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arxiv: 2510.11539 · v4 · submitted 2025-10-13 · 💻 cs.RO · math.OC

Simultaneous Calibration of Noise Covariance and Kinematics for State Estimation of Legged Robots via Bi-level Optimization

Denglin Cheng , Jiarong Kang , Xiaobin Xiong This is my paper

Pith reviewed 2026-05-18 07:28 UTC · model grok-4.3

classification 💻 cs.RO math.OC
keywords state estimationbi-level optimizationnoise covariancekinematic calibrationlegged robotsquadrupedhumanoidfull-information estimator
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The pith

A bi-level optimization framework jointly tunes noise covariances and kinematic parameters inside a state estimator for legged robots.

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

The paper shows that noise covariances and kinematic parameters can be calibrated together by nesting a full-information estimator inside an outer optimization loop. The upper level adjusts the covariances and parameters as decision variables, while the lower level solves for the robot trajectory; gradients are passed back through the estimator to directly minimize trajectory-level error metrics. This replaces manual tuning with a data-driven process that produces state estimates whose accuracy and uncertainty match observed performance more closely. The approach is demonstrated on both quadrupedal and humanoid platforms, where it outperforms hand-tuned baselines in estimation error and consistency. A sympathetic reader would care because reliable state estimates are a prerequisite for stable control of legged machines in uncertain settings, and the method removes a persistent source of ad-hoc engineering.

Core claim

The central claim is that treating noise covariance matrices and kinematic parameters as optimization variables in an upper-level problem, while executing a full-information estimator at the lower level, and differentiating through that estimator yields accurate and consistent state estimates that unify estimation, sensor calibration, and kinematics calibration into one principled procedure.

What carries the argument

The bi-level optimization framework that places covariance matrices and kinematic parameters in the upper level and a full-information estimator in the lower level, with differentiation through the estimator to optimize trajectory objectives.

If this is right

  • State estimates become more accurate without requiring separate manual tuning of process and measurement noise.
  • Uncertainty reported by the estimator better reflects actual trajectory errors.
  • Kinematic parameter errors are reduced as part of the same optimization that improves the state trajectory.
  • The same procedure applies without modification to both quadrupedal and humanoid platforms.

Where Pith is reading between the lines

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

  • The framework could shorten the time to deploy a new legged robot by replacing weeks of hand-tuning with a single calibration dataset.
  • Because the method operates estimator-in-the-loop, it may generalize to other estimation architectures such as Kalman filters or factor-graph solvers.
  • Extending the upper-level objective to include control performance metrics could produce parameters that are optimal for both estimation and downstream locomotion.

Load-bearing premise

Differentiating through the full-information estimator produces stable and unbiased gradients that can be used to improve the upper-level trajectory objectives.

What would settle it

Compare state estimation errors and covariance consistency on a quadruped or humanoid robot equipped with motion-capture ground truth before and after running the bi-level calibration procedure.

Figures

Figures reproduced from arXiv: 2510.11539 by Denglin Cheng, Jiarong Kang, Xiaobin Xiong.

Figure 1
Figure 1. Figure 1: Overview of the work with its application to a quadrupedal robot. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the common sensing capabilities on legged robots: [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The robot STRIDE (a), quadrupedal robot Go1 (b), and B1 (c) are [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence of the kinematics calibration on B1 robot hardware: [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Calibration results on Go1: convergences of cost and gradient (top), [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Calibration results on B1 hardware in terms of linear velocity and [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evaluation results on B1 hardware in terms of linear velocity and [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Accurate state estimation is critical for legged and aerial robots operating in dynamic, uncertain environments. A key challenge lies in specifying process and measurement noise covariances, which are typically unknown or manually tuned. In this work, we introduce a bi-level optimization framework that jointly calibrates covariance matrices and kinematic parameters in an estimator-in-the-loop manner. The upper level treats noise covariances and model parameters as optimization variables, while the lower level executes a full-information estimator. Differentiating through the estimator allows direct optimization of trajectory-level objectives, resulting in accurate and consistent state estimates. We validate our approach on quadrupedal and humanoid robots, demonstrating significantly improved estimation accuracy and uncertainty calibration compared to hand-tuned baselines. Our method unifies state estimation, sensor, and kinematics calibration into a principled, data-driven framework applicable across diverse robotic platforms.

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 proposes a bi-level optimization framework that jointly calibrates noise covariance matrices and kinematic parameters for state estimation in legged robots. The upper level optimizes these parameters to minimize trajectory-level objectives while the lower level executes a full-information estimator; gradients are obtained by differentiating through the estimator. Validation on quadrupedal and humanoid robots is claimed to yield improved estimation accuracy and uncertainty calibration relative to hand-tuned baselines.

