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Force Control of Shape Memory Alloy Wire Using Fuzzy Controller

Omid Rohani∗a, Aghil Yousefi-Komaa, Ayyoub Rezaeeiana, Alireza Doosthoseinia aAdvanced Dynamic and Control Systems Lab. (ADCSL), School of Mechanical Engineering,

University of the Tehran, Tehran, Iran

ABSTRACT

An experimental setup is designed and fabricated to measure the force induced by voltage in an SMA wire. Using autoregressive model with exogenous input (ARX) method for system identification of the experimental data, two appropriate transfer functions of the force in SMA wire versus the applied voltage during each of heating and cooling processes were derived. Afterwards, a conventional PID controller and a self-tuning fuzzy PID controller were designed to control the force in SMA wire. The latter control algorithm is used by tuning the parameters of the PID controller thereby integrating fuzzy inference and producing a fuzzy adaptive PID controller, which is used to improve the force control performance. The responses of the system with the both designed controllers for different inputs are simulated and compared to each other. At the end, simulation results show that in force control of the SMA wire, self-tuning fuzzy PID controllers are more efficient than conventional PID controllers.

Keywords: SMA, force control, system identification, fuzzy controller, PID

1. INTRODUCTION Shape Memory Alloys (SMAs) are a group of metallic materials that exhibit shape memory effect which occurs through a solid-state temperature and stress dependent shift in their crystalline structure between two different phases called Martensite and Austenite. At higher temperatures, the material is in the austenite phase and is relatively hard. As the temperature is lowered, the material changes to the martensite phase which is relatively soft and grows until sufficiently low temperatures. This unusual characteristic of SMA actuators has a wide variety of applications in control and actuation systems1, 2. SMA actuators have several advantages for miniaturization such as excellent power to mass ratio, maintainability, reliability, and clean and silent actuation. The disadvantages are low energy efficiency and limited bandwidth due to heating and cooling restrictions, degradation and fatigue.

Although there have been various kinds of grippers3, recently considerable attention is being given to the small-sized or miniaturized grippers. Some of the tasks of these types of grippers include the assembly of tiny work pieces in the semiconductor industry, the sample collection of microscopic observations in biochemical laboratory, and also in surgical tools and dentistry. Currently, substantial attention is given to micro-grippers that use SMAs in particular SMA wire as a means of actuation4, 5, and 6.

One of the most significant problems to develop such grippers is controlling the force induced in the SMA wire. The transformation between the phases of SMA exhibits a hysteretic effect, an inherent nonlinear phenomenon, in that the transformations on heating and on cooling do not overlap7. The hysteresis associated with SMAs makes the development of a mathematical or numerical model as well as designing a tracking controller for SMA materials and actuators a challenging task. This has motivated some researchers to conduct research into tracking control of SMA actuators with hysteresis compensation.

Compensation of hysteresis effect for SMA and piezoceramics8 has been addressed in some reports. Most of these works require a precise system model. This makes the controller synthesis complicated and time consuming. Usually the Preisach model of hysteresis based on phenomenological nature8-10is developed. The Preisach model is also inverted and then incorporated in an open-loop control system to achieve the desired output11. Song and Quinn12 have demonstrated the use of sliding-mode based robust controller for tracking control of an SMA wire. Elahinia and Ashrafiuon13 applied variable structure control to an SMA actuated manipulator.

∗ E-mail: [email protected]; Telephone: +1 (949) 981-0303

Modeling, Signal Processing, and Control for Smart Structures 2008, edited by Douglas K. LindnerProc. of SPIE Vol. 6926, 692611, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.776405

Proc. of SPIE Vol. 6926 692611-1

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S. L

Bending7rBeam

I I II 1111 LoadreIlI I

Semor

k ire

1

Choi et al.14 employed ∞H -controller for robust force tracking control of SMA-driven flexible gripper system. Shameli et al.15 suggested a PID and a novel PID-P3 controller to perform a position control of a miniature SMA actuator for precise application. It has been shown that fuzzy control provides a convenient method for constructing nonlinear controllers via the use of heuristic information16-18. Ahn and Nguyen19 presented a development of a SMA position control system by using a self-tuning fuzzy PID controller. This control algorithm was used by tuning the parameters of the PID controller in that way integrating fuzzy inference and producing a fuzzy adaptive PID controller.

In this paper, an ordinary PID controller and a self-tuning fuzzy PID controller are designed for controlling the force induced by voltage in a SMA wire clamped at two ends. The transfer function of the SMA wire has been found by system identification of data from the experimental setup made by the authors.

2. EXPERIMENTAL SETUP The test stand is shown in Fig. 1. The experimental setup includes a Nickel-Titanium (NiTi) SMA wire, a load-cell sensor, data acquisition terminal, a power amplifier and computer for recording data.

As for SMA wire specifications its diameter and length are 0.1mm, 32cm respectively, and its electrical resistance is 37.8Ω .

Fig. 1. Experimental test stand

For precise measuring of the tension force in SMA wire a setup as shown in Fig. 2 is designed. In order to obtain more precision, the employed load cell is of the bending beam type. Since the output signal power from DAC terminal is weak for heating the proposed SMA wire, a power amplifier should be used.

