Radar-Inertial Odometry with Online Spatio-Temporal Calibration via Continuous-Time IMU Modeling

Vlaho-Josip Štironja, Luka Petrović, Juraj Peršić, Ivan Marković, Ivan Petrović

University of Zagreb, Faculty of Electrical Engineering and Computing
Laboratory for Autonomous Systems and Mobile Robotics (LAMOR)

Paper Code arXiv

Temporal misalignment between discrete radar ego-velocity measurements and the continuous-time motion estimated via a uniform cubic B-spline fit to IMU acceleration signals.

Abstract

Radar-Inertial Odometry (RIO) has emerged as a robust alternative to vision- and LiDAR-based odometry in challenging conditions such as low light, fog, featureless environments, or in adverse weather. However, many existing RIO approaches assume known radar-IMU extrinsic calibration or rely on sufficient motion excitation for online extrinsic estimation, while temporal misalignment between sensors is often neglected or treated independently. In this work, we present a RIO framework that performs joint online spatial and temporal calibration within a factor-graph optimization formulation, based on continuous-time modeling of inertial measurements using uniform cubic B-splines. The proposed continuous-time representation of acceleration and angular velocity accurately captures the asynchronous nature of radar-IMU measurements, enabling reliable convergence of both the temporal offset and extrinsic calibration parameters, without relying on scan matching, target tracking, or environment-specific assumptions.

Methodology

Proposed factor graph structure. The graph comprises an IMU factor (f_IMU), a radar ego-velocity factor (f_R), a constant time-offset factor (f_CT), and a constant extrinsic calibration factor (f_CE).

The proposed method, LC-RIO-ET (Loosely Coupled RIO with online Extrinsic and Temporal calibration), is formulated within a factor graph optimization (FGO) framework. LC-RIO-ET jointly estimates the temporal offset and extrinsic parameters between radar and inertial measurements in an online manner, without relying on environmental assumptions such as ground-plane or Manhattan-world constraints. The key contribution is the use of uniform cubic B-splines to model inertial signals in continuous time, enabling acceleration and angular velocity estimation at arbitrary timestamps with C²-continuous derivatives. Given a window of discrete IMU measurements, independent B-splines are fitted to the raw acceleration and angular velocity signals. This enables reliable convergence of the temporal offset even under large initialization errors, and simultaneous online convergence of the extrinsic calibration parameters. Velocity increments over the temporal offset interval are computed via 5-point Gauss-Legendre quadrature, and a Huber loss is applied on the radar ego-velocity factor for robustness to outliers. The system is implemented using GTSAM with iSAM2 within a one-second sliding window, using a two-thread architecture separating optimization and navigation.

Results

Temporal offset convergence

Convergence of the temporal offset on Sequence 5 from the EKF-RIO-TC dataset, initialized from six different values (0, 25, 50, 75, 100, and 125 ms). In all cases the estimate converges robustly to a consistent steady-state of approximately 113.3 ms, demonstrating that the B-spline-based radar factor provides a well-conditioned gradient signal across a wide initialization range.

Comparison of temporal offset trajectories of LC-RIO-ET, EKF-RIO-TC, and RIO-T on Sequence 5. EKF-RIO-TC converges to 115.5 ms, RIO-T to 118.0 ms after several iterations, and LC-RIO-ET to 113.6 ms. Due to the locally constant acceleration assumption, RIO-T requires multiple iterations to resolve large temporal offsets, whereas LC-RIO-ET achieves online convergence directly within the FGO. A key distinction is that LC-RIO-ET simultaneously converges the extrinsic calibration parameters alongside the temporal offset, whereas EKF-RIO-TC and RIO-T assume fixed extrinsics throughout.

Odometry accuracy — EKF-RIO-TC dataset (unsynchronized)

LC-RIO-ET achieves the best mean performance across all four metrics, with a mean translational RPE of 0.138 m and rotational RPE of 2.377°, representing improvements of 18.3% and 4.5% over EKF-RIO-TC. Relative to the plain FGO baseline LC-RIO, the translational RPE improves by 33.0% and the translational APE by 43.9%, confirming that jointly estimating both calibration parameters is substantially more beneficial than either alone. The improvements are most pronounced in Sequences 4 and 5, which feature the richest rotational motion. In Sequence 5, LC-RIO-ET reduces translational RPE by 49.1% relative to LC-RIO.

