Sensors and Actuators A 253 (2017) 188–197

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Sensors and Actuators A: Physicaljournal homepage: www.elsevier.com/locate/sna

Multi-mode strain and curvature sensors for soft robotic applications
Edward L. White, Jennifer C. Case, Rebecca K. Kramer ∗Department of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088, USAa r t i c l e

i n f oArticle history:
Received 16 June 2016
Received in revised form
17 November 2016
Accepted 23 November 2016
Available online 29 November 2016
Keywords:
Soft robotics
Soft sensors
Flexible sensors
Stretchable sensors
Liquid metal
Sensor arraysa b s t r a c t
In this paper we describe the fabrication and testing of elastomer-based sensors capable of measuring
both uniaxial strain and curvature. These sensors were fabricated from Sylgard 184, which is a transparent silicone elastomer. We created microchannels directly in silicone elastomer substrates using a laser toablate material. The sensing element was an alloy of gallium and indium, which is liquid at room temperature, contained within the laser-created microchannels. As the substrate deformed, the microchannel
deformed within it, resulting in a measurable change in electrical resistance. By fabricating two matching resistive strain-sensing elements on opposite sides of the sensor, we were able to unambiguously
measure uniaxial strain and curvature by observing the common mode and differential mode changes inresistance, respectively. There was very little coupling between modes, demonstrating the utility of thedifferential sensor design. We characterized the sensor in terms of its response to strain and curvature,
its noise, and its stability over time. We believe that this type of sensor has application in soft sensory
skins and can observe pose in soft robotic systems.
© 2016 Elsevier B.V. All rights reserved.

1. Introduction
Soft robotics is a new class of intelligent systems that move inways very different than traditional rigid devices. Proposed applications of research in the ﬁeld are roughly grouped into two relatedclasses of soft robotic systems: mobile soft robots [1–4] and wearable robots [5,6]. Instead of being comprised of rigid joints and links,
where motion is predominately localized at the joint, soft robots
rely on continuum deformations to achieve motion [7–9]. This
deformation is the source of the unique capabilities of soft robots.
The drawback is that the distribution of deformation throughout
the body signiﬁcantly complicates the state observation problem.
In order to achieve control of these soft systems, we must observe
the current state [10–13]. In order to place sensors on the bodies ofthese systems, we need sensors which are materially compatible.
Traditional sensors have high stiffness compared to the materials
used in soft robots, necessitating the development of a new classof soft sensors made from the same low-stiffness materials foundin the soft robots themselves.
In this paper, we present a multi-mode sensor which uniquelydetermines strain and curvature, as shown in Fig. 1. Our sensors
were fabricated from Sylgard 184 silicone elastomer substrates
containing microchannels ﬁlled with a gallium–indium alloy that

∗ Corresponding author.
E-mail address: rebeccakramer@purdue.edu (R.K. Kramer).
http://dx.doi.org/10.1016/j.sna.2016.11.031
0924-4247/© 2016 Elsevier B.V. All rights reserved.

is liquid at room temperature. Substrate deformation resulted inchanges in the geometry of the liquid-metal-ﬁlled microchannels,
resulting in a change in resistance. By measuring this change inresistance, we were able to determine the strain and curvature stateof the sensor element. The liquid metal components described here
and in previous work are strain sensors with an added stress concentrator to enhance their response to curvature. By combining twoof these strain/curvature sensors, we demonstrate a device which
can differentiate between positive curvature, negative curvatureand strain. Individually, each element of the devices in this paper
behaves according to the theory described in [14,15], which werevisit in our theoretical discussion below. In previously publisheddevices, uniaxial strain and bending produced the same sensor output, resulting in ambiguous measurements. In the current case, theoutputs from two sensing elements are used together to measure“common-mode” and “differential-mode” signals, which correspond to strain and curvature, respectively. The theoretical basis
for this capability is described in a later section. In addition, thesensor described in this work is able to measure curvature without
relying on a model of the underlying host object.
We anticipate that dual-mode strain and curvature sensors will
be applicable across a wide range of soft systems, but will be particularly useful in thin devices such as soft sensory skins where
signiﬁcant strain and bending occur simultaneously. Our long-termgoal is to develop multi-modal active sensory skins. In these devices,sensors and actuators are combined into a fabric or elastomer substrate which can then be attached to a deformable body to impart

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189

sensing modality [27–30]. Changes in magnetic ﬁeld have also beenused to measure curvature [11]. These approaches to sensing curvature provide a range of sensitivities and accuracy. Depending on
the application, the relative importance of accuracy, repeatability,
integration and stretchability will vary. We believe that elastomerbased, and in particular silicone-elastomer-based sensors, have thegreatest potential for integration into active sensory fabrics andsensory skins due to their low stiffness, high stretchability, lack ofrigid components, and chemical resistance.