Fig. 2. Schematic experimental setup



SMA wire is clamped at two ends. When heated, due to transformation from martensite to ausenite phase, SMA wire tends to be shortened, but because it is clamped at two ends, a tension force is induced in it. By cooling the SMA, it returns to the martensite phase and so the induced force is released.

Using Matlab XPC target toolbox and D/A data acquisition terminal, different pulse inputs are applied to SMA wire and the response is recorded. Besides, the experimental setup provides the capability of force control of the SMA wire.

Thus, input of system is voltage and output is force. By Calibration, relation between voltage and force of the load-cell is obtained as Eq. (1) and shown in Fig. 3.

0.083+6.9664V = F (1)

FORCE_VOLT

y = 6.9664x + 0.083

00.5

11.5

22.5

33.5

44.5

0 0.1 0.2 0.3 0.4 0.5 0.6

Volt

Forc

e

Fig. 3. Load cell calibration equation

3. SYSTEM IDENTIFICATION The system identification problem is to estimate a model of a system based on observed input-output data. Several ways to describe a system and to estimate such descriptions exist. Parametric models can describe the true process behavior exactly with finite number of parameters20. In this paper, Autoregressive model with exogenous input (ARX) is considered, which is described by:

)()()()()( kekuqBkyqA += (2) Where )(ku is the input signal, )(ky is the output signal, and )(ke is the white noise. )(qA and )(qB are defined as,

( ) annaqaqaqA −− +++= ...1 1

1 (3)

bb

nn qbqbqB −− ++= ...)( 1

1 (4)

Where an and bn determine the model order.

The input signal )(ku of the process under consideration plays an important role in system identification. Clearly, the input signal is only possibility to influence the process in order to gather information about its behavior. Step input is well suited for identification 20.

It is necessary to test the system response of the SMA wire, as an open loop for both heating and cooling21.The experimental result of the pulse input response is shown in Fig. 4.

The raw signal from the load cell consists of significant noise component. In order to remove the noise, a low pass digital filter is designed to filter the output force signal of the data collected. However, the parameters of filter may introduce new errors or bias into the measurements.

By tuning the parameters of low pass filter, the filtered data is as closer as possible to the effective signal. The effect of the filter is shown in Figs. 4, 5.



0.8

0.5

0.4

03

0.2

UI

0 5 10 IS 20 28 30 38 40 48 50Time(s)

0.8

0.5

0.4

03

0.2

UI

0 5 10 IS 20 28 30 38 40 48 50Time(s)

Fig. 4. Raw experimental result of the response to pulse input

Fig. 5. Filtered experimental result of the response to pulse input

4. MODEL SELECTION To determine the estimation accuracy of models, we define the mean square of the prediction error as a loss function. This provides a metric to compare estimated and real outputs and hence choose the model with the best performance22.

The prediction error is given by:

)|(~)(),( θθ tytyte −= (5)

Where )|(~ θty is the prediction of )(ty given parametersθ .

The loss function is thus:

∑=

=N

tN te

NV

1

2 ),(211)( θθ (6)

Where N is the number of the points.

Since SMA transformation exhibits a hystersis effect on heating and on cooling, the modeling was performed for both heating and cooling separately21.The ARX model structure for both heating and cooling was increased and the loss function was recorded. It was obvious that the model in the larger model structure will automatically yield a smaller value of the criterion of fit (loss function). As the model structure increases, the loss function behaves as depicted in



Figs. 6 and 7, for heating and cooling, respectively. The loss function decreases since the model picks up more of the relevant features of the data. In order to select the optimum model the "knee" in the curve should be considered. Beyond the knee, additional parameters adjust the model which is known as over-fitting, in which extra improved fit is of no value due to the model will be applied to data with different noise realization.

The model selected for heating and cooling are arx331 (for heating) and arx310 (for cooling).

-1.00E-09

1.00E-09

3.00E-09

5.00E-09

7.00E-09

9.00E-09

1.10E-08

Model

Loss

Fun

ctio

n

arx111 arx551arx441arx331arx221

Fig. 6. Loss function for ARX models in case of SMA wire heating.

-0.00001

0.00001

0.00003

0.00005

0.00007

0.00009

0.00011

0.00013

0.00015

Model

Loss

Fun

ctio

n

arx11 arx22 arx33 arx441 arx55

Fig. 7. Loss function for ARX models in case of SMA wire cooling.

5. ADAPTIVE SELF-TUNING FUZZY PID CONTROLLER DESIGN

5.1 Structure of self-tuning fuzzy PID controller

PID controllers are widely used in industry due to their simple control structure and easy design. Nevertheless, there are certain problems that are encountered in practical control systems. The parameters of the conventional PID controller are not often properly tuned for the nonlinear plant with unpredictable parameter variations. For this reason, it is necessary to automatically tune the PID parameters.

