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Compressed delayed-information projection for six-degree-of-freedom underwater vehicle navigation under delayed acoustic positioning

Shuyue Li, Miguel López-Benítez, Eng Gee Lim, Fei Ma, Qian Dong, Mengze Cao, Limin Yu, Xiaohui Qin

arXiv:2609.27439v1cs.RO

TL;DR

Delayed acoustic packets constrain historical navigation states, but a current-time update evaluates them against a mismatched state. CDIP retains a source snapshot and historical-to-current coupling to project the correction directly to the current state; in the reported fixed-delay evaluation, it reduced mean trajectory RMSE relative to the current-time baseline while approaching replay with lower runtime.

  • Problem

    Delayed acoustic packets constrain a historical state after it has passed, creating a timing mismatch for current-time updates.

    Source: §I · p. 4

  • Method

    CDIP is a causal delayed-information projection that retains the source snapshot and historical-to-current cross-covariance to project the correction to the current state.

    Source: §VI · p. 33, §VII · p. 35

  • Results

    Across 154 paired recordings at a fixed 1.5-s delay without outage, CDIP reduced mean trajectory RMSE by 57.1% relative to the current-time baseline; its mean RMSE was 1.03% higher than replay, with 99.2% lower measured mean per-update runtime.

    Source: §VI · p. 33, §VII · p. 35

  • Takeaways & Limitations

    CDIP provides a causal delayed-information mechanism that approaches the replay reference in mean trajectory RMSE while requiring less measured per-update runtime.

    Source: §VI · p. 33

    The evidence is limited to a single-UUV, recorded/simulation-based evaluation.

    Source: §VI · p. 34

Abstract

from arXiv · show

Delayed acoustic positioning packets constrain historical navigation states, but a current-time update evaluates them against a mismatched state, whereas exact rewind/replay re-executes the intervening estimator history. This paper introduces compressed delayed-information projection (CDIP), a causal 15-state error-state Kalman filter (ESKF) treatment for delayed-acoustic unmanned underwater vehicle (UUV) navigation. CDIP retains a source-epoch snapshot and the historical-to-current cross-covariance, then projects the delayed source-epoch acoustic correction directly to the current state without full rewind/replay. Exact fixed-lag rewind/replay out-of-sequence-measurement (OOSM) processing serves as a high-fidelity accuracy reference. In 154 usable paired recordings at a fixed 1.5-s acoustic delay without an outage, CDIP reduced mean trajectory-position root-mean-square error (RMSE) from 1.062 m for the baseline to 0.456 m (57.1%). Its 0.456-m mean was 1.03% higher than the 0.451-m replay mean, while its measured mean per-update runtime was 99.2% lower (approximately 127-fold). A separate predeclared sweep across six fixed delays, with 30 paired recordings per delay, and a truth-supported 9-D consistency analysis bound the interpretation. Additional targeted experiments showed near-replay trajectory accuracy across 50-300-s acoustic outages while preserving sub-millisecond update cost. CDIP therefore provides a compact delayed-information treatment with an empirical accuracy-computation trade-off under the evaluated configuration; the evidence does not establish statistical equivalence or non-inferiority relative to replay.

I. INTRODUCTION

Delayed acoustic packets constrain historical states, but current-time updates evaluate them against the current state while exact rewind/replay reprocesses intervening events. CDIP projects the source-epoch correction to the current state using retained cross-covariance, and the paper evaluates its empirical accuracy–computation trade-off against these alternatives.

I. INTRODUCTION

  • Delayed acoustic observations can arrive after their generating state has passed, so current-time updates apply them to a mismatched state despite accumulating inertial drift.Exact fixed-lag rewind/replay instead applies the packet at its generation epoch and replays subsequent events.Source: §I · p. 4
  • CDIP retains a source-epoch snapshot and historical-to-current cross-covariance, then projects the delayed correction to the current state without replaying intervening events.The method is a causal treatment for a 15-state error-state Kalman navigation filter.Source: §I · pp. 5–6
  • CDIP is distinct from current-time updates, the authors’ earlier VHD treatment, and exact rewind/replay, and is not presented as formally equivalent to replay.Rewind/replay remains the high-fidelity accuracy reference.Source: §I · p. 6
  • The comparison is framed as an empirical accuracy–computation trade-off, not as evidence of statistical equivalence or non-inferiority to replay.The validated main experiment is distinguished from the targeted outage study.Source: §I · p. 6
  • The controlled evaluation uses 154 usable recordings at a fixed 1.5-s delay, a separate six-delay sweep with 30 paired recordings per delay, and truth-supported 9-D consistency diagnostics.The main experiment has no acoustic outage; a separate targeted study examines 50–300-s permanent-loss outages.Source: §I · p. 6

II. PROBLEM FORMULATION AND SYSTEM MODEL

The section formulates causal six-degree-of-freedom underwater navigation with a 15-dimensional error state and strapdown inertial dynamics. It also distinguishes the fixed-delay, no-outage main experiment from a separate packet-loss study.

