Organizations: Shanghai Key Laboratory of Navigation and Location Based Services, School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China
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
td
R
p
bg
ba
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.
Figure 4
Fig. 4: Unitree Go2 Platform with Noitom PN Studio suite, Intel RealSense D435i and Livox MID-360.
Duration
A 2 I-Calib++
iKalibr
Search
SVD
LS
Others
Total
Total
3.200
0.036
0.005
0.004
2.000
2.045±0.068
2.941±2.105
6.400
0.043
0.005
0.006
2.030
2.084±0.070
3.555±1.123
TABLE III: Comparison of Algorithm Runtime
Motion
Slip
Contact
VIMU Error *
Condition
Track.
( m/s )
( % )
∥Δω∥
∥Δv∥
κR/κt
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 ± STD)
Calibration Accuracy (RMSE ± STD)
Motion
Trace Corr.
Rot. Cond.
Trans. Cond.
Time Offset
Extri. Rot.
Gyro. Bias
Extri. Trans.
Accel. Bias
Style
rmax ( % ) ↑
κR↓
κt↓
∣δtd∣ ( ms ) ↓
∥δθ∥ ( deg ) ↓
∥δbg∥ ( rad/s ) ↓
∥δp∥ ( cm ) ↓
∥δba∥ ( cm/s 2 ) ↓
A 2 I-Calib++
99.9 ±
0.0
1.46 ±
0.01
1.640 ±
0.006
0.19 ±
0.05
0.177 ±
0.062
0.002 ±
0.001
0.287 ±
0.066
0.66 ±
0.26
w/o NIT
99.3 ±
0.1
280.64 ±
3.73
9.862 ±
0.066
0.38 ±
0.15
0.675 ±
0.266
0.001 ±
0.001
3.390 ±
0.342
0.54 ±
0.20
w/o AMP
99.9 ±
0.0
1.19 ±
0.01
2.033 ±
0.004
0.24 ±
0.05
0.191 ±
0.065
0.001 ±
0.001
0.335 ±
0.044
0.47 ±
0.19
Sinusoid
97.3 ±
0.1
349.89 ±
3.07
29.492 ±
0.602
8.38 ±
2.19
0.410 ±
0.129
0.002 ±
0.001
2.267 ±
1.231
0.50 ±
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
td ( ms )
Rx ( deg )
Ry ( deg )
Rz ( deg )
px ( cm )
py ( cm )
pz ( cm )
A 2 I-Calib++
A 2 I-Calib++
105.532 ±
0.777
-124.375 ±
0.052
1.468 ±
0.003
-91.738 ±
0.017
34.611 ±
0.054
0.878 ±
0.126
3.378 ±
0.145
iKalibr
110.534 ±
0.274
-125.797 ±
0.054
2.206 ±
0.030
-92.320 ±
0.196
35.114 ±
0.054
1.399 ±
0.251
4.974 ±
0.081
w/o NIT
A 2 I-Calib++
105.370 ±
0.940
-124.102 ±
0.069
1.577 ±
0.047
-91.769 ±
0.023
32.163 ±
0.619
-2.290 ±
0.358
0.086 ±
0.326
iKalibr
106.452 ±
0.179
-126.286 ±
0.013
2.355 ±
0.027
-91.884 ±
0.039
35.217 ±
0.338
0.292 ±
0.368
5.203 ±
0.347
Walking
A 2 I-Calib++
103.309 ±
1.906
-124.285 ±
0.346
1.371 ±
1.122
-96.528 ±
0.797
20.304 ±
1.150
-3.803 ±
2.758
-4.828 ±
0.364
TABLE VII: Comparison with Conventional Motions and SOTA Solver of Spatiotemporal Calibration Repeatability (Mean ± STD) for Camera IMU in Real-World Experiments
Solver
Evaluation
SR ↑
∣δtd∣↓
∥δθ∥↓
∥δp∥↓
A 2 I-Calib++
All Trials
9/9
0.19 ±
0.05
0.18 ±
0.06
0.29 ±
0.07
iKalibr
All Trials
6/9
226.13 ±
183.42
93.05 ±
75.82
34.83 ±
28.29
Succeeded Only *
–
2.95 ±
0.41
0.20 ±
0.09
0.27 ±
0.06
TABLE VIII: Comparison of Spatiotemporal Calibration Accuracy (RMSE ± STD) in Simulation Experiments
Solver
Motion
dt (ms)
dR (deg)
dp (cm)
A 2 I-Calib++
w/o NIT
0.16
0.30
5.18
Walking
2.22
4.80
17.15
iKalibr
w/o NIT
4.08
0.68
1.14
Walking
6.33
3.17
4.98
TABLE IX: Cross-Motion Consistency of the Estimated Parameters in Real-World Experiments
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).
Yibin Wu, Jian Kuang, Shahram Khorshidi +4
Center for Robotics, University of Bonn, Germany · GNSS Research Center, Wuhan University, China · Humanoid Robots Lab, University of Bonn, Germany +1
Robust proprioceptive state estimation is crucial for legged robots, particularly in challenging environments where exteroceptive sensors are unreliable. While traditional leg odometry typically relies on a single body-mounted inertial measurement unit (IMU) fused with leg kinematics via an extended Kalman filter (EKF), such methods are vulnerable to linearization errors and hard-contact assumptions, leading to severe drift during slippage or dynamic locomotion.In this paper, we propose LAG-IO, a factor graph-based multi-IMU state estimation framework that tightly couples measurements from a main body IMU, multiple leg-mounted IMUs, and joint encoders within a sliding-window optimization. To handle non-stationary foot-ground interactions, we introduce a continuous probabilistic contact detection module based on kernel density estimation (KDE), which adaptively scales the covariance of zero-velocity update (ZUPT) factors. Furthermore, kinematic relative position factors are formulated to structurally anchor the independent leg inertial states to the robot's main body. Experimental results on a quadruped robot across various unstructured terrains demonstrate that, compared to existing baselines, LAG-IO achieves higher planar trajectory accuracy, reduces unobservable vertical divergence, and provides smoother trajectory transitions during unstable contacts, while maintaining competitive attitude observability.
Reliable odometry for legged robots without cameras or LiDAR remains challenging due to IMU drift and noisy joint velocity sensing. This paper presents a purely proprioceptive state estimator that uses only IMU and motor measurements to estimate body pose and velocity, with a unified formulation applicable to quadruped and wheel-legged robots and extensible to other legged morphologies. The key idea is to treat each reliable contact as a kinematic anchor: joint-torque--based foot wrench estimation selects stance contacts, and the corresponding footfall records provide intermittent world-frame constraints that suppress long-term drift. To prevent elevation drift during extended traversal, we introduce a lightweight height clustering and time-decay correction that snaps newly recorded footfall heights to previously observed support planes. For wheel-legged platforms, the recorded contact is further propagated by effective wheel rolling displacement with shank-motion compensation and a slope-aware rolling direction. To improve foot velocity observations under encoder quantization, we retain an inverse-kinematics cubature Kalman filter as an optional velocity-enhancement module that filters foot-end velocities from joint angles and velocities. The implementation further mitigates yaw drift through multi-contact geometric consistency, which is injected as a soft heading prior rather than as a hard reset of the attitude state. The method is evaluated on four quadruped platforms.
Minxing Sun, Yao Mao
Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, China · Institute for Infocomm Research (I2R), Agency for Science, Technology and Research, Singapore. · Shenzhen Astralldynamics Technology.