SOFT ROBOTICS
Volume 5, Number 1, 2018ª Mary Ann Liebert, Inc.
DOI: 10.1089/soro.2017.0033

ORIGINAL ARTICLE

A Soft Parallel Kinematic Mechanism
Edward L. White,1 Jennifer C. Case,1,2 and Rebecca Kramer-Bottiglio1,2

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Abstract

In this article, we describe a novel holonomic soft robotic structure based on a parallel kinematic mechanism.
The design is based on the Stewart platform, which uses six sensors and actuators to achieve full six-degree-offreedom motion. Our design is much less complex than a traditional platform, since it replaces the 12 sphericaland universal joints found in a traditional Stewart platform with a single highly deformable elastomer body andflexible actuators. This reduces the total number of parts in the system and simplifies the assembly process.
Actuation is achieved through coiled-shape memory alloy actuators. State observation and feedback is accomplished through the use of capacitive elastomer strain gauges. The main structural element is an elastomer
joint that provides antagonistic force. We report the response of the actuators and sensors individually, thenreport the response of the complete assembly. We show that the completed robotic system is able to achieve fullposition control, and we discuss the limitations associated with using responsive material actuators. We believethat control demonstrated on a single body in this work could be extended to chains of such bodies to create
complex soft robots.
Keywords: SMA actuator, deformable body, parallel kinematics, soft body control, elastomer strain sensors
distribution of deformations throughout the body, rather thanat discrete joints, complicates the sensor placement problem.5
One simplification we make in this work is the considerationof only discrete points on a soft body. In this scheme, theexact deformation field within the body away from the points
is unimportant, and only the deformations of the points relative to one another are considered.
This work demonstrates a holonomic three-dimensional
soft robot module, as shown in Figure 1. A single piece ofelastomer will naturally have nontrivial compliance in all
DoFs. In three-dimensional space, this means that six DoFs
are required to fully define the deformation of an elastomer
body, and six independent channels of sensing and actuation
are required to observe and control the deformation. One
potential configuration of sensors and actuators to achieve
holonomic control is suggested by the Stewart platform.6
Although there are many solutions to this challenge, we believe that the Stewart platform configuration used in this article is a highly efficient way to position sensors and actuators
around a soft body. Our selection of a Stewart platform
configuration is motivated by our long-term goal of completely integrating all of the sensing, actuation, and structuralelements into a single body. A Stewart-like configuration

IntroductionS

oft robotic systems have unique characteristics, such as
high deformability and impact resistance, that make thempotential alternatives to traditional robots in applications
such as mobility in unstructured environments and operationin proximity to humans.1 Unlike in traditional rigid robots,
where deformations are sources of error to be avoided, in soft
robots the change in geometry is essential to the functioningof the robot.2 Further, deformations are distributed throughout the body in soft robots, unlike in traditional rigid robots
where articulation is localized at joints with limited degrees
of freedom (DoF).3 Distributed deformations allow soft robots to forgo bearings and joints for articulation, which has
the potential to greatly reduce cost and complexity while
increasing durability, particularly in dusty or corrosive
environments.
The same characteristics that make soft robots interestingalso make these systems difficult to design and control.4 To
take advantage of the deformations of these bodies, they must
be controlled, which requires both sensing and actuation. The
large deformations present in soft robotic systems necessitate
the use of stretchable state observation sensors whereas the1
2

School of Mechanical Engineering, Purdue University, West Lafayette, Indiana.
School of Engineering and Applied Science, Yale University, New Haven, Connecticut.

36

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A SOFT PARALLEL KINEMATIC MECHANISM

could use sensors and actuators located on the outer surfaceof the structure, with no overlapping elements. Both these
considerations are extremely important to facilitate
manufacturing of an integrated soft system.
The deformable body at the core of the soft robot module
comprised silicone elastomer cast into a spherical shell structure. We selected this geometry to make the stiffness in each
DoF comparable. For example, had a solid structure been used,the compressive stiffness would have been at least an order ofmagnitude higher than the bending stiffness. Acting against
this body are six nickel-titanium shape memory alloy (SMA)
actuators configured into spring-like coils; the elastomer bodyand actuators act as an ‘‘antagonistic pair.’’ The actuators are
only capable of contracting when actuated, and they require an
external restoring force to extend. In this robot module, thatforce is provided by a combination of the deformable elastomer body at the core of the structure and the other actuators.
Conductive composite elastomer-based strain gauges are used
to observe the state of the system.
By integrating highly deformable elastomer structures,
responsive material actuators, and large deformation strain
sensors into a single module, we have created a robotic architecture that can be applied to a wide range of soft robotic
applications and structures. In addition to the particular materials selected for this application, other types of deformablebodies, actuators, and sensors could be used in other applications while retaining the same robot topology. In the future,
this same basic structure could be used to create dexterous
soft robotic limbs from chains of these segments.
In addition to holonomic systems, we believe that the same
components and design philosophy could be used to create
either over- or under-actuated robotic systems as well. Underactuated octopus-inspired cable-driven limbs were describedby Calisti et al.7 Since control of under-actuated systems is
nontrivial, these structures have also seen additional work
focused on control.8,9 Bridging the gap between our conceptof modular fully actuated structures and under-actuatedcontinuum bodies, fluid-actuated multi-segment limbs were
demonstrated by Marchese et al.10,11 We believe that a combination of fully- and under-actuated components will provide
the best performance to complex soft robotic systems. Because of this, we see our work as complementing, rather thanreplacing, the ongoing work on under-actuated soft robots.

37

hybrid rigid/soft systems.17 In this work, we expand on what
has been published previously by using position observationand feedback in a multi-DoF system to achieve closed-loop
control, which we believe is a critical next step to enabling
more complex soft robots.
Parallel kinematic systems

The parallel kinematic architecture of the robotic structure
we describe in this article was first described by Stewart.6
Over the past five decades, there has been significant researchon these parallel kinematic platforms, although most of thefundamental work occurred during the 1980s and 1990s.18
The Stewart platform provides high stiffness and full six DoF
motion within a smaller volume than a similarly sized serial
kinematic system. For this reason, Stewart platforms havefound narrow yet deep success in areas such as full-motion
flight simulators and micropositioning systems. Creating a
traditional Stewart platform with spherical and universal
joints results in a complex assembly with many components.
In contrast to a jointed system, a soft system such as the one
we present in this work relies on the naturally low stiffness ofelastomer components to achieve motion through bending,which results in a more mechanically robust structure withfewer failure modes.
In addition to using traditional actuators, parallel kinematic systems with SMA actuators have been describedwith two-axis,19 three-axis,20 and six-axis motion.21 Further,
two of these approaches were extended into multi-segment
chains.21,22 The two major differences between the previous
work and the current work are the use of soft sensors for state
feedback and highly deformable elastomers for structuralelements. Although previous examples of SMA-based systems have demonstrated material state feedback by measuring
electrical resistance of the actuator wire, this does not provide
information on the configuration of the overall structure. To
overcome this limitation, we have added elastomer-based
strain sensors to provide measurements of the deformed stateof the platform. The previous approaches also rely on a conventional joint or joints. In our work, we eliminate the use of
mechanical joints by using deformable structures. This significantly decreases the part count and complexity of theoverall design, and it moves the state of the art forward toward
integrated soft robotic systems.

