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JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 1
Design and Experiments of Flexible Ultrasonic
Motor using a Coil Spring Slider
Ayato Kanada, Student Member, IEEE, and Tomoaki Mashimo, Member, IEEE
©2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other
uses, in any current or future media, including reprinting/republishing this material for advertising or promotional
purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted
component of this work in other works. DOI: 10.1109/TMECH.2019.2959614
Abstract—This paper proposes a flexible ultrasonic motor that
consists of a single metal cube stator with a hole and an elastic
elongated coil spring inserted into the hole. When voltages are
applied to piezoelectric plates on the stator, the coil spring moves
back and forward as a linear slider. The use of the coil spring
brings flexibility for the motor and enables a long stroke to
access to deeper sites. Furthermore, the coil spring provides an
appropriate pre-pressure between the stator and the coil spring
to enhance the motor output. We formulate the relation between
the coil spring parameters and the pre-pressure to clarify the
design methodology of the flexible ultrasonic motor. We model
the linear motion of the coil spring by an equation of motion
and compare it with the transient response by experiments. The
flexible ultrasonic motor prototype achieved a translation at a
speed of 120 mm/s and demonstrated a force of 0.45 N, and a
stable motion even when the coil spring is being bent.
Index Terms—Ultrasonic motor, soft actuator, pre-pressure,
coil spring, continuum robot.
I. INTRODUCTION
CONTINUUM robots that can access hard-to-reach targets
in unpredictable environments have a wide range of
potential applications from rescue to medicine [1]–[3]. For
example, a typical place where continuum robots apply is the
interior of gastrointestinal tracts which are soft and possess
considerable curvatures. A flexible and elongated continuum
robot goes inside our body via the mouth and enables the
diagnosis and treatment of digestive diseases. With the in-
crease in the expectation of such continuum robots, many
hardware designs have been proposed and demonstrated. The
important characteristics of these robots to explore deeper and
inaccessible sites are flexibility and stroke. High flexibility
is necessary to conform to surroundings and a long stroke
expands reachable work areas. One challenge of continuum
robot design is to have both high flexibility and long stroke
in the elongated body with a small diameter (e.g., concentric
tube robots [4]). However, it is difficult to have both the char-
acteristics because the designable space inside the elongated
tube robots is limited.
This work was supported by JSPS KAKENHI Grant Number 16H06075
and the Leading Graduate School Program 03.
A. Kanada and T. Mashimo are with the Department of Me-
chanical Engineering, Toyohashi University of Technology, 1-1 Hiragi-
gaoka, Tenpaku-cho, Toyohashi, Aichi, 441-8580, Japan e-mail: (see
http://eiiris.tut.ac.jp/mashimo/wordpress/en/top-page/).
Typical driving methods for continuum robots are tendon-
driven and pneumatically driven [5]–[7]. For example, tendon-
driven robots are controlled by wires that are connected to
electromagnetic motors or shape memory alloys [8]–[12].
These robots can obtain a relatively slender body and good
accessibility into narrower spaces, but they require many com-
plicated mechanical components. Pneumatically driven robots
have an inherent compliance. There are several pneumatic
actuators with a large stroke (e.g., origami structures [13] and
folded balloons [14]), but they require a large and complex
system using a compressor, air tubes, and valves at external
sites.
Ultrasonic motors based on piezoelectric phenomena are
known as the actuators used for the autofocusing system in
camera lenses [15]. None of us may have thought that these
ultrasonic motors can be used as flexible actuators because
they are in general known as rigid actuators without softness
[16]– [19]. Before talking about the flexible ultrasonic motor
that we propose, let us summarize several advantages of the
existing ultrasonic motors. (i) The high torque density that
generates a high torque from a small volume stator is a key
characteristic to miniaturize the stator. (ii) The simple structure
consisting of a stator and a rotor reduces the number of me-
chanical components. (iii) The design flexibility that can make
the stator hollow is suited for applications in narrow spaces.
(iv) They have an MRI compatibility [15] and might be useful
for medical applications such as MRI compatible concentric
tube robots. All these advantages are desired characteristics to
design new continuum robots.