2. Liquid metal embedded elastomer sensor bodyfabrication

Fig. 1. Overview of a differential multi-mode sensor. (A) The top of the sensor. (B)The sensor with backlighting, highlighting the location of the stress concentrator
channels behind the liquid-metal-ﬁlled microchannel. (C) The high deformabilityof the device resulting from the material properties of the elastomer substrate. The
scale bar in the upper right corner of each ﬁgure is 6.25 mm.

motion. The sensor element presented in this work is one element
which could be included in these future devices. In order to measurecurvature across a surface, an array of curvature sensing elements,
such as those discussed in this paper, would be required. The exact
design of that sensor array would require an understanding of
the spatial characteristics of the curvature ﬁeld to optimize sensor placement. In a surface-based sensory array, only the deformed
state of the surface of a deformable body will be known, and evenwith that the deformation ﬁeld will only be known at the location of
the sensors in the array. We suggest that boundary element methods from computational mechanics would be applicable to solving
this problem, but this is outside the scope of the current work.
Elastomer-based sensors with encapsulated liquid metals suchas those described in this work are well represented in theliterature. Previously reported soft sensory devices based on elastomers substrates include joint angle and curvature sensors [14,15],
pressure sensors [16,17] capacitance-based multi-element force
sensors [18,19], liquid-metal/conductive ﬂuid hybrid strain sensors [20–22], and multi-mode resistance-based devices measuringin-plane strain and out-of-plane pressure [23]. The current work
furthers the previously developed devices by being able to differentiate between positive curvature, negative curvature and strain,
while still being highly stretchable and capable of undergoing large
deformations. Additionally, the current work is able to measurecurvature without knowing the geometry of, or even requiring, an
underlying host. This is in contrast to previous work which useddeﬂection of a mechanical joint to induce strain in a liquid metal
sensor, and required re-calibration on a per-host basis [14,15].
Finally, the current work expands upon the previous example ofmulti-layer liquid metal sensors by placing strain gauges in parallel,rather than orthogonally [23]. This allows us to measure curvature,
while the previous work was used to measure biaxial strain. Thesesensing modalities could be combined in devices with even moresensor layers in the future.
Beyond elastomers, other types of materials can be used to sense
curvature based on strain measurement. These include multilayercomposites based on carbon ﬁlm/polymer electrolyte [24], conductive polymers [25], and piezoceramic systems [26]. Extending
even further, Bragg ﬁber gradings are another common curvature

