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IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. MARCH, 2020

1

Spinal Helical Actuation Patterns for
Locomotion in Soft Robots
Jennifer C. Case1,2,3 , James Gibert1 , Joran Booth2 , Vytas SunSpiral4 , and Rebecca Kramer-Bottiglio2

Abstract—Spinal-driven locomotion was first hypothesized toexist in biological systems in the 1980’s; however, only recently
has the concept been applied to legged robots. In implementing
spinal-driven locomotion in robots to-date, researchers havefocused on bending in the spine. In this paper, we propose
an additional mode of spinal-driven locomotion: axial torsion
via helical actuation patterns. To study torsional spinal-driven
locomotion, a six-legged robot with unactuated legs is used. This
robot is designed to be modular to allow for changes in thephysical system, such as material stiffness of the spine and legs,
and has actuators that spiral around the central elastomericspine of the robot. A model is provided to explain torsional
spinal-driven locomotion. Three spinal gaits are developed toallow the robot to walk forward, through which we demonstratethat the speed of the robot can be influenced by the stiffnessof the spine and legs. We also demonstrate that a single gait
can be used to drive the robot forward and turn the robot left
and right by adjusting the leg positions or foot friction. The
results indicate that the inclusion of helical actuation patterns
can assist in movement. The addition of these actuation patterns
or active axial torsion to future, more complex robots with active
leg control may enhance the energy efficiency of locomotion or
enable fast, dynamic maneuvering.
Index Terms—Soft Robot Materials and Design, Legged
Robots, Biologically-Inspired Robots

I. INTRODUCTION
He study of locomotion in legged robots is typicallyfocused on the legs. Yet, it is becoming increasingly
evident through the observation of animals that the spine playsa large role in legged locomotion as well [1], [2]. Gracovetsky
theorized and later demonstrated that the spine is fundamental
to locomotion and can itself be used as an engine to propel
an animal without requiring motion from the legs [3], [4].
This hypothesis of the spinal engine was later expanded to
explain how the spine and legs work in unison for locomotion

T

Manuscript received: October 15, 2019; Revised February 5, 2020; Accepted March 4, 2020.
This paper was recommended for publication by Editor Cecilia Laschi
upon evaluation of the Associate Editor and Reviewers’ comments. This
work was supported by the National Aeronautics and Space Administration(NASA) under the Early Career Faculty program (80NSSC17K0553). J.C.
Case was supported by a NASA Space Technology Research Fellowship(NNX15AQ75H).
1 J.C. Case was and J. Gibert is with School of Mechanical Engineering,Purdue University, West Lafayette, IN, USA jgibert@purdue.edu
2 J.C. Case was and J. Booth and R. Kramer-Bottiglio are with Schoolof Engineering & Applied Science, Yale University, New Haven, CT, USA

joran.booth@yale.edu, rebecca.kramer@yale.edu
3 J.C. Case is with the Intelligent Systems Division, National
Institute of Standards and Technology, Gaithersburg, MD, USA

jennifer.case@nist.gov
4 V. SunSpiral is with Zymergen, Emeryville, CA, USA

Digital Object Identifier (DOI): see top of this page.

Fig. 1. (a) The legged robot with actuator elements aligned in a spiralconfiguration to enable twisting along the spine. (b) The structural robot
design where the pink elements are elastomer, the light gray elements are
end caps on either end of the elastomer to enable connections between otherelements, and the dark gray elements are the leg connectors to which the legs
are attached.

