Bayesian Learning-Enhanced Navigation with Deep Smoothing for Inertial-Aided Navigation
Authors: Nadav Cohen, Itzik Klein
Organizations: N. Cohen and I. Klein are with the Hatter Department of Marine Technologies, Charney School of Marine Sciences, University of Haifa, Israel.
Accurate post-processing navigation is essential for applications such as survey and mapping, where the full measurement history can be exploited to refine past state estimates. Fixed-interval smoothing algorithms represent the theoretically optimal solution under Gaussian assumptions. However, loosely coupled INS/GNSS systems fundamentally inherit the systematic position bias of raw GNSS measurements, leaving a persistent accuracy gap that model-based smoothers cannot resolve. To address this limitation, we propose BLENDS, which integrates Bayesian learning with deep smoothing to enhance navigation performance. BLENDS is a a data-driven post-processing framework that augments the classical two-filter smoother with a transformer-based neural network. It learns to modify the filter covariance matrices and apply an additive correction to the smoothed error-state directly within the Bayesian framework. A novel Bayesian-consistent loss jointly supervises the smoothed mean and covariance, enforcing minimum-variance estimates while maintaining statistical consistency. BLENDS is evaluated on two real-world datasets spanning a mobile robot and a quadrotor. Across all unseen test trajectories, BLENDS achieves horizontal position improvements of up to 63% over the baseline forward EKF.
Figures & tables
Fig. 1 : Estimated trajectory with zero-mean GNSS noise ( μ=0 [m], σ=0.5 [m]). All estimators closely follow the GT, confirming standard smoothing performance under unbiased measurements.
Fig. 2 : Estimated trajectory with biased GNSS noise ( μ=1.5 [m], σ=0.5 [m]). The EKF, RTSS, and TFS solutions all exhibit a systematic offset from the GT, demonstrating that smoothing cannot correct a position bias.
Fig. 3 : Estimated trajectory with biased GNSS noise ( μ=3 [m], σ=0.5 [m]). The systematic offset is clearly amplified relative to Fig. 2 , further confirming that the bias is inherited by the smoothed solutions regardless of the smoothing algorithm used.
Fig. 4 : Overview of the BLENDS architecture. The forward EKF and backward information filter outputs are concatenated and passed through a transformer encoder, which produces covariance modification matrices Df,k , Db,k and an additive correction vector ck . These are used to augment the Two-Filter Smoother fusion, yielding the final BLENDS smoothed state δxs,kBLENDS .
Fig. 5 : Experimental platform used for data collection. The mobile robot is equipped with two sensor suites: the Arazim system, comprising a MEMS IMU and two GNSS antennas mounted 40 cm from the vehicle center on a rigid crossbar, and the Xsens system, comprising a consumer-grade IMU and a single GNSS antenna mounted at the vehicle body. A cellular RTK receiver provides centimeter-level reference trajectories via a network RTK correction service, supported by two cellular antennas and a power supply.
Parameter
Arazim EX-300
Gyroscope
In-run bias stability
2 °/hr
Angle random walk
0.1 °/ hr
Accelerometer
In-run bias stability
20 μ g
Velocity random walk
0.05 m/s/ hr
TABLE I : Arazim EX-300 IMU Specifications.
Fig. 6 : 2D position trajectory comparison across all evaluation trajectories for the mobile robot (top) and quadrotor (bottom) datasets. Each subplot shows the GT alongside the estimated trajectories of EKF, TFS, RTSS, and BLENDS.
Mobile robot
Quadrotor
State
mwide
mbase
mwide
mbase
δpx,y [rad]
3×10−7
2×10−7
3×10−7
2×10−7
δpz [m]
50.0
1.0
50.0
1.0
δvx,y,z [m/s]
2.0
0.50
15.0
5.0
δϕx,y,z [rad]
π
π/180
π
π/180
δbax,y,z [m/s 2 ]
0.5
0.2
0.5
0.1
TABLE II : Per-state correction bounds for the BLENDS correction term ck for the mobile robot and quadrotor datasets. The bound contracts during training from mwide to mbase via the curriculum ramp ρ .
Position [m]
Velocity [m/s]
Orientation [rad]
Traj.
Method
pN
pE
pD
vN
vE
vD
ϕ
θ
ψ
MR-Val (1)
EKF
0.824
2.373
0.953
0.195
0.247
0.139
0.027
0.027
1.795
RTSS
0.824
2.373
0.951
0.146
0.148
0.114
0.017
0.018
1.777
TFS
0.825
2.374
0.951
0.172
0.206
0.137
0.018
0.019
1.786
BLENDS
0.793
1.335
0.958
0.177
0.198
0.137
0.018
0.019
1.786
MR-Tes (2)
EKF
1.182
2.729
1.403
0.172
0.220
0.130
0.039
0.018
2.099
TABLE III : Mobile robot dataset per-axis RMSE for position, velocity, and orientation across validation and test trajectories.
Position [m]
Velocity [m/s]
Orientation [rad]
Traj.
