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Underwater Navigation in Unsteady Flows Using Measurement Histories from a Single Sensing Unit

Linhao Jin, Qimin Feng, Peter Gunnarson, Qiang Zhong

arXiv:2609.26753v1cs.ROphysics.flu-dyn

TL;DR

Spatial measurements aid underwater navigation, but simultaneous distributed sampling is constrained by robot size and sensor placement. The paper estimates current lateral velocities from a single sensor’s history for a fixed controller, achieving strong success in held-out cylinder wakes.

  • Problem

    Simultaneous spatial flow measurements can benefit underwater navigation, but their sampling-point spacing is constrained by robot dimensions and sensor placement.

    Source: §I · p. 1

  • Method

    A causal observer estimates current lateral velocities from center-flow history and supplies them, alongside measured center flow, to a fixed navigation controller.

    Source: §I · p. 1

  • Results

    At Re = 205 and 240, Flow history reaches 84.38% and 80.60% success, versus Direct spatial sensing’s 91.80% and 85.16% and Current only’s 53.65% and 47.27%.

    Source: A. Transfer to Held-Out Cylinder Wakes · p. 6

  • Takeaways & Limitations

    For this controller, observers should preserve the lateral velocity difference and limit persistent errors, with closed-loop evaluation complementing reconstruction metrics.

    Source: §VII · p. 8

    The study uses simulated two-dimensional wakes, one fixed controller, and body-frame ambient-velocity observations; hardware would require vehicle-motion estimation and compensation.

    Source: B. Spatial Flow Inputs · p. 2, §VII · p. 8

Abstract

from arXiv · show

Spatial flow measurements support underwater navigation, but distributed sensing is constrained by robot size and sensor layout. We use a causal observer to estimate current lateral velocities from a finite history of measurements collected by a single sensing unit, supplying the inputs of a fixed navigation controller. In two-dimensional wake simulations with access to body-frame ambient velocity, this virtual sensing interface reduces simultaneous flow sampling from three points to the robot center. Trained only in a circular-cylinder wake at Re = 100, the flow-history observer achieves 84.4% and 80.6% success at held-out Re = 205 and 240 without retraining. These rates are 7.4 and 4.6 percentage points below direct spatial sensing and more than 30 points above a matched current-only observer. Past flow remains beneficial when past goal and yaw information is available. Across obstacle geometries, performance remains close to direct sensing in square-prism wakes but declines in triangular-prism wakes. Component replacement identifies the lateral velocity difference as control-relevant, while controlled perturbations reveal sensitivity to error persistence. The results demonstrate the closed-loop utility of single-point flow histories under the assumed observation model.

I. INTRODUCTION

The paper asks whether a causal observer can recover spatial cues from a single sensor’s history, then evaluates that virtual interface as replacement lateral inputs to a fixed controller.

I. INTRODUCTION

  • Spatial measurements benefit navigation, but simultaneous sampling points are constrained by robot dimensions and sensor placement.The motivating question is whether a moving robot’s single sensor history can recover cues otherwise obtained simultaneously at multiple points.Source: §I · p. 1

A Flow and trajectory with a single sensing unit

  • A causal virtual sensing interface estimates the controller’s current lateral inputs from single-point flow history while retaining the directly measured center flow.The interface substitutes estimated lateral measurements into the fixed controller.Source: §I · p. 1, A Flow and trajectory with a single… · p. 1
  • 84.4% and 80.6% success at held-out Re = 205 and 240, respectively, followed training only at Re = 100; both exceed a matched current-only observer by more than 30 percentage points.The flow-history observer is evaluated without retraining.Source: A Flow and trajectory with a single… · p. 1
  • The evaluation matches historical input channels and tests transfer across Reynolds numbers and obstacle geometries with learned models held fixed after training.It also examines whether navigation depends on lateral-error structure and persistence beyond aggregate reconstruction error.Source: A Flow and trajectory with a single… · pp. 1–2

II. RELATED WORK

Related work addresses hydrodynamic navigation, sparse-field reconstruction, and history-based control; this paper studies whether a moving robot’s measurements can substitute for simultaneous lateral sensing.

