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

cs.ROOct 21, 2026

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

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

Organizations: Shanghai Key Laboratory of Navigation and Location Based Services, School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China

Abstract

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.

Figures & tables

Fig. 1: A 2 I-Calib++ performs kinematics-driven spatiotemporal calibration of body IMUs on legged robots using anti-noise calibration motion, AMP-based execution, and a contact-constrained VIMU reference, thereby improving whole-body multi-IMU calibration and downstream proprioceptive odometry.

Method	Target	Mode	$t_{d}$	$\mathbf{R}$	$\mathbf{p}$	$\mathbf{b}_{g}$	$\mathbf{b}_{a}$
iKalibr [ 11 ]	Body IMUs	Passive Calibration	✓	✓	✓	✓	✓
A 2 I-Calib [ 14 ]	Calf IMUs	Active Calibration	✓	✓	✗	✓	✗
A 2 I-Calib++	Body IMUs	Active Calibration	✓	✓	✓	✓	✓
* In iKalibr, IMU biases are jointly estimated only in configurations involving exteroceptive sensors [ 11 ] .

TABLE I: Comparison of Calibration Capabilities of Different Methods

Fig. 2: System overview of A 2 I-Calib++. (a) summarizes the Noise-Isotropy Proposition (NIP) and the resulting anti-noise reference excitation. (b) shows AMP-based contact-aware motion execution and multi-leg VIMU reference construction under a rolling-contact model. (c) presents search-and-solve parameter estimation and downstream proprioceptive-odometry evaluation.

Fig. 4: Unitree Go2 Platform with Noitom PN Studio suite, Intel RealSense D435i and Livox MID-360.

Duration	A 2 I-Calib++					iKalibr
Duration	Search	SVD	LS	Others	Total	Total
3.200	0.036	0.005	0.004	2.000	$2.045\pm 0.068$	$2.941\pm 2.105$
6.400	0.043	0.005	0.006	2.030	$2.084\pm 0.070$	$3.555\pm 1.123$

TABLE III: Comparison of Algorithm Runtime

Motion	Slip	Contact	VIMU Error *				Condition		Track.
Motion	( $\mathrm{m/s}$ )	( $\%$ )	$\\|\Delta\bm{\omega}\\|$		$\\|\Delta\mathbf{v}\\|$		$\kappa_{\mathbf{R}}/\kappa_{\mathbf{t}}$
Simulation Experiments: Ground Truth Available
A 2 I-Calib++	0.005	100.00	0.005 /	45.3	0.003 /	41.4	1.46 /	1.64	1.89
w/o NIT	0.001	100.00	0.005 /	37.3	0.002 /	27.5	275.70 /	9.85	0.98
w/o AMP	0.021	83.25	0.008 /	48.7	0.012 /	45.9	1.19 /	2.14	0.10
Sinusoid	0.032	76.15	0.010 /	42.9	0.020 /	35.4	337.70 /	26.47	0.08

TABLE V: Foot-Slip, Contact, VIMU Metrics, and Motion Quality of Different Calibration Motions

Settings

Trajectory Quality (Mean

\pm

STD)

Calibration Accuracy (RMSE

\pm

STD)

Motion

Trace Corr.

Rot. Cond.

Trans. Cond.

Time Offset

Extri. Rot.

Gyro. Bias

Extri. Trans.

Accel. Bias

Style

r_{max}

( % )

\uparrow

\kappa_{\mathbf{R}}

\downarrow

\kappa_{\mathbf{t}}

\downarrow

|\delta t_{d}|

( ms )

\downarrow

\|\delta\bm{\theta}\|

( deg )

\downarrow

\|\delta\mathbf{b}_{g}\|

( rad/s )

\downarrow

\|\delta\mathbf{p}\|

( cm )

\downarrow

\|\delta\mathbf{b}_{a}\|

( cm/s 2 )

\downarrow

A 2 I-Calib++

99.9

\pm

0.0

1.46

\pm

0.01

1.640

\pm

0.006

0.19

\pm

0.05

0.177

\pm

0.062

0.002

\pm

0.001

0.287

\pm

0.066

0.66

\pm

0.26

w/o NIT

99.3

\pm

0.1

280.64

\pm

3.73

9.862

\pm

0.066

0.38

\pm

0.15

0.675

\pm

0.266

0.001

\pm

0.001

3.390

\pm

0.342

0.54

\pm

0.20

w/o AMP

99.9

\pm

0.0

1.19

\pm

0.01

2.033

\pm

0.004

0.24

\pm

0.05

0.191

\pm

0.065

0.001

\pm

0.001

0.335

\pm

0.044

0.47

\pm

0.19

Sinusoid

97.3

\pm

0.1

349.89

\pm

3.07

29.492

\pm

0.602

8.38

\pm

2.19

0.410

\pm

0.129

0.002

\pm

0.001

2.267

\pm

1.231

0.50

\pm

0.14

TABLE VI: Comparison with Conventional Motions of Trajectory Quality and Calibration Results in Simulation Experiments

