Bioinspiration & Biomimetics

PAPER

Variable stiffness morphing limb for amphibious legged robots inspiredby chelonian environmental adaptations
To cite this article: Robert Baines et al 2020 Bioinspir. Biomim. 15 025002

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Bioinspir. Biomim. 15 (2020) 025002

https://doi.org/10.1088/1748-3190/ab68e8

PAPER

RECEIVED

19 September 2019

Variable stiffness morphing limb for amphibious legged robots
inspired by chelonian environmental adaptations

RE VISED

19 December 2019
ACCEP TED FOR PUBLICATION

8 January 2020
PUBLISHED

13 February 2020

Robert Baines1 , Simon Freeman2, Frank Fish3 and Rebecca Kramer-Bottiglio1,4
1

2
3
4

Department of Mechanical Engineering & Materials Science, Yale University, 10 Hillhouse Avenue, New Haven, CT 06520,
United States of America
Naval Undersea Warfare Center, 1176 Howell St, Newport, RI 02841, United States of America
Department of Biology, West Chester University, 730 S High St West Chester, PA 19383, United States of America
Author to whom any correspondence should be addressed.

E-mail: rebecca.kramer@yale.edu
Keywords: soft robotics, amphibious robotics, amphibious locomotion, variable stiffness, morphing limb, turtle, tortoise
Supplementary material for this article is available online

Abstract
Robotic vehicles capable of transition from aquatic to terrestrial locomotion face considerable
challenges associated with propulsive efficiency and performance in each environment. Here we
present a morphing amphibious robotic limb that combines the locomotor adaptations of sea turtles
for swimming and tortoises for walking. The limb can transform between the streamlined morphology
of a sea turtle flipper and the load-bearing geometry of a tortoise leg using a variable stiffness material
coupled to a pneumatic actuator system. Herein, we describe the fabrication and characterization of themorphing limb, and quantitatively show how morphing between hydrodynamic and axial-load bearing
states can enhance the locomotive performance of a single design over land and in water.

1. Introduction
1.1. State of the art: amphibious robot propulsive
mechanisms
Bioinspired amphibious robotics is a field that may
offer transformative performance in navigating theparticularly challenging transition between movement
in water and on land, including a zone that often
encompasses high-energy wave action, heterogeneous
substrate types, and dynamically fluidized sediment.
The paucity of literature concerning amphibious
robots is due in part to the fundamental challengesof designing and integrating together propulsive
mechanisms for effective water and land-based
locomotion. For instance, a conventional design for
amphibious robot propulsive mechanisms utilizes
propellers or fins and a separate wheel transmission
system to navigate on land and in water, respectively[1–3]. Although functional, this design strategy isburdened by mechanism weight and complexity,
inadequate ground clearance, the inability of wheels
to traverse unstructured terrain, and the inabilityof propellers to function effectively in very shallow,
fluidized sediment environments.

© 2020 IOP Publishing Ltd

An examination of biological solutions to transiting
between aquatic and terrestrial locomotion suggeststhat morphological compromise for high performancelocomotion in both environments has not yet occurred[4]. However, highly derived species are well-adapted toa specific environment, often at the expense of locomotion in the other (i.e. fish in water, cheetahs on land).
In designing a synthetic system, there is an opportunity
to combine disparate locomotion modes specialized
for a certain environ­ment. Here, we propose that the
combination of lift-based swimming and legged walking are promising for bioinspired amphibious robotic
design since efficiency in water, terrestrial maneuverability, and the ability to carry significant weight could
be achieved simultaneously.
Robotic propulsive mechanisms inspired by thebodily structure and locomotion strategies of snakes,
salamanders, and stingrays leverage coupled oscillation of interconnected modules to move both in waterand on land [5–8]. However, such robots do not havethe capacity to bear large payloads due to size limitations of the modules composing their bodies and themechanical drawbacks of undulatory propulsion overland.

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Robots utilizing limbs as propulsive mechanisms
possess the capacity to bear larger payloads than oscillating-type robots and the ability to navigate unstructured or uneven terrain, as opposed to wheeled robots.
One amphibious quadruped robot featured limbs with
passively retracting flippers that enabled both dragbased swimming and a traditional quadruped-style
walking [9]. Similarly, another robot possessed limbs
that consisted of flexible, flat plates constrained within
thin wire cages that served both as a wheel and a paddle [10]. Though they stand as benchmarks for clevermechanism design, these limbs had fixed structuresand stiffness that precluded the capability to tune thepropulsive mechanism for more effective locomotion
in a particular environment. In addition, though paddling can be advantageous for navigation in shallow
water, it is not an efficient mode of submerged swimming, as thrust is only generated over half of the stroke[4].
The ability to vary overall limb stiffness and structural shape confers substantial benefits in nature. The
two environments in question here (i.e. terrestrial andaquatic) fundamentally require propulsive limb types
to yield at different stiffnesses through variation inmaterial properties and/or structural forms. On land,
limb systems must be stiff enough so as to enable thetransfer of body weight to a point on the ground without buckling. In water, limbs serve primarily as hydrodynamic surfaces, and must at least partially yield topressure gradients created by propulsive movements
so as to limit stall and the reduction of lift-based thrust.
The optimal structural shape and response to loadingin each environment is thus fundamentally different.
Forms capable of supporting point loads are optimal for terrestrial locomotion, as they maximize grip
through pressure friction. Conversely, sleek forms thatcan sustain distributed hydrodynamic loads and maintain lift-based propulsion are optimal underwater. We
therefore contend that for a single limb to approach
optimality for both environments, some controllable variation in shape, and thus geometric stiffness, isrequired.
One group explored changing a propulsive mech­anism’s structure from a walking configuration to aswimming configuration using toggle-locking rigid
segments [11]. While the reported mechanism wascapable of structural variation, it was restricted to a set
stiffness in its leg and flipper modes and consequently
could not leverage the full energetic benefits that could
be derived from variable stiffness, as animals do whenlocomoting and interacting with obstacles [12, 13].
Toward this end, recent work showcased a robot withvariable stiffness limbs that allowed it to move effectively across different types of terrain [14]. A humanwas required to manually adjust the limbs and thus
the robot could not change the stiffness with which itinteracted with the environment autonomously and in
situ.

