# Fin ray-inspired, Origami, Small Scale Actuator for Fin Manipulation in
Aquatic Bioinspired Robots

Minh Vu1,2∗ Revathy Ravuri1 Angus Muir1 Charles Mackie1 Andrew Weightman3,4
Simon Watson1,4 Tim J. Echtermeyer1,2,5∗
1Department of Electrical and Electronic Engineering, The University of
Manchester
2Photon Science Institute, The University of Manchester
3School of Engineering, Faculty of Science and Engineering, The University of
Manchester
4Centre for Robotics and Artificial Intelligence, The University of Manchester
5National Graphene Institute, The University of Manchester
Manchester, M13 9PL, United Kingdom
∗Correspondence<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Fish locomotion is enabled by fin rays—actively deformable boney rods, which
manipulate the fin to facilitate complex interaction with surrounding water
and enable propulsion. Replicating the performance and kinematics of the
biological fin ray from an engineering perspective is a challenging task and
has not been realised thus far. This work introduces a prototype of a fin ray-
inspired origami electromagnetic tendon-driven (FOLD) actuator, designed to
emulate the functional dynamics of fish fin rays. Constructed in minutes using
origami/kirigami and paper joinery techniques from flat laser-cut
polypropylene film, this actuator is low-cost at £0.80 ($1), simple to
assemble, and durable for over one million cycles. We leverage its small size
to embed eight into two fin membranes of a 135 mm long cuttlefish robot
capable of four degrees of freedom swimming. We present an extensive kinematic
and swimming parametric study with 1015 data points from 7.6 hours of video,
which has been used to determine optimal kinematic parameters and validate
theoretical constants observed in aquatic animals. Notably, the study explores
the nuanced interplay between undulation patterns, power distribution, and
locomotion efficiency, underscoring the potential of the actuator as a model
system for the investigation of energy-efficient propulsion and control of
bioinspired systems. The versatility of the actuator is further demonstrated
by its integration into a fish and a jellyfish.

###### keywords:

Bio-inspired, Actuator, Aquatic

## Introduction

Fish are extraordinary swimmers. Their ability to navigate and manoeuvre
underwater with precision and efficiency has long fascinated scientists and
engineers seeking to further understand the hydrodynamic interaction[1, 2] and
develop novel aquatic propulsion systems [3, 4]. Central to their remarkable
locomotion is the intricate movement and control of their fins, which act as
multi-directional control surfaces that can efficiently manipulate the
surrounding water [5, 6, 7, 8]. In particular, the fin rays (_lepidotrichia_)
in ray-finned fishes (_Actinopterygii_) demonstrate a high level of individual
morphing ability [9, 10, 11], which collectively allows for three-dimensional
deformation of the fin. Replicating this elaborate functionality in a compact
and easily manufacturable actuator poses a significant engineering challenge,
requiring a combination of innovative design, suitable material properties,
advanced manufacturing techniques, and precise control mechanisms. Overcoming
these challenges would enable the creation of versatile robotic systems that
mimic biological fluid dynamics, opening up possibilities for environmentally
adaptive robots capable of complex, autonomous underwater tasks and
interactions.

Among ray-finned fishes, the fin ray serves two important purposes—stiffening
the fin under hydrodynamic load and bending the fin to interact with the
water, utilised to achieve complex fin movements and propulsion. The fin ray
is composed of two parallel halves (_hemitrich_); convex segmented flexible
beams that are tapered and joined at their tips[11, 9, 10]. They are linked
together along the length by a compliant core, which also connects them to the
fin’s membrane (Fig. 1A). Groups of antagonistic muscles are attached to the
base of each hemitrich; as they contract, one hemitrich is displaced relative
to the other, creating a shear moment, resulting in flexural deformation of
the fin ray [9, 11, 12, 10]. The complex interaction between the segmented
hemitrichs and the musculature generates a finely tuned mechanical response,
allowing the fins to achieve intricate deformation[9, 12, 13]. The
amalgamation of these individual deformations results in the emergence of
three-dimensional shapes from the two-dimensional fin membrane. While much
research has concentrated on the hydrodynamic advantages derived from the
passive curvature and flexibility (uniform and non-uniform) of fins[14, 15,
16], the areas of active curvature, tunable stiffness, and controlled
deformation remain underexplored[17, 18, 19].

The pursuit of emulating the biomechanical principles and characteristics of
fish fins has been driven by the increasing need for minimally invasive
automation in aquaculture [20, 21], marine monitoring [22, 23], and the study
of efficient mode of propulsion [24, 25, 26]. This breath of need has given
rise to a diverse range of approaches. Notable work such as SoFi [22], TunaBot
Flex [27], dolphin robots [28, 29], FinBot [30] chose to simplify the fin to a
homogeneous water foil and instead focus on the musculature to mimic the
movement seen in aquatic creatures such as tuna, salmon, and shark employing
body-caudal-fin (BCF) locomotion (Fig. 1C). Others, placing more emphasis on
fin kinematics, used motor-connected rods embedded in a membrane to replicate
the articulation of the fin, a technique often seen in robots using median-
paired-fins (MPF) locomotion, inspired by the undulating fins of rays,
knifefish, and cuttlefish[31, 32, 15, 33, 34, 35]. Despite these advancements,
achieving the nuanced fin control observed in nature remains elusive, relying
on the passive curvature resulting from the flexibility of the materials used
to construct the fin. Approaches using novel actuators like dielectric
elastomer actuators, shape memory alloys, ionic polymer–metal composites have
yielded varying results. Due to their inherent characteristics, the bending
motion is naturally achieved and closely mirrors the dynamic bending of fin
rays [36, 37, 38, 39, 40, 41, 42, 43]. However, they present their own set of
challenges in power and control, which limits the configuration and number of
actuators that can be implemented, ultimately constraining the design and
performance of the robot. This dichotomy between complexity in function and
simplicity in design poses a substantial challenge in developing dexterous
bioinspired aquatic robots. The high entry cost of equipment, materials, and
expertise creates a significant barrier for researchers, a problem that
becomes even more pronounced as the size of the robot decreases.

