IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2020 1
Self-Sensing and Feedback Control for a Twin Coil
Spring-Based Flexible Ultrasonic Motor
Yunosuke Sato1, Ayato Kanada2, and Tomoaki Mashimo1
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Abstract—We propose a twin coil spring-based soft actuator
that can move forward and backward with extensibility and can
bend left and right with ﬂexibility. It is driven by two ﬂexible ul-
trasonic motors, each consisting of a compact metallic stator and
an elastic elongated coil spring. The position of the end effector
is determined by the positional relationship of the two coils and
can be kinetically controlled with a constant curvature model.
In our design, the coil springs act not only as a ﬂexible slider
but also as a resistive positional sensor. Changes in the resistance
between the stator and the coil spring end are converted to a
voltage and used for position detection. Each ﬂexible ultrasonic
motor with the self-sensing is experimentally evaluated, and it
has shown good response characteristics, high sensor linearity,
and robustness, without losing ﬂexibility and controllability. We
build a twin coil spring-based ﬂexible ultrasonic motor prototype
and demonstrate feedback control of planar motion based on the
constant curvature model.
Index Terms—Soft sensors and actuators, ﬂexible robots,
continuum robots, piezoelectric actuators.
IMPROVING the ﬂexibility and compliance of actuators
is essential for extending the use of continuum robots in
applications such as medicine and rescue –. One typical
driving method for continuum robots uses electromagnetic
motors placed at external sites to control end effectors by
mechanisms such as wires (tendons) that transmit traction
forces , . In such robots, commercial rigid sensors
are mostly attached to the positions of the actuators. These
systems are regarded as having low controllability because
their complicity increases with the number of joints between
the sensors and the end effectors. The sensors can be placed
directly at the joints of the end effectors; however, conven-
tional sensors lack the softness and enlarge the joints.
Manuscript received: February, 24, 2020; Revised May, 21, 2020; Accepted
June, 23, 2020.
This paper was recommended for publication by Editor Cecilia Laschi upon
evaluation of the Associate Editor and Reviewers’ comments. This work was
supported in part by the JSPS KAKENHI under Grant 19H02110.
1Yunosuke Sato and Tomoaki Mashimo are with the Department
of Mechanical Engineering at Toyohashi University of Technology, 1-
1 Hibarigaoka, Tempaku-cho, Toyohashi, Aichi, 441-8580, Japan.
2Ayato Kanada is with the Department of Mechanical Engineering at
Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan.
Digital Object Identiﬁer (DOI): see top of this page.
With the rapid increase in the study of soft actuators, new
internal sensors with softness are required, and they are de-
signed according to the driving principles of soft actuators. The
largest class of prospective soft actuators is ﬂuidic elastomer
actuators (FEA), in which soft materials (e.g., gels and elas-
tomers) enclose ﬂuids such as air and functional ﬂuids –.
They are deformed by the expansion and compression of the
ﬂuid when a voltage or air pressure is applied. For the control
of FEAs, many intrinsic sensing methodologies have been
proposed using a variety of principles, such as strain gauges
using liquid metals (e.g., eutectic gallium-indium (eGaIn) ,
), conductive rubbers , capacitive strain sensors ,
, optical ﬁber sensors , , hall effect sensors ,
, and inductive sensors –. These sensors have
compliance and extensibility but lower resolution and linearity.
Another class of soft actuators involves deformable materials
that can produce a strain, such as dielectric elastomer ,
. Such materials can function as self-sensing to detect their
deformation , , but they have similar difﬁculties to
other types of sensors.
We have studied a “ﬂexible ultrasonic motor,” a new kind
of soft actuator designed for the use in soft continuum robots
. It consists of a single cubic stator with a center hole
threaded with an elastic elongated coil spring. When voltages
are applied to piezoelectric elements on the stator, the coil
spring moves linearly. One advantage of using a coil spring
is that its ﬂexibility and stroke (the traveling distance) are
designable by selecting the dimensions of the coil, such as
its diameter, cross-section, and the number of turns. In this
design, the coil spring has another signiﬁcant advantage in
terms of improving motor performance. The coil is designed
to have a slightly larger diameter than the stator hole through
which it passes. The force acting on the inner surface of the
stator produces an optimal pre-pressure at the stator-spring
interface to enhance the motor’s thrust force. In our previous
experiments, the actuation of a single ﬂexible ultrasonic motor
was demonstrated and evaluated. The motor succeeded in
actuating a coil spring that was being sharply bent under
curved constraint conditions . This result have shown
potential as a new soft actuator. However, none of the control
strategies have yet been studied, and sensors have not yet been
used to achieve control.