The devices we present in this paper are soft, ﬂat, transparent
devices comprised of layers of patterned silicone elastomer. The
complete device is shown in Fig. 1, which shows the liquid-metalﬁlled microchannels and stress concentrator features. The bodiesof the devices were manufactured from Sylgard 184 (Dow Corning) ﬁlm with gallium indium alloy (EGaIn, Sigma-Aldrich) ﬁlled
microchannels. This conﬁguration contained two strain gauges
placed back-to-back. The devices described in this work were
fabricated in four major steps: substrate preparation, substratepatterning, microchannel ﬁlling, and interfacing. The fabrication
sequence is shown in Fig. 2.
Sylgard 184 (polydimethylsiloxane, PDMS) substrates were prepared by spin-coating the uncured polymer onto 3 in. × 2 in. glass
slides (Fig. 2:A1 and B1). Before applying liquid polymer, a ﬁlm ofmold release (Ease Release 200, Mann Technologies) was applied
to the slide. Four layers of elastomer were applied to achieve uniformity. These layers were applied at 500 rpm, and spun for 180 s
using a Specialty Coating Systems Spincoat G3-8. The elastomer
layers were allowed to cure for at least 4 h at 60◦ between applications. The resulting elastomer substrates were 273 ± 8.25 ␮m (95%
conﬁdence) as measured by a Zeta Instruments Zeta 20 3D microscope.
The blank substrates were patterned using a Universal Laser Systems VLS 2.30 laser system ﬁtted with a 30W CO2 laser operating
at 10.6 ␮m (Fig. 2:A2 and B2). The pattern created by the laser
is shown in Fig. 3. This image contains a mixture of “thru” and“blind” features. Thru features are cuts made by the laser that passcompletely through the elastomer layer into the glass substrate.
Blind features only remove part of the thickness of the elastomer
layer. The depth of the cut is controlled by adjusting laser power.
Our approach of directly patterning features into the Sylgard 184
ﬁlm is different than previously published approaches that useda mold to create channels. This direct approach has the advantage of not requiring the fabrication of a mold, which removes
several processing steps and decreases design iteration time. However, the laser ablation process results in deposits of soot and debris
on the surface of the substrate. Unless this material is removed, itinterferes with bonding between elastomer layers. Further, small
particles remaining in the microchannels can either cause wickingof liquid elastomer into the channel, resulting in a ﬁlled channel, or can block the channels themselves. For these reasons, athorough cleaning process is required after patterning. First, thesubstrates were cleaned by sonication in a Liquinox detergent solution (Alconox) for 10 minutes in a Branson Bransonic 1800 bathultrasonicator to remove the bulk of the soot left over from the patterning process. Second, we used a Kimwipe (Kimberly-Clark) withtoluene to manually remove soot from the microchannels. We notethat toluene aggressively swells Sylgard 184, and so the minimal
amount required to wet the Kimwipe was used. Third, we sequentially rinsed the substrates in acetone, isoproponol, ethanol, anddistilled water to remove ﬁlm contaminants. Finally, we dried theclean patterned substrates at 60 ◦ C to remove moisture.

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Fig. 3. Laser cut pattern. The upper image shows the “middle” layer with microﬂuidic channels. The lower shows the “bottom” layer with stress concentrator features.
Black outline represents a complete vector cut through the material, gray regions
are only partially ablated through the substrate. The two scale bars at lower left are
20.0 mm and 1.00 in. long, respectively.

Fig. 4. Photographs of patterned Sylgard 184 elastomer ﬁlm on glass substrate. (A)Shows three basic modes of laser patterning: through-ﬁlm patterning (red highlight), thin feature (blue) and thick feature (green). Through-ﬁlm and thin features
were patterned using a continuous movement of the laser, similar to a plotter, while
the thick features were patterned using pixel-based approach where the laser rasterscanned over the surface. (B) Shows the ability of the laser to create boss features(green) by removing material around the feature (blue). (For interpretation of thereferences to color in this ﬁgure legend, the reader is referred to the web version ofthis article.)
Fig. 2. Complete fabrication sequence. Steps A1–A3 proceed in parallel with steps
B1–B6. Steps C1–C7 follow after A3/B6.

Once cleaned, we measured the geometry of the laser-cut features using a Zeta Instruments Zeta 20 3D microscope. Since thematerials involved in this study were soft, a non-contact (i.e. optical) method was selected to perform the characterization. There
were two types of features created by the laser system: rasterand vector patterns, as shown in Fig. 4. These patterns only differed in the way the laser was actuated. In raster-mode, the laser
sweeps back and forth, pulsing the laser when it passes over a
spot to ablate the surface. In vector mode, the laser moves likean XY-plotter, tracing a continuous path over the surface while
the laser pulses. The impact of these two modes is manifested in
the roughness of the resulting feature. Vector mode results in asmoother feature than raster mode. However, for features requiring larger lateral dimensions than a single laser spot, raster modemust be used. Thus, vector mode is used for thin microchannels,
while raster mode is used for the wider microchannels, ﬁll ports
and stress concentrators. The cross sections, along with measured
dimensions, are shown in Fig. 5. The letters in parentheses in thefollowing sentence refer to the labels in that ﬁgure. Over all the sensors fabricated for this study, the depth of the stress concentrator(B) 156 ± 10.4 ␮m, the width (C) was 1323 ± 40.0 ␮m and the pillar