[5]. Understanding that locomotion goes beyond the legs inbiology has influenced how researchers have thought about
and designed robots.
Legged robots are slowly transitioning away from theirrigidly designed bodies to biologically inspired bodies thatincorporate a passive [6]–[8] or an active spine [9]–[12]. These
new robots with flexible spines (either through traditional
hinges or soft materials) have shown many advantages. First,
the use of compliant, passive spines can increase the energy
efficiency of a robot [6], [7], which has also been shown forbiological systems [13]. Second, the use of active spines canincrease the speed [10] and energy efficiency [11], [14] of arobot. While these results are very promising and show thebenefits of incorporating spines into robots, the spines are
generally designed to bend along one axis [6]–[8], [10], [12]or two axes [9], [11].
While bending is an important part of spinal motion, thecurrent designs for spines in robots neglect axial torsion, whichis experienced by biological spines [15]. Although it is known
that axial torsion is an important component to human walking
and running gaits [3], [16], it is less studied in legged animals.
In exploring how axial torsion may affect four-or-more-legged
animals, we can consider research on lizards and salamanders.
When lizards and salamanders walk, they alternate their feet insuch a way that an axial torsion is applied along the spine [17],[18]. Lizards and salamanders have helical muscle patterns
that spiral along their body, like many animals [19], that are
believed to resist this axial torsion [17], [20]. However, when2377-3766 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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those same muscles are observed in lizards and salamanders
moving at high velocity, those muscles do not appear tofunction as torsional control [21], [22]. It is unclear if othermuscles resist the axial torsion or if, at high velocities, axial
torsion may be beneficial to locomotion.
While axial torsion has not been shown as a contributing
factor to locomotion in four-or-more-legged vertebrates, it iswell known that helical muscle patterns enabling coupled axial
torsion and bending that exist along the spine in vertebrate
animals are also observed in octopus tentacles and elephanttrunks, where they have been shown to be beneficial for othertasks [19], [23], [24]. Additionally, highly coupled actuation
patterns, including helical tendons, have been demonstrated
to reduce the actuator requirements for robotic continuumarms [25], [26]. Helical tendons have been applied to both
continuum arms [26], [27] as well as the legs of a walking
robot that was inspired by the octopus [28]. It follows that
helical actuation patterns will be beneficial in other robotic
applications.
This paper studies the applicability of helical actuation
patterns to locomotion of a legged robot. Herein, we test the
hypothesis that torsional spinal-driven locomotion is possible
through helical actuation patterns alone via a six-legged robot
with a soft, elastomeric spine, shown in Fig. 1. The legson this robot do not actuate, isolating the motion of therobot to the spiralling actuators around the spine. A modelis provided to explain the principles of operation for thisrobot and why it is capable of walking. On the physicalsystem, experiments are conducted with multiple spinal gaits
to determine if locomotion is possible. Additionally, we sought
out a single spinal gait that is capable of forward motion as
well as rotating left and right by altering the leg positions
or foot friction. Altering the leg positions or foot friction isused to demonstrate how actuated legs could be used alongsidetorsional spinal-driven locomotion for fully actuated robots inthe future and having a single gait for the spine will simplify
the control complexity of such a robot. Finally, stiffnessesof the spine and the legs are changed to understand theircorrelation to speed of the robot. This work provides insight
into how torsional spinal-driven locomotion of spines can beused to augment the design of robots.
II. SYSTEM DESIGNIn order to explore torsional spinal-driven locomotion, a
modular and reconfigurable approach was taken to enable thetesting of various design choices. Specifically, we looked at
a two-segment, six-legged robot manufactured with a passive elastomeric spine, fabricated from either EcoFlex 00-50(Shore hardness: 00-50; Smooth-On, Inc.) or Dragon Skin 10(Shore hardness: 10A; Smooth-On, Inc.)1 , and actuated withrobotic skins [29]. Robotic skins allow assembly of different
continuum robots by easily swapping the robotic skin, which
1 Certain commercial materials are mentioned in this paper to specify
the experiment adequately. Such identification is not intended to imply
recommendation or endorsement by the National Institute of Standards andTechnology, nor does it imply that the material is necessarily the best available
for the purpose.