II. RELATED WORK

  • Prior work studies hydrodynamic cues,, sparse or moving-sensor reconstruction,,, and history-based control,; this paper predicts lateral velocities for a fixed controller.The related work also includes spatial sensing for navigation and task-driven estimation.Source: §II · p. 2

A. Navigation Task and Dynamics

  • The navigation task places a robot in a recorded two-dimensional obstacle wake, where self-propulsion combines with local flow advection and heading changes are bounded by yaw rate.The robot swims at constant speed relative to the fluid.Source: A. Navigation Task and Dynamics · p. 2

B. Spatial Flow Inputs

The spatial controller uses center and two symmetric lateral flow readings, while its virtual-sensing counterpart can substitute estimated lateral readings. The mean and difference decomposition separates common flow from transverse variation and reconstruction error.

B. Spatial Flow Inputs

  • The controller samples flow at the center and two lateral points 2ℓ apart, and its inputs include center flow, target displacement, remaining time, and the preceding yaw rate.Direct sensing supplies all three measurements simultaneously; virtual sensing measures only at the center and predicts both lateral readings.Source: B. Spatial Flow Inputs · p. 2
  • Both sensing interfaces assume ambient Eulerian velocity in body coordinates, so hardware use requires motion compensation rather than center-relative probe flow alone.The experiments assess temporal sampling with access to these ambient-flow inputs.Source: B. Spatial Flow Inputs · p. 2

C. Mean and Lateral-Difference Components

  • The lateral-reading mean represents common flow, while their difference divided by 2ℓ approximates the transverse directional derivative of body-frame velocity.The two readings can be varied independently through their mean and difference.Source: C. Mean and Lateral-Difference… · pp. 2–3
  • The lateral reconstruction error decomposes into mean and difference errors, with Eside = Em + Ed/4; closed-loop interventions assess their control relevance.The controller continues to receive the two lateral readings individually.Source: C. Mean and Lateral-Difference… · p. 3

D. Frozen Spatial Controller

The study holds the spatial navigation controller fixed when comparing sensing interfaces. Its pursuit prior and learned residual together determine the controller’s actions.

D. Frozen Spatial Controller

  • The spatial controller combines a flow-compensated pursuit prior with a residual policy trained using PPO at Re = 100.The prior chooses a desired swimming direction from target displacement and center flow, then converts heading error into a bounded turn.Source: D. Frozen Spatial Controller · p. 3
  • The actor adds a learned residual before the final 4 tanh(·) action map.The reported checkpoint follows 223 PPO training episodes, approximately 8.4 million transitions.Source: D. Frozen Spatial Controller · p. 3
  • Across sensing interfaces, the pursuit prior, policy weights, normalization, and action map remain fixed, so performance differences reflect lateral-input substitution and its downstream effects.Each interface supplies shared inputs along its own closed-loop trajectory.Source: D. Frozen Spatial Controller · p. 3

IV. VIRTUAL SPATIAL SENSING

The virtual observer uses causal center-flow history to estimate current lateral velocities for the fixed controller. Its finite history and strictly causal inputs constrain what it can use at decision time.

A. Predicting Lateral Flow from Causal History

  • The observer predicts four additive corrections from causal history and the current center measurement to estimate current lateral velocities.Its outputs correct the two body-frame lateral velocity vectors.Source: A. Predicting Lateral Flow from Causal… · p. 3
  • The observer uses 32 observations spanning 1.55D/U∞, preserving sampled noise and marking unavailable entries until a full history window forms.When enabled, relative-yaw metadata is reconstructed from executed yaw-rate history; past flow remains in the body frames in which it was acquired.Source: A. Predicting Lateral Flow from Causal… · p. 3
  • The current-only and flow-history readouts hold past goal and yaw channels at current values, while only the flow-history readout retains past flow measurements.At inference, the observer receives no future measurements, absolute position, Reynolds-number label, lateral measurements, or teacher actions.Source: A. Predicting Lateral Flow from Causal… · p. 3
  • The observer is a 416 →128 →128 →4 MLP with tanh activations, trained using equally weighted squared error over four lateral velocity components.Feature normalization uses source-domain statistics held fixed at inference.Source: A. Predicting Lateral Flow from Causal… · p. 3

B. Local Analytic Readouts

Local analytic readouts infer lateral variation by fitting causal single-point measurements expressed in the current body frame. Matched neural readouts vary access to past flow and goal/yaw history to isolate the value of those inputs.