Figure 1: Fin ray-inspired actuator for aquatic robots
(A) A fin ray (lepidotrichium) at rest and while bending. The yellow parts
represents the collagen fibrils. The red arrow represents the pulling force
(B) FOLD actuator mimicking bending in fin ray. Pulling the tendon causes
deflection in the fin ray actuator. (C) Aquatic locomotion, the yellow
highlight indicates the propulsion mechanism. From top to bottom: Median
paired fin (MPF) propulsion, body-caudal fin (BCF) propulsion, jet propulsion.
(D) FOLD actuator integrated in three distinct bioinspired robot morphologies:
cuttlefish, fish, jellyfish.

We seek to unite these seemingly contradictory criteria by presenting the
initial proof of concept design and characterisation of fin ray-inspired
origami electromagnetic tendon-driven (FOLD) actuators (Fig. 1B and Movie S1).
The design is small, allowing for dense placement of actuators to enable
dexterous manipulation of a bionic fin while remaining adaptable in various
robotic systems. Constructed using origami/kirigami and paper joinery
techniques from a flat laser cut polypropylene film, the actuator is easily
and cheaply fabricated (£0.80/$1.00), without specialised equipment or
materials. These design principles are further extended to holistically design
three distinct underwater robot prototypes with a monolithic body with
actuators: a cuttlefish robot, a tuna robot, and a jellyfish robot (Fig. 1D),
demonstrating the efficacy and versatility of the actuator. Among these, the
cuttlefish robot demonstrates unprecedented capabilities in an aquatic robot
of that size, performing various swimming modes to achieve four independent
degrees of motion. Using videos, we were able to collect a large dataset to
identify the optimal kinematic parameters and confirm theoretical numerical
constants (such as Strouhal number [44, 45] and Specific Wavelength [46, 47]),
as well as to investigate the power-thrust relationship. Moreover, facilitated
by independent control over each fin ray, we explored the effects of wave
envelope on propulsion[48, 49]. These findings indicate that the platform
holds tremendous potential for enhancing biomimetic locomotion models and
studying complex hydrodynamic interactions.

## RESULTS & DISCUSSION

### Working principle and design overview

We propose a flat-constructed push-pull tendon-driven actuator to emulate the
antagonistic bending of fish fin rays. Measuring 33 mm long, 6 mm wide, and
consisting of a backbone (Fig. 2A) and tendon (Fig. 2B) made from laser-cut
polypropylene (PP) joined at the tip represent the structure of the
hemitrichs. The tendon transmits tension and compression force. Any buckling
of the tendon is limited by the series of hooks along the backbone, which act
like the collagenous core in a fin ray to maintain the separation between the
two halves while still allowing sliding motion. The bidirectional linear force
is generated using a voice coil actuator (VCA), utilising Lorentz force, the
magnetic field generated from a current running through a moving solenoid
repels/attracts a permanent magnet. Depending on the direction of the current,
the VCA pushes and pulls on the tendon, deflecting the fin ray. The design
does not feature any sealed chambers, allowing water to flow around the coil
and magnets. This allows for the coil to be water-cooled and the VCA to be
water-lubricated. This design further minimises the need for any additional
components and makes the actuator inherently pressure-tolerant.

### Laser-cut fabrication

We found that common polypropylene film folders with a thickness of 0.2 mm
match the required range of flexibility and durability. Laser cutting allows
for an affordable and quick method of iterative prototyping. Two different
settings are used during laser-cutting: a cut mode with approximately 0.6 W
laser power to fully cut through the PP sheet and an engraving mode with a
reduced 0.18 W laser power to only partially cut the PP sheet and facilitate
folds. These cuts and folds then allow the application of Origami/Kirigami,
pop-up and paper joinery design principles to construct out-of-plane elements
that can be mechanically joined without the use of adhesives[50]. The total
material cost of the actuator is less than £0.80 ($1), the full breakdown is
shown in Table S.1.

### Origami/Kirigami assembly

The assembly starts with step A (Figure 2A), the support structure is folded
into a T-shape, creating a spine that introduces rigidity along the length.
Subsequently, the rib-shaped flaps are folded up 90° and attached to the spine
with a clip mechanism, adding span-wise rigidity. The beam features two main
elements: T-shaped loops and a tab at the end of the beam to which a tendon
will be secured. The tab is in the shape of a sideways D, which folds up 90°
to insert into the slit on the tendon. The T-loops have rectangular horizontal
slits, measuring 1-by-2 mm, at the horizontal section of the T. They fold up
to create evenly distributed standing loops along the backbone (inset Fig.
2A). These T-loops act as guides for the tendon, suspending it 2 mm above the
backbone and ensuring that the tendon and backbone will remain parallel as the
fin ray flexes. Similar to the segmentation in a hemitrich, the cut-outs in
the T-loops contribute to the flexibility of the backbone and ultimately the
actuator’s bending profile. In step B, laser cut tendons (200 µm thick PP) are
slotted through the T-loops and secured to tab at the distal end of the
backbone. A slit in the tendon slightly smaller than the width of the D-shape
facilitates latching of the tendon with the D-shaped tab, as shown in Fig. 2B.

Figure 2: Assembly of paired fin ray.
(A) Folding assembly: 1. The polypropylene sheet is laser cut and engraved to
create the desired profile; 2. The flat laser cut profile is folded to create
the spine and ribs; 3. Tabs and slits are used to secure the structure. Tendon
hooks are folded up. (B) The tendon: Laser-cut from the PP sheet and attached
to the tab at the end of the backbone. (C) The assembly process of the coil:
1. The straw is laser cut to size; 2. The rings are glued on to create the
coil bobbin; 3. Copper windings are added. The tendon is attached using the
slits. (D) Magnets are attached on either side of the spine. (E) The coil is
slid over the limiter (black) and magnets to complete the assembly.