Previous WorkThe unique characteristics of soft robots come from theirmaterial composition, unlike traditional robots where materials are less of a concern than the geometry of the structure.
To develop the robotic module described in this article, wehave had to not only integrate existing elements but also
further develop the state of the art at the component level,
particularly in the area of soft sensing, as a part of our integration effort. In our previous work, we demonstrated thecontrol of a single DoF elastomer body by using SMA actuators and liquid metal sensors.12 This work builds on ourprevious system by increasing the number of DoF, sensors,
and actuators, all of which result in significant challenges inreliability and integration. Previous work by other groups
has demonstrated controlled motion of a snake-like robot13
and uncontrolled motion in other elastomer systems, such as
crawling robots,14 grippers,15 tentacles,16 and actuators in

Sensors

There are many potential solutions for proprioceptive
feedback in soft robotic systems. Our group recently presented
a review of planar sensor-embedded structures that we referto as ‘‘sensory skins.’’23 Both resistive and capacitive sensors
made from a range of materials have been demonstrated. Two
common material approaches to making very high deformation electrical conductors are elastomer-based composites24
and liquid metals.25 Liquid metals are very common in the soft
robotics literature, and have been used to fabricate a rangeof soft systems, including keypads,26 curvature,27,28 pressure,29 contact force,30 and multimodal pressure and strain
sensors.31,32 Proprioceptive liquid metal sensors embedded in soft actuators have been described, which is a very
similar application to what we describe in this work.33–35
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38

measure the state of an SMA-actuated soft robotic system12 and to create modular sensory skins.36
An alternative approach to resistive sensing is to usechanges in capacitance to measure strain. Previous elastomerbased capacitive strain sensors have been demonstrated with
gold37 and carbon nanotube38 electrodes. Both these approaches
may enable smaller capacitive sensors than we constructed inthis study, which we believe will facilitate integration ofsensing into soft robots and systems with distributed deformations. We chose to employ capacitive conductive composite sensors due to ease of manufacture, robustness, andstability over time. We selected a graphene-based conductive
composite as our electrode, which is a class of materials reviewed in Ref.39 In particular, our electrode material was
inspired by the work of Kujawski et al.40 However, in theoriginal work, the expanded graphene material was used as a
resistive strain sensor, whereas we use it as an electrode in acapacitive sensor. This mitigates the drift and degradation
observed when conductive composites are employed as resistors, at the cost of more complex signal conditioning.
Actuators

There are many types of responsive material or soft actuators that could be used in this application, including
pneumatics, dielectric elastomers, shape memory materials,
hydrogels, and ionic polymer composites.41 A frequently
reported type of actuator in the recent soft robotics literatureis the pneumatic actuator.42 McKibbon actuators are acommonly used type of pneumatic soft actuator that could
directly replace our SMA actuators.43 These alternatives
have lower power and energy densities than SMA actuatorsand require more complex interface hardware consisting ofa pressure source and valves. In addition to McKibbon actuators, which are designed to replace traditional pneumatic
pistons, using pressure to directly deform soft bodies has
also been demonstrated.14,44 In our robotic system, oursensors and actuators are individual components. However,direct integration of liquid-metal-based sensors into actuators has already been demonstrated in McKibbon actuators33 and deformable body pneumatic systems.34,35
Shape memory materials, and particularly alloys thatcomprised nickel and titanium, have received significant attention45 since the discovery of the shape memory effect byJackson et al.46 The mechanics behind metallic SMAs havebeen extensively studied, and many different approaches
exist to explain the underlying physical phenomena.47 Metallic SMAs have been used in many applications as actuators for macro-scale48 and micro-scale systems.49 Shapememory materials have been used in soft robots in the formof sheets50 and wires.51 SMAs have been used extensively inmobile soft robotic applications, including in robots inspiredby worms,52–54 fish,55 manta rays,56 starfish,57 turtles,58 and in
more traditional systems such as underactuated cylinders59
and jumping robots.60 As mentioned in the Introduction, coiled
SMA actuators are only capable of producing tension. Thisposes design challenges not frequently found in traditional
actuators. Liang and Rogers presented an analysis of single,
passively biased, and antagonistic SMA actuators.61 This basic
concept, of using an external force to provide a counter-acting
force to the actuator, has been reused in almost all subsequent
SMA work.

WHITE ET AL.

SMA antagonists fall into the two categories described by
Liang and Rogers: passive springs or active SMA actuatorpairs. An early example of SMA antagonist pairs was demonstrated by Ikuta et al., who combined multiple SMA actuator pairs into a continuum manipulator for endoscopic
surgery.62 This application is very similar to our previous
work12 but with rigid components and traditional joints instead of a deformable elastomer body. In this work, we haveextended control of deformable bodies into three-dimensional
space.
Researchers have had to address the complex thermomechanical control problem inherent in working with SMA.
Two approaches have been used: to measure some property,
typically resistance, of the SMA actuator to estimate its
present internal state, or to measure the displacement outputof the actuator. Ikuta et al. demonstrated both methods concurrently, using a material model that was later refined byMajima et al., who demonstrated that measurements of resistance could be used to control SMA-actuated systems.62–65
In our own previous experiments, we found that resistancebased state estimation is highly dependent on loading condition and ambient temperature, resulting in large errors
without additional measurements. For this reason, we use
direct measurement of actuator displacement in this work.
Designing the shape of the SMA actuators remains aheuristic process, although attempts are being made to provide rigor to the design process. De Aguiar et al. presented amodel of coiled SMA actuator performance based on themechanics of helical springs and a one-dimensional constitutive equation.66 An et al. provided a review of SMA coilsfrom the perspective of providing design rules.67 In designing
an SMA actuator, particularly one that will be used with anelastomer antagonist, there are three major criteria that must
be considered. First, the actuator must apply enough force inthe ‘‘on’’ state to perform the desired motion. Second, theantagonist must have sufficient strength to return the actuatorto the desired ‘‘off’’ position, while not being so strong as tointerfere with the first objective. Third, when a majority of theactuators are ‘‘on’’ or are still cooling, the antagonist must be
sufficiently strong to resist bucking or collapse. In this work,
our design process started with the desired size of the finishedrobotic system, and then considered the minimum, neutral,and maximum lengths of the actuators. Next, we comparedstretchability of a handful of samples we manufactured withdifferent wire diameters and coil diameters, where both parameters were constrained by commercially available materials. Finally, we were able to quickly converge on a usable,
although not optimal, actuator design. Our experience demonstrates that better actuator and soft system design tools
would be beneficial to further developments in this field.
Platform Kinematics

One of the barriers to wider adoption of parallel kinematic
systems is the non-linear mathematics that govern the kinematics of the platform. The purpose of the Stewart platform is
to achieve full six DoF motion relative to the base platform byusing six sensors and six actuators. In our formulation, we
consider two different sets of states: a configuration statevector that describes the location of the moving platform
relative to the stationary platform, and a length state vector
that describes the lengths between nodes on the platforms.

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A SOFT PARALLEL KINEMATIC MECHANISM

39

We use two sets of states because we want to control the
motion of the system in a natural coordinate system withmotions such as ‘‘move right’’ or ‘‘pitch down,’’ yet we mustobserve and actuate the system by using length-based sensors
and actuators. We define a configuration vector describing
the transformation from the base to the work platform as:

Stewart platform, which is known as a ‘‘6–6’’ configuration.
This signifies that there are six nodes on the upper surface,and six on the lower surface. Odd-numbered nodes are on thebase platform, and even-numbered nodes are located on themoving platform. The mapping between the internal statevector and the resulting configuration vector is defined as:

c ¼ [x1 , x2 , x3 , qr , q1 , q2 , q3 ]T ,l ¼ T (c),(1)

where x1, x2, x3 represent translations in an orthonormal coordinate system e and qr, q1, q2, q3 represent a rotation described in terms of quaternions. All three of the classic
rotation representations, Euler angles, direction cosine matrices, and quaternions, appear in the parallel kinematics
literature, with no representation being clearly superior tothe others. We selected a quaternion representation to eliminate the possibility of gimble lock and to take advantage ofcomputational efficiency. In addition, quaternions can readily
be used to describe rotations about the local coordinate system attached to the moving plane. This is useful when
the robot is used as a part of a human-in-the-loop system,
for example, allowing an operator to request a ‘‘pitch up’’ or‘‘turn left’’ motion relative to the current location. As with allquaternion formulations, four parameters are used to represent a rotation in three dimensions, necessitating the additionof an auxiliary constraint that the rotation direction be a unit
vector:q21 þ q22 þ q23 ¼ 1:

(2)

