Sensors and Actuators, 20 (1989) 57-64
57
Millimeter-sized Joint Actuator Using a Shape Memory Alloy KATSUTOSHI KURIBAYASHI
Department of Mechanical Engineering, College of Engineering, University of Osaka Prefecture. 804 Mozu-Umemachi 4-cho. Sakai 591 (Japan)
Abstract A millimeter-sized actuator for driving a rotary joint for a small robot is designed and fabricated using a shape memory alloy ( = S M A ) ; it has the advantages of a strong force/weight ratio and can be extended down to micron size. The actuator is of push-pull type composed of two 0.05 mm x 0.5 mm x 3 mm SMA sheets. First, a theoretical model of the dynamics of the SMA actuator is derived based on an experimental analysis of the dynamics of large SMA sheets. Using this model, the design method of the above type of SMA actuator is established. Finally, the theoretical torque versus angular displacement of a millimeter-sized rotary joint driven by the above milimeter-sized SMA actuator is obtained, with a maximum torque of 4 gf mm.
(1) A SMA actuator can generate a large force/weight ratio. (2) The control method for a SMA actuator has been established [8]. (3) It can be expected that the smaller the SMA actuator, the faster the response. (4) A micron-sized SMA actuator can be made of thin film using micromachining technology. The smallest SMA products on the market are a 70 # m diameter wire produced by a drawing method, and a 50 # m thick sheet produced by a cold rolling mill, .ychich will guarantee long life for an actuator duiing its cyclic use. Using the above materials, we have designed and fabricated a millimeter-sized SMA actuator for the rotary joint of a small manipulator whose structure is like an industrial robot.
Several Types of Millimeter-sized SMA Actuators Introduction A millimeter-sized actuator is necessary for a small size robot that works under a microscope and for assembling small mechanical parts. Moreover, the development of this actuator will give us a lot of information for designing a micron-size actuator, whose necessity has been discussed elsewhere [1]. To date, some papers [2-7] on microactuators have been reported, however the rotor speed of the rotary variable-capacitance micromotor [2] has been too fast to mount it on a robot, although its 100/~m size is small enough. It is difficult to use an electrostatic 200 # m linear actuator [3] for a robot. A l0 mm ROMAC made of rubber [4], a 20 mm rubber tube actuator [5] and a 50 mm shape memory alloy actuator [6] are too large. Although a 5 mm SMA actuator of torsional type [7] is quite small, it is difficult to mount it on a rotary joint. In this paper, focusing on the following merits of the SMA actuator, we will design and fabricate a small SMA actuator for a robot.
To develop a millimeter-sized SMA actuator for a rotary joint, the SMA actuator should be made from 70/~m diameter SMA wire or a 50/~m thick SMA sheet, and the joints from a 300/~m thick ceramic plate. The SMA sheet and ceramic plate should be cut by using a laser beam in order to form the SMA actuator and the joint mechanism. To make the SMA actuator, it is necessary to memorize the original profile of the SMA in a furnace by constraining its shape, for example, at 500 °C for an hour. There are many types of SMA actuators related to the original SMA profile. Some typical SMA actuators are shown in Fig. I. To aid in selecting one of them as a millimeter-sized SMA actuator, these actuators were compared by their ability to memorize their original profile, to connect the SMA to the joint and on other points. The results are shown in Table 1. From the results in Table l, the SMA actuator in Fig. l(f) was selected as being the best for proceeding with development of a good small robot. Elsevier Sequoia/Printed in The Netherlands
58 TABLE 1. Comparisons among several SMA actuators Type of SMA actuator
Material
Profile of SMA
Memorizing profile of SMA
Fig. l(a) Fig. l(b)
coil sinusoidal wave spiral
x
0
×
0
Fig. l(d) Fig. l(e)
wire wire, sheet wire, sheet sheet sheet
x O A
Fig. l(f)
sheet
flat
O
Fig. l(c)
flat circular
Strength
Setting SMA to link
Making joint
Extendability to micron size
x
0
x
x
0
x
0
x
0
x
x
A
A
x
0
0
/x
A
©
0
0
0
arc
O:
easy A: moderate
x:
difficult force
1 <
4
t/)
Ca) (
"I
sheet
(b)
Fig. 2. Schematic diagram of the experimental method for deforming a SMA sheet. (d (e) I I (f) Fig. 1. Several types of SMA actuators for rotary joint mechanisms. (a) Coil; (b) sinusoidal wave; (c) spiral; (d) rectangular wave; (e) bending beam (I); (f) bending beam (II). D y n a m i c s o f S M A Sheet and its Mathematical Model
Since the dynamics o f the S M A a c t u a t o r in Fig. l(f) depend on the dynamics o f the S M A sheet, the latter are analysed experimentally. However, these millimeter-sized S M A sheets are too small for their dynamics to be analysed experimentally with ordinary measurement equipment without getting large measurement errors. Thus, larger-sized S M A sheets were used instead as test pieces for the experimental analysis. Based on these experiments, a mathematical model o f the dynamics o f the S M A sheet is proposed. In a later Section, the theoretical model is used to design a millimeter-sized S M A actuator.
