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Contact-Constrained Lower-Limb Joint-Offset Calibration for Humanoid Robots

Kaixiang Lu, Haiyu Lan, Chunxiao Qiao, You Li, Chengyuan Luo, Enyu Li, Peiwen Lin, Chuang Wang

arXiv:2609.02306v1cs.RO

TL;DR

The paper addresses lower-limb encoder-offset calibration without external measurement systems. It uses inter-foot contact consistency, IMU-aided flat-ground constraints, and Hessian analysis to estimate offsets and characterize pitch coupling. Experiments show improved contact consistency, while individual offsets in weak pitch chains remain prior-dependent.

  • Problem

    Existing calibration commonly requires external motion-capture systems or fiducial targets, while parallel pitch axes create lower-limb observability challenges.

  • Method

    The framework estimates a shared 12-dimensional offset vector from fixed-foot inter-foot consistency and IMU-aided flat-ground constraints, with regularization and mechanical priors.

  • Results

    A2 pitch sums are more stable than their individual decomposition, and tested A3 ordering produces better conditioning than A2 ordering.

  • Takeaways & Limitations

    Static double-support stances provide contact-consistent corrections for well-excited directions using only onboard sensing.

  • Takeaways & Limitations

    The method assumes fixed, approximately coplanar double support, and individual pitch offsets can remain prior-dependent.

Abstract

from arXiv · show

Accurate joint encoder offsets are essential for kinematic consistency in humanoid lower limbs, yet existing calibration methods typically require external motion-capture systems or fiducial targets. We present a self-contained calibration framework exploiting only onboard joint encoders and a pelvis-mounted IMU during static double-support contact. The inter-foot transform from forward kinematics must stay constant when both feet are fixed; minimizing its posture-dependent dispersion yields a nonlinear least-squares problem over the 12-dimensional offset vector. A Hessian eigenstructure analysis shows that parallel pitch axes induce a rotational coupling. Orientation residuals then observe only the pitch-offset sum, while translation and posture diversity set the remaining numerical observability. For the A3 pitch-to-roll-to-yaw ordering, hip-roll and hip-yaw excitation reduce hip-pitch coupling. A standing-posture knee prior then anchors the remaining weak pitch-chain decomposition. Simulation and real-machine injection tests show consistent recovery, and on held-out recordings calibration reduces foot-height RMS residuals from 4.26 to 2.20 mm on A3 and from 8.03 to 1.43 mm on A2. An independent LiDAR-inertial reference checks the pitch-coupled channel. Removing an injected pitch offset moves the leg-odometry vertical drift back toward the LiDAR trajectory. A few static double-support stances thus provide contact-consistent corrections for well-excited directions. Individual offsets in the weak pitch chain remain prior-dependent.

I. INTRODUCTION

Humanoid joint-offset calibration is important for kinematic consistency but commonly depends on external measurements. This paper uses fixed-foot contact, onboard sensing, and Hessian analysis to address calibration and observability in lower-limb chains.

  • Accurate encoder offsets support locomotion, whole-body control, and contact planning, while small per-joint biases can create centimeter-level foot errors.
  • Classical calibration relies on laser trackers, motion-capture systems, or fiducial targets, limiting practicality for repeated in-field recalibration.
  • Onboard camera markers are unreliable for leg calibration because the feet rarely enter the head-camera field of view during normal stances.
  • Stable double support provides a self-contained signal: the inter-foot transform should remain constant across postures, so its dispersion can estimate joint offsets.
  • Parallel pitch axes make rotational Jacobian columns collinear, leaving only the pitch-offset sum observable from orientation and motivating Hessian-based coupling analysis.
  • The framework combines inter-foot consistency, IMU-aided flat-ground constraints, and mechanical priors, and validates the approach on A3 simulation and both real platforms.

B. Contact-Based Constraints for Legged Robots

The paper reframes foot-contact information from online state estimation as an offline calibration constraint and studies how humanoid joint ordering shapes offset observability.

  • Prior legged estimators use IMU, encoder, and foot-contact information for online state estimation, whereas this work estimates persistent encoder offsets offline.
  • Earlier contact-based humanoid calibration used joint encoders, an IMU, and flat-foot constraints, but did not formally characterize parallel-pitch coupling.
  • The paper formulates calibration through inter-foot transform dispersion, formalizes pitch coupling with Hessian eigenvectors, and analyzes ordering effects.
  • Observability analysis is extended beyond condition numbers and singular-value rankings to identify weak offset combinations.
  • A3 uses pitch→roll→yaw→knee→ankle-pitch→ankle-roll, while A2 uses roll→yaw→pitch→knee→ankle-pitch→ankle-roll.

B. Inter-Foot Consistency Constraint

During stable double support, the right-foot pose relative to the left foot should remain constant. The method measures deviations from a segment mean and adds IMU-based flat-ground constraints.

