NEW DEVELOPMENT ON SHAPE MEMORY ALLOYS

ACTUATORS

Roberto Romano

Instituto de Pesquisas Tecnológicas do Estado de São Paulo and Telecommunications and Control Department

Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

Eduardo Aoun Tannuri

Instituto de Pesquisas Tecnológicas do Estado de São Paulo and Mechatronics Engineering Department

Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

Keywords: Modeling, shape memory alloy, actuator, robotics, sliding mode control.

Abstract: The present paper presents the development of a mechanical actuator using a shape memory alloy with a

novel cooling system based on the thermo-electric effect (Seebeck-Peltier effect). Such a method has the

advantage of reduced weight and requires a simpler control strategy as compared to other forced cooling

systems. A complete mathematical model of the actuator was derived, and an experimental prototype was

implemented. Several experiments are used to validate the model and to identify all parameters. A robust

and nonlinear controller, based on sliding-mode theory, was derived and implemented. Experiments were

used to evaluate the actuator closed-loop performance, stability, and robustness properties. The results

showed that the proposed cooling system is able to improve the dynamic response of the actuator.

1 INTRODUCTION

Shape Memory Alloys (SMAs) consist of a group of

metallic materials that demonstrate the ability to

return to some previously defined shape or size

when subjected to the appropriate thermal

procedure. The shape memory effect occurs due to a

temperature and stress dependent shift in the

materials crystalline structure between two different

phases called Martensite and Austenite. Martensite,

the low temperature phase, is relatively soft whereas

Austenite, the high temperature phase, is relatively

hard. The change that occurs within SMAs

crystalline structure is not a thermodynamically

reversible process and results in temperature

hysteresis. SMAs have been used in a variety of

actuation applications. The key feature of this

material is its ability to undergo large seemingly

strains and subsequently recover these strains when

a load is removed or the material is heated. SMA

actuators have several advantages such as excellent

power to mass ratio, maintainability, reliability, and

clean and silent actuation. The disadvantages are

low energy efficiency due to conversion of heat to

mechanical energy, inaccurate motion control due to

hysteresis, nonlinearities, parameter uncertainties,

difficulty in measuring variables such as temperature

and the slow response due to the thermal process

evolved in the working principle.

To operate quickly, the SMA must be cooled

rapidly. Some researchers have proposed static

methods, in which the SMA wires are continually

cooled by means of an air stream (Tanaka and

Yamada, 1991). In a similar way, Furuya and

Shimada (1990) used a cooling system based on

water immersion. In such case, cooling time was

reduced in 10 times compared to a non-cooled wire.

However, the power consumption of such actuator

has increased by a factor of 20, since in the heating

phase it is necessary much more power to

compensate the heat that is lost by the cooling

system. Golbert and Russel (1995) used a mobile

metallic heat sink and a complex mechanism

guaranteed that the sink was only in contact to the

wire being cooled, which minimizes the power

consumption of the actuator and increases its

dynamic behavior. Asada and Mascaro (2002)

developed an actuator with a cooling system based

on flowing water around the wire. A complex

55

Romano R. and Aoun Tannuri E. (2008).

NEW DEVELOPMENT ON SHAPE MEMORY ALLOYS ACTUATORS.

In Proceedings of the First International Conference on Biomedical Electronics and Devices, pages 55-61

DOI: 10.5220/0001047800550061

Copyright

c

SciTePress

system guarantees that water flows only when wire

must be cooled. The dynamic response of the

actuator is expressively better, and the power

consumption is also acceptable.In order to increase

the dynamic characteristics of SMA actuators,

keeping a simple mechanical and electrical design,

the present work proposes a novel cooling system

based on thermoelectric effect (Seebeck-Peltier

effect). A complete mathematical model of the

actuator is derived and an experimental prototype is

used to validate the model and identify all

parameters. A sliding-mode control is also derived

and some preliminarily results of its application in

the experimental system are obtained and discussed.