Profile of SMA Sheet Deformed and Recovered The experiments on the deformation f r o m the original profile and its recovery to that o f the
Experiment
f--
Theory
/
Experiment
Theory
\
I
(a) (b) 10 mm I I Fig. 3. Profile of SMA sheets (l = 50 mm). (a) Deforming profile from the original flat profile at 18 °C. (b) Recovering profile to the original fiat profile at 84 °C S M A sheet were carried out by the m e t h o d shown in Fig. 2. The results are shown in Fig. 3 as the profiles during the processes o f deforming the S M A sheets from their original flat profiles (a) and recovering to them (b). The S M A sheets used in the experiments are 0 . 5 2 m m × 4 . 9 8 m m , in 30 mm, 40 m m and 50 m m lengths, and their original profiles are linear and fiat, with the austenite finish temperature Af = 44(°C).
59 TABLE 2. Flow charts for the numerical calculations (numbers in parentheses are equation numbers)
X
(4), (6) ,~/-
0
(XE, YE)
\
bending moment M0
(a)
xE
(b)
YE
(5), (6) ' 0
' YE
ar part (7)
Circular ~, arc part /
] (8), (9)
0
\ , . 7 ~ F '
> Mo
' F
_2
"/ (14)
(c)
7
>
0tI
(12)
(10)
>
0~2
)
rt --F-+T
I (11)
A theoretical model o f these profiles is derived as follows: since a serious model is difficult to derive due to the complex properties o f elasticity and plasticity o f the material, an approximate model is derived instead. As can be easily understood from Fig. 3, the profiles can be separated into two parts. One is a circular arc and the other is linear, as shown in Fig. 4. Moreover, the radius of curvature is constant independent o f the deformation during the deforming and recovering process o f the S M A sheet. Based on the above facts, the relationship between the radius o f curvature, f, and m a x i m u m strain, em, on the surface o f the S M A sheet can be expressed by (h/2)
(15)
F
Fig. 4. Dynamic model and profile of a SMA sheet.
?-
> 2X E
(d)
(c) ")'A (16y ~ 3'. (16)~
rA'
(~
XEA
(c)
(4) (9))FA_ "/'L (17) TR
(c) (4), (9) 713-----~ XEB ~ FB"
( c ) ~ rB'
1.0
(1)
~m
where h is the thickness o f the S M A sheet. The linear parts are assumed to be linear, and the small elastic strain is neglected, as it is very small. Therefore, a profile model o f S M A sheet is derived as shown in Fig. 4. Next, the relationships between the end point (XE, YE) and the circular arc angle 0 are obtained. Since the total length o f S M A sheet is constant, l f0 + [(XE -- r sin 0) 2 + (YE -- r(1 -- COS 0))2] 1/2 = -2
0.5
0
0
0.5
1.0
1.5
2.0
XE -- r sin 0 YE - ~( 1 - cos 0)
( -
tan 0)
Fig. 5. Numerical solutions of eqn. (4).
(5)
y = sin 0 + ct( 1 - 0 sin 0 - cos 0) =
-
1
(3)
where
is obtained. F r o m eqns. (2) and (3), we get ct(0 cos 0 - sin 0) - cos 0 + fl = 0
3.0 0 (rad)
(2) is obtained. As the linear parts are tangential to the circular arc,
2.5
XE (4)
ct = (l/2)'
fl
(1/2)'
YE Y
(l/2)
(6)
60 Referring to the experimental results in Fig. 6, the theoretical model of the relationship between the force and the displacement of the SMA sheets can be obtained. First, we propose the following equations for the bending moments Mo, for the martensite phase,
Given XE, we can get 0 by eqns. (4) and (6) and YE from eqns. (5) and (6) by the flow chart shown in Table 2(a). However, it is difficult to solve 0 from eqn. (4) given XE, because 0 is included implicitly in eqn. (4). Therefore, we solve it graphically by using Fig. 5, which is the numerical solution of eqn. (4). Using the above theoretical model, the theoretical profiles shown in Fig. 3 are calculated with em = 0.0443 for l = 50, 40 and 30 mm. These theoretical profiles show rather good agreement with the experimental ones.