In this paper, we propose a coil spring-based soft actuator
2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2020
Fig. 1. Twin coil spring-based ﬂexible ultrasonic motor (Twin-coil USM).
using two ﬂexible ultrasonic motors, the coil ends of which
are connected by a plastic component, as shown in Fig. 1.
This actuator has been named “Twin coil spring-based ﬂexible
ultrasonic motor (Twin-coil USM)” to distinguish it from a
single ﬂexible ultrasonic motor. In addition to the two noted
roles of the coil spring, its ﬂexibility and pre-pressure, we also
used it as a linear variable-resistance sensor. For this linear
sensor, a thin wire is attached to either end of the coil spring.
When a voltage is applied, a voltage drop occurs between one
end of the coil and the stator’s ground, and the position can
be estimated. There are three advantages to this sensing. The
ﬁrst one is that no additional sensing component is required.
It means that the motor can achieve control without reducing
output power, ﬂexibility, and range of motion. Secondly, this
principle has linearity and results in a low calculation cost in
comparison with non-linear soft sensors. The third advantage
is that the sensor output determined by the distance between
the stator and the end of the coil is very stable, even when an
external force acts on the coil and increases its deformation.
The rest of this paper is organized as follows. Section
II explains the principles of the ﬂexible ultrasonic motor
and the coil spring-based position sensor. A prototype sensor
system embedded in a ﬂexible ultrasonic motor is built, and
the performance parameters, such as a sensing accuracy and
frequency response, are examined in Section III. Finally,
Section IV demonstrates the Twin-coil USM with a feedback
II. DE SI GN A ND FABRICATION
A. Overview of Drive Principle and Design
We introduce the driving principle of the single ﬂexible
ultrasonic motor brieﬂy . Fig. 2(a) shows a schematic
of the stator. Four piezoelectric elements adhere to the four
sides of the metallic cube. The metallic part of the stator is
a phosphor bronze cube of edge length 14 mm with a center
hole of 10 mm in diameter. The inside of the hole is coated
by electroless nickel plating with a thickness of 10 µmto
prevent damage and wear being caused by contact with the
coil spring. Each piezoelectric plate has a length of 14 mm, a
width of 10 mm, and a thickness of 0.5 mm, and has two silver
electrodes on one side for applying two different voltages. As
shown in Fig. 2(b), two vibration modes are used as the driving
Fig. 2. Driving principle of the ﬂexible ultrasonic motors. (a) Schematic of
the motor. (b) Two vibration modes (Mode 1 and Mode 2) generated by stator
and an elliptical motion. (c) Applied voltages for the motor. (d) Pre-pressure
mechanism by using coil spring slider.
principle to move the coil spring slider linearly. Mode 1 is a
vibration that repeats expansion and contraction symmetrically
about the center cross-section, and Mode 2 is asymmetrical.
When both vibration modes are excited simultaneously at the
same resonance frequency, the stator produces an elliptical
orbit (lower right in Fig. 2(b)). This elliptical orbit moves the
coil linearly by friction between the stator and the coil. When
two voltages are applied to the piezoelectric elements (Fig.
2(c)), both modes are excited at the same driving frequency.
These voltages are expressed as
where AEand fEare the amplitude and frequency of the
voltages, respectively, and ϕis the phase between the two
voltages. The frequency fEcan be tuned to the resonance
frequency of the two vibration modes. The coil moves forward
when the phase ϕis set to π/2, and backward when ϕis −π/2.
In general, ultrasonic motors that use friction as their driving
principle require a pre-pressure between the stator and the
slider/rotor to enhance the force/torque. They need an addi-
tional mechanism and components to generate and optimize
the pre-pressure. In our design, the coil spring works not only
as the coil slider but also as the pre-pressure mechanism.