In this paper, we propose a flexible ultrasonic motor that
uses an elastic elongated coil spring as a linear slider. This
motor simply consists of a single cube stator with a through-
hole and a coil spring inserted into the hole. The coil spring
has a slightly larger diameter than the stator hole and contacts
the inner surface of the stator. Not only does the coil spring
provide flexibility to the slider, but also acts as a pre-pressure
mechanism to improve motor output. When voltages are
applied and the stator generates a driving force, the coil spring
moves back and forth as shown in Fig. 1. This motor can
obtain both a flexibility and stroke (travelling distance) by
designing the dimensions of the coil spring. The flexibility is
determined by coil spring parameters, such as the diameter, the
cross-section dimensions, and Young’s modulus. The travelling
distance of the flexible spring slider can be designed to be
JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 2
almost the same as the length of the coil spring.
The main contribution of this paper is the design and
experimental characterization of the flexible ultrasonic motor.
It reveals the relation of the motor performance to the pre-
pressure determined by the coil spring dimensions. (The
motion of a flexible ultrasonic motor was partially shown in a
conference proceeding [20], but the design was not mentioned
and the experiments were primitive.) The rest of the paper is
organized as follows. In Section II, we describe the driving
principle of the flexible ultrasonic motors and clarify how
to formulate the relationship between the stator and the coil
spring. Section III shows a prototype flexible ultrasonic motor
and evaluates basic performance such as the thrust force and
velocity in experiments. In Section IV, the linear motion of
the coil spring presented in Section II is compared with the
transient response experiment.
II. PRINCIPLE AND MODELING
A. Driving Principle
Fig. 2 shows the driving principle of the flexible ultrasonic
motor for a linear motion. The stator is composed of a single
metallic cube with a through hole and two piezoelectric plates
adhered on the sides, as shown in Fig. 2(a). The piezoelectric
plates are polarized in the thickness direction and their outside
is positive pole. The negative pole bonded to the metallic
cube is equivalent to ground. To move the coil spring slider
linearly, two vibration modes are simultaneously excited by
the piezoelectric plates as the driving principle. We call
the two vibration modes T1 and T2 modes for translation.
Fig. 2(b) shows how these vibration modes vibrate. The T1
mode is symmetric about the stator center, whereas the T2
mode is asymmetric. When two voltages are applied to the
piezoelectric plates, both modes are excited at the same driving
frequency. These two voltages are expressed as
E1=AEsin (2πfEt)(1)
E2=AEcos (2πfEt)(2)
where AEand fEare the amplitude and the frequency of the
applied voltage, respectively. The frequency fEis adjusted to
be equal to the natural frequency of both vibration modes.
When the symmetric and asymmetric modes are excited, the
stator vibration describes an elliptical trajectories, as shown
in Fig. 2(c). These elliptical motions generate throughout the
stator hole, and they become the maximum at both ends of
the stator. They move the coil spring slider inserted into the
stator hole by a friction between the stator and the slider. This
driving principle that uses both symmetric and asymmetric
modes is similar to that used in linear ultrasonic motors [21],
[22].
B. Design of the Coil Spring and Pre-pressure
In the flexible ultrasonic motor that uses the friction drive
as the principle, the most important parameter for optimizing
its output is the pre-pressure between the stator and slider.
The magnitude of the pre-pressure can be designed from the
dimensions of the coil spring and the diameter of the stator
Fig. 1. Flexible ultrasonic motor. The coil spring inserted to the stator can
move back and forth when voltages are applied.
hole. The coil spring slider is composed of a single metallic
wire formed into a helix. It has a slightly larger diameter than
the stator hole. Fig. 3(a) shows an original coil spring and the
coil spring inserted to the stator hole. The coil spring with an
outer radius r1shrinks to the hole radius r2. The shrinkage
of the outer radius is defined as ∆r(=r1−r2). The coil
spring has a rectangular cross-section with a width band a
thickness h, as shown in the detailed view in Fig. 3(a). The
median centerline of the coil spring exists at the cross-section
center vertically, and the median centerline length that spirals
inside the stator hole is defined as L. In other words, Lis
the product of 2π, the radius r2, and the number of turns N
between both the edges of the stator after the coil insertion:
L= 2πr2N(3)
When the coil spring is inserted into the stator hole, the shrunk
coil generates the pre-pressure Pat the interface between the
stator and the coil spring as shown in the right of Fig. 3(a).
