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Water Surface Swimming in a Centipede and its Robophysical Model

Zhaochen J. Xu, Delfin Aydin, Abdullah Mustafa, Margarita B. Levin, Jianfeng Lin, Tianyu Wang, Daniel I. Goldman

arXiv:2609.27088v1cs.ROphysics.bio-ph

TL;DR

Multi-legged robots’ bulky, non-streamlined morphology complicates water-surface swimming, and how their distributed legs contribute to direct-wave swimming is not established. The paper combines centipede kinematics with a robophysical model to test body-wave direction, leg coordination, and leg morphology. Direct body waves produce forward motion under tested conditions, while directional compliance increases displacement under matched anti-phase actuation.

  • Problem

    How multi-legged robots’ body and leg morphology supports water-surface swimming remains poorly understood, limiting the basis for extending their terrestrial capabilities to aquatic environments.

    Source: §I · p. 1

  • Method

    The study combines centipede kinematics with robophysical experiments that vary body-wave direction, leg coordination, and leg design.

    Source: §I · p. 2, B. Directionally Compliant Limb Design · p. 4, D. Leg-Wave Kinematics · p. 4

  • Results

    Directionally compliant legs increase displacement from approximately 0.08 to 0.21 body lengths per cycle relative to rigid legs under matched anti-phase actuation.

    Source: C. Directional Leg Compliance… · p. 5

  • Takeaways & Limitations

    The experiments identify body–leg coordination and directional leg compliance as gait and design guidance for the tested multi-legged platform.

    Source: §IV · p. 5, §V · pp. 6–7

Abstract

from arXiv · show

Elongate multi-legged robots use coordinated body waves and distributed legs to move through cluttered terrestrial environments. However, as housing actuators for independent leg control can require bulky body segments, their non-streamlined body and limb structure makes it difficult to achieve swimming capability comparable to their terrestrial locomotor performance. At the water surface, we found that the multi-legged robots we tested unexpectedly moved backward: their body waves traveled in the same direction as their displacement, i.e., swimming with a direct wave. We found similar behavior in the centipede \textit{Lithobius forficatus}, which swims with a direct body wave and periodic leg movement. To study how distributed legs contribute to direct-wave swimming, we analyze animal kinematics and develop a multi-legged robophysical model that allows independent variation of leg morphology and stiffness, body-wave direction, and leg coordination. Robophysical experiments show that direct body waves produce consistent forward motion under the tested conditions and that swimming performance depends on body--leg coordination. Additionally, directionally compliant legs increase displacement from approximately 0.08 to 0.21 body lengths per cycle relative to rigid legs under matched anti-phase actuation. These findings clarify how distributed appendages contribute to surface swimming and establish gait and morphology design principles for extending multi-legged field robots from terrestrial locomotion into aquatic environments.

I. INTRODUCTION

The paper asks how multi-legged robots’ terrestrial body-and-leg morphology can support water-surface locomotion. Animal kinematics and robophysical experiments test how body waves, leg coordination, and leg design shape direct-wave swimming.

I. INTRODUCTION

  • The tested multi-legged robot moved backward relative to its heading under a forward terrestrial command, with its body wave traveling in the direction of displacement.The paper terms this relationship direct-wave swimming.Source: §I · p. 1
  • Bulky body segments and protruding legs may alter swimming response relative to smooth, slender undulatory swimmers.Independent leg actuators can require bulky body segments, constraining streamlining.Source: §I · p. 1
  • Prior work does not establish how distributed legs contribute to direct-wave swimming in laterally undulating robots at the water surface.Body bending shifts leg attachment points, while leg actuation and passive deformation change leg motion relative to the water.Source: §I · p. 2
  • The study combines animal kinematics with a six-link robophysical model to test how body–leg coordination affects surface swimming.Animal body waves and bilateral leg timing guide controlled tests of robot gait and leg design.Source: §I · p. 2
  • Directionally compliant legs increase displacement from approximately 0.08 to 0.21 body lengths per cycle relative to rigid legs under matched anti-phase actuation.The contribution also reports obstacle-array locomotion.Source: §I · p. 2
  • The paper identifies body–leg coordination and leg morphology as design variables for extending terrestrial multi-legged robots into aquatic environments.The paper reports that direct body waves produce forward motion under tested conditions and that preferred bilateral leg phase depends on body-wave spatial frequency.Source: §I · p. 2

