Robust State Estimation for Legged Robots with Dual Beta Kalman Filter

Chang Liu; Jiangtao Li; Shengbo Eben Li; Tao Zhang; Tianyi Zhang; Wenhan Cao

REVIEW 4 major objections 4 minor 24 references

Robust State Estimation for Legged Robots with Dual Beta Kalman Filter

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.11483 v1 pith:QGJIW7WK submitted 2024-11-18 cs.RO

Tianyi Zhang , Wenhan Cao , Chang Liu , Tao Zhang , Jiangtao Li , Shengbo Eben Li This is my paper · ORCID

classification cs.RO

keywords leggedrobotstateestimationproprioceptiveodometrydualKalmanfilterbeta-divergencefootslippageleglengthcalibrationoutlierrobustnessUnitreeGO2

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The 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.

The 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.

Second 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.

Third, 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.

The 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.

Best,

[You]

Referee Report

4 major / 4 minor

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.

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 (4)

[Section V-A and Table I] 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 V-B and Eq. (19)] 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 IV-C, Eqs. (14)-(16)] 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).
[Algorithm 1 and Eq. (21)] 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.

minor comments (4)

[Remark 1] 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.
[Eq. (16)] 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 IV-B] 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 V-B] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: leg length estimation is state-decoupled and validation is external; the only self-citation is the beta-divergence loss from prior work, which is not a circular reduction.

full rationale

The paper's central validation is against external motion capture, so the ATE improvements are not forced by the estimator's own fitted parameters. The parameter filter (Eqs. 12-16) estimates calf length from joint torques and normal ground-reaction forces only; it does not use the state estimates, and the state filter (Eqs. 9, 22) uses the estimated lengths as inputs to a kinematic measurement model. No equation in the paper defines the output metric in terms of the fitted leg lengths or beta; the dual-loop is explicitly decoupled to avoid error accumulation. The only load-bearing self-citation is [13] for the beta-divergence loss and its bounded influence, and Remark 3 explicitly defers a proof of the robustness enhancement to future work. This is a minor self-citation rather than a circular reduction, because the empirical benchmark (ATE/MPD/DR vs. motion capture) is external and the beta-divergence robustness is also supported by independent references [23], [24]. One non-circular concern is that Table I's real-world QEKF row appears inconsistent with the paper's own metric definitions: MPD must be at least ATE, yet QEKF is reported with ATE = 0.412 m and MPD = 0.121 m, and DR = 7.12% over roughly 20 m implies a final error near 1.42 m, which exceeds the reported MPD. That is a correctness/consistency issue affecting the headline 42% improvement, but it is not circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The method leans on the leg-statics approximation, normal-force-only sensing, and the assertion that foot slippage is rare and treatable as outliers. The beta-KF robustness machinery is inherited from prior work, including the authors' own reference [13]. No new physical entities are introduced.

free parameters (4)

beta (beta-KF hyperparameter) = 3e-6 (Gazebo), 1e-3 (real world)
Tuned per environment according to assumed slip frequency; no sensitivity study, and at these values the divergence-based downweighting is very weak.
Process noise covariance Q = not reported
Set by hand as an adjustable parameter (Remark 2); directly controls the state filter and parameter filter random walk, and is not disclosed.
Measurement noise covariance Sigma = not reported
Set by hand as an adjustable parameter (Remark 2); directly affects both filters and is not disclosed.
Initial covariance P0 = not reported
Initial state is set to ground truth; the initial covariance is not specified, so reimplementation cannot match the transient behavior.

assumptions (6)

domain assumption Leg statics equation tau = -J^T F replaces leg dynamics in the parameter filter
Sec. IV-C: inertial, Coriolis, and gravity torques are neglected because leg mass is small; no quantification of the approximation error.
domain assumption Foot contact force sensors provide only the normal component F^z
Sec. IV-C: only the last component of the transformed Jacobian equation is used in Eq. (16); tangential GRF components are discarded.
domain assumption Jacobian J(phi_i; lc_i) is invertible for a three-joint leg
Sec. IV-C relies on invertibility to transform Eq. (15) into Eq. (16); no singularity analysis for the motion range of the GO2.
domain assumption Foot slippage is rare and can be modeled as measurement outliers
Sec. IV-B: slip probability below about 5% is asserted from preliminary tests; the state filter omits slip velocity in Eq. (9) and treats slips as outliers.
standard math Propagation and measurement noises are Gaussian
Remark 2 and Eq. (10): distributions are assumed Gaussian with adjustable covariances, following prior work.
domain assumption IMU frame coincides with the body frame
Sec. III-A: this simplifies the rotation model; in practice the IMU has an extrinsic offset, which is not modeled.

