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Sensor enabled closed-loop bending control of soft beams

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Smart Materials and Structures
Smart Mater. Struct. 25 (2016) 045018 (10pp)

doi:10.1088/0964-1726/25/4/045018

Sensor enabled closed-loop bending controlof soft beams
Jennifer C Case, Edward L White and Rebecca K Kramer
Department of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
E-mail: rebeccakramer@purdue.edu
Received 20 October 2015, revised 8 February 2016
Accepted for publication 9 February 2016
Published 14 March 2016
Abstract

Control of soft-bodied systems is challenging, as the absence of rigidity typically implies
distributed deformations and inﬁnite degrees-of-freedom. In this paper, we demonstrate closedloop control of three elastomer beams that vary in bending stiffness. The most stiff beam iscomprised of a single prismatic structure made from a single elastomer. In the next beam,increased ﬂexibility is introduced via an indentation in the elastomer, forming a joint. The most
ﬂexible beam uses a softer elastomer in the joint section, along with an indentation. An
antagonistic pair of actuators bend the joint while a pair of liquid–metal-embedded strain sensors
provide angle feedback to a control loop. We were able to achieve control of the system with aproportional–integral–derivative control algorithm. The procedure we demonstrate in this work
is not dependent on actuator and sensor choice and could be applied to to other hardware
systems, as well as more complex multi-joint robotic structures in the future.
S Online supplementary data available from stacks.iop.org/sms/25/045018/mmedia
Keywords: closed-loop control, soft structures, responsive materials(Some ﬁgures may appear in colour only in the online journal)
1. Introduction

This complicates the dynamics of soft systems and makes
them challenging to control.
We can look towards vertebrates for a potential solution
to this problem. Generally, the human body is considered asoft system compared to rigid robots. However, complex
joints which can be approximated as rotary or hinge joints still
exist inside of this ‘soft’ system. This is because humans andother vertebrates have stiff skeletal structures. The skeletons
give a supporting structure to vertebrates while still being
composed of relatively soft materials (compared to materials
used in traditional robotics) that still experience viscoelasticity [1]. While adding rigid or stiff components to soft systems can seem limiting, it has allowed vertebrates to operate
on a larger scale than purely soft animals, such as caterpillars,
worms, and jellyﬁsh. Skeletal structures have also developed
to allow vertebrates to accomplish speciﬁc tasks. For example, rats have a hinged rib cage that will compress allowing
them to squeeze through small holes [2]. Animals withendoskeletons use harder components (bones) held in place
by a softer material (cartilage) and moved by contractile
actuators (muscles) to operate and move various joints.

Robots manufactured from elastomers possess unique functionalities due to their highly deformable structures. This
deformability is the source of both the most compelling
advantages and most daunting challenges of these systems.
The design process for a soft robot involves developing
materials, geometries, and control algorithms simultaneously.
That interaction is the fundamental contribution of this present work. We describe the interplay between the mechanicaldesign of a soft structure, the application of soft andresponsive material actuators and sensors to that structure,and the integration of control algorithms to the resulting soft
assembly. This work is a prototype of the design process thatwould accompany a full soft robot.
Traditional, rigid robotic structures have two general
types of joints: rotational and prismatic. The motions of thesejoints are clearly deﬁned and make ﬁnding the forward andinverse kinematics relatively simple. Soft robots, on the other
hand, are generally continuously deformable systems andhave the potential to deform in any direction and at any point.
0964-1726/16/045018+10$33.00

1© 2016 IOP Publishing Ltd Printed in the UKSmart Mater. Struct. 25 (2016) 045018