# Sensor/Actuator ## Sensor #

4
3
2
1

5

6

1

2

4

3

Fig. 2. Basic layout for the twisting skin and how it fits around the cylindricalstructures. The skin is shown unwrapped. In the physical skins, the sensor andactuator pairs with the blue and green dots are sewn at the locations on theskin with matching blue and green dots. The actuators and sensors are labeled
for reference. This skin is stretched around the cylindrical structure and leg
connectors are applied on either side.

contains the actuating and sensing components, from structureto-structure. For this work, a new robotic skin was designedwith sensors and actuators arranged such that they spiralledaround a cylindrical structure, shown in Fig. 2. While sensors
were included on the skin, they were not utilized for thiswork, although previous works have used sensors in a different
skin design for state estimation and closed-loop control [30],[31]. The skins in this work were operated with open-loop
control using pressure regulators [32] to control the pressurein each actuator. The pressure regulators were connected to a
138 kPa line and a vacuum line, which quickly deflated theactuators. Additionally, two types of legs were tested: rigid
legs fabricated from brass tubing and compliant legs fabricatedfrom copper wire. These legs were also modular, allowing
us to change out the legs as needed. Further informationon component fabrication is provided in the Supplementary
Materials.
Several robot configurations and spinal gaits were considered: twist (i.e., both skins twist in the same direction),counter-twist (i.e., skins twist in opposite directions), andbend-and-twist. The inflation patterns of the two robotic skins
for the twist gait are shown in Fig. 3, with the other gaitsshown in the Supplementary Materials. Additionally, the various robot configurations that are used with each gait anddiscussed in this section are outlined in Table I. All tests
occurred with the robot walking on felt fabric to provide
friction for the feet.

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and Automation Letters
CASE et al.: SPINAL HELICAL ACTUATION PATTERNS FOR LOCOMOTION IN SOFT ROBOTS

1

2

1

2

4

3

4

3

3

gait used the Ecoflex 00-50 spine and was only demonstrated
moving forward. The full cycle of this gait was about 35 %faster than the twist gait.
InflateC. Bend-and-Twist

Actuator 1
0

1

2s

0

1

Deflate2s

InflateActuator 2
0

1

2s

0

1

Deflate2s

InflateActuator 3
0

1

2s

0

1

Deflate2s

InflateActuator 4
0

1

2s

0

1

Deflate2s

Fig. 3. Inflation and deflation patterns for the twist spinal gait. The left andright columns represent the front and back skins of the robot, respectively.
TABLE IS UMMARY OF DIFFERENT CONDITIONS TESTED WITH THE SPINAL GAITS .
* INDICATES THE CONFIGURATION THAT EXPLORED ROTATION .

Legs
Spine
Material
Foot
Friction

Rigid
Compliant
Ecoflex 00-50
Dragon Skin 10
Constant
Varied

Twist
X
X
X*
X
X

CounterTwist

The final gait tested was a bend-and-twist gait. The bendand-twist gait differentiates from the other two because it
inflates a single actuator on each skin at a time rather thanactuating pairs, which changes the system behavior. The bendand-twist gait causes the robot to essentially pick up a leg
and twist it forward in the air to take a step. The leg is thenplaced on the ground and twisted back to propel itself forward.
Since this gait has a very clear direction and stepping patternunlike the simple twist and counter-twist gaits, the inflation
pattern can be reversed to drive the robot backwards (shown in
Fig. S3). However, in order to achieve this backwards motion
in practice, the foot friction of the middle legs had to be
reduced through the use of smooth plastic caps on the bottomof the feet. The bend-and-twist gait was demonstrated with theuse of the rigid legs and the Dragon Skin 10 spine.

Bend-andTwist
X

X
X
X
X
X

A. Twist
The twist gait was the most rigorously studied to demonstrate the capabilities of torsional spinal-driven locomotion inthis paper and is used in the model described in the following
section. This gait involves inflating antagonistic actuator pairs
one after the other to twist both segments in the same direction
back and forth to achieve forward locomotion (shown in
Fig. 3). The twist gait required the rigid legs and was not
functional with the compliant legs. It was tested with both the
Ecoflex 00-50 and Dragon Skin 10 spine in forward walking.
Turning was demonstrated using the Dragon Skin 10 spine
in two manners: (1) maintaining foot friction while changing
the leg positions, and (2) maintaining the leg positions (suchthat they match what is shown in Fig. 1b) while changing thefoot friction. To change the leg positions, the legs were bent
by hand to bias the robot into rotating in either a clockwise
or counter-clockwise direction, effectively turning the robot tothe left or right. To change the foot friction, an acrylic sheet
was slid under either all the legs on the left or right side of
the robot to encourage slipping of those feet and, thus, turningof the robot.
B. Counter-Twist
The counter-twist gait is a slight variation on the twist gait.
Rather than having both robotic skins twist in unison, thecounter-twist gait has the skins twist in opposite directions(shown in Fig. S2). This gait required the compliant legs
and was not functional with the rigid legs. The counter-twist

