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A2I-Calib++: A Kinematics-Driven Anti-noise Active Calibration Framework for Whole-Body Multi-IMU Systems on Legged Robots

Fangyu Jiang, Chaoran Xiong, Kehui Ma, Zhen Sun, Zeyu Zhang, Ling Pei

arXiv:2610.29438v1cs.RO

TL;DR

Legged-robot multi-IMU calibration is challenging because locomotion may provide insufficient excitation while slips, impacts, and vibrations disturb measurements. A2I-Calib++ combines anti-noise trajectory design, contact-aware motion, and a contact-constrained virtual IMU reference to estimate whole-body sensor parameters. The paper reports lower simulation calibration errors, repeatable real-world calibration, and improved downstream proprioceptive odometry.

  • Problem

    Legged-robot multi-IMU calibration must provide sufficient excitation despite locomotion-related measurement disturbances, and body-mounted IMUs require calibration beyond the parameters addressed by calf-IMU methods.

    Source: §I · p. 1

  • Method

    A2I-Calib++ combines noise-isotropy-inspired excitation, an AMP controller with a rolling-contact reward, and a contact-constrained VIMU built from pre-calibrated calf IMUs and joint kinematics.

    Source: §I · pp. 1–2, D. Contact-Constrained Virtual IMU… · p. 4

  • Results

    The evaluations report reduced simulation calibration errors, repeatable real-world calibration, and lower downstream proprioceptive-odometry errors.

    Source: 1) Ablation Study on NIT · p. 6, §VI · p. 8

  • Takeaways & Limitations

    The framework extends active spatiotemporal calibration to body-mounted IMUs covered by a legged robot’s kinematic model and reports downstream odometry benefits.

    Source: §VI · p. 8

Abstract

from arXiv · show

Recently, multi-node inertial measurement unit (multi-IMU) systems have shown promise for robust odometry in legged robots. Multi-IMU fusion can improve legged robot odometry but requires accurate spatiotemporal calibration among the sensors. Existing multi-IMU calibration methods, designed for aerial or handheld platforms, face challenges on legged robots because routine locomotion may provide insufficient excitation, while foot slips, impacts, and vibrations introduce measurement disturbances. To address these challenges, we propose A2I-Calib++, a kinematics-driven anti-noise active calibration framework for body-mounted IMUs covered by the robot kinematic model. It estimates temporal offsets, extrinsic rotations and translations, and inertial biases through temporal correlation search followed by analytic spatial and bias estimation, without artificial calibration targets or complex parameter tuning for the target IMU. A2I-Calib++ comprises three core modules: 1) a Noise-Isotropy-Inspired Trajectory supported by a finite-horizon conditioning proposition that generates sufficient excitation to limit noise sensitivity; 2) a constraint-aware Adversarial Motion Priors controller that combines the anti-noise excitation with a rolling-contact reward for contact-aware execution; and 3) a contact-constrained virtual IMU that fuses pre-calibrated calf IMUs and joint kinematics under a rolling-contact model to construct the calibration reference. Furthermore, evaluations on both simulation and real-world quadruped platforms demonstrate notably reduced noise sensitivity and simulation calibration errors compared to conventional methods, together with repeatable real-world calibration and improved proprioceptive odometry accuracy for legged robots.

I. INTRODUCTION

Multi-IMU fusion can improve legged-robot state estimation, but calibration is difficult because locomotion may provide insufficient excitation and introduces disturbances that increase noise sensitivity. A2I-Calib++ extends active calibration to whole-body IMUs, combining anti-noise excitation, contact-aware motion, and a fused reference.

I. INTRODUCTION

  • Legged-robot calibration must handle insufficient excitation and noise from foot slips, impacts, and vibrations, while body-mounted IMUs also require translation and accelerometer-bias estimation.A2I-Calib calibrates calf-mounted IMUs but does not address those additional body-IMU parameters.Source: §I · p. 1
  • A2I-Calib++ proposes active, kinematics-driven calibration for whole-body multi-IMU systems on legged robots.Its contributions include temporal correlation search and analytic spatial estimation without artificial targets or complex target-IMU parameter tuning.Source: §I · p. 1
  • The framework combines anti-noise reference excitation and a contact-constrained, multi-leg VIMU with an AMP controller that learns the reference motion’s state-transition distribution and uses a rolling-contact reward.These components are designed to reduce calibration noise sensitivity and generate contact-aware motions.Source: §I · pp. 1–2
  • The paper reports reduced noise sensitivity, high simulation calibration accuracy, repeatable real-world calibration, and lower proprioceptive-odometry drift than a state-of-the-art calibration method.Multi-IMU misalignment can introduce inconsistent kinematic constraints and systematic odometry drift.Source: §I · pp. 1–2

II. RELATED WORK

  • Existing pure-inertial methods are sensitive to measurement noise and excitation quality, while active trajectory design has mainly addressed industrial manipulators rather than agile quadrupeds,,,.Related approaches include exteroceptive-aided multi-IMU calibration, and legged-robot methods with more limited calibration scope.Source: §II · p. 2

III. PROBLEM FORMULATION

The formulation describes target and reference IMU frames, measurement disturbances, temporal offset, and extrinsic translation, then frames calibration as reducing noise sensitivity through trajectory-dependent conditioning.