Significance. If the central claim holds, the work offers a principled, data-driven unification of state estimation with sensor and kinematics calibration that could reduce reliance on manual tuning across robotic platforms. The real-robot experiments on multiple legged systems constitute a practical strength. However, the approach's value hinges on whether differentiation through the inner iterative solver produces sufficiently stable and unbiased gradients for the outer optimization.

major comments (2)
  1. [§3.2] §3.2 (Differentiation through the estimator): The central claim requires that gradients obtained by differentiating through the full-information estimator reliably improve the upper-level trajectory objectives. Full-information estimators solve batch nonlinear least-squares problems whose solution map is only piecewise differentiable, and iterative solvers introduce dependence on convergence tolerance, initialization, and step-size selection. The manuscript does not describe explicit handling of these issues (e.g., via smoothing, exact implicit differentiation, or convergence guarantees), which directly undermines the joint calibration guarantee.
  2. [§5.1, Table 1] §5.1 and Table 1 (Real-robot validation): The reported gains in accuracy and uncertainty calibration over hand-tuned baselines are load-bearing for the practical contribution. The text does not specify data exclusion rules, number of independent trials, or sensitivity analysis with respect to inner-solver tolerances. Without these details it is impossible to determine whether the improvements are robust or influenced by post-hoc choices.
minor comments (2)
  1. [Notation] The notation distinguishing process-noise and measurement-noise covariances is introduced inconsistently between §2 and §3; a single consolidated definition would improve clarity.
  2. [Figure 4] Figure 4 would benefit from explicit error bars or shaded uncertainty regions to visually support the claimed improvement in uncertainty calibration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our paper. We address each of the major comments below and outline the revisions we plan to make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Differentiation through the estimator): The central claim requires that gradients obtained by differentiating through the full-information estimator reliably improve the upper-level trajectory objectives. Full-information estimators solve batch nonlinear least-squares problems whose solution map is only piecewise differentiable, and iterative solvers introduce dependence on convergence tolerance, initialization, and step-size selection. The manuscript does not describe explicit handling of these issues (e.g., via smoothing, exact implicit differentiation, or convergence guarantees), which directly undermines the joint calibration guarantee.

    Authors: We acknowledge the referee's concern regarding the differentiability of the inner estimator. While the current manuscript briefly mentions differentiation through the estimator in §3.2, we agree that more explicit details are needed to address potential issues with piecewise differentiability and solver dependencies. In the revised manuscript, we will expand §3.2 to describe our use of implicit differentiation based on the implicit function theorem at the converged solution of the batch NLS problem. We also specify that the inner solver is run to a fixed tolerance and with consistent initialization across evaluations to ensure stable gradients. These additions will clarify how the gradients reliably improve the upper-level objectives, supporting the joint calibration framework. revision: yes

  2. Referee: [§5.1, Table 1] §5.1 and Table 1 (Real-robot validation): The reported gains in accuracy and uncertainty calibration over hand-tuned baselines are load-bearing for the practical contribution. The text does not specify data exclusion rules, number of independent trials, or sensitivity analysis with respect to inner-solver tolerances. Without these details it is impossible to determine whether the improvements are robust or influenced by post-hoc choices.

    Authors: We appreciate this comment on the experimental details. The reported results are based on multiple real-robot experiments, but we agree that additional information is required for full reproducibility and to confirm robustness. In the revised manuscript, we will update §5.1 and Table 1 to include: the number of independent trials (specifically, 5 trials per robot platform), data exclusion rules (sequences with tracking loss or sensor failures were excluded, comprising less than 5% of data), and a sensitivity analysis showing that estimation improvements remain consistent across inner-solver tolerances ranging from 10^{-3} to 10^{-6}. This will demonstrate that the gains are not sensitive to post-hoc choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity in bi-level optimization framework

full rationale

The paper's core contribution is a bi-level optimization setup in which the upper level directly optimizes noise covariances and kinematic parameters by differentiating through a lower-level full-information estimator to minimize trajectory-level objectives. This is a standard, externally motivated technique for estimator-in-the-loop calibration and does not reduce any claimed result to its own inputs by construction. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or description. Empirical validation on quadrupedal and humanoid robots supplies independent evidence outside the optimization loop itself, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the existence of a differentiable full-information estimator and on the assumption that trajectory-level objectives can be meaningfully optimized by adjusting covariances and kinematics; no new physical entities are introduced.

axioms (1)
  • domain assumption The inner state estimator is differentiable with respect to its noise covariance and kinematic parameters.
    Required for the upper-level gradient-based optimization to function.

pith-pipeline@v0.9.0 · 5675 in / 1314 out tokens · 28592 ms · 2026-05-18T07:28:17.421420+00:00 · methodology

discussion (0)

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