Improvement of a Parallel Type Two-axial Actuator Controlled by a Multi-layered Neural Network

Abstract

Our parallel typed two-axial actuator was composed of two bimorph piezoelectric elements and two small links connected by three joints. We formulated kinematics for the parallel typed two-axial actuator because the endpoint is controlled in the two-dimensional coordinate. Since relationship between applied voltage and displacement cause by the voltage shows a hysteresis loop in the bimorph piezoelectric element used as components of the two-axial actuator, we produce a control system for the two-axial actuator based on a multi-layered artificial neural network to compensate the hysteresis. The neural network is comprised of 4 neurons in the input layer, 10 neurons in the hidden layer and ones neuron in the output layer. The output neuron emits time derivative of voltage; two bits signal expressing increment or decrement condition is generated by two input neurons; one of the other two input neurons and the other calculate current values of voltage and displacement, respectively. In the learning process, the network learns the hysteresis including minor loops. In the verification test, the endpoint of the two-axial actuator traces the desired circular trajectory in the two-dimensional coordinate system. After learning hysteresis loops including minor loops, the neural network simulates these hysteresis phenomena with very high accuracy.

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Improvement of a Parallel Type Two-axial Actuator Controlled by a Multi-layered
Neural Network

Kazuya Esumi*,Masahiro Ohka*, Yasuhiro Sawamoto**, Shiho Matsukawa***,
Tetsu Miyaoka***
*Department of Complex Systems Science, Graduate School of Information Science, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, 464-8601
**Molex Co. , 1-5-4 Fukami Higashi, Yamato, 242-8585
***Olympus, Co., 2-3-1 Nishishinzyuku, Shinzyuku-ku, Tokyo 163-0914
**** Department Information System, Shizuoka Institute of Science and Technology,
Toyosawa 2200-2, Fukuroi, 437-8555

Abstract:
Our parallel typed two-axial actuator was composed of two
bimorph piezoelectric elements and two small links connected
by three joints. We formulated kinematics for the parallel
typed two-axial actuator because the endpoint is controlled in
the two-dimensional coordinate. Since relationship between
applied voltage and displacement cause by the voltage shows a
hysteresis loop in the bimorph piezoelectric element used as
components of the two-axial actuator, we produce a control
system for the two-axial actuator based on a multi-layered
artificial neural network to compensate the hysteresis. The
neural network is comprised of 4 neurons in the input layer, 10
neurons in the hidden layer and ones neuron in the output
layer. The output neuron emits time derivative of voltage; two
bits signal expressing increment or decrement condition is
generated by two input neurons; one of the other two input
neurons and the other calculate current values of voltage and
displacement, respectively. In the learning process, the
network learns the hysteresis including minor loops. In the
verification test, the endpoint of the two-axial actuator traces
the desired circular trajectory in the two-dimensional
coordinate system. After learning hysteresis loops including
minor loops, the neural network simulates these hysteresis
phenomena with very high accuracy.

1. INTRODUCTION

Simultaneous presentation of pressure distribution and
shearing force is effective to enhance presentation reality of
a tactile display[1]-[7] because there are pressure and
shearing force accepting points (mechanoreceptive units)
distributing on human palm and finger surfaces[8][9]. Thus,
we are studying a two-axial micro-actuator[10][11] for
development of a tactile display pad of two-axial actuators.
Our parallel typed two-axial actuator was composed of
two bimorph piezoelectric elements and two small links
connected by three joints. We formulated kinematics for it
due to two-dimensional control of its endpoint (movable
end, hereafter) in the two-dimensional coordinate system.
The movable end of the two-axial actuator does not follow
same route in increment and decrement of applied voltage
because the present two-axial actuator utilizes piezoelectric
elements possessing a hysteresis phenomenon.
In the previous study, a new control method for the
piezoelectric actuator is established on the basis of a
multi-layered artificial neural network [12]-[14] to achieve the
sensor-less control of the actuator. Since the network scale
becomes huger without a new idea to learn the hysteresis
loop, we apply causality to formulate the modified neural
network. In the causality which is a basic idea of the
classical physics, a certain current time derivative of
physical variable is determined by current physical variables
if all physical variables can be measured at a certain instant.
On the basis of the causality, we assumed that the current
time increment of the voltage is determined by the current
voltage, the current displacement and flags indicating
increment or decrement condition. The present network is
comprised of 4 neurons in the input layer, 10 neurons in the
hidden layer and one neuron in the output layer. The current
values of voltage and displacement are input to the two of
the 4 neurons in the input layer; two bits signal expressing
increment or decrement condition calculated from time
derivative of displacement are input to the other two
neurons; the output neuron emits time derivative of voltage.
After time integration operation is performed to the output,
the integrated output is used as voltage value, which should
be applied to the actuator. Simultaneously, the voltage is
applied to the input neuron as feedback signal. Although in
the other researcher's study the idea of using the neural
network has been also used to control the piezoelectric
actuator [15], it is noted that the integration operation is
included in the abovementioned feedback loop to reduce the
network scale.
In this study, both of hardware and software are
improved on the basis of the previous study[16]. Although
the movable joint of the two-axial actuator roughly traced
the desired circular trajectory in the two-dimensional
coordinate system, the movable joint was apart from the
desired trajectory in the last half. This is caused by loose
fitting between the pin and hole, which play as a role of a
joint. We designed and developed a revised version of the
978-1-4244-2919-6/08/$25.00 ©2008 IEEE
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two-axial micro actuator using a micro sliding bearing. 4.
Additionally, learning procedure of the multi-layered neural
network was improved by adding some minor loops to
leaning data. The abovementioned improvements are
continued at the present. In the conference, we will show
these results.

