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

cs.ROSep 23, 2026

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

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

Organizations: Academy of Artificial Intelligence and Advanced Technology, Xi’an Jiaotong-Liverpool University, No. 111, Ren’ai Road, Suzhou Industrial Park, Suzhou, Jiangsu Province 215123, China · Department of Electrical Engineering and Electronics, School of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GJ, United Kingdom · Jiangsu JITRI Tsingunited Intelligent Control Technology Co., Ltd., D1 Building, No. 999 Gaolang East Road, Wuxi, Jiangsu Province, China · School of Computer Science and Informatics, University of Liverpool, Ashton Building, Liverpool L69 3DR, United Kingdom · ARIES Research Centre, Polytechnic School, Universidad Antonio de Nebrija, C/ Pirineos, 55, 28040 Madrid, Spain · School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University, No. 111, Ren’ai Road, Suzhou Industrial Park, Suzhou, Jiangsu Province 215123, China · College of Mechanical and Vehicle Engineering, Hunan University, Lushan Road (S), Yuelu District, Changsha, Hunan Province 410082, China

Abstract

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.

Figures & tables

Figure 1: Causal sensing and delayed-acoustic context. IMU, DVL, and depth observations are locally available to the causal ESKF. An acoustic position packet generated at source epoch

j

becomes available at reception epoch

k

and therefore constrains the historical state rather than the current state.

Figure 2: Conceptual contrast between the two historical treatments. (a) M2 (CDIP) retains a source snapshot and the historical-to-current cross-covariance needed to project the source-epoch correction directly to the current state. (b) M3 (Exact rewind/replay) restores a checkpoint before the source epoch, applies the historical acoustic update, and replays the intervening causal event sequence. The replay path is an accuracy-reference contrast, not an equivalence claim.

Method	Traj. RMSE (m)	Final err. (m)	Peak err. (m)	Vel. RMSE (m/s)	Runtime (ms/update)
M1 (Baseline)	$1.062\pm 0.421$	$1.469\pm 0.625$	$1.825\pm 0.682$	$0.095\pm 0.035$	$0.0273$
M2 (CDIP)	$0.456\pm 0.171$	$0.346\pm 0.220$	$1.116\pm 0.401$	$0.057\pm 0.020$	$0.0743$
M3 (Exact rewind/replay)	$0.451\pm 0.171$	$0.347\pm 0.218$	$1.088\pm 0.371$	$0.057\pm 0.019$	$9.469$

Table 1: Final-main descriptive performance at a fixed 1.5-s acoustic delay without an outage. Accuracy and error metrics are mean

\pm

standard deviation; runtime/update is reported as a mean only.

N=154

paired recordings per method; all failure counts were zero.

Comparison	Metric	Mean difference (95% CI)	$p$ ( $t$ /Wilcoxon)	Cohen’s $d_{z}$
M1–M2	Traj. RMSE (m)	$0.606$ $[0.549,0.664]$	$<0.001/<0.001$	$1.68$
M1–M2	Final err. (m)	$1.123$ $[1.025,1.222]$	$<0.001/<0.001$	$1.81$
M1–M2	Peak err. (m)	$0.709$ $[0.616,0.802]$	$<0.001/<0.001$	$1.21$
M1–M2	Vel. RMSE (m/s)	$0.038$ $[0.033,0.042]$	$<0.001/<0.001$	$1.34$
M2–M3	Traj. RMSE (m)	$0.005$ $[-0.001,0.010]$	$0.0912/0.0187$	$0.137$
M2–M3	Final err. (m)	$-0.001$ $[-0.005,0.003]$	$0.615/0.906$	$-0.041$

Table 2: Paired final-main comparisons (

N=154

). For M1 minus M2, positive differences favor CDIP. For M2 minus M3, negative differences favor CDIP. Trajectory RMSE is the primary performance metric in the manuscript presentation. The paired

t

-test and Wilcoxon signed-rank

p

values are the saved unadjusted values from the frozen final-main analysis; no multiplicity-adjusted

p

values are claimed, and secondary endpoint tests are interpreted descriptively. No equivalence or non-inferiority inference is made.

Figure 3: Final-main trajectory-position RMSE at the fixed 1.5-s acoustic delay without an outage. M1 denotes the baseline, M2 CDIP, and M3 exact rewind/replay. Open markers show all

N=154

paired recordings per method; boxes show the interquartile range with median bars, whiskers extend to the non-outlier range, and filled diamonds denote means.

Figure 4: Final-main accuracy–runtime display at the fixed 1.5-s acoustic delay without an outage. Points are saved means and horizontal and vertical error bars are the saved 95% confidence intervals of those means. The horizontal axis is logarithmic. This is an implementation-specific accuracy–computation comparison, not a significance or equivalence display.