2 EXPERIMENTAL SET-UP

An experimental prototype of the SMA actuator,

cooled by a thermoelectric element, was built. It was

used to validate the mathematical model and to

evaluate the control algorithm. Figure 1 shows a

simple scheme of the actuator. The thermoelectric

tablet is assembled in a heat sink and a small blower

is also used to dissipate the heat. The SMA wire is

attached to the structure, by means of an electric

connector C1. The other end of the wire is attached

to the intern pulley (radius r1=0.45cm). A 40g load

is supported by a wire connected to the external

pulley (r2=4.5cm). The length of the SMA wire is

15cm, and its typical 4.0% deformation is amplified

by a factor 10. So, it is expected a 6cm elevation of

the load, which is equivalent to a 76

0

pulley rotation.

Such rotation is measured by a potentiometer

directly attached to the pulley axis.

Figure 2 shows a picture of the actuator. The signal

conditioning module is composed by a constant

current amplifier that supplies up to 1A the electric

current to heat the SMA wire and by a voltage

amplifier/filter connected to the potentiometer.

Load

Common wire

Potentiometer

Pulley

SMA

wire

+5V

Current input

C

1

Thermoelectric

tablet

r

2

r

1

Figure 1: Diagram of the SMA actuator.

The module is connected to a computer (Pentium

100MHz) by mean of a 10bits AD/DA board. The

thermoelectric tablet is constantly powered by a 5V

tension.

A Matlab/Simulink software was used to acquire and

process the data from the potentiometer, and to send

the command to the current that must be imposed to

the SMA wire. Such software is flexible, and several

control algorithms can be easily implemented.

Furthermore, all graphical and mathematical tools

provided by Matlab/Simulink can be used. The

interface with AD/DA board was developed by

means of a low-level code included in the software.

Figure 2: Picture of the actuator.

3 MATHEMATICAL MODEL

The model developed in the present work is based

on the formulations proposed by Ikuta, Tsukamoto

and Hirose (1991); Grant, Hayward and Lu (1997);

Ashrafiuon and Elahinia (2002), Hoder, Vasina and

Solc (2003) and Dutta and Ghorbel (2005). It is

composed by a thermal model, a phase

transformation model and a description of the

mechanical properties and dynamics of the system

(Figure 3).

Figure 3: SMA actuator model.

The main variables used in the model are:

i – electric current in the SMA wire (A)

T – SMA wire temperature (ºC)

ξ

- martensite fraction (0 a 100%)

σ

- mechanical stress in the SMA wire (N/m

2

)

ε

- deformation (strain) of the SMA wire (ΔL/L)

BIODEVICES 2008 - International Conference on Biomedical Electronics and Devices

56

3.1 Thermal Model

Thermal model was base on the system shown in

Figure 4 in which the SMA wire touches the

thermoelectric element. The temperature of the

element is considered to be constant, equal to 15

0

C.

Figure 4: SMA wire and cooling element.

Considering several simplification hypothesis

(Grand, Hayward and Lu, 1997), the thermal model

can be written as (thermomechanical coupling is also

not included since the deformation rate of the SMA

wire is small, and such effect becomes important for

fast or highly oscillatory deformations):

).(.).(....

2

pambp

TTCTTAhRi

d

t

dT

cm −−−−=

(1)

In the previous equation, T

p

is the temperature on the

surface of the cooling element (15

o

C), T

amb

is the

ambient temperature (considered to be 20

o

C), h is

the natural convection coefficient per unity length of

SMA wire (in W/m

2

ºC/m) and C is the conduction

coefficient per unity length (in W/

ºC/m).

Furthermore, technical specifications of the SMA

wire Flexinol FLX 00870, 0.008’’, 70ºC) are given

by (Dynalloy, 2005):

m

- mass per unity length (2.10

-4

Kg/m)

p

c - specific heat (837 J/Kg.K)

R - electrical resistance per unity length ( 32Ω/m)

A

- external area per unity length ( 6.28.10

-4

m

2

/m)

d -diameter (2.10

-4

m)

The parameters h and C are very difficult to estimate,

since they depend on several variables. A rough

estimation of such parameters, based on the theory

exposed in Incropera and Witt (1998), are C = 0.4

W/

o

C/m and h=6.55 W/m

2

ºC/m. In the sequel, an

identification procedure will be used to obtain values

closer to the real ones.