2. TWO-AXIAL ACTUATOR

2.1 Kinematics of Parallel Type Two-axial Actuator
Figure 1 shows mechanism and photograph of the
two-axial actuator, respectively. It is composed of two
piezoelectric elements, three joints and two small links to
generate two-dimensional displacement as shown in Fig. 1.
The displacement of movable end (the center joint C) is
controlled by controlling the displacement of right and left
bimorph type piezoelectric elements. First, to formulate
kinematic equation of the two-axial actuator, nomenclatures
used in the formulation are shown as follows:
a : length of the small link.
b : distance between joint A and B in reference
configuration.
x u : x-directional displacement of movable end C.
yu : y-directional displacement of movable end C.
u : bending displacement of right piezoelectric element.
u : bending displacement of left piezoelectric element.
V : applied voltage of right piezoelectric element.
V : applied voltage of left piezoelectric element.
θ : establishment angle of piezoelectric element.
x & : the time derivative of the variable x .
Coordinates of points A’ and B’ are calculated from
geometrical relationship shown in Fig 1 as follows. That is,

+−
θ
,sin
2
R
L
R
L
Point A’:








−−
θ
cos
4
1
2
2
LL
u
b
aub

Point B’:







−−−
θθ
cos
4
, sin
2
1
2
2
RR
u
b
aub

where,
2
ba >
.
When the vector of movable end is expressed as
ji
yx
uu
+
, the following expressions are obtained from
the condition of link length being constant.

−+



u
=
2
2
2
2
2
cos
4
sin
2
1
au
b
auubu
LyLx
=







+−



−+
θθ
.................................................................... （１）

−+



2
2
2
2
2
cos
4
sin
2
1
au
b
auubu
RyRx
=







+−



+−
θθ
............................................................................... （２）

Fig. 1 Principle of the parallel typed two-axial actuator

The following equations are obtained from
differentiating Eq.(1) and (2).
1
2


()
()
0coscos
4
2
sinsin
2
2
2
=+








+−−+
−



−+
θθ
θθ
LyLy
LxLx
u
&
u
&
u
b
au
u
&
u
&
ubu
… （３）
()
()
0coscos
4
2
sinsin
2
1
2
2
2
=+








+−−+
+






+−
θθ
θθ
RyRy
RxRx
u
&
u
&
u
b
au
u
&
u
&
ubu
… （４）
Simultaneous equation composed of Eq.(3) and (4) is
written as following matrix expression:

−
=


R
AA
θθ
cossin
22 21

















+
−
+
−
−
A



y
x
L
u
&
u
&
AA
A
AA
A
AA
A
u
&
u
&
θθ
θθθθ
cossin
cos sincossin
22
21
2221
12 11
12
1211
11
（５）
θ
sin
2
1
11Lx
ubuA
−+=
.............................. （６）
θ
cos
4
2
2
12Ly
u
b
auA
+−−=
.................. （７）
θ
sin
2
1
21Rx
ubuA
+−=
............................. （８）
θ
cos
4
2
2
22Ry
u
b
auA
+−−=
（９）