2

Examples in literature point to a common theme:
the viability of amphibious robotic designs hinges on
the mechanical capacity to efficiently adapt to land,transitional environments, and water. No reported
propulsive mechanism has fully addressed the multiple factors—payload capacity, unstructured terrain
navigation, variable stiffness, minimal complexity—that together make a successful design an elusive challenge. New propulsive mechanisms must be designed
to accommodate both changes in structure and stiffness without the need for hands-on intervention.
1.2. Chelonian limb adaptations as biologicalinspiration
Considering the challenge of developing anamphibious propulsive mechanism that is capableof significant structural and stiffness changes, weobserved the distinct limb morphologies of twogroups of turtles in the order Testudines: testudinidea(Tortoise) and Chelonioidea (Sea Turtle) (figures
1(a) and (b), respectively). Each of these animals’
limbs exhibits unique locomotor adaptations for theirprimary habitation environments (i.e. terrestrial/
shallow-water and open ocean, respectively).
For example, the turtle leg in the top view of the 3D
scan in figure 1(a) features an elliptical cross-section
and low aspect ratio (l2/t  =  1.77). This adaptation enables the limb to bear greater loads when upright during walking, when turtles and tortoises experience a
localized force applied by the ground, Fenv . A combination of geometric stiffness (the cylindrical shape) andmaterial stiffness (bone and tissue) in the limb allows
tortoises to bear and walk under Fenv energeticallyefficiently, albeit slowly, over varying terrain [15, 16].
In fact, freshwater turtle bones were shown to be excessively reinforced relative to the loads they and othertetrapods would normally encounter [17]. However,land-based tortoises are incapable of diving and submerged swimming. Freshwater turtles, with legs ofsimilar aspect ratio to that of the tortoise, have diminished top locomotor speed and energetic efficiency incomparison to sea turtles, though this comes with thegain of locomotor maneuverability [18, 19].
In contrast to freshwater turtle and land tortoiselimbs, the connective tissue and bone structures in
sea turtle flippers are functionally fused into a single
high aspect ratio (l/t  =  5.98 from the Front View of
figure 1(b)) hydrofoil ideal for locomotion throughwater with minimal drag and efficient thrust production [20]. During normal swimming, sea turtles
utilize a lift-based locomotion scheme by varying
the angle of attack of their flippers relative to oncoming flow to dynamically generate forward thrust andactively mitigate stall [19–21]. While a degree of stiffness is retained in the flipper in order to transmit theload applied by oncoming flow, more mechanicalflexibility is inherent in this structure both geometrically (thin, hydrofoil-like shape) and materially (thinner, longer bones and cartilaginous structures). Such

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Figure 1. Morphing limb. (a) A Galapagos tortoise and below, a 3D scan of a freshwater turtle leg. Notable morphological adaptions
include shortened digits and a relatively thick, round limb suitable for walking on land. (b) A green sea turtle, and below, a 3D scanof a loggerhead turtle flipper, a limb that appears to be well-adapted for efficient lift-based locomotion due to its high aspect ratio,
airfoil-like geometry and elongated, functionally fused musculo-skeletal structure. (c) The morphing limb transitions from a flipper
shape to a leg shape when its addressable variable stiffness material is Joule heated and consequently softened, and the antagonistic
pneumatic actuators on its surface are inflated. The limb transitions back to the flipper shape via Joule heating and pneumatic
deflation. Scale bar: 40 mm. Inset scale bar: 30 mm.

relative flexibility enables the turtle limb to respond ina mechanically compliant manner to distributed loads,
and delay the onset of flow separation in hydrodynamic scenarios such as an accelerative stroke. In contrast to the propulsive mechanism’s prowess in water,
however, sea turtles have been described to be clumsy
and labored when moving on land [22]. Flipper musculoskeletal structure is not well suited to support theconcentrated weight due to gravity of a massive bodyoff of the ground. For instance, the length of the flip3

per would create a large moment under such load andwould only be supported by the comparatively small
area at the end of the flipper. As a result, sea turtles mustdrag themselves along the ground on land and demonstrate reduced ability to traverse obstacles [22, 23].
Such flipper-based dragging locomotion, especiallyover granular media, is prone to slips that impact costof transport.
Beyond their morphologically distinct propulsors, sea turtles and tortoises share a similar body plan;

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Figure 2. Fabrication of morphing limb. Fabrication can be broken down into two main tracks: elastomeric actuators (top), andJoule-heating variable stiffness material (bottom). We fabricate components for each of the limbs’ two halves, A and B. Actuatorsand variable stiffness material components for A and B come together in the final step, in which they are hinged together via a sewed
joint. The limb cross section is displayed in the inset.

