Research, vol. 33, no. 7, pp. 947–964, Jun. 2014.DOI: 10.1177/0278364913515286

    work page doi:10.1177/0278364913515286 2014

  3. [3]

    A Question of Time: Revisiting the Use of Recursive Filtering for Temporal Calibration of Multisensor Systems,

    J. Kelly, C. Grebe, and M. Giamou, “A Question of Time: Revisiting the Use of Recursive Filtering for Temporal Calibration of Multisensor Systems,” inProceedings of the IEEE International Conference on Multisensor Fusion and Integration (MFI), Karlsruhe, Germany, Sep. 23–25, 2021.DOI:10.1109/ MFI52462.2021.9591176. [Online]. Available:https://arxiv.org/ab...

    work page arXiv 2021

  4. [4]

    Degenerate Motion Analysis for Aided INS With Online Spatial and Temporal Sensor Calibration,

    Y. Yang, P. Geneva, K. Eckenhoff, and G. Huang, “Degenerate Motion Analysis for Aided INS With Online Spatial and Temporal Sensor Calibration,”IEEE Robotics and Automation Letters, vol. 4, no. 2, pp. 2070–2077, Apr. 2019.DOI:10.1109/LRA.2019.2893803

    work page doi:10.1109/lra.2019.2893803 2070

  5. [5]

    Online Self-Calibration for Visual-Inertial Navigation: Models, Analysis, and Degeneracy,

    Y. Yang, P. Geneva, X. Zuo, and G. Huang, “Online Self-Calibration for Visual-Inertial Navigation: Models, Analysis, and Degeneracy,”IEEE Transactions on Robotics, vol. 39, no. 5, pp. 3479–3498, Oct. 2023.DOI:10.1109/TRO.2023.3275878

    work page doi:10.1109/tro.2023.3275878 2023

  6. [6]

    State Estimation Based on the Concept of Continuous Symmetry and Observability Analysis: The Case of Calibration,

    A. Martinelli, “State Estimation Based on the Concept of Continuous Symmetry and Observability Analysis: The Case of Calibration,”IEEE Transactions on Robotics, vol. 27, no. 2, pp. 239–255, Apr. 2011.DOI:10.1109/TRO.2011.2109210

    work page doi:10.1109/tro.2011.2109210 2011

  7. [7]

    On the observability and identifiability of nonlinear structural and mechanical systems,

    M. N. Chatzis, E. N. Chatzi, and A. W. Smyth, “On the observability and identifiability of nonlinear structural and mechanical systems,”Structural Control and Health Monitoring, vol. 22, no. 3, pp. 574– 593, Mar. 2015.DOI:10.1002/stc.1690

    work page doi:10.1002/stc.1690 2015

  8. [8]

    On Structural Identifiability,

    R. Bellman and K. J. ˚Astr¨om, “On Structural Identifiability,”Mathematical Biosciences, vol. 7, no. 3, pp. 329–339, Apr. 1970.DOI:10.1016/0025-5564(70)90132-X

    work page doi:10.1016/0025-5564(70)90132-x 1970

  9. [9]

    Identifiability of Linear and Nonlinear Dynamical Systems,

    M. S. Grewal and K. Glover, “Identifiability of Linear and Nonlinear Dynamical Systems,”IEEE Transac- tions on Automatic Control, vol. 21, no. 6, pp. 833–837, Dec. 1976.DOI:10.1109/TAC.1976.1101375

    work page doi:10.1109/tac.1976.1101375 1976

  10. [10]

    Rigid body mechanics in Galilean spacetimes,

    A. Bhand, “Rigid body mechanics in Galilean spacetimes,” Master’s thesis, Queen’s University , Kingston, Ontario, Canada, Sep. 2002. 23 STARS-2026-001 Rev. 1.07

    2002

  11. [11]

    All About the Galilean Group SGal(3),

    J. Kelly, “All About the Galilean Group SGal(3),” University of Toronto, Toronto, Ontario, Canada, Tech. Rep. STARS-2023-001, Nov. 26, 2023.DOI:10.48550/arXiv.2312.07555. [Online]. Available: https://arxiv.org/abs/2312.07555

    work page doi:10.48550/arxiv.2312.07555 2023

  12. [12]

    Towards consistent filtering for discrete-time partially-observable nonlinear systems,

    G. Huang, “Towards consistent filtering for discrete-time partially-observable nonlinear systems,”Sys- tems & Control Letters, vol. 106, pp. 87–95, Aug. 2017.DOI:10.1016/j.sysconle.2017.06.006

    work page doi:10.1016/j.sysconle.2017.06.006 2017

  13. [13]

    Galilean Symmetry in Robotics,

    R. Mahony, J. Kelly, and S. Weiss, “Galilean Symmetry in Robotics,”arXiv preprint arXiv:2510.10468, Oct. 12, 2025.DOI:10.48550/arXiv.2510.10468. 24

    work page doi:10.48550/arxiv.2510.10468 2025