We also define an internal state vector:l ¼ [l0, 1 , l2, 3 , l4, 5 , l6, 7 , l8, 9 , l10, 11 ]T ,(3)

where li, j is the length between the ith and jth nodes, which
are shown in Figure 2. This formulation describes a general

c¼T

1

(l),

(4a)
(4b)

1

are the mapping and its inverse, respecwhere T and T
tively, between the configuration and state vectors. As notedin our review of the literature on parallel kinematic systems,
the complexity of the inverse transformation has been the
motivation for much of the work in this area. Fortunately, theforward transformation is considerably more straightforward,
even if it is highly nonlinear. In computing both the forward
and reverse transformations, we frequently need to rotate avector by using our quaternion representation. To compute arotation, we use the computationally efficient expressionx¢ ¼ x þ 2q · ðq · x þ qxÞ, where x¢ and x are the rotated andunrotated vectors, respectively. In this notation, the scalar q
represents the angle of rotation (qr in Eqn. 1), whereas thevector q represents the direction of the rotation ([q1 , q2 , q3 ] in
Eqn. 1).
We compute the forward transformation T (c) in threesteps. First, we compute the location of the nodes in the top
and bottom plates relative to the center of each plate. Second,
we rotate the points in the top plate based on the rotation
components in the state vector in Equation (1). Third, we translate the rotated points in the top plate according to the translation components in state vector in Equation (1). Starting withthe points in the lower platform, which does not move, thenodal locations are:

FIG. 1. Completed soft robotic module with support electronics. Not shown is a power supply and PC hosting the control
software and user interface. The robotic structure is the blue and red object at center right. An Arduino used to communicate with the PC is behind and to the left of the robotic structure. Closed-loop current controllers for the SMA actuators arein the foreground at right. The robotic structure consists of two moving rigid plates (blue) connected by a deformablesilicone body (red). There are six sensors and six actuators connected between the upper and lower rigid plates. Thesesensors and actuators are shown in the inset. For a schematic illustration of the robot with components labeled, please refer
to Figure 2. SMA, shape memory alloy. Color images available online at www.liebertpub.com/soro40

WHITE ET AL.

where r is the nominal radius of the attachment points, o is theattachment point offset, and h is defined as:
 

iþ2
2p
mod3
:
(6)
h¼
4
3
Similarly, for the top plate, before the application oftranslation or rotation:hii1
ri ¼ r cos h þ (  1) 2 mod2 o sin h e1 þhii1r sin h þ (  1) 2 mod2 þ 1 o cos h e2 þ(7)

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0e3 ,i 2 (1, 3, 5, . . . , 11):
Based on the configuration vector and the known geometry, we define a vector from the center of the lower plate to thecenter of the upper plate:t ¼ fx1 e1 þ x2 e2 þ (x3 þ h)e3 g,(8)where xi are the translation components of the state vector c,and h is the nominal separation between the two plates. To
compute the location of a node on the upper plate, we use:ri¢ ¼ qri q  1 þ t:

(9)

The rotation vector is defined as:n q q orrþ ðq1 i þ q2 j þ q3 kÞ sin,
q ¼ cos22(10)

where q1, q2, and q3 are the rotation components of the statevector c. Finally, we compute the distance between nodes:
li ¼ jri  r¢i þ 1 j2 , i 2 (0, 2, 4, . . . , 10),

FIG. 2. Schematic illustrations of the robot showing criticalelement labels. Sensors and actuators are shown as green andorange lines. (a) Shows an isometric perspective of the robot,
with nodes and actuators labeled. Actuators are labeled ‘‘A0’’ to‘‘A5’’ in orange. Orange lines represent the nominal location of
the actuators. The nodes are located at the center of the attachment locations of the sensors. The attachment locations of theactuators are not specifically labeled. (b) Shows a top perspective of the robot, showing nodes, actuators, and sensors. Sensors
are shown in green and labeled ‘‘S0’’ to ‘‘S5.’’ Figures 10 and 11
show side views of the robot undergoing deformations. Color
images available online at www.liebertpub.com/sorohiiri ¼ r cos h þ (  1)2mod2 þ 1 o sin h e1 þhiir sin h þ (  1)2mod2 o cos h e2 þ0e3 ,i 2 (0, 2, 4, . . . , 10),(11)

where jvj2 denotes the 2-norm of a vector and li are the elements of the internal state vector l. Equations (8)–(11) are theforward transformation T . The forward transformation isused to convert user inputs into the space used for control.
The reverse transformation is more complex, and we solve
this problem by using an iterative numerical scheme. We
begin by assuming a state vector (Eqn. 1), which we denotecest . We assume a zero-configuration vector, although using
the previous solution as a starting point improves the performance of the algorithm at the cost of increased memorystorage. Based on this initial estimate, and using the forward
transformation described earlier, we compute a set of estimated sensor lengths:
lest ¼ T (cest ):(12)

Next, we compute the error in the sensor lengths:
e ¼ lmes  lest ,
(5)

(13)

where lmes is the measured internal state vector provided fromthe sensor outputs. We use the Newton-Raphson algorithm
to minimize the error in lengths. Since the system is fully
defined, we can achieve a solution with arbitrary numericalA SOFT PARALLEL KINEMATIC MECHANISM

41

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precision, although the accuracy of the solution is limited by theaccuracy of the sensor observations. To compute the required
Jacobian, we perturb the configuration vector and determine
the change in error due to the perturbation. In the case of thetranslation components, this is trivial. In the case of the rotations, we apply perturbations of 0:001rad in the local yaw,
pitch, and roll directions in the local coordinate system of themoving platform based on the current estimate. We denote these
perturbations as q. Formally, in quaternion notation in the local
coordinate system attached to the moving platform, they are:q1 ¼ f0:001, 1, 0, 0g,(14a)q2 ¼ f0:001, 0, 1, 0g,(14b)q3 ¼ f0:001, 0, 0, 1g:(14c)

Based on the error calculations, we compute a Jacobianmatrix of the form:
2 qe0
qx1

6
6 qe1
6 qx1
6
6 qe2
6 qx
6 1
J¼6
6 qe3
6 qx1
6
6 qe4
6 qx1
4
qe5
qx1

qe0
qx2

qe0
qx3

qe0
qq1

qe0
qq2

qe0 3
qq2

qe1
qx2

qe1
qx3

qe1
qq1

qe1
qq2

qe1 7
qq2 7

qe2
qx2

qe2
qx3

qe2
qq1

qe2
qq2

qe3
qx2

qe3
qx3

qe3
qq1

qe3
qq2

qe4
qx2

qe4
qx3

qe4
qq1

qe4
qq2

qe5
qx2

qe5
qx3

qe5
qq1

qe5
qq2

7
7

qe2 7
7
qq2 7

7,

qe3 7
qq2 7

(15)

7

qe4 7
qq2 7
5
qe5
qq2

where the error terms ei are the components of the error
vector defined in Equation (13). We use lower-upper (LU)
decomposition to solve the system of equations: þ 1)
J c(n c(n)est
est ¼  ,

(16)

where the (n þ 1)and (n)superscripts denote the iteration
number.
þ 1)
The required step between c(nand c(n)est
est for the translation components x1 , x2 , x3 is trivial to apply.
However, rotations do not exhibit superposition, and sothey must be added sequentially. We apply the largest rotation step first (as measured by the magnitude of the rotation
angle), followed by the second and third largest. This approach minimizes the error between the step determined byusing the iteration scheme and the actual step applied. Ultimately, as the solution converges and the step size decreases,
this effect becomes negligible.
We used a convergence criterion of 1 · 10-6 cm based on
the 2-norm of the error vector . Convergence occurred in lessthan five iterations in most cases. We implemented this entire
algorithm in Cython, which is a language based on Python
with the static variable typing of C. This was imported into
our overall interface and control software, which was written
in regular Python. Running on a 2.60 GHz 64-bit computerwith Ubuntu Linux, convergence required less than 1 ms,
which was sufficiently brief so that the numeric inversion
could be integrated into a control loop.

