SOFT ROBOTICS
Volume 5, Number 1, 2018ª Mary Ann Liebert, Inc.
DOI: 10.1089/soro.2017.0030

ORIGINAL ARTICLE

Reducing Actuator Requirements in Continuum Robots
Through Optimized Cable Routing

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Jennifer C. Case,1 Edward L. White,1 Vytas SunSpiral,2 and Rebecca Kramer-Bottiglio1,*

Abstract

Continuum manipulators offer many advantages compared to their rigid-linked counterparts, such as increased
degrees of freedom and workspace volume. Inspired by biological systems, such as elephant trunks and octopus
tentacles, many continuum manipulators are made of multiple segments that allow large-scale deformations to be
distributed throughout the body. Most continuum manipulators currently control each segment individually. For
example, a planar cable-driven system is typically controlled by a pair of cables for each segment, which implies
two actuators per segment. In this article, we demonstrate how highly coupled crossing cable configurations can
reduce both actuator count and actuator torque requirements in a planar continuum manipulator, while maintaining
workspace reachability and manipulability. We achieve highly coupled actuation by allowing cables to crossthrough the manipulator to create new cable configurations. We further derive an analytical model to predict theunderactuated manipulator workspace and experimentally verify the model accuracy with a physical system. We
use this model to compare crossing cable configurations to the traditional cable configuration using workspace
performance metrics. Our work here focuses on a simplified planar robot, both in simulation and in hardware, withthe goal of extending this to spiraling-cable configurations on full 3D continuum robots in future work.
Keywords: continuum robot, cable driven, bioinspired
IntroductionT

he design of soft robots is often inspired by soft
biological creatures and components. These creatures andcomponents generally have hydrostatic skeletons (jellyfish,1,2
starfish,3 etc.) or muscular hydrostats (elephant trunks,4,5 tongues,6,7 tentacles,8,9 etc.), which are highly deformable. However, most hydrostatic skeletons and muscular hydrostats exist
either on a small scale or in a pressurized environment, suchas water or underground. Those that do not exist in such anenvironment (elephant trunks, tongues, etc.) are generally
attached to larger structures that contain a skeletal support
system and, hence, are not constantly load bearing. ‘‘Purely’’soft robots can only operate efficiently either at small scales,
where gravitational forces are insignificant, or in pressurizedenvironments, where the environment provides support. Accordingly, there is consensus that for soft robots to operate ona large scale, there will need to be structural support.10
An important structural element to many large-scale species
is the spine, which can be thought of as a multi-jointed system.

Skeletal spines allow large-scale motions to be discretized,
which reduces local deformation. In humans, our spine holds
us upright and contributes to movement.11 The musculature of
the human spine is composed of spirals of muscle chains that
help move and stabilize the spine.12 In addition, recent studies
have suggested notable increases in locomotion efficiency due
to inclusion of a spine in quadruped robots.13–15 Continuum
manipulators and hyper-redundant manipulators are ideal for
approximating spines due to their high flexibility and distributed deformations.16–18 In particular, cable-driven continuum
manipulators have been demonstrated and modeled both
statically and dynamically.4,19–23
In this study, we explore the concept of a planar cabledriven soft spine, which mimics the disks (soft elastomer
segments) and vertebrae (rigid PLA parts) in biological spinesand borrows concepts from the highly coupled nature ofmuscular spirals. Previous planar cable-driven continuumarms typically include a pair of cables controlling each segment,19 such that the total number of system actuators is
twice the number of system segments. While this approach is1

Department of Mechanical Engineering, Purdue University, West Lafayette, Indiana.
SGT, Inc., NASA Ames Intelligent Robotics Group, Moffett Field, California.
*Current address: School of Engineering & Applied Science, Yale University, New Haven, Connecticut.

2

109

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CASE ET AL.

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FIG. 1.

