Methods to correct for creep in elastomer-based sensors

Kim Le Phan Research / Corporate I&T

NXP Semiconductors High Tech Campus 04

5656AE Eindhoven, the Netherlands Email: kim.le.phan@nxp.com

Abstract— Elastomer materials such as PDMS and SU-8 have been used as mechanical springs in sensors and actuators. A disadvantage of elastomer-based structures is that they slightly creep with time, especially for low frequency and static applications, which causes hysteresis up to a few percents in the sensor signal. In this paper, by thoroughly studying the creep process and model, we propose some novel signal processing methods to correct for the creep in real time. The methods have been successfully applied in an elastomer-based magnetoresistive accelerometer, which shows that the error due to creep has been reduced from 3% down to 0.08%.

I. INTRODUCTION Nowadays elastomer materials have become increasingly

popular in modern micromachine technologies as an alternative to silicon, due to their ease of fabrication and low cost and biological compatibility in some applications [1]. Some research groups including ours have reported using those materials as mechanical springs in some kinds of sensors such as accelerometers, pressure and tactile sensors, and in actuators [2-7]. We have previously reported a biaxial magnetoresistive accelerometer using polydimethylsiloxane (PDMS) structures as the mass-spring system and magnetoresistive (MR) sensors for signal readout [2].

Fig. 1 briefly explains the construction and working principle of the device. On a Si substrate, two Wheatstone bridges of MR sensors for detecting acceleration in X and Y directions are arranged in radial directions. On top of the sensor substrate, a tiny mushroom-shaped structure made of PDMS is mounted at the center of the MR sensor

configuration. The mushroom consists of two parts: the upper massive part (the mushroom cap) is a permanent polymer magnet made of PDMS filled with Sr/Ba-ferrite particles and the lower thin part (the mushroom stem) is made of pure PDMS. The cap and the stem of the mushroom thus form a mass-spring system. When there is no acceleration the mushroom stands upright. The projection of its magnetic stray field on the sensor substrate forms a symmetrical radial pattern, and the Wheatstone bridges give zero output voltage. When a lateral acceleration (in the X-Y plane) is applied to the accelerometer, the cap tilts elastically a fraction of a degree. The symmetry is therefore broken and consequently non-zero X and Y signals are obtained indicating the 2D acceleration vector on the sensor plane.

Elastomer materials such as PDMS belong to the group of materials called viscoelastic polymers. This material behaves partly as an elastic solid at short time range and partly as a viscous liquid at long time range. PDMS is famous for its excellent elasticity and flexibility among other types of elastomers [8]. However, by nature PDMS also possesses some small amount of viscous liquid behavior, and exhibits a creeping behavior with time, which may result in hysteresis up to 3-7 % of full signal scale. By studying the creep behavior and creep model, we developed in this work several methods to correct for creep in real time for static and low frequency applications. In this paper, the elastomer-based MR accelerometer is used as a vehicle to demonstrate the methods. Our generic correction methods can be used not only for this device but also for any type of sensor that employs elastomer structures as deflection elements (such as pressure, force, tactile sensors, etc.).

II. CREEP MODEL In order to develop correction methods, the creep phenomenon should be well understood. In Fig. 2 a schematic creep phenomenon is presented. Suppose that an elastomer structure is subjected to a multi-step loading program, in which incremental stresses Δσ1, Δσ2… are added at times T1, T2,… respectively. The total strain at time t, e(t), is a sum of an instantaneous term (first term) and a creep term which is time dependent [9]:

Elastic magnet (ferrite filled PDMS)

MR sensorsSubstrate

N

S

MM

MM

MM

MM

Pure PDMS

Bridge X

Bridge Y

Bridge X

Bridge Y

(a) Top view of MR sensors (b) Side view of the device

Fictitiousforce

Acceleration

Magnetization

Center of radial fieldat zero acceleration

Figure 1. Principle of the elastomer-based magnetoresistive accelerometer.

1-4244-2581-5/08/$20.00 ©2008 IEEE 1119 IEEE SENSORS 2008 Conference

)()()( teJtte creepins +σ= (1)

The first term is proportional to the stress (load) σ(t), which may be varied with time. Jins is the instantaneous compliance (reciprocal of elastic modulus) which is time-independent. The second term ecreep(t) is the time-dependent (creep) part of the strain even when the load is fixed. According to Boltzmann’s superposition principle, the creep term is a function of the entire past loading history of the structure. Each loading step makes an independent contribution to the final deformation (strain) and the amount of creep at each loading step is proportional to the increment of load at that step. To a first approximation, the creep term can be expressed in terms of exponential functions of time (the Voigt model):