Experimental

The robot was fabricated in three steps. The capacitive
sensors and SMA actuators were fabricated in two separate
parallel processes, then integrated together into a complete
system with passive elements. The capacitive sensor fabrication process is novel and presented in more detail, whereas
the SMA programming procedure is well known and only
summarized.
Capacitive Sensor Fabrication

Capacitive sensors were fabricated from three-layer structures of conductive composite elastomer electrodes coating a
non-conductive dielectric layer. All of the layers were based on
Dragon Skin 10 elastomer (Smooth-On, Inc.). The conductive composite material was made from 10 wt% expanded
intercalated graphite (EIG). The EIG was made by soaking
graphite flakes (Sigma-Aldrich) in a 4 : 1 nitric acid:sulfuric
acid solution, followed by roasting at 800C for 5 min. Thissoaking and roasting process caused the graphene plates
within the graphite to partially pull apart (‘‘exfoliate’’), resulting in a loose connection between plates. We then soaked
the expanded material in cyclohexane and sonicated by using a tip sonicator (Q700 with 1/4† tip, Qsonica) for 1 h at an
amplitude of 36 lm to further separate the plates. Finally, wedried the resulting slurry to a concentration between 3 wt%
and 5 wt%. To make the composite material, we combined
pre-mixed Dragon Skin 10 with EIG slurry and additional
cyclohexane to achieve a concentration of 3.00 wt% ofgraphite in the wet material, and 10.0 wt% in the finishedcomposite.
Three-layer blank substrates were fabricated by using a
rod-coating process, which we have previously described.36
A 1/4†-10 Acme threaded rod (McMaster-Carr) was used
to apply a uniform layer of liquid composite elastomer to apolyethylene terephthalate (PET) substrate. Liquid conductive elastomer was poured onto the PET film, then scraped
across the surface by using the threaded rod. The liquid
elastomer owed through the threads in the rod, forming auniform film. This electrode layer was allowed to cure, which
generally took between 4 and 8 h due to the evaporation of
cyclohexane. Once cured, we applied a layer of normal Dragon Skin 10 elastomer without the inclusion of EIG as a dielectric. We poured the liquid dielectric layer directly onto thecured electrode layer and used an identical rod-coating procedure to create a uniform film. Immediately after casting thefilm, we applied elastomer-impregnated muslin fabric to thecuring film at the ends of the sensor elements as a reinforcement tab. The dielectric layer, with fabric tabs, was allowed
to cure for *45 min until it achieved a ‘‘tacky’’ consistency.
At this point, we folded the sheet in half and bonded the tacky
dielectric layer to itself. This resulted in a six-layer structure:
PET—conductive composite—dielectric— dielectric—conductive composite—PET. We allowed this layered structure
to sit at room temperature for at least 4 h to complete curing.
Using the fabric tabs as alignment guides, we cut the final
sensor shapes from the completed elastomer substrate byusing a Universal Laser Systems VLS 2.30 patterning systemfitted with a 30 W CO2 laser module. The backing PET layer
was removed from the completed sensors, and soot from thelaser patterning process was cleaned by using a soap solution(Liquinox, Alconox).

42

WHITE ET AL.

Once cut from the elastomer substrate, the deformablecomponent of the capacitive sensor was attached to polystyrene attachment tabs, interface electrodes, and electronics.
To begin this attachment, we sewed two laser-cut polystyrene(PS) tabs to the front and back of one end of the sensor byusing cotton thread. We cut matching holes in the PS andsensor structure to facilitate sewing. On one side of the sensor, we attached only the PS, whereas on the other side we
inserted Pyralux (Adafruit) leads to provide electrical contact
to the two electrodes on the sensor. We soldered these leads
to a printed circuit board containing the electronics, thenfinished by sewing the circuit board to the PS tabs.

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Capacitive Sensor Functionality

The capacitive sensor is shaped like a ‘‘dog bone’’ material
testing sample, and it consists of an inner deformable region
with two nondeformable regions at the ends. These nondeformable regions are made rigid through the inclusion of
fabric into the dielectric material and are included to provide
a mechanical attachment. In addition to the capacitance of
these regions of the sensor, there is also a parasitic contribution from the electrical interface and the signal conditioning electronics. In operation, all of these sources combine
into a baseline capacitance. As we show in our later discussion on calibration, the baseline capacitance can be removed,with the result being that only the change in capacitance isrelated to strain. Since the body of the sensor is a parallelplate capacitor, the capacitance of the active region is related
to the geometry by:
CActive ¼ 0 rawl¼ 0 r ,
t
t

w0 l0
ð1 þ eÞ:t0

RD
Vc (t) ¼
RC þ RD

1e

dVc
1
1
1
Vc þ
Vc ¼
VIn ,
þ
dt
RD
RC
RC

where C is the capacitance of the sensor (which is the sum ofboth parasitic and active components), Vc is the voltageRC þ RD
RC RD C

!
t

VIn :

(19)

Using a similar process, in the discharging case, the voltage
across the sensor is:

Vc (t) ¼ e





1
RD C


t

Vh ,

(20)

where Vh is the initial voltage across the sensor, which we callthe ‘‘high threshold voltage.’’ In operation, we charge thecapacitor from 0V to Vh, then discharge back to a lower
threshold voltage Vl. During both the charge and discharge
operations, we drive a status pin high, which generates thevariable pulse width signal that is fed into the filter. Theoretically, we can set Vh 2 (0V, RC RþDRD VIn ) and Vl 2 (0V, Vh ).
We will discuss the practical issues associated with thesevalues later. From Equation (19), the time required to charge
the sensor is:(18)

To measure the capacitance, we utilize a fixed-frequency
oscillator. The critical timing events of the oscillation are
shown in Figure 3a, and the charge and discharge circuit isshown in Figure 3b. The concept is to control the duty cycleof a fixed frequency square wave by using the capacitance ofthe sensor, then filter that signal through a low-pass filter torecover the average output. The governing equation for thevoltage across the capacitor while charging is:C

across the sensor, RD is the fixed discharge resistor, RC is the
fixed charge resistor, and VIn is the input voltage to the RC
network. In the time domain, the solution is:(17)

where 0 is the permittivity of free space, r is the relative
permittivity of the dielectric material, a is the planform areaof the device (as opposed to the cross-section area), t is the
thickness of the sensor, w is the width, and l is the length.
Assuming the material is incompressible and homogeneous,and by introducing a linear strain e such that l ¼ l0 (1 þ e)
where l0 is the initial length, the deformed width and thickness are w0 ¼ (1 þ e)  0:5 and t ¼ t0 (1 þ e)  0:5 . Substituting
the deformed geometry, we have the capacitance as a function of strain:
CActive ¼ 0 r

FIG. 3. (a) Voltage across capacitive sensor as a functionof time. Solid black line represents voltage history acrossthe sensor for a nominal strain state. The dashed black line
represents the voltage history for a higher strain state. Timing
events are labeled under the horizontal axis. Threshold voltages are labeled on the vertical axis. The meaning of thesevalues is described in the text. (b) Capacitive sensor charging
and discharging circuit.