3.2 Phase Transformation

During heating, occurs the transformation between

Martensite (M) to Austenite (A), and during the

cooling phase, the opposite transformation occurs.

Basic equations that model such transformations, as

a function of temperature, are given below (Ikuta,

Tsukamoto and Hirose, 1991):

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

+

−

−

+

=

2

2.6

exp1

AfAs

T

AsAf

m

ξ

ξ

(heating)

A

A

MfMs

T

MsMf

ξ

ξ

ξ

+

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

+

−

−

+

−

=

2

2.6

exp1

1

(cooling) (2)

where As and Af are the initial and final temperature

of austenite transformation respectively; Ms and Mf

are the initial and final temperature of martensite

transformation respectively,

ξ

m

is the highest

martensite fraction during cooling and

ξ

A

is the

initial value of martensite fraction during cooling.

Typical values are As=68ºC, Af= 78ºC, Ms=52ºC and

Mf= 42ºC (Dynalloy, 2005). However, such values

may present variations up to ±15

o

C, and an

identification procedure will be applied to evaluate

the correct values for the wire used in the

experimental actuator. Phase transformation is

shown in Figure 5, and the hysteresis gets evident

(Holder; Solc and Vasina, 2003).

Cooling

Heating

Histeresis

Figure 5: Phase transformation plot.

3.3 Mechanical Properties and

Dynamics

Mechanical properties of shape memory alloys are

obtained by means of a multiple layer model. The

austenite phase is characterized by an elastic

behavior. The martensite phase presents a behavior

that seems to be plastic, deformed by a small stress

(Ikuta, Tsukamoto and Hirose, 1991). So, for

ξ

= 0

(full austenite), the stress-strain relation is given by:

ε

σ

.

AA

E

=

(3)

where

A

σ

is the mechanical stress in the austenite

portion of the alloy and E

A

is the austenite Young’s

Modulus. In the other way, when ξ = 1 (full

martensite), the stress-stain relation is given by:

mymymm

mymm

E

E

εεεσ

εεεσ

>=

≤=

if .

if .

(4)

NEW DEVELOPMENT ON SHAPE MEMORY ALLOYS ACTUATORS

57

E

m

is the martensite Young’s Modulus ,

ε

my

is the

martensite maximum elastic deformation,

σ

m

is the

maximum stress in the martensite. The martensite

mechanical behavior is illustrated in the fig. below:

σ

ε

ε

my

Figure 6: Martensite mechanical behaviour.

Considering then the case with 0<ξ< 1, the stress-

strain relation is given by:

⎪

⎪

⎩

⎪

⎪

⎨

⎧

>

−

−

=

≤

−+

=

⇒−+=

my

A

my

my

AM

AM

E

Em

EE

εε

ξ

εξσ

ε

εε

ξξ

σ

ε

σξσξσ

for

).1(

..

for

.).1(.

).1(.

(5)

The dynamics of the actuator is now considered. A

simplified diagram of the actuator and the main

forces are shown in the Figure 7, where the

coordinate x represents the position of the load.

T

1

T

1

T

2

T

2

r

2

x

=

ω

x

m

L

.g

Figure 7: Diagram of the actuator and main forces.

The traction T

1

holds the load, T

2

is the traction

acting on the SMA wire and

ω

is the angular

velocity of the pulley. So, being J the moment of

inertia of the pulley and m

L

the mass of the load,

using the basic laws of Mechanics one obtains:

2

1221

..

r

x

JrTrT

=+−

; xmTgm

LL

.