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191

After measurement, the patterned “middle” layer was removed
from the glass slide and cleaned again in acetone, isoproponal,
ethanol, and distilled water to remove mold release from the backside of the substrate (Fig. 2:B3). To fabricate the top layer, we prepared another blank PDMS substrate by coating a glass slide withmold release, and spinning a layer of Sylgard 184 onto the slide at
500 rpm for 180 s (Fig. 2:B4). This resulted in a layer 52.4 ± 1.42 ␮mthick, as measured by a Zeta Instruments Zeta 20 3D microscope.
We placed this in an incubator at 60 ◦ C for ≈50 min, until it was“tacky.” At this point, we placed the middle PDMS layer, patternedside down, onto the tacky PDMS, effecting a strong bond between
the layers (Fig. 2:B5). Curing overnight in an incubator at 60 ◦ C ﬁnalized the bond. We used the “tacky” bond approach instead of themore traditional plasma-based approaches due to poor bonding
between laser-patterned layers of Sylgard 184 with plasma. Thisapproach has been described previously [31].
To bond the middle/top layer assembly to the bottom layer, weapplied a thin layer of liquid PDMS to the bottom layer throughspin coating at 2000 rpm for 180 s (Fig. 2:A3). The middle/top layer
assembly was placed on top of the wet bottom layer, pressed
into place to remove bubbles, and placed in an incubator at 60 ◦ Covernight to cure (Fig. 2:C1). This process resulted in a three-layer
stack that comprised one half of the complete device.
After curing, we removed the completed subassemblies fromthe glass substrates (Fig. 2:C2). We injected liquid gallium indium
alloy using a syringe into the micochannels between the top andmiddle layers (Fig. 2:C3). We inserted stripped 34 ga copper wires
into the injection ports to provide an electrical interface with ourmeasurement equipment. Finally, to seal the wires to the devices
and to fully encapsulate the liquid gallium indium, we poured asmall amount of uncured Sylgard 184 over the back of the device,
and cured it in an incubator at 60 ◦ C.
Once each subassembly was completed, we tested the resistanceof the sensor element. Subassemblies found to be functional (i.e.
all four leads were connected) were combined into full devices, as
shown in Fig. 1. To bond the two halves of the device together,
we spun a thin layer of liquid Sylgard 184 onto a clean glass slide
at 2000 rpm for 180 s (Fig. 2:C4). We placed the halves, side to
be bonded down, onto this slide to “wet” the bonding side, andthen removed them from the slide (Fig. 2:C5 and C6). Thus wetted,
the two halves were aligned and pushed together, placed undera weight, and placed in an incubator at 60 ◦ C overnight to cure(Fig. 2:C7).
In order to improve the stability of the sensors, during these tests
we bonded Sylgard-184-infused muslin fabric squares to the endsof the device. There were two purposes for these reinforcements.
The ﬁrst was to stabilize the electrical interface by encapsulating
the lead wires. The second was to better distribute the force into thebody of the sensor. Preliminary testing showed that stress concentrations caused by handling the sensors caused a marked increasein variability. These reinforcement pads were only required during
these tests because of the isolated nature of the sensors during testing. When implemented, the sensors would likely be integrated intoa larger sensory skin, changing the requirements on the interface.
Fig. 5. Histograms showing manufacturing variation associated with lasermanufacturing parameters. All histograms show deviation from mean as apercentage, with the mean () reported in the box in the upper right in ␮m. The
top row shows the data for the thick elastomer substrates. The second and thirdrows show data from the features generated by the laser in raster mode. The fourthand ﬁfth rows show data from the features generated in vector mode. The sixth row
shows data for the thin elastomer substrate.

width (D) was 943 ± 16.0 ␮m. For the microchannels, the depth (E)was 154 ± 22.8 ␮m, the width (F) was 589 ± 17.3 ␮m and the crosssectional area (G) was 69,700 ± 6700 ␮m2 . All bounds represent
95% conﬁdence intervals.

3. Electrical interface and characterization
The devices presented in this paper rely on room-temperatureliquid alloy of gallium and indium as a sensing element. This liquid
is embedded in microchannels within an elastomer matrix. As thematrix deforms due to externally applied stress, the microchannels deform. This deformation results in a change in the resistanceof the embedded liquid metal. In order to observe this effect, wemeasured the voltage drop across the resistor while supplying a
constant 100mA current through the device. This “four terminal”

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Fig. 6. Noise measurements of short and long time domains. The short time domain data represents 1000 samples captured at approximately 22 Hz across three sensors.
The long time domain data represents 3600 samples captured at 1 Hz. Both sets of data are across all three sensors. The data reported show the deviation of observed values
from the mode of all observations. The 95% conﬁdence intervals are 0.0335 mV and 0.0546 mV, respectively.