RC RD C
Vh RC þ RD
th ¼ ln 
þ1 ,RC þ RD
VIn RD
where th denotes the time to charge to Vh. Likewise,from Equation (20), the time required to discharge from Vh to
Vl is:
 
Vl:
tl ¼  RD Cln
VhAdding these times together into a single expression:tTot ¼ th þ tl ¼ aC,
where:(21)

A SOFT PARALLEL KINEMATIC MECHANISM

a¼ 

43



 
RC RD
Vh RC þ RDVlln 
þ 1  RD ln,
RC þ RD
VIn RDVh

in which all the terms are constants. Substituting in Equation(18), we obtain a relationship between the cycle time andstrain:
w 0 l0ð1 þ eÞ þ CParasitic ,tTot ¼ a 0 r
t0
where CParasitic is added to account for the capacitance of theinactive (nondeformable) part of the sensor and the signal
conditioning board. The timing is ultimately simplified to:tTot ¼ b1 e þ b0 ,(22)

SMA Actuator Programmingwhere:
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discharge resistance RD. This has the effect of increasing thevoltage to which the sensor can be charged, which, in turn,has the effect of reducing noise. Second, the cycle period
T needs to be long enough to allow for complete charging
and discharging of the sensor, but it should be as short as
possible to maximize sensitivity. Third, the high thresholdvoltage Vh to which the sensor is charged should be balanced
between maximizing sensitivity and minimizing charge
time. In practice, we have found that Vh  12 VIn works formost cases. Finally, Vl should be selected while considering
the noise in the system. As the sensor discharges and thevoltage becomes lower, the sensitivity to noise increases.
Finally, the limits on the electronics used in a particular application, and in particular the rail offset voltage of any op
amps, must be considered.

w 0 l0b1 ¼ a0 r
t0

w 0 l0b0 ¼ a 0 r
þ CParasitic
t0which are both constants.
To understand the effect of the low-pass filter on the pulse
train, we consider the Fourier series expansion of the sensor
charge signal. By setting the cutoff frequency of the filter below(ideally at least two decades below) the fundamental frequencyof the sensor charging frequency, we can neglect the harmonic
terms, and are left with only the direct current (DC) contribution:
VDC ¼

1
T

Z T
VSig (s)ds,

Integration and Passive Components

0

where T is the fixed period of the charging process, and VSig isa charging status indicator. VSig is high whenever the sensor ischarging or discharging and low otherwise; the output voltage becomes:
1
VDC ¼
T

Z tTotVIn (s)ds þ

Z t

0


0(s)ds :tTot

Substituting in for the total charge cycle time tTot from
Equation (22):VDC ¼

The SMA actuators were manufactured from 0.508 mm(0.02†) diameter nickel-titanium alloy wire (McMasterCarr). The actuators were shaped into a ‘‘counter-coil’’ design, where half of the coiled actuator was oriented clockwise, whereas the other half was oriented counterclockwise.
This was done to cancel the effects of torque generated during
actuation. We secured 280 mm (11†) lengths of SMA wire in
the desired shape on a 3.18 mm (0.125†) 316 stainless steel
shaft by using split shaft collars (McMaster-Carr). We typically required two to three annealing steps at 390C for *1 hfollowed by air cooling to reach the final shape. Once in the
final shape, we programmed the coils by a process similar tothat described in Ref.68 The coils were heated to 390C for
10 min followed by a water quench to room temperature. This
heating and quenching process was repeated 10 times. With
this, the coils were ready for installation and activation.

b1 e þ b0
:
T

(24)

The conclusion is that the output voltage is a linear functionof strain. The coefficients b1 and b0 contain many terms, someof which are difficult to measure individually in practice, such
as CParasitic . However, this is easily overcome since the terms
appear as groups that are easy to measure collectively, such as
b1 and b0 . As we discuss later in our Results section, using a
regression analysis with experimental data yields a very accurate relationship between strain and output voltage.
In designing a capacitive sensor signal conditioning system for a given application, there are a few issues to consider. First, the charge resistance RC should be lower than the

The passive components of the Stewart platform included
the top and bottom plates, to which the sensors and actuators
were attached, and a central elastomer core that works as amechanical antagonist to the SMA actuators. The top andbottom plates were identical pieces of 3D-printed polylactic
acid (PLA). We used a Printrbot Metal Plus to manufacture
all 3D-printed parts. All mechanical connections were madewith M3 316 stainless steel machine screws. The central
elastomer core was manufactured from Dragon Skin 10 silicone rubber (Smooth-On, Inc.), which is the same materialas the elastomer used in the capacitive strain sensors. To castthis structure, we 3D-printed a PLA mold that comprised anouter shell and an inner core. Liquid elastomer was poured
into the space between the shell and the core, then allowed todegas for 30 min at room temperature, followed by curing at
60C for at least 1 h. The column was then removed from themold and stretch-fit over a boss on the end plates, resulting ina secure but removable connection.
The elastomer structure was designed to provide equal
stiffness in all DoF, including both translation and bending. A
solid prismatic element would have much lower stiffness inshear and bending than in twist and compression. Althoughequal stiffness in all directions is not essential to the basic
functionality of the system, for purposes of testing and demonstrating the capabilities, we elected to make this design
choice. We initially made prismatic tubes, but found that they44

tended to collapse when made thin enough to enable axial
compression. The structure used in this work was a thinwalled (5 mm) ball with a diameter of 50 mm. Reducing the
thickness to 3 mm caused the ball to buckle when SMA actuators on opposing sides of the structure were activated.
Since this work was focused on demonstrating integrationand closed-loop control, the structure presented was sufficientfor our purposes. To expand this work to create a functioning
manipulator, an analysis of the forces to be generated by thesystem would be required.

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Electronics

The overall architecture of the electronics of the system,and the ow of data between different modules, is shown in
Figure 4. There were two custom-fabricated printed circuitboards used in this project. One was used to provide signal
conditioning for the capacitive sensors, and one was used toprovide a closed-loop current supply to the SMA actuators.
Six of each board were used to control the Stewart platform. Both the sensor signal conditioning and current amplifier
modules were based on the PIC16F1825 8-bit microcontroller
and MCP6281T quad op amp (Microchip Technologies). All
analog filters were implemented by using multiple feedback
typology to enable stable operation with greater than unity gain
in noninverting operation, which was a requirement imposedby the use of a single-sided voltage supply.
In the case of the SMA current amplifier, the microcontroller hosted a bang-bang control algorithm. This electronics module contained three filters. The first and second
filter were used to filter the analog voltage input and current
through the resistive load. The third filter was a differential
amplifier that was used to measure the voltage drop across theresistive load. This last filter is not presently implemented, but
could be used in the future to control power, rather thancurrent as is presently the case. Current through the SMA wire
was controlled via an N-channel depletion-mode MOSFET.
To maximize the efficiency of the current driver, this MOSFET was run either fully open or fully closed to reduce ohmic
losses within the transistor. The control algorithm applied gate
voltage when the filtered SMA current was below the setpoint
and turned on the gate voltage when the current was above the

WHITE ET AL.

setpoint. This combination of filtering and bang-bang control produced a stable response with near-optimal performance for a purely feedback algorithm. The user interface,kinematics solution, and control algorithm were all implemented on a Linux-based PC. Communication between thecontrol software and the electronics was accomplished byusing an Arduino Uno R3 microcontroller (Adafruit) and a
USB connection. This intermediate controller was responsible for reading the output from the sensor signal conditioning boards by using a 16-bit ADC (ADS1115; Adafruit)and generating eight-bit pulse width modulated (PWM) command signals for the current control boards. The electronics
configuration was designed to be stationary and so relies on arelatively bulky PC, individual current drivers, and signal
conditioning boards. In the future, an application-specific
PCB could be designed and integrated into the structure of therobot to make the overall system more compact.
Testing

To determine the coefficients in Equation (24), we needed
to collect output voltage as a function of strain. To do so, weused an Instron 3345 fitted with a 50 N load cell to measure
the response of the sensors. As described earlier, the active
elastomer sensing element was integrated with rigid tabs toprovide an attachment point. The sensors were held by theattachment points in an identical way as they were held when
attached to the robot to ensure consistency of results between
the Instron testing and on-robot operation. Tests were conducted over operationally representative strain values. Sensors were put through 10 strain cycles to ensure consistent
operation.
SMAs have complex thermomechanical responses. We
performed two sets of tests to characterize the as-fabricated
SMA actuators. The first test was designed to quantify theeffect of performance degradation with different appliedcurrent and set an upper safe limit on current. Immediately
after programming the SMA wire into coils, we used an Instron 3345 materials testing machine fitted with a 50 N loadcell to measure the response of the actuators. We applied a
30 mm displacement to simulate pulling the coil to the neutral
length when integrated into the robot (i.e., the length shown

FIG. 4. Architecture of the complete soft robotic system. Commercially available components are shown in blue. Customer
fabricated circuit boards are shown in green. Fabricated hardware is shown in red. The labels in the arrows describe the type offlow between boxes. Beginning clockwise from left, six PWM voltage signals are passed from the interface Arduino to the SMA
drivers. This voltage input signal is converted into a current (i) using a 1A=V transfer function. The current causes the SMA wire
to heat, resulting in a force (F) that is applied to the structure. This force results in a deformation and strain () that is measured by
the strain sensors, which exhibit a capacitance (C). This capacitance is measured with the signal conditioning electronics, which
produce a voltage (V) that is read by ADC. Data from the ADCs are read by the Arduino using the I2C protocol. ADC, analog-todigital converters; PWM, pulse width modulated. Color images available online at www.liebertpub.com/soroDownloaded by YALE UNIVERSITY from www.liebertpub.com at 01/16/19. For personal use only.