notably, they have bony or cartilaginous shells spanning the majority of their bodies and are quadrupeds. Morphological similarity between the animals’
bodies suggests that a single turtle-inspired robot
chassis could employ either form of propulsor. With
these observations in mind, we introduced a proofof-concept amphibious robot limb which sought tomarry the locomotor benefits of sea turtle and land
tortoise limbs—streamlined geometry and load bearing capacity, respectively—by morphing back andforth between semblances of these two fundamental
shapes [24]. Building on our previous work, this paper
details a number of improvements to the original limb,
including refined geometry for optimized hydrodynamics in the flipper state, cyclic load stability in theleg state, and a streamlined fabrication process which
renders a cohesive, demonstrably water-resistant limb.
In this study, the morphing limbs were developed andassessed independently of a robotic platform. We arepresently building a robot equipped with the morphing limbs; its locomotive performance will be thesubject of future work.
The morphing limb (figure 1(c)) utilizes soft fluidic actuators in an antagonistic configuration and athermoset epoxy, which serves as a variable stiffness
material, bonded to the strain limiting layer between
the actuators to permit morphing between flipper
and leg states (supplementary video 1 (stacks.iop.org/
BB/15/025002/mmedia)). The limb’s default state isa flipper. Within the thermoset is an embedded thinfoil copper heater which can Joule heat the thermoset
above its glass transition temperature (Tg) to soften
it on-demand. Then, the fluidic soft actuators inflate,causing the limb to deform from flipper to leg. Pres­sure is held until the thermoset cools. Upon releasingpressure, the limb retains the leg shape. The significant
cross-sectional area and structural change is captured
4in figure 1(c)) insets for both leg and flipper states.
Heating the limb back up again returns it to the flipper
state by thermally relaxing the thermoset. Stored elastic energy in the elastomer actuators also help return itback to the flipper state.

2. Materials and experimental
2.1. Fabrication of morphing limb
Figure 2 illustrates the manufacturing process of themorphing limb. The limb consists of two halves that
we call part A and B. Pneumatic elastomeric actuators
reside on each side of the morphing limb. The
deformation trajectories of the actuators were tailoredby changing the geometry of the acrylic molds inwhich they were cast. For instance, the mold for part A
contained an array of chambers with a larger length towidth ratio. This forced bending about the length-wiseaxis, or the long portion of the limb. Part B featured thesame chambers as part A, but with additional chamberswith greater width to length ratio at the distal end of
the actuator to induce curling to make a foot pad forextra stability. The actuators were cast from a silicone
rubber (Dragon Skin 10A, Smooth-On) and degassed
in a vacuum chamber to eliminate defects.
After curing, the actuators were bonded to a bidirectional strain-limiting fabric on the bottom and anelastomer lamina with unidirectionally embedded fibers, (introduced in our previous publication [25]), on
the tops to further direct their deformation. For part A,
the truncated actuator, only a single piece of unidirectional elastomer lamina at 0° was used to facilitate further axial-wise bending. For part B, the unidirectionallamina was placed at 90° at the distal end to facilitate
curling and 0° for axial-wise bending.
To create the self-heating addressable variable
stiffness material, first, the thin foil copper heater wasBioinspir. Biomim. 15 (2020) 025002

R Baines et al

etched in a ProtoLaser U4 (LPKF laser and electronics)
from 0.127 mm thick copper-clad kapton (DigiKey)
in a serpentine pattern of traces 2 mm thick to yield aresistance of 1.2 and 2.2 Ω, for parts A and B, respectively. Two wires for positive and negative electrodes
were soldered onto the heater. Then, the two-partthermoset epoxy system was cast by mixing 29% wt.
Jeffamine D400 (Hunstaman International, LLC),
71% wt. EPON 828 (Momentive Performance Mat­erials Inc.). After mixing and de-gassing the epoxy, we
poured it into molds corresponding to actuator parts A
and B lined with the thin copper foil heaters to enableon-demand Joule heating. The mold was then placedin an incubator for 12 h at 60 °C, with weights on top to
ensure smooth variable stiffness specimens.

compression force (estimated force experienced by
one leg of a four-legged robot weighing 45 kg). We
marked the edges of the limb and tracked them with ahigh-definition camera, recording Euclidean distance
between the markers every 10 s.
Uni-axial quasi-static compression-to-failure tests(Instron 3345 equipped with a 5 kN load cell) wereconducted on the morphing limb in both leg and flipper states, while monitoring electrical continuity. We
also conducted the same test on the limb in the flipper
state while uniformly heated above its Tg in order tostudy the effects of both geometric and material softening. The loading rate was 40 mm/min. Cyclic compression tests (Instron 3345 equipped with a 5 kN load cell)were conducted on the limb in leg state, reminiscentof conditions it would experience on a legged robot
weighing 45 kg. We set the loading rate to 100 N s−1.
We performed trials on two specimens: one was subjected to 30% of its ultimate failure force and the other
60% (260 N and 520 N, with lower bounds 26 N and
52 N, respectively).

