{"id":"f8f76baa-2340-49fe-bdac-de3b8b52c8a5","arxiv_id":"2411.11483","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Dual beta-KF estimates calf length from torque and contact force, and uses beta-divergence to downweight foot-slip outliers, improving proprioceptive state estimation on a quadruped.","lead":"A legged-robot state estimator is proposed that combines a dual Kalman filter for online leg length estimation with a beta-divergence robust filter against foot slippage. On a Unitree GO2 robot the method reports about 40% lower trajectory error than a quaternion EKF baseline.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table I's real-world metrics contradict the definitions in Sec. V-A: the QEKF row has MPD (0.121 m) below its ATE (0.412 m), and its DR implies a final error of about 1.42 m for a 20 m trajectory. The headline 42% ATE improvement is therefore not supported by the reported numbers.","rationale":"The reader's weakest_assumption concerned the statics-based leg-length filter (Eqs. 15-16), which is a real modeling risk: ignoring inertial and Coriolis terms and lacking stance-phase gating could bias the estimated leg length and thereby the state filter. That concern is legitimate but would weaken the method's accuracy, not directly invalidate the headline measurements. The Table I inconsistency is more load-bearing because it attacks the empirical evidence for the central claim itself. It is internal to the paper's own definitions and cannot be blamed on external conventions. I do not take this as evidence of misconduct; a transcription error in one cell is plausible, but the burden is on the authors to supply corrected trajectories and recomputed metrics. The appropriate verdict remains conditional pending correction and release of the raw data. I partially agree with the reader: the reader noted the data inconsistency in the rationale but selected the statics model as the weakest assumption; my analysis elevates the Table I contradiction to the primary concern.","tokens_in":15815,"tokens_out":6808,"duration_ms":66670,"concrete_test":"Recover the raw ground-truth and estimated trajectories used for the real-world Table I rows (at minimum QEKF and Dual beta-KF). Recompute ATE, MPD, and DR from the same logged trajectory using the formulas in Sec. V-A, with the actual trajectory length. If the recomputed QEKF row does not satisfy MPD >= ATE and MPD >= DR * l_traj, the reported comparison is invalid and the experimental support for the central claim fails; if the values reconcile after correction, rerun the full comparison to determine whether the 42% improvement persists.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section V-A defines ATE = sqrt((1/T) sum ||p_t - p_hat_t||^2), MPD = max_t ||p_t - p_hat_t||, and DR = ||p_T - p_hat_T|| / l_traj. For any trajectory, the RMS of a nonnegative sequence cannot exceed its maximum, so MPD >= ATE must hold. In the real-world column of Table I, QEKF is reported with ATE=0.412 m and MPD=0.121 m, which is impossible. Moreover, with l_traj approximately 20 m, DR=7.12% implies a final position error of roughly 1.42 m, which also contradicts MPD=0.121 m (the maximum must be at least the final error). The Dual beta-KF row is also borderline: DR=2.72% and l_traj approximately 20 m imply a final error of about 0.54 m, slightly larger than the reported MPD of 0.499 m. Since the paper's central claim of a 42% ATE reduction rests on this table, at least one of the three metrics must have been computed on a different run or with a different convention. Without the raw trajectory logs, the headline result cannot be verified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Dual β-KF, a proprioceptive state estimator for legged robots. The contributions are a measurement model that accounts for time-varying leg length and foot slippage, a dual estimation architecture in which a statics-based parameter filter estimates calf lengths independently of the state filter, and a β-divergence-based robust Kalman filter to downweight slip-induced outliers. Experiments in Gazebo and on a Unitree GO2 with motion capture report that Dual β-KF reduces ATE by roughly 42% relative to QEKF in the real world.","tokens_in":16104,"tokens_out":6837,"duration_ms":66964,"significance":"If the claims were fully supported, the paper would make a useful contribution to proprioceptive legged-robot state estimation. The dual-filter idea is sensible: decoupling parameter estimation from state estimation via a statics-based measurement model avoids the error-accumulation problem of joint filters, and β-divergence is a principled way to handle outliers without hard thresholding. The paper also includes real-robot experiments with motion-capture ground truth and an ablation that separately evaluates the dual framework and the β-KF. However, the experimental evidence as reported is internally inconsistent, and the ablation does not isolate the proposed robustness mechanism. The central claims are therefore not yet supported by the presented data.","major_comments":[{"comment":"The real-world metrics in Table I are mutually inconsistent with the