III. MODEL
In order to achieve torsional spinal-driven locomotion, theactuators must exert a moment along the spine. The angle, θ,that the actuator is placed on the robotic skin will translate
to the angle of the force applied to the cylindrical structure,
as shown in Fig. 4a. A single actuator force will result inbending, twisting, and compression of the cylindrical structure.
When θ = 0◦ , the actuator will cause only bending andcompression, which was demonstrated in previous works [29]–[31]. When 0◦ < θ < 90◦ , the actuator will cause acombination of bending, twisting, and compression. If two
ideal actuators work together in opposite directions (such as
Actuators 1 and 3 in Fig. 2, which are shown as F1 and F3
in Fig. 4a, respectively), they cause twisting and compression
while eliminating the bending in the cylindrical structure. If we
let F1 (t) = F3 (t) = F (t), the force and moment experienced
by the cylindrical structure can be written as,
Fcomp. (t) = 2F (t) cos θ,
Mtwist (t) = 2F (t)r sin θ,

(1a)
(1b)

where Fcomp. is the axial compression force, Mtwist is themoment that causes the cylindrical structure to twist, t is
the time, and r is the radius of the cylindrical structure. In
this section, we explore how walking is achievable throughalternating pairs of actuators (1 & 3 and 2 & 4), such that theskin only causes twisting and compression.
A. Theory of Walking
1) Complex, Local Deformation of the Elastomer Sections:
As the robot walks, the torsional and compressive inputs fromthe robotic skins alongside the forces from the feet result incomplex deformations along the spine that shrink the lengthof the robot along the e2 -axis. In this paper, we make several
simplications to the spinal deformation in order to explain theprinciples of motion for this robot. We assume the twisting2377-3766 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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r12 vα + cα
r1 r2 vα + r3 sαr1 r3 vα − r2 sα
0


a

F3

b

F2

e3
MtwistMtwistMtwiste2

e1
F4

Fcomp. F1

r1 r3 vα + r2 sα
r2 r3 vα − r1 sα
r32 vα + cα
0

r (2)


1 0
0 1
T(λX2 ) = 0 00 0α(X2 ) = |X2 |αmax /L,

r (1)

c
t=0 s

r2 r2 vα − r3 sα
r22 vα + cα
r2 r3 vα + r1 sα0

t=3.3 s


0
0
,
0
1
(2b)


0
0
0 λX2 
,
1
0 
0
1
X2 ∈ [−L, L],

(2c)
(2d)

T

t=6.7 swhere x = {x1 , x2 , x3 } is the deformed point given thereference point of X = {X1 , X2 , X3 }T , cα = cos(α),
sα = sin(α), and vα = 1 − cos(α), αmax is the maximum
angle of twist experienced by a single elastomer section, and
L is the length of a single elastomer section. The point P is
defined in the reference frame and the definition depends on if
the point is on the elastomer sections or the legs. If the pointis on the elastomer sections,

t=10 sP = {λ−ν X1 , 0, λ−ν X3 , 1}T ,(3)where ν is Poisson’s ratio (assumed to be 0.5 for elastomers).
To define the position of the feet on the legs, we usedRotating Left