2.2. Thermal characterization
The stoichiometric ratio of epoxy to amine, as well as
cure temperature, influence thermoset properties likeelastic modulus and Tg [26, 27]. We chose to explore
stoichiometric ratio as a way to tune the mechanicalproperties of a thermoset. We conducted dynamic
mechanical analysis (DMA) on samples of three 2.4. Hydrodynamic testing
different stoichiometric weight ratios (DMA Q800, TA We measured lift and drag forces on the limb in a
Instruments), holding cure temperature constant at laminar flow tank. The flow tank comprises a 1503
60 °C, to determine which ratios yielded appropriate Gal recirculating system with dimensions 225 cm (L)stiffness changes for our morphing limb system.
× 45 cm (W) × 45 cm (H). All of the walls of the test
Sample geometries were ensured to be within the section are smooth optical glass and had minimal
ideal measurement range of the DMA; namely rectan- impact on the flow development. We mounted thegular prismatic specimens having 30 mm × 12.5 mm × limb on a sturdy scaffolding that straddled the top of
5 mm dimensions were cast in a mold, and placed in an the flow tank, placing it out of the wall boundary layer
incubator at 60 °C for 12 h. After curing, samples were region and in the direct center of oncoming flow. We
removed and placed into a single-cantilever clamp of adjusted the angle of attack of the limb with respect
the DMA with 9.6 in-lbs torque. A 5 Hz sinusoid force to the incoming flow. Lift and drag force data were
input was applied to the samples with strain amplitude acquired by a 6-axis Gamma DAQ F/T Transducer (ATIof approximately 0.01%. A temperature sweep was industrial), considering the component of force in the
conducted between approximately 20 ° C and 80 °C at direction of flow for drag, and the component of force
1 °C min−1. Storage modulus (E), loss modulus (E), normal to flow as lift. A National Instruments DAQ
and damping (tan(δ)), were recorded every 1 s.
(USB-6212) was used with a Matlab data acquisition
We assessed the capacity of each heater to evenly toolbox to record force data. We acquired three dataraise the temperature of the thermoset epoxy in which points for each flow rate and angle in both the flipper
it was embedded above Tg when Joule heated. While and leg states. The tested angles were 0 through 60° ineach heater was subjected to 50 W, measurements of increments of 5°. Flow rates ranged from 0.3 to 0.6 m s−1
the material system were taken with a infrared cam- in increments of 0.1 m s−1.
era (FLIR) over 30 s. Since the epoxy layer is very thin
We conducted a numerical analysis using ANSYS(1.8 mm) and has a high surface area (13 533 mm2), its to obtain the 3D flow in the vicinity of the limb as wellsurficial heat distribution as captured by the IR cam- as ratio of lift to drag forces on the limb. To provide
era gives a close approximation of the internal temper­ data to compare to experimental results, we conducted
ature distribution.
tests across the same angles of attack and flow rate conditions as in the experimental setup. This simulation
2.3. Mechanical testing
analysis was conducted using a steady state approach.
We tracked how well the morphing limb held its The morphing limb model was imported as an .IGS
morphed leg shape in ambient temperature of file that was originally constructed in SolidWorks and
23 °C. Two cases were studied: radial and axial loading. represents very closely the dimensions and angles of
For radial loads, experiments were conducted in free the physical prototype. We modeled the neighborhood
resting conditions (no weight) and with 0.5 kg and flow domain of interest as a cylindrical control volume
1 kg weights applied downward onto the limb. For with inlet and outlet areas 3.1416 m2 and length 2 m.
axial loading, we placed the limb in an Instron 3345 The limb model was positioned directly in the center
equipped with a 5 kN load cell and applied a 110 N of the cylinder. Three computational subdomains,5Bioinspir. Biomim. 15 (2020) 025002

R Baines et al

Figure 3. Thermal characterization of morphing limb. (a) DMA results. On the left axis in blue: log plot of storage modulus, log(E ),indicates the change in stiffness of the material as a function of temperature, T. On the right axis in black: plot of tan δ helps pinpoint
Tg of three distinct stoichiometric ratios of the two-part epoxy mixture. Tg may be identified as the peak of the tan(δ) curve; for thethree samples’ lines reported here, Tg occurs on average at 46 °C for the 10:4 ratio, 56 °C for the 10:5 ratio, and 58 °C for the 10:5.8
ratio. (b) Thermoset epoxy with embedded thin foil copper heater heated at 50 W for 30 s in free convection. The maximum, averageand minimum temperature inside a bounded area of the material was tracked. (c) Snapshots at moments of interest during heating.
The color scale bar on the left hand side of the figure indicates temperature. Notably, the entire heater uniformly saturates past Tg
within 30 s for both geometries of heater.

the inlet, limb surface, and outlet were designated. We
utilized a tetrahedral mesh for both the limb and thedomain, calculated via the ANSYS 14.0 mesh tool. The
entire computational domain consisted of between
500 000 and 1000 000 cells depending on the param­
eters, with the density of these cells increasing near thesurface of the limb geometry. We utilized the Fluent
CFD solver and the solution method scheme was chosen as SIMPLEC. We set the boundary conditions inthe simulation to mass flow rate in equal to mass flowrate out of the two faces of the cylindrical domain. The
relationship between mass flow rate and the velocity ofthe experimental test was calculated via:ṁ = ρAv.
(1)Here, ρ is the density of the fluid medium (water at
998 kg m−3), A is the area of the face of the cylinder
through which the fluid flows (3.1416 m2), and v is thefluid velocity. We initialized the solution with v = 0 m s−1
in the direction of flow. Lastly, for convergence criteria,
we monitored force of lift and force of drag, as well as
the velocity in the flow direction. We set the program toconverge and report the ratio of lift to drag forces (FL /FD )if all of these metrics reached 0.001 residual value.