definitions in Section V-A. For any single trajectory, the RMS of a nonnegative sequence cannot exceed its maximum, so ATE ≤ MPD must hold. The QEKF row reports ATE=0.412 m and MPD=0.121 m, which is impossible. Moreover, with trajectory length ≈20 m, DR=7.12% implies an endpoint error of about 1.42 m, while MPD=0.121 m contradicts this because the endpoint error at t=T is one of the per-timestep errors and therefore cannot exceed the maximum. The Dual β-KF row is also borderline: DR=2.72% implies an endpoint error of about 0.54 m, larger than the reported MPD of 0.499 m. These numbers cannot all be produced by the same run and convention. Since the paper's headline 42% ATE improvement is derived from this table, the authors must provide corrected metrics, the raw trajectory logs, or clarify how each metric was computed on different segments or runs.","section":"Section V-A and Table I"},{"comment":"The chosen β values, 3e-6 in simulation and 1e-3 in the real world, are so small that the β-divergence loss is effectively the standard quadratic loss. In Eq. (19), for small β one has exp(β/2 ||·||^2) ≈ 1 + β/2 ||·||^2, so the objective in (22) reduces to the usual Kalman MAP objective and the influence function is close to that of KL divergence. Consequently, the reported improvements of β-KF and Dual β-KF over QEKF cannot be attributed to the robust downweighting of outliers. To support the robustness claim, the paper should report results for a range of β values (for example 0.01, 0.1, and 0.5) and show that the estimates remain accurate and that outlier measurements actually receive lower weight. Remark 3 states that theoretical support is future work; the empirical evidence must therefore be unambiguous, and currently it is not.","section":"Section V-B and Eq. (19)"},{"comment":"The parameter filter uses the static equation (15) as if it held at every time step, with no stance-phase gating. During swing phases the contact force is zero, and during a dynamic trot the inertial, Coriolis, and centrifugal terms neglected from (14) are not necessarily negligible. Applying Eq. (16) indiscriminately will bias the leg-length estimates. The estimated range 0.182–0.253 m (Fig. 7) is never validated against a ground-truth leg length; it is only indirectly tied to state-estimation accuracy. Since the state filter's measurement model (9) uses these estimated calf lengths, any bias in lc is inherited by the state estimate. The authors should gate the parameter filter to stance phases, account for the approximation error, or provide direct validation of the estimated leg lengths (for example, against kinematic calibration or CAD/measured values).","section":"Section IV-C, Eqs. (14)-(16)"},{"comment":"The covariance P_{t+1|t} is computed using the standard Kalman Riccati equation (21), but the state update is obtained by solving the β-divergence objective (22). The posterior covariance of the β-divergence-based estimate is not, in general, the Kalman covariance, so the prior loss (20) that appears in the optimization is not derived from the same objective. This inconsistency means the weighting between the prior and the measurement in the optimization is not justified by the stated robust Bayesian formulation. The authors should either derive the correct covariance for the β-KF update or explicitly state that Eq. (21) is an approximation and empirically justify that choice.","section":"Algorithm 1 and Eq. (21)"}],"minor_comments":[{"comment":"The sentence 'we explicitly consider the values of ϱt and ˙si,t = 0 in our measurement model (9)' appears to contain a typo: the entire motivation of the paper is to avoid assuming foot non-slippage, so the expression should likely be ˙si,t ≠ 0, or the authors should clarify that the model initially includes slippage but the implemented filter sets it to zero and treats slippage as outliers.","section":"Remark 1"},{"comment":"The definition of λ is unclear. The notation 'λ = [0, 0, 1, ..., 0, 0, 1]' with an 'N ×' superscript does not specify the dimensions or the exact positions of the ones; please clarify whether λ is a row vector of length 2N or a matrix, and give its explicit form.","section":"Eq. (16)"},{"comment":"The paper states that foot slippage occurs with probability less than about 5% in preliminary tests. It would be helpful to specify how this probability was measured and whether the same rate holds in the reported simulation and real-world trials, since this assumption underlies the treatment of slippage as a rare outlier.","section":"Section IV-B"},{"comment":"The statement that 'a larger β is required for real-world experiments' is not derived from any quantitative relationship; providing a heuristic or a small sensitivity study would make the hyperparameter choice more reproducible.","section":"Section V-B"}],"recommendation":"major_revision","confidential_remarks":"The paper is at the boundary between major revision and rejection. The central claims rest on Table I, and the internal inconsistencies in that table are severe enough that the current results cannot be verified. However, the issues appear fixable within the scope of a revision: the authors can correct the metrics, rerun or re-analyze the experiments, and provide a proper β-ablation. If the correct numbers no longer support the 42% improvement, or if the parameter-filter validation cannot be supplied, I would recommend rejection. I also note that the robust-loss machinery is taken from the authors' own prior work [13], so the incremental novelty is mainly the application and the statics-based dual-filter design."