measurement scheme allows us to negate the effects of interface
resistance between electrical leads. Our selection of four-terminal
versus the more common two-terminal measurement was motivated by a preliminary series of tests that indicated that the contact
resistance was not only variable, but accounted for ≈10% of thetotal resistance. The electronics required for signal conditioning
were designed and integrated onto a custom PCB, which we havedescribed previously [32]. The sensitivity of the signal conditioning
circuit is 0.0433 V −1 . We measured the output from the signal conditioning electronics using an ADS1115 analog-to-digitalconverter (ADC) (Adafruit). This device allowed for differential
voltage measurements with 16-bit resolution, combined with aprogrammable gain ampliﬁer (PGA) on the device. The gain in the
PGA of the ADC was set to 16, resulting in an overall resolution of
the system of 0.00122 V bit−1 (0.0282  bit−1 ).
One measure of performance is the noise in the sensor. To quantify the noise and drift in the sensor, we measured the unloaded
response of the device in two time domains. The electronic noise
measured in this process will be compared to the overall error in thesensor response in a later section. Compared to the uncertainty inthe overall measurement, the electrical noise is a fairly small contribution to the overall error. We observed the high-frequency noiseby sampling as rapidly as possible for 1000 samples, and observed
the low-frequency noise by sampling at 1 Hz for 1 h. The measurement time and communication bandwidth effectively limited thehigh-frequency measurement to ≈22 Hz. This process used two different time domains to compare the effects of high-frequency noise,such as noise in the electronics and analog-to-digital converter, tothe lower-frequency noise, such as thermal effects in the elastomer.
The long-term measurements show a slightly larger deviation than
the short term test, suggesting that there is a low-frequency component of the noise in addition to the high-frequency component.
We speculate this could be due to thermal drift in the environment, as this test was conducted in ambient conditions. Further,
these tests provide a noise baseline which we can compare to thesensitivity of the sensor in the next section. The distribution ofobserved values is shown in Fig. 6. The 95% conﬁdence intervalsof the short and long-term tests were 0.0335 mV and 0.0546 mV,
respectively.
4. Strain and curvature sensing
We conducted two different types of mechanical tests on thesensors: curvature testing and uniaxial strain testing. All electricalmeasurements were made with the signal conditioning electronics
described in the previous section. We tested the curvature responseby wrapping the sensors around various solid cylindrical objects ofknown radius. The objects used in the study are shown in Table 1. Inorder to measure both positive and negative curvature, the sensors
were ﬂipped over and tested in both “up” and “down” conﬁgurations. We tested the strain response by stretching the sensor to
known lengths. We marked two lines on the sensor body usinga Sharpie permanent marker such that the insides of the lines
were 20 mm apart. We placed a printed guide with lines 2 mmapart on a table surface. While looking through the clear Sylgard
184 body of the sensor, we extended it until the inside of the
Sharpie lines matched the lines printed on the guide for a given
extension. We tested extensions of 2 mm, 4 mm, 6 mm, and 8 mm.
We found during testing that 10 mm extension, corresponding to
50% strain, caused unreliable operation and promoted failure. This
Table 1Known diameters of cylindrical items used as known curvature objects. Measurements performed with hand-held digital calipers. The sensors were manuallywrapped around these items to impose a known curvature.
ObjectKnown diameter and 95%
conﬁdence interval (mm)1/2 in. PVC pipe
1 in. PVC pipe
1 1/4 in. PVC pipe
2 in. PVC pipe
3D printed cylinder