Six-segmented multi-jointed spine. Color images available online at www.liebertpub.com/soro

useful for decoupling segments in the system, it misses advantages that come from having a highly coupled and underactuated system. A highly coupled system allows us toreduce the maximum cable tension, which allows us to reduce motor requirements and make more energy efficientsystems. An underactuated system allows us to reduce thenumber of actuators in a system. As an example, we demonstrate a six-segment planar spine shown in Figure 1, in
which cables are able to cross through the spine at the rigid
segments in patterns, which we refer to as crossing cableconfigurations. A computational workspace analysis for anumber of crossing cable configurations shows the feasibilityof this approach and the ability to tune the workspace basedon user requirements. The model is experimentally verified
by showing that the behavior of the physical system follows
the behavior of the computational model. While we confine
the complexity of the system to 2D in this work, we plan toextend this concept to 3D systems in the future.
Fabrication

The spine system described in this article consists of twoparts as follows: soft elastomer segments and rigid PLA
vertebrae. While any elastomer can be used in the elastomer
segments, we chose to use Dragon Skin 10 Slow (SmoothOn, Inc.) due to its low hardness of 10 on the Shore A scale,
which reduces tension requirements in the cables and, thus,
requires smaller motors. In addition to this, Dragon Skin 10
Slow also exhibits linear material properties below 40%
strain, which simplifies the model. Figure 2 shows stressstrain data from three samples of Dragon Skin 10 Slow. Thesesamples were tested in an Instron 3345 with a 50 N load celland pulled at a rate of 100 mm/min. The stress and strain were
calculated using the method described in Reference.24
The elastomer segments were created using a mold madefrom acrylic sheets. Both ends of the Dragon Skin 10 Slowsegments were inserted into 3D printed end caps made with a
Printrbot Metal Plus. The segments were then mechanicallylocked into the end caps by sealing with more Dragon Skin 10
Slow. The dimensions of the spine segment are shown in
Figure 3a. In addition to the 40 mm segments shown in the
Figure 3a, we also made 20 mm long elastomer segments,
which were used for the experimental validation described ina later section.

The rigid vertebrae were 3D printed using a Printrbot Metal
Plus. Steel rods were used as cable guides in the rigid vertebrae
to reduce friction as the fishing line cables passed through thevertebrae, as shown in Figure 3b. The spine was held together
with M2 bolts and nuts and fixed to an acrylic box that housed
the motors. Eight NEMA-17 stepper motors were used tocontrol the length of the cables. Each motor was fitted to aspool which controlled the length of each cable. The spools
were sized such that each step of the motor either lengthened or
shortened the cable lengths by 0.5 mm. An Arduino Uno with
Adafruit motor shields was used to control the motors. The
complete spine with motors attached can be seen in Figure 1.
Model

We model our multi-jointed soft spine using similar approaches to those proposed by Camarillo et al.25,26 Our soft
spine system differs from these previous works in two ways.
First, we use external cables rather than internal cables,
which means that the equations must be recalculated tohandle this; external cables are a standard cabling method forcontinuum manipulators.19 Second, we introduce a crossing
cable configuration rather than a parallel cable configuration,
as shown in Figure 3b, which complicates the coupling of thesegments and requires a redefinition of a multi-segment
manipulator.
For this model, we make several assumptions as follows: (1)linear material properties; (2) constant curvature/bending
moment of elastomer segments (note that the system is modeled as a beam27); (3) friction is negligible (i.e., sliding friction

FIG. 2. Stress-strain data for Dragon Skin 10 Slow. Threetrials are shown in gray dotted lines, and a linear approximation is shown by the black dashed line.

REDUCING ACTUATOR REQUIREMENTS IN CONTINUUM ROBOTS

111

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We begin our analysis by considering a single segment thatis fixed on one end and free on the other. We can define oursegment in two frames, e and e¢, as shown in Figure 4. Notehere that we have neglected to include the rigid componentsthat are also part of the segment, which simplifies the visualization of the segment and does not alter any of the equations
that follow, since we assume that the rigid components transfer
the forces and are not geometrically altered by them. The eframe is on the fixed end of the segment while the e¢ frame ison the free end of the segment. To transform from the e¢ frameto the e frame, we can use the following rotation matrix:cosh  sinh
e
Re¢ ¼
(1)
sinh
cosh
where Ree¢ is the rotation matrix, h is the angle of the rotation
defined by h ¼ js, j is the curvature, and s is the arc length(note that the arc length changes as the segment compresses).
Using the e¢ frame, we can define our force due to a single
cable tension to be:




cosa
sin h2FTi ¼  Ti
(2)
¼  Tisina
cos h2

FIG. 3. (a) Labeled drawing of two segments. The relevant
dimensions are shown, as well as how segments are added toeach other. The medium gray represents rigid PLA components,
the light gray represents elastomer segments, the black represents cables, and the dark gray represents M2 nuts and bolts or
steel rods. (b) Demonstration of how cables were patterned
through the rigid vertebra with crossing cable and non-crossing
cable. Steel pins were pushed through the PLA components tocreate a reduced-friction contact for the fishing line cables. Notethat I = 1.63 · 10-9 m4 and A = 0.2 · 10-3 m2. A more detailed
CAD drawing with dimensions is given in the Supplementary
Data (Supplementary Data are available online at www.liebertpub.com/soro).
and cable friction); (4) the cables do not contact the body ofthe spine (this stems from the fact that we have externalcables19); and (5) external loads are negligible, including gravity. The experimental setup is designed to eliminate gravity effects and the spine itself moves orthogonal to gravity.

where FTi is the force due to the ith cable, Ti is the tension in
the ith cable, and a is the angle the cable forms due to thecurvature, such that a ¼ p 2 h. Taking a static modeling approach, which we assume is valid due to slow operating
speeds, we find our sum of forces to be:+F ¼ 0n1+ Ree¢ FTi þ FR ¼ 0

i¼0

n1



F R ¼ + Ti
i¼0

 sin h2
cos h2(3)where FR is the reaction force in both the x- and y-directions at
the fixed end of the elastomer segment, and n is the number ofcables. To look at the moments caused by forces, we must
consider the moment arm, which can be defined as:

 1

 j1
þ di cosh
j


(4)
rTi ¼
1
j þ di sinhwhere rTi is the moment arm to the force, and di is the distancefrom the center to the cable as measured in the e¢ frame.
Looking at the sum of moments about the center of the fixedend of the elastomer segment, we find:+M ¼ 0n1

MR ¼  + Ti di cos
i¼0

FIG. 4.

Representation of single segment bending.

h
2

(5)

Using our assumptions of constant curvature and linear
material properties, we can also state that:

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CASE ET AL.

MR ¼ Kb j(6)

where Kb is the bending stiffness of the material, such thatKb ¼ EI, where E is the nominal Young’s modulus of thematerial, and I is the second moment of inertia of the segment. Combining Equations 5 and 6, we get:n1j¼

 +i ¼ 0 Ti di cos js2
Kb(7)

Equation 7 shows how the cable tensions relate to the curvature. If we allow our material to deform due to compressive
forces caused by the cable tension, we must consider theeffects on strain:

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n1b ð xÞ ¼

 +i ¼ 0 Ti di cos h2
x
Kb

a ¼

 +i ¼ 0 Ti cos h2
Ka

(8a)

n1

(8b)

y ðxÞ ¼ b ðxÞ þ a(8c)where b is strain due to bending moments, a is strain
due to axial forces, y is total strain in the y-direction, xis the distance from the central axis, and Ka is axial
stiffness, such that Ka ¼ EA, where A is cross-sectional
area. A more in-depth discussion on the strain equations
can be found in Camarillo et al.,25 and is not repeated inthis study. From this, we can see that the neutral axis isgoing to shift due to the combined axial compression
and bending. We can find the new location to be a
distance from the center as:n1

xn:a: ¼

 K b + i ¼ 0 Tin1
Ka +i ¼ 0 Ti di

¼

an
ad

(9)

where xn:a: is the distance of the neutral axis from the center,
an is the numerator of the neutral axis calculation, and ad isthe denominator of the neutral axis calculation. The neutral
axis shifts as the spine compresses. These equations can betaken and expressed in shortened form:
K m q ¼ Ds

FIG. 6. Flowchart of simulation andphysical spine control. The kinetics portionof the robot model (I) computes the deformed configuration based on cable tensions
obtained from the random sample space. The
kinematics portion of the robot model (II)takes the deformed configuration and determines the required change in cable length.
When working with the physical robot, thesechanges in cable length are fed to the motor
control, which drives the physical robot into
the required deformed configuration.