The VCA serves as the “muscle" to flex the fin ray. Unlike the biological
counterpart, which has antagonistic pairs of muscles to flex the fin ray, the
voice coil actuator performs both push and pull motion. As such, a single VCA
and a single tendon, located only on one side of the backbone, is sufficient
to induce upward and downward bending, further simplifying the implementation
of the fin ray. The stiffness of the tendon, originating from the PP sheet and
its geometry, in combination with the tightly spaced T-hooks that limit
buckling similar to the collagenous core in a fin ray, allows the application
of both tensile and compressive forces to the tendon. The coil bobbin is
manufactured using a 5 mm diameter polypropylene straw, which has a smooth low
friction surface finish (Fig. 2C). Slits are cut into the straw to attach the
tendon, and two lines are engraved 4 mm apart to attach two 9 mm diameter
rings, which secure the windings. The position of the windings on the bobbin,
relative to static magnets assembled in the subsequent step, has been
carefully chosen to ensure operating the VCA at its maximum mechanical output
power point and with maximum travel range. More details on the manufacturing
and characterisation of the VCA can be found in Supplementary Materials. 100
µm diameter polyimide-coated copper wire was used to wind the coil with the
maximum possible windings of $\approx$ 450, resulting in a resistance of
$\approx$ 25 $\Omega$ and inductance of $\approx$ 1 mH. The resulting coil
measures 12.5 mm in length with an outer diameter of 9 mm.

In the final step, three circular 4 mm diameter Neodymium magnets of 2 mm
thickness are attached to each side of the spine (Fig. 2D). Here, the
actuators are configured in a back-to-back arrangement of fin rays, i.e.
symmetry of the design allows securing the magnets to the spine by magnetic
forces between adjacent magnets on the left- and right-hand side of the spine.
To prevent the coil from sliding off and away from the magnet, a limiter
extends from underneath the magnets to limit the actuation stroke to ±2 mm. A
semicircular arrowhead allows the coil to slide over during the assembly but
prevents the coil from sliding back out. The VCA is capable of producing a
peak force of 490 mN at a current of 480 mA, the force profile follows a bell
curve shape along the stroke distance [51]. Full characterisation of the
force-displacement curves is shown in Supplementary Materials.

### Deflection characterisation

To characterise the deflection achieved by the FOLD actuator, we investigated
three main relationships: current-deflection, transient response, and
frequency response. Understanding these parameters is crucial for optimising
the actuator’s performance and limitations, especially under varying loads and
dynamic conditions. By comparing the deflection behaviours when unloaded,
loaded with weights at the tips, and embedded in a membrane, we aim to
simulate realistic operational scenarios. All experiments are conducted
underwater to replicate the intended operational context and to provide
relevant data for potential aquatic applications.

To characterise the deflection profile of the fin ray actuator in water, the
distal tip was tracked by processing videos of the motion at 240 frames per
second (see Supplementary Materials). Figure 3A illustrates the transition
from maximum push to pull deflection at an excitation frequency of 0.5 Hz. At
this low frequency, the fin ray achieves a maximum bending angle close to 90°,
settling fully at the extremes and displaying a homogeneous bending profile
with constant curvature along its length during the transition. At higher
frequencies, such as 4 Hz, the deflection profile changes significantly,
showing an asymmetric bending resembling a cupping motion due to the reactive
mechanical force distribution along the fin ray (Movie S2). The tendon exerts
force initially at the tip, with inertia causing the rest of the fin ray to
lag, creating a cupping effect. This behaviour can potentially be exploited to
increase the interaction of the fin ray with the surrounding water, as is
commonly found in fish fin and has been associated with increased hydrodynamic
performance[52, 12, 16].

Figure 3: Characterisation of the fin ray in water.
(A) Left: Composite side-view stills of the fin ray oscillating at a frequency
of 0.5 Hz. Right: Asymmetric curvature profile at a frequency of 4 Hz. (B)
Peak-to-peak deflection amplitude as a function of the current of the fin ray
loaded with different weights at the tip at a frequency of 0.5 Hz. The dashed
line at a current of 212 mA marks the upper current limit before the actuator
enters the saturation regime due to the mechanical limit.(C) Step response of
the fin ray loaded with different weights, excited with a current step of 212
mA amplitude and 1 s duration (the white region indicates when the signal is
on).(D) Frequency response of the fin ray under various load conditions
(unloaded, loaded at the tip, enclosed in a membrane) under sinusoidal
excitation with a peak current of 212 mA.

Figure 3B presents the peak-to-peak deflection of the fin ray under different
loads as a function of driving current. The VCA was driven using a step
signal, applying alternating negative and positive current steps with
instantaneous transitions and varying hold times at the extremes to observe
maximum push/pull deflection of the fin ray. The unloaded fin ray exhibits a
maximum deflection of 47 mm, nearing to ±90° bending. This upper limit is
imposed by the VCA’s travel range of ±2 mm, corresponding to an 11.75:1
transmission ratio. Additional weights shift the neutral position towards the
pull direction due to gravity and reduce the overall peak-to-peak deflection,
maintaining an approximately linear deflection-current relationship. The
majority of the deflection data show a small spread, highlighting the
repeatability and controlability of the actuator, and simultaneously the
robustness of the video capture setup. Currents above 212 mA lead to
deflection saturation for both unloaded and lightly loaded (116 mg) fin rays,
hence 212 mA was selected for subsequent characterisations to avoid
saturation.

To determine the transient response of the fin ray, we captured the
experiments on videos and used image processing to locate the tip. The fin
ray, under different load conditions, was actively pulled to the maximum pull-
deflection and then pushed with a current step of 212 mA of one second
duration (Fig. 3C). The weights of [116, 333, 396, 667] mg were achieved using
combinations of plastic/metal screws and nuts (further details in
Supplementary Materials). When unloaded, the fin ray exhibits a rise-time (10
to 90$\%$) of 100 ms, decreasing to 54 ms when a load of 667 mg is applied.
However, this reduction in the rise time can be attributed to the diminished
actuation range under load. The velocity of the tip is independent of the
load, displaying constant velocities of 0.37 m/s. When the beam is unloaded, a
slight overshoot of 0.4 mm is observable, followed by rapid settling within
150 ms. Conversely, at the highest load of 667 mg, the overshoot becomes more
pronounced at 9.5 mm with an increased settling time of 400 ms.