II. PROBLEM FORMULATION AND SYSTEM MODEL

  • The problem formulation treats delayed acoustic observations separately from locally available onboard measurements.The packet-loss intervals are a separate targeted experiment rather than the main fixed-delay, no-outage evaluation.Source: §II · p. 7
  • The modeled vehicle is a six-degree-of-freedom underwater vehicle operating with a causal navigation filter.The main experiment uses fixed acoustic delay without outage; a separate experiment evaluates bounded packet-loss intervals.Source: §II · p. 7

A. System state definition

  • The navigation state includes position, velocity, unit-quaternion attitude, and gyroscope and accelerometer biases in the global navigation frame.Position and velocity are three-dimensional, and attitude is represented by a four-component quaternion.Source: A. System state definition · pp. 7–8
  • Attitude errors are represented in the tangent space of SO(3), mapping the 16-dimensional nominal state to a minimal 15-dimensional error state.This representation is described as avoiding singularities and preserving physical consistency.Source: A. System state definition · p. 8

B. Motion model

  • The continuous-time vehicle dynamics follow a standard strapdown inertial navigation model.The model uses measured specific force and angular velocity, body-to-navigation rotation, and the navigation-frame gravity convention.Source: B. Motion model · p. 8
  • Noise and bias-driving terms are represented in the ESKF process covariance.Source: B. Motion model · p. 8

C. Measurement models

The measurement models combine onboard DVL and depth aiding with delayed external acoustic positioning. Acoustic observations constrain their source epoch, while reception occurs later and is treated causally in the experiments.

C. Measurement models

  • The DVL measures three-dimensional vehicle-frame velocity, while the depth sensor provides vertical position at locally available sampling times.Source: C. Measurement models · p. 9
  • The DVL prediction includes translational and lever-arm rotational velocity, using bias-corrected IMU angular rate in the lever-arm term.The frozen estimator implementation applies that bias correction before DVL lever-arm prediction.Source: C. Measurement models · p. 9
  • Acoustic updates come from an external underwater reference and provide horizontal two-dimensional position information.The acoustic observation model extracts horizontal position.Source: C. Measurement models · p. 10
  • The final experiments preserve recorded DVL input and generate depth and horizontal acoustic observations from ground truth at DVL timestamps under the declared measurement policy.The declared offline DVL measurement-noise realization is 0.05 m/s and is distinct from raw Gazebo DVL noise.Source: C. Measurement models · p. 10

D. Delayed acoustic observation model

  • Because acoustic packets arrive asynchronously, each observation constrains its source epoch rather than its reception epoch.The main and delay-sweep experiments assign reception causally from generation time; the outage study adds scheduled permanent packet-loss intervals while retaining the fixed 1.5-s delay outside outages.Source: D. Delayed acoustic observation model · pp. 10–11

E. Problem statement

The problem is real-time 6-DOF underwater navigation that incorporates delayed acoustic packets at their generation epochs. CDIP projects historical acoustic information to the current state using retained source-to-current statistics, between a current-time baseline and exact rewind/replay reference.

E. Problem statement

  • The objective is real-time estimation of a UUV's 6-DOF navigation state while incorporating each received acoustic packet according to its generation epoch.The comparison includes delayed acoustic communication, locally available onboard aiding measurements, and limited computational resources.Source: E. Problem statement · p. 11

A. Compared delayed-acoustic treatments

  • The three treatments share a causal 15-state ESKF and differ in delayed-packet handling: current-state update, CDIP projection, or historical update followed by replay.CDIP uses maintained historical/current cross-covariance; rewind/replay restores the filter before packet generation and replays causally to reception.Source: §III · p. 12, A. Compared delayed-acoustic treatments · p. 13