21.4 ± 0.157
33.4 ± 0.0594
42.1 ± 0.350
60.4 ± 0.237
79.4 ± 1.00

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was likely due to separation of the liquid metal from the interface electrodes. Our previously published work has investigated the
dynamics of the response of liquid-metal-based elastomer sensors,which showed that the response was generally decoupled fromloading rate, and so we limit our experiments here to static loading conditions [33]. Each loading condition was tested three times
in a random sequence which included both curvature and strain
measurements. There were a total of 12 measurements of strain (4conditions with 3 repetitions) and 30 measurements of curvature(10 conditions with 3 repetitions) for a total of 42 measurements
per device. We fabricated three identical devices for this study. In
the following sections, the data from all three devices is presented
in aggregate, unless otherwise noted. Although we measured thevoltage output from the two sensors, we were actually interested in
the summation of these two signals, which we call sigma (), andthe difference between the output of the two sensors, which we calldelta (). These two derived responses are ideally only responsive
to strain and curvature, respectively, while being insensitive to theother. In actuality, cross-coupling occurs due to variations in gauge
factor.
Between each measurement, we returned the sensor to an
unstrained state and took a baseline measurement. We report two
different approaches to using this baseline resistance, which wecall “averaged” and “updated.” In the averaged mode, only the initial and ﬁnal baseline unstrained resistance measurements wereused. We believe this is more representative of a case where adevice might be zeroed at the beginning of operation, and thennot returned to an unstrained state. In the updated mode, we used
the baseline measurements before and after each measurement tocompute the change in resistance. This more accurately reﬂects
the dynamic response of the sensor, but the operation is less likea real-world application. Our previous work showed that liquidmetal-based devices are stable and respond primarily to strain, anddo not experience stress relaxation effects [33]. To evaluate thestability of the devices, we compared the measured performanceusing the averaged and updated baseline approaches. This comparison is shown in Fig. 7, which shows data from the experiments
described above. Based on these two ﬁgures, we see that there isno signiﬁcant difference between using the average baseline resistance and continually updating the resistance. We believe this isa strong positive indicator to using these devices in practical soft
systems.
We can analyze the output from the sensors by considering
their resistance and basic mechanical properties. The resistance ofa homogenous material is R = L/A, where  is the resistivity of thematerial, L is the length, and A is the cross sectional area. Further,
we can compute the volume of this same piece of material as V = AL,
where V is the volume. If we assume that all of the materials areincompressible, A = A0 L0 /L, where the subscript indicates the initial condition. Substituting, the current and initial resistances are,R = L2 /A0 L0 and R0 = L02 /A0 L0 , respectively. We are interested in
the change in resistance, and so we deﬁne:R = R − R0 =


(L2 − L02 )
A0 L0

(1)

The engineering strain in the sensor is deﬁned as  = L − L0 /L0 . Substituting this into the proceeding:R =
((L0 (1 + ))2 − L02 ) = R0 (2 + 2 )
A0 L0

(2)

Next, we deﬁne two values,  and , which are the sum and difference of the outputs from the two sensors. This results in two
measured properties:

193

where G is the gain discussed in the previous section and the subscript t refers to the top sensor and b to the bottom sensor. and  are the two quantities we are interested in, since they arevery nearly decoupled from each other, and highly correlated tostrain and curvature measurement, respectively. Next, we need toconsider the strain in each sensor as a function of the loading condition. In the uniaxial strain case, the solution is trivial:  = t = b .
Substituting these strains into the proceeding:Strain = GR0t (2 + 2 ) + GR0b (2 + 2 ) = G(R0t + R0b )(2 + 2 )(4a)Strain = GR0t (2 + 2 ) − GR0b (2 + 2 ) = G(R0t − R0b )(2 + 2 )(4b)
In the case of curvature, we assume that the neutral axis of thedeformation is in the middle of the sensor body, which is reasonable
due to the symmetry of the device (see Fig. 2). The deformed lengthof the sensor element as a function of the curvature is:Lt =L0 (r + T )
r

(5a)

Lb =

L0 (r − T )r(5b)where r is the radius of curvature to the neutral axis, and T is the
thickness from the neutral axis to the plane of the sensor. The
strains in the upper and lower sensors, along with the substitutionof the proceeding, results in:t =Lt − L0T= = T
L0
r

(6a)

b =

Lb − L0T= − = −T
L0
r

(6b)

where  is the curvature, deﬁned as r−1 . We can substitute these
expressions into Eq. (3):Curvature = G(R0t + R0b )(T 2 2 )(7a)

Curvature = G(R0t + R0b )(2T)(7b)Eqs. (4) and (7) represent the ideal response of the sensors. At thispoint, we assume that the initial resistance of all devices is similar.
This is warranted since all of the sensor elements have nominallyidentical geometry. Making this assumption, the response due tostrain and curvature becomes:
Strain = 2GR0 (2 + 2 )(8a)Strain = 0

(8b)
2 2

Curvature = 2GR0 (T  )(8c)Curvature = 2GR0 (2T)(8d)These theoretical relationships are shown Fig. 7 along with theexperimental observations. As discussed previously, the devices
are very thin. Therefore, we can neglect quadratic thickness terms.
Further, if we linearize about zero strain, we are left with an approximate form that we recast as a system of equations:











≈ 2GR0

 


2 0
0

2T

(9)



Since we are interested in determining the deformed conﬁguration
based on voltage measurements, we invert the system to obtain:

 = GR0t (2t + 2t ) + GR0b (2b + 2b )

(3a)

 

 = GR0t (2t + 2t ) − GR0b (2b + 2b )

(3b)







1
≈
4GR0

1 0
1
0
T







(10)

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Fig. 7. Measured response of sensors to strain (top row) and curvature (bottom row). (A) Shows the response using the baseline resistance from the start and end of thetest, while (B) shows the response with continual baseline updating. (A) is more representative of what would be observed in operation. The vertical axes show the sum() and the difference () of the signals from the upper and lower strain gauges. The solid lines show the theoretical response of the sensors based on Eq. (8). In this case,GR0 = 22.6 mV and T = 784 ␮m. Both of these values were based on measurements across all of the devices used in the study. We suspect that the deviation between thedeviation in predicted and actual value in the  response to curvature is due to elasticity of the sensor.

This expression demonstrates that this type of sensor has thecapacity to simultaneously measure both strain and curvature.
We would also like to point out that the assumption of identical initial resistance and the linearization step are not required
to achieve this capability. Without the assumptions, Eq. (10)would be non-linear and contain four known initial resistanceterms. As we show later, the capability of our sensors is limitedby measurement noise, rendering this additional complication
unnecessary.
Next, we wished to perform a regression analysis to quantify the
response of the sensors. We began by assuming a linear response,

which we believe is justiﬁed by Fig. 7, and the result in Eq. (10), of
the form: = a0 + a1  + a2 

(11a)

 = b0 + b1  + b2 

(11b)where  is the strain,  is the curvature,  is the sum of the outputs from the resistive strain sensors,  is the difference in the
outputs, and ai and bi are coefﬁcients of the ﬁt. Since we had an

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195

Fig. 8. Comparison of the known and measured strain and curvature in the left and right ﬁgures, respectively. The scattered points represent experimental observations, theshaded box represents the 95% conﬁdence interval, and the solid line is an ideal 1:1 line. The errors between the experimental observations and ideal line are shown as ahistogram in Fig. 9.

overconstrained system, we used a generalized least squares
regression. To do so, we constructed a design matrix of the form:

⎡1 
0
⎢1 
1
⎢
⎢
⎢ .
⎢ ..
D=⎢
⎢1 
⎢
i
⎢
⎢ .
⎣ ..
1

0
1

i

126

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(12)

126

where each row corresponds to a single measurement, and a
response matrix of the form:

⎡

0

⎢
⎢ 1
⎢
⎢
⎢
R=⎢
⎢
⎢ i
⎢
⎢
⎣

0
1
..
.
i
..
.

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(13)

126 126
The system of equations we wished to solve was therefore:
DC = R(14)

where C is a matrix of the unknown coefﬁcients. Using a pseudoinverse transformation, we determined the unknown coefﬁcients:−0.399C = 1.07
0.124

3.86
−0.16927.6(15)

We note that these coefﬁcients assume strain in units of %, curvature with units of m−1 and voltages in mV. These are the same units
shown on the corresponding ﬁgures.
In order to determine the quality of this model, we used Eq. (14)and the values of the coefﬁcients in Eq. (15) to determine the estimated strain and curvature for each measurement. We have plotted
these results in Fig. 8. Additionally, we have plotted histograms of
the error between the experimental data and reconstructed data
in Fig. 9. Based on this distribution, we found that the 95% conﬁdence intervals for strain was 2.80% and for curvature was 17.5 m−1 .
These two values corresponded to 7.00% and 8.75% of the full-scaleof strain and curvature, respectively.
We previously stated that there was low coupling between
the two measurements of strain and curvature. Looking at the
coefﬁcients in Eq. (15), the ﬁrst row corresponds to the offset terms.
The coupled terms (i.e. the strain response to  and curvature
response to ) are C21 and C32 , respectively. These terms are O(1)and O(10). The cross-coupled terms are C22 and C31 . These terms
are O(0.1) or less, which is one to two orders of magnitude lower
than the coupled terms, demonstrating the low cross-couplingof the current device. Although this low cross-coupling is not arequirement of device operation, it greatly simpliﬁes the overall functionality. In practice, the cross-terms could be eliminated,
resulting in fewer mathematical operations to convert from resistance measurements to strain and curvature estimates.
Finally, we compared the model uncertainty to the noise
measured in the system. The previously discussed noise and stability measurements were based on single channel measurements.
To convert to two-channel measurements, such as our  and parameters, we assumed independent, identically distributed
noise. Under this assumption, we can combine noise from two
2E12 , where ET ischannels using the root sum of squares, or ET =the total error, and E1 is the error in a single channel. This resulted ina value of 0.0772 mV. Using the coupled terms in Eq. (15), we foundthis noise would result in variations of 0.0826% and 2.13 m−1 for196