A SOFT PARALLEL KINEMATIC MECHANISM

in Fig. 1). Second, we applied a series of 100 current pulses tothe coil, and measured the force generated. The pulses consistof a 20 s on period, followed by a 120 s off period. The resultsof this test are shown in Figure 6. We conducted these tests at
two different current levels, 0.98 and 1.57 A, which correspond to PWM values of 50 and 80 bits, respectively. The
second test we conducted on the SMA actuators was to determine the quasi-static response as a function of displacement
and applied current. We were interested in displacements from10 to 50 mm, and applied current values between 10 and 50
bits, which equates to 0.20–0.98 A. To perform this test, we
fixed the SMA actuator in the materials testing machine as in
the previous test. We created a test grid with all possiblecombinations and randomly selected trials from this list to
mitigate the effects of drift or degradation during the test. For
each test condition, we pulled the SMA actuator to the required
displacement, and applied a 50-bit pulse of current to activate
the coil and remove any slack in the actuator. The data fromthis initial pulse were not considered in the analysis. Subsequently, we applied three pulses at the required current, using a
20 s on time followed by a 120 s off time. The results of thistest are shown in Figure 7.
Once unit testing was complete on the sensors and actuators, we integrated six of each into the complete robotic

45

system. We conducted two tests on the complete robot: an
open-loop step response test and a closed-loop response test.
In the case of the open-loop test, we applied the same current
levels of 10, 20, 30, 40, and 50 bits to each of the six actuators, one actuator at a time, in a random order, and observed
the response in the six sensors. As in the component actuator
tests, the actuators were turned on for 20 s, and allowed tocool for 120 s. However, unlike the previous actuator tests,
there was no ‘‘pre-heating’’ step to remove slack. The actuators were kept taut by the central elastomer structure. Eachtest condition was evaluated three times. The results of thisexperiment are shown in Figure 8.
The second trial conducted with the integrated system wasa closed-loop test. In this test, a simple per-channel bangbang controller was implemented with a maximum commandof 50 bits. As previously discussed, the controller ran on a PC
connected to the robot through an Arduino interface. Eachposition was held for 60 s. The sequence of commanded
states (in absolute coordinates, not relative to the previous
state) was neutral, down 3 mm, down 3 mm + roll right 5.62,down 3 mm + roll right 5.62 + pitch forward 5.62, down
3 mm + roll right 5.62 + pitch backward 5.62, down 3 mm,
and neutral (for 120 s). This sequence was repeated threetimes. The results are shown in Figure 9.

FIG. 5. Sensor calibration data. The top
two charts show measured voltage and force
responses, respectively, as functions of displacement. The bottom chart shows thereconstructed displacement based on a quadratic model. Zero displacement corresponds
to the robot in a neutral configuration. Experimental data are shown as dark gray dots,
the mean is shown as a black line, and the
95% confidence intervals are shown as grayregions. Data show responses from six sensors over 10 loading/unloading cycles.

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46

WHITE ET AL.

FIG. 6. Degradation testing on SMA actuators. (a, b) shows the mechanical responseof the first 10 pulses of current at 50 and 80
bits. (c, d) shows 100 pulses, including thefirst 10. (e, f) shows the maximum force
generated as a function of pulse number. Top
two rows show data from one actuator coileach, whereas the bottom row shows summary data from three sensors. Color images
available online at www.liebertpub.com/soro

Results and Discussion
Sensor performance

One of our goals with this project was to create sensors thatwere mechanically compatible with the SMA actuators. We
have demonstrated that capacitance-based conductive elastomer composite sensors are highly deformable, and therefore have minimal impact on the mechanical characteristics
of soft structures to which they are attached. As shown in theforce curve of Figure 5, the maximum force observed over therange of operation was less than 1.5 N, which is lower thanthe maximum force output of the SMA actuators shown in
Figure 7 of 4 N. To further decrease the mechanical impact,
the sensors could be reduced in width and thickness. This
would be easier to implement in integrated sensors than in
free-standing sensors such as those we used, since the body towhich they are attached would be able to provide mechanicalsupport.
We have also demonstrated that a simple film-based approach can be used to easily produce capacitive sensors thatcan measure strains of the same scale as the SMA actuators.
The voltage response of the sensors shown in Figure 5 illustrates the stability of the sensors. The data represent theoutput from three different sensors, demonstrating that thereis good consistency across devices. In addition, we demonstrated that the response of the sensors is stable across many
cycles, as shown by the small confidence intervals in boththe output voltage and reconstructed displacement curves of
Figure 5. Moreover, the voltage output is actually more stable
and repeatable than the mechanical response. Based on theobserved voltage response, we assumed a quadratic relation
between the displacement and voltage. Although Equation(24) shows that the ideal response is linear, we have added aquadratic term to account for non-ideal effects, particularly atlarge deformation values. We suspect these deviations from a
linear response are mostly due to geometric effects within thesensors. To eliminate the effects of buckling, we only considered data in the range d 2 [  10, 20] mm. Using a generalized least-squares approach, we found the response of thesensors to be:
d ¼ 0:152  76:9DV þ 59:4DV 2  1:16 mm,(25)

where d is the displacement, measured from the neutral position in mm, DV is the change in measured voltage, measured in V, and the error represents the 95% confidence
interval.
SMA Performance

The limited lifespan of SMA actuators is well establishedin the literature.45 In addition, SMA actuators are known tohave a short transient in performance during the first few
cycles. We confirmed both these effects experimentally in theDownloaded by YALE UNIVERSITY from www.liebertpub.com at 01/16/19. For personal use only.

A SOFT PARALLEL KINEMATIC MECHANISM

47

FIG. 7. SMA actuator force as a function of
displacement and current. (a) Shows the time
history response of a single actuator over three
trials in each condition. The top row of images
show displacements of 10, 20, 30 (neutral), 40,
and 50 mm. Within each figure, data for current commands of 10, 20, 30, 40, and 50 bits
are shown as different shades of red. In thesecond row of images, each plot shows a single
current, whereas the different displacements
are plotted as different shades of blue. (b, c)Shows the maximum and minimum force exerted by an actuator coil in each condition. The
points show the observations from three trials
with each of three coils, whereas the surface
represents average of the experimental observations at each condition. (d, e) shows thesame experimental points as (b, c), but withthe surface replaced by the linear approximation. Color images available online at www.liebertpub.com/soro

data shown in Figure 6a and b. Moreover, these data showthat multiple SMA actuators behave very similarly in Figure 6e
and f. To determine the maximum permissible current, we
compared the results in Figure 6e and f. In Figure 6e, we saw apronounced degradation in performance over the first 40 cycles. However, after this initial transient, the force was approximately constant for the remaining cycles. In contrast, inFigure 6f, the force continues to decay over the entire 100
cycle test. Further, the force generated by the higher current
test actually produces less force than the lower current test at
high numbers of cycles. Therefore, we concluded that a currentof 0.98 A (Fig. 6e) was safe for continuous operation. Further
supporting this conclusion, we noticed that the response of
the first current pulse at the higher 1.57 A setting resulted indegradation during the heating phase as shown in Figure 6b.
Instead of increasing in force during the entire heating phase,the trace shows that maximum force was achieved around 10 s,
and then began to decay for the remaining 10 s of the pulse.
This effect was not observed in any other pulse.
Overheating is a significant challenge in SMA actuator
design. We want to achieve rapid activation of the coil, which