J C Case et al

could be combined with a representative soft body to make amobile soft robot, as shown conceptually in ﬁgure 1(a), or asoft gripper; however, in this work, we conﬁne our study tosingle joint systems.
In the present work, the three structures were fabricatedfrom Smooth-Sil 935 (Shore hardness: 35A), a relatively stiff
silicone elastomer, with the exception of one structure that
contains a soft joint fabricated from Dragon Skin 10 Slow
(Shore hardness: 10A). This design improves the controllability of the structure while successfully localizing thedeformation and maintaining the beneﬁts from using softpolymer materials.
To build a complete closed-loop control system, we
integrated elastomer strain gauges to measure joint angle andshape memory alloy (SMA) coils to provide actuation. Using
this approach, we are able to directly measure the angle of thejoint in the structure and use this information as an input toour control system. A proportional–integral–derivative (PID)
control algorithm was used to control the angle of the soft
robotic joint. The strain gauges were only sensitive todeformations at the soft joint, although the SMA was connected across a much larger portion of the body (see ﬁgure 1
for an overview of the system). Conﬁrmation of the accuracyof the measured angle was obtained by visually observing therobotic joint. We note here that the primary contribution ofthis work is design for control of soft joints and that our
results are independent from speciﬁc actuator and sensor
choice.

2. Previous work
There is growing interest in control of soft systems with manydifferent approaches [3, 4]. Researchers have experimented
with several control strategies to date, including open-loop
control [5–8], model-based control [9–11], model-free control[12], and closed-loop control enabled by the inclusion of soft
sensors to the robotic system [13, 14]. In this work, we takethe latter approach of employing sensors that provide state
feedback and enable correction. Given that elastic materials
have nonlinear responses at high strains and exhibit viscoelastic behaviors [15, 16], we believe this sensor-enabledapproach can be used in conjunction with modeling control
approaches to allow for simpliﬁed models and to correct formodeling errors.
A further measure to simplify the control problem for soft
robots is to localize deformations. The objective of localizing
deformation has been achieved by various groups using different means, including ﬁlm-like shape memory hinges
between rigid plates [17, 18], particle jamming [19], hinged
shape memory polymer ﬁlms [20, 21], localized heating ofhydrogels [22], gradiated elastomers [23], and localized
melting of metal alloys [24]. None of these methods are
particularly well-suited to the application we foresee offreestanding elastomer robots, either due to complexity in
manufacturing or actuation. Therefore, we elected to passively localize deformation by modifying the structure. These

Figure 1. This ﬁgure shows (a) a conceptual soft robot based on the

deformation localization presented in this work; (b) the threeelastomer beams with attached sensors and actuators used in thiswork; and (c) an image showing a beam in neutral and activatedconﬁgurations.

Borrowing these concepts, we can utilize geometry andmaterial properties of structures to localize deformation on asoft robotic structure, and therefore make it easier to track the
movement of soft structures.
Our goal in this work is to demonstrate three soft structures undergoing localized deformations. Deformation localization serves two purposes. First, it simpliﬁes the kinematic
model of the soft structure. Second, it simpliﬁes the proprioceptive feedback problem, since state only needs to besensed at the soft ‘joints,’ rather than continuously across theentire structure. We focus on the design fabricated from two
dissimilar materials that exhibits motion conﬁned primarily toa soft ‘joint’ between two harder ‘bones.’ Multiple joints
2

Smart Mater. Struct. 25 (2016) 045018

J C Case et alvascular stents, morphing structures, and wearable devices.
All of these applications could potentially beneﬁt from theintegration of position sensing to achieve ﬁner control during
actuation. However, we note here that SMA has several disadvantages as a responsive actuator such as slow deactivation
times, thermal sensitivity, and poor energy efﬁciency, andother soft and responsive actuators such as ﬂuidic actuators,
cable driven systems, electroactive polymers, and shapememory polymers could replace the SMA actuators depending on the target application of the system [4].
In this work, we ampliﬁed the natural deformation of
SMA by introducing a coiled geometry. Nickel–titanium
alloys are capable of sustaining strains of 1% ∼ 2% overmany cycles. The use of a coiled shape allows us to achieve
the required ∼60% deformation required overall, while
keeping the local deformation in the material small [47]. Thisrequirement was dictated by our target range of motion for thesoft system of 60 and the location of the attachment pointsof the SMA actuators. The trade-off is that force output fromthe actuator is reduced proportionally. Since the elastomer
structure used in this work was very deformable, we wereable to easily construct actuators capable of providing therequired force. The response of the actuator is also a functionof the imposed conditions during manufacturing. In this work,
we used the results of Seok et al to inform our manufacturing
approach [48].
As noted previously, SMAs exhibit a complex thermomechanical response. They have a nonlinear response totemperature and large hysteresis effects. Despite this, progresshas been made in controlling these materials [49–54]. In thepresent work, we took the approach of Ikuta et al who used
tuned PID controllers to overcome the complexities associated with the thermomechanical response. Given the relatively slow actuation and motions required for our work, aproperly tuned PID controller resulted in satisfactory control.
In summary, the individual elements used in this work
were either inspired by or directly borrowed from previous
work. Our focus has been on integrating these elements
together in such a way that we can demonstrate closed-loop
control of a structure comprised entirely of soft materials. Our
long-term goal is to develop the results of this work into more
complex soft robotic systems with multiple joints and degrees
of freedom that are capable of operating in unstructured
environments.

