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REVIEW 3 major objections 4 minor 30 references

Multi-Sensor Fusion for Quadruped Robot State Estimation using Invariant Filtering and Smoothing

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that LiDAR odometry and GPS position readings can be written as right-invariant observations, letting a quadruped's invariant filter and smoother fuse them and cut position drift.

desk verdict The experimental results are plausible, but the LiDAR/GPS observation models in equations (19)-(20) swap the measurement and constant vector; an absolute position is left-invariant, not right-invariant, so the claimed group-affine guarantees do not hold. read the letter →

arxiv 2504.20615 v1 pith:PCNUA7CL submitted 2025-04-29 cs.RO

Ylenia Nisticò , Hajun Kim , João Carlos Virgolino Soares , Geoff Fink , Hae-Won Park , Claudio Semini This is my paper · ORCID
classification cs.RO
keywords sensorfusionstateestimationquadrupedrobotsleggedinvariantextendedKalmanfiltersmootherLiDARodometryGPS
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Legged robots that estimate their state from leg kinematics and an IMU drift, especially along the vertical axis, because absolute position and yaw are unobservable. This paper tries to fix that by fusing LiDAR odometry and GPS into the invariant extended Kalman filter (InEKF) and invariant smoother (IS) frameworks that have become standard for legged state estimation. The authors' core claim is that a LiDAR or GPS position measurement can be cast in the right-invariant observation form $y = X^{-1}b + w$ with a constant vector $b$, which is exactly the condition that preserves the group-affine property and its state-independent error dynamics. They implement this on the HOUND2 quadruped with the LiDAR odometry computed in a parallel thread, and report that the fused estimators reduce position error, particularly z-drift, and beat LiDAR-inertial odometry baselines in indoor and outdoor tests. If true, the result matters because it brings drift-free global measurements into a theoretically principled fusion framework without sacrificing real-time efficiency.

What carries the argument

The machinery that carries the argument is the group-affine observation condition of equation (7), $y_t = X_t^{-1}s + w_t^{\text{obs}}$ with $s$ constant, together with the proposed encodings of LiDAR and GPS positions as right-invariant observations: $Y_{\text{lid}} = X^{-1}b_{\text{lid}} + V_{\text{lid}}$ and $Y_{\text{gps}} = X^{-1}b_{\text{gps}} + V_{\text{gps}}$, where $b_{\text{lid}} = [p_{\text{lid}}, 0, 1, 0]^\top$ and $b_{\text{gps}} = [p_{\text{gps}}, 0, 1, 0]^\top$. These write absolute position output as if it were a right-invariant measurement, which is the format the invariant filter/smoother needs to keep the innovation a function of the invariant error alone. A supporting mechanical element is the parallel LiDAR thread that runs a point-to-point ICP algorithm to produce $p_{\text{lid}}$ at roughly 10 Hz while the estimator thread continues at high rate, which is what keeps the added cost to about 0.06 ms per iteration for the filter.

What would settle it

Feed a recorded run back through the estimator with a fixed, time-invariant GPS position while the robot moves, and check whether the innovation and covariance evolve according to the state-independent linear equation (6); if they depend on the robot's pose, the observation is not group-affine and the central claim collapses.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is that global position measurements—the output of a LiDAR odometry system and a GPS receiver—can be expressed as right-invariant observations of the form $Y = X^{-1}b + V$, with $b$ a vector that packs the measured position together with zeros and ones, as in equations (19) and (20). Because the observation inherits the right-invariant form of equation (7), the authors argue, the log-linear error dynamics of the invariant filter and smoother are preserved, so the fused estimators keep the convergence guarantees of the invariant filtering theory. The contact-aided kinematic observation is aggregated with these new LiDAR and GPS observations into a single update for the InEKF and a single residual for the IS, and the LiDAR odometry is produced by a point-to-point iterative closest point algorithm running in a separate thread so the main estimator loop stays fast. Experiments on the HOUND2 quadruped, indoors without GPS and outdoors with GPS, show reduced absolute trajectory error and relative position error compared to established LiDAR-inertial odometry systems and to the proprioceptive-only baselines, with the largest visible gain in the z-axis position.

