Smart Braid: Air Muscles that Measure Force and Displacement

Wyatt Felt and C. David Remy, Member, IEEE

Abstract— In this paper, we introduce a novel method tomeasure the contraction length and force output of PneumaticArtificial Muscles. Our key innovation is to make the braidof these fiber-reinforced actuators out of insulated wires thatform a single electric circuit. Changes in the inductance andresistance of this circuit can be related to muscle contractionlength and muscle force. We implemented the proposed ap-proach in two McKibben muscles that were evaluated undera variety of air pressures (0-200 kPa) and loads (0-45 N). Theexperiments confirmed our theoretical prediction of a roughlylinear relationship between the measured inductance and thecontraction length, as well as a strong correlation between themeasured resistance and the actuator force. We were able todetermine force with a resolution of 5 N and contraction lengthwith a resolution of 0.5 mm. This technique can be used tocreate flexible, precise, and robust self-sensing actuators thatbenefit a multitude of robotic applications.

I. INTRODUCTION

Pneumatic Artificial Muscles (PAMs or McKibben mus-cles [1]) have become a very popular method of actuationfor robotics and assistive devices. PAMs are typically madefrom an elastomeric bladder that is enclosed in a braidedfiber mesh [2]. When a PAM is pressurized, it expandsin volume and the fibers of the braid force it to contractalong its longitudinal axis, creating large contraction forcesin the process. Air muscles can only act in one directionand so are often used in antagonized pairs [3]. Despitethis drawback and the difficulty of controlling these highlynonlinear actuators [4], they are a popular choice becausethey are force dense, easy to construct, and compliant [5].Furthermore, the force profile of PAMs are similar to thatof biological muscles [6]. This allows them to be readilyused in biomimetic devices [7], [8], [9] and rehabilitation[10], [11] applications. PAMs are inexpensive, and since theyare hermetically sealed against the environment [12] theycan operate in harsh environments [13]. Yet, to be usefulin robotics, their motion and force must be measured andcontrolled in a closed loop fashion. While force control canbe achieved through modeling and feedback from relativelyinexpensive and robust pressure transducers [4], positioncontrol typically relies on sensors that are both expensiveand fragile. Additionally, these position sensors requirewell-defined joints and rigid segments in order to workproperly. It would thus be highly desirable if PAMs did nothave to rely on external sensors at all, but could self-sensetheir contraction and force output. PAMs that could sensetheir own state could be used with compliant links and in

The authors are with the Robotics and Motion Laboratory (RAMlab),Department of Mechanical Engineering, University of Michigan, Ann Arbor,MI (wfelt@umich.edu, cdremy@umich.edu)

Fig. 1: The presented self-sensing air muscle (a) is essentiallya McKibben muscle in which the reinforcing mesh is madeout of conductive fibers. These fibers are connected in sucha way that the net current circles the long axis of the musclein a consistent direction (b). When the muscle is pressurizedit contracts and increases in diameter (c). The wires alignmore closely and the inductance of the circuit increases. Anexternal force is creating a strain on the wires (d) and thusincreases the resistance of the circuit.

configurations more complex than direct antagonization suchas as tri-articulated muscles, which are very common in thehuman musculoskeletal system.

It is thus no surprise that a considerable amount ofresearch has been focused on the development of suchself-sensing actuators. Most of the proposed designs ex-ploit changes in resistance and/or capacitance in strainedelastomers. Those that rely only on resistance can exhibitvery high sensitivities and undergo large strains [14], [15].Those that additionally exhibit changes in capacitance areknown as Dielectric elastomers. The changes in their com-plex impedance can be used to characterize their strain[16]. Many of these elastomers also function as actuators,generating force when subjected to extremely high voltages(e.g. 3kV) [17]. These elastomers could be used to measurethe contraction of PAM. Goulbourne and Son, for example,have shown how dielectric elastomers could be used on thebladder of McKibben muscles to sense contraction [18].

In this paper, we present a novel self-sensing air musclethat uses changing properties in the braided fiber mesh tomeasure contraction and force output. To this end, we madethe braid of a pneumatic muscle from insulated conductivefibers that form a single electric circuit. The inductance

and resistance of this circuit changes in reaction to theforce on the muscle and its degree of contraction. Externalforces introduce a strain on the wires, thereby changing theresistance of the circuit. The change in length causes thewires to become more aligned and results in an increase ininductance. This allows the PAM to be used without externalencoders. In this paper, we present a prototype, introducethe theory of measurement, and experimentally evaluate thefunctionality of the muscle.