Figure 2. Different elastomer structure designs. From top to bottom:
a non-homogenous beam, a non-prismatic beam, and a prismatic
beam. The bones of the non-prismatic and non-homogeneous beams
are 80 mm×10 mm×15 mm with an 8 mm joint connecting thebones together. The prismatic beam has the same overall length as
the other two beams. All dimensions are in mm.

modiﬁcations, both geometric and material, are easy to apply
to any elastomer-based robotic system.
Another design goal is to minimize the mechanicalimpact of the sensors on the response of the system. Therefore, we wish to minimize the stiffness of the sensors relative
to the host structure, which undergoes strains on the order of
100%. In general, soft sensing is a diverse ﬁeld with many
demonstrated approaches of fabrication [25–29]. We take theapproach of liquid–metal-embedded elastomer strain gauges
because they have very low stiffness and minimal impact on
the mechanical properties of the system. Additionally, thistype of strain sensor exhibits low noise and consistent
response to strain [30]. Room temperature liquid metal alloys
made from a combination of gallium and indium are an ideal
material to use in this application. Filling microchannels withgallium–indium alloy was ﬁrst reported by Chiechi, Dickey,
and colleagues [31, 32]. Since then, the concept of ﬁlling
microchannels with liquid metal has been applied to createpressure, force, and touch sensors [33–35], curvature andjoint angle sensors [27, 36], and combined strain and pressure
sensors [37]. Our strain sensors were very similar to those
produced previously, except that we used a different elastomer substrate. Previous groups typically used EcoFlex 00-30(Smooth-On) as a very soft elastomer. We have found that
this material is incompatible with our laser-based approach tofabricating sensors, which we describe below. Instead, weused Dragon Skin 10 Slow (Smooth-On), which was slightly
harder, but more compatible with our fabrication process.
We selected SMA as the actuator because it required very
little interface hardware compared to pneumatics, which
require pumps, valves, pressure sensors, and the like. In SMA
systems, the thermally responsive actuators are activated by
Joule heating. This results in a very simple interface. Nickeltitanium SMA such as those used in this work were ﬁrst
described by Jackson, et al [38]. The thermomechanicalresponse behind the shape memory effect is both complex andwell studied [39–46]. This class of materials has been used ina wide range of applications, including robotics, endoscopes,

3. Design
3.1. Elastomer beams

The objectives of this system are to localize bending in a
speciﬁed location along an elastomer beam and to determine
if that localized bending facilitates control of the deformationof the elastomer beam. We refer to this speciﬁed location as
the ‘soft joint’, which will connect ‘soft bones’.
To localize bending, we tested two methods: (1) reducing
the width of the joint, and (2) replacing the joint with a softer
elastomer. To test the effectiveness of these techniques, we
3Smart Mater. Struct. 25 (2016) 045018

J C Case et al

looked at: (1) a homogeneous elastomer beam with constantwidth (prismatic beam), (2) a homogeneous elastomer beamwith reduced joint width (non-prismatic beam), and (3) anelastomer beam with a softer elastomer at the joint with
reduced width (non-homogeneous beam). These conﬁgurations are shown in ﬁgure 2.