Load-bearing premise

The load-bearing premise is that a LiDAR or GPS position fix can be modeled as a right-invariant observation with a constant vector $b$; if the measured position entering $b$ makes the vector time-varying, the group-affine convergence guarantee does not apply.

Editorial extensions

If this is right

  • The fused estimators cut position drift, especially along the z-axis, reporting up to 28% lower error indoors and 40% lower outdoors relative to two established LiDAR-inertial odometry systems.
  • The right-invariant encoding of absolute position keeps the estimator's error dynamics state-independent, preserving the convergence behaviour of the invariant filter and smoother.
  • The loosely coupled parallel LiDAR thread keeps the filter's computation at about 0.06 ms per iteration, while the smoother's cost grows with its window size, making the accuracy-versus-speed trade-off explicit.
  • Ablations show each exteroceptive source contributes: removing GPS increases the outdoor error, and removing LiDAR reverts to the larger drift of the proprioceptive-only configuration.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The parallel-thread design suggests a generic recipe for invariant estimators: any odometry source that emits a position—camera, radar, or learned—could be fused without touching the group-affine core.
  • A direct testable extension is to compare the right-invariant encoding against a left-invariant or plain measurement model for absolute position; this would show whether the invariance structure itself drives the reported accuracy gains.
  • Since the GPS covariance is treated as fixed in the paper, modulating it with satellite geometry or dilution of precision is a natural next step that could further cut the outdoor error.
  • Because the smoother's window size controls both accuracy and computation time, an adaptive window that grows when the robot speed is low or when drift is detected could give the best of both filter and smoother.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes two state-estimation frameworks for quadruped robots, E-InEKF and E-IS, which fuse leg kinematics and IMU with LiDAR odometry (from KISS-ICP) and GPS position measurements. The central claim is that the LiDAR and GPS measurement models, given in Eqs. (19) and (20), satisfy the group-affine property of Eq. (7), thereby preserving the invariant-filtering and invariant-smoothing guarantees from Barrau and Bonnabel and from Yoon et al. The methods are evaluated on the KAIST HOUND2 robot in indoor and outdoor experiments against LIO-SAM, FAST-LIO2, P-InEKF, and P-IS, reporting lower ATE and RPE and reduced z-axis drift.

Significance. If the group-affine observation models were correctly derived, the paper would make a useful contribution: it would extend invariant filtering/smoothing for legged robots to include exteroceptive sensors while retaining the computationally attractive proprioceptive pipeline. The experimental study is also valuable, with comparisons against strong LiDAR-inertial baselines and an ablation of the LiDAR/GPS contributions. However, the theoretical centerpiece of the paper, the claim that Eqs. (19) and (20) are group-affine observations of the form of Eq. (7), is not correct as written; this must be fixed for the contribution to stand.