Seq.	Method	APE Trans. (m)	APE Rot. (°)	RPE Trans. (m)	RPE Rot. (°)
1	EKF-RIO	0.583	2.997	0.354	2.707
	EKF-RIO-TC	0.333	1.907	0.175	2.412
	RIO-T	0.435	2.102	0.183	2.618
	LC-RIO	0.411	1.915	0.196	2.992
	LC-RIO-E	0.586	1.794	0.224	3.081
	LC-RIO-T	0.380	1.614	0.171	2.181
	LC-RIO-ET	0.324	1.570	0.161	2.131
2	EKF-RIO	0.870	6.150	0.306	2.684
	EKF-RIO-TC	0.372	2.323	0.153	2.582
	RIO-T	0.412	2.046	0.175	2.238
	LC-RIO	0.379	1.996	0.173	1.890
	LC-RIO-E	0.376	2.078	0.171	1.997
	LC-RIO-T	0.278	1.967	0.143	2.324
	LC-RIO-ET	0.219	1.992	0.140	2.284
3	EKF-RIO	0.772	3.619	0.314	2.299
	EKF-RIO-TC	0.216	2.291	0.131	1.876
	RIO-T	0.720	3.915	0.205	2.226
	LC-RIO	0.657	3.769	0.199	2.033
	LC-RIO-E	0.719	4.885	0.209	2.037
	LC-RIO-T	0.206	2.003	0.131	1.977
	LC-RIO-ET	0.219	1.985	0.127	1.861
4	EKF-RIO	0.827	16.523	0.316	4.844
	EKF-RIO-TC	0.264	2.730	0.167	2.991
	RIO-T	0.268	3.764	0.221	3.214
	LC-RIO	0.247	3.362	0.176	3.156
	LC-RIO-E	0.329	3.281	0.138	3.279
	LC-RIO-T	0.320	3.253	0.183	3.002
	LC-RIO-ET	0.304	3.074	0.116	3.114
5	EKF-RIO	1.311	6.122	0.386	3.391
	EKF-RIO-TC	0.570	2.601	0.221	2.591
	RIO-T	0.614	2.958	0.264	3.337
	LC-RIO	0.713	2.812	0.287	3.584
	LC-RIO-E	0.378	1.948	0.174	3.061
	LC-RIO-T	0.620	2.500	0.222	2.548
	LC-RIO-ET	0.284	1.693	0.146	2.494
Mean	EKF-RIO	0.873	7.082	0.335	3.185
	EKF-RIO-TC	0.351	2.370	0.169	2.490
	RIO-T	0.490	2.957	0.210	2.726
	LC-RIO	0.481	2.771	0.206	2.731
	LC-RIO-E	0.477	2.797	0.183	2.691
	LC-RIO-T	0.361	2.267	0.170	2.406
	LC-RIO-ET	0.270	2.063	0.138	2.377

Quantitative comparison on the EKF-RIO-TC dataset. APE evaluated under SE(3) alignment, RPE computed over 10 m intervals.

Odometry accuracy — ICINS dataset (hardware-synchronized)

On the hardware-synchronized ICINS dataset (IWR6843AOP radar + ADIS16448 IMU, VI-SLAM pseudo ground truth), the estimated temporal offset converges to approximately −10 ms. LC-RIO-ET performs on par with LC-RIO, confirming the method does not degrade performance in synchronized settings. Notably, LC-RIO-ET achieves the lowest mean rotational APE (4.864°), suggesting a residual benefit of the continuous-time inertial model even when temporal offsets are small.

Sequence	Method	APE Trans. (m)	APE Rot. (°)	RPE Trans. (m)	RPE Rot. (°)
carried 1	EKF-RIO	0.995	5.553	0.134	0.857
	EKF-RIO-TC	0.922	5.102	0.139	0.798
	RIO-T	0.754	4.098	0.126	0.641
	LC-RIO	0.745	4.037	0.128	0.655
	LC-RIO-E	0.863	5.228	0.141	0.687
	LC-RIO-T	0.763	4.053	0.128	0.665
	LC-RIO-ET	0.768	4.038	0.126	0.666
carried 2	EKF-RIO	1.761	10.743	0.150	1.083
	EKF-RIO-TC	1.594	9.613	0.152	0.992
	RIO-T	1.519	9.258	0.126	0.841
	LC-RIO	1.513	9.094	0.126	0.845
	LC-RIO-E	2.091	12.385	0.182	0.886
	LC-RIO-T	1.526	9.095	0.128	0.863
	LC-RIO-ET	1.518	9.071	0.128	0.871
flight 1	EKF-RIO	0.736	4.944	0.213	1.401
	EKF-RIO-TC	0.764	5.274	0.219	1.493
	RIO-T	0.699	4.725	0.189	1.428
	LC-RIO	0.699	4.783	0.191	1.424
	LC-RIO-E	0.698	4.713	0.180	1.424
	LC-RIO-T	0.695	4.727	0.189	1.419
	LC-RIO-ET	0.698	4.685	0.190	1.423
flight 2	EKF-RIO	0.100	1.938	0.176	0.888
	EKF-RIO-TC	0.098	1.815	0.161	0.914
	RIO-T	0.098	1.727	0.153	0.817
	LC-RIO	0.101	1.759	0.157	0.836
	LC-RIO-E	0.102	1.725	0.160	0.796
	LC-RIO-T	0.099	1.720	0.155	0.792
	LC-RIO-ET	0.096	1.663	0.160	0.805
Mean	EKF-RIO	0.898	5.795	0.168	1.057
	EKF-RIO-TC	0.845	5.451	0.168	1.049
	RIO-T	0.768	4.952	0.149	0.932
	LC-RIO	0.764	4.918	0.151	0.940
	LC-RIO-E	0.939	6.013	0.166	0.948
	LC-RIO-T	0.771	4.899	0.150	0.935
	LC-RIO-ET	0.770	4.864	0.151	0.941

Quantitative comparison on the ICINS dataset. APE evaluated under SE(3) alignment, RPE computed over 10 m intervals.

Impact of spatio-temporal calibration on baseline estimators

Trajectory visualization of EKF-RIO with original and LC-RIO-ET estimated parameters on Sequence 5 from the EKF-RIO-TC dataset.

To isolate the value of the calibrated parameters independently of the FGO formulation, we re-initialize EKF-RIO and LC-RIO with the extrinsic and temporal offset parameters estimated by LC-RIO-ET and re-run both estimators with these fixed values. On Sequence 5, calibrated parameters reduce translational RPE of EKF-RIO by 68.4% and of LC-RIO by 51.2%, with translational APE reductions of 80.5% and 58.9% respectively. Even on the synchronized flight 2 sequence, re-injecting calibrated parameters consistently reduces rotational errors and translational RPE. This confirms that the value of spatio-temporal calibration extends beyond the online estimation window and directly benefits simpler downstream estimators.

Acknowledgment

This research has been supported by the European Regional Development Fund under grant agreement PK.1.1.10.0007 (DATACROSS) and the Croatian Science Foundation under grant agreement DOK-NPOO-2023-10-6705.