Aim 1 — Soft Robotic Fish

Nebula

A continuously flexible, solid-state robotic trout actuated by HASEL artificial muscles — designed, fabricated, and characterized at three damped natural frequencies in quiescent water.

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Design & Fabrication

Solid-state soft robotic trout with embedded HASEL actuators

Nebula is a continuously flexible robotic propulsor inspired by cartilaginous BCF fish. It is solid-state — no rigid linkages — with a flexible TPU skeleton encased in Ecoflex 00-10 silicone. HASEL muscles embedded antagonistically across the spine produce lateral contractions that passively induce BCF swimming kinematics.

Skeleton

3D-Printed TPU Spine

E ≈ 2.41 GPa. Variable cross-section (trout profile equation). ABS standoffs position HASELs away from neutral axis to maximize moment arm.

Body

Ecoflex 00-10 Silicone

High-compliance cast around actuated skeleton. Two-part ABS mold, vacuum degassed. Silpoxy adhesive bonds silicone to TPU. Wiring routed internally through the head.

Actuation

HASEL Artificial Muscles

Peano-HASEL pouches in antagonistic configuration. Alternating square-wave signals at 8 kV, offset by half-period. Artimus Robotics portable HV supply (PS1-10-005-01).

DIC Finish

Speckle Paint

NoVOCs Matte Psycho Paint (Smooth-On). Black dots on white base by airbrush. Sealed with clear coat. Enables full-field 3D Digital Image Correlation.

Mounting

Cantilever Mount

Aluminum tether bolted through anterior body. ASA plastic head halves (Silpoxy-bonded, Ecoflex-filled). Fixed–free cantilever boundary condition.

Dimensions

Key Geometry

Body ≈ 306 mm (without head), 456 mm with head. Caudal fin width ≈ 132 mm. Target: ≥ 4 in. pk-pk deflection at first natural frequency.

Design Rationale — Anterior Actuation: EMG studies show that at cruising speeds fish use mainly anterior muscles while the posterior body passively carries the traveling wave. Nebula places two HASEL actuators anteriorly with a passive tail.

Fabrication Timeline

Jul 2022

Fish Jr. prototype

Smaller test of silicone casting methods. LDV showed minimal deformation — body too thick. Prompted slimming Nebula.

Oct 2022

Nebula fabrication complete

Pre-cast LDV tests (Aug), post-cast air tests (Oct).

Nov 2022 – Feb 2023

Aquarium testing

Modal, LDV, and DIC experiments in quiescent water.

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Kinematic Testing

DIC + LDV characterization in quiescent water at three natural frequencies

Characterized at three damped natural frequencies in an aquarium. A Laser Doppler Vibrometer (LDV) measured single-point out-of-plane velocity/displacement at the caudal fin tip. 3D Digital Image Correlation (DIC) gave full-field surface deformation. The two systems were in agreement, cross-validating both measurements.

1.12 Hz

Mode 1 natural frequency

3.5 Hz

Mode 2 natural frequency

11 Hz

Mode 3 natural frequency

74 mm/s

Peak tip velocity (Mode 1)

9.6 mm

Tip displacement (Mode 1)

1D Tailbeat Data

Excitation Freq (Hz)	Tip Velocity (mm/s)	Tip Displacement (mm)	ω·w (mm/s)	Traveling Index Ti
1.12	74, 70	9.6, 9.4	67.5	Ti₁ = 0.065
3.5	51, 54	1.36, 1.40	31	Ti₂ = 0.28
11	47	0.65	45	Ti₃ = 0.14

Key insight — velocity vs. displacement: Despite a ~15× drop in displacement from Mode 1 to Mode 3, tail-beat velocity only roughly halves because ẇ = ωw. Higher modes maintain good propulsive velocity with significantly less drag-inducing body deviation from streamline shape.

MAC Matrix — Sim vs. Experiment

	Exp Mode 1	Exp Mode 2	Exp Mode 3
Sim Mode 1	0.989	0.098	0.077
Sim Mode 2	0.086	0.755	0.152
Sim Mode 3	0.186	0.027	0.559

High diagonal = strong agreement. Mode 1 near-perfect (MAC > 0.90); Mode 3 weaker.