  • The active inter-foot transform maps coordinates from the right-foot frame to the left-foot frame and must remain constant during a contact segment.
  • The calibration uses the intrinsic mean inter-foot transform on SE(3) as the reference for retained frames.
  • The contact-consistency residual measures each frame’s deviation from the segment mean, with translation followed by rotation in the logarithmic representation.
  • The pelvis-mounted IMU supplies gravity in the base frame, enabling world-frame flat-ground residuals from foot positions and normals.
  • The flat-ground residual enforces equal foot heights and penalizes deviations of both foot normals from gravity; tilt error or uneven contact can bias it.

D. Optimization Objective

The optimization combines contact consistency, flat-ground constraints, regularization, and optional mechanical priors over shared joint offsets. An alternating scheme updates the contact mean and offset estimate.

  • The objective combines contact consistency, IMU-aided flat-ground constraints, and Tikhonov regularization over the shared offset vector.
  • A3 adds a quadratic knee prior, whereas A2 uses no knee prior; matched comparisons omit the mechanical prior.
  • The platform-specific settings use different scalar weights and per-joint weight vectors, with A2 applying stronger ridge regularization to its pitch-axis near-nullspace.
  • The real-machine protocol uses A2 and A3 platforms with only onboard encoders and a pelvis IMU.
  • The solver alternates recomputation of the contact mean with offset optimization while holding the mean fixed during the inner solve.
  • All reported experiments converge within 3–4 outer iterations, with offsets initialized at zero and bounded to ±0.08 rad.

F. Data Preprocessing

The preprocessing and observability analysis filter static-contact data, assemble the data Hessian, and characterize weak offset directions caused by parallel pitch axes and limited translational excitation.

  • Data filtering: Frames violating fixed-contact assumptions are rejected using local temporal consistency, rotation-residual, and gyroscope thresholds.A 2.5 s window is used; frames with rotation residual above 5° are discarded, with a permissive 0.5 rad/s gyroscope guard.
  • Data filtering: The retained encoder data are clustered and subsampled to 500 representatives per stance, then the first solve removes the largest 10%, leaving 450 configurations.
  • Observability: Structural observability depends on kinematic topology, whereas numerical observability depends on posture diversity and data volume.
  • Observability: The data Hessian is formed from contact and flat-ground residual Jacobians, while regularization and mechanical priors add curvature without adding information to the data Jacobian.Matched four-stance Jacobians have numerical rank 12; small eigenvalues identify offset combinations that are unreliable.
  • Pitch-axis coupling: Parallel pitch-axis columns are collinear in the rotational Jacobian, so orientation observes only one pitch-offset combination and leaves an (n−1)-dimensional rotational nullspace.Translation and full SE(3) residuals can weakly break this degeneracy through lever-arm differences, but compact geometry and sagittal-dominated stances produce near-null directions.

C. Effect of Kinematic Ordering

Kinematic ordering changes how posture can decouple the pitch chain, while matched real-platform spectra show that conditioning differences are not attributable to ordering alone.

  • Ordering effects: In A3’s pitch→roll→yaw ordering, nonzero hip-roll and hip-yaw make the hip-pitch axis nonparallel to the knee and ankle-pitch axes in the foot frame.
  • Ordering effects: In A2’s roll→yaw→pitch ordering, hip-pitch, knee, and ankle-pitch are consecutive parallel-axis joints, leaving only weak translational information in recorded stances.
  • Decoupling strategies: The framework separates geometric decoupling from algebraic resolution: posture diversity breaks axis parallelism, whereas priors constrain the remaining nullspace.
  • Matched comparison: Figure 4 compares Hdata eigenvalues with pitch components of the three weakest eigenvectors across matched four-stance real data.The physical-platform conditioning gap is not ordering-only.

1) Geometric Decoupling via Roll and Yaw Excitation:

Hip-roll and hip-yaw excitation geometrically decouple A3 hip-pitch from the distal pitch chain, after which a knee prior selects a decomposition that remains prior-dependent.

  • Geometric decoupling: Nonzero hip-roll and hip-yaw rotate the A3 hip-pitch axis out of alignment with knee and ankle-pitch, reducing the pitch nullspace from two dimensions to knee–ankle-pitch coupling.
  • Algebraic resolution: A roughly 5 s straight-leg standing phase supplies the posture used to construct a knee prior.
  • Algebraic resolution: The quadratic knee penalty anchors the knee, allowing the pitch-sum constraint to determine the remaining coupled pitch component.
  • Algebraic resolution: The knee prior selects one decomposition in the coupled subspace but does not make individual pitch offsets observable.Its mechanical-zero reference comes from the URDF and CAD model, so model error propagates into selected individual offsets.
  • Platform boundary: On A2, three consecutive parallel-axis joints prevent geometric decoupling, so Tikhonov regularization suppresses near-nullspace drift but can bias individual estimates.The regularization weights and centers must therefore be reported and tested.

V. EXPERIMENTS

Experiments evaluate offset recovery and conditioning in simulation and on A2 and A3 real machines using static double-support stances, matched ordering comparisons, and separate consistency metrics.

