methods to correct for creep in elastomer-based sensors

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: [email protected] 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 Elastic magnet (ferrite filled PDMS) Fictitious Magnetization force

Bridge X M M

M

N

M

Pure PDMS Bridge Y

S

Bridge Y

M

M M

Substrate

MR sensors

M Center of radial field

Acceleration at zero acceleration Bridge X (a) Top view of MR sensors (b) Side view of the device

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

1-4244-2581-5/08/$20.00 ©2008 IEEE

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 elastomerbased 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]:

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IEEE SENSORS 2008 Conference

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):

ecreep (t ) = ∑ Δσ i J (t − Ti ) = β∑ Δσ i [1 − exp(− (t − Ti ) / τ)] ,(2) i

i

where J(t-Ti) is the creep compliance function which is timedependent, 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:

⎡ ⎤ Scor(t) = Sraw(t) − β' ⎢Scor(t) − exp(−t / τ)∑ΔScor(Ti ) exp(Ti / τ)⎥ , (5) i ⎣ ⎦ 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:

⎡ S cor (t i ) = S raw (t i ) − β' ⎢S cor (t i ) ⎢⎣ i ⎤ − exp(−t i / τ) ∑ (S cor (t j ) − S cor (t j − 1 ) ) exp(t j / τ) ⎥ , (6) j =0 ⎦

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:

S cor (t i ) = S raw (t i ) − β' S cor (t i ) i −1

j =0

⎡ ⎤ e(t ) = σ(t ) J ins + β⎢σ(t ) − exp(−t / τ)∑ (Δσ i exp(Ti / τ))⎥ (3) i ⎣ ⎦

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:

β ⎡ ⎤ ecor (t ) = e(t ) − ecor (t ) − exp(−t / τ)∑Δecor (t ) exp(Ti / τ)⎥ (4) J ins ⎢⎣ i ⎦

And finally we derive:

⎡ S cor (t i ) = S raw (t i ) − β' ⎢ S cor (t i −1 ) ⎣

Δσ3

i −1 ⎤ − exp(−ti / τ)∑ (Scor (t j ) − Scor (t j −1 ) )exp(t j / τ) ⎥ (7) j =0 ⎦ 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.

Δσ4

Δσ2

Loading steps

Δσ5

Signal (a.u.)

Stress σ

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:

Δσ6

Δσ1 Strain e

time

Unloading steps

(a) Raw signal Corrected signal Time (a.u.)

creep instantaneous strain

T1

T2 T3

T4 T5

T6

time

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

]

+ β' exp(−t i / τ)[(S cor (t i ) − S cor (t i −1 ) ) exp(t i / τ)]

CREEP CORRECTION USING ITERATIVE LOOPS

III.

[

+ β' exp(−t i / τ)∑ (S cor (t j ) − S cor (t j −1 ) ) exp(t j / τ)

Loading steps Signal (a.u.)

(1)

e(t ) = σ(t ) J ins + ecreep (t )

Unloading steps

(b) Raw signal Corrected signal Time (a.u.)

Figure 3. (a) Simulations of the subtractive correction method. The creep part is trimmed off from the raw signal, leaving the corrected signal (dashed line) directly proportional to the loads. (b) Simulation of the supplemental correction method. The corrected signal always tries to predict the saturation state (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:

J (t ) = β∑ ck [1 − exp(−t / τ k )]

(8)

k

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

i −1

+ β' exp(−ti−1 / τ)∑ (S cor (t j ) − S cor (t j −1 ) )exp(t j / τ)

⎡ S cor (t i ) = S raw (t i ) − β' ∑ c k ⎢ S cor (t i − 1 ) k ⎣

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:

S cor (t ) = S raw (t ) + β' exp( −t / τ)∑ (ΔS cor (Ti ) exp(Ti / τ) ) (10) i

The sensor signal can also be corrected in iterative loops:

Scor (ti ) = S raw (ti )

[

]

+ β' exp(−ti / τ)∑ (S cor (t j ) − S cor (t j −1 ) )exp(t j / τ) (11) j =0

1e-4

Raw

Signal (V)

Unloading

4e-5

Signal (V)

Corrected

6e-5

Zoomed-in graph

8.0e-5 7.8e-5

Raw

7.6e-5

Corrected

7.4e-5 7.2e-5 7.0e-5 0

Loading

2e-5

50

100 150 200 250 300

Time (s) 0 -2e-5 0

100

200

300

400

500

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:

⎡ S cor (ti ) = S raw (ti ) + β' ∑ ⎢ck exp(−ti −1 / τ k )∑ k ⎣ i −1 (S cor (t j ) − S cor (t j −1 ) )exp(t j / τ k )⎤⎥ ∑ j =0 ⎦

(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.

8.2e-5

8e-5

(12)

j =0

i −1 ⎤ − exp(−ti / τk )∑ (Scor (t j ) − Scor (t j −1 ) )exp(t j / τ k )⎥ , (9) j =0 ⎦ 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.

i

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: Scor (ti ) = S raw (ti )

600

Time (s)

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

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:

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Vout = Vin

R2 R1 + Vin exp(−t / τ RC ) , R1 + R2 R1 + R2

(14)

Corrected

7e-4

Signal (V)

Raw

5e-4 Unloading

Signal (V)

6.7e-4

6e-4

C

6.8e-4

4e-4

6.6e-4

6.4e-4

Zoomed-in graph

R1

6.2e-4 0

50 100 150 200 250 300

Loading

2e-4

Vout = S ins

Raw

6.5e-4

6.3e-4

3e-4

Vout

Corrected

Time (s)

R2 (1 + β 0 ) R1 + R2

Conditions: τ RC = τ R1 / R2 = β 0 /(1 + β 0 )

Vin

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.

1e-4 0 0

100

200

300

400

500

600

V.

Time (s)

CONCLUSIONS

with the time constant τ RC = CR1 R2 /( R1 + R2 ) .

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%.

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:

[1]

8e-5 6e-5 4e-5 2e-5 0 -2e-5 -4e-5 -6e-5 -8e-5 -1e-4 -1e-4

(a) Uncorrected

Hysteresis ~3%

-10

-5

0

5

2

Acceleration (m/s )

Signal (V)

Signal (V)

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

10

1e-4 8e-5 6e-5 4e-5 2e-5 0 -2e-5 -4e-5 -6e-5 -8e-5 -1e-4

(b) Corrected (supplemental)

Hysteresis 0.08% -10

-5

0

5

2

10

Acceleration (m/s )

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

Vin = Sins + β0 Sins [1 − exp(−t / τ)] ,

(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:

Vout = S ins

R2 (1 + β0 ) , R1 + R2

REFERENCES [2]

[3]

[4]

[5]

[6]

[7]

(16)

with conditions that τRC = τ and R1 / R2 = β0 /(1 + β0 ) . 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.

[8]

[9]

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