E.L. White et al. / Sensors and Actuators A 253 (2017) 188–197

Fig. 9. Histogram of the error between the applied and measured strain and curvature. The histogram shows the distance from each observation to the ideal line in Fig. 8.

strain and curvature, respectively. These values are approximately
an order of magnitude smaller than the uncertainty in the model,
indicating that electrical noise is not a signiﬁcant contributor to
overall sensor performance.
In summary, the sensors exhibit good linearity and low coupling between summation and differential channels in response tostrain and curvature. The electrical noise was a minor contributor to overall sensor performance. The primary limiting factor insensor performance was scatter in the observed data, which we
believe is primarily due to variation in how the sensors are held
across different tests. Although the observed variation in the sensor is worse that what is expected in commercial strain sensors, theperformance is acceptable for observing many soft systems.
5. Conclusion
As the soft robotics community continues to progress as a ﬁeld
towards more realistic applications and devices, we must become
more tolerant of realistic loading cases. In our present study, we
have fabricated a sensor that can measure both strain and curvature using soft materials that are compatible with soft robotic
systems and wearable devices. The use of two coupled sensing
elements allows the sensor presented in this work to differentiate between positive curvature, negative curvature, and strain.
Previous curvature sensors required either knowledge of the kinematics of the host on which they were placed, or in situ calibration
to translate strain into curvature. With this device, we use thedifference in output from two paired resistive strain sensors to produce measurements of curvature and strain without requiring any
knowledge of the host. In the future, this two-element sensor could
be used as the basis for more complex soft robotic skins that arecapable of measuring their state across their entire surface.
One of the challenges associated with wider adoption of thisclass of device is manufacturing. At the end of the fabrication
sequence, the device consists of nine layers of silicone elastomer
containing two different liquid metal ﬁlled microchannel structures (Fig. 2:C7). This results in a complex process with many steps

and a low overall device yield rate. We intend to integrate ourother research on manufacturing liquid metal-based stretchable
electronics using approaches such as ink-jet printing to simplify
the design and manufacturing processes.
Acknowledgements
This work was supported by the National Aeronautics andSpace Administration under the Early Career Faculty program(NNX14AO52G). ELW is supported by the National Science Foundation Graduate Research Fellowship Program (DGE-1333468). JCC
is supported by the NASA Space Technology Research Fellowship(NNX15AQ75H). Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the authors anddo not necessarily reﬂect the views of the National Aeronautics andSpace Administration or the National Science Foundation.
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Biographies

Edward L. White is a Ph.D. student in the Faboratory in
the School of Mechanical Engineering at Purdue University, West Lafayette, IN, USA. He has an M.S. is mechanicalengineering and an MBA from the University of Arizona,
Tucson, AZ, USA. His research is focused on making soft
robotic systems more manufacturerable and robust.

Jennifer C. Case is currently pursuing her Ph.D. degree
in Mechanical Engineering from Purdue University, West
Lafayette, IN, USA. She received her B.S. degree in Mechanical Engineering from Northern Illinois University, DeKalb,
IL, USA, in 2013. Her research is focused on adding closedloop control systems to highly deformable robots.

Rebecca K. Kramer is an Assistant Professor of MechanicalEngineering at Purdue University. She holds the degreesof B.S. from Johns Hopkins University, M.S. from the University of California at Berkeley, and Ph.D. from Harvard
University. At Purdue, she founded The Faboratory, whichcontains a leading facility for the rapid design, fabrication,
and analysis of materially soft and multifunctional robots.
Her research interests include stretchable electronics,
responsive material actuators, soft material manufacturing, and soft-bodied control. She is the recipient of the NSF
CAREER Award, the NASA Early Career Faculty Award, the
AFOSR Young Investigator Award, and was named to the2015 Forbes 30 under 30 list.