6

3. Results and discussion
3.1. Variable stiffness material thermal
characterization
3.1.1. Dynamic mechanical analysis
We wanted to create a variable stiffness material that
could be softened on demand to facilitate geometry
and stiffness changes of the morphing limb. Thus,
we turned our attention to a two-part thermosetting
polymer mixture consisting of epoxy and amine.
The thermoset provides sufficient stiffness in its cold
phase, but when heated beyond its Tg becomes quite
compliant.
Figure 3 shows results of the DMA studies we carried out on different stoichimometric ratio samples to
identify an appropriate ratio for the morphing limb.
Our selection criteria emphasized high stiffness change
between soft and rigid phases for distinct limb states, yeta low Tg in efforts to minimize required power input formorphing. Figure 3(a) (left axis) displays the log of thestorage modulus (log(E )) of the epoxy specimens as afunction of temperature (T), indicating relative stiffness changes between unheated and heated states. Storage modulus (E) quantifies a mat­erial’s ability to storeBioinspir. Biomim. 15 (2020) 025002

R Baines et alenergy of deformation elastically. Higher E means amaterial is more stiff. On average, the largest stiffnesschange was attributed to the samples with the 10:4
epoxy to amine ratio, as they started at E = 1.2 × 109
at 25 °C and fell to 2.7 × 106 by 80 °C: nearly a threeorder-of-magnitude decrease. The specimens withother ratios saw about two orders-of-magnitude in
stiffness change. Figure 3(a) (right axis) shows tan(δ)
as a function of T. The peak ratio of loss modulus to
storage modulus, E /E , otherwise known as damping (tan(δ)), is a metric often used to pinpoint Tg inmaterials whose mechanical response changes withtemperature [28]. The peaks of tan(δ) reside at 46 °C
for the 10:4 ratio, 56 °C for the 10:5 ratio, and 58 °C for
the 10:5.8 ratio.
Considering figure 3(a), we opted to use the 10:4
epoxy ratio for the morphing limb variable stiffness
material.We made this choice because a lower Tg of 46 °C
requires less electrical power input to the heaters forlimb transformations. Also, the three three order-ofmagnitude stiffness change between unheated andheated states for the 10:4 ratio was the largest witnessed, giving us the widest number of possible stiffness adaptations for the limb.
3.1.2. Joule heating the variable stiffness material
After selecting a stoichiometric ratio for the thermoset
epoxy, we cast it in shapes (part A and B) of two sidesof the morphing limb and embedded thin copper
foil heaters previously discussed in section 2.2. We
heated the heater at 50 W to ascertain the baseline
speed we could expect morphing operations to takeplace. Figure 3(b) plots the maximum, average, andminimum temperature as a function of time within
the contours of the material shown in figure 3(c) for
both parts. Tg is indicated by a horizontal dashed redline. The average temperature for each part surpasses
Tg within 20 s. Part B reached a higher maximum
temperature than did part A, and this occurred at afaster rate.
More insight into the heating over time behavior
can be gleaned by observing figure 3(c) and snapshots
taken at moments of interest during heating. Despite
having consistent material composition, the variable
stiffness material cast into parts A and B exhibited
unique and localized hot spots, which helps explain
the difference between the recorded values of each. For
instance, at 5 sec, part A witnessed a steady perimeterwise aggregation of heat all below Tg, whereas part B
had small patches forming that were already well above
Tg. Moving on in the sequence to 15 s, parts of bothdesigns exceeded Tg, but were still patchy. We believe
heating non-uniformities arose due to slight thickness
variations and geometric changes of the copper traces
within the material, given they were subject to thesame boundary conditions and power input. In spiteof variations, by the end, both designs had an averagetemper­ature above Tg. We note that extreme edges ofthe material did not exceed Tg.
7With the variable stiffness material fully functional, we moved on to understand the complete limb
system’s capacity to sustain loads analogous to those it
would experience if attached to a walking robot.
3.2. Limb characterization for land-based
locomotion
3.2.1. Leg state shape relaxation over time
Maintaining as close as possible to a circular crosssection improves the load bearing properties of thelimb’s leg state. This fact is underscored by observing
the ratio between two Euler critical buckling loads, Pcr,
one for a beam of a hollow circular cross section and
one for a hollow rectangular one of equivalent
perimeters and thicknesses (representing the flipper
and the leg state, respectively):
π 2 EI
Pcr =
.
(2)
(kL)2

Where E is elastic modulus of the variable stiffness
material, I is the cross sectional area moment of inertia,
k is the length factor (1 for pin boundaries on eitherside), and L is the unsupported length of the limb. All ofthese factors except I happen to be the same regardlessof limb state. Thus:
Pcr,circle /Pcr,rectangle = Icircle /Irectangle
(3)πIcircle = (r24 − r14 )(4)4bh3 − b1 h31
(5).
Irectangle =12

Solving each in terms of the same set of parameters b,h, and t, we have the ratio:
π 2(b+h) 4(( 2π ) − ( 2(b+h)− t)4 )
2π
Icircle /Irectangle = 4.
(6)bh3 −(b−t)(h−t)312

From this relation, it is clear that:
Icircle /Irectangle > 1 ∀(b, h, t) ∈ R > 0.
(7)Which implies:
Pcr,circle > Pcr,rectangle .
(8)Due to the importance of geometric stiffness in
our limb system, we inspected the rate at which the leg
state’s radius decreased as a function of time underconstant radial and axial loading. Shape relaxation
over time as a result of radial loading could be relevant
to, say, a robot floating on the surface of the water,
having morphed its limbs into the leg state in prep­
aration for transitioning to land. Radial loads might
be induced by water currents or an obstacle pressing
against the legs. On the other hand, shape relaxationfrom axial loading is pertinent to a robot’s stability
while it is upright, bearing weight.
The results of the radius relaxation tests are summarized in figure 4(a). In the radial loading condition
without any weight on top, the leg relaxed by about
40% of its initial radius over a 2.5 h period and had