Pf ront = {±d1 , d3 sin β, −d2 − d3 cos β, 1}T , X2 = −L,(4a)Rotating Right

Pmid = {±d1 , d3 sin β, −d2 − d3 cos β, 1}T , X2 = 0,(4b)
TPrear = {±d1 , d3 sin β, −d2 − d3 cos β, 1} , X2 = L,
(4c)Fig. 4. (a) Model of cylindrical structure with forces from the actuators.
(b) Model of a robot showing an example of how the axis of rotation isdefined for both legs when the front, right and rear, left feet remain on the
ground. This model also shows how the local coordinate frame, ei , is defined.
(c) Example simulation demonstrating a robot walking with θ = 26.5◦ . (d)Diagram showing how to adjust the legs for rotating as seen by the front of
the robot.

behavior can be described in terms of an angle and axis
homogeneous transformation [33], where the angle is definedby Mtwist and the axis is defined such that a given footremains on the e1 -e3 plane. Additionally, the deformation
is defined such that the middle feet also remain on the e1 e3 plane. To simplify the model further, we assume ideal
actuators, which enables the use of Eqn. 1 and remove therigid parts for the model and simulation. Thus, our robot canbe described with q(t) = {λ(t), α(X2 , t), r(1) (t), r(2) (t)}T ,where q is the state of the robot, λ is axial stretch, and α(X2 )is the angle of twist where X2 is the point along the length of
the robot in the reference configuration, and r(i) is the axis ofrotation along the ith segment for i = 1, 2. The deformationof the system can be written as,{xT , 1}T = R(α(X2 ), r(i) )T(λX2 )P,R(α, r) =(2a)where Pf ront refers to the front legs, Pmid refers to the middle
legs, Prear refers to the rear legs, the ±d1 term is positive
for the left foot and negative for the right foot. The terms
d1 = 73.8 mm, d2 = 25.9 mm, d3 = 13.5 mm, and β = 45◦
are found with Fig. S1.
2) Boundary Conditions: The walking gait of the robot canbe divided into four steps based on which foot is assumed to
be immovable due to high friction on the foot and resistanceto slide backwards. The feet are assumed to be planted inplace in the following order: (1) front, right foot; (2) rear, left
foot; (3) front, left foot; and (4) rear, right foot. We assume
the feet cannot slide backwards, as they were designed toonly slide forwards. This foot pattern results in the front legs
pulling the body forward as it twists the body upward and theback feet pushing the body forward as it twists downward.
To hold the feet in place, we assume xplanted,1 = Pplanted,1and xplanted,3 = Pplanted,3 where xplanted is the deformed
location of the planted foot. Using Eqn. 2a to determine theseequations along with kr(i) k = 1, the solutions for r(i) fori = 1, 2 can be solved for using Newton’s method. An exampleof how r(i) is defined when the front right foot is planted in
the given configuration is shown in Fig. 4b.
3) Solving the Deformation of the Elastomer Sections: To
explain the general mechanics behind how the twisting of theskins assist in locomotion, we simplify the force relationshipwith the state of the deformation such that it can be writtenas,λ(t) = 1 − Fcomp. (t)/Ka ,(5a)2377-3766 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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and Automation Letters
CASE et al.: SPINAL HELICAL ACTUATION PATTERNS FOR LOCOMOTION IN SOFT ROBOTS

αmax (t) = Mtwist (t)λ(t)L/Kt ,(5b)where Ka is the axial rigidity and Kt is the torsional rigidityof the elastomer sections.
To determine how far the robot has walked in a given time
step (u(t)), we can write,u(t) = xplanted (t) − xplanted (t − 1).