( )[ ]τ−−−σΔβ=−σΔ= ∑∑ /)(exp1)()( ii

ii

iicreep TtTtJte ,(2)

where J(t-Ti) is the creep compliance function which is time-dependent, Δσi is incremental stress at time step i, β is a proportional factor and τ is the retardation time. Combining (1) and (2) we finally derive:

( )⎥⎦

⎤⎢⎣

⎡ τσΔτ−−σβ+σ= ∑i

iiins TttJtte )/exp()/exp()()()( (3)

III. CREEP CORRECTION USING ITERATIVE LOOPS

A. Subtractive method Based on the creep model, a correction method using

iterative loops during post signal processing is proposed. In this method, the creep part ecreep(t) is subtracted from the measured strain e(t). That means the corrected strain, ecor(t), only reflects the instantaneous strain, σ(t)Jins, which is directly proportional to the load. From (3) we have:

⎥⎦

⎤⎢⎣

⎡ τΔτ−−β−= ∑i

icorcorins

cor TtetteJ

tete )/exp()()/exp()()()( (4)

Due to the principle of the sensor, the signal of the sensor is proportional to the strain of the elastomer structure. That means signal S(t)=αe(t), in which α is a calibration factor. From (4), we can obtain the following expression:

⎥⎦

⎤⎢⎣

⎡ τΔτ−−β−= ∑i

iicorcorrawcor TTSttStStS )/exp()()/exp()(')()( , (5)

where β’ = αβ/Jins and Sraw(t) is the raw signal before correction. The sensor signal can be corrected in iteration loops. At every iteration i, the raw signal is recorded and corrected:

⎢⎢⎣

⎡β−= icorirawicor tStStS )(')()(

( ) ⎥⎦

⎤τ−τ−− ∑

=−

i

jjjcorjcori ttStSt

01 )/exp()()()/exp( , (6)

in which j is the index of the sum that accumulates over the present and past iterations. A problem of (6) is that Scor(ti) appears in both sides of the equation. To resolve this, we proceed further by splitting the last term of the sum:

( )[ ]( )[ ])/exp()()()/exp('

)/exp()()()/exp('

)(')()(

1

1

01

τ−τ−β+

τ−τ−β+

β−=

−

−

=−∑

iicoricori

i

jjjcorjcori

icorirawicor

ttStSt

ttStSt

tStStS

And finally we derive:

⎢⎣

⎡β−= −1 )(')()( icorirawicor tStStS

( ) ⎥⎦

⎤τ−τ−− ∑

−

=−

1

01 )/exp()()()/exp(

i

jjjcorjcori ttStSt (7)

Note that in (7) the second terms (correction term) contains all values taken from j=0 to j=i−1, i.e. signals from past (known) iterations.

Due to the fact that the creep behavior depends on the loading history, the measurement has to start from a known state. The best known state for this method is the unloaded state. In this state the corrected signal equals the raw signal. Starting from this initial state, the sensor signal is continuously sampled and corrected using (7). A simulation to demonstrate the correction method is given in Fig. 3a. The correction method works for both loading and unloading situations.

timeT1 T2 T3 T4 T5 T6

Δσ1

Δσ2

Δσ3 Δσ4

Δσ5

Δσ6

Stre

ss σ

Stra

in e

creepinstantaneous strain

time

Figure 2. Schematic creep phenomenon of an elastomer structure. Theupper/lower graphs show the stress/strain profiles with time, respectively.

Time (a.u.)

Sig

nal (

a.u.

)

Raw signalCorrected signal

Loadingsteps

Unloadingsteps

(a)

Time (a.u.)

Sig

nal (

a.u.

)

Raw signalCorrected signal

Loadingsteps

Unloadingsteps

(b)

Figure 3. (a) Simulations of the subtractive correction method. The creeppart is trimmed off from the raw signal, leaving the corrected signal (dashedline) directly proportional to the loads. (b) Simulation of the supplementalcorrection method. The corrected signal always tries to predict the saturationstate (i.e. the state when creep saturates).

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The Voigt model as explained in (2) represents creep to the first approximation. In reality, creep compliance is best described as a superposition of many exponential functions:

[ ])/exp(1)( kk

k tctJ τ−−β= ∑ (8)

Using the creep compliance in (8), the correction as given in (7) can be rewritten as:

∑ ⎢⎣

⎡β−= −

kicorkirawicor tSctStS 1 )(')()(

( )∑ ⎥⎦

⎤τ−τ−−

−

=−

i

jkjjcorjcorki ttStSt

1

01 )/exp()()()/exp( , (9)

which describes better the saturation state of creep. An example of experimental results is given in Fig. 4. In this experiment, the signal of the accelerometer has been corrected using a sum of three exponential functions of different coefficients ck and retardation times τk, which had been derived in advance from fitting the time-dependent creep curve of the same device. The shorter the measurement time scale, the higher number of exponential functions in the superposition is needed to ensure a good correction. However, in practice, it has been found that the number of exponential functions does not have to exceed about five.