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Figure 4. Mechanical Characterization of morphing limb. (a) The morphing limb, when in leg mode, loses its round shape slowly
over time due to strain relaxation in the variable stiffness material. The burden of radial load on the morphing limb expedites its
relaxation, as evidenced by the decreasing initial slope values. Axial load enforces the leg shape. (b) The leg mode is shown by theblack line and corresponds to the left axis. Two stiffness were observed in the leg state, k1 and k2, and are marked by linear trend lines.
Past around 23 mm, the red region of the leg’s force curve segregates where electrical continuity, and therefore morphing function,
was lost. The flipper at room temperature and heated past Tg are shown by solid and dashed blue lines, respectively, corresponding to
the right axis. The flipper deformed much more for lower forces, and as such, demonstrates lower stiffness than the leg. The heatedflipper’s response sheds light on the contribution of material stiffness to overall system stiffness. (c) Cycling to 0.3Fmax, the limb
breaks into a highly repeatable regime after 50 cycles, and settles into a highly repeatable regime demarcated by the upper and lower
bound lines. (d) Cycling to 0.6Fmax, the limb exhibits heavy break in after the first cycle. After eight cycles, preliminary cracks develop
from fatigue. After 15 cycles, the limb fails catastrophically.

an initial slope of  −0.004. With the addition of external load, the rate increased. With F  =  4.9 N, we notea steep initial decrease in radius over the first 30 min
with a slope of  −0.006, followed by a period of relaxation behavior. Similarly, for F  =  9.8 N, a steep initial
decrease in radius occurred over the first 15 min with aslope of  −0.008. Overall, it decreased by about 60% of
its initial radius.
Under axial compression, the rate of radius decreaseof the limb was much slower. Even after 12  +  h of compression, the axially-loaded leg relaxed by only 20% of
its initial radius, and had an initial slope an order ofmagnitude less than radial load curves:  −0.0009. This
result is actually advantageous considering realistic
operational conditions—it suggests a robot equippedwith the morphing limb and standing upright would
have to undergo fewer re-morphing operations tomaintain the leg shape. Another phenomenon evident
at the beginning of the axial loading curve is a brief
increase in leg radius from its starting value. We attribute the initial radius increase of the leg to ‘settling in’
8

due to viscoelastic creep under the compression load.
The subsequent decrease is characterized by a gentler
slope than the radially loaded conditions.
Physically, the fact that axial compression slowed
the radius relaxation could be influenced by two factors: first, being under compression, higher normal
forces increased friction at the base of the limb andlikely slowed the collapse of the radius as a result of theconstrained boundary condition (compare this to thefree-end that did not have to overcome any friction bysliding across a substrate in the radially loaded condition). Second, stresses oriented radially due to thecompression of the leg combat the relaxation of thematerial.
The results of radius relaxation experiments suggest the morphing limb will have to be periodicallyre-heated and re-inflated during normal operation tomaintain its optimal leg geometry. The optimal timescale of the repeated re-heat and re-inflate operations
depends on the robot’s task, environment, and loading
conditions.