A. IMU Measurement Model and Kinematics

  • The measurement model distinguishes the target IMU frame from the reference-sensor frame and represents target measurements using angular velocity, specific force, biases, and Gaussian white-noise terms.Transmission and synchronization delays contribute a temporal offset; the model also defines the reference-to-target translation in the reference frame.Source: A. IMU Measurement Model and Kinematics · p. 2
  • The target and reference IMU frames provide the coordinate-frame basis for expressing the target measurements and the reference-to-target translation.Source: A. IMU Measurement Model and Kinematics · p. 2
  • The IMU biases are assumed constant over the short observation window.Source: A. IMU Measurement Model and Kinematics · p. 2

B. Conditioning Objective for Calibration Trajectory

  • For a fixed, full-column-rank overdetermined system, the spectral condition number and an observation-perturbation bound characterize conditioning.The stated setup has H ∈ R^m×n with m ≥ n.Source: B. Conditioning Objective for… · p. 2
  • A large condition number permits stronger amplification of observation noise into parameter error, whereas a small condition number tightens the bound; κ2(H) = 1 minimizes amplification.Source: B. Conditioning Objective for… · p. 2
  • The calibration matrix H(T) depends on the trajectory, so the formulation seeks an anti-noise trajectory T* that minimizes calibration noise sensitivity.Source: B. Conditioning Objective for… · p. 2

IV. METHODOLOGY

The methodology constructs a contact-aware calibration reference and estimates parameters through a search-and-solve pipeline. Temporal alignment precedes rotation and gyroscope-bias estimation from aligned measurements.

A. System Overview

  • The pipeline uses NIP to guide anti-noise excitation, a preliminary A2I-Calib step for calf-IMU offsets and rotations, AMP for contact-aware execution, and rolling-contact multi-leg VIMU fusion as the calibration reference.It then applies search-and-solve estimation and evaluates corrected IMU measurements through proprioceptive odometry against an LIO reference.Source: A. System Overview · p. 2, Figure 2 · p. 3

B. Search-and-Solve IMU Calibration

1) Temporal and Rotational Calibration:

  • The temporal offset is selected by maximizing the trace correlation coefficient over candidate delays; singular value decomposition then estimates extrinsic rotation and gyroscope bias from temporally aligned sequences.The empirical means used in the calculation are computed from the aligned sequences.Source: 1) Temporal and Rotational Calibration · p. 3

2) Translation and Accelerometer Bias Calibration:

Translation and accelerometer bias are estimated through a velocity-residual least-squares solve under a standstill initialization. The NIT design uses orthogonal harmonic excitation to improve conditioning and limit noise sensitivity.

2) Translation and Accelerometer Bias Calibration:

  • The translation is recovered from a pseudoinverse least-squares estimate of the stacked velocity residuals, together with the previously estimated parameters.The derivation assumes calibration begins at strict standstill, with reference velocity and angular velocity both zero.Source: 2) Translation and Accelerometer Bias… · p. 3

C. Noise Isotropy Inspired Trajectory (NIT) Generation

  • Orthogonal harmonic excitation yields the Noise Isotropy Proposition, which guides generation of an anti-noise calibration trajectory.The proposition states isotropy properties for finite trajectory-family members, while its normalized infimum is approached only as frequency increases and is not attained by a finite nonzero member.Source: C. Noise Isotropy Inspired Trajectory… · pp. 3–4
  • Within each micro-period, the sign-reversed, time-reversed second half makes the attitude retrace its first-half rotation and return to its initial value.The construction supports the trajectory's isotropy analysis over complete periods.Source: C. Noise Isotropy Inspired Trajectory… · p. 4
  • The deployed 3.2-s trajectory has theoretical condition numbers κR = 1.000 and κt ≈ 1.872, with κt changing by less than 1% across 200–1000 Hz processing and up to 100-ms truncation.The trajectory repeats a 1.6-s waveform twice to reduce the influence of the less-stable startup transient.Source: C. Noise Isotropy Inspired Trajectory… · p. 4

D. Contact-Constrained Virtual IMU Fusion

The contact-constrained VIMU reference is built by fusing calf IMUs that have been pre-calibrated with A2I-Calib. Their calibrated parameters support time-aligned, bias-corrected angular-velocity inputs for kinematic propagation.