Setting

Time Calibration

Rotation Calibration

Translation Calibration

Motion

Solver

t_{d}

(

\mathrm{ms}

)

R_{x}

(

\mathrm{deg}

)

R_{y}

(

\mathrm{deg}

)

R_{z}

(

\mathrm{deg}

)

p_{x}

(

\mathrm{cm}

)

p_{y}

(

\mathrm{cm}

)

p_{z}

(

\mathrm{cm}

)

A 2 I-Calib++

105.532

\pm

0.777

-124.375

\pm

0.052

1.468

\pm

0.003

-91.738

\pm

0.017

34.611

\pm

0.054

0.878

\pm

0.126

3.378

\pm

0.145

iKalibr

110.534

\pm

0.274

-125.797

\pm

0.054

2.206

\pm

0.030

-92.320

\pm

0.196

35.114

\pm

0.054

1.399

\pm

0.251

4.974

\pm

0.081

w/o NIT

A 2 I-Calib++

105.370

\pm

0.940

-124.102

\pm

0.069

1.577

\pm

0.047

-91.769

\pm

0.023

32.163

\pm

0.619

-2.290

\pm

0.358

0.086

\pm

0.326

iKalibr

106.452

\pm

0.179

-126.286

\pm

0.013

2.355

\pm

0.027

-91.884

\pm

0.039

35.217

\pm

0.338

0.292

\pm

0.368

5.203

\pm

0.347

Walking

A 2 I-Calib++

103.309

\pm

1.906

-124.285

\pm

0.346

1.371

\pm

1.122

-96.528

\pm

0.797

20.304

\pm

1.150

-3.803

\pm

2.758

-4.828

\pm

0.364

TABLE VII: Comparison with Conventional Motions and SOTA Solver of Spatiotemporal Calibration Repeatability (Mean

\pm

STD) for Camera IMU in Real-World Experiments

Solver	Evaluation	SR $\uparrow$	$\|\delta t_{d}\|$ $\downarrow$		$\\|\delta\bm{\theta}\\|$ $\downarrow$		$\\|\delta\mathbf{p}\\|$ $\downarrow$
A 2 I-Calib++	All Trials	9/9	0.19 $\pm$	0.05	0.18 $\pm$	0.06	0.29 $\pm$	0.07
iKalibr	All Trials	6/9	226.13 $\pm$	183.42	93.05 $\pm$	75.82	34.83 $\pm$	28.29
iKalibr	Succeeded Only *	–	2.95 $\pm$	0.41	0.20 $\pm$	0.09	0.27 $\pm$	0.06

TABLE VIII: Comparison of Spatiotemporal Calibration Accuracy (RMSE

\pm

STD) in Simulation Experiments

Solver	Motion	$d_{t}$ (ms)	$d_{R}$ (deg)	$d_{p}$ (cm)
A 2 I-Calib++	w/o NIT	0.16	0.30	5.18
A 2 I-Calib++	Walking	2.22	4.80	17.15
iKalibr	w/o NIT	4.08	0.68	1.14
iKalibr	Walking	6.33	3.17	4.98

TABLE IX: Cross-Motion Consistency of the Estimated Parameters in Real-World Experiments

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DogLegs: Robust Proprioceptive State Estimation for Legged Robots Using Multiple Leg-Mounted IMUs

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Open paper AskPDF arXiv

Robust and accurate proprioceptive state estimation of the main body is crucial for legged robots to execute tasks in extreme environments where exteroceptive sensors, such as LiDARs and cameras, may become unreliable. In this paper, we propose DogLegs, a state estimation system for legged robots that fuses the measurements from a body-mounted inertial measurement unit (Body-IMU), joint encoders, and multiple leg-mounted IMUs (Leg-IMU) using an extended Kalman filter (EKF). The filter system contains the error states of all IMU frames. The Leg-IMUs are used to detect foot contact, thereby providing zero-velocity measurements to update the state of the Leg-IMU frames. Additionally, we compute the relative position constraints between the Body-IMU and Leg-IMUs by the leg kinematics and use them to update the main body state and reduce the error drift of the individual IMU frames. Field experimental results have shown that our proposed DogLegs system achieves better state estimation accuracy compared to the traditional leg odometry method (using only Body-IMU and joint encoders) across various terrains. We make our code and datasets publicly available to benefit the research community (https://github.com/YibinWu/DogLegs).

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