(10a)FIG. 5. Two-segment spine for (a) traditional and (b)crossing cable patterns. The numbers represent the cable
number, and the black lines represent the cable pattern.
2

Kb
6 0
Km ¼ 6
4 0
0

0
Ka
0
0

0
0
Kb 1
0

3
0
0 7
7
0 5

(10b)

Ka 1

q ¼ fj, a , an , ad gT(10c)

s ¼ fT0 , T1 , . . . , Tn g T(10d)where K m is the stiffness matrix, q is the configuration-spacedescription, D consists of the multipliers necessary to solve
for K m q, and s is the cable tensions. For a single segment, wefind D to be:23
 d0 cos h2  d1 cos h2 . . .  dn cos h2
6  cos h cos h2 . . .
 cos h2 7
7
2
D¼6
(11)
4 1
1
...
1 5
d1
...
dn
d0
This model of a single segment is important, but coupling of
the spine segments becomes relevant when there are multiple
segments. Camarillo et al. introduced an analysis for multisegment continuum arms with parallel cables,26 which serves
as the basis for our crossing cable model. A full derivation of
the multisegment continuum arm with crossing cables can beREDUCING ACTUATOR REQUIREMENTS IN CONTINUUM ROBOTS

113

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FIG. 7. Representation of possible cable
patterns. Note that cables come in pairs andonly half are shown. The cable’s pair is
opposite and symmetrical to what is shown.
The cables shown in the blue box are ourinitial guess at a harmonic crossing cable
pattern. The cables shown in the green box
are the cables for the traditional parallel
cable pattern. For each spine tested, four
configurations are selected from this representation. Color images available online atwww.liebertpub.com/soro

found in Supplementary Data (Supplementary Data are
available online at www.liebertpub.com/soro). To illustrate
the difference between the two cabling patterns (traditionaland crossing), let us consider a two-segment spine. For thisexample, there are four cables in both the traditional andcrossing cable configurations shown in Figure 5. For all cables,
we set jdi j ¼ lc for all i. Consider first a simple two-segment
example for a traditional cable configuration:½K m f q g ¼ ½D f sg


(12a)

 
 
q0
D0 D1
s0
¼
(12b)
q1
0 D1
s1


where the subscripts in qj , Dj , sj refer to these respective
vectors and matrices in the jth segment. This can be expanded
for any number of segments. Note that the tension is split intos0 and s1 ; this highlights the concept of having specific cables
for each segment of the manipulator. In a traditional cable
pattern, [D] indicates the decoupling. This can be shown by
the upper triangular matrix since the cables from the 0th
segment do not influence the outcome of the 1st segment (i.e.,
the tensions s0 do not affect the 1st segment’s configuration
q1 ), but the cables for the 1st segment do influence the outcome for the 0th segment (i.e., the tensions s1 affect the 0th
segment’s configuration q0 ). In this study, D0 corresponds tothe cables for the 0th segment and D1 corresponds to thecables for the 1st segment.
If we consider the cable pattern shown in Figure 5a andreduce the complexity of Equation 12 by simplifying suchthat qj ¼ jj , we find:


"

Kb
0

Km
0

0
Kb

lc cos h2
0

0
Km





j0
j1

cannot be broken up and cable tensions cannot be divided per
segment. The system must be taken as a whole:   
Km 0
q0
D0
¼
(14)
fsg
0 Km
q1
D1
where all the cable tensions are described by fsg and are
relevant to all segments. In this study, D0 and D1 keep trackof which side the cable is on for a specific segment. That is, d0
can be positive in the 0th segment and negative in the 1st
segment if it crosses between them.
If we consider the cable pattern shown in Figure 5b andreduce the complexity of Equation 14 by simplifying suchthat qj ¼ jj , we find:

 
Kb 0
j0
¼
j1
0 Kb
8 9
T
>
> 0>
"
#>
> (15)
<
h
h
h
h >
T1 =
lc cos 2  lc cos 2
lc cos 2
 lc cos 2T2 >lc cos h2  lc cos h2  lc cos h2
lc cos h2 >
>
>
>
;
: >
T3


¼

 lc cos h20lc cos h2
lc cos h2

8 9
T
>
> 0>
#>
>
<
h >
T =
 l cosc2
 lc cos h2

(13)

1

>
T2 >
>
>
>
;
: >
T3

Note that, in this case, s0 ¼ fT0 , T1 gT and s1 ¼ fT2 , T3 gT
since cables 0 and 1 are used to move the 0th segment, andcables 2 and 3 are used to move the 1st segment.
In contrast, cables in the crossing cable configurations cancross from one side to the other and are designed to influence
every segment rather than decouple. Therefore, segments

