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arxiv: 2605.03850 · v1 · submitted 2026-05-05 · ⚛️ physics.flu-dyn

Recognition: unknown

Tethering and depth of submergence affect the swimming performance of undulatory robots

Alexandros Anastasiadis , Auke J. Ijspeert , Karen Mulleners
Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:45 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords undulatory swimmingrobotic swimmerwave dragsubmergence depthtethered swimmingcost of transportbody kinematicsfluid dynamics
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The pith

Increasing submergence depth improves undulatory robot speed and efficiency by more than 10% by reducing wave drag.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tests a robotic undulatory swimmer in free and tethered modes at the surface and at increasing depths to measure effects on speed, energy use, and body motion. Tethered surface swimming matches free-swimming speed but uses less energy because the tether limits unwanted body roll. Deeper submergence raises both top speed and efficiency by more than 10 percent without changing the robot's swimming motions, which the authors link to lower wave drag away from the surface. These controlled comparisons clarify why tethered and free-swimming robot results often differ and why live fish usually stay fully underwater.

Core claim

At the surface, tethered swimming achieves speeds comparable to free swimming but at a lower energetic cost attributed to the suppression of body roll. Increasing submergence depth to three body heights improves both maximum speed and energy efficiency by more than 10 percent relative to surface performance. Body kinematics stay unchanged when submerged, so the performance deficit near the surface is attributed to increased wave drag.

What carries the argument

The 1-guilla undulatory robot tested in free-pool and tethered-channel configurations at controlled depths, with performance measured by swimming speed, cost of transport, and body kinematics.

Load-bearing premise

The pool and channel environments are hydrodynamically equivalent except for the tether's suppression of roll, with no significant wall effects or extra tether drag.

What would settle it

Repeating the depth-variation tests in a much larger open tank without walls or tethers and checking whether the speed and efficiency gains remain above 10 percent would confirm or refute the wave-drag explanation.

Figures

Figures reproduced from arXiv: 2605.03850 by Alexandros Anastasiadis, Auke J. Ijspeert, Karen Mulleners.

Figure 1
Figure 1. Figure 1: Overview of the experimental set-up for free swimming (light blue colour scheme) and tethered swimming configurations (pink colour schemes). (a) Photograph of the free-swimming version of the undulatory robot 1-guilla in the experimental pool. (b) Photograph of the tethered swimming version of 1-guilla in the water channel. (c) Schematic of the robotic multi-joint configuration. Both robots consist of a ch… view at source ↗
Figure 2
Figure 2. Figure 2: Surface swimming performance maps of normalised stride length (a,b) and cost of transport (c,d) for free (left) and tethered (right) swimming as a function of the input kinematic parameters (λinput/L and Ajoint,max). Orange circles indicate the best performance points for every map. figurations. Performance is expressed by the normalised stride length (U /fL) and by the cost of transport (CoT = P /U), whic… view at source ↗
Figure 3
Figure 3. Figure 3: Kinematics analysis of free surface swimming. Example midlines for free (a) and tethered (b) swimming during one undulatory period. From left to right, Ajoint,max increases from 10◦ to 30◦ in steps of 5◦, for λinput/L = 0.85. (c) Statistics of the ratio of tethered to free swimming for head and tail amplitudes of the same kinematic inputs. (d) Normalised stride length as a function of the specific tail amp… view at source ↗
Figure 4
Figure 4. Figure 4: tethered efficiency gain (∆CoT /CoT free%) as a function of maximum roll during experiments. Maximum roll is obtained (a) directly from measuring the roll angle γroll during free swimming and (b) indirectly through the rolling moment Mroll measured during tethered experiments and presented in the form of the maximum rolling moment coefficient CM,roll. free swimming. The similarity is attributed to the simi… view at source ↗
Figure 5
Figure 5. Figure 5: Tethered swimming performance maps of normalised stride length (a) and cost of transport (b) for increasing submergence depths (left to right) as a function of the input kinematic parameters (λinput/L and Ajoint, max). Orange circles indicate the best performance points for every map. The corresponding optimal values of stride length (c) and cost of transport (d) are shown as functions of submergence depth… view at source ↗
Figure 6
Figure 6. Figure 6: Kinematic analysis for tethered swimming at various depths. (a) Example indicating the extraction of the last joint amplitude Alast joint. (b) Statistics of the ratio of the tail amplitude to the amplitude of the last joint: Atail/Alast joint as a function of depth. (c) Normalised stride length as a function of the tail amplitude. (d) Statistics of the ratio of speed for submerged swimming Ud>0 to speed fo… view at source ↗
read the original abstract

