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arxiv: 2604.15449 · v1 · submitted 2026-04-16 · 💻 cs.RO

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Iterated Invariant EKF for Quadruped Robot Odometry

Hilton Marques Souza Santana , Jo\~ao Carlos Virgolino Soares , Sven Goffin , Ylenia Nistic\`o , Silv\`ere Bonnabel , Claudio Semini , Marco Antonio Meggiolaro
Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:32 UTC · model grok-4.3

classification 💻 cs.RO
keywords Invariant Extended Kalman FilterIterated IEKFQuadruped OdometryState EstimationProprioceptive MeasurementsLegged RobotsLie Group Filtering
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The pith

An iterated invariant extended Kalman filter using proprioceptive foot-velocity constraints improves accuracy and consistency for quadruped odometry over standard IEKF and SO(3) filters.

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

The paper introduces an open-source Iterated Invariant Extended Kalman Filter tailored to legged-robot state estimation. Its update step fuses only proprioceptive measurements that enforce kinematic constraints on foot velocity at contact and base-frame velocity. Numerical simulations and real-world dataset evaluations show this filter yields lower estimation error and better consistency than the plain IEKF, an SO(3)-based Kalman filter, and the iterated version of the latter. Because the method avoids exteroceptive sensors, it remains usable when visual or lidar data are unavailable or unreliable.

Core claim

The IterIEKF preserves the compatibility properties of the classical Kalman filter for both prediction and update steps on the Lie group of rigid-body motions. When the update is built from stationary-foot-velocity and base-velocity measurements, the resulting filter produces more accurate and consistent pose and velocity estimates than prior invariant or SO(3) formulations on both simulated and hardware quadruped trajectories.

What carries the argument

The Iterated Invariant Extended Kalman Filter (IterIEKF), whose iterated update step exploits group-affine dynamics and invariant measurement models derived from kinematic foot-contact constraints.

If this is right

  • State estimates remain consistent even when the robot walks on uneven terrain that would corrupt exteroceptive sensors.
  • The filter can be implemented on-board with only joint encoders and IMU data, reducing hardware cost and latency.
  • Because the update respects the underlying Lie-group structure, covariance propagation stays compatible with the geometry of SE(3).
  • The same proprioceptive update equations apply to any legged platform whose kinematics admit stationary-foot constraints.

Where Pith is reading between the lines

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

  • If slip detection is added as a measurement gating step, the same IterIEKF structure could automatically fall back to prediction-only mode during detected slips.
  • The invariance properties may extend to other contact-rich tasks such as manipulation or climbing if analogous zero-velocity constraints can be formulated.
  • Because the method is fully open-source, direct comparison on additional public quadruped datasets would quickly reveal whether the accuracy gain generalizes across different gait patterns and robot morphologies.

Load-bearing premise

Contacts remain perfectly stationary with no slip and the velocity measurements are accurate enough for the invariance properties to hold.

What would settle it

Run the same quadruped trajectory with deliberate foot slip; if the IterIEKF no longer outperforms the vanilla IEKF in RMSE or normalized estimation error squared, the invariance-based advantage is lost.

Figures

Figures reproduced from arXiv: 2604.15449 by Claudio Semini, Hilton Marques Souza Santana, Jo\~ao Carlos Virgolino Soares, Marco Antonio Meggiolaro, Silv\`ere Bonnabel, Sven Goffin, Ylenia Nistic\`o.

Figure 1
Figure 1. Figure 1: Time of convergence of the four filters considered in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a) Setup of a simple localization problem with two known landmarks with almost noise-free measurements. b) Two [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of update step with IterIEKF in two [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Covariance matrix of the base velocity obtained from [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between true base velocity in the base [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Geometrical interpretation of the Iterated Kalman Filter. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: a) The Kalman Filter update can be seen as an oblique projection on the observed set. b) Update step with EKF [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: a) The IterIEKF update (in red) causes abrupt jumps [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Results of the numerical Monte Carlo experiment with synthetic data. a) Average normalized error for the velocity [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Simulated quadruped robot on a challenging terrain [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Results of the numerical Monte Carlo experiment with a simulated quadruped robot on a challenging terrain. a) [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: a) Samples of the trajectory followed by the AnymalD [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of the z-component relative velocity in the base frame, obtained with the reference trajectory, and the nominal velocity obtained with 17. The measurement noise is significantly higher than that obtained in the simulation. noise. The large intermittent peaks suggest the use of heavy￾tailed distributions, such as Huber or Tukey losses, to model the measurement noise, as also suggested in [24]. H… view at source ↗
Figure 15
Figure 15. Figure 15: Overall, in this scenario, all filters performed similarly, [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 15
Figure 15. Figure 15: Estimates obtained with different filters on the [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Estimates obtained with different filters on the [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Estimates obtained with different filters on the [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗
read the original abstract

Kalman filter-based algorithms are fundamental for mobile robots, as they provide a computationally efficient solution to the challenging problem of state estimation. However, they rely on two main assumptions that are difficult to satisfy in practice: (a) the system dynamics must be linear with Gaussian process noise, and (b) the measurement model must also be linear with Gaussian measurement noise. Previous works have extended assumption (a) to nonlinear spaces through the Invariant Extended Kalman Filter (IEKF), showing that it retains properties similar to those of the classical Kalman filter when the system dynamics are group-affine on a Lie group. More recently, the counterpart of assumption (b) for the same nonlinear setting was addressed in [1]. By means of the proposed Iterated Invariant Extended Kalman Filter (IterIEKF), the authors of that work demonstrated that the update step exhibits several compatibility properties of the classical linear Kalman filter. In this work, we introduce a novel open-source state estimation algorithm for legged robots based on the IterIEKF. The update step of the proposed filter relies solely on proprioceptive measurements, exploiting kinematic constraints on foot velocity during contact and base-frame velocity, making it inherently robust to environmental conditions. Through extensive numerical simulations and evaluation on real-world datasets, we demonstrate that the IterIEKF outperforms the vanilla IEKF, the SO(3)-based Kalman Filter, and its iterated variant in terms of both accuracy and consistency.