B. Baseline error-state filtering architecture

  • The ESKF error state represents position, velocity, attitude, gyroscope-bias, and accelerometer-bias deviations from the nominal navigation state.At DVL/depth epochs, the filter performs the common onboard update and right-error reset before handling delayed acoustic packets.Source: B. Baseline error-state filtering… · p. 13
  • The baseline evaluates a received acoustic innovation using the current state and covariance, without retaining the source-epoch state relation needed for a delayed update.CDIP instead retains source-state information and cross-covariance for its current-state projection.Source: B. Baseline error-state filtering… · pp. 13–14, C. Compressed delayed-information… · p. 14

C. Compressed delayed-information projection

  • CDIP retains a source-epoch snapshot and historical-to-current cross-covariance to project delayed acoustic information to the current state without replaying historical events.The paper distinguishes this projection from both the current-time delayed update and exact rewind/replay; it does not claim equivalence to replay.Source: C. Compressed delayed-information… · p. 14

D. Exact fixed-lag rewind/replay reference

Exact rewind/replay is the fixed-lag reference: it applies a delayed acoustic update at its historical generation epoch and replays subsequent causal events to reception. The implementation also specifies a 15-dimensional error state and linearized error dynamics.

D. Exact fixed-lag rewind/replay reference

  • Exact rewind/replay restores the causal filter trajectory before packet generation, applies the acoustic update historically, and replays subsequent events to reception.The replay pass includes subsequent IMU propagation, DVL and depth updates, ESKF resets, and already received acoustic events available by current wall-clock time.Source: D. Exact fixed-lag rewind/replay… · p. 15

E. Algorithm summary

  • Algorithm 1 summarizes causal CDIP processing: propagate and update the filter, then retrieve a delayed packet's source snapshot and cross-covariance to update the current state.The algorithm forms the delayed residual, innovation covariance, and gain, then updates the current state, covariance, and retained cross-covariances.Source: D. Exact fixed-lag rewind/replay… · p. 16, E. Algorithm summary · p. 16

IV. ERROR-STATE MODELING AND IMPLEMENTATION DETAILS

  • The section places the rewind/replay reference within the paper's error-state modeling and implementation details for delayed-acoustic updates.The section's stated focus is estimator consistency, covariance propagation, and practical integration of delayed-information updates.Source: §IV · p. 16

A. Error-state definition

  • The 15-dimensional error state comprises position, velocity, attitude, gyroscope-bias, and accelerometer-bias errors, with attitude represented as a three-dimensional right error.The attitude error is parameterized in the tangent space associated with SO(3); the nominal quaternion is normalized after propagation and error injection.Source: A. Error-state definition · pp. 16–17

B. Linearized error dynamics

  • Linearizing the nonlinear motion model around the nominal trajectory yields continuous-time error dynamics driven by the state transition matrix, noise input matrix, and process noise.The detailed 15 × 15 transition structure follows standard inertial-navigation linearizations.Source: B. Linearized error dynamics · p. 17

C. Covariance propagation and reset consistency

The implementation propagates covariance and retained CDIP cross-covariances through measurement updates and right-error resets. The experiments assess trajectory accuracy, numerical calibration, and computational cost under fixed acoustic delays, with a separate outage study.

C. Covariance propagation and reset consistency

  • Covariance propagation advances both the current covariance and each active CDIP historical/current cross-covariance, with right-error reset maps applied to keep their relation synchronized.The same measurement update, error-state injection, and reset operations are applied to retained cross-covariances.Source: C. Covariance propagation and reset… · pp. 17–18

D. Discrete-time implementation

  • The real-time implementation discretizes continuous-time error dynamics using a first-order approximation and the measured IMU interval.The discrete process-noise covariance is configured for the implementation.Source: D. Discrete-time implementation · p. 18

E. Measurement update

  • Onboard aiding measurements use standard Kalman updates, followed by nominal-state correction, error-state reset, and covariance update.The measurement Jacobian is evaluated at the nominal state with respect to the 15-dimensional error state.Source: E. Measurement update · pp. 18–19

F. Computational considerations

  • CDIP stores and propagates snapshots and cross-covariances for packets awaiting reception, while rewind/replay re-executes the affected fixed lag after each packet.The runtime comparison is an empirical property of the frozen implementation and hardware environment, not a claim of asymptotic equivalence.Source: F. Computational considerations · p. 19

V. EXPERIMENTS

  • The experiments use the frozen RexROV recording corpus to assess trajectory accuracy, numerical calibration, and computation under fixed delays, plus a targeted prolonged-packet-loss study.The final-main evaluation and fixed-delay sweep are distinct predeclared paired experiments.Source: §V · p. 19

A. Compared estimation methods

The evaluation compares a current-time delayed-update baseline, CDIP, and exact rewind/replay under shared inputs and measurement realizations. It specifies fixed-delay experiments, performance metrics, paired recording selection, and sensor-measurement generation.