Measurement Methods

LDVLaser Doppler Vibrometer — Doppler-based out-of-plane velocity at caudal tip

DIC3D Digital Image Correlation — full-field surface deformation from speckle images

TiTraveling Index (Complex Orthogonal Decomp.) — 0 = standing, 1 = traveling wave

MACModal Assurance Criterion — correlation between sim and experimental mode shapes

3D finding — caudal cupping: Full-field DIC revealed passive cupping of the caudal fin about the rostro-caudal axis, increasing with excitation frequency. Negligible at Mode 1, prominent at Mode 3. Cupping may alter hydrodynamic length, stiffness, and trailing-edge vortex shedding.

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Electro-Hydro-Elastic (EHE) Simulation

Nonlinear beam model of the submerged soft robotic propulsor

The EHE model treats Nebula's body as a nonlinear Euler-Bernoulli beam. Fluid coupling uses Lighthill's elongated body theory (quiescent) or linearized potential flow (uniform flow). Actuator forcing uses a linearized HASEL constitutive relation (Kellaris 2019). The Galerkin method with N = 20 basis functions gives the frequency response.

Structural

Nonlinear Euler-Bernoulli Beam

Variable cross-section piecewise EI(x), m(x) tuned to fabrication drawing. Skeleton + silicone + HASEL contributions. ABS standoffs added. Fluid added mass C_a = 0.5.

Fluid — Quiescent

Aureli Nonlinear Damping

Nonlinear fluid damping from vortex shedding. Lighthill slender-body theory for added mass. Linearized about harmonic motion; root-finding for the nonlinear eigenvalue.

Fluid — Uniform Flow

Linearized Potential Flow

Validated against LDV/DIC data from a piezo-actuated aluminum propulsor in the water tunnel. Good agreement at low reduced frequencies.

Actuator

Linearized HASEL Model

Voltage → force → beam deflection. Silicone mass and stiffness added directly to substrate matrices. Piecewise functions along body length for both actuator and body geometry.

Model Corrections to Match Experimental Data

1Silicone width/thickness via piecewise functions (not trout equation alone)

2Skeleton mass reduced at punched holes (modified density function)

3ABS standoff mass + stiffness added to skeleton

4Silicone mass multiplier 0.75; stiffness multiplier 1.3

5Fluid added mass coefficient reduced to C_a = 0.5

6Basis function count increased to 20 for convergence

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Nebula v2

Redesigned robot — presented at RoboSoft 2025, Lausanne

After a broken HASEL in Nebula v1 (Dec 2023) halted water tunnel testing, Nebula v2 was designed with three independently controllable muscle sets, embedded strain gauges, and a 6-axis load cell. Results at RoboSoft 2025 (Lausanne, Apr 2025) demonstrated a 44% thrust increase via sequential phase offset actuation at Mode 2.

RoboSoft 2025 paper: K. Soto, I. Hess, B. Schrader, S. He, and P. Musgrave, "Improving Swimming Performance in Soft Robotic Fish with Distributed Muscles and Embedded Kinematic Sensing," IEEE RoboSoft 2025, Lausanne. doi:10.1109/RoboSoft63089.2025.11020876

7.9 mN

Max thrust — Mode 1 (2 Hz), all in phase

5.0 mN

Thrust — Mode 2 (8 Hz), synchronized

7.2 mN

Thrust — Mode 2 with phase offset

+44%

Thrust improvement with phase offset

4×

Traveling wave content increase

Key Upgrade

3 Distributed Muscle Sets

Three axially-distributed HASEL sets (vs. one pair in v1). Sequential phase offset increases traveling wave content and thrust at Mode 2.

Sensing

Embedded Strain Gauges

Gauges along the TPU spine, 10 channels. Strain-to-displacement via spline interpolation between sensor locations. Verified fixed broken gauge before experiments.

DAQ

dSpace + Artimus ARDI

All signals through dSpace. Artimus 8-channel ARDI power supply. Python phase-offset code for 6 channels. BNC-to-pigtail for strain gauges.

Key Finding

Mode 2 Advantages

Mode 2 (8 Hz) generates nearly as much thrust as Mode 1 (2 Hz) with only ~25% tail displacement — advantageous for confined environments requiring a small kinematic envelope.

Note on dry testing: TPU-only dry testing is not fully representative — the number of ground wires changed between tests and wires are slightly off the neutral axis, affecting mass distribution.