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.