II. THEORY

The basic structure of a PAM includes a flexible, innertube and a braided sheath. The braid is made of fibers inan alternating helical pattern. When pressurized, the musclecontracts axially as it grows in volume. Because the fibersof the braid are practically inextensible, the volume increaseis accomplished through the radial expansion of the braid.During this expansion, the fibers of the braid become moreperpendicular to the long axis. To make the braid into anelectromagnetic coil, we replaced the fibers in the braid withinsulated wires. The end of each wire is connected with theend of an adjacent, opposite hand wire until a single circuit isformed. When connected to a voltage source, current travelsup the muscle along one helix and down on another helix ofthe opposite hand, creating a net current that flows aroundthe muscle axis in a single direction (Fig. 1). The reinforcingmagnetic fields of the wires are thus analogous to those ofa solenoid.

As an approximation, we can assume that this circuitbehaves like a long, thin solenoid. Its inductance will thusbe proportional to the cross-sectional area A and inverselyproportional to its length l:

L ∝Al. (1)

As the muscle contracts, the cross-sectional area increasesand the length decreases. We consequently expect the induc-tance to increase as the muscle contracts.

Approximating the shape of the PAM to be cylindrical,we can use the following equations to characterize the lengthand diameter of the actuator [19]:

l = bcosθ (2)

D =bsinθ

nπ. (3)

The length of the muscle l and its diameter D are therebyexpressed as a function of the winding angle θ that isexpressed with respect to the long-axis. b is the total lengthof the fibers, and n is the number of turns about the axis.We can express b with respect to the initial length l0 and theinitial fiber angle θ0, and compute θ from the length of themuscle:

b =l0

cosθ0(4)

θ = cos−1(ll0

cosθ0). (5)

Substituting eqs. (4) and (5) into eq. (3) allows us toexpress the cross-sectional area A as a function of the musclelength l:

A = π

4 D2 =l02(1− l2

l02 cos2θ0)

4n2πcos2θ0. (6)

The complexity of this expression comes from the fact thatthe volume increases as the muscle contracts. Substitutingthis expression into (1) yields:

L ∝

l02

l

(1− l2

l02 cos2θ0

)4n2πcos2θ0

=l0

4n2π

(1

ll0

cos2θ0− l

l0

), (7)

and thus

L ∝

(1

ll0

cos2θ0− l

l0

). (8)

According to [19], maximum contraction occurs when θ is54.7 . For a typical initial winding angle θ0 of 20 , thepressurized muscle will contract from a fully extended state,l = 1lo, to a fully contracted state, l = 0.615lo. Over thisdomain, the change in inductance predicted by eq. (8) isroughly linear with respect to l (r2 = .9935).

The change in resistance comes from a mechanism verysimilar to that of a strain gauge. If we assume constantresistivity, the resistance of the wires R should vary linearlywith the stain ε:

∆RR

= (1+2ν)ε, (9)

where ν is Poisson’s ratio [20]. This equation can be rewrit-ten with the initial resistance R0, the stress in the fibers σ ,and the elastic modulus E:

R =(1+2v)R0

Eσ +R0. (10)

Davis [2] showed that the tensile stress in the wires is afunction of the gauge pressure P, the diameter of the actuatorD, the length L, the number of turns each fiber makes n, thecross-sectional area of each fiber A, and the total number offibers N:

σ =PDL2nNA

. (11)

By using the expressions for D and l, (eqs. (3) and (2)) thisequation can be written in terms of the winding angle θ andthe fiber length b:

σ =Pb2 sinθ cosθ

2πn2A. (12)

Using a form of the force equation reported by Schulte [19],we can define the gauge pressure P in the muscle in termsof the force output F :

F =Pb2(3cos2θ −1)

4πn2 (13)

Substituting eq. (13) for the presssure term P in eq. (11)yields an expression for the stress σ that is linear with respectto the output force F and explicitly defined in terms of thewinding angle θ . Because the winding angle θ and the length

l are directly related (2), we can write this as a function g(l)of the muscle length:

σ = F 2NA

cosθ sinθ

3cos2θ −1= F 2

NA g(l) (14)

This allows us to rewrite (10) in terms of F and l.

R =2(1+2ν)R0

ENAFg(l)+R0 (15)

This predicts a linear relationship between resistance andforce output at a given contraction length. The gradient ofthis relationship depends on g(l), and increases as the musclecontracts.