1

=+− (6)

Traction T

2

may be estimated using a linear spring

analogy. Figure 8 shows the wire in three possible

states. In the austenite phase, the “spring” presents

its initial length l

0

. In the martensite phase, the wire

reaches its maximum length l

wire

, that correspond to

the situation in which the load position is x=0. The

difference between l

0

and l

wire

is approximately 4%

of the l

wire

. In an intermediate phase, the “spring”

elongation

lΔ is given by nxl

fio

−%.4 .

Austenite

0

l

Martensite

(x=0)

Intermediate

nx

0

l

wire

l

wire

l%.4

≅

l

Δ

Figure 8: Spring analogy.

So, the traction is given by

(

)

nxlKlKT

wire

/.04,0..

2

−

=

Δ

=

, where K is the

equivalent stiffness coefficient of the wire. Including

a damping term, the equation of motion of the

actuator becomes:

(

)

nKlgmxnKxcxmrJ

wireL

L

)04.0(...

2

2

2

+−=+++

(7)

The stiffness coefficient K may be evaluated using

(5). Assuming elastic behavior, being A

wire

the

sectional area of the wire, one obtains:

[

]

ε

ξ

ξ

σ

.).1(.

AM

EE

−

+

=

or

[]

0

2

.).1(.

l

l

EE

A

T

AM

wire

Δ

−+=

ξξ

(8)

Finally, considering that martensite stress will be

higher than its elastic limit, it may be assumed that it

will present a full plastic behavior, with a very small

stiffness. So, E

M

may be excluded from (8), resulting

the approximation:

[]

0

.).1(

l

A

EK

wire

A

ξ

−= (9)

4 PARAMETER

IDENTIFICATION

Thermal parameters C and h, as well as the

transformation temperatures (Ms, Mf, As and Af)

must be accurately evaluated using a proper

experimental procedure. A ramp excitation was

induced in the wire, inducing a displacement of the

load, as shown in Figure 9.

Using an optimization algorithm based on

Quadratic Sequential Programming (SQP), the

model parameters were adjusted in order to make the

model to recover the measured position accurately.

The following parameters were obtained,: C=0.255

W/

o

C/m, h=7 W/m

2

ºC/m, Ms = 66

o

C , Mf = 34ºC

As = 53ºC, Af = 93ºC. The comparison between the

measured position ant that obtained by the model

simulation is given in Figure 10. A good accuracy of

the model is verified.

BIODEVICES 2008 - International Conference on Biomedical Electronics and Devices

58

0 100 200 300 400 500 600 700

-0. 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Tem po (s )

Corrente (A)

Time (s)

Current (A)

(a)

0 100 200 300 400 500 600 700

-1

0

1

2

3

4

5

6

7

Tempo(s)

Deslocamento(cm)

Experimento

Time (s)

x position (cm)

Experiment

(b)

Figure 9: (a) Current ramp excitation; (b) Position of the

load.

Figure 10 Comparison between simulation and

experimental results

5 CONTROL SYSTEM DESIGN

In order to apply the sliding mode control to the

SMA actuator, the model previously developed must

be adapted, following the formulation exposed in

Slotine and Li (1991). After some algebraic

manipulation using (1) and (2), the dynamics of

phase transformation can be written as:

ubf ).()(

ξξξ

+=

(10)

with:

()

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

−

+−+−−−

−

= b

ka

chAcThATkk

mc

a

f

Pamb

P

1ln

1

)()()()(

0

2

0

0

ξ

ξ

ξξξ

ξ

ξ

()

Rkk

mc

a

b

P

2

0

0

)()()( −−−

−

=

ξξξ

ξ

ξ

;

2

Iu =

and

⎪

⎪

⎩

⎪

⎪

⎨

⎧

<=−=

+

=

−

=

>==

+

=

−

=

0 if ;1;

2

;

2.6

0 if 0;;

2

;

2.6

0

0

Tk

MM

b

MM

a

Tk

AA

b

AA

a

AA

fs

fs

M

fs

sf

ξξξ

ξξ

Using (1) and (9), the dynamics of the motion can

also be written as:

Δ

−

+−=

−

++

n

K

gmx

n

K

xcxI

L

)1()1(

0

2

0

ξξ

(11)

With

(

)

L

mrJI +=

2

2

;

00

lAEK

wireA

=

and

wire

l04.0=Δ

. In equations (10) and (11), the variable

x is measured, but the variable

ξ

must be estimated.