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Cite this review

Pith. "Pith review of Robust State Estimation for Legged Robots with Dual Beta Kalman Filter." pith.science (2026). https://pith.science/paper/QGJIW7WK

@misc{pith2026241111483,
  author       = {Pith},
  title        = {Pith review of: Robust State Estimation for Legged Robots with Dual Beta Kalman Filter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QGJIW7WK}},
  note         = {Machine review of arXiv:2411.11483}
}

read the original abstract

Existing state estimation algorithms for legged robots that rely on proprioceptive sensors often overlook foot slippage and leg deformation in the physical world, leading to large estimation errors. To address this limitation, we propose a comprehensive measurement model that accounts for both foot slippage and variable leg length by analyzing the relative motion between foot contact points and the robot's body center. We show that leg length is an observable quantity, meaning that its value can be explicitly inferred by designing an auxiliary filter. To this end, we introduce a dual estimation framework that iteratively employs a parameter filter to estimate the leg length parameters and a state filter to estimate the robot's state. To prevent error accumulation in this iterative framework, we construct a partial measurement model for the parameter filter using the leg static equation. This approach ensures that leg length estimation relies solely on joint torques and foot contact forces, avoiding the influence of state estimation errors on the parameter estimation. Unlike leg length which can be directly estimated, foot slippage cannot be measured directly with the current sensor configuration. However, since foot slippage occurs at a low frequency, it can be treated as outliers in the measurement data. To mitigate the impact of these outliers, we propose the beta Kalman filter (beta KF), which redefines the estimation loss in canonical Kalman filtering using beta divergence. This divergence can assign low weights to outliers in an adaptive manner, thereby enhancing the robustness of the estimation algorithm. These techniques together form the dual beta-Kalman filter (Dual beta KF), a novel algorithm for robust state estimation in legged robots. Experimental results on the Unitree GO2 robot demonstrate that the Dual beta KF significantly outperforms state-of-the-art methods.

Figures

Figures reproduced from arXiv: 2411.11483 by the authors.

**Figure 2.** Important frame definitions and leg length parameters of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Framework overview of the Dual β-KF. Proprioceptive sensor data are input to the Dual β-KF, where a state-independent parameter filter first estimates the leg length parameters. These estimates are then used by a robust state filter to determine the robot’s state. B. Dual Robust Estimation Framework The proposed dual robust estimation framework consists of two concurrently running filters, a parameter filter and a s… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: A Unitree Go2 robot walks with a trot gait in a laboratory [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 6.** Figure 6: Position estimation results for different algorithms in the simulation (left) and the real-world experiment (right) using the Unitree [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Above: The estimated length of each calf leg obtained from the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 21 canonical work pages

[1]

Mit cheetah 3: Design and control of a robust, dynamic quadruped robot,

G. Bledt, M. J. Powell, B. Katz, J. Di Carlo, P. M. Wensing, and S. Kim, “Mit cheetah 3: Design and control of a robust, dynamic quadruped robot,” in 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2018, pp. 2245–2252

2018
[2]

State estimation for legged robots: Consis- tent fusion of leg kinematics and imu,

M. Bloesch, M. Hutter, M. A. Hoepflinger, S. Leutenegger, S. Gehring, C. Remy, and R. Siegwart, “State estimation for legged robots: Consis- tent fusion of leg kinematics and imu,” in Robotics: Science and Systems Conference (RSS 2012) . RSS 2012: Robotics: Science and Systems Conference, 2012

work page 2012
[3]

State estimation for legged robots on unstable and slippery terrain,

M. Bloesch, C. Gehring, P. Fankhauser, M. Hutter, M. A. Hoepflinger, and R. Siegwart, “State estimation for legged robots on unstable and slippery terrain,” in 2013 IEEE/RSJ International Conference on Intel- ligent Robots and Systems . IEEE, 2013, pp. 6058–6064

work page 2013
[4]

Proprioceptive sensor fusion for quadruped robot state estimation,

G. Fink and C. Semini, “Proprioceptive sensor fusion for quadruped robot state estimation,” in 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2020, pp. 10 914–10 920

work page 2020
[5]