As fish commonly employ sinusoidal movements for locomotion, characterising
the frequency response of the fin ray under sinusoidal excitation is vital.
The fin ray’s frequency response was investigated by applying a sinusoidal
input signal with frequencies ranging from 0.5 to 15 Hz in 0.25 Hz increments.
The supply current was set at 212 mA, just below the saturation region, to
ensure sinusoidal movement of the fin ray and prevent clipping of the
actuation. As shown in Figure 3D, when normalised to the static amplitude, the
frequency response is fairly uniform. The increase in loading of the fin ray
has minimal impact on the cutoff frequency, with a slight shift from 4.5 Hz
for the unloaded case to 4 Hz under the heaviest weight. Resonance peaks
emerge around 1-2 Hz, becoming more pronounced as the load increases.
Increased weight of the loaded fin ray contributed to increased momentum,
facilitating overshooting and resonances, as previously observed in Figure 3C.

When embedded in a fin, the amplitude and frequency response are highly
dependent on the material and geometry used. To exemplify the performance in a
robotic system, a FOLD actuator was embedded in a square 33 x 33 x 0.8 mm
silicone membrane. The added stiffness reduces the maximum amplitude achieved.
The step response of the fin ray within the membrane, with a rise time of
approximately 400 ms, confirms significant damping from water interaction
(Fig. 3C). The frequency response of the integrated fin shows a reduced cutoff
frequency of 1 Hz due to water drag and damping effects (Fig. 1D).

When operated at 3 Hz, the presented actuator has a lifetime of more than a
million cycles. It is noteworthy that at failure, no mechanical failure
occurred in the PP, but the point of failure was the copper wire used for the
coil. The region where the thin wires exit the coil is subjected to repeated
stress due to coil movement, eventually leading to the breakage of the wire.
This further demonstrates that PP is a suitable material choice for the fin
ray as the deformation is well within the elastic region, and the forces are
low enough to not introduce creeping.

### Proof-of-concept integration of the fin ray: The CuttleBot

A bioinspired aquatic robot was designed and fabricated to evaluate the
performance of the fin rays in a robotic system. We explored a dual undulating
fin design, previously investigated by [53, 54, 55, 56, 34] and demonstrated
impressive manoeuvrability. We leverage the small size of the FOLD actuator to
explore combinations of frequency, wavelength and amplitude in a palm-sized
robot.

Inspired by cuttlefish, rays and skates, the CuttleBot has two medial fins
that run along the side of the body. Undulating motion of the fins, generating
a travelling wave, propels water enclosed within the wave and generates a net
thrust forward [54, 57, 58, 34]. The design and assembly of the CuttleBot is
depicted in Figure 4. The CuttleBot measures 135 mm in length and 125 mm in
width, consisting of eight single actuators in four pairs. Four fin rays
extend from each side of the body and are connected by a flexible membrane to
complete the fin. This monolithic construction allows for a sturdy structure
and quick assembly.

Figure 4: Assembly of the CuttleBot.
(A) Folding of the body: The laser-cut skeleton is folded to create spine and
ribs, tendons are inserted through the T-hooks and secured to the fin rays.
(B) Blade casting of the 0.8 mm thick silicon elastomer film. (C) The
CuttleBot skeleton is embedded into the liquid silicone film, cured for 30
minutes and the outline defined using a scalpel. (D) The coils and magnets for
the VCAs are added to complete the assembly.

The assembly process consists of four main steps. Starting with a flat laser-
cut skeleton structure consisting of PP that is folded to create the spine and
ribs, similar to the singular fin ray (Fig. 4A). After inserting and securing
the tendons, the arrowhead limiters on each fin ray are bent upward (the
folding steps are shown in Movie S3). In the next step, shown in Figure 4B,
two glass spacers are used for blade-casting/doctor-blading to create a 0.8 mm
thick layer of silicone elastomer (Smooth-on Dragon Skin™ 10 VERY FAST) [59]
on a glass surface. The CuttleBot skeleton is then embedded into the silicone
elastomer to cure (Fig. 4C). In its liquid state, the silicone flows through
cut-out holes in the skeleton and fin rays, enclosing them. After
approximately 30 minutes of curing, the outline of the CuttleBot is cut with a
scalpel and stencil before it is lifted from the glass surface. Excess
silicone is removed from the cavities to make space for the VCAs. To assemble
the VCA, the coils are attached to the tendons and the magnets to the spine,
completing the CuttleBot (Fig. 4D).

### Characterisation of the CuttleBot

#### Undulating fin characterisation

Figure 5A shows an image of a CuttleBot. Weighing 20 grams, it has been
equipped with polystyrene floats, symmetrically positioned at the front and
rear of the CuttleBot, to achieve close to neutral buoyancy. Similar to its
biological counterpart, a travelling sine-wave is imposed on the fins on the
left- and right-hand side of the CuttleBot, respectively, to facilitate
swimming. The kinematics of an ideal travelling sine wave formed at the fin
edges in terms of its amplitude $A(x,t)$ as a function of position $x$ along
the fin edge and time $t$ can be described as

$A(x,t)=A_{\text{max}}(I)\ \sin(kx\pm\omega t)$ (1)

, where $A_{\text{max}}(I)$ is the maximum amplitude determined by the input
current $I$ driving the VCAs, $k=2\pi/\lambda$ is the wave vector as a
function of the wavelength $\lambda$, $\omega$ is the angular frequency, and
$\pm$ resembling a forward ($-$) and backward ($+$) travelling wave. Since the
positions $x_{i}$ between the fin rays actuating the fin are determined by the
design, the relative phase-offsets between the fin rays can be determined in a
straightforward manner and each fin ray driven accordingly.

Figure 5: Characterisation of the fin membrane.
(A) Completed CuttleBot. (B) Side view of travelling wave on one fin. (C) Wave
amplitude at wavelengths and frequencies combinations.