B. Local Analytic Readouts

  • In the local flow fit, displacement and time-offset terms model spatial and temporal variation, and coefficient b determines the predicted lateral difference.The past center-flow estimate and current center reading anchor the fitted deviations.Source: B. Local Analytic Readouts · p. 3
  • The spatial ridge and space–time ridge readouts fit past center-flow deviations from the current center reading after expressing samples in the current body frame.The space–time fit includes local temporal variation, while the spatial fit uses only spatial terms.Source: B. Local Analytic Readouts · p. 3
  • Both analytic readouts use causal center-flow and goal/yaw histories without absolute world poses, approximate lateral mean by current center flow, and revert to that reading when history is insufficient or the fit is ill-conditioned.They recover the two lateral readings as the mean plus or minus half the predicted difference.Source: B. Local Analytic Readouts · p. 3

C. Isolating the Contribution of History

  • Matched neural readout comparisons measure the incremental value of past flow given past goal/yaw and of past goal/yaw given past flow, though their physical information may overlap.Repeated center removes lateral variation, while pursuit only removes the learned residual and uses the fixed pursuit prior.Source: C. Isolating the Contribution of History · pp. 3–4

D. Source-Domain Training

Observers are trained on circular-cylinder wake data at Re = 100, with data aggregation to account for trajectory shifts caused by virtual sensing. Evaluation conditions vary wake geometry and Reynolds number while keeping the learned readouts fixed.

D. Source-Domain Training

  • Two rounds of data aggregation collect trajectories from four source-domain mission families and train on clean lateral velocities, preserving each history from the mission start.The families are standard, reversed start/target, jointly translated, and shifted-target missions; all readout variants use the same sampled states.Source: D. Source-Domain Training · p. 4

A. Flow Fields and Navigation Missions

  • All learned readouts are trained in the circular-cylinder wake at Re = 100; development and confirmation wakes are evaluated without gradient updates.Confirmation uses new missions at Re = 205 and 240 after models and primary analysis are fixed.Source: A. Flow Fields and Navigation Missions · p. 4, D. Source-Domain Training · p. 4
  • The circular-cylinder wakes are simulated on a 1024 × 512 lattice, warmed up for 240D/U∞, then recorded for 60D/U∞ at 0.05D/U∞ intervals.The robot operates within [−3.8D, 11.8D] × [−3.8D, 3.8D].Source: A. Flow Fields and Navigation Missions · p. 4
  • Geometry-transfer tests use triangular- and square-prism wakes at Re = 100, 205, and 240, transferring readouts and the spatial controller without retraining.Each geometry–Reynolds-number condition uses 256 missions shared across sensing configurations.Source: A. Flow Fields and Navigation Missions · p. 4

B. Navigation and Reconstruction Evaluation

Navigation is evaluated through complete closed-loop missions, while reconstruction compares predictions at shared states. Separate interventions test the contribution of lateral-error components and the effect of error persistence.

B. Navigation and Reconstruction Evaluation

C. Component Replacement and Error Persistence

  • Component replacement substitutes one lateral component with its current noisy measurement while the other remains predicted; the observer is not retrained.The intervention uses lateral measurements at evaluation time and is a diagnostic, not part of the single-point interface.Source: C. Component Replacement and Error… · p. 5, Figure 3 · p. 6
  • Perturbation tests compare common-mode and differential errors of fixed norm, with their direction either resampled each decision or held fixed in the body frame for a mission.The tests run at Re = 100, 175, and 225; performance difference is defined as common-mode success minus differential success.Source: C. Component Replacement and Error… · p. 5

D. Statistical Analysis

The analysis uses paired mission-level comparisons to quantify history-channel contributions and uncertainty. In held-out cylinder wakes, past flow improves success over current-only inputs, while its incremental contribution depends on which other historical channels are available.

D. Statistical Analysis

  • Eight prespecified contrasts are assessed with paired mission-bootstrap samples and simultaneous 95% intervals; intervals are conditional on fixed observers and controller, not retraining variability.Individual success rates use pointwise intervals, while geometry-transfer and intervention analyses use descriptive pointwise intervals.Source: D. Statistical Analysis · p. 6