FIG. 8. Intensity versus span of all 31,465 crossing cablecombinations with 105 random tension samples shown in gray.
The cyan dots represent the 170 selected configurations. The
black dots represent those selected configurations run with 106
random tension samples. The red stars indicate the crossing
cable configurations with the highest (i) span and (ii) intensity.
Color images available online at www.liebertpub.com/soro114

Note the differences in the simplified ½D matrix between
Equations 13 and 15, which demonstrate the differences between the two cabling patterns. In particular, the cables canbe shown switching between positive and negative signs in
the last two columns of the simplified ½D matrix in Equation
15, which is due to the crossing cables and is not present in
the representation of the traditional cabling method.
Simulation Results

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To capture the behavior of this analytical model, we developed a computational model using C++. Because Equation 10a is nonlinear, we solved for the state q using an
iterative solver, using the root finding secant method. The
overview of our simulation and experimentation process is

FIG. 9. (a) Intensity versus
span of configurations withchanges to elastomer lengths.
Note that the cable patterns ofcrossing span and crossing intensity are represented bythose shown in Figure 8. (b)Representation of the workspace for the crossing cableconfiguration with the best
performance in intensity[configuration (ii) in Fig. 8].
(c) Representation of theworkspace for the traditional
parallel cable configuration.
(d) Difference between thetwo cable configurations with
40 mm elastomer segments.
The base of the spine starts at(0,0). The shaded region iswhere the spine can reach. The
shading scale denotes how
many angular position bins thespine is able to reach. Note thatthe positive numbers indicatethat the system performs betterin the crossing cable configuration, while the negative
numbers indicate that the systems performs better in the
traditional cable configuration. Color images available
online at www.liebertpub.com/soro

CASE ET AL.

given as flowchart in Figure 6. Note that in this section, weonly used the blocks up to the Simulation Results, while therest of the flow chart is applicable to the experimental validation, which is detailed in the next section.
First, we sought to show that crossing cable configurations
can be designed such that their workspaces closely match theworkspace of the traditional cable configuration. To do this, we
needed to further define the different cable patterns. For the
traditional cabling method, cables terminate at each segment,
as shown in the green box in Figure 7. For a six-segment spine,
this implied a total of 12 cables. Our goal was to reduce thetotal number of cables (and subsequently actuators) required toachieve a comparable system workspace using crossing cableconfigurations. Our first intuition for a crossing cable configuration was to use the harmonics of a six-segmented system,

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REDUCING ACTUATOR REQUIREMENTS IN CONTINUUM ROBOTS

115

FIG. 10. Demonstration of different workspaces achievable through different crossing cable configurations. The base of
the spine starts at (0,0). The shaded region is where the spine can reach. The shading scale denotes how many angular
position bins the spine is able to reach.
which resulted in a spine with eight total cables, as shown inthe blue box in Figure 7. Using a reduced number of eightcables, we sought to compare all the possible crossing cableconfigurations from the cables shown in Figure 7 (31,465
configurations) to the traditional 12-cabled configuration. Notethat symmetry was assumed for all these cable configurations(i.e., there are four pairs of cables).
To determine the difference in performance of the cableconfigurations, we explored the workspace of a six-segment
system with 40 mm long elastomer segments. This consisted ofrunning samples of randomly generated cable tensions throughthe model to determine final end position and orientation. The
tensions were randomly selected using C++’s rand() command. End position was discretized into a workspace grid, with
1 cm resolution. End orientation was discretized into 8 bins
divided evenly between 0 and 360. We were most interested
in two criteria: span (the number of position bins the spine
reached, regardless of orientation) and intensity (the number ofangular bins reached among all locations).
Because of the large number of possible crossing cableconfigurations, we ran a cursory pass over all the combinations
using a total of 105 random tension samples (Ti 2 ½0, 1N),

FIG. 11. Experimental setup with spine floating in waterbath. Color images available online at www.liebertpub.com/soro