Figure 5B exemplary shows half an actuation cycle of the fins of a stationary
(clamped) CuttleBot in water. A travelling wave propagates from the right- to
the left-hand side, indicated by the highlighted region. In order to
investigate the kinematics in more detail, thresholding techniques were used
to reconstruct the amplitude of the magenta dyed fin edge from recorded
videos. The excitation wavelength is varied over a range of 52.5 to 420 mm,
equivalent to 0.5 to 4 body (fin) lengths (BLs) and the frequency is varied
from 0.5 to 7 Hz with increments of 0.25 Hz. The heat map in Figure 5C shows
that a maximum wave amplitude is achieved at a driving wavelength of
$\lambda=$ 367.5 mm and a frequency of $f=$ 1.75 Hz. Frequencies greater than
2 Hz result in a drop-off of the amplitude due to hydrodynamic drag, in
agreement with the observations for the singular fin ray (Fig. 3). Similarly,
wavelengths shorter than 157.5 mm also result in a drop-off of the maximum
achievable amplitude. As the wavelength approaches the body length of the
CuttleBot, the minimum sustainable wavelength is limited by the number of fin
rays in the system. The Nyquist criterion dictates that a travelling wave of
wavelength $\lambda$ requires base points not to be spaced more than
${\lambda}/{2}$ apart to resemble it correctly. Additionally, cross-coupling
between the individual fin rays is increased for shorter wavelengths. Shorter
wavelengths result in an increased phase offset between fin rays, i.e. a
greater amplitude difference. This increased amplitude difference leads to
greater stretching of the silicone membrane connecting adjacent fin rays,
resembling the coupling of neighbouring fin rays via a spring that exerts a
force between them proportional to the strain applied. Overall, both the
number of fin rays and cross-coupling between fin rays limit the minimum
wavelength that can be sustained and the maximum amplitude of the wave. This
is further emphasised by the local maximum in amplitude formed at a wavelength
of 52.5 mm and frequency of 2.25 Hz. Examination of the video footage of this
peak reveals that a standing, instead of a travelling, wave is formed in the
fin. This is attributed to the extreme difference in amplitude of adjacent fin
rays and thus maximum cross-coupling, resulting in counter-propagating waves
originating from both ends of the fin that interfere and form a standing wave.

#### Swimming characterisation

The locomotion capabilities of the CuttleBot were systematically investigated.
The ability to control each fin ray individually allows for different swimming
modes and exploitations of various degrees of freedoms (DOFs). We investigated
the effect of the fin kinematic on propulsion by inducing travelling sine
waves along the robot’s fins, varying in wavelength and frequency.

When the sine waves on the fins were synchronized in phase, the CuttleBot
moved in the body-fixed surge direction. The backward travelling waves led to
forward movement and vice versa. An automated video analysis system was used
to track the position of the robot over time, from which we deduced the
velocity corresponding to each combination of wavelengths and frequencies.
Movie S4 shows the robot swimming at four different wavelength-frequencies
combinations. As depicted in Figure 6A, the maximum mean velocity achieved was
60.4 mm/s (0.45 body lengths per second (BL/s)) at a frequency of 2 Hz and a
wavelength of 262.5 mm. Notably, the CuttleBot’s velocity is adjustable over a
range of 10 to 60 mm/s and the thrust profile is periodic (see Supplementary
Materials), opening possibilities for precise operations such as controlled
hovering in aquatic environments.

Diving deeper into the swimming dynamics, it was observed that velocity rose
almost linearly with increased frequency up to a certain threshold where it
reached a peak, beyond which it began to diminish (Fig. 6B), consistent with
literature[49]. Notably, the peak fin amplitude (Fig. 5C) does not coincide
with the maximum swimming velocity (Fig 6A), suggesting a complex relationship
between fin movement and thrust generation. The literature consistently states
that thrust is expected to be directly related to the volume of water
displaced by the fin within a given time frame[8, 48, 49]. Theoretically,
increasing the amplitude of the undulatory wave should maximise the enclosed
volume of water displaced, just as elevating the frequency should accelerate
the rate of displacement. However, a larger amplitude also increases drag due
to a greater projected area, and the frequency is limited by the mechanical
cutoff frequency of the fin[52]. Therefore, achieving optimal propulsion
requires a nuanced balance between these interrelated parameters.

Figure 6: Quantitative Swimming Characterisation.
(A): The mean horizontal swimming velocity of the CuttleBot at different
frequency and wavelengths combinations; (B): Peak horizontal(surge) swimming
speed; (C) Peak rotation(yaw) speed; (D) Peak horizontal translation(sway)
speed; (E) Peak ascending/descending(heave) speed.

A comparison with marine animals sheds further light on the observed
horizontal swimming peak velocity of the bioinspired CuttleBot. Fish fins are
often modelled as water foils and are described using a dimensionless figure
called the Strouhal number $St=f\cdot A/U$, where $f$ is the frequency of
oscillation, $A$ is the width of the wake (often approximated as the peak-to-
peak oscillation amplitude of the fin), and $U$ is the swimming velocity[45,
60, 2, 44]. This ratio could be thought of as the speed of the lateral
movement of the fin versus the velocity produced from the thrust. Studies of
biological species state Strouhal numbers 0.2 $\leq St\leq$ 0.4 for
oscillatory animals with swimming motion wavelength longer than body
lengths[60, 61, 52], and 0.6 $\leq St\leq$ 2 for undulatory swimming at
wavelength shorter than body length [62]. This suggests an optimum frequency-
amplitude combination. Another dimensionless figure is the specific wavelength
$SW=\lambda/A_{\text{mean}}$, the ratio of the wavelength $\lambda$ of an
undulation to its mean amplitude $A_{\text{mean}}$. The finding of [46] is
that there is a convergent evolution towards an Optimal Specific Wavelength
(OSW) of 20, for speed optimisation (equivalent to maximising propulsive
force), irrespective of frequency, length, height or overall shape of the fin.
This has been observed for 22 species of fish of varying lengths (spanning two
orders of magnitude), different body and fin morphology, and no geographical
or taxonomical correlation and also substantiated experimentally with a
robotic knife fish. The observed swimming behaviour of the CuttleBot is
correlated with a Strouhal number of $St=$ 0.8 and a specific wavelength of
$SW=$ 21.88 at peak velocity. These numbers are in good agreement with the
Strouhal number range observed in marine animals and also the observed OSW.