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3.2.2. Leg and flipper states compression to failure
Important factors to consider when implementing themorphing limb on a robot system are the maximum
loads it can withstand in its leg state before the leg fails
entirely and before it loses morphing functionality.
To determine the maximum load the leg state of
the morphing limb could tolerate, we conducted
a compression to failure test. Results are detailed
in figure 4(b) as the black and red line. As the leg
state consists of a foot pad at its distal end, the initial
compression response of the limb was similar to a lowforce spring, providing a compliant response with spring
constant k1  =  19.53 N mm−1. Once the foot pad had
been compressed to the point where it made full contact
with the cylindrical walls of the leg, a new stiffness
regime more than twice as stiff, k2  =  44.25 N mm−1,
emerged. Around 23 mm the foot pad cracked andelectrical continuity of the heater was destroyed.
Finally, at 864 N the limb catastrophically failed because
the walls supporting the load fractured and the seamholding the two halves of the limb together ripped out
from the fabric.
The max stress seen by the artificial leg can be calculated via:
σmax = Fmax /Amin .
(9)Here, Amin is the cross sectional area of the limb at its
narrowest point and Fmax is the max force seen during
compression. Note the minimum area on which
compression force acts is a thin shell since the interiorof the limb in the leg state is hollow. Moreover, thecompressive stiffness of the elastomer is negligible incomparison to that of the epoxy. The minimum area
can be found by multiplying the limb width by the
epoxy thickness and accounting for both halves of thelimb with a scaling factor of 2: Amin  =  2lw  =  2*0.070
m∗0.0018 m = 0.000 252 m2.
Plugging in the corresponding Fmax gatheredfrom the force-displacement graph figure 4(b), yields
3.4 MPa. For a biological comparison, animal bones
can have a compressive strength ranging from thesame order of magnitude to two orders of magnitude
higher than that of our artificial limb [29]. Freshwater cooter turtle leg bones experience peak combined
compressive and bending stresses of 31.1±9.1 MPa
while walking [17]. Although not as strong as its biological counter­part, the maximum sustained load of
864 N by the leg state of the morphing limb is sufficient
for a sizable payload-bearing robot.
We also conducted a compression to failure teston the flipper state at 23 °C (room temperature) andheated to 80 °C, indicated by the solid and dashedblue lines in figure 4(b), respectively. The results suggest that the flipper would not perform well in loadbearing applications. Specifically, the flipper at 23 °C
exhibited significantly reduced stiffness in compariso­n
to the leg: initially 1.71 N mm−1, and after 5 mm, 0.24
N mm−1. The reduced stiffness caused the flipper todeflect up to a large 70 mm and achieve a relatively
9small maximum force of 21 N. Though there was noevidence of catastrophic failure in the variable stiffness
material at 70 mm, at this point the flipper was nearlyfolded in half and slipped substantially in the setup,causing a rapid drop in force. The flipper was plastically deformed to the extent that it did not recover its
initial, flat shape when it was extracted from the test
setup. At 80 °C, the heated flipper was even less stiff;
its initial slope of 0.41 N mm−1 is a quarter of that of
the flipper at room temperature and underscores thecontribution of material stiffness to the overall system
stiffness. The heated flipper deflected up to 80 mm, atwhich point the test was stopped because the flipper
had folded onto itself. Electrical continuity was maintained and minimal plastic deformation was evident
after extracting and cooling the flipper.
3.2.3. Leg state cyclic compression
Having measured the ultimate load bearing capacity of
the leg state of the limb under compression, we sought
to study repeated compression emulating a walking
motion. Thus, we conducted cyclic compression tests
in two conditions: one to 30% and another to 60%of the maximum force seen in figure 4(b). The two
conditions were intended to replicate normal working
conditions for a 45 kg robot which stands on at least
two legs at all times and extreme conditions such as
falls or balancing on one leg.
The leg fared for a long duration—the entire 500
cycles of the test—at 30% of the max force. Figure 4(c)displays the results of cyclic loading up to 30% of
the max force (260 N). The limb underwent a steadybreak-in period from cycle 1 up to 50. The rate of hysteresis slowed significantly past 50 cycles, and the driftof the upper and lower bounds of δ was negligible (lessthan 5%) past 200 cycles. Electrical continuity was stillpresent after the 500 cycles.
Figure 4(d) shows that cyclic loading up to 60%of the max force (520 N) induced substantial plastic
deformation within just three cycles. The limb was
unable to sustain this higher loading over a long time.
Namely, initial crack development occurred at cycle
eight, shortly after the initial region of plastic deformation, and is evident as the drop in the upper displacement bound (red dashed lines). After the initial crack,electrical continuity was lost. Crack propagation insubsequent cycles eventually caused the limb to buckle
at 15 cycles. Upon inspection of the limb after the test,
we noticed the variable stiffness material fractured in
multiple locations, including the upper top of the walls
by the solder leads and the footpad. Unlike in the compress-to-failure test, the seams mechanically fixing thetwo halves of the limb together were intact—even after
catastrophic failure of the variable stiffness material.
This fact suggests the strength of the material was significantly reduced from the high-amplitude cycling so
it failed well in advance of the seam.
Land-faring tortoises walk with a duty factor of
0.75, meaning a currently moving limb resides on theBioinspir. Biomim. 15 (2020) 025002

R Baines et al

Figure 5. Experimental and simulation hydrodynamic analysis. (a) Experimental setup. We placed the limb in the center of the flow
section, as indicated by the call-out box. We varied the angle φ and measured the components of force parallel to and normal to theflow direction around the limb. (b) Studying flow characteristics at points of interest at 0.5 m s−1 flow. Each color, detailed in theParticle ID key, represents a particle’s path over 200 steps in the simulation. Here, we have chosen to visualize the mesh of the limb aswell as particle flow. Note the increase in flow separation as well as a turbulent wake with increasing angle of attack from the point of
maximum FL /FD , φ = 10◦. (c) Experimental and simulated flipper plotted against experimental leg mode. The actual flipper’s lift
regime (FL /FD > 1) is shaded blue, while that of the CFD analysis is shaded cyan.

ground 75% of the stride duration. Their cadence isalso very slow; Galapagos tortoises in particular wereshown to take between three and five seconds for a single stride [30, 31]. Consequently, tortoises exert comparable forces when walking as they do when standing
still [15, 32]. In the case of our artificial leg, we foundthat it was able to carry a reasonable robot payload,
under assumed normal operation conditions, for 500
cycles, at comparable or faster loading than actual tortoises. Furthermore, it could tolerate up to 60% of its
compressive strength to eight cycles, which bodes well
for realistic operation conditions, i.e. a robot equipped10with the limb may experience a number of impacts orfalls without losing morphing functionality. Having
validated the mechanical efficacy of the leg state of themorphing limb, we moved on to assess its hydrodynamics in both leg and flipper states.
3.3. Limb characterization for water-based
locomotion
3.3.1. Limb lift and drag measurements
Figure 5(a) illustrates the experimental setup for testing
the hydrodynamic properties of the morphing limb.
Experiments were conducted between 0.3 m s−1 andBioinspir. Biomim. 15 (2020) 025002