(6)An example of this model used in a simulation is shownin Fig. 4c. A complete derivation of this model, example
simulation, and code for its simulation are provided in the
Supplementary Materials.
B. Theory of RotatingWith an understanding of how the robot walks, we cancloser analyze what changes can be made to get the robot torotate, or turn in place. To achieve directional change in the
robot, we considered two methods: (1) changing leg positionand (2) changing foot friction. If the legs are bent either tothe left or right, as shown in Fig. 4d, as the robot twists, itwill take larger steps to one side than the other, due to the
distance of the foot from the body center-line. This results in
the robot effectively pivoting, or rotating, to the left or right.
If the legs are held in their initial unbent position, altering
the friction under the feet will also enable the robot to rotate.
If there is a large enough difference in foot friction between
the right and left side of the robot, the side with lower footfriction will allow the feet to slip on that side while the otherfeet will not slip. This slipping will result in the robot favoring
one direction (left or right) over the other and will also resultin the robot rotating.
IV. RESULTS & DISCUSSION
In testing the robot, we sought to determine what conditions
made torsional spinal-driven locomotion possible and what
factors could influence the behavior of the system. For every
robot configuration and gait tested, we measured the speed
to determine how the choice of gait, legs, spine material,
actuator angle, and foot friction may influence the behaviorof the robot. Since the robot does not move at a constantvelocity, speed was averaged together after a full step cycle
was performed (e.g., half the cycle shown in Fig. 3 for the twist
gait). For the experimental data, the speed was measured by
pixel distance the robot had moved forward over the lengthof time from the initial position to the measured position. The
pixel distance was converted to millimeters through a known
pixel-millimeter measurement. Additionally, for the twist gait,
we did additional analysis with a simulation using the model
provided in Section III.
A. Speed is influenced by the spine stiffness and actuator angle
To explore how the spine stiffness affects the speed at whicha robot walks, the twist gait was tested on the physical robot
with both spine materials (Ecoflex 00-50 and Dragon Skin 10).
For both materials, the twist gait successfully drove the robot
forward (shown in Fig. 5), but differed in speed. For the stiffer
Dragon Skin 10 spine, the robot walked at 1.9±0.2 mm/s

5with a corresponding simulation speed of 1.8 mm/s. The
errors reported for the experimental measurements are thestandard deviation over three measurements. For the softer
Ecoflex 00-50 spine, the robot walked at 4.1±0.1 mm/s witha corresponding simulation speed of 4.0 mm/s. By reducing
the stiffness of the spine, an increase of 115 % in the speed
was observed. This is due to the increased ability in twistingand deforming the spine given the same actuator outputs.
To get a better understanding of how the spine material
and the actuator angle affect the speed of the robot, an
additional study was done in simulation. The input force
into the simulation was chosen such that the corresponding
speed of a simulated EcoFlex 00-50 robot matched that ofthe physical robot. We tested seven different elastic moduli
ranging from 100 kPa to 500 kPa, including the elastic moduli
of Ecoflex 00-50 (122 kPa - calculated from [34]) and Dragon
Skin 10 (256 kPa [30]), along with a range of actuator angles
from 0◦ to 60◦ . From Fig. 6, we can observe that the material
stiffness has a larger influence in the speed of the robot than
the actuator angle. However, isolating actuator angle, robots
with an actuator angle of θ = 0◦ were found to be the fastest
for each stiffness. This makes sense since all of the workfrom the actuators would go into shortening the length ofthe elastomer sections, rather than deforming them out-ofplane. However, purely compressing the segments increases
the likelihood that the segment will buckle, which will affectthe overall behavior of the system.
To study buckling in this robot, we used Timoshenko beams
under the fixed-free boundary condition [35]. As an example,
we looked at the case where the force from the actuators was
set to 1 N. Fig. 7 shows that softer segments are more likely
to buckle, but as the actuator angle increases, the likelihoodof buckling decreases; this is best shown when the elastic
modulus is 200 kPa. Fig. 7 shows that our EcoFlex 00-50 robot
falls within the buckled region for that material, and we didobserve the EcoFlex 00-50 robot buckling during experimental
testing, which compromised its overall performance. While therobot was able to walk forward sufficiently well, the EcoFlex
00-50 robot was unable to turn left or right when we changed
the leg positions. Additionally, it should be noted that if theactuator angle is set to 0◦ , the robot will be unable to turnby simply adjusting the leg positions. Turning would requirea change in the spinal gait as demonstrated in [29]. A balanceneeds to be struck between the stiffness of the spine andstrength of the actuators to prevent issues due to buckling.
B. Rotation is possible by adjusting leg positions
The same spinal gait that is able to drive the robot forward
can also be used to turn the robot left and right by adjusting
the position of the legs. We demonstrate this ability throughthe use of the twist gait with the Dragon Skin 10 spine(shown in Fig. 8a-b). The robot was able to rotate left
at a rate of 1.3±0.1◦ /s and right at a rate of 1.2±0.1◦ /s,essentially rotating in either direction at the same rate. This
result indicates that a consistent twisting spinal gait can beused to drive a robot with the position of the legs and feet
serving as steering for the robot. Therefore, a simple twisting2377-3766 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. MARCH, 2020