Multi-imu propri- oceptive odometry for legged robots,

S. Yang, Z. Zhang, B. Bokser, and Z. Manchester, “Multi-imu propri- oceptive odometry for legged robots,” in 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2023, pp. 774–779

2023
[6]

Invariant smoother for legged robot state estimation with dynamic contact event information,

Z. Yoon, J.-H. Kim, and H.-W. Park, “Invariant smoother for legged robot state estimation with dynamic contact event information,” IEEE Transactions on Robotics , 2023

work page 2023
[7]

Legged robot state estimation in slippery environments using invariant extended kalman filter with velocity update,

S. Teng, M. W. Mueller, and K. Sreenath, “Legged robot state estimation in slippery environments using invariant extended kalman filter with velocity update,” in 2021 IEEE International Conference on Robotics and Automation (ICRA) . IEEE, 2021, pp. 3104–3110

work page 2021
[8]

Online kinematic calibration for legged robots,

S. Yang, H. Choset, and Z. Manchester, “Online kinematic calibration for legged robots,” IEEE Robotics and Automation Letters , vol. 7, no. 3, pp. 8178–8185, 2022

work page 2022

Show all 24 references

[9]

Vilens: Visual, inertial, lidar, and leg odometry for all-terrain legged robots,

Wisth, David and Camurri, Marco and Fallon, Maurice, “Vilens: Visual, inertial, lidar, and leg odometry for all-terrain legged robots,” IEEE Transactions on Robotics , vol. 39, no. 1, pp. 309–326, 2022

2022
[10]

Legged robot state estimation with dynamic contact event information,

J.-H. Kim, S. Hong, G. Ji, S. Jeon, J. Hwangbo, J.-H. Oh, and H.- W. Park, “Legged robot state estimation with dynamic contact event information,” IEEE Robotics and Automation Letters , vol. 6, no. 4, pp. 6733–6740, 2021

2021
[11]

Probabilistic contact estimation and impact detection for state estimation of quadruped robots,

M. Camurri, M. Fallon, S. Bazeille, A. Radulescu, V . Barasuol, D. G. Caldwell, and C. Semini, “Probabilistic contact estimation and impact detection for state estimation of quadruped robots,” IEEE Robotics and Automation Letters, vol. 2, no. 2, pp. 1023–1030, 2017

2017
[12]

Kinematic batch calibration for legged robots,

M. Bloesch, M. Hutter, C. Gehring, M. A. Hoepflinger, and R. Sieg- wart, “Kinematic batch calibration for legged robots,” in 2013 IEEE International Conference on Robotics and Automation . IEEE, 2013, pp. 2542–2547

2013
[13]

Robust bayesian inference for moving horizon estimation,

W. Cao, C. Liu, Z. Lan, S. E. Li, W. Pan, and A. Alessandri, “Robust bayesian inference for moving horizon estimation,” arXiv preprint arXiv:2210.02166, 2022

2022 arXiv
[14]

Legged robot state estimation using invariant kalman filtering and learned contact events,

T.-Y . Lin, R. Zhang, J. Yu, and M. Ghaffari, “Legged robot state estimation using invariant kalman filtering and learned contact events,” arXiv preprint arXiv:2106.15713 , 2021

2021 arXiv
[15]

Unsupervised contact learning for humanoid estimation and control,

N. Rotella, S. Schaal, and L. Righetti, “Unsupervised contact learning for humanoid estimation and control,” in 2018 IEEE International Conference on Robotics and Automation (ICRA) . IEEE, 2018, pp. 411–417

2018
[16]

Probabilistic contact state estimation for legged robots using inertial information,

M. Maravgakis, D.-E. Argiropoulos, S. Piperakis, and P. Trahanias, “Probabilistic contact state estimation for legged robots using inertial information,” in 2023 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2023, pp. 12 163–12 169

2023
[17]

Dual extended kalman filter for the identification of time-varying human manual control behavior,

A. Popovici, P. Zaal, and D. M. Pool, “Dual extended kalman filter for the identification of time-varying human manual control behavior,” in AIAA Modeling and Simulation Technologies Conference, 2017, p. 3666

2017
[18]

Dual extended kalman filter methods,

E. A. Wan and A. T. Nelson, “Dual extended kalman filter methods,” Kalman filtering and neural networks , pp. 123–173, 2001

