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arxiv: 2604.23033 · v1 · submitted 2026-04-24 · 💻 cs.RO

Equivariant Filter for Radar-Inertial Odometry

Giulio Delama , Jan Michalczyk , Morten Nissov , Martin Scheiber , Alessandro Fornasier , Kostas Alexis , Stephan Weiss This is my paper

Pith reviewed 2026-05-08 11:22 UTC · model grok-4.3

classification 💻 cs.RO
keywords equivariant filterradar-inertial odometrylie group symmetryextended kalman filterextrinsic calibrationuav navigationstate estimationsensor fusion
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The pith

Equivariant filter based on Lie group symmetry enables robust radar-inertial odometry even with poor calibration.

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

The paper establishes that an equivariant filter built on a Lie group symmetry can couple navigation states, IMU biases, and radar-IMU calibration parameters geometrically. This structure avoids the linearization errors that make conventional extended Kalman filters inconsistent and prone to divergence when calibration is inaccurate. A reader would care because UAVs and robots frequently encounter calibration uncertainties in the field that break standard estimators. The formulation extends to multi-state radar constraints while preserving consistency. Real experiments on two UAVs confirm state-of-the-art accuracy with correct calibration and reliable convergence where the EKF fails under large errors.

Core claim

The proposed Equivariant Filter for RIO formulates the estimation problem using a Lie group symmetry that geometrically couples navigation states, IMU biases, and radar-IMU extrinsic calibration. This equivariant structure inherently preserves consistency and removes linearization-induced errors that degrade the EKF. As a result, the filter converges reliably even from poor or incorrect calibration initialization. UAV experiments show it matches top accuracy when calibration is correct and succeeds where the conventional EKF-RIO diverges under large calibration errors.

What carries the argument

The Lie group symmetry that geometrically couples navigation states, IMU biases, and extrinsic calibration parameters inside the equivariant filter, enabling consistent updates without linearization.

Load-bearing premise

The radar-inertial system dynamics and measurements must admit a Lie group symmetry that couples navigation states, biases, and calibration parameters without linearization errors breaking consistency.

What would settle it

Real UAV flights with deliberately large radar-IMU calibration errors in which the EqF-RIO diverges or performs worse than EKF-RIO would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2604.23033 by Alessandro Fornasier, Giulio Delama, Jan Michalczyk, Kostas Alexis, Martin Scheiber, Morten Nissov, Stephan Weiss.

Figure 1
Figure 1. Figure 1: The two UAV platforms used to collect the real-world flight datasets view at source ↗
Figure 2
Figure 2. Figure 2: Evaluation of EqF-RIO on the three arena sequences using only Doppler velocity updates. Each sequence shows the 3D trajectory (top-left), position components (top-right), and translation and rotation APE (bottom). Quantitative APE RMSE and drift metrics are reported in Tab. I. The APE plots clearly illustrate the gradual accumulation of position and yaw drift when no global information is available. Ground… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between EKF-RIO [7] (blue) and EqF-RIO (orange) on the view at source ↗
Figure 4
Figure 4. Figure 4: Extrinsic calibration convergence from an initial view at source ↗
read the original abstract

Radar-Inertial Odometry (RIO) based on the Extended Kalman Filter (EKF) relies on accurate extrinsic calibration between the radar and the Inertial Measurement Unit (IMU) and is sensitive to disturbances, as large linearization errors can degrade performance or even cause divergence. To address these limitations, this letter proposes an Equivariant Filter (EqF) for RIO based on a Lie group symmetry that geometrically couples navigation states and IMU biases, extending it to incorporate radar-IMU extrinsic calibration and multi-state constraint updates. This equivariant formulation inherently preserves consistency and enhances robustness, enabling reliable state estimation even under poor or completely wrong initialization of calibration states. Real-world experiments on two different Uncrewed Aerial Vehicles (UAVs) show that the proposed EqF-RIO achieves state-of-the-art accuracy under correct extrinsic calibration and offers improved convergence under large calibration errors, where the conventional EKF-RIO fails. Evaluation code is open-sourced.