Beyond forward and backward swimming, the CuttleBot can yaw when the
travelling waves on the left- and right-hand-side propagate in opposite
directions. Angular momentum is induced, leading to rotation on the spot
(zero-degree turning radius) (Fig. 6C). Keeping the wavelength fixed at
$\lambda=262.5$ mm, for which a peak in forward and backward swimming has been
observed, variation of the frequency allows control of the angular speed with
a peak of 35 ∘/s observed at a frequency of $f=1.5$ Hz. We attribute the
difference in observed peak rotational speed for clockwise and
counterclockwise rotations to the influence of the tethers powering the
actuators.

Sway can be achieved by flapping either the left- or right-hand side fins.
Sinusoidal operation of all fin rays on one side in phase leads to a side-ways
motion of the robot, pushing the robot away from the fin that is flapping,
shown in Figure 6D. Similar to horizontal forward/backward swimming and
rotation, a peak in speed occurs at a frequency of $\approx$ 1.25 Hz. However,
unlike previously described swimming modes, as the frequency increases, the
motion of the actuator transitions from flapping to cupping (Fig. 3A). In
contrast to flapping, this cupping mechanism pulls the robot, producing thrust
towards the same side as the fin. These two mechanisms compete, which results
in two peaks, the first at 1.25 Hz indicates when the normal flapping motion
is dominant, and the second smaller peak at 5 Hz when the flapping motion
diminishes and the cupping motion takes over, pulling the robot.

Heave can be achieved by synchronously flapping the fins on both sides of the
CuttleBot up- and down-wards (Fig. 6E). Ascending motion is realised by
actively pulling the fins down from their horizontal plane and then allowing
them to return passively to their neutral state when the actuators are
switched off. The descent is facilitated by the inverse process. For these
vertical movements, the fin rays are driven by a rectangular waveform with a
50% duty cycle at varying frequencies, rather than the sinusoidal waveforms
used for horizontal locomotion. A velocity peak is noted around 1 Hz for both
ascending and descending, though the absolute values for each are not
identical. This discrepancy can be attributed to the challenges in attaining
perfect neutral buoyancy and inherent asymmetries in the up-and-downward fin
strokes. These three additional swimming gaits can be seen in Movie S5.

Beyond the four independent DOFs demonstrated, the CuttleBot can perform
complex trajectories and dynamic manoeuvres by transitioning between these
isolated swimming modes. For instance, as illustrated in Figure 7A and Movie
S6, the CuttleBot initially descends by synchronously flapping its fins, then
shifts to forward motion near the tank’s bottom through sinusoidal excitation.
At T = 6 s, altering the phase shift from $\Delta\phi=0^{\circ}$ to
$\Delta\phi=20^{\circ}$ between the left and right fins prompts the CuttleBot
to roll. This manoeuvre illustrates the critical relationship between
instability and manoeuvrability [63], specifically when generating rotation
around the anteroposterior body axis as observed by [64].

Figure 7: Free Swimming and Comparative Performance
(A) CuttleBot free swimming, the images are at two seconds intervals. First,
it descends to the bottom of the tank, then swims diagonally upwards and rolls
right. (B) The CuttleBot in relation to existing biomimetic and bioinspired
aquatic robots.

Figure 7B provides a comparative analysis of recent advances in bioinspired
aquatic robots, focusing on their size, top speed, and agility. The proposed
agility metric for aquatic bioinspired robots evaluates multiple dimensions of
locomotive proficiency beyond degrees of freedom (the scoring methodology is
detailed in the Supplementary Materials). The ubiquitous BlueROV2 from
BlueRobotics offers a practical performance benchmark, capable of traversing
five degrees of freedom independently and swims at 1.5 m/s (3.3 BL/s) [65].
The combination of performance, configurability, and affordability have
rightfully justified its popularity in aquatic robotics research. In
comparison, robots modelled after BCF locomotion (green) are noted for their
remarkable top speeds; the TunaBotFlex [27] swims at 4.7 BL/s. However, they
exhibit limited agility, typically confined to forward movement with a
relatively large turning radius due to the propulsion mechanism of BCF motion.
In contrast, robots inspired by median-paired fin (MPF) locomotion (purple)
theoretically offer greater agility, but experimental demonstrations have
shown limited ranges of motion and degrees of freedom (DOFs). A notable
exception is the design by [34], which uses two preloaded rubber fins with a
permanent sinusoidal shape, moved by a series of brushless motors. By varying
the phase difference between the motors, the robot can independently traverse
four DOFs and even move over land. However, this design imposes limitations on
the fin’s workspace (wavelength and amplitude) as well as the size,
complexity, and cost of the robot. The CuttleBot contribute to this diverse
landscape by demonstrating similar maneuvrability to [34] at one-quarter the
size. Leveraging the small size of the FOLD actuator, four independent
actuators can be fitted, which allows the fin to adopt various arbitrary
shapes.

#### Power and efficiency characterisation

The experiments presented were performed at a peak current of 212 mA per VCA
to avoid amplitude clipping of the fin rays, as determined for a singular fin
ray (Fig. 3B). In this configuration, the entire CuttleBot consumes
approximately 11.8 W of electrical power, which we refer to as 100% power
consumption in the following. Every fin ray of the CuttleBot had been operated
under identical conditions, i.e. each VCA has been supplied with the same
power. However, since the fin rays driving each fin constitute a coupled
system, it is of interest to understand how an uneven power distribution
influences the performance of the CuttleBot.