3.2. SMA coils

The transition temperature of the nickel titanium alloy used inthis work was approximately 70 °C at no load. However, thistransition temperature is a function of the stress of the system.
Higher stress causes the transition temperature to increase.
Because of this, creating a purely temperature-based control
algorithm would not be able to directly control either stress ordeformation in the system. Some mechanical feedback isrequired. Additionally, because we used an antagonistic
conﬁguration of SMA, there was an interaction between thetwo coils that is not present in more traditional actuator systems. As one coil activated, it pulled on and deformed the
opposite coil. The force required to complete this was afunction of the temperature of the other coil, which itself waschanging over time due to free convection. Our approach was
to neglect these complexities and treat the system as a
black box.
We tested several different combinations of SMA wire
diameter and coil diameter before selecting the coil we usedin this work. Our goal was to balance the force produced byan active coil with the force required to restore an inactive
coil. As expected, tighter coils with larger wire resulted inlarger blocking force when active, but also required larger
force to return to their neutral length. In the case of a 0.5 mm
wire coiled into a 1.6 mm diameter coil, the resulting active
force was approximately 7 N with a current of 1.5 A. These
forces were sufﬁcient to buckle the soft structure, and wasconsiderably higher than required to meet our objectives. We
will point out here that SMA coils could be designed forlarger applications by increasing wire diameter and decreasing coil length, although the latter becomes difﬁcult for larger
wires.

Figure 3. An assembled robotic joint with a non-homogenous beam.
The mounting brackets are placed over the beam and locked in placewith pins. The elastomer struts are visible on either side of the joint.
The pins also lock in the sensors on either side of the beam (seen in
the lower left inset) and the SMA is crimped through the holes on themounting brackets (seen in the upper right inset).

the SMA coil length and allowing multiple soft structures tobe joined together in a chain.
During initial testing, we noticed instabilities in thesystem due to both the sensors and actuators. The sensors,
being pre-strained, would cause the system to snap from one
side to the other, which prevented system control over theentire range of motion. To ﬁx this snap-through instability, weadded elastomer struts (seen in ﬁgure 3) to either side of thejoint to discourage this snapping behavior by applying a
restoring force on the opposing sensor. Snapping also
occurred when the actuators were placed directly next to theelastomer beam. By moving the connection point farther
away from the elastomer beam, we were able to eliminate that
effect. However, we note that this instability could be leveraged to achieve rapid motion between two regions. For
example, this could be used to deploy a structure and lock it
in position.

4. Fabrication
A detailed discussion of the fabrication process is provided in
the supplemental information. We will only brieﬂy summarize the process here. For the elastomer structure fabrication,
three different types of elastomeric beams were fabricated as apart of this study using a combination of stiff elastomer,
Smooth-Sil 935, and a softer elastomer, Dragon Skin 10, bothfrom Smooth-On. These elastomers were cast in 3D printed
molds, shown in ﬁgure 4. For the liquid–metal-embedded
elastomer sensor fabrication, we created microchannels inelastomer substrates using a laser. These microchannels werethen ﬁlled with liquid metal and sealed, completing the sensor. The sensors were fabricated from Dragon Skin 10 elastomer, which is the same soft elastomer used the beam
structures. This process is illustrated in ﬁgure 5. Additional
information on this class of sensors can be found in ourprevious work [30]. For the actuator fabrication, nickel–titanium wires were programmed into coils at a high temperature.

3.3. Sensor and actuator placement

We placed our sensors and actuators over the joint of theelastomer structure. A sensor and actuator pair was placed oneither side of the elastomer structure as seen in ﬁgure 3. The
sensors must be pre-strained along the length of the elastomer
joint so that when the soft structure reaches its maximum
angles (in this case, -60 to 60°), neither of the strain sensors
buckle. The sensors and actuators are attached to the elastomer structure via mounting brackets, which are pinned inplace. The mounting brackets for sensor attachment are pinned 40 mm from the ends of the bones. As shown in ﬁgure 3,
the SMA coil is attached to one of the sensor mounting
brackets. The SMA is also attached to a mounting bracket thatis pinned 10 mm from the end of a bone. The mounting
locations were selected as a compromise between maximizing
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J C Case et al

The programming procedure was similar to that reported in[55]. In this work, we have used a ‘counter-coil’ design for
our SMA actuators, which eliminates the torque produced by
the coils by combining equal-length clockwise and counterclockwise segments.

