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

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 (11)

• Figure 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.
• Figure 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.
• Figure 3: Estimated trajectory with biased GNSS noise ($\mu = 3$ [m], $\sigma = 0.5$ [m]). The systematic offset is clearly amplified relative to Fig. \ref{['fig:traj_1_5']}, further confirming that the bias is inherited by the smoothed solutions regardless of the smoothing algorithm used.
• Figure 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 $\delta\boldsymbol{x}^{\text{BLENDS}}_{s,k}$.
• Figure 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.

Theorems & Definitions (2)

• Proposition 1
• proof