major comments (3)
  1. [Sec. III-B, Eqs. (19)-(20)] The LiDAR and GPS observation models are not in the group-affine form of Eq. (7). In Eq. (7), the observation y_t is the measurement (possibly a nonlinear function of the raw measurement) and s is a constant vector. In Eqs. (19) and (20), Y_lid and Y_gps are set to constant vectors, while b_lid and b_gps contain the time-varying measured positions p_lid and p_gps. This is exactly reversed: the measurement enters through s, the supposedly constant vector, and the observation variable is constant. Consequently, the models as written do not satisfy the right-invariant condition y_t = X_t^{-1} s + w_t with constant s, and the log-linear error and convergence guarantees of [7] do not transfer. An absolute position measurement is a left-invariant output of the state, y = X s (with s constant), not a right-invariant one; the paper provides no derivation showing how such a measurement can be cast in the right-invariant form while keeping s constant.
  2. [Sec. III-D] The E-IS residual and Jacobian definitions inherit the same problem. The residual is r_lid = X Y_lid - b_lid, with b_lid containing the measured position p_lid, and the Jacobian is J_lid = [b_lid^⊙, 0]. Because b_lid is measurement-dependent, the Jacobian depends on the noisy measurement, which is inconsistent with the invariant-smoothing derivation in [12] where the Jacobian is evaluated at a constant vector. The measurement dependence enters the linearized cost and breaks the group-affine structure that the smoother is supposed to exploit. In addition, the text in Sec. III-D says "plid ... represents the LiDAR position in the robot's body frame," which contradicts Sec. III-A where plid is the LiDAR position in the world frame; the paper must clarify which definition is used and reconcile it with the proposed residual.
  3. [Sec. III-C, Eq. (21)] The aggregated observation model of Eq. (21) does not rescue the formulation. Writing the observation as X_aug^{-1} b with b = [b_kin; b_lid; b_gps] is only valid if each b component is a constant; here b_lid and b_gps contain the measurements, so the right-hand side is not a linear function of a fixed vector. The update equations of Sec. II-C, which were derived for observations of the form of Eq. (7), therefore cannot be applied to this model. The Jacobians H_lid = H_gps = [0,0,-I_3,0] suggest a direct position measurement in world coordinates, which is a left-invariant observation, not the right-invariant form the paper claims. The paper needs to state which invariant structure it actually uses and re-derive the filter and smoother updates accordingly.
minor comments (4)
  1. [Table II] The column headers for the outdoor results table are run together and missing spaces, making the table hard to read; please reformat.
  2. [Sec. II-B] The notation "SEN+2(3)" should be written as SE_{N+2}(3) to avoid confusion with SEN as a separate symbol.
  3. [Sec. III-B] The dimensions of Y_lid and b_lid are not explicitly stated, and it is unclear how the 5-vector [0_{3,1}; 0; 1; 0] is embedded in the higher-dimensional state X_t. This should be spelled out.
  4. [Sec. I] The statement that this is "the first work to incorporate LiDAR odometry and GPS into InEKF and IS" is broader than the evidence presented; please qualify the novelty claim against the existing invariant-filtering literature.

Circularity Check

1 steps flagged · score 6.0 of 10

LiDAR/GPS 'group-affine' observation models place the measured position inside the supposedly constant vector b, so the right-invariant property is asserted, not derived.

  1. self definitional [Section III-B, Eqs. (19)-(20); propagated to Sec. III-D residuals]
    "we express the LiDAR position estimate plid with respect to the state Xt in the right-invariant form as given in (7): Ylid = X−1 t blid + Vlid (19) where blid = [plid 0 1 0]⊤ is the LiDAR observation vector, Ylid = [03,1 0 1 0]⊤ is a constant vector... The introduced observation models for LiDAR and GPS of Equations (19) and (20) fulfill the group-affine property required for the invariant state estimation, because our measurement models follow the right-invariant observation form [7], given in Equation (7)."

    Equation (7) defines group-affine observations by y = X^{-1} s + w with s a constant vector. In Eqs. (19)-(20), the paper puts the measured positions p_lid and p_gps inside b_lid and b_gps, so those vectors vary at every time step and the models are not of the form (7). The paper then concludes the models 'fulfill the group-affine property' solely because they are written in the algebraic shape y = X^{-1} b. That is a definitional assertion: the measurement is smuggled into the 'constant' vector, while the declared observation Y is constant and carries no data. The same non-constant b propagates to Sec. III-D, where the residuals r = X Y - b and Jacobians J = b^⊙ become measurement-dependent, contrary to the invariant-smoothing derivation.