5. System integration
We used rigid mounting brackets to attach sensors andactuators to the outside of the soft structure. These brackets
were printed from PLA ﬁlament using a printrbot simple
FDM printer. Two different styles of mounting brackets wereused, one with and one without horns to attach SMA. The
mounting brackets served two functions. First, they provided
a rigid attachment point for the SMA to the soft structure.
Second, they held the SMA far enough away from the body tolimit the snapping instability observed at large deﬂections.
Given the low forces observed in this structure, the strength of
the 3D printed parts was not a factor.
These mounting structures were held in place with pinsthat passed through the mounting bracket and the soft bonesof the structure. This resulted in a very mechanically stableattachment scheme. In the case of the brackets holding thesensors, these pins passed through the fabric reinforcement
pads on the sensor body as well, holding them securely to thebody. SMA was attached to the mounting brackets by passing
the wire through holes in the brackets, then joining the SMA
with a copper wire using a metal crimp. The crimp not only
ensured electrical contact between the copper lead wire andthe SMA, it also provides mechanical attachment since thecrimp is larger than the hole in the bracket.

Figure 4. Fabrication process for elastomer beams: (a) Smooth-Sil

935 is poured into a mold and leveled with the ﬂat side of a plastic
knife and cured at room temperature overnight (length of molds varyfor beam type: 168 mm for a prismatic beam and 80 mm for nonprismatic and non-homogeneous beams); (b) cured beams and bones
are removed from the molds (this is the ﬁnal fabrication step for theprismatic beam); (c) two bones are placed in compression ﬁtted joint
mold; (d) either Smooth-Sil 935 or Dragonskin 10 was poured into
joint mold and cured overnight in room temperature for the nonprismatic or non-homogeneous beam, respectively; (e) ﬁnished joint
system removed from the mold (a non-homogeneous beam isshown here).

6. Experimental setup
The experimental setup consisted of the soft structure with
attached sensors and actuators and the interface electronics.
The structure was controlled, and data collected, via an
Adruino Uno microcontroller connected to a laptop computer.
We designed the soft structure to be modular so that the threedifferent structure conﬁgurations could easily be secured withthe use of two bolts. Each soft structure had twelve electricalconnections: two on each SMA (four total) and four on eachsensor (eight total). We were able to remove and replacestructures in the ﬁxture within a few minutes.

Figure 5. The elastomer sensors are fabricated from two ﬁlms
sandwiching a microchannel pattern. The ﬁrst ﬁlm (gray) is createdby rod coating liquid elastomer on a polymer substrate (a).
Elastomer-infused fabric reinforcement pads (green) are bonded tothe upper surface of the ﬁlm (b). This ﬁlm is removed from thepolymer backing (not shown) and inverted. A laser is used to pattern
the microchannels into, and cut the sensor from, the ﬁlm (c). The
resulting half-sensors are then cleaned, inverted so that the channels
are facing down, and bonded to a second ﬁlm of elastomer (d). The
complete assembly is then cut from the ﬁlm to ﬁnal shape with alaser (e). A schematic of the ﬁnished sensor is shown in (f).
Dimensions are in millimeters.