  • Simulation setup: A3 simulation uses one continuous-posture double-support squat with 450 retained frames and a physically plausible injected offset pattern.The injection is large enough to exceed sensor noise while remaining within the feasible static-contact regime.
  • Real-machine setup: A2 and A3 real-machine experiments use the same four-stance protocol, while A2 receives stronger offset regularization because its ordering has weaker pitch observability.
  • Controlled comparison: A same-geometry MuJoCo comparison changes only proximal joint ordering, isolating the ordering effect under otherwise matched conditions.
  • Metrics: The paired real-recording error tests estimator consistency rather than native mechanical-zero accuracy, and data-only and regularized condition numbers are reported separately.
  • Simulation results: 0.27° is the peak A3 simulation injection-recovery error, concentrated along the weak pitch directions.The contact residual constrains the pitch sum, while the anchored knee fixes the selected decomposition within the weak pitch chain.
  • Interpretation: The injection test does not establish that every individual offset is observable from the data.
  • Simulation results: Ideal zero-noise recovery is below 1.2 × 10^-13 degrees, while weld and settling produce RMS floors of 0.092° and 0.121°, respectively.

C. A2 and A3 Real-Machine Results

Real-machine tests show improved conditioning and contact-consistency outcomes, while A2 pitch-chain decomposition remains sensitive to regularization and prior choices.

  • Conditioning: A3 has better data-only conditioning than A2 under common weights, with κ(Hdata) of 333.23 versus 4340.79.The matched comparison still reflects platform differences in geometry, excitation, IMU mounting, and model error.
  • Conditioning: For the tested same-geometry simulation, P-R-Y improves conditioning over R-Y-P, reducing κ(Jdata) from 348.90 to 90.69 and κ(Hdata) from 121728.69 to 8224.49.This supports an ordering effect for the tested geometry and excitation.
  • Pitch-chain observability: A2 individual pitch errors have 0.366° RMS, whereas pitch combinations sL, sR, and sR−sL have errors of −0.0053°, +0.0165°, and +0.0218°.Changing knee-prior centers shifts individual estimates by 0.59° to 1.90°, but pitch combinations by at most 0.0071°.
  • Method comparison: The comparison with Yamane’s method gives similar A2 results, while Yamane’s method has lower individual-offset RMS on A3.The reported results do not support a claim of universal superiority.
  • Held-out validation: Held-out foot-height RMS residuals decrease from 4.26 to 2.20 mm on A3 and from 8.03 to 1.43 mm on A2 after calibration.The validation uses recordings excluded from optimization; sole-normal residuals change in the same direction on both platforms.

F. External Validation

External checks test the calibrated pitch–vertical channel against LiDAR-inertial odometry, while closed-loop simulation evaluates nominal tracking after correction.

  • Scope of validation: The external tests share the forward-kinematics and flat-ground model, so they measure internal consistency rather than absolute external accuracy.The LiDAR-inertial cross-check specifically addresses the pitch–vertical channel.
  • LiDAR-inertial validation: A symmetric hip-pitch offset produces vertical drift linear in the offset, with sensitivity of 1.04 m per degree.This experiment checks the pitch–vertical channel rather than absolute encoder zeros.
  • LiDAR-inertial validation: Calibration moves the A3 leg-odometry trajectory toward the LiDAR reference, reducing vertical RMS error by 25.5% over 16 held-out segments.FAST-LIO2 uses neither leg encoders nor forward kinematics, providing an external reference for this channel.
  • Independent orientation check: Held-out inter-foot orientation error decreases from 5.223° to 4.085° in an independent registered optical reference recording.The second placement is evaluated without fitting forward kinematics or offsets.
  • Closed-loop simulation: At 3°, corrected closed-loop simulation reduces foot-strike RMS deviation from 1190.61 to 26.57 mm and pelvis-trajectory RMS deviation from 1224.10 to 27.16 mm.All trials remain upright, but torque and power do not improve consistently.

H. Stance Diversity and Saturation

Diverse static stances improve conditioning quickly, but additional stances saturate and regularization or priors mainly stabilize weak directions rather than add data information.

  • Stance diversity: κ(Hreg) drops from 1.27 × 10^5 to 6.19 × 10^3 after adding the first diverse A2 stance, with later gains saturating.Added contact geometries can introduce stance-dependent contact and flatness errors.
  • Regularization: Varying λq barely changes A2 data fit but moves weak pitch offsets, selecting a stable solution without creating observability.The effect is solution selection in poorly observed directions, not additional information from the measurements.
  • Constraint ablation: Removing the flat-ground term shifts frontal and transverse hip offsets by 2° to 5°, confirming its out-of-plane information.The flat-ground constraint therefore affects hip-offset estimates beyond the pitch-coupled channel.
  • Mechanical priors: Under the A3 operating setting, the knee prior improves κ(Hreg) from approximately 333 to 72.27 through added prior curvature.This conditioning improvement does not add information to Hdata.
  • Scope: Static double-support stances improve contact consistency using onboard sensing, while individual pitch offsets remain prior-dependent under the stated assumptions.The method assumes fixed, approximately coplanar double support and interprets estimates as contact-consistent corrections.
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