2.2 Neural Network Including Feedback Loop
In the present research, we are attempting to achieve the
sensor-less control of the actuator, which is established by a
new control method of piezoelectric actuator using a neural
network model. The present structure of network is featured
with causality as a basic idea, in which time increment of
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Fig.2 Neural network with a feedback loop

physical variable is determined at a certain instant, if all
current physical variables can be measured at the moment.
Generally, it is well known that in piezoelectric actuator
slope of the voltage-displacement curve is different
noticeably between increment and decrement sequences of
applied voltage. If the above-mentioned causality is applied
to control method of piezoelectric actuator, the time
increment of voltage at a certain instant can be determined
by current voltage, current displacement and information
related to whether increment or decrement condition.
Figure 2 shows structure of the present neural network;
it is comprised of 4 neurons in the input layer, 10 neurons in
the hidden layer and one neuron in the output layer. The
output neuron emits time derivative of voltage; two bits
signal expressing increment or decrement condition is
generated by two input neurons; one of the other two input
neurons and the other show current values of voltage and
displacement, respectively. After time integration the output
time derivative of displacement is not only used for voltage
that should be applied to piezoelectric element, but also fed
to the neuron in the input layer through a feedback loop.
The neural network is featured with the feedback loop
including an integral unit to reduce number of neurons.
In the learning process, synaptic connection’s weight
)(s
ij
w
is adjusted by using the back propagation algorithm
for neural networks after the feedback loop shown in Fig.3
is removed. Where, the suffix s shows layers and s = 0 and 1
are input layer to the hidden layer and the hidden layer to
the output layer, respectively. The suffix i and j denote the
neurons number and depend on layer; if s = 0, then i =
0,1,2,3 and j = 0,1, …,9; if s = 1, then i= 0,1, …,9 and j = 0.
Since the error back propagating method to adjust synaptic
connection’s weight is introduced at many other references
[12]-[14], expressions related to it are abbreviated in the
present paper.


Fig. 3 Block diagram for controller equipped with neural networks

Voltage history including increment and decrement
process is applied to piezoelectric element to obtain data for
network learning. In the network learning, we used data
composed of about one hundred patterns of two bits
expressing increment or decrement, applied voltage,
displacement and displacement derivative dV/du of voltage.
Therefore, the value of displacement derivative dV/du of
voltage will be output from the output neuron. Since du/dt is
given from the trajectory planning, the time derivative of
voltage is calculated by multiplying du/dt by dV/du.
In addition, usual error back propagation is applicable
to the present neural network with removing the feedback
loop from the present network. This is different from the
RTRL [15] of recurrent neural networks.

2.3 Control System
In the present two-axial actuator, the positional error will
be caused by individual differences about not only the
inclination of a linear portion in the hysteresis loop but also
non-linearity and width of the loop. The influence of the
individual difference is modified by individually putting in
the characteristic of right and left actuators according to the
neural network model.
Figure 3 shows block diagram of the control system
designed on the basis of the above-mentioned idea. The
neural network is incorporated into this control system to
control right and left piezoelectric elements.
At first,
y
u & are decided from designed
trajectory in two-dimensional area. The displacement rates
of right and left type piezoelectric elements
x u & and
L
u & and
R
u &
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Fig. 4 Newly designed and developed new 2D micro
actuator and mechanical elements (the left panel shows the
2D micro actuator; a grain of rice displayed for comparison)

are calculated by substituting
kinematic equation Eq. (5). Then,
by numerical integral of
L
u & and
The 0th and the 1st neurons of right and left neural networks
in the input layer accept binary number 1 or 0 judging
increment or decrement condition. The condition of
increment or decrement is simply decided according to sign
of
L
u & and
R
u & . If the sign is positive or negative, the
judgment is increment or decrement. For example, if the
input vector of the left piezoelectric element is
()T
LL
Vu01
L
u10
specified as increment or decrement.
Since the displacement derivative of the voltage is output
from the output neuron of the neural network as previously
mentioned, the time derivative of voltage is calculated by
multiplying the time derivative of displacement by it. In
addition, after integration the voltage is applied to the
piezoelectric element, and is fed to the input layer through
the feedback loop.

3. EXPERIMENTAL PROCEDURE

3.1 Prototype of the Two-Axial Actuator
The parallel typed two-axial actuator is comprised of
two bimorph type piezoelectric elements, two small links
and three joints as shown previously in Fig. 4. The
piezoelectric element (length: 31mm, width: 2.0mm,
thickness: 0.50mm) is disassembled from Braille cell
SC9[5], which has been developed for the visually
handicapped person by KGS Ltd. The small links of 5mm in
length made of aluminum alloy are used in this actuator.
x u &
u and
u & .
and
y
u &
into the
u is obtained
LR
R
or ()T
L
V
, the condition is
0.51
0
0.5
1
Left piezoelectric element
: simulation
: experimental
Normalized displacement
Fig. 5 Simulated result of left piezoelectric element

Normalized voltage

0.51
0
0.5
1
Left piezoelectric element
: simulation
: experimental
Normalized displacement
Normalized voltage

Fig. 5 Simulated result of right piezoelectric element

Since the two links or the link and the piezoelectric
element end are connected with a joint of the aluminum
alloy, they are able to rotate mutually. The center of the
three joints was functioned as the movable end. In these
joints, micro sliding bearings are used to remove gap
between shaft and bearing. The bearing has the same size of
rice grain as shown in Fig. 4.