As shown in Fig. 2(d), the diameter of the coil is slightly
larger than that of the stator hole. When the coil is twisted
appropriately in the circumferential direction, its diameter
decreases, and it can be smoothly inserted into the stator hole.
When the torsional force is removed, the coil expands in the
radial direction, generating a pre-pressure without the need for
B. Self-Sensing Using the Coil Spring
We present a new sensing methodology to detect the dis-
placement of the coil spring slider. As mentioned above, the
SATO et al.: SELF-SENSING AND FEEDBACK CONTROL FOR A TWIN COIL SPRING-BASED FLEXIBLE ULTRASONIC MOTOR 3
coil spring inserted into the stator hole has two essential roles:
ﬂexibility and pre-pressure. In this study, we also uses the coil
as a linear resistive sensor. In other words, this single ﬂexible
ultrasonic motor behaves like a linear resistive potentiometer,
which is a kind of three-terminal resistor consisting of an
electrical resistance and a sliding contact. Fig. 3(a) shows the
self-sensing design concept for the coil. The coil and the stator
are treated as the resistance element and the sliding contact of
a linear potentiometer, respectively. When a voltage is applied
to the ends of the coil, a voltage drop occurs between each
end and the stator, which is at ground potential. When the coil
moves, the voltage drop changes continuously in proportion to
the displacement of the coil, and its position can be measured.
One advantage of using such a potentiometer is the stability
inherent in the electrical connection between the resistance
element and the stator. In the design of the coil, the coil
expands in the radial direction and makes ﬁrm contact with the
inner surface of the stator hole. Wherever the stator is located
along the resistance element, the electrical connection remains
Fig. 3(b) shows an electrical model of the potentiometer. We
deﬁne the resistance of the whole coil as R0, and the stator
divides it into R1and R2. The resistances of the wires are
denoted as R3and R4. When a voltage Ein is applied to both
ends of the coil, the output voltage Eout is obtained as
This is the output voltage from the potentiometer. With the
cross-sectional area Sand electrical resistivity ρof the coil
spring, the relative position of the coil spring to the stator is
When (4) is substituted into (3), the position pis obtained
from the measured voltage Eout.
Since all variables in (5) are constant, it can be rewritten using
the arbitrary constants Cand D, as follows:
p=CEout +D. (6)
This equation shows that the relationship between the mea-
sured voltage Eout and the position pis linear.
C. Constant Curvature Model
We model the motion of the Twin-coil USM to estimate the
position of the end effector. The constant curvature model is a
well-known forward kinematics formula for continuum robots
. A Twin-coil USM with two ﬂexible ultrasonic motors can
move and bend the end effector by the relationship between
the two coils. Considering that the coils will move on a plane
in the experiments, as described in a later section, we use a
planar constant curvature model to express the motion.
Fig. 4 shows a schematic of the constant curvature model.
The position of the end effector is expressed as Px=r(1 −
Fig. 3. Principle of self-sensing using the coil. (a) Mechanical components
and the simpliﬁed electrical model. (b) Detailed electrical model.
Fig. 4. Constant curvature model for the twin coil spring-based ﬂexible
cos θ)and Py=rsin θ. Here, ris the bend radius and θis
the angle between the x-axis and the line P Q. The solid lines
represent the coil springs of the Twin-coil USM. The springs
are held at a distance of 2dfrom each other. The arc lengths
of the coil springs (i.e., the dashed lines in the range y > 0)
are set to l1and l2. Using the arc lengths l1and l2and the
distance d, the end effector’s position Px=r(1 −cos θ)and
Py=rsin θcan be expressed as follows:
These equations are the forward kinematics equation
f(l1, l2)=(Px, Py). The solution of the inverse kinematics
equation f−1(Px, Py)=(l1, l2)can be solved numerically.
Note that this constant curvature model ignores the inﬂuence
of disturbances, such as external forces.