II. CENTIPEDE SWIMMING

The authors quantify centipede body kinematics to identify coordination patterns that guide robot experiments. In a representative sequence, a tail-to-head body wave accompanies forward swimming.

II. CENTIPEDE SWIMMING

  • Centipede body and leg kinematics are quantified to identify coordination patterns that can guide robot experiments.Earlier measurements reported only 0.05 ± 0.02 body lengths of coasting after body undulation ceased.Source: §II · p. 2

A. Direct Body-wave Propagation in Centipede Swimming

  • The analysis projects mean-subtracted curvature onto the first two spatial modes to represent progression around the gait cycle separately from instantaneous animal position.The cyclic body-shape phase is defined from the two modal coefficients.Source: A. Direct Body-wave Propagation in… · p. 2
  • A curvature trough advances from tail to head, with fitted phase changing by 351.2° and a dominant frequency of 4.99 Hz in the representative 23.0 mm specimen.The phase is nearly linear along the body (R2 = 0.994); the period is 0.200 s.Source: A. Direct Body-wave Propagation in… · p. 2

B. Traveling Limb Waves

The section quantifies traveling leg waves in the centipede and relates them to the direct body wave. A figure shows the body wave’s tail-to-head progression across tracked configurations and curvature over time.

B. Traveling Limb Waves

  • The number of leg waves spanning the array is calculated from the mean unwrapped phase lag between adjacent legs and the number of legs.Positive phase lag uses the tail-to-head convention.Source: B. Traveling Limb Waves · pp. 2–3
  • Figure 2 tracks body configurations over two periods, plots the first two PCA modes, and shows curvature over four periods, with diagonal bands indicating tail-to-head wave travel.Clockwise rotation in the PCA profile denotes tail-to-head phase progression.Source: Figure 2 · p. 3
  • The analyzed sequence shows 14 tracked legs per side oscillating at 4.98 Hz, with a mean neighboring-leg phase lag of 31.6° corresponding to 1.14 waves across a 14-leg side.Left and right phase lags are 30.3° and 32.9°, respectively.Source: B. Traveling Limb Waves · p. 3
  • Regional leg phases progress from approximately 0° anteriorly to 151° centrally and 309° posteriorly, forming an earlier-peaking posterior pattern that travels tail to head.The phase convention treats positive neighboring-leg lag as tail-to-head.Source: B. Traveling Limb Waves · p. 3

C. Contralateral Leg Coordination

The centipede’s left and right legs move nearly in anti-phase, and their coordination is quantified from corresponding leg-peak timing.

C. Contralateral Leg Coordination

  • Contralateral phase is calculated from the mean delay between left and right leg peaks relative to the gait period.For each leg pair, the mean peak-time delay is compared with that pair’s gait period.Source: C. Contralateral Leg Coordination · p. 3
  • The leg wave travels from posterior to anterior, with spatial frequency ξleg ≈ 1.14.The figure shows successive leg-angle peaks propagating from tail to head.Source: Figure 3 · p. 3
  • The mean contralateral phase was 170.2°, just 9.8° from perfect anti-phase at 180°.The figure reports a mean of 170.2°; the text gives the difference from perfect anti-phase as 9.8°.Source: C. Contralateral Leg Coordination · p. 3, Figure 3 · p. 3

III. ROBOPHYSICAL MODEL AND GAIT SETUP

The robophysical platform uses a modular six-link body with independently actuated legs and foam buoyancy to operate at the water surface.