2001
[19]

A novel approach for vehicle inertial parameter identification using a dual kalman filter,

S. Hong, C. Lee, F. Borrelli, and J. K. Hedrick, “A novel approach for vehicle inertial parameter identification using a dual kalman filter,” IEEE Transactions on Intelligent Transportation Systems , vol. 16, no. 1, pp. 151–161, 2014

2014
[20]

Introduction to robotics

J. J. Craig, “Introduction to robotics.” 2005

2005
[21]

External force estima- tion of legged robots via a factor graph framework with a disturbance observer,

J. Kang, H.-B. Kim, K. H. Choi, and K.-S. Kim, “External force estima- tion of legged robots via a factor graph framework with a disturbance observer,” in 2023 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2023, pp. 12 120–12 126

2023
[22]

Simultaneous state estimation and contact detection for legged robots by multiple-model kalman filtering,

M. Menner and K. Berntorp, “Simultaneous state estimation and contact detection for legged robots by multiple-model kalman filtering,” arXiv preprint arXiv:2404.03444, 2024

2024 arXiv
[23]

Variational inference based on ro- bust divergences,

F. Futami, I. Sato, and M. Sugiyama, “Variational inference based on ro- bust divergences,” in International Conference on Artificial Intelligence and Statistics. PMLR, 2018, pp. 813–822

2018
[24]

An optimization-centric view on bayes’ rule: Reviewing and generalizing variational inference,

J. Knoblauch, J. Jewson, and T. Damoulas, “An optimization-centric view on bayes’ rule: Reviewing and generalizing variational inference,” Journal of Machine Learning Research , vol. 23, no. 132, pp. 1–109, 2022

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Mit cheetah 3: Design and control of a robust, dynamic quadruped robot,

G. Bledt, M. J. Powell, B. Katz, J. Di Carlo, P. M. Wensing, and S. Kim, “Mit cheetah 3: Design and control of a robust, dynamic quadruped robot,” in 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2018, pp. 2245–2252

2018

[2] [2]

State estimation for legged robots: Consis- tent fusion of leg kinematics and imu,

M. Bloesch, M. Hutter, M. A. Hoepflinger, S. Leutenegger, S. Gehring, C. Remy, and R. Siegwart, “State estimation for legged robots: Consis- tent fusion of leg kinematics and imu,” in Robotics: Science and Systems Conference (RSS 2012) . RSS 2012: Robotics: Science and Systems Conference, 2012

work page 2012

[3] [3]

State estimation for legged robots on unstable and slippery terrain,

M. Bloesch, C. Gehring, P. Fankhauser, M. Hutter, M. A. Hoepflinger, and R. Siegwart, “State estimation for legged robots on unstable and slippery terrain,” in 2013 IEEE/RSJ International Conference on Intel- ligent Robots and Systems . IEEE, 2013, pp. 6058–6064

work page 2013

[4] [4]

Proprioceptive sensor fusion for quadruped robot state estimation,

G. Fink and C. Semini, “Proprioceptive sensor fusion for quadruped robot state estimation,” in 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2020, pp. 10 914–10 920

work page 2020

[5] [5]

Multi-imu propri- oceptive odometry for legged robots,

S. Yang, Z. Zhang, B. Bokser, and Z. Manchester, “Multi-imu propri- oceptive odometry for legged robots,” in 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) . IEEE, 2023, pp. 774–779

2023

[6] [6]

Invariant smoother for legged robot state estimation with dynamic contact event information,

Z. Yoon, J.-H. Kim, and H.-W. Park, “Invariant smoother for legged robot state estimation with dynamic contact event information,” IEEE Transactions on Robotics , 2023

work page 2023

[7] [7]

Legged robot state estimation in slippery environments using invariant extended kalman filter with velocity update,

S. Teng, M. W. Mueller, and K. Sreenath, “Legged robot state estimation in slippery environments using invariant extended kalman filter with velocity update,” in 2021 IEEE International Conference on Robotics and Automation (ICRA) . IEEE, 2021, pp. 3104–3110

work page 2021

[8] [8]

Online kinematic calibration for legged robots,

S. Yang, H. Choset, and Z. Manchester, “Online kinematic calibration for legged robots,” IEEE Robotics and Automation Letters , vol. 7, no. 3, pp. 8178–8185, 2022

work page 2022