A. Transfer to Held-Out Cylinder Wakes

  • At Re = 205 and 240, Flow history achieves 84.38% and 80.60% success, versus 91.80% and 85.16% for Direct spatial sensing and 53.65% and 47.27% for Current only.Source: A. Transfer to Held-Out Cylinder Wakes · p. 6
  • 21.61 pp [15.49, 27.74] and 22.01 pp [15.76, 28.26]: adding past flow (FM −M) increases success at Re = 205 and 240.Both simultaneous 95% intervals lie above zero.Source: A. Transfer to Held-Out Cylinder Wakes · p. 6
  • 30.99 pp [24.18, 37.80] and 26.30 pp [19.80, 32.81]: the combined-history contrast (FM −C) at Re = 205 and 240.The interaction contrasts are −9.11 pp [−15.04, −3.19] and −11.33 pp [−17.65, −5.00], indicating a smaller incremental past-flow benefit when past goal/yaw is available.Source: A. Transfer to Held-Out Cylinder Wakes · p. 7

B. Transfer Across Obstacle Geometries

Transfer depends on wake geometry: Flow history stays close to direct sensing in square-prism wakes but declines in triangular-prism wakes. Replacing its predicted lateral difference with the measured difference restores much of the lost performance, including under geometry shift.

B. Transfer Across Obstacle Geometries

  • In square-prism wakes, Flow history stays within 3.4 pp of direct spatial sensing at all three Reynolds numbers, while its gain over Current only never exceeds 3.2 pp.Current center-flow and task observations already support high success on these missions, leaving limited additional benefit from flow history or direct lateral sensing.Source: B. Transfer Across Obstacle Geometries · p. 7
  • At Re = 205 and 240 in triangular-prism wakes, Flow history achieves 53.26% and 47.40% success, versus 68.36% and 64.84% for direct spatial sensing.At Re = 100, Flow history reaches 83.98%, 21.22 pp above Current only but 14.46 pp below direct spatial sensing.Source: B. Transfer Across Obstacle Geometries · p. 7

C. Lateral-Difference Repair Across Flow Conditions

  • In held-out circular-cylinder wakes, difference repair raises Flow history success to 92.06% at Re = 205 and 85.03% at Re = 240, near direct sensing at 91.80% and 85.16%.Full history and Spatial ridge show the same qualitative benefit of difference repair despite different unmodified performance.Source: C. Lateral-Difference Repair Across… · p. 7
  • In the triangular-prism wake at Re = 240, difference repair raises Flow history success from 47.40% to 66.28%, versus 64.84% for direct spatial sensing.Mean repair yields 46.61%, close to the unmodified estimate; the interventions identify lateral-difference accuracy as a major performance determinant across Reynolds-number and geometry shifts.Source: C. Lateral-Difference Repair Across… · p. 7

D. Effects of Error Structure and Persistence

At equal added-error magnitude, persistent differential perturbations impose much larger navigation penalties than common-mode perturbations, while resampled-direction gaps are small. The authors recommend preserving lateral differences and limiting persistent errors, but the evidence is limited to simulated two-dimensional wakes, one fixed controller, and body-frame ambient-velocity observations.

D. Effects of Error Structure and Persistence

  • At Re = 205 and 240, Spatial ridge achieves 80.86% and 78.12% success versus 55.86% and 54.69% for Space–time ridge, despite larger lateral-difference MSE.Spatial ridge lateral-difference MSE is 11.180 and 11.179, compared with 10.542 and 9.629 for Space–time ridge.Source: D. Effects of Error Structure and… · p. 7
  • 25.00 pp [19.53, 30.47], 20.31 pp [14.84, 25.78], and 19.92 pp [14.06, 25.78] lower success under persistent differential than common-mode perturbations at Re = 100, 175, and 225.Each added lateral-error vector has magnitude 0.10U∞; intervals are pointwise 95% paired-bootstrap intervals.Source: D. Effects of Error Structure and… · p. 8
  • With resampled perturbation directions, the success gap between differential and common-mode perturbations is 0.78, 2.34, and 3.52 pp at Re = 100, 175, and 225, respectively.Source: D. Effects of Error Structure and… · p. 8

VII. DISCUSSION AND CONCLUSION

  • For this controller, the findings motivate observers that preserve the lateral difference and limit persistent errors, with closed-loop evaluation complementing reconstruction metrics.Source: §VII · p. 8
  • The study is limited to simulated two-dimensional wakes, one fixed controller, and body-frame ambient-velocity observations; hardware deployment also requires motion compensation and body-flow characterization.Broader flow conditions, independently trained controllers, and hardware experiments are needed to assess generality and practical sensing requirements.Source: §VII · p. 8
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