III. ROBOPHYSICAL MODEL AND GAIT SETUP

  • The platform has six rigid cabins joined by five actuated yaw joints that generate body undulation.Each cabin carries one leg on each side, for twelve distributed legs.Source: A. 6-link Robophysics Platform · p. 3, Figure 4 · p. 4
  • Each self-contained cabin allows link count and leg configuration to be changed independently.The cabin modules connect through the body-servo horns.Source: A. 6-link Robophysics Platform · pp. 3–4
  • Foam sections attached along the body provide buoyancy and keep the robot at the water surface.Buoyancy is provided externally rather than integrated into the cabin geometry.Source: A. 6-link Robophysics Platform · p. 3
  • Each of the five body yaw joints has a dedicated servo, while smaller servos independently actuate each leg at its root.An ESP32 computes gait targets and sends them to the servo-driver board.Source: A. 6-link Robophysics Platform · p. 4

B. Directionally Compliant Limb Design

The robot’s four-link legs combine an actuated root with passive, directionally compliant distal joints that bend under load and reduce resistance during recovery.

B. Directionally Compliant Limb Design

  • Each leg has a servo-driven proximal link and three passive revolute joints with torsional springs.The body-to-leg connection is actuated, not spring-driven.Source: B. Directionally Compliant Limb Design · p. 4
  • Passive-joint stiffness decreases distally, supporting fluid load near the root while allowing greater distal reconfiguration.The gradient is k1 > k2 > k3; the leg bends as a distributed structure rather than mainly at one hinge.Source: B. Directionally Compliant Limb Design · p. 4
  • One-sided stops keep the leg extended during the power stroke, then disengage during recovery so springs bend the leg and reduce projected area and hydrodynamic resistance.The springs restore the leg toward its extended configuration when external load decreases or reverses.Source: B. Directionally Compliant Limb Design · p. 4

C. Body-wave Kinematics

The platform reproduces the centipede’s planar lateral body undulation by prescribing a serpenoid traveling wave across its actuated yaw joints.

C. Body-wave Kinematics

  • The platform reproduces planar lateral body undulation with a serpenoid traveling wave across the actuated yaw joints.The wave is prescribed for each joint angle αi over time.Source: C. Body-wave Kinematics · p. 4
  • The body-wave prescription specifies amplitude Abody, spatial frequency ξb, temporal frequency f, and angle offset φ.The spatial frequency denotes waves along the body.Source: C. Body-wave Kinematics · p. 4
  • The wave is indexed by yaw-joint index i and total joint count Ni.These quantities identify each joint’s position in the prescribed traveling wave.Source: C. Body-wave Kinematics · p. 4

D. Leg-Wave Kinematics

The leg command specifies how leg-pair angles vary, while paired-leg phase sets their bilateral coordination. Figure 5 relates body-wave direction to signed swimming displacement.

D. Leg-Wave Kinematics

  • The leg-angle command is parameterized by amplitude, spatial frequency, and temporal frequency, with six leg-pair locations and a neutral angle.The paired-leg phase difference is defined as right phase minus left phase; a difference of π gives anti-phase coordination.Source: D. Leg-Wave Kinematics · p. 4
  • Figure 5 compares commanded joint-angle heatmaps and representative configurations for opposing body-wave directions.Its signed displacement traces are normalized by body length over 5 s; head-to-tail waves produce backward motion and tail-to-head waves produce forward motion.Source: Figure 5 · p. 5
  • Both wave-direction conditions show displacement in the direction of wave propagation under the same gait.Curves and error bars report means and standard deviations across three trials.Source: Figure 5 · p. 5

IV. ROBOPHYSICAL EXPERIMENTS

The experiments test body-wave direction and bilateral leg coordination as determinants of swimming displacement. Direct waves produce forward motion, while the displacement benefit of active leg coordination varies with body-wave spatial frequency.