which had a low enough resolution to reduce computation
time, but high enough resolution to indicate significant trends.
The span and intensity were tracked for each combination, andthe results are shown in Figure 8. While there was a wide
spread in performance for the crossing cable patterns, we
wanted to focus on the specific configurations that performed
well in both span and intensity to match the performance of the
traditional cable pattern. Therefore, we selected the top 100 interms of span and the top 100 in terms of intensity, which
resulted in 170 ‘‘selected’’ cable configurations with an overlap of 30 configurations, as shown in cyan in Figure 8.
We then ran these 170 configurations at a higher resolutionof 106 tension samples (Ti 2 ½0, 1N). The results from thisare shown in black in Figure 8. In this study, we have shownthat the higher resolution yields even more favorable span
and intensity for the selected configurations. The best performing cable configurations are shown in Figure 8, whereconfiguration (i) represents the best performing configurationin terms of span and configuration (ii) represents the best
performing configuration in terms of intensity. Both of thehighest performing crossing cable configurations included acable pair that goes all the way up the sides without crossing,similar to the traditional parallel cable configuration.
In addition, we investigated the impact of elastomer
segment length on workspace span and intensity. In thisexperiment, we compared the workspace of our selectedcrossing cable configurations to the traditional parallel cable configuration with the high resolution of 107 tension
samples (Ti 2 ½0, 2N). As shown in Figure 9a, the crossing
cable configurations (8-cables) perform comparably to the
traditional configuration (12-cables) in terms of both span
and intensity. We note that shortened lengths are morebiomimetic, as biological spines have short elastic disks
between their rigid vertebrae. We also note that the span
and intensity values decrease because the spine cannot
reach as far with smaller segments. Figure 9b–d shows one
workspace comparison between the crossing cable configuration with the best performance in intensity [configuration (ii) from Fig. 8, shown in Fig. 9b] and the traditional
parallel cable configuration shown in Figure 9c, for 40 mm
elastomer segments. The difference between the workspaces is shown in Figure 9d. All of the workspaceDownloaded by YALE UNIVERSITY from www.liebertpub.com at 01/16/19. For personal use only.

116

comparisons shown in Figure 9a demonstrate that segment
coupling can be used to reduce the total number of actuators
in segmented manipulators while maintaining workspace
performance.
We also sought to show that crossing cable configurations
require less cable tension than the traditional cable configuration. In the crossing cable configuration, all cables run thefull length of the spine and therefore extend to each segment.
In the traditional cable configuration, only a subset of cables isavailable for any given segment. As the crossing cable configuration enables more cables per segment, tension requirements are therefore distributed between a larger number ofcables, and the tension requirements per cable are relatively
reduced. To demonstrate this, we compared the tensions of thecrossing cable configuration given by configuration (ii) in
Figure 8 with the tensions of the traditional cable configuration
across a range of spine positions. The spine positions we used
for comparison are the same positions used in the experimental
validation. For this testing, we can consider two quantities asfollows: (1) the maximum cable tension for each cable configuration for a given spine position (Tmax, k ¼ maxfTi, k g fori 2 [0, n], where k denotes a given position) and (2) the peak
cable tension of each cable configuration, which is quantified
as the maximum value of the cable tensions across all thetested spine positions (Tpeak ¼ maxfTmax, k g for all k). The data
showing the results can be found in Supplementary Table S1.
While we found some isolated spine positions for which thetraditional cable configuration generated lower maximum cable tensions than the corresponding crossing cable configuration, the crossing cable configuration had a lower peak cabletension over the entirety of the workspace (i.e., all tested spine
positions).
In addition, in contrast to attempting to find crossing cableconfigurations with workspaces comparable to the traditional
cable configuration, we also sought to identify crossing cableconfigurations with workspaces that differ drastically fromthe traditional cable configuration. Such designs are notuniversally useful, but would be application driven. Some
examples of other workspaces are shown in Figure 10. In thisstudy, we again use the high resolution of 107 tension samples(Ti 2 ½0, 2N). These examples demonstrate that the workspace can be limited by cable configuration, rather than bysoftware, which could be used as an additional safety measure in robot manipulator design.

CASE ET AL.
Experimental Validation

To determine the accuracy of the model, we tested it on aphysical system, as shown in Figure 11. The system was placedsuch that it was floating on water with Styrofoam platforms toeliminate friction caused by it sliding on a surface. For the experimental validation, we selected the crossing cable configuration shown in Figure 12a, which matches configuration (ii) in
Figure 8, and used 20 mm long elastomer segments. Spine positions yielded from the kinetic model can be translated to cabledisplacement values using the following kinematic relationship:
5

