Ensuring accurate and stable state estimation is a challenging task crucial to safety-critical domains such as high-speed autonomous racing, where measurement uncertainty must be both adaptive to the environment and temporally smooth for control. In this work, we develop a learning-based framework, LACE, capable of directly modeling the temporal dynamics of GNSS measurement covariance. We model the covariance evolution as an exponentially stable dynamical system where a deep neural network (DNN) learns to predict the system's process noise from environmental features through an attention mechanism. By using contraction-based stability and systematically imposing spectral constraints, we formally provide guarantees of exponential stability and smoothness for the resulting covariance dynamics. We validate our approach on an AV-24 autonomous racecar, demonstrating improved localization performance and smoother covariance estimates in challenging, GNSS-degraded environments. Our results highlight the promise of dynamically modeling the perceived uncertainty in state estimation problems that are tightly coupled with control sensitivity.
Figures & tables
Fig. 1: LACE applied to IAC racecars. Top : Illustration of the learned GNSS covariance evolution as the racecar passes under a concrete bridge at Laguna Seca Raceway. The structure degrades GNSS measurement quality, causing the covariance to broaden. For clarity, the covariance distribution is visualized in 2D. Bottom : Caltech IAC AV-24 racecar.
Fig. 2: LACE architecture overview. The core model produces a symmetric positive-definite matrix, while during both training and inference, LACE propagates this output through its dynamical model to guarantee stability and enforce temporal smoothness.
Fig. 3: Illustration of the robot’s position x(t) , reparameterized into arc-length coordinates s(t) and processed by a low-temperature attention mechanism to capture localized environmental features.
Fig. 4: Dense LiDAR map of the Laguna Seca Raceway from running offline FGO based SLAM framework with color coded by altitude.
Fig. 5: Train and evaluation loss of our method (LACE) and MLP, yellow stars indicate the lowest part of the evaluation loss. LACE shows a much faster training convergence rate.
Fig. 6: Guaranteed exponential convergence of covariance evolution for LACE, demonstrated by different initial covariances R(0) .
Fig. 7: Covariance model comparison. Left : satellite view of a practice lap at Laguna Seca Raceway, with bridges that potentially degrade GNSS measurements labeled. Top right : traces of the covariance matrices from different models. Our method successfully encloses the residuals while maintaining smoothness. Bottom right : negative log likelihood of evaluated models, LACE overall has the lowest loss suggesting a better estimation of the GNSS uncertainty.
Fig. 8: Effects of different choice of eigenvalues on the predicted covariance. λ∗ denotes the learned value. Lower λ biases the learned progress noise matrix Q(t) more resulting overall more conservative R(t) .
Model
Avg Loss
Std
LACE ( λ∗ )
2.9694
5.3995
LACE ( λ=−0.1 )
5.3642
0.5913
LACE ( λ=−0.2 )
4.4280
1.1541
LACE ( λ=−0.3 )
3.9281
1.7011
LACE ( λ=−0.7 )
3.1205
3.8769
LACE ( λ=−2.0 )
3.2145
11.8982
TABLE I: Average loss across multiple laps. While λ∗ achieves the lowest average loss, a smaller λ (such as λ=−0.1 ) can help robustify the model performance by biasing towards the degradation signals picked up by Q(t) .
Fig. 9: Trajectories estimated from MINS with different covariance models. As the racecar passes under the bridge (illustrated by the gray slabs), the GNSS quality becomes strongly disturbed, as indicated by the spread of measurements. Our method, with an optimal λ , successfully captures the rapid changes in noise distribution, whereas other methods produce overconfident covariance estimates that mislead the estimator.
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
Reliable localization in GNSS-denied environments remains a fundamental challenge for intelligent vehicles, as inertial navigation systems accumulate unbounded drift without external correction. Existing approaches provide drift correction through dedicated infrastructure, expensive external sensors, or complex multi-sensor fusion, each introducing practical deployment barriers. We propose Evidential Velocity Correction using Mamba (EVC-Mamba), a learning-based architecture that transforms onboard vehicle sensor data into a virtual velocity sensor for IMU drift correction without additional hardware. A Mamba-based selective state space model captures the temporal dynamics of vehicle motion, while evidential deep learning with a Normal-Inverse-Gamma distribution provides principled uncertainty quantification. The resulting uncertainty-aware velocity estimate is incorporated as a virtual correction measurement into an Error-State Extended Kalman Filter to reduce position drift. Evaluation on real-world vehicle data demonstrates that inertial navigation using the proposed velocity correction achieves localization accuracy within 10% of a dedicated external velocity sensor across different outage durations. The proposed architecture supports real-time onboard deployment at 40 Hz on edge hardware, enabling reliable localization during prolonged GNSS outages.
Abinav Kalyanasundaram, Karthikeyan Chandra Sekaran, Wolfgang Utschick +1
AImotion Bavaria, Technische Hochschule Ingolstadt, Germany · Technische Universität München, Germany
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.