R Baines et al

0.6 m s−1 free-stream flow velocity. As a comparison,sea turtles swim normally around 0.278 m s−1 but canengage in fast bursts of 0.53 m s−1 or higher [19, 33].
Other work has found that leatherback sea turtles
swim most commonly between 0.2 and 0.8 m s−1, andwith a maximum value of 2.8 m s−1 [34].
From simulation, we visualized stream lines acrossthe morphing limb at particular angles of interest at a
fixed velocity of 0.5 m s−1, shown in figure 5(b). These
included: φ = 10◦, 25◦, and 60◦, representing angles ofmaximum FL /FD , stall onset, and bluff body dynamics(lowest tested FL /FD ), respectively. Figure 5(b) qualitatively displays the effect of the flippers’ limited chord
length on the flow structure. As the φ is increased, tip
vortices above and below the main hydrofoil test section become more pronounced. At higher φ values,
these vortices interact with turbulent flow originating from the low pressure face of the flipper, creating
a post-stall flow field broadly similar to what is produced by a delta-shaped airfoil [35].
Figure 5(c) compares glide ratio (FL /FD ) against
angle of attack (φ) for both flipper and leg geometries experimentally, and includes numerically modeled glide ratio estimates from the flipper geometry.
There is good agreement between experiment andsimulation, likely owing to the relatively low freestream velocity magnitudes tested, fine resolution of
the CFD mesh, and closely matched geometry of the
CAD model to the test hardware. The flipper state of
the morphing limb approximated the performanceof a conventional hydrofoil, with the optimal angleof attack for lift-based propulsion occurring around
10°–15° for all tested velocities. Stall, defined here as
when FL /FD reduces to 1 or less, occurred consistently
around 20°. As the velocity of flow increased, the positive glide ratio area (AL) grew, and the maximum valueof the ratio also increased. Optimal free-stream velocity, likely higher than 0.6 m s−1, was not determined
due to experimental limitations in testing higher
flow speeds. However, stall angle remains relatively
unchanged with speed, and FL /FD reduced below
unity above 30° in all tests regardless of speed.
Post-stall, the relationship between glide ratio andφ for the flipper was broadly similar to that of the leg
geometry, with the exception of a peak at around 30°.
When in the leg state, we observe that there is some variation of FL /FD with φ, likely due to the fact that the leg
does not become perfectly circular in cross-section but
retains a ‘thick hydrofoil’ shape. While FL /FD neverexceeded 1, a glide ratio maxima consistently occurredat 30°. This observation is consistent with the mechanics of flow around thick airfoils. While these foils are
not capable of producing high FL /FD values, theircurvature helps retain boundary flow over the low
pres­sure face and delays separation until larger angles
of attack [36]. Nevertheless, the design of the limb
when in the leg state is intended to support terrestrial

11

locomotion rather than hydrodynamic propulsion. It
does not perform hydrodynamically as efficiently as
the flipper state; even the highest glide ratio is a factorof two reduced from that produced by the flipper at agiven flow speed.
In summary, experimental and simulation flow test
results for the morphing limb show that at this scale ofsize and flow speeds, the flipper state is well-suited forlift-based locomotion, while the hydrodynamic natureof the leg state can be characterized as a thick hydrofoil.
An understanding of the stall properties and optimal
angle of attack of the flipper, combined with expected
FL /FD , can assist in the design of a servomotor actuation sequence that optimizes lift-based propulsion at
higher speeds.

4. Conclusion and future work
We have introduced an improved morphing limb
intended for amphibious legged robots. The limb isfundamentally different than existing amphibious
propulsive mechanisms: a controllable ability tovary shape and stiffness enables optimization for
drastically different locomotor styles. Considered intandem, the performance of the morphing limb in theleg and flipper states holds promise in addressing thechallenges faced by robotic vehicles transiting between
marine and terrestrial environments.
Just as the tortoise leg exemplifies musculo-skeletal
evolutionary adaptations ideal for walking bearing aheavy carapace, the robotic limb’s leg state was able
to sustain large compression forces over hundreds ofcycles without buckling. During compression, the limb
maintained morphing functionality, which serves asa testament to its efficacy as a propulsive mechanism
for walking on land and subsequently transitioning
to marine environments. Moreover, like the sea turtle
flipper with hydrofoil-like geometry evolved for efficient lift-based propulsion underwater, the flipper
state of the morphing limb demonstrated favorable
lift-to-drag forces at a variety of angles of attack andflow velocities that are similar to what a real turtle
would experience during normal activities. Simulation
validated that lift-dominated areas had tight and surface-conforming boundary layer flow and wake turbulence typical of a hydrodynamic shape rather thana bluff body, suggesting the flipper would perform well
as a propulsive mechanism for lift-based swimming.
Future work entails the construction of a quadruped turtle- and tortoise-inspired amphibious robot
utilizing the morphing limb presented herein. Challenges associated with such a robot include autonomously adapting both the morphology and stiffnessof the morphing limb to the terrain or medium therobot senses it is on/in, and controlling the gait actuation sequence of the robot’s limbs as it moves on/in themedium.

Bioinspir. Biomim. 15 (2020) 025002

R Baines et al

Acknowledgments
This work was supported by the Office of Naval
Research under the Young Investigator Program(N00014-17-1-2604). Certain images (figure 1(b)) inthis publication have been obtained by the author(s)from the Pixabay website, where they were madeavailable under the Pixabay License. To the extent thatthe law allows, IOP Publishing disclaims any liability
that any person may suffer as a result of accessing,using or forwarding the image(s). Any reuse rights
should be checked and permission should be sought ifnecessary from Pixabay and/or the copyright owner (asappropriate) before using or forwarding the image(s).

ORCID iDs
Robert Baines
https://orcid.org/0000-0002-90231536
Frank Fish https://orcid.org/0000-0001-5973-3282
Rebecca Kramer-Bottiglio
https://orcid.org/00000003-2324-8124

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