By disabling (0% power per fin ray) or partially powering (20% power per fin
ray) different pairs of fin rays, the fin can form various wave envelopes
(Fig. 8A and Movie S7). We evaluate and compare their effect on vertical
swimming performance. In the first experiment, the two pairs of fin rays at
the back of the CuttleBot were disabled. The amplitude at the anterior was
unaffected but we observed a noticeable reduction in the posterior amplitude.
This formed a wave envelope that is posteriorly decreasing. However, partially
powering the two fin rays at the back leads to an almost complete recovery of
the envelope, indistinguishable to that of uniformly powered fin rays at 100%
power consumption. Figure 8B shows frames of the CuttleBot operating at an
overall power of 50% and 60%, respectively, corresponding to the cases of both
fin rays at the back completely disabled and partially powered, highlighting
the difference in wave envelopes during locomotion. In the other cases, where
the first and last fin rays are underpowered (diamond-shaped), the first two
(posteriorly increasing), and in the centre of the fin (bowtie-shaped), this
recovery is less pronounced.

The difference in the wave envelopes formed translates into the swimming
speed. Figure 8C shows the swimming speed for the four different cases of
posteriorly decreasing envelope, diamond-shaped envelope, posteriorly
increasing envelope, and bowtie-shaped envelope at 50% and 60% total power,
respectively. The swimming speed for a uniform sinusoidal envelope at
different total power levels is shown as a benchmark for comparison.
Surprisingly, the posteriorly decreasing envelope at 60% total power results
in swimming speeds almost comparable to that of a uniformly powered sinusoidal
wave. The others perform significantly worse than their uniform sinusoidal
envelope counterparts. The bowtie-shaped wave performs the worst, unable to
lift the robot from the floor.

Figure 8: Power Velocity Relationship.
(A) Power distribution at each fin ray and the approximate resulting wave
envelope. (B) Posteriorly decreasing wave envelope at 50% and 60% total power.
The wave envelopes are outlined in red. (C) Swimming performance of each wave
envelope benchmarked against standard uniform sinusoidal swimming. (D) Uniform
sinusoidal swimming speed. (E) A tuna-inspired fish robot with undulating
tail. (F) An ephyra-stage jellyfish robot with four flapper.

At this stage, the mechanism behind the observed phenomenon is not fully
understood and is beyond the scope of this study. However, we speculate that
for our case of a posteriorly decreasing envelope, the wave induced by the fin
rays at the front propagates towards the fin rays at the back, facilitated by
the surrounding water and formed vortices, requiring little excessive energy
for amplification, akin to keeping a pendulum or children’s swing in motion.

Limited research in this area has not arrived at a general consensus on the
effect of wave envelope on thrust and efficiency [49, 48, 66]. The CuttleBot
constitutes a promising platform for studying the correlation of localised
power distribution and waveforms/envelope with the overall performance of the
biomimetic system, which is a challenging task in biological species.

### Versatility in morphology

Lastly, we wish to emphasise the versatility and integration capability of the
presented actuator through the integration into two other bio-inspired robotic
systems; the T.U.N.A. (Tendon-driven UNdulating tAil) and the jellyfish. In
Figure 8E, the T.U.N.A robot has a pronounced caudal fin powered by two
actuators, which oscillate side-to-side to provide forward thrust achieving a
maximum speed of 48.5 mm/s (0.34 BL/s) at 2.75 Hz. The jellyfish has four fins
that can be powered individually to achieve control over more than one DOF
(Fig. 8F). It is capable of vertical swimming at 15.6 mm/s (0.14 BL/s). As
such, the demonstrated actuator clearly exhibits potential for integration
into future biomimetic and bioinspired aquatic robotic systems.

## CONCLUSION

In this work, we have demonstrated a bioinspired hybrid actuator with active
curvature control that emulates the functionality of a fish fin ray, enabling
complex manipulation of a fin. FOLD actuators can be rapidly fabricated by
laser cutting ordinary polypropylene film and hand-assembled in minutes,
resulting in an accessible and rapid manufacturing process. The cost of
materials is low (approximately £0.80/$1), and the actuators can be controlled
with simple electronics. Additionally, the design allows FOLD actuators to be
integrated into various robotic systems, as demonstrated with the CuttleBot,
T.U.N.A., and the jellyfish robot. We highlighted the CuttleBot, where
multiple actuators were embedded and collectively actuated to create 3D
deformations of two fins, smoothly transitioning between propagating wave
patterns, gliding, and flapping. Current approaches mostly utilize the
flexibility of the fin ray structure to achieve some degree of passive
curvature [67, 16], or neglect this aspect completely by modeling the fin ray
as a stiff rod [68, 54]. We improve upon this by actively controlling the
curvature of the fin. Moreover, the CuttleBot qualitatively and quantitatively
demonstrated and supported biological observations such as the Strouhal
number[44, 45] and Optimal Specific Wavelength[46, 47]. The FOLD actuator and
the presented platform hold high potential for the study of the principles of
biomimicry and robotics, serving as an ideal tool for instructional purposes,
enabling students and researchers to directly engage with these concepts.

Future work will focus on optimising the shape of the actuator to achieve the
desired deflection characteristics, ultimately working towards automated
inverse design. Numerous opportunities exist to explore new materials and
configurations to enhance the efficiency and control of the actuator, as well
as to enable new modes of actuation. Additionally, we aim to investigate the
CuttleBot’s application in evaluating and improving energy efficiency within
biosinpired systems, and to develop its system modeling through a combination
of fluid-structure interaction and computational fluid dynamics. The
simplicity of the CuttleBot’s deployment and the ease of data collection offer
great potential for fine-tuning simulations governed by differentiable physics
[69, 70, 71]. Efforts will also be directed towards refining the manufacturing
process to establish a catalogue of readily deployable robotic models for on-
site use. The next phase of development anticipates the CuttleBot’s transition
to wireless autonomy, setting the stage for fully autonomous operation.