This may be done considering a quasi-static

approximation to (11), making

0== xx

, that

results:

⎟

⎠

⎞

⎜

⎝

⎛

−

Δ

−=

2

0

1

ˆ

n

x

n

K

gm

L

ξ

(12)

Differentiating (11), and using (10), one obtains the

SMA model adequate to apply sliding mode control,

as proposed by Slotine and Li (1991):

ubfx ).()( xx +

=

(13)

with

(

)

T

xxx

ξ

=x

,

)(

)1(

)(

0

2

0

2

0

ξ

ξ

f

In

K

In

xK

x

In

K

x

I

c

f

⎟

⎠

⎞

⎜

⎝

⎛

Δ

−+

−

−

−

=

x

and

)()(

0

2

0

ξ

b

In

K

In

xK

b

⎟

⎠

⎞

⎜

⎝

⎛

Δ

−=x

.

So, the control action

2

iu = is given by:

(

)

)/(

~~

2)(

ˆ

)(

ˆ

1

2

φλλ

ssatKxxxf

b

u

SMd

−−−+−=

x

x

(14)

with

xxxs

~

~

2

~

2

λλ

++=

. )(

ˆ

xf and

)(

ˆ

xb

are the best

estimates of the functions in model (13), considering

the approximate values for the parameters and the

estimate of the variable

ξ

given in (12).

d

x is the

desired position of the actuator load (set-point),

λ

is

a positive constant related to the cut-off frequency of

the closed-loop system,

φ

is the boundary layer

thickness to avoid control chattering and K

SM

is a

control gain related to the modeling and parameter

estimation errors. A detailed description of the

control design may be found in Slotine and Li

(1991).

6 EXPERIMENTAL RESULTS

The control logic previously developed was applied

to the experimental prototype, and the performance

NEW DEVELOPMENT ON SHAPE MEMORY ALLOYS ACTUATORS

59

of the closed loop system could be evaluated. Figure

11 shows a reference step of 2.5cm applied at t=10s.

Control parameters are λ=40; φ=1,6 and K

SM

=64.

The 5% settling time obtained is 0,23s, and the

maximum overshoot is 0,6%.

5 10 15 20

1.5

2

2.5

3

3.5

4

4.5

5

Tim e(s )

X P os it i o n ( c m )

Position

Reference

Figure 11: Step response of the SMA actuator.

A harmonic set-point was applied to the actuator,

with amplitude of 1cm and periods of 20s and 5s

(Figure 12. It can be seen that, despite a small

oscillation around the set-point, the system follows

the reference with good accuracy. Tests with

decreasing periods indicated a 0.69Hz cut-off

frequency, despite of a 0.37Hz obtained with a

conventional PID controller. Finally, a cooling

disturbance was applied, created by a computer

cooler fan directed toward the SMA wire (Figure

13). The robustness of the controller can be verified,

by comparing the response with the open-loop

response.

0 5 10 15 20 25 30 35 40 45 50

0.5

1

1.5

2

2.5

3

3.5

Tim e(s )

X Position(cm)

Position

Reference

(a)

0 5 10 15 20 25 30 35 40 45 50

0.5

1

1.5

2

2.5

3

3.5

Tim e(s )

X Position(cm)

Position

Reference

(b)

Figure 12: Harmonic set-point response of the SMA

actuator (cm amplitude). (a) 20s; (b) 5s period

0 5 10 15 20 25 30 35 40 45

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Tim e(s)

X Position(cm)

Position

Reference

cooling

(a)