A. Direct Body-waves Produce Forward Swimming

  • Tail-to-head body waves produce 0.14 ± 0.01 BL forward displacement over 5 s, while head-to-tail waves produce −0.04 ± 0.03 BL (n = 3 per condition).The reported mean displacements differ by 0.18 BL, and both conditions move in the direction of wave propagation.Source: A. Direct Body-waves Produce Forward… · p. 5

B. Leg Coordination Modulates Performance

  • At ξb = 0.8, in-phase, quadrature, and anti-phase legs yield 0.15±0.04, 0.11±0.01, and 0.09 ± 0.01 BL/cycle, respectively, versus 0.06 ± 0.02 BL/cycle passive.At ξb = 1.0, anti-phase and quadrature each reach approximately 0.21 BL/cycle; at ξb = 1.2, anti-phase yields 0.20 ± 0.02 BL/cycle, exceeding quadrature and in-phase motion.Source: B. Leg Coordination Modulates… · p. 5
  • Across active-leg conditions, displacement peaks locally near ξb = 1.0, where anti-phase and quadrature produce the greatest measured displacement.The reported phase-dependent performance indicates that leg motion must be coordinated with body-wave spatial structure rather than selected independently.Source: B. Leg Coordination Modulates… · p. 5

C. Directional Leg Compliance Increases Performance

Directionally compliant legs outperform rigid legs under matched physical anti-phase actuation, and the selected direct-wave gait also progresses through tested obstacle arrays.

C. Directional Leg Compliance Increases Performance

  • One-sided stops keep legs extended during the power stroke, while spring joints permit bending during recovery to reduce projected area and hydrodynamic resistance.This passive reconfiguration supports a more effective stroke without adding sensing or another actuator.Source: C. Directional Leg Compliance… · p. 5
  • The robot’s local performance optimum near ξb = 1.0 is close to the animal’s measured spatial frequency, ξb = 0.94±0.07.The animal reference displacement is 0.24±0.02 BL/cycle (n = 5 trials from one specimen).Source: C. Directional Leg Compliance… · p. 6

D. Locomotion Through Hydro-Clutter

  • The direct-wave gait progresses through both compliant and rigid post lattices during repeated obstacle contact.The robot advances over a 30 s sequence in the compliant array and a 60 s sequence in the rigid array.Source: D. Locomotion Through Hydro-Clutter · p. 6

C. Directional Leg Compliance Increases Performance

  • At ξb = 1.0, directionally compliant legs produce 0.21 ± 0.01 BL/cycle versus 0.08 BL/cycle for rigid legs under physical anti-phase actuation.Across ξb = 0.8, 1.0, and 1.2, compliant legs produce more displacement than rigid legs; Figure 7C plots displacement per cycle against body-wave spatial frequency.Source: C. Directional Leg Compliance… · p. 5, Figure 7 · p. 6

V. CONCLUSION AND DISCUSSION

The conclusion links surface-swimming performance to body-wave direction, leg coordination, and directional compliance, and reports that the selected gait traversed hydro-clutter. It identifies body size and morphology as areas for future study.

V. CONCLUSION AND DISCUSSION

  • Opposite body-wave commands were associated with opposite mean swimming directions; leg phase changed displacement, and directional compliance increased swimming performance.The experiments identify body–leg coordination and leg mechanics as ways to improve direct-wave swimming.Source: §V · p. 6
  • The selected gait enabled hydro-clutter locomotion; Fig. 8 shows configurations in compliant and rigid post lattices at different times.The caption identifies sequences at 0, 10, and 30 s for the compliant lattice and 0, 14, and 60 s for the rigid lattice.Source: §V · p. 6, Figure 8 · p. 7
  • The robot’s highest measured displacement occurred near ξb = 1.0, close to the animal’s measured ξb = 0.94 ± 0.07 despite their size difference.The findings provide gait and leg-design guidance for the tested platform.Source: §V · p. 7
  • Future work will test how benefits from leg coordination and directional compliance depend on body size and morphology.Comparisons with other centipedes and polychaetes are proposed to assess how broadly the strategies apply.Source: §V · p. 7
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