A. Compared estimation methods

  • The comparison evaluates the current-time ESKF baseline, proposed CDIP, and exact rewind/replay reference using the same input data and measurement realization.The methods differ in their treatment of delayed measurements, without adding a learning component or changing the nominal navigation model.Source: A. Compared estimation methods · p. 20

B. Communication-delay design

  • The final-main experiment uses a fixed 1.5-s acoustic delay without outage, while a separate sweep examines six fixed delays from 0 to 10 s.The sweep characterizes measurement age without changing measurement realization; it is not a random 5–30-s delay design.Source: B. Communication-delay design · p. 20

C. Performance metrics

  • Performance measures include trajectory-position RMSE, other position and velocity errors, truth-supported 9-D NEES/ANEES, DVL and acoustic NIS diagnostics, and mean update computation time.Trajectory RMSE is the primary performance metric.Source: C. Performance metrics · p. 21

D. Simulation setup

  • The final-main experiment includes 154 recordings after 46 of 200 candidate RexROV recordings were deemed input-unusable for lacking the required DVL stream.Each retained recording is evaluated by all three methods with identical input and deterministic measurement/noise realizations; the separate sweep uses 30 paired recordings per delay.Source: D. Simulation setup · pp. 21–22

1. Sensor and measurement generation

  • Recorded IMU and DVL observations and ground-truth states are used, while depth and horizontal acoustic-position observations are declared simulation measurements generated from ground truth.The same complete realization is supplied to all three methods within each recording, and the final DVL measurement policy adds deterministic Gaussian noise with standard deviation 0.05 m/s.Source: §1 · p. 22

E. Final main-experiment results

At a fixed 1.5-s acoustic delay without an outage, CDIP reduced trajectory RMSE relative to the baseline and approached exact rewind/replay accuracy with lower measured runtime. These results describe the evaluated recordings and do not establish statistical equivalence or non-inferiority to replay.

E. Final main-experiment results

F. Consistency and numerical integrity

The separate fixed-delay sweep found that CDIP and replay had delay-dependent accuracy patterns, while CDIP runtime increased with delay. The truth-supported 9-D diagnostics indicated average calibration for both delayed-information treatments, but not perfect consistency.

F. Consistency and numerical integrity

  • Numerical-integrity checks found zero covariance asymmetry, quaternion-norm error below 10^-15, and a positive minimum covariance eigenvalue above 10^-9.DVL and acoustic NIS means were close to their expected dimensions for all methods.Source: F. Consistency and numerical integrity · p. 27
  • For the 9-D ANEES analysis, M1 was above the 95% upper bound at 88.3% of grid points, while M2 and M3 means were near the expected range.M2 and M3 were conservative at 50% of evaluated grid points and inside the bounds at 40%; the diagnostics support average calibration, not perfect consistency.Source: F. Consistency and numerical integrity · p. 26, Figure 5 · p. 28

G. Separate fixed-delay sweep

  • The fixed-delay sweep does not support uniform CDIP superiority, statistical equivalence, or non-inferiority relative to replay.The paired RMSE-difference intervals included zero at every nonzero delay.Source: G. Separate fixed-delay sweep · p. 29, Table IV · p. 29
  • CDIP runtime rose from 0.0416 ms/update at 0.5 s delay to 0.6525 ms/update at 10 s, while replay remained slower at each predeclared delay.Replay runtime decreased at longer delays because fewer packets remained within the fixed evaluable horizon, so that trend is not evidence of a computational advantage.Source: G. Separate fixed-delay sweep · pp. 29–30
  • Across the separate six-delay sweep, CDIP had lower mean trajectory RMSE than replay at 0.5, 1.5, and 5 s, while replay was lower at 3 and 10 s.The paired confidence interval for the RMSE difference included zero at every nonzero delay; each delay used 30 paired recordings.Source: G. Separate fixed-delay sweep · pp. 27–29, Table IV · p. 29, Figure 6 · p. 30

H. Robustness under prolonged acoustic outages

The targeted outage study found substantially lower CDIP trajectory error than the baseline across 50–300-s permanent acoustic outages, with small mean gaps from replay and much lower measured update cost. The discussion bounds these findings to the tested design and notes that they do not establish equivalence with replay.

