An Asynchronous Two-Speed Kalman Filter for Real-Time UUV Cooperative Navigation Under Acoustic Delays

Abstract: In GNSS-denied underwater environments, individual unmanned underwater vehicles (UUVs) suffer from unbounded dead-reckoning drift, making collaborative navigation crucial for accurate state estimation. However, the severe communication delay inherent in underwater acoustic channels poses serious challenges to real-time state estimation. Traditional filters, such as Extended Kalman Filters (EKF) or Unscented Kalman Filters (UKF), usually block the main control loop while waiting for delayed data, or completely discard Out-of-Sequence Measurements (OOSM), resulting in serious drift. To address this, we propose an Asynchronous Two-Speed Kalman Filter (TSKF) enhanced by a novel projection mechanism, which we term Variational History Distillation (VHD). The proposed architecture decouples the estimation process into two parallel threads: a fast-rate thread that utilizes Gaussian Process (GP) compensated dead reckoning to guarantee high-frequency real-time control, and a slow-rate thread dedicated to processing asynchronously delayed collaborative information. By introducing a finite-length State Buffer, the algorithm applies delayed measurements (t-T) to their corresponding historical states, and utilizes a VHD-based projection to fast-forward the correction to the current time without computationally heavy recalculations. Simulation results demonstrate that the proposed TSKF maintains trajectory Root Mean Square Error (RMSE) comparable to computationally intensive batch-optimization methods under severe delays (up to 30 s). Executing in sub-millisecond time, it significantly outperforms standard EKF/UKF. The results demonstrate an effective control, communication, and computing (3C) co-design that significantly enhances the resilience of autonomous marine automation systems.

• The paper introduces an asynchronous two-speed Kalman Filter with variational history distillation to decouple real-time state propagation from delayed acoustic updates.
• It employs a fast thread for high-rate inertial sensor data and a slow thread for asynchronous acoustic measurements, ensuring robust performance in GNSS-denied settings.
• Quantitative results demonstrate low position RMSE and linear computational scaling, outperforming conventional EKF and FGO approaches under variable delay conditions.

An Asynchronous Two-Speed Kalman Filter for Real-Time UUV Cooperative Navigation Under Acoustic Delays

Background and Motivation

The paper addresses the fundamental challenge of achieving robust, real-time state estimation in swarms of unmanned underwater vehicles (UUVs) operating in Global Navigation Satellite System (GNSS)-denied environments. Accurate localization of UUVs is a prerequisite for marine exploration, inspection, and defense operations. Collaborative navigation (CN) using acoustic communication is mandatory, but the slow propagation and variability of underwater acoustics introduce significant, highly variable delays (5–30 s) and intermittent packet loss. Standard estimation frameworks such as Extended Kalman Filters (EKF), Unscented Kalman Filters (UKF), and even classical Out-of-Sequence Measurement (OOSM) handling methods either block the control pipeline waiting for delayed data or are unable to scale computationally to long delay windows, resulting in severe real-time performance degradation and divergence. State-of-the-art batch methods, such as factor graph optimization (FGO), yield superior estimation accuracy but require computational resources beyond the scope of embedded UUV microcontrollers.

Architecture: Two-Speed Kalman Filtering with Variational Projection

The authors propose the Asynchronous Two-Speed Kalman Filter (TSKF) framework, which explicitly decouples the estimation process into a synchronous, high-rate (fast thread) and an asynchronous, event-driven, low-rate (slow thread) pipeline, integrated via a finite-length state buffer and a novel projection mechanism termed Variational History Distillation (VHD).

• Fast Thread: Operates at the rate of onboard inertial and Doppler velocity log sensors (nominally 100 Hz), sustaining uninterrupted state propagation for real-time control. This thread incorporates a Sparse Gaussian Process (GP) regression module to compensate for non-parametric, residual hydrodynamic variabilities, tightly controlling prediction drift during long intervals with no new acoustic information.
• Slow Thread: Activated only upon asynchronous reception of delayed collaborative acoustic measurements. A precise $\mathcal{O}(1)$ buffer lookup retrieves the predicted state at the measurement's actual time of origin ($t-T$). A Kalman update is applied at this historical state, and VHD is employed to project the probabilistic state correction forward to the real-time state, leveraging the local linearity assumption ensured by GP compensation.

This design strictly prevents blocking or reprocessing in the main estimation/control loop. The state buffer's depth is set to the maximum anticipated acoustic delay interval, enabling accurate, asynchronous updates without state augmentation or full-trajectory optimization.

Quantitative Results

The framework is rigorously validated within a high-fidelity integrated simulation environment employing the Aqua-Sim FG toolchain, incorporating physically-realistic acoustic propagation, multipath effects, bandwidth constraints, and non-Gaussian, dynamic network-induced delays and packet loss. Evaluation includes Monte Carlo trials across 500 runs with dynamically simulated hydrodynamic disturbances.

Results demonstrate:

• Tracking accuracy: Under maximal tested delay ($T = 30$ s), TSKF achieves a position RMSE of $1.92$ m, closely matching FGO's $0.94$ m benchmark, while standard EKF exceeds $50$ m divergence and Augmented EKF collapses due to out-of-memory. At moderate delays ($T = 10$ s), TSKF's RMSE is $1.27$ m, within $0.4$ m of FGO.
• Computational efficiency: TSKF executes in $0.0027$ to $t-T$0 ms per step (MATLAB reference implementation), remaining constant with delay magnitude. FGO's per-step cost increases nonlinearly with delay, peaking at $t-T$1 ms at $t-T$2 s, while Augmented EKF is infeasible beyond $t-T$3 s due to cubic scaling with delay window length.
• Resilience to dropouts and delay variations: TSKF maintains stringent consistency and suppresses divergence throughout all tested network conditions, illustrating robust practical deployability for real-time microcontroller-based UUV clusters.

Theoretical Analysis

Mathematical analysis reveals that, unlike EKF/UKF or FGO approaches with $t-T$4 cost (where $t-T$5 is delay and $t-T$6 is state dimension), TSKF presents strictly linear scaling ($t-T$7), with all heavy computation relegated to infrequent, event-driven, low-rate fusion threads. The GP-based compensator ensures that residual error dynamics are tightly bounded even under prolonged communication outage, and the VHD projection compresses asynchronous corrections into minimal-variance, forward-propagated updates without reprocessing entire state histories.

Implications and Future Prospects

This work significantly advances practical cooperative navigation under highly intermittent, high-latency communication. By structurally decoupling state propagation and asynchronous sensor fusion, the TSKF enables strict real-time, low-drift control even on constrained embedded platforms, while approaching the accuracy of offline batch optimization methods. The combination of data-driven GP compensation with principled variational projection offers a new architecture for scalable, resilient fusion under severe OOSM scenarios.

Potential future directions include extension to higher-dimensional or bias-augmented states, adaptation to dynamic sensor selection and topology changes within UUV swarms, and deeper integration with online path planning and mission autonomy layers. The framework could also inform state estimation for other robotics domains characterized by acute communication latency, such as space robotics or deep subterranean operations.

Conclusion

The Asynchronous Two-Speed Kalman Filter (TSKF) architecture, with Variational History Distillation, represents a practical and theoretically sound solution to the acute OOSM bottleneck in UUV cooperative navigation over acoustic networks. It decisively addresses both computational scalability and estimation fidelity, enabling real-time embedded deployment without accuracy sacrifice seen in lower-complexity filters. The results suggest meaningful improvements for the reliability and operational range of autonomous underwater missions.