requires rapid heating of the coil, which, in turn, requires highcurrents. However, high currents also result in steady-state
temperatures that are well above the threshold for damaging
the actuator coil. Since we cannot directly measure thetemperature of the coil in this configuration, we are limited
in what information we can provide to the control system.
Further, since our response time is dominated by the freeconvective cooling of the actuators, we conclude that attempting to run the coils with higher currents substantially
increases the risk of damage with little improvement to theoverall response of the system.
As has been previously reported in the literature,69 we findthat the quasi-static force output from the SMA actuators iswell modeled by a linear spring model. Figure 7a shows thatthe force output from the actuator is a function of displacement (top row) and current (bottom row). Further, the figure
shows that the effects are approximately linear. Finally, thefigure also shows that the slow cooling rate dominates the
dynamics of the system. To improve the dynamics responseof the system, some way must be found to enhance thecooling rate. We speculate that this might be accomplishedDownloaded by YALE UNIVERSITY from www.liebertpub.com at 01/16/19. For personal use only.

48

WHITE ET AL.

FIG. 8. Step response from on-board sensors. Each column represents a different actuator. Each row represents a different
sensor. Each image contains data from three independent trials. All figures have identical vertical and horizontal axes.
Colors represent different current levels, with black being 10 bits &0.20 A and red being 50 bits &0.98 A. All datacollected with 11-tap rolling FIR filter. Red vertical bars at 30s and 50s show the times at which the SMA actuators wereturned on and off, respectively. The sensor and actuator names are consistent with the labels in Figure 2. FIR, finite impulse
response. Color images available online at www.liebertpub.com/soro
by placing the SMA actuators in a thermally conductive environment or by using forced convection.
Returning to Figure 7b, we found that the maximum force
generated by the SMA actuator coils was an approximately
linear function of the product of displacement and current. In
Figure 7c, we found that the minimum force (i.e., residual
force in the actuator after the current was turned off) was afunction of the displacement, but not the on current. Together, these observations suggested a model of the form:
F(i, d, d0 ) ¼ (Ki i þ K0 )(d  d0 ),(26)

where F(i, d, d0 ) is force applied by the actuator as a function of the applied current i, displacement d, and initial length
d0 and ki and k0 are parameters of a linear spring whose
stiffness is a function of current. Based on both the maximum and minimum observed force, we determined thatthese coefficients were Ki ¼ 1:86 · 10  3 N=mm:bit, K0 ¼
5:06 · 10  3 N=mm, and d0 ¼ 2:74mm. Using this model, we
generated estimated maximum and minimum forces in
Figure 7d and e, respectively.
The good agreement between the model and the experimental data validates that this approach is sufficient for
capturing the quasi-static performance of an SMA actuator
without considering the full thermomechanical behavior.
Further, this is similar to the approach taken by Holschuh
et al., who investigated SMA actuators in a compression
garment.69 Given the similar results from two different applications of SMA actuators, we conclude that the reduction

to a quasi-static model has utility across a range of applications. At the very least, a quasi-static approach provides a starting point for the design process. Althoughquasi-static models cannot account for the hysteresis effects in SMAs, they do provide upper bounds on the force
that can be generated.
Robot Performance

The open-loop test of robotic system performance is shownin Figure 8. The numbering scheme used in the figure isshown in Figure 2. The sensor data in the vertical axes indicate the internal state vector l from Equation (3). Sensors
are arranged into rows, and actuators are arranged into columns. For example, the data in the third row and second
column (labeled ‘‘S2,A1’’) show the response of sensor 2(‘‘S2’’) to the activation of the actuator 1 (‘‘A1’’). The sensors and actuators are numbered counterclockwise whenlooking down at the top of the robotic structure. The data
show two effects. First, we see coupling between axes of therobotic system. The diagonal subplots of Figure 8 show eachsensor’s response to the actuator attached between the samenodes. As expected, the subplots along the diagonal show the
maximum deflection. Moving from the diagonal three columns to the left or right, we see the data from the sensor on
the opposite side of the structure to the actuator. In these
subplots, we see a response of opposite direction to the ondiagonal plots with lower amplitude. Comparing Figure 8
with Figure 6a and c and Figure 7a, we find that the time

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A SOFT PARALLEL KINEMATIC MECHANISM

49

FIG. 9. Closed-loop tests of the integrated robot system. In all subplots, the red line represents the setpoint, the thin blueline represents the mean of the observation, and the light blue region represents the 95% confidence interval across three
trials. The upper row of subplots shows the length states in the vector l. The middle row shows the configuration states of thevector c. The bottom row shows the errors in the configuration states. The first three subplots in the middle and bottom rows
are translations in x, y, and z, whereas the second group of three subplots are the rotation components. Color images
available online at www.liebertpub.com/soro
constants are approximately equal in all situations tested.
Since the heating and cooling rates are dictated by the current
and free convection, which are constant between tests, thisis as expected and indicates that the viscoelasticity of thecentral elastomer structure plays a negligible role in the dynamics of the system.
The closed-loop test of robotic system performance isshown in Figure 9. The data for the internal state vector l are
shown in the top row, data for the configuration state vector c
are shown in the middle row, and error data are shown in thebottom row. Commands were received from the operator in
configuration space, then converted into internal space as an
input to the control loop. The data reported in Figure 9 represent the mean and 95% confidence from three independent
trials. The relatively small confidence interval indicatesthat the robotic system performance is highly repeatable. The
relatively large excursions from ideal performance are,therefore, due to system dynamics and material limitations,which are repeatable, rather than from random noise.
The workspace of the robot was limited by the nature of the
SMA actuators, which could only apply tension. The effect of
this limitation is clearly seen in the first and second internal
state vector data between 240 and 300 s (‘‘S0’’ and ‘‘S1’’ inthe top row of Fig. 9), as well as in the fifth and six internal
state vector data between 180 and 240 s (‘‘S4’’ and ‘‘S5’’ inthe top row of Fig. 9). At these points in the command history,
the required lengths of the actuators exceeded the neutral
position length (i.e., >67 mm). For the actuators to havecontributed to reducing the error in the system, they would
have had to provide extensive force, which is not possible

with coiled SMA wire actuators. This situation is illustrated
in Figure 10. In this example, the left-hand actuator is fully
off and completely cooled. To achieve the commanded position, the actuator would have to extend to ll, c . However,
due to the contractive force of the antagonistic actuator on
the right side of the figure working in conjunction with theelastomer body, it is only able to extend to ll, a . The extensionof the left-hand actuators from their neutral length of ll, n toll, a is due to the antagonistic effect of the right-hand actuators
working in conjunction with elastomer structure. This same
effect is observed in both the open- and closed-loop tests.
Based on this limitation, we conclude that successful operation of the system is limited to cases where shorter-thanneutral lengths of the actuators are required. To achieve this,
one needs to pre-compress the system by using uniform activation of all the actuators. This requirement was anotherfactor that influenced our desire to reduce the stiffness of thecentral elastomer structure, in addition to the goal of achieving
equal stiffness in all axes.
Another limitation of the workspace is imposed due to thegeometry of the system. Looking at the open-loop tests in
Figure 8, it appears that the actuators are able to cause acontraction of >10 mm. However, looking at the closed-loop
state data, and in particular in the internal state vector data forsensors S2 and S3 (‘‘S2’’ and ‘‘S3’’ in the top row of Fig. 9),
the actuators are only able to achieve *5 mm of compression. This is because A2 and A3 are activated simultaneously
in the closed-loop tests, as opposed to one at a time in the
open-loop test. As a result, in the open-loop test, the actuatoris able to pull the two nodes to which it is attached directlyDownloaded by YALE UNIVERSITY from www.liebertpub.com at 01/16/19. For personal use only.