## MATERIALS AND METHODS

### Materials and Constructions

Sheets of polypropylene (PP) (Value 400135897 A4 Display Folder - Black,
Cartridge People, UK) of thickness 0.2 mm have been laser cut with a laser
cutter (Neje Master 7W [2 W optical power]). The black colour of the PP
facilitates the absorption of the laser light with a wavelength of 450 nm. The
PP sheets are glued with a thin layer of liquid, water-soluble PVA glue onto
transparent acrylic (Perspex) sheets of 3 mm thickness. The high transparency
of the acrylic sheets in the visible wavelength range prevents interaction
with the laser cutter. Adhesion of the PP sheets to the acrylic carrier
prevents buckling and curling of the PP sheets. Further, cut-outs in the PP
sheets remain in place during the laser-cutting process. Engraving is used to
create folds in the PP sheet with a laser power of 9$\%$ and 9 ms dwell time.
A laser power of 30$\%$ and a dwell time of 30ms are used to fully cut through
the PP sheet. After the cutting process, parts can be lifted from the acrylic
plate using a scalpel or a pair of tweezers. The parts were then cleaned with
water in an ultrasonic cleaner and a brush to remove glue and any remaining
debris.

Black drinking straws with a diameter of 5 mm (Polypropylene Bendy Drinking
Straws, Buzz Catering Supplies, UK) were mounted on a 3D-printed stage to
fully suspend them into the air for laser cutting. During the cutting process,
the top and bottom surfaces of the straw will be cut. The suspension of straws
leads to thermal decoupling from the underlying surfaces and promotes enhanced
symmetrical cutting of the top and bottom surfaces. A laser power of 9$\%$
($\approx$ 0.18 W) and a dwell time of 9 ms is used for engraving. A laser
power of 30$\%$ ($\approx$ 0.6 W) and a dwell time of 30 ms with two passes
are used to cut the straws.

Washers laser cut from PP sheets were secured to the straw with small amounts
of superglue (Loctite SuperGlue Liquid Precision Max) to assemble a bobbin for
the coil, Fig. 2C. The bobbin is slid onto a rod that is mounted in a low RPM
hand drill. Subsequently, an enamelled copper wire of 100 $\mu$m diameter
(BNTECHGO 38 AWG Magnet Wire, Amazon) is wound on the bobbin using the hand
drill and distributed manually along the length of the bobbin. Approximately
11 m of copper wire is used for a single bobbin, where 9 m is wounded and 1m
is slack at each end. The slack wires are secured together along the length
with tape to prevent entangling. Standard 2.54 mm pitch 2-by-1 header pins are
soldered onto the ends of the wires for electrical connection. The enamel of
the wires evaporates during the soldering process at a temperature of 350°C,
i.e. it is not necessary to remove the enamel beforehand.

After assembling the body, a thin layer of DragonskinTM 10 Very Fast is casted
onto a glass plate employing a blade-casting/doctor-blading approach. Glass
spacers with a thickness of 0.7 mm are secured to the plate using 0.1 mm
double-sided tape on the left- and right-hand side allow producing a silicone
film of 0.8 mm thickness. The body of the fish is then immersed into the
silicone layer using tweezers. Cut-outs defined in the skeleton allow the
silicone to fully enclose the PP sheets, ensuring a stable mechanical
connection between the silicone and PP skeleton. After curing the silicone at
room temperature, the outline of the fish is cut with a scalpel, using the
stencil mask outline defined during the laser cut. The CuttleBot is then
removed from the glass plate.

Overall assembly times totalled 3.5 hours, with 2.5 hours spent on making the
coils, 20 minutes folding the body and attaching the tendons, and 40 minutes
to mix and form the silicone around the body.

### Electronics and Firmware

The actuator is driven by an L298N H-bridge, controlled by pulse width
modulation (PWM) signals from an Arduino Mega 2560. The sine values are stored
in a lookup table and are modified to become the PWM values.

### Experimental Setup

Experiments were conducted within a glass aquarium of dimensions 100 cm by 40
by 50 cm. All parameters, barring voltage and current, were measured
optically. For timing and synchronisation, we used a setup with three LEDs
(see Supplementary Materials), which emitted flashes at designated moments
within the actuation sequence or upon alteration of any variable (such as
amplitude, frequency, wavelength, etc.).

The video recordings were made with an iPhone 11 Pro Max with the Yamera app,
which enabled customisation of shutter speed and ISO settings to minimise
motion blur. Videos of single actuators were shot at 240 frames per second,
ensuring at least 16 frames per actuation cycle for the maximum tested
frequency of 15 Hz. This high frame rate and the stark contrast between the
dark actuators and the white backdrop facilitated straightforward segmentation
through intensity thresholding in MATLAB. For comprehensive fin
characterisation, one edge of the fin was tinted magenta to stand out
distinctly, enabling detection during processing via binary masking of
specific colour values. When characterising free swimming, the Physlets
Tracker software [72] was utilised to track a group of moving pixels—manual
placement of tracking points are sometimes required due to occlusions caused
by water reflections, and/or distortions from surface waves and motion blur.
Detailed methodology is provided in the Supplementary Materials.

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Acknowledgments: The authors would like to thank Que Anh Dang, Rawan Elsayed,
Kinjiro Amaro, and Andrew West for providing valuable feedback on the early
drafts of this paper. The icons used in the figures were created by
Smashicons, GregorCresna, NT Sookruay, Freepik, surang, JunGSa from
www.flaticon.com.
Funding: Minh Vu acknowledges support by the UK Engineering and Physical
Sciences Research Council (EPSRC) [Grant number EP/T517823/1].
Author Contribution: M.V. and T.J.E. conceptualised the actuator and
cuttlefish robot, and developed the fabrication method. M.V., T.J.E, S.W. and
A.W. developed the design, electronics, control software and visualised the
data. M.V. performed the experiments and collected the data. T.J.E., M.V., and
C.M. conceptualised the jellyfish robot. R.R. further improved the design and
developed the control, electronics, and performed experiments. A.M. and M.V.
conceptualised, designed, and experimented with the fish robot. M.V. and
T.J.E. wrote the initial draft of the manuscript. All authors contributed to
the manuscript, provided feedback, and approved the final draft. T.J.E., S.W.
and A.W. were responsible for the overall research supervision and objectives.
T.J.E. secured the funding.
Competing Interest: All author declares no competing interest.
Data and materials availability: Design files and source code can be found at
https://github.com/nhatminh2h/FOLDer

See pages - of Supplementary.pdf