6.1. Test apparatus

The soft structure was clamped on one end and free to rotate
on the other end. We used a printed angular scale attached tothe table under the robot to facilitate calibration and monitoring of the robot during operation (see ﬁgure 1(c)). We
used double-sided tape to afﬁx 1 cm square pieces of PTFE
ﬁlm to the bottom of the 3D printed attachment brackets to
reduce friction and facilitate smoother motion.
The two 3D printed attachment brackets on the nonmoving side of the structure are bolted to another 3D printed
5

Smart Mater. Struct. 25 (2016) 045018

J C Case et albracket, which is in turn clamped to a standard laboratory ringstand. This rigidly held the non-moving part of the structure
in place.
6.2. Electrical setup

The system uses three power supplies. The Arduino used forcommunication and control is powered with a 12 V laptopstyle power supply. This is not required, but we found that itprovided more reliable performance than using power fromthe USB. The SMAs are powered with an independent Korad
KA3005D power supply set at 6.00 V with a maximum output current of 1.00 A. A second Korad KA3005D is used toprovide a 2.50 V reference for the sensor signal conditioning
electronics.
The angle of the robotic structure was measured using
two liquid metal strain sensors attached at the joint. In orderto read these two sensors, we used a pair of custom-built
signal conditioning circuits. These circuits provide a constant
100 mA current to the sensors while measuring the voltage
drop across the resistive sensor element. The resulting voltage
difference is then read by a ADS1115 16-bit ADC andcommunicated to the Arduino Uno. This circuit is shown inﬁgures 6(a) and (b).
The SMA coils are powered by controlled-current supplies. These were built from IRF 510 N-channel MOSFETs
controlled by OPA347 operational ampliﬁers. The ampliﬁers
sensed the voltage drop across a shunt resistor, and drove the
MOSFETs such that a desired shunt voltage was achieved.
We selected components such that the MOSFETs would
nominally be able to supply 700 mA when VGS = 5 V. Thiseffectively limited the maximum current that could passthrough the SMA actuators. The setpoint voltage was provided via an DAC output from the Arduino Uno. Since the
DAC on the Arduino uses a 1 kHz signal to produce
approximate analog outputs, we passed that signal through alow-pass ﬁlter with a cutoff frequency of 30 Hz before
sending it to the current control ampliﬁer. This reduced
electrical noise in the system since the MOSFET operated as
an analog device, instead of in a PWM mode. This circuit isshown in ﬁgure 6(c).

Figure 6. The electronics were fabricated from commerciallyavailable components. There were three basic modules: a constantcurrent supply for the strain sensors (a), a differencing ampliﬁer tomeasure the voltage across the resistive strain sensor (b), and acurrent supply for the shape memory alloy (c). Modules (a) and (b)are integrated on a custom-built PCB (see ﬁgure 1(b)). Module (c) isfabricated on a breadboard.

7. Sensor calibration
The robotic structure used two Dragon Skin 10 Slow strain
sensors to determine the the bend angle of the system. In
order to accomplish this, the soft structure was held at angles
between –60°and 60°at 15° intervals and the sensors were
read. Data were collected at each test angle a total of threetimes in a randomized sequence to eliminate the effects ofhysteresis. Generalized least squares regression was used todetermine the coefﬁcients for the following equation:q = a 0 + a1V1 + a 2 V2,coefﬁcients of the ﬁt. Sensor calibration was performed foreach elastomer beam. The ﬁt for the non-homogeneous beam
is shown in ﬁgure 7; it has a 95% conﬁdence interval of 7.53°
or 6.28% of the full scale. The ﬁts for the prismatic and nonprismatic beams can be found in ﬁgures S1 and S2(supplemental information). The prismatic beam has a 95%
conﬁdence interval of 10.6° or 8.83% of the full scale; thenon-prismatic beam has a 95% conﬁdence interval of 5.13° or
4.28% of the full scale.

(1 )where θ is the bend angle of the joint, V1 and V2 are thevoltages across the strain gauges, and a0, a1, and a2 are the6Smart Mater. Struct. 25 (2016) 045018

J C Case et alpresented in ﬁgure 8. Only the current-on data is shown. The
current-off response showed roughly a ﬁrst-order response,
which we expect from free convection. We performed thesame test on three coils, cycling the power on and off ﬁvetimes per coil. We discard the data from the ﬁrst cycle due tostart-up transient effects. We believe this is a valid approachsince in operation the coils will never be fully cooled orinactive as they were at the start of these tests. The error
clouds in ﬁgure 8 show the 95% conﬁdence intervals overthose four cycles. The 95% conﬁdence interval of the meanvalues at the end of the test was 0.0764 N, or 10.1% of theaverage. The non-zero initial value was due to the fact that theactuator does not completely ‘turn off’ once cooled to room
temperature and continues to exhibit purely elastic stress.