a

b

5s

0s

10 s

0s

5s

10 s

Normalized Average Speed (v/vE=100kPa, =0°)

Fig. 5. Forward locomotion with the twist gait demonstrated with (a) Dragon Skin 10 and (b) Ecoflex 00-50 spine.

spinal gait is capable of generating a variety of movements
when working with the legs, which means that complex gaits
are not required to achieve rotating of the robot. Having asingle, simple gait lowers the barrier to including this type of
locomotion on future robots.

E = 100 kPa
E = 200 kPa
E = 300 kPa
E = 400 kPa
E = 500 kPa
EcoFlex 00-50
Dragon Skin 10

1
0.82

C. Rotation is possible by adjusting foot friction
0.50
0.38
0.33
0.25
0.20

0

10

20

30

40

50

60

Actuator Angle (°)

Fig. 6. Simulation of the normalized average walking speed of robots withdifferent elastic moduli as the actuator angle is changed from 0 to 60◦ . The
speed is normalized by the speed of the robot with an elastic modulus of 100
kPa and an actuator angle of 0◦ . The dots indicate the speed of the robots
tested experimentally.

In addition to rotating by adjusting the leg positions, it isalso possible to achieve rotating by adjusting the friction on theright or left legs [36]. Rotating by adjusting foot friction wasdemonstrated through the use of acrylic sheets and the twist
gait with the Dragon Skin 10 spine (shown in Fig. 8c-d). Using
the acrylic sheets to alter foot friction, the robot rotated left
at 1.5±0.2◦ /s and right at 0.8±0.2◦ /s. The large variability inthe rotational speeds between rotating left and right is likely
due to variability in placing the acrylic sheets under the feetof the robot. Essentially, this demonstration highlights that theability to rotate is also achievable by changing the foot friction
behavior, enabling one side of the robot being more prone toslipping, which facilitates rotating.
D. Compliant legs can increase the speed of the robot
In addition to testing torsional spinal-driven locomotion
with rigid legs, compliant legs were also tested to see if theirenergy storage capabilities would enhance walking. To test
the compliant legs, the Ecoflex 00-50 spine was used withthe counter-twist gait (shown in Fig. 9). While using theselegs, the robot drifted to the left. We believe this was due todifferences in the legs caused by the manufacturing processrather than the introduced compliance in the legs. With thisconfiguration and spinal gait, the robot walked forward at a
speed of 8.7±1.0 mm/s. While the counter-twist gait is run
35 % faster than the twist gait, the increase in speed is 110 %faster, which indicates that this increase in speed is likely
due to the compliance in the legs in addition to the fasterrate of the gait. The compliant-legged robot appears to havemore of a bounce to its gait than its rigid legged counterpart(see Supplementary Video), which helps it move faster. Thisbouncier gait is more reminiscent of biological systems.

Fig. 7. Simulation comparing the compression force with each actuator at 1 N
to the buckling force of robots with varying stiffnesses. The shaded region
highlights the combinations of actuator angle and elastic modulus that would
buckle under the applied force, and the two dots indicate where the physicalrobots fall on the graph.

E. Additional gaits allow bi-directional motion without changing the angular offset of the footWhile the simplicity of the twist and counter-twist gaits areappealing, they are not the only gaits that enable walking of2377-3766 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LRA.2020.2982352, IEEE Robotics
and Automation Letters
CASE et al.: SPINAL HELICAL ACTUATION PATTERNS FOR LOCOMOTION IN SOFT ROBOTS

a

7

c

b

d

10 s

0s

10 s

0s

10 s

0s

0s

7s

7s

Fig. 8. Demonstration of the robot turning (a) left and (b) right by adjusting the leg positions and (c) left and (d) right by adjusting foot friction.