B. Supplemental method An alternative method is to predict the saturation states of

creep, rather than to trim the creep off from the raw signal. That means the supplemental part of the creep is added to the raw signal in order to reach, at any time, the saturation states:

( )∑ τΔτ−β+=i

iicorrawcor TTSttStS )/exp()()/exp(')()( (10)

The sensor signal can also be corrected in iterative loops:

)()( irawicor tStS =

( )[ ]∑=

− τ−τ−β+i

jjjcorjcori ttStSt

01 )/exp()()()/exp(' (11)

A problem of (11) is that Scor(ti) appears in both sides of the equation and this is not possible to eliminate as in the subtractive method. To resolve this, we assume that the correction at any iteration step (i) is calculated for all previously loading steps (0 to i−1), except for the current one. Hence the corrected signal at any iteration step i can be approximated as:

)()( irawicor tStS =

( )∑−

=−− τ−τ−β+

1

011 )/exp()()()/exp('

i

jjjcorjcori ttStSt (12)

This means that the correction term (second term) lags one step behind the real signal. When the sampling frequency is notably larger than the frequency range of the measurement, this approximation can be made without significant error. A simulated illustration is given in Fig. 3b.

Correction using a superposition of multiple exponential functions now becomes:

∑ ∑⎢⎣

⎡τ−β+= −

kkikirawicor tctStS 1 )/exp(')()(

( )∑ ⎥⎦

⎤τ−

−

=−

i

jkjjcorjcor ttStS

1

01 )/exp()()( (13)

This correction method has several advantages compared to the subtractive method. Firstly, at saturation states, Scor(t) = Sraw(t), meaning that the sensor can start at any saturation state. For instance for the tilt measurement, the accelerometer can start at any position, including the unloaded position, provided that the creep has been saturated at that position. Secondly, in general the difference between Scor and Sraw is smaller than the subtraction method, which results in minimal error if the previous condition cannot be (completely) met. This method has been successfully implemented (see Fig. 5). Using this correction method, characteristic curves of the sensor now exhibit hysteresis of merely 0.08 %, compared to more than 3% without correction (Fig. 6).

IV. CREEP CORRECTION USING RC CIRCUIT In a circuit consisting of a resistor R and a capacitor C

connected in series, voltage on the capacitor during charging can be written as: V = VB[1-exp(-t/τRC)], in which τRC=RC is the time constant of the RC circuit, VB is the bias voltage on the circuit. It is straightforward to recognize that charging a capacitor in a RC circuit is analogous to the creep behavior according to the Voigt model (2). The time constant τRC=RC is analogous to the retardation time of the elastomer material. This suggests that creep can be corrected using a RC circuit.

The simplest form of an RC correction circuit is the subtractive method, using a single exponential function. Fig. 7 presents such a circuit. The output of this circuit can be written as:

)/exp(21

1

21

2RCininout t

RRRV

RRRVV τ−

++

+= , (14)

Time (s)0 100 200 300 400 500 600

Sign

al (V

)

-2e-5

0

2e-5

4e-5

6e-5

8e-5

1e-4

Time (s)

0 50 100 150 200 250 300

Sig

nal (

V)

7.0e-5

7.2e-5

7.4e-5

7.6e-5

7.8e-5

8.0e-5

8.2e-5

Corrected

Raw

Corrected

Raw

Loading

Unloading

Zoomed-in graph

Figure 4. Experimental results using the subtractive correction methodimplemented in the accelerometer. The raw signal clearly shows creep afterloading and unloading steps, that takes minutes to saturate. The correctedsignal remains substantially flat (no significant time dependence).

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with the time constant )/( 2121 RRRCRRC +=τ .

For the sake of simplicity, suppose that the input signal, which is the raw signal from the sensor, is a signal due to a single loading step. The signal can be written as:

[ ])/exp(10 τ−−β+= tSSV insinsin , (15)

where the first term, Sins, is the signal due to the instantaneous deflection of the elastomer structure, and the second term is the creep part; β0 is the ratio of creep at infinite time to the instantaneous deflection. In reality , β0 is in the order of a few percents. To correct for creep, the creep part should be precisely subtracted from the measured signal by tuning the charge-discharge behavior of the circuit to match the creep behavior. To do that, the time constant of the RC circuit (14), τRC , is chosen (by tuning R1, R2 and C) to be equal to the retardation time τ in (15). In practice, the retardation time τ is in the order of 20-100 s. Substituting (15) into (14) we finally derive:

)1( 021

2 β++

=RR

RSV insout, (16)

with conditions that τRC = τ and )1/(/ 0021 β+β=RR . This formula states that the output signal from the correction circuit, when the input signal exhibits creep, is proportional to the instantaneous signal and is not dependent on time.