0 50 100 150 200 250

2

2.5

3

3.5

4

4.5

5

Tim e (s )

X Position (cm)

cooling

cooling

(b)

Figure 13: Cooling disturbance response of the SMA

actuator (a) closed-loop; (b) opened-loop

7 CONCLUSIONS

A novel SMA actuator was proposed, using the

thermoelectric effect for cooling the SMA wire. An

experimental prototype was built, and a

mathematical model was developed. The model

parameters were adjusted by means of an

experimental identification procedure. The validated

model was then used to design a sliding mode

controller. Such controller is able to deal with model

and parameter uncertainties, and also with non-linear

effects. Closed-loop preliminary results obtained in

the experimental set-up showed that the proposed

actuator presents a good dynamic response and low

sensibility to disturbs of load and ambient cooling.

So it is recommended in application with SMA

devices.

ACKNOWLEDGEMENTS

The authors would like to express their gratitude for

the IPT (Institute for Technological Research - São

Paulo, Brazil) for supporting this research and for

CNPq (National Council for Scientific and

Technological Development), Research Process

Number 484232/2006-1.

BIODEVICES 2008 - International Conference on Biomedical Electronics and Devices

60

REFERENCES

Asada, H.H. and Mascaro S., 2002, “Wet Shape Memory

Alloy Actuators”, MIT Home Automation and

Healthcare Consortium, Phase 3 Final Report, Boston,

USA.

Ashrafiuon, H. and Elahinia H.M., 2002, “Nonlinear

Control of a Shape Memory Alloy Actuated

Manipulator”, ASME Journal of Vibration and

Acoustics, Vol. 124, pp.566-575.

Dutta, S.M. and Ghorbel, F.H., 2005, "Differential

Hysteresis Modeling of a Shape Memory Alloy Wire

Actuator," IEEE/ASME Transactions on

Mechatronics, Vol. 10, Issue 2, April , pp. 189-197.

Dynalloy, Inc., 2005, “Flexinol – Wire specifications”.

Available at: http:// www.dynalloy.com. Acess: 20

Dec. 2005.

Furuya, Y. and Shimada H., 1990, “Shape Memory

Actuator for Robotic Applications, Engineering

Aspect of Shape Memory Alloys”, Butterworth-

Heinemann, London, pp 338-355.

Gorbet, B.R. and Russel A.R., 1995, “Improve the

Response of SMA Actuators”, IEEE International

Conference on Robotic and Automation, Vol. 3, May,

pp. 2299-2303.

Grant, D., Hayward V. and Lu A., 1997, “Design and

Comparison of High Strain Shape Memory Alloy

Actuators”, International Conference on Robotic and

Automation, Albuquerque, New Mexico, 260-265.

Hoder K., Vasina, M. and Solc F., 2003, “Shape Memory

Alloy- Unconventional Actuators”, International

Conference on Industrial Technology ICIT, Maribor,

Slovenia, pp. 190-193.

Ikuta K., Tsukamoto M. And Hirose, S., 1991,

“Mathematical Model and Experimental Verification

of Shape Memory Alloy for Designing Micro

Actuator”, Proc. of the IEEE on Micro

Electromechanical Systems, an Investigation of

Microstructures, Sensors, Actuators, Machines, and

Robots, pp.103-108.

Incropera, P.F. and Witt, D.P.D., 1998, “Fundamentos de

transferência de calor e de massa”, 4 ed., Rio de

Janeiro, LTC (In Portuguese).

Slotine, J.J.E. and Li, W., 1991, “Applied nonlinear

control”, Englewood Cliffs: Prentice –Hall, USA.

Tanaka,Y. and Yamada Y., 1991, “A Rotary Actuator

Using Shape Memory Alloy for a Robot, and Analysis

of the Response with Load”, IEEE/RSJ International

Workshop on Intelligent Robots and Systems IROS

’91, Osaka, Japan, pp1163-1168.

NEW DEVELOPMENT ON SHAPE MEMORY ALLOYS ACTUATORS

61