III. METHODS

A. PAM Construction

While typical PAMs are made with nylon fiber braids[5], our unique design requires the reinforcing fibers to beconductive and insulated from one another. Because ourprototypes were manufactured by hand, the right-handedand left-handed helices were not braided (i.e. interwoven).Instead, the two sets were layered and embedded in siliconeto maintain their form. This technique has been used withother McKibben muscles and does not substantially changetheir function [21]. The muscles were constructed by winding32 AWG, stranded copper wire with silicone insulationaround a silicone tube with a nominal 9.5 mm diameter. Twomuscles were constructed for use in this study: one with aninitial angle of 20 and a relaxed length of 29 cm and anotherwith an initial angle of 30 and a relaxed length of 30 cm.The initial angle is the angle between the wires and the longaxis in the relaxed configuration. The ends of the wires wereconnected so that the path of the current always circled theaxis of the muscle in the same direction. For simplicity ofconstruction, the PAMs we wound consist of only 8 wiresper layer.

B. Test Stand and Electrical Measurements

To test the contraction of the PAMs, a custom test-standwas created. This test stand (Fig. 2) allows us to automati-cally control the pressure in the PAM and measure musclecontraction. To measure the inductance and resistance weused a 4320 Wayne Kerr LCR Meter connected to a standarddesktop computer. All electrical properties measurementswere taken at 100 kHz in the series mode. The 4-wire leadsfrom the LCR meter were connected to 2-wire test leadsconnected to the muscles with a nominal inductance andresistance of 0.36 µH and 123.49 mΩ for the 20 muscle and0.43 µH and 125.18 mΩ for the 30 muscle. A transducer(WIKA A-10, 60 PSI range) was used to measure the internalpressure of the PAM, and a string potentiometer (UniMeasureLX-PA, 10” range) was used to characterize the contraction.The string potentiometer created a nominal downward forceof 4.45 N.

(a) (b)

(c)

Pre

ssur

e tr

ansd

ucer

Srt

ing

Pot

enti

omet

er

PAM

SlidingCarriage

ElasticBands

Weights

Fig. 2: A custom made test stand (a) allows us to pressurizeand exhaust the PAM via Solenoid Valves, and automaticallyrecord the contraction and pressure of the air muscle. Watercontainers (b) create an adjustable load to test the actuatorunder different loading conditions. Similarly, elastic bands(c) create an arbitrary downward force to test the sensorcalibration.

C. Sensor Characterization

We tested the muscles under a load created by the weightof water we added to containers attached to sliding carriage(Fig. 2b). The mass of the water was increased to create adownward force that varied between 0 and 44.2 N. At eachforce level, the pressure in the PAM was gradually increasedfrom 0 to 200 kPa. Higher pressures were not attempted toavoid rupturing the PAM prototypes. The resulting lengths,inductance and resistance values were recorded at roughly 3Hz. When a force is applied to the unpressurized muscle, theforce causes the muscle to stretch and deform. This causesthe inductance to decrease (up to 2.5%). Once the muscle hasenough pressure to balance the load, the inductance returnsto normal values. For this reason, the electrical properties re-ported here are taken after the muscle has sufficient pressureto support the applied load.

Inductance: The inductance characterization of our twoprototypes is shown in Fig. 3. As our model predicts, theinductance increases linearly as the muscle contracts. Thesensitivity of the inductance to the contraction was found tobe 0.069 µH/mm for the 20 muscle and 0.072 µH/mm forthe 30 . This sensitivity was not strongly influenced by theapplied force. The largest recorded contractions were 18 mm(6% of total length). The contraction is small because of thepressure limitations of our hand-built prototypes.

Contrary to our predictions, the characterization showedthat inductance was also sensitive to the force that wasapplied to the muscle. This relationship was linear, with amuch lower sensitivity. The sensitivity was determined to be0.016 µH/N for the 20 muscle and 0.0088 µH/N for the30 muscle. The r2 values for linear regression of length andforce on inductance are 0.987 for the 20 muscle and 0.998for the 30 muscle. This sensitivity could come from the gapbetwen the two layers of wire compressing under high forces.

5.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

PAM contraction [mm]

Induct

ance

[µ

H]

−10 −5 0 5 10 15 204

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

Inducta

nce [µ

H]

(b) 30° muscle

45

0A

pp

lied

Fo

rce

[N

]

−10 −5 0 5 10 15 20

(a) 20° muscle

Fig. 3: Shown is the characterization of the inductance overcontraction for various applied loads. The inductance valuesof our prototypes show a strong, linear relationship betweenthe contraction state and the measured inductance.