mMo = (mO'E+ mkpO)Z and for the austenite phase, am0 = (aO'E+ akpO)Z
Design Method of a Rotary Joint Using SMA Sheets We can establish the design method of the rotary joint shown in Fig. l(f) by using the above equations. The characteristics ( t o r q u e - a n g u l a r displacement) of the rotary joint shown in Fig. 7(a) have been obtained theoretically. These characteristics depend on both the characteristics (force-linear displacement) of the SMA sheet and the geometry of the joint. First, the geometrical relationship between the pushing points P and Q of the SMA sheet and the angular displacement ? is derived. Referring to Fig. 7(b), which presents
(7)
and because the bending moment Mo is constant for each 0, independent of I. The larger YE is (which is caused by a larger l), the smaller F is.
F (kgf)
'1-
Fgf 0.4 "]
~,(u// ~ ~
0.2 ,- Y ~
o (a)
.
-
(9)
where Z is the modulus of section of the SMA sheet. In these equations fiE stands for the elastic stress and kpO for the plastic stress in the SMA sheet. Using the flow chart in Table 2(b) to calculate F, the theoretical curves can be obtained as shown by the dotted lines in Fig° 6. Figure 6 shows good agreement between the theoretical curves and the experimental ones. Therefore, the theoretical model proposed here is proved to be valid.
Force and Displacement of the SMA Sheet A force F was loaded experimentally onto t h e SMA sheet along vertical directions as shown in Fig. 2 at r o o m temperature (18 °C), until the circular angle 0 becomes ~/2 in the deforming processes of the SMA sheet. Then the SMA sheet was heated up to 84°C under a vertical displacement kept constant at the end of the above deforming process. The vertical displacement was then released to increase slowly, maintaining the temperature of 84°C. The displacement and the force of the SMA sheets during the above processes were recorded on a X - Y recorder, as shown in Fig. 6 for 1 = 50, 40 and 30 mm. F r o m Fig. 6, it can be seen that the larger the l value, the larger the displacement, though it makes the force smaller. This is because the bending moment M 0 at the centre of the SMA sheet can be expressed as Mo = FyE
(8)
/,1
F (kgf)
1.0
a FI,I
o
08
// P
0.6
_
...... 0.4
o0
[ ' _.. e "
0.2
?J, , ' ~ ~ , , o 0 5 10 0 l - 2XE (ram) (b)
~
0.6 0.4 0.2
5
o o 10 l - 2XE (ram) (c)
5
1o l-2XE (mm)
Fig. 6. Force and linear displacement of SMA sheets (solid line experiments; dotted line theory). (a) l = 50 mm; (b) l = 40 mm; (c) l = 30 mm.
61 (rad) 3.0
N ~
R/r = 1 ~ 3
2.5
Step: 0.2
2.0 1.5
R /r = 1 \ ~ ' ~ ' ~ 1.0 0.5 0 link
0
i
r
0.5
1.0
1.5
2.0
2.5
3.0
CtI
(rad) Fig. 7. (a) A joint mechanism shown in Fig. l(f) and (b) its geometrical model.
Fig. 8. Numerical solutions of eqn. (14).
the counter-clockwise direction can be expressed by the above geometric relationships, the following equations, are derived: 2X E
=
R
cos
~ 2 -If- r c o s
0~ I
(10)
r ' = r sin ~1
(11)
7 -~-(X1-]-~2
(12)
=~
r sin ctI = R sin ~2
(13)
From eqns. (12) and (13), we get R
-- sin{~ - (~ + ~l )} + sin ~q = 0 r
(14)
Next, we can easily get the torque r of the joint as follows: z = F . r'
----aTa - -
mTA
(17)
Given ~, the torque ZL can be calculated by the flow chart in Table 2(d).
D e s i g n o f a Joint M e c h a n i s m with a Millimeter-sized S M A Actuator
The elements of a millimeter-sized rotary joint mechanism and link were made from a 300 p m thick ceramic plate by a laser beam cutting method. They are shown in Fig. 9. The elements and the 100 p m diameter shafts were assembled to the joint and links by glue, as shown in Fig. 10.