A. Evaluation of Self-Sensing
The self-sensing apparatus is built and experimentally eval-
uated. During the experiments, a constant voltage Ein of
140 mV is applied to the coil spring. The output voltage
Eout is ampliﬁed to 55 times by an ampliﬁer circuit because
the original signal is very low. This value is converted by
a 10-bit analog-to-digital (AD) converter with a reference
voltage of 5 V. The voltages obtained are averaged over 10
measurements to reduce noise. Fig. 5(a) shows the behavior
4 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2020
Fig. 5. Change in the sensor output when (a) the coil moves linearly, (b)
the coil bends, and (c) the coil extends and contracts. (Error bars indicate SD
from ﬁve tests of one condition).
of the sensor output when the coil moves. In the abscissa
axis, the displacement between one end of the coil spring and
the front surface of the stator is taken from 0 mm to 100
mm in 10 mm steps. In this experiment, the coil spring is
manually moved using a scale. The error bars indicate the
standard deviation from ﬁve tests at each position. The results
show that the relationship between the displacement and the
output voltage is linear, and the maximum standard deviation
is 28.1 mV. The constants in (6) are obtained by approximating
this result by a least-squares method (C= 26.6 and D= 34.5).
We examine how the sensor output changes when applying
external forces such as those experienced when bending,
extending, or contracting the coil spring. In the experiments,
the output voltages in each condition are measured ﬁve times.
The coil spring is set to a displacement of 50 mm and is
ﬁxed by insulating tape. The coil springs are set in constraint
components with a radius of 5, 10, 15, or 20 mm. Fig. 5(b)
shows the voltage change for each bending radius. The voltage
change is slight at all bending radii, and the maximum is less
than 4.0 mV even at a bending radius of 5 mm. This value is
as small as the resolution of the AD converter.
The sensor output with the expansion and contraction of
the coil is evaluated. One end of the coil spring is ﬁxed to
a force gauge to measure the restoring force. The voltage is
measured while the restoring force changes from −0.5 N to
+0.5 N in 0.1 N steps (negative values indicate compression).
Fig. 5(c) shows the variation in voltage with changing force.
When the restoring force is 0 N in the coil, the output voltage
is deﬁned as 0 V. The variation in the output voltage occurs
by the change in the contact condition between the stator
(electrical contact) and the coil spring (resistance). As the coil
Fig. 6. Change in the sensor output from coil B when the displacement when
the displacement of coil A increases (a) and decreases (b). The displacement
difference (abscissa) is the subtraction of coil A from coil B. (Error bars
indicate SD from ﬁve tests of one condition).
spring compresses, the gap of the coils reduces. At −0.5 N,
the coils contact and lower the electric resistance. The output
voltage is approximately −27.5 mV, which is equivalent to a
displacement of 0.73 mm in this displacement sensor. Because
it is smaller than the maximum standard deviation of the
linear movement (Fig. 5(a)), the resulting sensor has sufﬁcient
robustness against disturbance.
Another important aspect is the implementation of the
sensing methodology into the Twin-coil USM. Let us examine
how the output signal of the sensor varies when the two coils
form a curve in the Twin-coil USM. Fig. 6 shows the sensor
output from coil B when coil A moves and coil B is stationary.
The output voltage slightly increases at a larger displacement
of coil A (Fig. 6(a)), and vice versa (Fig. 6(b)). The voltage
change is 26.6 mV (a displacement of 0.71 mm) at maximum.
B. Feedback Control Experiment
We build a feedback control system consisting of a single
ﬂexible ultrasonic motor and the self-sensing. Fig. 7 shows
the self-sensing feedback control loop. This circuit includes a
central processing unit (an Arduino Uno), a two-phase inverter,
a direct digital synthesizer (DDS), an ampliﬁer, and a PC.
To drive the ﬂexible ultrasonic motor, the two-phase inverter
converts a rectangular wave of 5 Vp−pfrom the DDS into
a sine wave of 120 Vp−pby a bridge circuit and an LC
ﬁlter circuit. To control the position and speed of the ﬂexible
SATO et al.: SELF-SENSING AND FEEDBACK CONTROL FOR A TWIN COIL SPRING-BASED FLEXIBLE ULTRASONIC MOTOR 5
Fig. 7. Feedback control circuit for the ﬂexible ultrasonic motor with self-
Fig. 8. Time table of the control cycle.