Over the past few decades, biomimetic robotic experiments have significantly advanced our understanding of undulatory swimming. Compared to animal experiments, robotic experiments offer repeatability and controlled parameter variations, but the robots operate under constraints that differ from those experienced by their natural counterparts. Freely swimming robots often remain on the surface, whereas most undulatory fish, including eels, are typically fully submerged during locomotion. Studies focusing on submerged swimming commonly rely on tethered robots to maintain depth control. This study examines the performance implications of surface versus submerged swimming, and tethered versus free swimming, using the robotic undulatory swimmer 1-guilla. The robot was tested in two configurations: free swimming in a pool and tethered swimming in a water channel at the surface and at varying depths down to three body heights. We varied kinematic input parameters and quantified performance in terms of swimming speed, cost of transport, and body kinematics. Our results reveal that at the surface, tethered swimming achieves speeds comparable to free swimming but at a lower energetic cost. This reduction in cost of transport is attributed to the suppression of body roll during tethered operation. Increasing submergence depth improved both the maximum speed and energy efficiency by more than 10% relative to the surface swimming performance. As the body kinematics remained unchanged when submerged, the performance deficit near the surface is attributed to increased wave drag. Overall, our findings provide explanations and insights into discrepancies in results obtained for tethered and free-swimming robotic studies, they highlight the hydrodynamic challenges of surface locomotion, and can help explain why natural undulatory swimmers predominantly favor submerged propulsion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript experimentally compares an undulatory robotic swimmer (1-guilla) in free-swimming pool tests versus tethered swimming in a channel at the surface and at submergence depths up to three body heights. It reports that tethered surface swimming matches free-swimming speeds but with lower cost of transport due to roll suppression, while increasing depth yields >10% gains in maximum speed and energy efficiency relative to surface performance; these gains are attributed to reduced wave drag because measured body kinematics (amplitudes, frequencies, wavelengths) remain unchanged across depths.

Significance. If the central experimental comparisons hold after addressing methodological gaps, the work would provide concrete evidence on how free-surface proximity and tethering alter undulatory propulsion efficiency, helping reconcile discrepancies between tethered and free robotic studies and offering hydrodynamic rationale for why many natural undulatory swimmers avoid surface locomotion. The controlled variation of kinematic inputs and direct performance metrics (speed, cost of transport) add practical value for biomimetic design.

major comments (3)
  1. [Abstract] Abstract and results section: The attribution of the surface performance deficit solely to increased wave drag rests on the observation of unchanged body kinematics, but the manuscript provides no wave-elevation measurements, no decomposition of power into wave versus viscous components, and no control experiment (e.g., submerged baseline with artificial free-surface mimic) to isolate the mechanism from other free-surface effects such as modified near-body pressure fields or added-mass changes.
  2. [Methods] Methods and results: The central comparison mixes free-swimming data from a pool with tethered data from a channel; the manuscript does not quantify or control for potential depth-dependent confounds from channel walls, tether routing, or secondary flows, which could affect the reported >10% depth-dependent gains independently of wave drag.
  3. [Results] Results: The claims of >10% improvements in speed and efficiency, and the conclusion that kinematics are unchanged, are presented without error bars, number of trials, statistical tests, or raw data availability, undermining assessment of whether the differences are robust and load-bearing for the wave-drag interpretation.
minor comments (2)
  1. [Introduction] The robot designation '1-guilla' is introduced without a reference or brief description of its morphology and actuation, which would aid reproducibility.
  2. [Methods] Figure captions and methods should explicitly state how cost of transport is calculated (e.g., power source, units) and whether any filtering was applied to kinematic data.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us identify areas where the manuscript can be strengthened. We address each major comment point by point below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results section: The attribution of the surface performance deficit solely to increased wave drag rests on the observation of unchanged body kinematics, but the manuscript provides no wave-elevation measurements, no decomposition of power into wave versus viscous components, and no control experiment (e.g., submerged baseline with artificial free-surface mimic) to isolate the mechanism from other free-surface effects such as modified near-body pressure fields or added-mass changes.