D. Contact-Constrained Virtual IMU Fusion

  • Pre-calibrated calf IMUs are fused to construct a contact-constrained VIMU reference.The preliminary calibration supplies each calf IMU's time offset, IMU-to-calf rotation, and gyroscope bias.Source: D. Contact-Constrained Virtual IMU… · p. 4
  • The reference construction uses calf sensors pre-calibrated by A2I-Calib rather than treating their measurements as uncalibrated inputs.Those preliminary parameters include temporal offset, rotation, and gyroscope bias.Source: D. Contact-Constrained Virtual IMU… · p. 4
  • The preliminary calf-IMU parameters provide time-aligned, bias-corrected angular velocity for kinematic propagation.This is the calibrated angular-velocity input used in constructing the reference.Source: D. Contact-Constrained Virtual IMU… · p. 4

1) Kinematic Propagation under Pure-Rolling Constraint:

Under the pure-rolling assumption, a supporting foot has zero instantaneous linear velocity, allowing calf velocity to be computed from foot angular velocity and the contact-to-foot-center lever arm. Forward kinematics propagates these measurements upward to form a leg VIMU.

1) Kinematic Propagation under Pure-Rolling Constraint:

  • Pure-rolling contact sets the grounded foot's instantaneous linear velocity to zero, so calf velocity follows from foot angular velocity crossed with the contact-to-foot-center vector.The figure contrasts pure rolling with foot slip.Source: 1) Kinematic Propagation under… · p. 4, Figure 3 · p. 5
  • Joint position and velocity, initialized calf velocity, and calibrated calf angular velocity feed recursive forward-kinematics propagation toward the robot base.The propagation produces a leg VIMU whose angular and linear measurements are mapped to the trunk.Source: 1) Kinematic Propagation under… · pp. 4–5
  • The resulting leg VIMU is constructed by propagating kinematic measurements through the robot's forward-kinematics chain.Fig. 3(b) depicts this propagation.Source: 1) Kinematic Propagation under… · p. 4, Figure 3 · p. 5

2) Multi-Leg VIMU Fusion and Noise Suppression:

Multiple supporting-leg VIMUs are fused by arithmetic averaging, but their errors may be correlated because they share measurements, kinematics, and contact modeling. Averaging suppresses uncorrelated errors, while cross-covariance limits the reduction.

2) Multi-Leg VIMU Fusion and Noise Suppression:

  • Leg-VIMU errors can be correlated because each estimate combines calf-IMU measurements, joint encoders, forward kinematics, and the contact model.The fused-error covariance therefore depends on cross-covariances as well as individual leg covariances.Source: 2) Multi-Leg VIMU Fusion and Noise… · p. 5
  • Four-leg averaging gives a 75% covariance-trace reduction when leg errors are mutually uncorrelated and have equal covariance.Cross-covariance limits the attainable reduction when errors are correlated.Source: 2) Multi-Leg VIMU Fusion and Noise… · p. 5
  • Pure-rolling contact is adequate when the stated slip-related bound on translation-error increment is met; the condition-number-based bound is not a strict threshold.The criterion uses window-RMS tangential slip speed and allowable translation-error increment.Source: 2) Multi-Leg VIMU Fusion and Noise… · p. 5

E. Constraint-Aware Adversarial Motion Prior Controller

The constraint-aware AMP controller preserves anti-noise excitation while encouraging rolling contact, addressing the gap between an ideal reference trajectory and feasible robot motion. Its training uses a reference-motion dataset and rewards that balance excitation retention against contact violations.

E. Constraint-Aware Adversarial Motion Prior Controller

  • AMP learns the reference motion’s state-transition distribution while encouraging pure-rolling contact to reduce disturbances in the kinematic reference.Directly tracking the theoretical NIT can leave a gap between ideal and feasible motion, risking foot slip, contact-induced noise, and instability.Source: E. Constraint-Aware Adversarial Motion… · p. 5

1) Reference Dataset:

  • A traditional controller tracks the reference trajectory on the Unitree Go2 model to collect robot states for the reference dataset.The dataset is shown in Fig. 2(b).Source: 1) Reference Dataset · p. 5

2) Reward Design:

  • The style reward preserves the reference distribution’s anti-noise excitation, while the task reward penalizes translational slip to encourage pure-rolling contact.The total reward combines the style and task terms with weights that balance them.Source: 2) Reward Design · p. 5

3) Observation and Action Spaces:

The experiments describe the AMP controller’s spaces, domain randomization, hardware platform, and comparison baselines. Search-and-Solve runs faster than iKalibr in the reported experiments, with preprocessing dominating runtime as motion duration doubles.