Resistance: Equation (15) provides an explicit relationshipbetween contraction, applied force and the resistance valuesthat we measure. We found, however, that the resistancevalues resulting from a direct evaluation of this equation donot provide a good approximation for the resistance valuesR that we measure. Even when we re-write (15) with acoefficient k and we allow R0 to be a free variable as in(16), the equation is not readily scaled to fit the data:

R = k2(1+2v)R0

ENAFg(l)+R0 (16)

We optimized k and R0 to minimize the squared-error withour data. We found, for the 20 muscle, optimal valuesof k = 20.48 and R = 3.136Ω (r2 = 0.8135). For the 30

muscle, optimal values of k = 8.255 and R = 3.282Ω (r2 =0.8672) were identified. The optimal fit for the 30 muscle ispresented alongside the data from our experiments in Fig. 4.It is apparent that the simple models for force, strain and

resistance are not sufficient. Over the range of contractionthat our experiments cover, we would expect the functiong(l) to be roughly linear (r2 = 0.998). This appears to matchthe trend in the data. Reesistance does not have a stronglinear relationship with the force. A much better predictioncan be obtained with a model that is quadratic in force:

R = c1F + c2F2 + c3L+R0 (17)

This leads to a much better characterization of the resistanceresponse of the muscle with an r2 value of .9805 for the 20

3.32

3.31

3.3

3.29

3.28

403020100-505 10

15

Re

sist

an

ce [Ω

]

Force [N]Contraction [mm]

Fig. 4: Measured resistance values are shown for the 30

muscle as a function of muscle contraction and muscle load.Due to nonlinearities in the strain-resistance relation, thetheoretical prediction (15) is not very accurate.

3.32

3.31

3.3

3.29

3.28

403020100-505 10

15

Re

sist

an

ce [Ω

]

Force [N]Contraction [mm]

Fig. 5: A model with a quadratic force-term (as in eq. (17))predicts resistance with a r2 value of 0.9882. (Shown for the30 muscle)

muscle and .9882 for the 30 muscle (Fig. 5).One possible explanation for the breakdown of the lin-

earizing assumptions in the resistance versus strain equation(10). The change in resistance we see in our experimentsis orders of magnitude higher than the changes one wouldexpect to see in a typical strain gauge. The disparity mayalso be tied to the high frequencies at which we characterizethe series resistance. Nevertheless, the resistance does changepredictably, even if not as predicted by the usual models.

D. Sensor Calibration

Because each of the electrical properties exhibits sensitiv-ities to both force and length, it was necessary to generate2D calibration surfaces that we could use to predict the state.The results from the weighted trials were processed with thecurve-fitting tool in MATLAB to generate polynomial fitsthat map the electrical values to the length of the muscleand the force on the carriage. For the length, we used alinear fit (Fig. 6). The r2 values for these fits are 0.9937 and

0.9994 for the 20 and 30 muscles respectively. For theforce calibration, we used a fit that was linear with respectto the inductance and quadratic with respect to the resistance(Fig. 7). This is because of the quadratic relationship weobserved between resistance and force during our sensorcharacterization. The r2 values for these fits are 0.9680 and0.9950 for the 20 and 30 muscles respectively.

20

10

0

-5Con

trac

tion

[mm

]

-10

5

15

3.31 3.3053.285 5.6 5.83.2953.3 3.29 6 6.2 6.4 6.6 6.8

Resistance [Ω]

Inductance [µH]

Fig. 6: A linear fit maps inductance and resistance tomuscle contraction nearly perfectly (r2 = 0.9994 for the 30

muscle.)

50

30

10

0

-203.31 3.305

3.285 5.6 5.8 Resistance [Ω]

For

ce [N

]

Inductance [µH]3.295

-10

3.3 3.29 6 6.2 6.4 6.6 6.8

20

40

Fig. 7: A fit that is linear in inductance and quadratic inresistance predicts muscle force with an r2 value of 0.9950for the 30 muscle.