9. Controller design optimization
Figure 7. Sensor calibration results for the non-homogeneous beam

showing how the actual angle compares to the estimated angle. The
shaded region represents the 95% conﬁdence interval.

To determine the optimal PID controller design, we tested thesystem at a number of points in the {k p, ki, kd} gain space. We
initially explored a larger space on the non-prismatic beam
where k p Î [2, 20], ki Î [0, 10], and kd Î [0, 10]. From this,
we found an optimal solution for all the beams existed in thefollowing subspace: k p = {10, 15, 20}, ki = {0.1, 0.4, 0.8},
and kd = {0.1, 0.4, 0.8}. At each test condition, we drove thesystem with ten signals, which are a combination of steps andramps seen in ﬁgure S3 (supplemental information). We used
these complex signals because we believe they are more
representative of actual command histories which might beseen in a soft robotic system, rather than simple step and ramp
commands. To evaluate the performance of each controller,
we took the sum of the error between the desired and actual
angle at 3.3 Hz. The total performance of the controller is thesum of the errors across all ten tests. Once the data werecollected, we ﬁt the observations to get an estimated error
value of the form given in equation (2) using least-squares
regression.

Figure 8. Actuator force generation averaged across three coils withfour trials each. Shaded region represents 95% conﬁdence interval.
The average data has a symmetric moving average ﬁlter across 11
points with uniform weight. The shaded region has a symmetricmoving average ﬁlter across 41 points with a uniform weight.

E (k p, ki , k d ) = A 0 + a1k p2 + a 2 ki2 + a3 kd2 + a4 k p ki

8. Actuator performance

Figure 9 shows the results matching the actual error to
the estimated error for the non-homogeneous beam. The
results are all clustered in one area and are relatively close to
the 1:1 mapping line shown in black. The inset of ﬁgure 9shows the cluster at the front. It can be seen that we reach aminimum error in the performance of the system. This meansthat we have reached the noise ﬂoor of the system and will not
be able to tune it past that ﬂoor. Therefore, we take our bestsolution as the optimal solution, and, thus, selected the optimal from the discrete set rather than ﬁnding the optimal on acontinuous space. The error clusters for the prismatic andnon-prismatic beams are shown in ﬁguresS4 and S5 (supplemental information), respectively.
The optimal gains were found to be kp = 20, ki = 0.4,
kd = 0.1 for the prismatic beam; kp = 15, ki = 0.4, kd = 0.1+ a 5 k p k d + a 6 ki k d + a7 k d + a 8 ki + a 9 k d .
(2 )Prior to integration into the soft structure, we measured theopen-loop performance of the SMA actuator coils. In this test,
we used identical coils to those used in the soft structure tests.
We ﬁxed the end of the coil in an single-column Instron 3345
materials testing machine, then pulled the coils to simulate thedeformation in the actual installation. The initial coil lengthwas 26.2 mm, and the deformed length was 97.0 mm. Thislatter length matched the nominal length of the coil in aneutral conﬁguration of the soft structure. We then applied aconstant current of 0.6 A to heat the wire and cause actuation.
We applied current for 60 s, then allowed the actuator to coolfor 120 s before repeating the cycle. The resulting blocking
force of the actuator was recorded as a function of time and is7

Smart Mater. Struct. 25 (2016) 045018

J C Case et al

Figure 9. Optimization of PID controller for non-homogeneous

beam. Blue dots represent represent test controllers. The line
represents the one-to-one mapping of actual sum of errors against thetheoretical sum of errors.

for the non-prismatic beam; and kp = 20, ki = 0.1, kd = 0.8
for the non-homogeneous beam. The optimal controller
response for the reference signals can be seen in ﬁgures S6–
S8 (supplemental information) for each of the beams.