(15)
If ~ and F are given, several variables can be calculated by using the flow chart in Table 2(c). The force F can be obtained from XE by eqns. (4)-(9). However, as ~ cannot be solved by eqn. (14) explicitly, it should be calculated by using Fig. 8, which shows the numerical solutions of eqn. (14). The above relations are for the B side of the joint, which can be seen in Fig. 7(a). The same relations can be obtained for the reverse side A of the joint in Fig. 7(a). The former variables and parameters will be assigned by subscript B, and the latter by A. In order to get the torque of the rotary joint, we define a new variable ~ whose origin is the central point of the angular displacement of the joint. Thus = 7B -- ~7= -- 7A + ~
ZL = , F n r ~ - - m F A r ' A
I
-~
.........
10mm
~j~
(16)
where ~7indicates the central position of the movable range of the joint. The maximum torque TL in
Fig. 9. Elements of millimeter-sized joints and SMA sheets.
62 SMA A
SMA A
,
,,
~ ~ _ r
7
",
;,/< ..×. ,." ) 8MA B
~ 5000 #m
. ~ - \
SMA
ceramic
a\~
LJ
SMA B >4 |
SMA
k\~
SMA
~
].
ul"r I I
SMA
Fig. 10. Joint mechanisms of a robot designed by using a millimeter-sized SMA actuator.
5 0 / ~ m x 500 # m x 3 0 0 0 / ~ m S M A sheets, w h i c h a r e s h o w n in Fig. 9, w e r e m a d e f r o m a S M A sheet by the laser b e a m c u t t i n g m e t h o d . T h e w h o l e j o i n t m e c h a n i s m s are s h o w n in Fig. 10 a n d t h e real j o i n t s are s h o w n in Fig. 11. T h e d y n a m i c s o f this S M A a c t u a t o r w e r e c a l c u l a t e d by e q n s . ( 4 ) - ( 1 5 ) .
r (gf mm) 4
Fig. 11. Assembled joints with millimeter-sized SMA actuators.
0 0
0.5
1.0
1.5
2.0
2.5
vn(rad)
I
>f
z (gf mm)
F(gf) 12
2 8
I a F
4
°
-2 m F
--4
0 0
0.3
0.6
0.9
1.2
1.5
.OR
0
0.'5
1.J0
' 1.5
' 2.0
J 2.5
"(B (rad)
l - 2XE (mm) Fig. 12. Theoretical force and linear displacement of a millimeter-sized SMA sheet.
Fig. 13. Theoretical torque and angular displacement of a rotary joint driven by millimeter-sized SMA actuators.
63
The results are shown in Figs. 12 and 13. Figure 12 shows the dynamics of the SMA sheet, and Fig. 13 shows the torque versus the angular displacement of the rotary joint. These results indicate that the maximum torque is about 4 gf mm and the range of the angular displacement is about _ 1.4 rad.
4.5% for a heat engine made from TiNi alloy. It is stated there that this value shows good agreement with the experimental result, and it corresponds to 35% of the ideal energy efficiency 12.8% of the Carnot cycle, which indicates high energy efficiency. It is estimated that the energy efficiency of the millimeter-sized SMA actuator is almost equal to the above value.
Discussion Conclusions The other characteristics of the millimeter-sized SMA joint actuator are now discussed.
The Time Response [9] The time response of this kind of actuator depends on the speed of the SMA heat transfer. For cyclic motion as the frequency response, the maximum frequency is determined from the point of the heat condensation in the SMA. Therefore the frequency response depends mainly on the SMA cooling response. From the theoretical analysis of the cooling response by thermal dynamics, the frequency response of the millimeter-sized SMA actuator is calculated as about 1.5 Hz without ventilation and 2.5 Hz with ventilation of 1 m s -1 at room temperature (20°C). On the other hand, the transient response (the step response) depends on the SMA heating response. Thus, the settling time of the step response can be expected to be less than 100 ms, from the experimental data found in ref. 9. The Fatigue Limit [10] An experimental test of the fatigue of the millimeter-sized SMA actuator under cyclic usage has not been carried out. However, it is estimated [10] that cyclic motion can occur 10000 times before the originally memorized profile is lost, with the maximum strain =0.0443, whose level is related to the structure of the stress induced martensite and the thermal martensite. If a maximum strain of less than 0.01, not related to the above martensite phase but to the R phase, is used for designing a millimeter-sized SMA actuator, a cyclic usage of over 100 000 times can be expected. The Power and Energy Efficiency We can perform a rough calculation of the input power of the millimeter-sized SMA actuator using the data on the experimental power of the SMA actuator in ref. [9]. The results show a maximum of 7 5 m W ( = m a x . 0.14V × max. 0.55 A), assuming the same input power per unit of electrical resistivity as in ref. [9]. The energy efficiency can be calculated by the equation proposed in ref. 11, where one example of the theoretical energy efficiency is calculated as
A millimeter-sized SMA actuator of the bending beam type was selected to drive a millimetersized rotary joint made from a ceramic plate, because of its ability to memorize the SMA profile easily, to be connected to a joint, to be extended down to micron size etc. The design method for this type o f SMA actuator was established by the following steps: (1) The dynamics of SMA sheets larger than a millimeter are analysed experimentally because of the experimental difficulties for a millimeter-sized SMA sheet. (2) To design a millimeter-sized SMA actuator, theroetical models of the dynamics of the SMA sheet are derived and proved to be valid. (3) The theoretical torque of a rotary joint driven by the SMA sheet is obtained by considering the geometry of the joint and the dynamics of the SMA sheet. (4) A millimeter-sized rotary j o i n t using a SMA actuator is designed and its theoretical torques versus angular displacement calculated, which indicates a maximum of 4 gf mm.