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper has a genuinely interesting idea buried under a results table that doesn't add up. The new element is the statics-based parameter filter: they estimate calf length from joint torques and normal foot forces via the leg static equation, decoupled from the state filter, so state errors don't corrupt the parameter estimates. That's a sensible way to handle leg deformation without external sensors, and the observability discussion is a real contribution. The beta-divergence robust filter is mostly borrowed from their own earlier work, but the combination with a dual framework is new and worth testing.\n\nThe problem is the evidence. Table I's real-world numbers are internally impossible. ATE is defined as RMS position error, MPD as the maximum, so MPD must be at least ATE. For QEKF they report ATE 0.412 m and MPD 0.121 m. That can't happen. Also DR 7.12% over a 20 m trajectory implies a final error around 1.42 m, which contradicts MPD 0.121 m. The Dual beta-KF row has a milder version of the same problem (DR 2.72% implies ~0.54 m final error vs MPD 0.499 m). The headline 42% improvement rests on this table, so as reported it's unsupported.\n\nSecond issue: the beta values. They set beta to 3e-6 in simulation and 1e-3 in real world. Beta-divergence only starts downweighting outliers when beta is not tiny; as beta approaches zero it converges to KL. At these values the robustness effect is negligible. So the ablation comparing beta-KF to QEKF isn't isolating the divergence robustness — it may just be showing that solving the MAP problem with a nonlinear optimizer helps. They need a sensitivity analysis and a beta value that actually triggers downweighting.\n\nThird, the statics approximation Eq. (15) ignores inertial and Coriolis terms and has no stance-phase gating. During a trot the legs are swinging; the equation only holds in stance. Yet the parameter filter appears to run on all data. Their estimated calf lengths swing from 0.182 to 0.253 m, which is a lot; without ground-truth leg length we can't tell if this is real deformation or a biased fit. This needs a contact-aware update or at least a justification.\n\nThe writing is clear and the method section is easy to follow. They're honest about not having a theoretical proof for beta-KF. But the reported results need to be corrected and the code/data released before the central claim can be trusted. I'd send it to review, but with the expectation of substantial revision and verification.\n\nBest,\n\n[You]","headline":"A clever statics-based leg-length estimator and beta-divergence state filter, undone by a self-contradictory results table that makes the headline 42% gain unverifiable.","tokens_in":16665,"tokens_out":3262,"would_cite":false,"duration_ms":30545,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a dual beta-Kalman filter, which learns variable leg length from joint torques and foot forces and downweights foot-slip outliers with beta-divergence, reduces the absolute trajectory error of a trotting Unitree GO2…","keywords":["legged robot state estimation","proprioceptive odometry","dual Kalman filter","beta-divergence","foot slippage","leg length calibration","outlier robustness","Unitree GO2"],"falsifier":"A direct test would attach motion-capture markers to the calf links of a trotting robot and compare the true leg length with the values estimated by the parameter filter across speeds and terrains. Systematic divergence of the estimates from the measured lengths as joint accelerations grow would falsify the statics approximation. A second test would induce a known foot slip, for instance with a low-friction patch, and check whether the beta-KF position estimate recovers the true trajectory; failure to downweight the slip would falsify the outlier model.","tokens_in":15605,"feed_emoji":"🦿","tokens_out":8204,"duration_ms":77748,"temperature":0.7,"pith_summary":"Legged robots that estimate their motion from onboard sensors usually assume the feet never slip and the legs keep a fixed length. This paper shows both assumptions can be abandoned using only proprioceptive sensors: leg length is an observable quantity that a dedicated parameter filter can learn from joint torques and foot contact forces, while foot slippage, being rare, can be treated as measurement outliers. The resulting dual beta-Kalman filter (Dual beta-KF) couples this parameter filter with a state filter whose loss function uses beta-divergence, which automatically downweights outlier measurements. On a Unitree GO2 robot, the method cuts the absolute trajectory error from 0.412 m (quaternion EKF) to 0.240 m, about a 42 percent reduction, without any