Gaussian Process Regression for Improved Underwater Navigation

Accurate underwater navigation is a challenging task due to the absence of global navigation satellite system signals and the reliance on inertial navigation systems that suffer from drift over time. Doppler velocity logs (DVLs) are typically used to mitigate this drift through velocity measurements, which are commonly estimated using a parameter estimation approach such as least squares (LS). However, LS works under the assumption of ideal conditions and does not account for sensor biases, leading to suboptimal performance. This paper proposes a data-driven alternative based on multi-output Gaussian process regression (MOGPR) to improve DVL velocity estimation. MOGPR provides velocity estimates and associated measurement covariances, enabling an adaptive integration within an error-state Extended Kalman Filter (EKF). We evaluate our proposed approach using real-world AUV data and compare it against LS and a state-of-the-art deep learning model, BeamsNet. Results demonstrate that MOGPR reduces velocity estimation errors by approximately 20% while simultaneously enhancing overall navigation accuracy, particularly in the orientation states. Additionally, the incorporation of uncertainty estimates from MOGPR enables an adaptive EKF framework, improving navigation robustness in dynamic underwater environments.

💡 Research Summary

The paper addresses the persistent challenge of accurate underwater navigation for autonomous underwater vehicles (AUVs) where Global Navigation Satellite System (GNSS) signals are unavailable. Conventional navigation relies on an inertial navigation system (INS) that suffers from drift over time, and a Doppler Velocity Log (DVL) that provides low‑rate but accurate velocity measurements relative to the seabed. The standard practice for extracting the vehicle’s 3‑D velocity from raw DVL beam data is a least‑squares (LS) estimator, which assumes unbiased Gaussian noise and perfect sensor calibration. In real oceanic conditions, however, DVL beams exhibit residual biases, scale factor errors, and non‑Gaussian disturbances, causing LS to become sub‑optimal and leading to degraded navigation performance.

To overcome these limitations, the authors propose a data‑driven alternative: Multi‑Output Gaussian Process Regression (MOGPR). MOGPR is a non‑parametric Bayesian regression technique that simultaneously models multiple correlated outputs—in this case, the three velocity components—while providing a full predictive covariance matrix for each estimate. The training set pairs raw beam velocity vectors (˜v_beam) with ground‑truth DVL velocities (v_DVL) obtained after the conventional pseudo‑inverse solution. The authors employ a composite kernel consisting of three Automatic Relevance Determination (ARD) components: squared‑exponential, Matérn 3/2, and rational‑quadratic. Each ARD kernel learns an individual length‑scale for every input dimension, effectively performing feature selection and handling the heterogeneous sensitivities of the four DVL beams.

Model parameters (signal variance, length‑scales, and noise variance) are optimized by maximizing the log‑marginal likelihood using the ADAM optimizer (learning rate 0.1, β₁ = 0.9, β₂ = 0.999) over 50 iterations. Once trained, the MOGPR yields for any new beam measurement a mean velocity estimate μ* and an associated covariance Σ*. Crucially, Σ* quantifies the measurement uncertainty and is fed directly into the measurement noise covariance Rₖ of a loosely‑coupled error‑state Extended Kalman Filter (EKF). The EKF prediction step integrates high‑rate INS data, while the update step incorporates the DVL measurement weighted by the Kalman gain Kₖ = Pₖ⁻ Hᵀ (H Pₖ⁻ Hᵀ + Rₖ)⁻¹. By adapting Rₖ in real time, the filter automatically reduces reliance on DVL data when uncertainty is high (e.g., during beam outages or strong currents) and trusts the measurement when confidence is high, thereby achieving an adaptive sensor‑fusion scheme.

The authors evaluate the approach on real‑world AUV missions covering diverse depths, seabed types, and flow conditions. Three pipelines are compared: (1) LS‑based DVL velocity feeding the EKF, (2) BeamsNet—a one‑dimensional convolutional neural network previously proposed by the same group, and (3) the proposed MOGPR‑EKF combination. Performance metrics include root‑mean‑square (RMS) velocity error and orientation (roll, pitch, yaw) error. Results show that MOGPR reduces velocity RMS error by roughly 20 % relative to LS and by about 10 % relative to BeamsNet. Orientation errors improve by more than 15 % on average. Moreover, the adaptive EKF using MOGPR‑derived covariances maintains stable navigation during periods of beam degradation, where LS and BeamsNet either over‑trust noisy measurements or lack uncertainty estimates entirely.

The paper’s contributions are threefold: (i) introduction of a multi‑output Gaussian Process framework tailored to DVL beam data, effectively learning and compensating for sensor biases and scale errors; (ii) seamless integration of learned measurement covariances into an EKF, enabling an uncertainty‑aware, adaptive fusion architecture; and (iii) empirical validation on extensive field data demonstrating superiority over both classical LS and state‑of‑the‑art deep‑learning baselines. The authors suggest future work on sparse or variational GP approximations for real‑time deployment, exploration of non‑diagonal output‑covariance structures to capture inter‑axis correlations, and hybrid models that combine physics‑based flow dynamics with data‑driven learning for even more robust underwater navigation.