Attribution and Uncertainty Behavior of Learned Residual Gyro Correction for Gyro-Stellar Estimation

AI summary

Overview

Research area: Machine learning for spacecraft guidance, navigation and control — specifically uncertainty quantification (UQ) and explainable AI (XAI) applied to a learned component inside a hybrid gyro-stellar attitude estimation pipeline.

Technical level: Advanced. The paper assumes familiarity with Kalman-style state estimation, Gaussian negative log-likelihood training, deep ensembles, and gradient-based attribution methods.

Scope: A systematic study of how a 1-D CNN that predicts residual gyroscope bias corrections, together with its aleatoric and ensemble-based epistemic uncertainty outputs, behaves under nominal conditions and under structured, temporally correlated sensor perturbations.

What This Paper Is About

Spacecraft attitude estimation often relies on a gyroscope (high-rate but drifting) fused with star trackers (drift-free but slow and noisy). Deep learning modules are increasingly inserted into these pipelines to fix residual gyro bias, but their internal behavior and their uncertainty estimates are poorly understood outside the conditions they were trained on. This paper trains a CNN to predict residual gyro corrections plus a per-axis uncertainty estimate, feeds the corrections into a flight-representative Gyro-Stellar Estimator (GSE), and then uses gradient-based attribution and uncertainty decomposition to explain what drives both the correction and the uncertainty when the sensors degrade.

Key Contributions

1. Attribution of a heteroscedastic output. The authors analyze input attributions for both the predicted mean correction μ and the aleatoric uncertainty log σ², under nominal and perturbed conditions — prior work is described as focusing mostly on deterministic outputs.
2. Attribution of uncertainty. They examine how uncertainty behaves with respect to calibration, entropy, and epistemic separation under out-of-distribution (OOD) response.
3. Structured perturbation analysis. Rather than i.i.d. noise only, they analyze multiple forms of temporally structured perturbations (autoregressive noise, periodic vibration, bias step and drift, star tracker dropout) intended to be realistic to sensor degradation.
4. Joint interpretation of uncertainty behavior. They provide a joint reading of aleatoric calibration and epistemic separation, highlighting attribution stability despite OOD conditions and increasing uncertainty without feature redistribution.

Main Findings

• Correction performance: On residual magnitude statistics in deg/s, the CNN reaches a mean of 0.000622 versus 0.00661 for raw measurements and 0.00341 for a naive mean-bias-removal baseline — a 90.6% improvement over raw and 81.8% over naive. RMSE is 0.000844 versus 0.00733 (raw, 88.5% improvement) and 0.00495 (naive, 82.9%); median is 0.000462 versus 0.00591 (92.2%) and 0.00220 (79%); the 95th percentile is 0.00166 versus 0.0121 (86.2%) and 0.0103 (83.8%).
• Aleatoric calibration under nominal conditions: The selected model achieves 1σ coverage of 63.81% and 2σ coverage of 94.75%, against nominal Gaussian expectations of approximately 68% and 95%, with a mean test entropy of −0.5407.
• Aleatoric uncertainty rises but does not separate cleanly: Predicted entropy and variance both increase with perturbation intensity, but the distributions across regimes overlap, and calibration is not consistent across regimes. Standardized residuals (z-scores) are wide under nominal conditions (under-dispersion) and narrow under OOD conditions (over-dispersion), meaning the model becomes increasingly conservative.
• Epistemic uncertainty separates regimes: Ensemble variance (M = 5 independently trained models) stays low and tightly clustered under nominal conditions and shifts progressively higher with perturbation strength, with very small overlap between regimes compared to aleatoric uncertainty. The authors conclude epistemic uncertainty is better at distinguishing nominal from perturbed operating conditions.
• Attribution is dominated by the gyroscope: Channel attribution is dominated by gyro inputs in all regimes, most strongly the y-axis gyro, for both μ and log σ²; star tracker channels contribute negligibly. This pattern remains stable from nominal to increasingly perturbed conditions, with only moderate redistribution across gyro axes.
• Temporal attribution differs by axis: Attribution for the x-axis prediction is spread broadly across the 1 s window, the y-axis prediction is dominated by a single timestep near the middle of the window, and the z-axis shows intermediate behavior with multiple contributing timesteps and a clear peak.
• Uncertainty uses the same inputs, slightly more diffusely: Temporal attribution for the aleatoric head mirrors the mean prediction but is less sharply focused, spreading importance over nearby timesteps. No substantial shift in temporal attribution is observed between in-distribution (ID) and OOD regimes; the model preserves its temporal dependencies while uncertainty increases.

Methodology in Plain English

The authors keep the existing estimator untouched and bolt a learned correction on the outside. A 1-D convolutional network takes a time window of gyroscope readings and angular rates derived from two star trackers, and predicts the residual gyro bias to subtract from the gyro signal before the Gyro-Stellar Estimator runs. The network has two outputs per axis: a mean correction μ and a log-variance log σ² that represents aleatoric (input-dependent) noise. It is trained with a Gaussian negative log-likelihood loss plus a weighted MSE term: L = ½[(y − μ)²e^(−s) + s] + α·MSE(y, μ), where s = log σ².