ultrasonic motor, the Arduino changes the frequency and phase
of the rectangular wave by communication through the Serial
Peripheral Interface (SPI). A USB cable connects the PC and
One of the technical problems in the control system is that
the actuation voltage and the sensing signal use the same
terminal of the stator, as shown in Fig. 2 and Fig. 3. The sensor,
therefore, suffers from noise due to the high driving voltage
applied while the motor is moving. To overcome this problem,
we implemented a program to divide the operating time into
two separate sensing and actuation periods in one control cycle
of 11.5 ms, as shown in Fig. 8. In the initial period of 3 ms,
the AD converter reads the output voltage from the sensor. For
the next period of 6.5 ms, the driving voltage is applied to the
ﬂexible ultrasonic motor. The remaining 2 ms is a waiting time
for the safety of the system. The proportion of actuation time
in one cycle is about 60%, and this reduces the speed of the
motor. These times were determined experimentally to obtain
Next, we consider how to control the motion of the ﬂexible
ultrasonic motor. The ﬂexible ultrasonic motor changes its
velocity and traveling direction by modulating the frequency
fEand the phase ϕof the applied voltages, respectively, as
described by (1) and (2). Fig. 9 shows the forward velocity
(ϕ=π/2) and the backward velocity (ϕ=−π/2) of the
motor when the frequency of the applied voltages is changed
from 81.0 kHz to 84.5 kHz. The error bars show the standard
deviations of ﬁve experiments because the coil vibrates in
the traveling direction during the motion. Although there is
a difference between the forward and backward velocities,
both velocities peak at the resonance frequency (81.5 kHz)
and gradually decrease at higher frequencies. Using these
characteristics, it is possible to control the motion of the
ﬂexible ultrasonic motor by adjusting the driving frequency
Fig. 9. Relationship between the velocity and the driving frequency. (Errorbars
indicate SD from ﬁve tests of one frequency).
Fig. 10. Feedback control scheme.
fEbetween 81.5 kHz and 84.5 kHz. The difference between
the forward and backward velocities is due to experimental
factors such as fabrication and adhesion of the piezoelectric
Fig. 10 shows the closed-loop position control scheme. The
proportional (P) controller determines the frequency fEand
the phase difference ϕbased on the displacement error e.
Since the motor velocity depends on the traveling direction,
the constant of P controller has different values in the forward
and backward directions. Although the relationship between
the voltage frequency and the velocity is non-linear, we assume
it as linear for simplicity. The displacement of the coil is
estimated by measuring the ampliﬁed voltage Eout. To reduce
noise, the output signal passes through a 10-sample moving-
average ﬁlter and a low pass ﬁlter with a cutoff frequency of
We investigate the frequency response of the feedback
control system. Fig. 11 shows the bode plot when reference
sine waves of between 0.1 and 5 Hz and a constant amplitude
of 60 mm are given as an input. The controller is able to follow
the inputs up to about 0.5 Hz without any delay. Although the
response depends on the reference displacement, the results
show a good response characteristic in comparison with other
linear motors because the inertia of the coil spring is very low
for a generated torque.
IV. DEM ON ST RATION OF A TWIN-CO IL U SM
We build a Twin-coil USM using two ﬂexible ultrasonic
motors and demonstrate its feedback control. Fig. 12(a) shows
a schematic diagram of the Twin-coil USM, in which the
two coils are aligned in parallel, and the ends of the coils
6 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JUNE, 2020
Fig. 11. Bode plot for the ﬂexible ultrasonic motor.
are connected to form an end effector. The two stators are
ﬁxed to a housing part, and the distance between their centers
is approximately 17 mm. As shown in Fig. 12(b), ﬂexible
bronze electrodes are attached to the housing to stabilize the
electrical contact with the piezoelectric elements on the stator.
Although the end effector ﬁxes one end of each coil, the other
end remains free. The end effector of the Twin-coil USM can
move and bend by controlling the displacement of the two
coils. To control the two coils, we added a two-phase inverter
and a DDS to the control circuit shown in Fig. 7.