Method
pN
pE
pD
vN
vE
vD
ϕ
θ
ψ
QR-Val (2)
EKF
3.703
1.100
3.819
0.287
0.175
0.275
0.041
0.024
0.408
RTSS
3.704
1.101
3.822
0.283
0.166
0.292
0.032
0.012
0.179
TFS
3.704
1.102
3.822
0.372
0.176
0.369
0.028
0.011
0.183
BLENDS
2.533
0.586
3.801
0.184
0.118
0.276
0.013
0.012
0.426
QR-Tes (12)
EKF
3.423
2.609
2.979
0.205
0.281
0.255
0.037
0.050
0.406
TABLE IV : Quadrotor dataset per-axis RMSE for position, velocity, and orientation across validation and test trajectories.
Fig. 7 : RMSE improvement [%] relative to the forward EKF for TFS, RTSS, and BLENDS across all navigation states on the mobile robot dataset test trajectories. Positive values indicate improvement over the EKF baseline.
Fig. 8 : RMSE improvement [%] relative to the forward EKF for TFS, RTSS, and BLENDS across all navigation states on the quadrotor dataset test trajectories. Positive values indicate improvement over the EKF baseline.
Fig. 9 : BLENDS navigation error and 2σ covariance bounds for a representative test trajectory from the mobile robot (top) and quadrotor (bottom) datasets. Each subplot shows the per-axis error (black) and the corresponding ±2σ uncertainty envelope (red dashed), demonstrating that the smoothed error remains consistent with the predicted covariance throughout the trajectory.
Fig. 10 : PCI over time for TFS, RTSS, and BLENDS relative to the forward EKF on the mobile robot dataset test trajectories.
Fig. 11 : PCI over time for TFS, RTSS, and BLENDS relative to the forward EKF on the quadrotor dataset test trajectories.
Maintaining accurate navigation during GNSS outages remains a significant challenge for autonomous systems relying on low-cost inertial sensors. While classical smoothing methods, such as the two-filter smoother and Rauch-Tung-Striebel smoother, exploit measurements collected before and after an outage, their performance remains limited by the accuracy of conventional GNSS measurements. This paper presents Bayesian learning-enhanced navigation with deep smoothing (BLENDS), a transformer-based framework that augments Bayesian smoothing with learned covariance adaptation and state correction. The proposed method preserves the statistical foundations of Bayesian estimation while leveraging data-driven learning to improve navigation accuracy. Evaluations on the quadrotor dataset with GNSS outages demonstrate that BLENDS consistently outperforms both model-based smoothers, achieving up to 25.6% improvement in the position root mean square error while also reducing estimation uncertainty. Furthermore, BLENDS learns to compensate for the systematic bias between conventional GNSS positioning and RTK ground truth, enabling accuracy beyond that achievable with conventional GNSS measurements alone. The results demonstrate the potential of learning-enhanced Bayesian smoothing for resilient and high-accuracy navigation in GNSS-challenged environments.
Nadav Cohen, Itzik Klein
The Hatter Department of Marine Technologies University of Haifa Haifa, Israel
Modern autonomous navigation for unmanned ground vehicles relies on different estimators to fuse inertial sensors and GNSS measurements. However, the constant noise covariance matrices often struggle to account for dynamic real-world conditions. In this work we propose a hybrid estimation framework that bridges classical state estimation foundations with modern deep learning approaches. Instead of altering the fundamental unscented Kalman filter equations, a dedicated deep neural network is developed to predict the process and measurement noise uncertainty directly from raw inertial and GNSS measurements. We present a sim2real approach, with training performed only on simulative data. In this manner, we offer perfect ground truth data and relieves the burden of extensive data recordings. To evaluate our proposed approach and examine its generalization capabilities, we employed a 160-minutes test set from three datasets each with different types of vehicles (off-road vehicle, passenger car, and mobile robot), inertial sensors, road surface, and environmental conditions. We demonstrate across the three datasets a position improvement of 12.7% compared to the adaptive model-based approach. Thus, offering a scalable and a more robust solution for unmanned ground vehicles navigation tasks.
Gal Versano, Itzik Klein
Autonomous Navigation and Sensor Fusion Lab, Hatter Department of Marine Technologies, Charney School of Marine Sciences, University of Haifa, Israel.
Neural inertial odometry has demonstrated strong potential for motion estimation in challenging environments, yet inertial-only preintegration remains sensitive to IMU bias and uncertainty. To this end, this paper introduces PLATO:Preintegration Learning from Accurate Trajectory Observations, a likelihood-based framework that leverages accurate trajectory observations to jointly learn IMU bias dynamics modeled by a neural ordinary differential equation(NODE) and gyroscope and accelerometer noise covariances. Optimization exploits the sparse structure of the negative log-likelihood, with IMU noise-parameter gradients computed by forward differentiation. A tailored double-adjoint scheme couples a discrete invariant-error adjoint with a continuous-time adjoint for the bias NODE, enabling memory-efficient likelihood optimization over the nested bias-dynamics and IMU-preintegration rollouts. Validation on EuRoC shows improved performance, and underwater robot experiments demonstrate applicability under intermittent lighting failures and visual degradation.