10. Results and discussion
To test the effectiveness of our controller, we tested thesystem with both open- and closed-loop controllers and stepand ramp inputs. For the open-loop steps, we sent a pulse
width modulated (PWM) signal (30 on a scale of 0–255) fromthe Arduino to one of the actuators at a time for 90 s. For the
open-loop ramps, we sent a growing PWM signal (0–30 on ascale of 0–255) from the Arduino to one of the actuators at atime that grew linearly over 90 s at an update rate of 10 Hz.
We then performed the same tests with the optimal controllers
we had found previously. Each test was performed three times
to look at the repeatability of the system. Figure 10 shows theresults of the tests for each of the elastomer beams. The
results in blue were driving the arm to 30° in either a step or
ramp and the results in red were –30°.
It can be seen for the open-loop step response, the arm
settled to a given angle (the same angle over multiple tests),
but the response could be slow and the controller did notknow the current state of the system. The open-loop ramps for
all the beams were signiﬁcantly worse than the open-loop
steps. This was because the SMA has a threshold temperature
where it begins actuating and we did not see actuation prior toreaching this temperature. Once it reaches its actuation
temperature, it can change quickly, as seen with the open-loop
ramps in ﬁgure 10(b). These results demonstrated that openloop control is not sufﬁcient for control of the system.
By closing the loop with our optimal controllers, wedrastically changed the response of the system. The settling

Figure 10. Open-loop and closed-loop results of 30°and –30° steps
and ramps for (a) prismatic, (b) non-prismatic, and (c) nonhomogeneous beam. Three trials are represented in each graph. The
shaded region represents the 95% conﬁdence interval with a moving
average ﬁlter across 11 points with uniform weight.

8

Smart Mater. Struct. 25 (2016) 045018

J C Case et al

Acknowledgments

time for the open-loop steps ranged from 7.5–42.3 s, while thesettling time for the closed-loop steps ranged from 4.5–7.5 s.
The closed-loop controllers were able to maintain the desired
step and ramp responses considerably better than the openloop controllers since they did not drift over time like some ofthe open-loop controllers. The closed-loop ramping responses
were able to track the desired ramp.
Looking at the closed-loop response of the different
beams, we found that the prismatic beam tracks the steps andramps very well with little variability in the response. By
introducing a joint in the non-prismatic beam, we found that
we have greatly altered the response of the system and made it
signiﬁcantly harder to control. We believe this was due toinstability in the joint caused by the reduced ratio of bending
stiffness to compressive stiffness. Snapping was most pronounced in the non-prismatic beam. The system oscillatedaround the desired angle rather than being able to maintainposition as in the case of a prismatic beam. However, when
we reduced the stiffness of the joint, we found that we wereable to reobtain a good response. The results from the nonhomogeneous beam (recall that this refers to the beam withtwo dissimilar materials) closely matched the results of theprismatic beam.
The non-homogeneous beam required less power tocontrol than the prismatic beam because it is less stiff and,thus, took less power to move to or maintain a position. Thiscan be seen in ﬁgure S8 (supplemental information) where thesystem did not return to 0° before it started the next test.
Another beneﬁt of the non-homogeneous beam was that wehave localized the deformation and, thus, know where
bending of the beam was taking place. This means we cancreate jointed soft robotic systems that reduce the degrees of
freedom and power consumption in the systems.

This work was partially supported by an Early Career Faculty
grant from NASA’s Space Technology Research Grants
program (NNX14AO52G). ELW is supported under the
National Science Foundation Graduate Research Fellowship
program (DGE-1333468). JCC is supported under theNASA Space Technology Research Fellowship program(NNX15AQ75H). 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.

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11. Conclusion
We have demonstrated closed-loop control of three soft
robotic systems. These systems used soft sensors to close thecontrol loop and provide sensory feedback. The degrees-offreedom of two of the systems were limited by introducing a
joint. In one case, the system was left materially homogeneous and, the other, the joint was made from a softer
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beginning to be integrated with soft robots and soft andresponsive material actuators, but are not actively being used
to control those systems. By integrating control into soft
robots, we begin to move towards intelligent, autonomous
soft systems.
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