E. Verification

To certify that the fit could indeed be used to determine thelength of the actuator under realistic loading conditions, weattached elastic tubing to the sliding carriage of the test standto create an arbrtirary, spring-like downward force (Fig. 2c).We compared the length determined from the calibration fitto the length measured by the string potentiometer. We testedthe results at two levels of tension. The verification tests werecarried out on the 30 muscle. Because of variations in theresistance of the meter connection, we calibrated for staticresistance at the beginning of each trial. The different forcelevels from the elastic tubes caused the muscle to contract todifferent lengths for the same input pressure (200 kPa). The

calibration fit allowed us to very closely track the contractionof the air muscle even under the varying external conditions(see Fig. 8). There are a couple ”spikes” in the data (Fig.8a, at 20 and 30 seconds). We believe these are caused bythe LCR meter failing to react to the dynamic changes inthe inductance. We believe that the resolution of the positionsensing is well within the precision requirements of PAMcontrol. PAMs, by their nature, are difficult to control withmilimeter accuracy.

0 40 80 120 160 200

0

5

10

15

−5

Co

ntr

actio

n [

mm

]

Actual MeasuresPredicted Values

Actual MeasuresPredicted Values

0

5

10

15

−5C

ontr

act

ion [m

m]

0 40 80 120 160 200time [s]

(a)

(b)

Fig. 8: Ssensor verification: After calibration, the sensor isable to very closely predict true muscle contraction, evenunder varying unkown loads. In this example, the 30 muscleis antagonized by (a) rubber bands with low tension and (b)rubber bands with high tension.

IV. DISCUSSION

The inductance and resistance of our custom-made PAMbraids provide very good estimates of muscle contractionand muscle force. As can be seen in Fig. 8, our calibrationallows us to predict the length of our prototypes to within.5 mm under a changing load. Similarly, the force outputof the muscle can be predicted. Our calibration curve forforce could predict the applied force values to within 5 N.The combination of these two sensing capabilities creates theopportunity for this technology to be used in both force andposition feedback control. This technology has the potentialto enable robots and assistive devices to be built without the

added weight, cost, and complexity of external encoders andforce transducers.

Though the absolute value of the inductance of oursimplified prototypes is very small, the sensitivity of ourmethod is comparable to other techniques. Table I is a briefcompilation of work in the field of self-sensing elastomersfor comparison. The table uses the sensitivity metric S, forcomparison:

S =∆p

∆ε

, (18)

where ∆p is the percent increase in the electrical propertyfrom one strain level to another and ∆ε positive strainbetween those two points. Those elastomers that rely onlyon changes in resistance are often able to achieve veryhigh sensitivities but, because of their dependence on themechanical integrity of the elastomer, are susceptible to longterm inaccuracy from creep and fatigue [14]. Dielectric Elas-tomers, or those relying on changes in capacitance, appearto be more stable but exhibit lower sensitivities to deflection.For example, experiments by Goulbourne et al. suggest thata Mckiben muscle with a dielectric elastomer in the bladdercould expect roughly a 25% increase in capacitance acrossthe extremes of a 25% strain [18]. The sensitivity metricof the 20 muscle is 4.63 and the sensitivity metric of the30 muscle 3.61. Though we were not able to characterizethe sensitivity of our method for very large strains, our twoprototypes lead us to believe that the sensitivity will notdecrease sharply as the winding angle increases. Our methodis thus more sensitive than elastomeric capacitive methodsand less prone to creep than elastomeric strain sensors basedon resistance. While the results of this work have focusedon McKibben muscles, the principles governing the changein inductance and resistance can extend to a broader class offiber-reinforced soft actuators [21], [22]. The low samplingrate of our instrument limited our ability to characterizethe dynamic response of our sensor, and prevented us fromapplying its output to a feedback control loop. Our futurework will thus focus on the development of the necessaryelectric circuitry that will allow us to quickly characterize thechanges in inductance and resistance, and use this sensingtechnique in the closed-loop control of robotic systems withsoft fluidic actuators.

TABLE I: Comparison of sensitivities of high-strain sensingtechniques. For fair comparison, only the positive propertychanges are used regardless of whether the property is pos-itively or negatively correlated with strain. When possible,the metrics use explicit values from the text. Otherwise, thevalues are taken from generous inspection of the figures.

Author Property SZhu[14] Resistance 600

Felt (this work) Inductance 4.6, 3.6O’Brien[23] Resistance 4.5Lorussi[24] Resistance 2.3

Benslimane[25] Capacitance 1.5Goulbourne[18] Capacitance 1StretchSense[26] Capacitance 0.79

ACKNOWLEDGMENT

Thanks to Joshua Bishop-Moser for his instruction onthe method of manually winding PAMs with different braidangles.

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