H. Robustness under prolonged acoustic outages

  • The outage stress test used five paired scenarios per duration at a fixed 1.5-s delay, with outage-generated packets permanently lost and no batch recovery.Figure 7 plots mean trajectory-position RMSE with saved 95% confidence intervals.Source: H. Robustness under prolonged acoustic… · p. 31, Figure 7 · p. 32
  • Across 50–300-s outages, CDIP mean trajectory-position RMSE was 0.412–0.470 m versus 1.816–1.920 m for baseline, a 75.5–77.4% reduction; replay gaps were 7.7–9.1 mm.The table summarizes means over five paired scenarios per duration.Source: Table V · p. 31, H. Robustness under prolonged acoustic… · p. 31
  • Paired confidence intervals included zero, so no statistically detectable trajectory-RMSE difference from replay was observed; this does not establish equivalence, while CDIP updates took 0.150–0.165 ms versus replay’s 325–456 ms.These runtime values are specific to the separate outage stress test.Source: H. Robustness under prolonged acoustic… · p. 32

VI. DISCUSSION

  • CDIP retains the source snapshot and historical-to-current cross-covariance to project the source-epoch correction directly to the current state, rather than evaluating it at current time or replaying history.The paper presents this as a distinct delayed-information treatment, not a replay surrogate.Source: §VI · p. 33
  • In the fixed 1.5-s main experiment, CDIP approached replay in mean trajectory RMSE with a 1.03% gap and 99.2% lower measured mean per-update runtime.The discussion describes this as an accuracy–computation trade-off within the evaluated design, not proof of statistical equivalence or universal replay dominance.Source: §VI · p. 33
  • Across the predeclared delay grid, baseline accuracy worsened as delay increased, while CDIP and replay remained closer; their mean ordering changed and paired confidence intervals included zero.The runtime contrast depends on the frozen implementation and evaluable horizon, not an asymptotic complexity claim.Source: §VI · p. 33
  • The 9-D diagnostics showed the baseline predominantly above the ANEES upper bound, while CDIP and replay had broadly reasonable mean calibration but were conservative at about half the grid points.Numerical-integrity checks do not replace external validation.Source: §VI · p. 34
  • The evaluation is limited to a single UUV with recorded or simulated data and frozen sensor, noise, initialization, and covariance assumptions; the targeted n = 5 outage study is not a network-level model.The tested outage evidence covers scheduled permanent loss, not mobility, propagation, connectivity, or stochastic packet loss.Source: §VI · p. 34

VII. CONCLUSION

The paper presents CDIP, which projects a delayed acoustic correction to the current state using a source-epoch snapshot and historical-to-current cross-covariance. In the fixed-delay benchmark it reduced trajectory RMSE relative to the current-time baseline, approached replay’s mean accuracy, and used substantially less measured per-update computation.

VII. CONCLUSION

  • CDIP retains the source-epoch snapshot and historical-to-current cross-covariance to project an out-of-sequence correction directly to the current 15-state ESKF without full rewind/replay.Exact rewind/replay remains the accuracy reference.Source: §VII · p. 35
  • Across 154 paired recordings at a fixed 1.5-s acoustic delay without outage, CDIP reduced mean trajectory RMSE by 57.1% versus the current-time baseline; its mean RMSE was 1.03% higher than replay’s, with 99.2% lower measured mean per-update runtime.The conclusion reports a separate 30-recording-per-delay sweep across 0–10 s and targeted 50–300-s outage experiments with sub-millisecond update cost.Source: §VII · p. 35
  • The results support CDIP as a replay-efficient delayed-information treatment within evaluated conditions, but do not establish statistical equivalence, non-inferiority, or universal superiority over replay.The conclusion describes the evidence as a fixed-configuration accuracy–computation trade-off.Source: §VII · p. 35
  • Future work will extend evaluation to distributed multi-UUV navigation, timing and clock errors, broader network and stochastic-loss effects, and field or sea trials.Source: §VII · p. 35

Conflict of Interest

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