V. CONCLUSIONS Starting from the creep model that states that creep of an

elastomer structure is basically an exponential function of time and depends on the load history, we have proposed some novel signal processing methods to correct for creep in elastomer-based sensors in real time. The first correction method is the subtractive method, in which the creep part of the signal is subtracted from the raw signal. The second correction method is the supplemental method, that is, the supplemental part of the creep is added to the raw signal in order to reach, at any time, the saturation states. The last method is to insert an RC circuit in the readout circuit of the sensor, since charging a capacitor in the RC circuit is analogous to the creep behavior. Those methods can be implemented in the signal processing algorithm or electronic circuit of the sensors. Two from the three methods have been successfully applied in an elastomer magnetoresistive accelerometer, which shows that the error due to creep has been reduced from more than 3% down to 0.08%.

REFERENCES [1] S.R. Quake and A. Scherer, “From micro- to nanofabrication with

soft materials”, Science, vol. 290, pp. 1536-1540, 2000. [2] K.L. Phan, A. Mauritz, and F.G.A Homburg, “A Novel Elastomer-

Based Magnetoresistive Accelerometer”, Sensors & Actuators A, vol. 145-146, pp. 109-115, 2008

[3] J.C. Lötters, W. Olthuis, P.H. Veltink and P. Bergveld, “Theory, technology and assembly of a highly symmetrical capacitive triaxial accelerometer”, Proceedings of MEMS, pp. 31-36, 1997.

[4] J. Kim, S. Yang, J. Baek, S. Park, H. C. Kim, E-S. Yoon, and K. Chun, “Cardiomyocytes self-powered polymermicrorobot”, Proceedings of Transducers 07, pp. 1405-1408, 2007.

[5] W. Wang, J. Yao, J. C. Chen, and J. Fang, “Composite elastic magnet films with hard magnetic feature”, J. Micromech. Microeng., vol. 14, pp. 1321–1327, 2004.

[6] T. Pan, S. J. McDonald, E.M. Kai, and B. Ziaie, “A magnetically driven PDMS micropump with ball check-valves”, J. Micromech. Microeng. vol. 15, pp. 1021–1026, 2005.

[7] M. Feldmann and S. Büttgenbach, “Novel microrobots and micromotors using Lorentz force driven Linear microactuators based on polymer magnets”, IEEE Trans. Magn., vol. 43, pp. 3891-3895, 2007.

[8] J.C. Lötters, W. Olthuis, P.H. Veltink and P. Bergveld, “The mechanical properties of the rubber elastic polymer polydimethylsiloxane for sensor applications”, J. Micromech. Microeng., vol. 7, pp. 145-147, 1997.

[9] I.M. Ward, J. Sweeney, “An introduction to the mechanical properties of solid polymers”, 2nd ed., John Wiley & Son, Ltd., 2005.

Time (s)0 100 200 300 400 500 600

Sig

nal (

V)

0

1e-4

2e-4

3e-4

4e-4

5e-4

6e-4

7e-4

Time (s)

0 50 100 150 200 250 300

Sig

nal (

V)

6.2e-4

6.3e-4

6.4e-4

6.5e-4

6.6e-4

6.7e-4

6.8e-4

Raw

CorrectedCorrected

Raw

Zoomed-ingraph

Loading

Unloading

Figure 5. Experimental results of the supplemental method, using threeexponential functions. At the loading step the sensor is loaded to 1 g. The raw signal was continuously creeping after changing load, while thecorrected signal was immediately stabilized at the equilibrium state.

Acceleration (m/s2)-10 -5 0 5 10

Sig

nal (

V)

-1e-4-1e-4-8e-5-6e-5-4e-5-2e-5

02e-54e-56e-58e-5

Acceleration (m/s2)-10 -5 0 5 10

Sign

al (V

)

-1e-4-8e-5-6e-5-4e-5-2e-5

02e-54e-56e-58e-51e-4

Hysteresis ~3%

(a) Uncorrected (b) Corrected (supplemental)

Hysteresis 0.08%

Figure 6. Signal of the accelerometer is recorded as a function ofacceleration (sweeping up and down between -1g and +1g). Withcorrection, hysteresis is reduced from more than 3% to 0.08%.

ττ =RCR1

C

Vin

Vout

Conditions:

)1( 021

2 β++

=RR

RSV insout

)1/(/ 0021 ββ +=RRRC

Figure 7. An RC circuit for creep correction. The raw signal is supposed to be fed to the input of the circuit and the corrected signal can be obtained from the output.

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