3.2 Evaluation Apparatus
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We used a microscope (OLYMPUS, inverted research
microscope: IX71) for measures displacements of the three
joints. The 1.25-power PlanApo×1.25 for objective lens and
the 0.35-power U-TV0.35×C-2 for a camera adaptor were
used. The movement trajectories of joint A, B and C were
measured by image data processing of image retrieved by a
CCD camera mounted in the microscope. In the image data
processing, centroid coordinates of these joints were
obtained with noise reduction and circular regression. The
above-mentioned operation was executed in each stepwise
voltage variation to measure trajectories of the three joints.
Multifunctional universal image analysis software (Library
Ltd., cosmos 32 Ver4.6) was used for the image processing.

3.3 Experimental Procedure
In order to adjust weight
the following experiment was performed. This experiment is
performed to examine the relationship between applied
voltage and displacements of right and left piezoelectric
elements. In this experiment, 100V datum voltage was
applied to right and left piezoelectric elements at first and it
was made a starting point. In discussion in the subsequent
sections, the voltage 100V will be expressed as 0V. In the
beginning the voltage is sequentially increased to 40V at
intervals of 10V, and then the applied voltage is decreased
to -40 V at intervals of 10V. Subsequently, it is decreased to
-60V after increased to 60V at intervals of 10V; it is
decreased to -80V after increased to 80V at intervals of
10V; it is decreased to -100V after increased to 100V at
intervals of 10V. In addition, it is increased to the starting
point 0V at intervals of 10V. Centroid displacements of
joint A, B and C were measured at every stepwise variation.

4. EXPERIMENTAL RESULT AND DISCUSSION

Hysteresis loops of the left piezoelectric element
obtained by the present experiment are shown in Fig. 5 by
●. The solid line in Fig. 5 is output result of the control
system shown in Fig. 3. The output result from the neural
network almost coincides with the experimental result for
not only a large loop of ±100 V but also a small inside loops
of ±40, ±60 and ±80 V.
Next, hysteresis loops of the right piezoelectric
element obtained by the present experiment are shown in
Fig. 6 by ●. The solid line in Fig. 6 is output result of the
control system. Inclination of the right element is the same
as that of the left element. The output result from the neural
network almost coincides with the experimental result.
Therefore, the high accuracy learning result is obtained as
abovementioned.
If the result of left piezoelectric element is compared to
that of right element, we can notice that displacement
)(s
ij
w
of synaptic connection,
amplitudes and the inclinations coincide on right and left
piezoelectric elements. Although
characteristics such as the loop width are considerably
different, the present neural system can follow the sight
difference.
In addition, 100 thousand times of iteration was needed
to obtain the learning result of Fig.6. The calculation was
executed on a notebook computer (Panasonic, CF-R4), and
required about 15 minutes to complete the learning
calculation.

5. CONCLUSION

The control method of two-axial actuator was presented
to enhance positioning accuracy and to apply it to the tactile
display. In order to realize sensor-less control of
piezoelectric actuators possessing obvious hysteresis
characteristic, we established a new neural network model
including feedback loop based on causality that the time
derivative of applied voltage was determined by increment
or decrement condition, current voltage and displacement.
The two-axial micro actuator was composed of the right
and left bimorph piezoelectric elements, two links and three
joints. The control system was also developed with
incorporating the neural network for compensation of the
hysteresis characteristic.
The learning was terminated within reasonable
calculation time; after the learning process, it was able to
reproduce hysteresis characteristics including several minor
loops in high accuracy.
In the future, it is necessary to enhance accuracy of in
software with increasing number of learning data and
accuracy joint bearings with exchanging the present
hand-made one for precise one.

ACKNOWLEDGEMENT

This study was supported by fiscal 2008 grants from the
Ministry of Education, Culture, Sports, Science and
Technology (Grant-in-Aid for Scientific Research in
Priority Areas, No. 1607807)

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precise hysteresis
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