Dli ¼ lneutral  + li, j þ Dltension, i

(16a)

j¼0

Dltension, i ¼

Ti ltotal
Kt

(16b)

where Dli is the change in length for the ith cable, lneutral is theneutral cable length (this does not account for the cable lengththat goes through the vertebrae since that is assumed to be aconstant length regardless of spine position), li, j is the length ofthe cable for the jth segment, Dltension, i is the change in length(stretch) due to the ith cable tension, ltotal is the total length ofthe cable (this accounts for the length going through vertebrae),and Kt is the cable stiffness of the fishing line (96.5 N/m).
To control the movement of the spine, we developed a
Python program that ran an Arduino with a slave program.
The Python program translated the raw cable displacements,
given in meters, to motor commands, given as steps in the
stepper motors. The motor commands were sent to the Arduino Uno, which used the Adafruit motor shield and AccelStepper libraries to actuate the spine. To determine theactual spine position, we used a vision tracking system consisting of a Logitech C270 Webcam and OpenCV. Red andyellow dots were placed on the spine to facilitate this and canbe seen in Figure 11. The computer vision (CV) systemproduced an estimate of the spine position based on the average position of the fiducial dots taken over 50 images. In
addition, we applied an outlier rejection filter to remove cases
where the CV system misidentified the location of the dots.
Figure 12 shows three different comparisons between thepredicted and actual spine positions. While the model

FIG. 12. Experimental tests for model validation. The recreation of the spine is shaded blue, while the model prediction isshown as a black outline. (a) Shows the cables used and their respective numbers. Changes in lengths for (b) are {11.5, 39.5,
46.5, -10, 8.5, -16.5, -24.5, -31}mm, for (c) are {-9.5, 2.5, -13.5, 11.5, 29.5, 15.5, 31.5, 5.5}mm, and for (d) are {7.5,
24.5, -2.5, 28, 12.5, -5, 22, -10}mm for the respective cables given in (a). Color images available online at www.liebertpub.com/soro

REDUCING ACTUATOR REQUIREMENTS IN CONTINUUM ROBOTS

predictions do not exactly match the physical outcomes, wefind the agreement in the trends of the deformations to beconvincing that the model is accurate. The error in end position for Figure 12b was 30.3 – 2.3 mm; for Figure 12c, theerror was 46.9 – 4.4 mm; and for Figure 12d, the error was
36.2 – 2.7 mm. Additional spine configurations can be seen in
Supplementary Data. The main sources of discrepancy between the model and physical system are cable friction andcumulative errors due to positional error in each segment.
Since the model does not account for a number of real
physical terms, such as friction, we expected the results to beclose, but not perfect. As it turned out, our errors were small(on the order of 30 mm) and there was good agreement between our models and the shape and behavior of the physicalsystem.

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Conclusion and Future Work

We have demonstrated that actuator count and maximum
cable tension can be reduced without loss to workspace
performance through the use of crossing cable configurations.
Furthermore, we have shown that through the use of crossing
cable patterns, we are able to tune the workspace for a given
application, which may be leveraged to increase safety during
manipulator operation. Use of highly coupled actuation allows for greater flexibility in design and more closely matches the behavior of biological systems. A supplementary
video is provided that gives a broad overview of the paper andshows the system deforming with a crossing cable pattern.
Moving forward, we plan to extend the 2D soft spine model
introduced herein to a 3D model. 3D soft spines using spiralcable patterns more closely mimic biological spines, as many
mammals exhibit spiral muscle patterns in their arms andlegs.12 In addition, spiral cable patterns will add torque to thesystem, which could be used for energy storage or more precise
control of the end location and orientation. We expect that thiswill give more profound improvement at the cost of an evengreater challenge in the design phase. We also expect to further
use our models in continuum manipulator control strategies.
Acknowledgments

This work was partially supported by an Early Career Faculty grant from NASA’s Space Technology Research Grants
program (NNX14AO52G). J.C.C. is supported under theNASA Space Technology Research Fellowship program(NNX15AQ75H). E.L.W. is supported under the National
Science Foundation Graduate Research Fellowship program(DGE-1333468). Any opinions, findings, and conclusions or
recommendations expressed in this material are those of theauthors and do not necessarily reflect the views of NASA or the
National Science Foundation. The authors also thank Jonathan
Bruce for his helpful input in getting the system running.
Author Disclosure Statement

No competing financial interests exist.
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Address correspondence to:Rebecca Kramer-Bottiglio
School of Engineering & Applied Science
Yale University
P.O. Box 208286
New Haven, CT 06520
E-mail: rebecca.kramer@yale.edu