Bayesian Learning-Enhanced Navigation with Deep Smoothing for Inertial-Aided Navigation

Nadav Cohen; Itzik Klein

cs.ROMar 26, 2026

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.

Abstract

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 ( $\mu=0$ [m], $\sigma=0.5$ [m]). All estimators closely follow the GT, confirming standard smoothing performance under unbiased measurements.

Fig. 2 : Estimated trajectory with biased GNSS noise ( $\mu=1.5$ [m], $\sigma=0.5$ [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 ( $\mu=3$ [m], $\sigma=0.5$ [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 $\mathbf{D}_{f,k}$ , $\mathbf{D}_{b,k}$ and an additive correction vector $\boldsymbol{c}_{k}$ . These are used to augment the Two-Filter Smoother fusion, yielding the final BLENDS smoothed state $\delta\boldsymbol{x}^{\text{BLENDS}}_{s,k}$ .

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 °/ $\sqrt{\text{hr}}$
Accelerometer
In-run bias stability	20 $\mu$ g
Velocity random walk	0.05 m/s/ $\sqrt{\text{hr}}$

TABLE I : Arazim EX-300 IMU Specifications.

Panel of Figure 7 — 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	$\boldsymbol{m}^{\text{wide}}$	$\boldsymbol{m}^{\text{base}}$	$\boldsymbol{m}^{\text{wide}}$	$\boldsymbol{m}^{\text{base}}$
$\delta p_{x,y}$ [rad]	$3\times 10^{-7}$	$2\times 10^{-7}$	$3\times 10^{-7}$	$2\times 10^{-7}$
$\delta p_{z}$ [m]	$50.0$	$1.0$	$50.0$	$1.0$
$\delta v_{x,y,z}$ [m/s]	$2.0$	$0.50$	$15.0$	$5.0$
$\delta\phi_{x,y,z}$ [rad]	$\pi$	$\pi/180$	$\pi$	$\pi/180$
$\delta b_{a_{x,y,z}}$ [m/s 2 ]	$0.5$	$0.2$	$0.5$	$0.1$

TABLE II : Per-state correction bounds for the BLENDS correction term

\boldsymbol{c}_{k}

for the mobile robot and quadrotor datasets. The bound contracts during training from

\boldsymbol{m}^{\text{wide}}

\boldsymbol{m}^{\text{base}}

via the curriculum ramp

\rho

		Position [m]			Velocity [m/s]			Orientation [rad]
Traj.	Method	$p_{N}$	$p_{E}$	$p_{D}$	$v_{N}$	$v_{E}$	$v_{D}$	$\phi$	$\theta$	$\psi$
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	$p_{N}$	$p_{E}$	$p_{D}$	$v_{N}$	$v_{E}$	$v_{D}$	$\phi$	$\theta$	$\psi$
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.

Panel of Figure 11 — 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.

Panel of Figure 12 — 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.

Panel of Figure 13 — Fig. 9 : BLENDS navigation error and $2\sigma$ 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 $\pm 2\sigma$ uncertainty envelope (red dashed), demonstrating that the smoothed error remains consistent with the predicted covariance throughout the trajectory.

Panel of Figure 14 — Fig. 10 : PCI over time for TFS, RTSS, and BLENDS relative to the forward EKF on the mobile robot dataset test trajectories.

Panel of Figure 15 — Fig. 11 : PCI over time for TFS, RTSS, and BLENDS relative to the forward EKF on the quadrotor dataset test trajectories.

Inertial Navigation

Inertial navigation estimates position, velocity, and orientation from motion sensors, often combining them with external measurements to limit drift.

See all 60 papers

BLENDS: Bayesian Learning-Enhanced Deep Smoothing for GNSS-Denied Environments
Jun 21, 2026Nadav Cohen, Itzik KleinBayesian Inference GNSS-Denied Localization
The Hatter Department of Marine Technologies University of Haifa Haifa, Israel
A Hybrid Neural-Assisted Unscented Kalman Filter for Unmanned Ground Vehicle Navigation
Mar 12, 2026Gal Versano, Itzik KleinRobot Navigation Unscented Kalman Filtering
Autonomous Navigation and Sensor Fusion Lab, Hatter Department of Marine Technologies, Charney School of Marine Sciences, University of Haifa, Israel.
PLATO: Preintegration Learning from Accurate Trajectory Observations for Neural Inertial Odometry
Sep 6, 2026Haoying Li, Qihang Liu, Yifan Peng +2Inertial Odometry Robot State Estimation

Jun 21, 2026J/K move · Enter open · S save

BLENDS: Bayesian Learning-Enhanced Deep Smoothing for GNSS-Denied Environments

Nadav Cohen, Itzik Klein

The Hatter Department of Marine Technologies University of Haifa Haifa, Israel

Bayesian Inference GNSS-Denied Localization Inertial Navigation

Open paper AskPDF arXiv

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.

Abstract

Inertial Navigation

Explore similar work

BLENDS: Bayesian Learning-Enhanced Deep Smoothing for GNSS-Denied Environments

A Hybrid Neural-Assisted Unscented Kalman Filter for Unmanned Ground Vehicle Navigation

PLATO: Preintegration Learning from Accurate Trajectory Observations for Neural Inertial Odometry