For the demonstration of feedback control, a circle with a
diameter of 25 mm at a position 35 mm away from the edge
of the stator’s housing is set as the desired trajectory. This
circle is approximated by a 36-sided polygon prepared from
an inverse kinematics correspondence table. The end effector
is made to draw the same circle four times at a constant speed
(11.5 s per lap) to evaluate repeatability. The motion of a
marker on the end effector is tracked by a camera with a
frame rate of 30 Hz (Fig. 12(c)). The sensor outputs from the
two coils are also recorded at the same time. The experimental
conditions to drive the motor and sense position are the same
as the previous section.
Fig. 13 shows the response of each coil spring as measured
by the self-sensing. The position of each coil spring shows a
good agreement with the desired trajectory from the constant
curvature model, without overshoot or delay. This result means
that the end effector should have drawn the desired circle, but
the actual motion showed an unexpected trajectory. Fig. 14
shows the motion of the end effector obtained by the camera.
The recorded trajectory appears as a distorted ellipse, and it
repeats the almost same trajectory four times. The difference
between the sensor and the camera is caused by external
forces and friction. It can be clearly seen in the x-direction,
even though the stators move the coil based on the constant
curvature model. There are two probable reasons for this: (1)
Fig. 12. Structure of Twin-coil USM. (a) Schematic diagram of Twin-coil
USM. (b) Housing and electrodes for the stator. (c) Image taken by 2D
tracking camera for tracking end effector.
Fig. 13. The signals from the two coil spring-based resistive sensors. They
are in good agreement with the curve from the constant curvature model.
friction between the end effector, and the ground restricts the
motion, and (2) the stiffness of the proposed actuator in the
x-direction is lower than that in the y-direction due to the
inherently elongated structure of the spring. In fact, when we
lift the end effector to remove the friction, the end effector
moves to the desired circle in the neighborhood of positions
of (b) and (d).
We proposed a self-sensing-based soft sensor for a ﬂexi-
ble ultrasonic motor and demonstrated the feedback control
of a Twin-coil ﬂexible ultrasonic motor. Hence, the elastic
elongated coil had three important functions: the ﬂexibility
(compliance), pre-pressure, and resistance sensor. This sensor
was able to obtain a good positioning accuracy less than an
error of 0.75 mm and linearity over a wide range of motion
from 0 mm to 100 mm. Furthermore, the system showed high
electrical stability even when the coil spring was sharply bent
with a minimum radius of 5 mm or pulled/pushed with a
maximum force of 0.5 N. A feedback control system was
constructed and evaluated experimentally. A single ﬂexible
ultrasonic motor with a resistive sensor showed a frequency
SATO et al.: SELF-SENSING AND FEEDBACK CONTROL FOR A TWIN COIL SPRING-BASED FLEXIBLE ULTRASONIC MOTOR 7
Fig. 14. Experimental circular motion obtained by the camera. Four snapshots
from (a) to (d) are accorded to the points in the ellipse trajectory during the
response that was able to follow an input of up to about
0.5 Hz without degradation of gain or phase delay. We
built a Twin-coil USM using two ﬂexible ultrasonic motors
and implemented a feedback control of tracking a desired
trajectory, but an unexpected error between the camera and
the resistive sensor occurs. In future work, we will derive a
correct model based on the modiﬁed constant curvature model
incorporating friction and external forces.
The proposed sensor-actuator system is still under develop-
ment, and there are many ways it can be improved. First, the
noise resistance, which is robustness against the inﬂuence of
external noise, can be enhanced. Since the coil spring is made
from stainless steel and has low resistance, the sensor requires
a very low voltage to minimize power consumption and heat
dissipation, which results in weak noise resistance. Increasing
the electrical resistance by an electrostatic coating can increase
the resolution of the sensor and its susceptibility to noise.
Second, the motor response can be improved. The controller
restricts the motor response by alternating the operation be-
tween sensing and actuation in a control cycle. Electrically
insulating a part of the stator to separate the sensing and
actuation grounds would allow the controller to drive both the
sensor and the actuator simultaneously, improving the motor
response. Third, it would be possible for the sensor to measure
more complex motion without changing its structure or adding
additional components. Although the proposed sensor only
measures the displacement of the coil, it is known that the
strain of a coil spring can be estimated by measuring its
inductance . Inductance-based self-sensing could also be
embedded in our proposed system without the need for extra
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