external sensors.","feed_headline":"Dual beta filter cuts legged-robot drift by 42%","feed_subtitle":"Learns true leg length from torque and force, and downweights foot-slip outliers, with no external sensors.","key_machinery":"The key machinery is a dual estimation loop with two decoupled filters. The parameter filter estimates calf-leg lengths $l_{c,i}$ from the leg static equation $\\tau = -J(\\phi; l_c)^T F$, multiplying by the inverse Jacobian transpose and keeping only the normal force component (Eq. 16) so the measurement uses only joint torques and foot contact force sensors, not state variables. The state filter uses the measurement model in Eq. (9), which includes the estimated leg length and omits foot-slip velocity except when it appears as an outlier. Robustness comes from replacing the canonical MAP loss, a Kullback-Leibler divergence, with $\\beta$-divergence (Eq. 22); $\\beta$-divergence has a bounded influence function, so anomalous innovations are automatically down-weighted rather than hard-thresholded. The dual structure is what prevents the two filters from corrupting each other: the parameter filter never sees state estimates, and the state filter receives only the current leg-length estimate.","core_discovery":"The paper's central claim is that the measurement model for proprioceptive legged-robot state estimation can be made faithful to physics by explicitly including variable leg length and foot-slippage velocity in the residual between forward kinematics and state-based foot position and velocity. Leg length is shown to be observable, so it can be recovered by an auxiliary filter whose measurement model is the leg static equation, $\\tau = -J(\\phi; l_c)^T F$, transformed through the inverse Jacobian to use only the normal ground-reaction force and joint torques. This statics-based parameter filter is deliberately independent of the state estimate, preventing error entanglement in the dual-loop structure. Foot-slippage velocity, in contrast, is not observable with the given sensors, but because slips are infrequent, the paper treats them as outliers and redefines the state filter's maximum-a-posteriori objective with $\\beta$-divergence instead of Kullback-Leibler divergence, giving the $\\beta$-Kalman filter. The combination, Dual $\\beta$-KF, is tested in Gazebo simulation and on a real Unitree GO2 robot in a trot gait, reporting ATE 0.400 m in simulation and 0.240 m in the real world, the lowest among the compared estimators in both cases.","pith_inferences":["Editorial inference: the beta-divergence robustification is a general recipe for any Kalman filter with sparse measurement outliers, and could be applied to wheeled-robot odometry, visual-inertial fusion, or GPS-denied navigation without structural changes.","Editorial inference: the statics-based parameter filter estimates only calf length, but the same torque-force residual could identify other parameters entering the Jacobian, such as foot position offsets or a center-of-mass shift caused by payload.","Editorial inference: the hyperparameter beta is hand-tuned per environment; an adaptive beta updated from the innovation distribution would make the method self-configuring across terrains, and the paper gives no guidance for such a rule.","Editorial inference: the outlier treatment implicitly assumes slippage is sparse; on persistently slippery ground where every stance phase slides, the low-frequency-outlier model would likely break, and a structured slip-velocity state or slip-probability input would be needed."],"forward_implications":["If the central claim is right, online kinematic calibration of leg length needs no motion-capture system: the static equation and torque and force sensors are enough, so the method transfers to field robots.","Beta-KF's soft weighting keeps measurement information that hard thresholding throws away, so the filter should outperform outlier-rejection UKF near the slip-detection boundary while staying robust to actual slips.","Because the parameter filter is state-independent, leg length estimates remain reliable even when the state filter's pose estimate is temporarily poor, breaking the error-accumulation loop typical of dual estimators.","The real-world gain from the dual component (ATE 0.270 m versus 0.412 m for QEKF) implies that leg deformation, not just slip, is a dominant error source in physical quadruped trotting.","The approach should extend to other legged platforms with joint torque and foot force sensing, since the measurement model and the static equation are agnostic to the number of legs."],"supporting_citations":[{"why":"Supplies the residual-based measurement model (foot position and velocity relative to the body) that this paper extends by adding slip velocity and variable leg length.","marker":"[2]"},{"why":"Provides the UKF with outlier rejection (UKF-OR) baseline, the threshold-based innovation gating, and the Gaussian noise convention adopted by the paper.","marker":"[3]"},{"why":"Defines the quaternion-based EKF (QEKF) baseline and the quaternion state propagation formulation used in the state filter.","marker":"[4]"},{"why":"Contributes