Acknowledgements This work was financially supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan, No. 62850043, and a Corning Japan Research Grant (1988). I would like to thank the Tokyo Precision M F G Co. Ltd. for its support in the laser beam machining, and Mr Sigenobu Kishi, an undergraduate student of the University of Osaka Prefecture, for performing the numerical calculation and the experiment.
References 1 W. S. N. Trimmer and K. J. Gabriel, Proc. 1EEE Micro Robots and Teleoperator Workshop, Hyannis, MA, U.S.A., Nov. 1987. 2 J. H. Lang, M. F. Schlecht and R. T. Howe, Electric micromotors: electromechanical characteristics, Proc. IEEE Micro Robots and Teleoperator Workshop, Hyannis, MA, U.S.A., Nov. 1987.
64 3 H. Fujita and A. Omodaka, Electrostatic actuators for micromechatronics, Proc. IEEE Micro Robots and
I1 T. Honnma, F. Kohno and M. Matsumoto, Efficiencyof energy conversion in solid state heat engine, Bull. Res. Inst.
Teleoperator Workshop, Hyannis, MA, U.S.A., Nov. 1987.
Mineral Dressing and Metallurgy, Tohoku University, Jpn., 39 (1983) 95-104.
4 G. B. Immega, ROMAC actuators for micro robots, Proc. IEEE Micro Robots and Teleoperator Workshop, Hyannis, MA, U.S.A., Nov. 1987.
5 K. Suzumori, S. Ikuta and H. Tanaka, Development of micromanipulator(1)-FRR-applied actuators, Prepr. 6th Ann. Conf. Robotics Soc., Japan, 1988, pp. 275-276. 6 P. Dario, M. Bergamasco, L. Bernardi and A. Bicchi, A shape memory alloy actuating module for fine manipulation, Proc. IEEE Micro Robots and Teleoperator Workshop, Hyannis, MA, U.S.A., Nov. 1987.
7 J. A. Walker, A small rotary actuator based on torsionally strained SMA, Proc. IEEE Micro Robots and Teleoperator Workshop, Hyannis, MA, U.S.A., Nov. 1987.
8 K. Kuribayashi, A new actuator of joint mechanism using TiNi alloy wire, Int. J. Robotics Res., 4 (1986) 47-58. 9 K. Kuribayashi, Improvement of the response of SMA actuator using temperature sensor, J. Robotics Soc. Jpn., 7 (1989) 39-46. 10 H. Tamura, Y. Suzuki and T. Todoroki, Fatigue properties of a Ni-Ti alloy in thermal cycling mode, Proc. Int. Conf. Martensitie Transformations, Nara, Japan, 1986, pp. 736 741.
Biography K a t s u t o s h i K u r i b a y a s h i was b o r n in E h i m e Prefecture, Japan, o n F e b r u a r y 6, 1938. He received his bachelor a n d d o c t o r degrees in engineering from the University of O s a k a Prefecture, i n 1961 a n d 1982, respectively. He is a n associate professor in the D e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g of the above University. He was a Visiting Scholar of H a r v a r d University in the fall semester, 1986. His c u r r e n t research interests are in robotics, especially the d e v e l o p m e n t o f new r o b o t actuators. He is a recipient of the O u t s t a n d i n g Technical Paper A w a r d from the I n d u s t r i a l Electronics Society, I E E E , in 1987 a n d the C o r n i n g Research G r a n t A w a r d in 1988.