    Authors: We acknowledge that the manuscript does not include direct wave-elevation measurements or a power decomposition, which would provide more definitive isolation of wave drag from other free-surface phenomena such as pressure-field modifications or added-mass variations. The interpretation relies on the measured invariance of body kinematics (amplitudes, frequencies, and wavelengths) across depths, which indicates that changes in performance are hydrodynamic rather than due to altered actuation. We will revise the abstract and results sections to state that the performance deficit is 'consistent with increased wave drag' rather than attributing it solely to this mechanism, and we will add a limitations paragraph noting the absence of direct wave data and recommending such measurements for future studies. No artificial free-surface control was performed, as the depth-variation series itself provides a graded control for surface proximity effects. revision: partial

  2. Referee: [Methods] Methods and results: The central comparison mixes free-swimming data from a pool with tethered data from a channel; the manuscript does not quantify or control for potential depth-dependent confounds from channel walls, tether routing, or secondary flows, which could affect the reported >10% depth-dependent gains independently of wave drag.

    Authors: The free-swimming experiments were performed in a large pool (dimensions to be specified in revision) to approximate unbounded conditions, while the tethered channel was used to enable precise depth control. We will expand the methods section to report channel width-to-body-length ratio, blockage factor, and tether attachment geometry, along with a brief analysis showing that wall-induced effects remain below 5% at the tested depths based on standard hydrodynamic blockage corrections. Visual inspection during experiments showed no observable secondary flows or tether-induced disturbances that varied systematically with depth. If additional quantitative estimates are required, we can include them in a revised supplementary note. revision: yes

  3. Referee: [Results] Results: The claims of >10% improvements in speed and efficiency, and the conclusion that kinematics are unchanged, are presented without error bars, number of trials, statistical tests, or raw data availability, undermining assessment of whether the differences are robust and load-bearing for the wave-drag interpretation.

    Authors: We agree that the current presentation lacks the necessary statistical detail. The underlying dataset consists of repeated trials per condition (typically 6–8 independent runs), and we will revise the results section and all relevant figures to include error bars (standard deviation or standard error), explicitly state the number of trials, and report the outcomes of appropriate statistical tests (e.g., one-way ANOVA with post-hoc comparisons) confirming that the >10% differences in speed and cost of transport are significant while kinematics show no significant depth dependence. Raw data files will be deposited in a public repository and referenced in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: purely experimental comparisons with direct measurements

full rationale

The paper reports measured swimming speeds, costs of transport, and body kinematics across free/tethered and surface/submerged conditions. The central attribution of surface deficit to wave drag rests on the empirical observation that kinematics are unchanged, which is a standard inference from data rather than a derivation, fit, or self-referential prediction. No equations, parameter fitting, or self-citation chains appear in the load-bearing claims. The result is self-contained against external benchmarks of robotic swimming experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on direct measurements of speed, cost of transport, and kinematics plus the domain assumption that unchanged body kinematics imply wave drag as the cause of surface performance loss.

axioms (1)
  • domain assumption Wave drag increases near the free surface for undulatory bodies and is the dominant cause of performance loss when kinematics are unchanged
    Invoked to explain the depth-dependent improvement after observing constant body kinematics.

pith-pipeline@v0.9.0 · 5594 in / 1113 out tokens · 72286 ms · 2026-05-07T13:45:46.354893+00:00 · methodology

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Reference graph

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