On the other hand, fuzzy control provides a formal methodology for representing, manipulating, and implementing a human’s heuristic knowledge about how to control a system. This is a convenient method for constructing nonlinear controllers by using heuristic information obtained from experience.

Therefore, the advantages of fuzzy and PID controllers can be incorporated into an adaptive self-tuning fuzzy PID controller in order to achieve high control performance. The structure of the self-tuning fuzzy PID controller is shown in Fig. 8.



FLiC

Fig. 8. Structure of adaptive self-tuning fuzzy PID controller to SMA wire.

The controller has the form of PID structure, but the PID parameters are tuned by fuzzy inference, which provides a nonlinear mapping from the error signal )(te (the difference between reference signal and system output), and derivation of error, )(tde to the PID parameters, ip KK , and dK . These parameters are changed within the initial parameter boundaries.

5.2 Normalization

In order to obtain feasible rule bases with high inference efficiency, the PID parameters must be normalized on the interval [ ]1,0 .The parameters ip KK , and dK are varying over [ ]maxmin, pp KK , [ ]maxmin , ii KK and [ ]maxmin , dd KK intervals, respectively. These intervals are obtained experimentally. The parameters are normalized as follow:

minmax

minp' K

pp

pp KK

KK

−

−= (7)

The above equation is similar for 'dK , '

iK . Above equations are used in Conversion block in Fig. 8.

5.3 Fuzzification

In this paper, the linguistic levels assigned to the input variables )(te and )(tde are as follows:

NB: negative big; NM: negative medium; NS: negative small; ZE: zero; PS: positive small; PM: positive medium; PB: positive big.

The membership functions of these fuzzy sets are shown in Figs. 9-13.The maximum error and the maximum derivation of error ranges are chosen from the specification of SMA wire. S, MS, M, B are assigned as the fuzzy sets of-

'pK , '

iK and 'dK which are the output variables. in this paper fuzzy rules are designed based on properties of SMA wire and

performance of PID controller. The fuzzy IF-THEN rules are defined as:

Rule l: IF e (t) is lA1

and de (t) is lA2 THEN 'pK is lB and '

dK is lC and 'iK is, lD ml ,...,2,1= (8)

Where liA is the input fuzzy set and lB , lC and lD are the output fuzzy sets.

The fuzzy rule sets that used in this paper are shown in Tables 14-16, respectively.



V

Fig. 9. Membership functions for error, e (t).

Fig. 10. Membership functions for derivation of error, de (t).

Fig. 11. Membership functions for '

pK .

Fig. 12. Membership function for '

iK

Fig. 13. Membership function for '

dK



Table 1. Fuzzy control rule of 'pK

)(te

NB NM NS ZO PS PM PB

NB B B B B B B B

NM S B B B B B S

NS M MS M M M MS MS

ZO S S S MS MS S S

PS MS MS MS M M MS S

PM B B B B B B B

)(te&

PB B B B B B B B

Table 2. Fuzzy control rule of '

iK

)(te


NB S S S S S S S

NM S S S S S S S

NS M MS M M M MS M

ZO B B M M M B B

PS M M MS MS MS M M

PM S S S S S S S

)(te&

PB S S S S S S S

Table 3. Fuzzy control rule of 'dK

)(te


NB M MS S S S MS M

NM MS S S S S S MS

NS M M MS MS MS M M

ZO B B M MS M B B

PS M M MS MS MS M M

PM MS S S S S S MS

)(te&

PB M MS S S S MS M

6. RESULTS The performance of the proposed controller was simulated for different reference inputs. Fig. 17 shows the performance of the control system with respect to multi-step reference input signal. The system response to step reference input is compared with conventional PID controller in Fig. 18 and shows that adaptive self-tuning fuzzy PID controller is better than the conventional PID controller without fuzzy tuning. Fig. 19 shows the performance of the control system with respect to sine reference input ( )Hzf 02.0= .



0 20 40 60 80 100 120 140 160 180-0.2

0

0.2

0.4

0.6

0.8

1

Time(s)Fo

rce(

N)

desiredsimulated output

Fig. 17. Multi Step response

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time(s)

Forc

e(N

)

desiredsimulated output(without fuzzy tunning)simulated output(with fuzzy tunning)

Fig. 18. Comparison of step response with and without fuzzy tuning.

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time(s)

Forc

e(N

)

desiredsimulated output

Fig. 19. System response with respect to sine reference input (f = 0.02Hz).

7. CONCLUSION The hysterisis effect in SMAs makes their force control a challenging task. In this paper an experimental setup for precise measuring of the force in SMA wire is presented. Two separate equations of force versus voltage applied to the SMA wire undergoing heating and cooling processes are obtained. Based on these equations, an adaptive self-tuning fuzzy PID controller of the force in SMA wire is investigated. The responses of the proposed control system to step, multi-pulse and sinusoidal inputs are simulated and compared to responses when just an ordinary PID controller is



used. In sum, simulation results showed the effectiveness of self-tuning fuzzy PID controllers in force control of the SMA wire, in comparison with conventional PID controller. The results achieved in this study can be used as fundamental and guidelines for force control of a gripper actuated by SMA wires.

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