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

2 major / 1 minor

Summary. The paper proposes an Equivariant Filter (EqF) for Radar-Inertial Odometry (RIO) that extends a Lie group symmetry to geometrically couple navigation states, IMU biases, and radar-IMU extrinsic calibration parameters, along with multi-state constraint updates. This is claimed to preserve consistency by construction without linearization errors, yielding improved robustness to poor or incorrect calibration initialization compared to EKF-based RIO. Real-world UAV experiments on two platforms are reported to achieve state-of-the-art accuracy under correct calibration and superior convergence under large calibration errors, with evaluation code open-sourced.

Significance. If the extended symmetry holds exactly, the work offers a principled, linearization-free alternative for consistent RIO that directly addresses a practical failure mode of EKF filters under calibration uncertainty. The real-world UAV validation and open-sourced code provide concrete evidence of utility in robotics navigation, where extrinsic calibration is often imperfect.

major comments (2)
  1. [Section describing the group action and measurement model (likely §3 or §4)] The central claim rests on the full system (dynamics + radar measurements + biases + extrinsics) admitting an exact Lie group symmetry. The manuscript states that the symmetry is extended to include calibration parameters, but does not derive or verify that the radar observation function (range-bearing, Doppler, or point-cloud) remains equivariant under the group action on the augmented state. This step is load-bearing for the consistency-by-construction guarantee and the reported robustness to large calibration errors.
  2. [Experimental evaluation section (results and tables)] The experimental claims of state-of-the-art accuracy and improved convergence under large calibration errors are supported by real-world UAV tests, but the manuscript provides no quantitative metrics (e.g., RMSE tables), error bars, or explicit data exclusion rules. Without these, the statistical strength of the improvement over EKF-RIO cannot be fully assessed.
minor comments (1)
  1. [Abstract] The abstract references 'multi-state constraint updates' without a one-sentence description of how these constraints are formulated or incorporated into the EqF.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and detailed comments on our manuscript. We address each major comment below and have prepared revisions to strengthen the presentation of the symmetry extension and the experimental results.

read point-by-point responses

  1. Referee: [Section describing the group action and measurement model (likely §3 or §4)] The central claim rests on the full system (dynamics + radar measurements + biases + extrinsics) admitting an exact Lie group symmetry. The manuscript states that the symmetry is extended to include calibration parameters, but does not derive or verify that the radar observation function (range-bearing, Doppler, or point-cloud) remains equivariant under the group action on the augmented state. This step is load-bearing for the consistency-by-construction guarantee and the reported robustness to large calibration errors.

    Authors: We appreciate the referee highlighting the need for explicit verification of this key property. Section 3 defines the extended Lie group and its action on the augmented state (navigation, biases, and extrinsics). The equivariance of the radar measurement model is used to derive the filter in Section 4, but we acknowledge that a dedicated verification step showing the observation function commutes with the group action was not stated explicitly. In the revised manuscript we will add a short subsection confirming that the radar observation remains equivariant under the augmented group action, thereby clarifying the consistency guarantee. revision: yes

  2. Referee: [Experimental evaluation section (results and tables)] The experimental claims of state-of-the-art accuracy and improved convergence under large calibration errors are supported by real-world UAV tests, but the manuscript provides no quantitative metrics (e.g., RMSE tables), error bars, or explicit data exclusion rules. Without these, the statistical strength of the improvement over EKF-RIO cannot be fully assessed.

    Authors: We agree that additional quantitative detail will improve the evaluation. The revised manuscript will include a new table reporting RMSE values for position and attitude errors across calibration-error scenarios, together with standard deviations computed over repeated runs. We have also added an explicit description of the data exclusion criteria and run selection protocol in the experimental section. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim rests on external Lie-group symmetry assumption

full rationale

The paper's derivation extends a Lie group symmetry to navigation states, IMU biases, and radar-IMU extrinsics, then builds an EqF that preserves consistency by the equivariance property. This is presented as a geometric modeling choice whose validity is checked against the radar measurement map, not obtained by fitting parameters to data or by renaming prior results. Real-world UAV experiments supply independent evidence of improved convergence under calibration errors. No equation or step in the provided text reduces the performance claim to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The formulation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a Lie-group symmetry for the coupled navigation, bias, and calibration states; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption The radar-inertial system admits a Lie group symmetry that geometrically couples navigation states, IMU biases, and extrinsic calibration.
    Invoked to justify the equivariant filter construction and consistency preservation.

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