0s

8s

16 s

24 s

Fig. 9. Demonstration of walking with compliant legs. The scale bar represents 100 mm.
a

b

0s

11 s

23 s

0s

11 s

23 s

Fig. 10. Demonstration of the bend and twist gait moving (a) forward and (b) backward by reversing the gait and altering the friction of the middle feet.

the robot. An additional gait (bend-and-twist) was found thatallowed the robot to walk both forward and backward (albeit
with a change in foot friction of the middle feet required)without the need to adjust angle of the feet. This is due to afundamental change in how the robot is driven. This gait wastested with the Dragon Skin 10 spine (shown in Fig. 10). Using
the new gait, the robot was able to walk forward at a speed of
1.4±0.3 mm/s and backward at a speed of 1.9±0.5 mm/s. The
speed achieved by the bend-and-twist gait is comparable to that
achieved by the twist gait with the same robot configuration.
V. CONCLUSION & FUTURE WORKIn this letter, we demonstrated that torsional spinal-driven
locomotion is achievable through the use of a helical actuation
pattern that is capable of twisting and bending the spinal
structure of the robot and provided a theory to explain theachieved locomotion. We believe this is the first demonstrationof a robot achieving locomotion through the use of active
torsion along the spine. In order to isolate locomotion behavior
originating from torsional motion about the spine, this study
focused on a robot with unactuated legs. We demonstrated how
solely twisting or bending-and-twisting motions of the spine
are sufficient for robot locomotion. Additionally, we demonstrated that through changes in leg position or foot friction, asingle spinal gait can be used to move the robot forward, or

rotate it left or right, which simplifies how torsional spinaldriven locomotion can be incorporated into future systems.
In addition to showing the basic abilities of torsional spinal
gaits, we considered how additional changes to the basic robot
structure affect system behavior. First, we demonstrated thatthe speed of the robot is influenced by the spine material. A
softer spine will be able to twist further and will, thus, drive
the robot further with each step. Second, we demonstrated
that the use of compliant legs can further increase the speedof the robot. This increase in speed is likely due to the factthat compliant legs can store energy and then release it as therobot walks forward. Finally, we showed that alternative gaits
are possible with the spiralling actuators in the forward andbackward driving bend-and-twist gait. While this gait enables
the robot to move forward at the same speed as the twist
gait, the exact mechanics in which the robot is driven forward
differs.
While we believe this work represents a significant contribution to locomotion of legged robots, there are additional areas
to continue studying in the future. One such area is in energy
efficiency. Animals have actuators distributed throughout their
entire body (i.e., muscles) that contribute to locomotion. We
believe that a distributed actuator system in robots could
increase their overall energy efficiency due to distributed
loads throughout the systems, and that this use of torsional

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LRA.2020.2982352, IEEE Robotics
and Automation Letters
8IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. MARCH, 2020

spinal gaits will contribute to that efficiency. Alongside thespiralling actuators presented here, actuators that run parallelto the neutral axis of the cylindrical structures could be
added to generate pure bending motions on the spine. This
would result in a system that more closely mirrors biologicalsystems and could potentially lead to more advantageous gaits.
Additionally, the sensors could be used to inform future control
strategies of the current state of the spine. We also believe thatthe twisting ability of the spine is relevant to fast, dynamic
maneuvers seen by animals, such as cheetahs when they chaseprey, and the inclusion of active twisting in spines will improvethe maneuverability of robots.
VI. ACKNOWLEDGEMENTS
The authors would like to thank Prof. Charles M. Krousgrillof Purdue University for the helpful conversations pertaining to
the deformation and model of the robot and Gabrielle Braninof Yale University for preliminary testing of the design of therobot.
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Authorized licensed use limited to: Yale University. Downloaded on April 06,2020 at 20:08:30 UTC from IEEE Xplore. Restrictions apply.