50

FIG. 10. Schematic illustration of the platform commanded to rotate beyond kinematic capability. The neutral
position (i.e., the position of the robot with all actuators off),the commanded position, and the achieved position are allshown. The rigid end plates are blue, the central elastomer
structure is red, and the actuators are shown schematicallyas orange lines. Sensors are not shown. In this example, theactuator on the right side is able to achieve the commanded length (i.e., lr, a ¼ lr, c ), whereas the actuator on theleft side is not able to achieve the commanded length (i.e.,ll, a < ll, c ). The left side actuator extended past the neutral
length (ll, a > ll, c ) because of the center elastomer structure
and antagonist force from the other actuator. Color images
available online at www.liebertpub.com/soro
toward one another. This results in maximum mechanicalefficiency and, hence, maximum contraction. However, in theclosed-loop case, when both A2 and A3 are working simultaneously, the corresponding nodes on the upper plate have arelative sideways motion since Nodes 5 and 7 are being
pulled down into the gap between Nodes 4 and 6 (see Fig. 2
for node numbers). At high contraction, this results in a nearsingular configuration, where the mechanical efficiency ofA2 and A3 is much less than unity.
These two configurations are illustrated schematically in
Figure 11. As a result, the contraction observed in the closedloop test is lower than that observed in the open-loop test. The
particular command sequence shown in Figure 9 was selected
to test this exact phenomenon and demonstrated the limitation of motion in this pathological case.
Looking at the Z translation data (Fig. 9 middle and bottomrows, third column), we can observe the effect of mechanicalcoupling between actuators. During the period between 120
and 300 s, we observe that the moving platform drops below
the setpoint value every 60 s, when a new angular configuration is commanded. This occurs because the cooling rate of
the SMA actuators is slower than the heating rate. Because ofthis, actuators on both sides of the structure were applying acontractive force: The actuators from the previous cycle werestill ‘‘on’’ and cooling to below their activation temperature,
whereas the actuators for the current cycle were also commanded ‘‘on.’’ As the deactivated SMA actuators cooled, theerror in Z translation decreased until the system achieved
zero error, which was typically in about 30 s. This effect was,

WHITE ET AL.

FIG. 11. Schematic illustration of the differences in mechanical advantage between open- and closed-loop tests. The
rigid end plates are blue, the central elastomer structure is red,and the actuators are shown schematically as orange lines.
Only the two actuators closest to the viewer are shown. In the
open-loop tests (upper image), only one actuator was activated at a time. With only one actuator active, the moving
plate rotates toward the active actuator attachment point,
maintaining mechanical advantage. In the closed-loop test
configuration shown, both actuators are on. Because bothactuators are pulling, the moving plate rotates down between
the actuator attachment points, reducing mechanical advantage relative to the open-loop test. Color images available
online at www.liebertpub.com/soro
in part, an artifact of our control strategy. Instead of controlling the translation and rotation states, we controlled thesensor lengths. As such, the control algorithm was unaware of
the deviations in Z translation, and no effort was made tocorrect the error.
We utilized a simple bang-bang controller architecturewith a single control loop attached to each internal state.
Outside of the workspace limitations discussed earlier, oncethe system achieves a zero-error condition, the controller isable to maintain the system at that location as shown by thestable regions in the bottom row of Figure 9. Controllers of
this type are subject to ‘‘chatter,’’ particularly in cases ofnoisy observations, which we observed in the small oscillations in the error traces. We believe that more advanced
controllers, particularly a control architecture employing
fuzzy logic, could be implemented in the future to enhance theperformance of the robotic system. Further, by implementing
a controller with lower noise sensitivity, we would be able toincrease the maximum current in the system without increasing the risk of overheating, which would improve performance. In addition, our current approach relies exclusively
on feedback. Given the long time delays associated with SMA

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A SOFT PARALLEL KINEMATIC MECHANISM

actuators, we suggest that model-based feedforward controllers could provide better performance, particularly if themechanical coupling between SMA actuators was consideredin the model.
The maximum current used in this system was limited by
the danger of overheating the SMA actuators. We chose amaximum current based on steady-state heating. However,pulse heating can also be used with SMA actuators where an
initial pulse is provided to rapidly heat the coils. Withoutstate feedback from the SMA wire, this approach can overheat the coils and cause degradation. More than just temperature feedback is required. In addition, the current loading
condition, and an estimate of the internal composition of the
SMA wire (i.e., percentage of martensite) are needed to ensure that the thermodynamic conditions remain below thedegradation point. This is very difficult to achieve in practice.
Further, this only solves the heating problem. As noted earlier, cooling rate is just as important to overall system performance. Since this is currently a passive process, we arelimited in our options. We suggest that improved heat transfer
rates, through either forced convection or enhanced conduction into a higher heat capacity medium, such as water,
could address both challenges simultaneously. However,adding a fluidic temperature control system would present a
new set of integration and fabrication challenges.
Conclusion

In this article, we have described the manufacturing, integration, and testing of a soft robotic structure. Like many
soft robots, the functionality of our system is based on the
responsive materials from which the robot is fabricated. We
relied on nickel-titanium SMA wire as an actuator material,
and we determined the maximum current that could be used
for actuation without damaging the actuators. We also collected data to support a quasi-static force and displacement
model of the coiled actuators, which we used for design
purposes. Finally, we determined the dynamic response of
the isolated SMA actuators and compared this with the dynamic response of the integrated robot. We concluded thatthe dynamics of the system was driven by the actuators, andnot by the material properties of the sensors or central elastomer structure. To provide state feedback, we also developed
a conductive elastomer composite material and created highly
deformable strain gauges from deformable capacitive structures. We developed an analytic model of these sensors, including the operation of the readout electronics, and compared
the results with experimental data. From our experimental data,
we characterized the repeatability of the sensors and the accuracy. Finally, we created an experimentally anchored calibration curve that we used in the integrated robotic system.
We integrated six sensors and actuators together to provide
full six DoF state control in an elastomer-based robotic system. We defined a series of internal states corresponding to thelengths of each sensor, then actively controlled these lengths
to achieve position control. Since these internal states are notintuitive and are difficult to use to express the kinematics ofthe system, we also described the mapping and its inverse
between the internal states and a more intuitive translationrotation state formulation. All of the commands we gave tothe robotic system were in terms of the intuitive states, whichwere then converted into the internal states for input to the51

control loops. We were able to demonstrate highly repeatable
controlled motion of the robotic system.
In the future, we intend to continue to refine the materiallevel models of the responsive materials, particularly of theactuator. We believe that the future of soft robot development
requires an enhanced understanding of the interactions between responsive materials elements, which, in turn, requires
the development of integrated models that can be used fordesign and analysis purposes. In this article, we assembled
our soft system from discrete responsive material components. Our long-term goal is to create complete soft robotic
systems from heterogeneous responsive and traditional materials through advanced manufacturing processes, such as
3D printing and molding, without any assembly steps. To
achieve this level of integration, we will need to progress
through our current assembled prototype phase into a fieldwith better models and design tools.
Acknowledgments

This work is partially supported under the NASA Early
Career Faculty Program, Space Technology Research Grants
Program (Grant NNX14AO52G). E.L.W. is supported by the
National Science Foundation Graduate Research Fellowship(Grant DGE-1333468). J.C.C. is supported by the National
Aeronautics and Space Administration Space Technology
Research Fellowship (NNX15AQ75H). Any opinion, findings, and conclusions or recommendations expressed in thismaterial are those of the authors and do not necessarily reflect
the views of the National Science Foundation or the National
Aeronautics and Space Administration.
Author Disclosure Statement

No competing financial interests exist.
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Address correspondence to:Rebecca Kramer-Bottiglio
School of Engineering and Applied Science
Yale University
10 Hillhouse Avenue
New Haven, CT 06250
E-mail: rebecca.kramer@yale.edu