To get epistemic (model) uncertainty, they train an ensemble of five independently initialized models and use the variance of the ensemble members' mean predictions at each timestep. Uncertainty quality is judged three ways: empirical coverage at 1σ and 2σ against the 68-95-99.7 normal rule, predictive entropy H = 0.5·log(2πeσ²) as a measure of sharpness, and standardized residuals z = (y − μ)/σ, which should look standard-normal if calibrated.

The data comes from a high-fidelity mission simulator at 32 Hz, with an onboard computer running at 8 Hz; gyro and star tracker signals at 16 Hz are downsampled to 8 Hz to match. The dataset has 100 logs of roughly 3.24 hours each, split 40-30-30 into training, validation and test. Hyperparameters are tuned with BOHB (Bayesian Optimization with HyperBand) over 150 epochs using validation NLL, with the ensemble using 5 different seeds.

Perturbations are synthesized by altering the statistical properties of the nominal signals while keeping the underlying dynamics: an AR(1) process for temporally correlated additive noise plus sinusoidal periodic vibration, slow bias step and drift applied only to the gyroscope, and dropout with a hold-last-value strategy applied only to star trackers. These come in three escalating regimes — OOD-L, OOD-M and OOD-H — with parameters listed in the paper's Table 1 (for example, AR noise strength 0.05 / 0.10 / 0.20, vibration scale 0.03 / 0.06 / 0.10, frequency range 0.5–3.8 Hz, dropout duration 2 s / 5 s / 15 s). The perturbation data is never seen in training.

Finally, integrated gradients are used to attribute each output (μ and log σ²) per axis (x, y, z) back to input channels and timesteps. Attributions are summed over time for channel importance and over channels for temporal importance, then averaged across samples separately by axis and regime.

Why This Matters

Research impact: The paper argues that calibration metrics such as coverage and likelihood measure predictive reliability but do not explain which inputs or model behaviors drive uncertainty. By combining attribution with aleatoric/epistemic decomposition in a hybrid deep-filter, it extends explainable-AI-for-time-series work into uncertainty-aware state estimation pipelines, and it moves beyond i.i.d. noise to temporally correlated perturbations characteristic of sensor degradation. It also shows that attribution structure can stay stable while uncertainty grows — a distinction relevant to anyone interpreting learned components in safety-critical loops.

Real-world applications:

• Spacecraft attitude determination and control, where gyro bias drift degrades pointing accuracy over a mission.
• Onboard fault detection and health monitoring, using epistemic uncertainty spikes as an indicator that sensors have left their nominal regime.
• Sensor degradation monitoring (gyro drift, star tracker tracking loss) in long-duration missions or in-orbit operations.
• Design and validation of hybrid learning-plus-classical estimator architectures for other domains where a model-based filter is already certified and cannot easily be replaced.

Industry relevance: The work is a collaboration between the German Research Center for AI (DFKI GmbH) and Airbus Defence and Space GmbH, situating it directly in the spacecraft GNC supply chain. The design choice to correct the gyro externally, rather than modify the estimator internals, preserves existing, qualified filter structures — which matters for deployment and certification in flight software.

Future Directions

• Whether the near-absence of star tracker contribution to attribution reflects a genuine property of the task or an artifact of how the input streams are homogenized into angular rates.
• Whether the conservatism observed under OOD conditions (over-dispersion of z-scores) can be corrected, since aleatoric calibration was not consistent across regimes.
• Whether attribution stability holds for perturbation types or distribution shifts beyond the ones tested; the authors note their perturbations do not exhaust the space of possible shifts and that results characterize behavior under controlled shifts rather than demonstrating general OOD detection capability.
• Turning the epistemic separation signal into an operational monitor for downstream fault detection, as the paper's stated motivation.

Target Audience

Researchers and engineers working on hybrid or learning-augmented state estimation, uncertainty quantification, and explainable AI for time series — particularly those in spacecraft GNC and safety-critical robotics. The paper is also useful to practitioners who need to know which uncertainty component to trust when deciding whether an onboard learned module is operating inside its training distribution.

Authors’ abstract

This work investigates uncertainty decomposition and explainability in a deep learning-based framework for gyroscope bias correction. A 1-D Convolutional Neural Network is trained to predict residual angular rate corrections from multi-sensor inputs, including gyroscope and star tracker measurements. The bias corrections are sent to a flight-representative Gyro-Stellar Estimator. The network produces both mean corrections and input-dependent (heteroscedastic) aleatoric uncertainty, while epistemic uncertainty is estimated via an ensemble of independently trained models. The proposed approach is trained under nominal conditions and evaluated in both nominal and structured perturbations that include additive and temporally correlated noise. Gradient-based attribution methods are applied to both the correction and uncertainty outputs, enabling a decomposition of the evidence that drives state updates and uncertainty estimates. By aggregating attribution patterns across rotational axes and regimes, we reveal axis-specific behaviors and characterize how structured perturbations influence the collaboration between aleatoric and epistemic uncertainty. Uncertainty analysis shows that aleatoric uncertainty increases with perturbation intensity, but the distributions overlap and the calibration is not consistent across regimes. On the other hand, epistemic uncertainty gives a clear signal that gets clearer as the distributional shift happens, showing that the models disagree more. These results show that aleatoric and epistemic uncertainty work well together and that epistemic uncertainty is better at distinguishing between nominal and perturbed operating conditions. The results provide insight into the behavior of hybrid learning-based state estimation components and motivate the use of uncertainty for downstream monitoring and fault detection.