3) Observation and Action Spaces:

1) Development of Experimental Platforms:

2) Comparison Baselines:

  • The comparisons vary reference motion, execution controller, and calibration solver; hardware trials use the two AMP-based motions and Walking for safety.The motion baselines include NIT + AMP, Sinusoid + AMP, NIT without AMP, Sinusoid without AMP, and conventional Walking.Source: 2) Comparison Baselines · p. 6
  • Search-and-Solve is faster than iKalibr in these experiments, and runtime changes little when motion duration doubles because preprocessing dominates computation.The comparison is between lightweight correlation search, SVD, and least squares, and iKalibr’s iterative nonlinear optimization.Source: 2) Comparison Baselines · p. 6, Table III · p. 6

B. Sensitivity to Preliminary Calf-IMU Calibration

NIT improves conditioning and calibration outcomes in simulation and hardware, while AMP reduces simulated VIMU errors and improves hardware reference-consistency proxies relative to Walking. Preliminary calf-IMU errors produce smaller camera-IMU parameter shifts than the injected residual magnitudes.

B. Sensitivity to Preliminary Calf-IMU Calibration

  • The reported shifts are 0.252 ms, 0.546°, and 0.0050 rad/s, respectively; all remain below the injected residual magnitudes, indicating limited sensitivity to leg-dependent preliminary-calibration errors.Source: B. Sensitivity to Preliminary Calf-IMU… · p. 6

C. Contact-Aware Motion and VIMU Validation

  • AMP reduces four-leg angular- and linear-velocity errors for both reference motions in simulation, while AMP-based motions yield smaller reference-consistency proxies than Walking on hardware.Hardware Slip and VIMU Error values are proxies because ground truth is unavailable; a large relative reduction can still accompany a large remaining error.Source: C. Contact-Aware Motion and VIMU… · p. 6, Table V · p. 6

D. Spatiotemporal Calibration Performance

  • Calibration results report the camera IMU relative to the trunk IMU, with translations expressed in the trunk-IMU frame.The results cover the estimated spatiotemporal parameters.Source: D. Spatiotemporal Calibration… · p. 6

1) Ablation Study on NIT:

  • κR/κt falls from 280.64/9.862 to 1.46/1.640 in simulation; NIT also lowers three calibration RMSEs and yields the smallest within-motion standard deviations for all seven parameters across hardware windows.Executed condition numbers differ from the theoretical 1.000/1.872, yet NIT improves conditioning over the no-NIT motion in simulation and hardware.Source: 1) Ablation Study on NIT · p. 6, Table VI · p. 7, Table VII · p. 7

2) Ablation Study on AMP:

The section reports AMP’s contact-aware motion trade-offs, compares Search-and-Solve with iKalibr, and evaluates cross-motion consistency and downstream odometry. The conclusion summarizes A2I-Calib++’s calibration pipeline and reported simulation and real-world outcomes.

2) Ablation Study on AMP:

  • With NIT fixed, AMP reduces slip and increases contact duty despite larger tracking error, and Table VI reports lower spatiotemporal calibration errors.AMP retains anti-noise excitation through the reference state-transition distribution rather than pointwise tracking; poor conditioning can mask its contact-disturbance benefit.Source: 2) Ablation Study on AMP · p. 7

3) Search-and-Solve vs. iKalibr:

  • Search-and-Solve succeeds in 9/9 simulation trials versus 6/9 for iKalibr, while the solvers show comparable hardware repeatability.iKalibr failures are initialization-sensitive; diagnostics trace them to gyro-based rotation–time coupling that can trap refinement in a local optimum.Source: 3) Search-and-Solve vs. iKalibr · p. 7

4) Cross-Motion Consistency and Interpretation:

  • Across motions, Search-and-Solve has smaller temporal, comparable rotational, and larger translational shifts than iKalibr; consistency alone does not establish accuracy.Rolling-contact mismatch can inject linear-velocity errors, while conditioning governs error amplification and estimation can propagate errors or introduce bias.Source: 4) Cross-Motion Consistency and… · p. 7, Table IX · p. 7

E. Real-World Downstream Odometry Validation

VI. CONCLUSION

  • A2I-Calib++ estimates body-mounted IMU temporal offsets, extrinsic rotations and translations, and biases after preliminary calf-IMU calibration and data collection.The conclusion reports lower simulation calibration errors, repeatable real-world calibration, and reduced downstream odometry errors; future work targets friction and ground variation, sim-to-real gaps, and exteroceptive modalities.Source: §VI · p. 8
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