Method	Mean 9-D ANEES	Below / inside / above	Mean DVL NIS	Mean acoustic NIS
M1 (Baseline)	$18.110$	$0.0\%/11.7\%/88.3\%$	$2.995$	$2.084$
M2 (CDIP)	$8.728$	$50.0\%/40.0\%/10.0\%$	$2.990$	$1.993$
M3 (Exact rewind/replay)	$8.691$	$50.0\%/40.0\%/10.0\%$	$2.990$	$1.992$

Table 3: Truth-supported 9-D consistency and innovation diagnostics for the final main experiment. Fractions denote the 60-point aggregate time grid below, inside, and above the 95% ANEES bounds

[8.342,9.682]

. The expected NIS dimensions are three for DVL and two for acoustic position.

Figure 5: Truth-supported 9-D consistency summary for the final-main experiment. M1 denotes the baseline, M2 CDIP, and M3 exact rewind/replay. (a) Saved mean 9-D ANEES values relative to the saved 95% expected range. (b) Fractions of the 60-point aggregate time grid below, inside, and above that range. This diagnostic complements the trajectory-error results and is not a 15-D ANEES analysis or a performance endpoint.

Delay (s)	M1 RMSE	M2 RMSE	M3 RMSE	M2–M3 difference (95% CI)	M2 runtime	M3 runtime
$0$	$0.455$	$0.455$	$0.455$	—	$0.0307$	$9.565$
$0.5$	$0.535$	$0.457$	$0.462$	$-0.005$ $[-0.015,0.005]$	$0.0416$	$9.207$
$1.5$	$1.053$	$0.469$	$0.477$	$-0.008$ $[-0.023,0.006]$	$0.0877$	$10.774$
$3$	$1.934$	$0.511$	$0.501$	$0.010$ $[-0.007,0.027]$	$0.1141$	$7.980$
$5$	$2.948$	$0.531$	$0.537$	$-0.006$ $[-0.035,0.023]$	$0.2409$	$7.664$
$10$	$4.354$	$0.674$	$0.631$	$0.043$ $[-0.008,0.094]$	$0.6525$	$6.834$

Table 4: Separate fixed-delay sweep (

N=30

paired recordings per delay). M1 denotes the baseline, M2 CDIP, and M3 exact rewind/replay. Entries are mean trajectory RMSE in metres and mean runtime in ms/update. The M2 minus M3 RMSE difference is shown with its 95% confidence interval for nonzero delays; negative values favor CDIP. All such intervals include zero.

Figure 6: Separate fixed-delay sweep (

N=30

paired recordings at each predeclared delay). M1 denotes the baseline, M2 CDIP, and M3 exact rewind/replay. (a) Mean trajectory-position RMSE with saved 95% confidence intervals. (b) Mean runtime/update with saved 95% confidence intervals; the vertical axis is logarithmic.

Outage (s)	M1 Baseline ESKF	M2 Proposed CDIP	M3 Exact rewind/replay	CDIP reduction
$50$	$1.816$	$0.412$	$0.421$	$77.3\%$
$100$	$1.825$	$0.412$	$0.421$	$77.4\%$
$200$	$1.826$	$0.435$	$0.443$	$76.2\%$
$300$	$1.920$	$0.470$	$0.478$	$75.5\%$

Table 5: Targeted prolonged-outage results at a fixed 1.5-s acoustic delay. Entries are mean trajectory-position RMSE in metres over five paired scenarios per duration. CDIP reduction is relative to M1 Baseline ESKF.

Figure 7: Targeted prolonged-outage study at a fixed 1.5-s acoustic delay (

n=5

paired scenarios per duration). Points denote mean trajectory-position RMSE, and bars show the saved 95% confidence intervals. Packets generated during each outage were permanently lost, with no batch recovery.

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Apr 3, 2026J/K move · Enter open · S save

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

Shuyue Li, Miguel López-Benítez, Eng Gee Lim, Fei Ma+4

School of Advanced Technology, Xi’an Jiaotong-Liverpool University, Suzhou, China · School of Computer Science and Informatics, University of Liverpool, Liverpool, UK · ARIES Research Centre, Universidad Antonio de Nebrija, Madrid, Spain+3

Kalman Filtering GNSS-Denied Localization Cooperative Localization Underwater Robotics Autonomous Underwater Navigation Robot State Estimation+1

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In Global Navigation Satellite System (GNSS)-denied underwater environments, individual unmanned underwater vehicles (UUVs) suffer from unbounded dead-reckoning drift, making collaborative navigation (CN) 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 (EKFs) or Unscented Kalman Filters (UKFs), usually block the main control loop while waiting for delayed data, or effectively discard Out-of-Sequence Measurements (OOSMs), 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 Circular State Buffer (FLCSB), the algorithm applies delayed measurements 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 a trajectory error 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.

Abstract

Error-State Kalman Filtering

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