the online kinematic calibration idea and the forward-kinematics and Jacobian expressions, while requiring motion capture that the proposed dual filter eliminates.","marker":"[8]"},{"why":"Justifies the bounded influence function of beta-divergence and forms the basis for replacing the KL divergence in the Kalman filter objective.","marker":"[13]"},{"why":"Supplies the approximation of the leg dynamic equation by the leg static equation, which is the measurement model of the parameter filter.","marker":"[21]"},{"why":"Establishes the truncation effect and outlier-robustness properties of beta-divergence used in the beta-KF loss.","marker":"[23]"},{"why":"Provides the optimization-centric robust Bayesian inference view that lets the paper reinterpret Kalman filtering as MAP estimation and swap the divergence.","marker":"[24]"}],"fun_headline_variants":["Torque and force reveal leg length, beta filter downweights foot slips","No external sensors: dual beta filter nails legged robot pose","Foot-slip outliers tamed by beta divergence in dual Kalman filter","Observable leg length, slip outlier rejection: dual beta KF"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on treating the leg static equation $\\tau = -J(\\phi; l_c)^T F$ as an exact measurement model for leg length during dynamic locomotion, with only the normal force component used after inverting the Jacobian. If inertial, Coriolis, or swing-phase forces are not negligible, the estimated leg lengths are biased and the state filter's measurement model inherits that bias.","fun_headline_variants_meta":{"raw":{"variants":["Torque and force reveal leg length, beta filter downweights foot slips","No external sensors: dual beta filter nails legged robot pose","Foot-slip outliers tamed by beta divergence in dual Kalman filter","Observable leg length, slip outlier rejection: dual beta KF"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000789,"raw_usage":{"total_tokens":3565,"prompt_tokens":1119,"completion_tokens":2446,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":735,"completion_tokens_details":{"reasoning_tokens":2370}},"tokens_in":735,"tokens_out":2446,"duration_ms":57479,"temperature":1.0,"reasoning_tokens":2370,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:28:34.683060+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test would attach motion-capture markers to the calf links of a trotting robot and compare the true leg length with the values estimated by the parameter filter across speeds and terrains. Systematic divergence of the estimates from the measured lengths as joint accelerations grow would falsify the statics approximation. A second test would induce a known foot slip, for instance with a low-friction patch, and check whether the beta-KF position estimate recovers the true trajectory; failure to downweight the slip would falsify the outlier model.","supporting_citations":[{"cited_title":"State estimation for legged robots: Consis- tent fusion of leg kinematics and imu,","cited_arxiv_id":null,"evidence_quote":"Supplies the residual-based measurement model (foot position and velocity relative to the body) that this paper extends by adding slip velocity and variable leg length."},{"cited_title":"State estimation for legged robots on unstable and slippery terrain,","cited_arxiv_id":null,"evidence_quote":"Provides the UKF with outlier rejection (UKF-OR) baseline, the threshold-based innovation gating, and the Gaussian noise convention adopted by the paper."},{"cited_title":"Proprioceptive sensor fusion for quadruped robot state estimation,","cited_arxiv_id":null,"evidence_quote":"Defines the quaternion-based EKF (QEKF) baseline and the quaternion state propagation formulation used in the state filter."},{"cited_title":"Online kinematic calibration for legged robots,","cited_arxiv_id":null,"evidence_quote":"Contributes the online kinematic calibration idea and the forward-kinematics and Jacobian expressions, while requiring motion capture that the proposed dual filter eliminates."},{"cited_title":"Robust Bayesian Inference for Moving Horizon Estimation","cited_arxiv_id":"2210.02166","evidence_quote":"Justifies the bounded influence function of beta-divergence and forms the basis for replacing the KL divergence in the Kalman filter objective."},{"cited_title":"External force estima- tion of legged robots via a factor graph framework with a disturbance observer,","cited_arxiv_id":null,"evidence_quote":"Supplies the approximation of the leg dynamic equation by the leg static equation, which is the measurement model of the parameter filter."},{"cited_title":"Variational inference based on ro- bust divergences,","cited_arxiv_id":null,"evidence_quote":"Establishes the truncation effect and outlier-robustness properties of beta-divergence used in the beta-KF loss."},{"cited_title":"An optimization-centric view on bayes’ rule: Reviewing and generalizing variational inference,","cited_arxiv_id":null,"evidence_quote":"Provides the optimization-centric robust Bayesian inference view that lets the paper reinterpret Kalman filtering as MAP estimation and swap the divergence."}],"review_version":1}