full rationale

The paper's real-robot comparisons against LIO-SAM, FAST-LIO2, and the proprioceptive-only invariant estimators are independent empirical evidence, so those experimental results are not circular. However, the stated first-principles contribution, namely deriving LiDAR and GPS observation models that 'satisfy group-affine properties,' reduces to a definitional maneuver: Eqs. (19)-(20) are written in the right-invariant shape of Eq. (7), but the required constant vector s is replaced by measurement-dependent b_lid and b_gps. The claim that the models fulfill Eq. (7) is therefore asserted by construction rather than demonstrated, and the same choice undermines the smoother residuals in Sec. III-D. Because this is a central theoretical claim, but the experimental drift-reduction results stand independently, the overall circularity score is 6.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central theoretical claim rests on the group-affine right-invariant form (7) and on the assertion that LiDAR and GPS positions can be expressed in that form. Both are assumptions rather than derivations, and the equations as printed do not satisfy the constant-b requirement.

free parameters (4)
  • LiDAR position noise covariance Sigma(w_lid)
    Controls how much the LiDAR odometry correction is trusted; value not reported in the paper.
  • GPS position noise covariance Sigma(w_gps)
    Controls GPS correction weight; not reported.
  • E-IS window size WS = 1, 5, 10, 15 in experiments
    Chosen by the authors to tune accuracy-compute trade-off.
  • Slip rejection covariance adjustment parameters
    Inherited from [6]; not specified here.
assumptions (4)
  • domain assumption The state propagation model of [8] is group-affine with log-linear error dynamics (equations 9-13).
    Borrowed from prior work [8]; the paper relies on it for the filter and smoother derivation.
  • standard math The right-invariant observation form y = X^{-1} b + w (equation 7) is sufficient to inherit the convergence guarantees of InEKF and IS.
    This is the main theorem of [7]; the paper uses it as the basis for the LiDAR and GPS models.
  • domain assumption LiDAR odometry and GPS provide position measurements with zero-mean Gaussian noise.
    Stated in the definitions of V_lid and V_gps; no validation of the noise model.
  • domain assumption The robot's contacts have zero velocity during stance, with slip events handled by a contact estimator and slip rejection [6].
    The kinematic observation (14) depends on this assumption, as in prior work.

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Pith. "Pith review of Multi-Sensor Fusion for Quadruped Robot State Estimation using Invariant Filtering and Smoothing." pith.science (2026). https://pith.science/paper/PCNUA7CL

@misc{pith2026250420615,
  author       = {Pith},
  title        = {Pith review of: Multi-Sensor Fusion for Quadruped Robot State Estimation using Invariant Filtering and Smoothing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCNUA7CL}},
  note         = {Machine review of arXiv:2504.20615}
}
read the original abstract

This letter introduces two multi-sensor state estimation frameworks for quadruped robots, built on the Invariant Extended Kalman Filter (InEKF) and Invariant Smoother (IS). The proposed methods, named E-InEKF and E-IS, fuse kinematics, IMU, LiDAR, and GPS data to mitigate position drift, particularly along the z-axis, a common issue in proprioceptive-based approaches. We derived observation models that satisfy group-affine properties to integrate LiDAR odometry and GPS into InEKF and IS. LiDAR odometry is incorporated using Iterative Closest Point (ICP) registration on a parallel thread, preserving the computational efficiency of proprioceptive-based state estimation. We evaluate E-InEKF and E-IS with and without exteroceptive sensors, benchmarking them against LiDAR-based odometry methods in indoor and outdoor experiments using the KAIST HOUND2 robot. Our methods achieve lower Relative Position Errors (RPE) and significantly reduce Absolute Trajectory Error (ATE), with improvements of up to 28% indoors and 40% outdoors compared to LIO-SAM and FAST-LIO2. Additionally, we compare E-InEKF and E-IS in terms of computational efficiency and accuracy.

Figures

Figures reproduced from arXiv: 2504.20615 by the authors.

Figure 1
Figure 1. The structure of the two proposed frameworks. The E [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The indoor experiment involved large impacts and [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The results for the indoor experiment, where the robot is prone to experience high contact impacts, show that the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The illustration of the outdoor experiment: Google Earth [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: The results for the outdoor experiment clearly show the improvement in the z-axis drift when using LiDAR and GPS, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Comparison of average computation times for E-IS, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reference graph

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Reviewed August 16, 2026 · model on record in the stance chip above.