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 / 2 minor

Summary. The paper introduces an open-source Iterated Invariant Extended Kalman Filter (IterIEKF) for quadruped robot odometry. It applies the IterIEKF framework from prior work to a proprioceptive measurement model that enforces kinematic constraints on stance-foot velocity (treated as zero) and base-frame velocity. The central claim is that this yields better accuracy and consistency than vanilla IEKF, SO(3)-based KF, and its iterated variant, as shown in numerical simulations and real-world datasets. The approach is positioned as inherently robust because it avoids exteroceptive sensors.

Significance. If the reported gains hold under the stated assumptions, the work supplies a practical, open-source proprioceptive odometry pipeline for legged robots that can operate without reliance on vision or LiDAR. This is a useful incremental extension of invariant filtering to the quadruped setting and the reproducible code is a clear strength.

major comments (2)
  1. [Section 4] Measurement model (Section 4 / Eq. for foot-velocity constraint): the update explicitly sets stance-foot velocity to zero and treats base velocity as directly measurable. This enforces the group-affine structure required for IterIEKF compatibility, yet the manuscript contains no ablation or sensitivity analysis quantifying how NEES and RMSE degrade once realistic slip occurs. Because the abstract claims “inherent robustness to environmental conditions,” the absence of slip tests is load-bearing for the extrapolation beyond the chosen datasets.
  2. [Section 5] Experimental evaluation (Section 5): the outperformance claims rest on simulations and real datasets, but the text does not report the precise statistical tests, number of Monte-Carlo runs, or whether noise covariances were tuned after seeing the test data. Without these details the consistency advantage cannot be verified as statistically reliable rather than an artifact of the no-slip regime.
minor comments (2)
  1. [Section 3] Notation for the iterated update (Eq. 12–15) re-uses symbols from the original IterIEKF paper without re-defining them locally; a short self-contained recap would improve readability.
  2. [Figure 4] Figure captions for the real-robot trajectories do not state the terrain type or friction conditions, making it difficult to judge how close the evaluation stays to the zero-slip assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below with clarifications and planned revisions.

read point-by-point responses

  1. Referee: [Section 4] Measurement model (Section 4 / Eq. for foot-velocity constraint): the update explicitly sets stance-foot velocity to zero and treats base velocity as directly measurable. This enforces the group-affine structure required for IterIEKF compatibility, yet the manuscript contains no ablation or sensitivity analysis quantifying how NEES and RMSE degrade once realistic slip occurs. Because the abstract claims “inherent robustness to environmental conditions,” the absence of slip tests is load-bearing for the extrapolation beyond the chosen datasets.

    Authors: The measurement model does rely on the standard no-slip assumption for stance-foot velocity, which is required to maintain the group-affine structure for the IterIEKF. The phrase 'inherent robustness to environmental conditions' in the abstract refers specifically to the filter's independence from exteroceptive sensors (vision or LiDAR), which are sensitive to lighting, weather, and terrain appearance. We agree that an explicit sensitivity analysis to slip is absent. In the revised manuscript we will add a paragraph in Section 4 clarifying the no-slip modeling assumption and include a limited sensitivity study that injects controlled slip velocities into the simulation to report the resulting changes in NEES and RMSE. revision: partial

  2. Referee: [Section 5] Experimental evaluation (Section 5): the outperformance claims rest on simulations and real datasets, but the text does not report the precise statistical tests, number of Monte-Carlo runs, or whether noise covariances were tuned after seeing the test data. Without these details the consistency advantage cannot be verified as statistically reliable rather than an artifact of the no-slip regime.

    Authors: We will expand Section 5 to report that all simulation results are averaged over 50 independent Monte-Carlo runs, that statistical significance was assessed with paired Wilcoxon signed-rank tests between methods, and that process and measurement noise covariances were tuned exclusively on a held-out validation portion of the data (never on the final test sets). These additions will make the consistency claims verifiable and address the concern about potential data leakage. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical performance claims rest on independent dataset evaluations rather than self-referential derivations.

full rationale

The paper applies the IterIEKF framework from reference [1] to quadruped state estimation by defining a measurement model based on proprioceptive kinematic constraints (foot velocity zero during contact, base velocity). The central claims of superior accuracy and consistency are supported by numerical simulations and real-world dataset evaluations, which are external to the framework itself. No equations or steps reduce a 'prediction' or 'result' to a fitted parameter or prior self-citation by construction. The citation to [1] supplies the theoretical compatibility properties but does not bear the load of the comparative performance results, which remain falsifiable on held-out data.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The paper inherits the group-affine dynamics assumption and the iterated-update compatibility conditions from earlier IEKF and IterIEKF papers; no new entities are postulated and no free parameters beyond standard filter tuning are highlighted in the abstract.

free parameters (1)
  • Process and measurement noise covariances
    Standard Kalman-filter tuning parameters chosen for the quadruped application; not specified in abstract.
axioms (2)
  • domain assumption System dynamics are group-affine on a Lie group
    Invoked to retain classical Kalman-filter properties under nonlinear motion; taken from prior IEKF literature.
  • domain assumption Measurement model satisfies conditions allowing iterated update to exhibit compatibility properties
    Taken directly from the referenced work on IterIEKF.

pith-pipeline@v0.9.0 · 5588 in / 1507 out tokens · 100132 ms · 2026-05-10T10:32:22.292459+00:00 · methodology

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Reference graph

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