To estimate the pre-pressure value from the coil’s parameters,
we consider two types of the elastic potential energy stored
in the coil spring: strain energies by a shrinkage in the radial
direction and by bending deformation. Assuming that these
two energies take the same value, we can estimate the pre-
pressure using this equivalence. First, we consider the energy
by the shrinkage in the radial direction. Deriving a rigid
solution of the radial deformation is too complicated because
the coil spring with a thick cross-section has a non-linearity. In
addition, the shrinkage of the coil spring results in the radial
and circular deformations. To simplify this radial deformation,
we regard the coil spring as a cylinder with an unknown
JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 3
(a) (b)
E
1
E
2
1 2 3 4
Piezo plate
Slider
Stator
T1 + T2 mode
(c)
Fig. 2. The driving principle of the flexible ultrasonic motors. (a) Schematic
of the stator. (b) Vibration modes (T1 and T2 modes) generated by the stator.
(c) When T1 and T2 modes are simultaneously excited, the stator generates
an elliptical motion and moves a slider.
elastic coefficient. When the pressure Pis applied in the
radial direction, the coil spring shrinks with a displacement
of ∆r. The work done by the pressure is equivalent to the
strain energy stored in the coil spring:
U=1
2P bL∆r(4)
where the product of the width band length Lis similar to
the outer surface area where the pressure acts. This is the
energy stored by the radial shrinkage, and the pressure Pis
still unknown in (4). The pressure Pcan be estimated after
the strain energy is solved from the bending deformation.
Second, we consider the bending deformation of the Euler–
Bernoulli beam, which is well known in the mechanics of
materials [23]. Fig. 3(b) shows an element of the coil spring
from the view of the axial direction of the stator hole. When
the coil spring is inserted into the stator hole and bends,
the upper part of the beam is in tension and the lower is
in compression. In somewhere between the top and bottom,
there is a neutral line, which is neither under tension nor
compression. An elemental length of the neutral line that
remains constant is defined as ds. Denoting the deformation
at a distance yfrom the neutral line as ∆ds, the strain εis
determined as ∆ds/ds (ε= ∆ds/ds). The strain energy by
the bending deformation is the integral over the volume of the
coil spring:
U=ZV
1
2Eε2dV =1
2EbL Zh/2
−h/2
ε2dy (5)
where Eis Young’s modulus and the coil spring volume Vis
the product of b,h, and L. How the strain εchanges with y
is geometrically determined when the dimensions of the coil
spring and the inner radius of the stator hole are determined.
The strain εis expressed as
ε=∆r
r1r2
y(6)
Substituting (6) into (5), the strain energy can be obtained.
Hence, these equations (4)-(6) show the relation between the
pre-pressure and the design parameters of the coil spring slider.
The pre-pressure Pcan be estimated by substituting the energy
Usolved in (5) into (4).
C. Modeling of Translational Motion
We build a mechanical model to estimate the linear motion
of the flexible ultrasonic motor. In general, the motion of
the ultrasonic motors is expressed as a first-order lag system
regardless of rotary and linear motions [24]. When the stator
generates a force Fand the slider translates with the velocity
˙x, the motion is expressed as
m¨x+c˙x=F(7)
where mis the mass of the slider and cis the damping
coefficient. This damping coefficient is determined by the axial
velocity of the elliptical motion generated by the stator [25].
This is the simplest model of the linear ultrasonic motor with
a rigid slider.
In the flexible ultrasonic motor, the coil spring exists at
both sides of the stator; therefore, the equation of motion
must incorporate the spring components of the coil spring
in addition to the above model. Fig. 4 shows the model of
the flexible ultrasonic motor with a coil spring slider, both
the sides of which are expressed as mechanical components:
mL,cL, and kLare mass, damper, and spring at left side,
respectively, and mR,cR, and kRare those at right side. The
sum of mRand mLis the mass of the whole coil spring
slider. The terms with cLand cRare mechanical loss in the
coil spring, regardless of the stator’s vibration. This model
has three degrees of freedom with xL,x, and xR. When the
stator generate a force F, the motion of the coil spring can be
expressed by the equation of motion:
mL0 0
0 0 0
0 0 mR
¨xL
¨x
¨xR
+
−cLcL0
−cLcL+cR−cR
0cR−cR
˙xL
˙x
˙xR
+
−kLkL0
−kLkL+kR−kR
0kR−kR
xL
x
xR
=
0
F
0
(8)
This three degrees of freedom model is usable at x∼
=0. i.e.,
the motion can be estimated when the displacement xis in
the neighborhood of the stator position that generates a force
F.
When the displacement xenlarges and the position that
generates the force Fis far from the stator (x0), the model
(8) is not accorded to the actual. The stator is rigidly fixed
in an experimental setup while the coil spring slider moves.
In this case, the parameters of the coil spring slider change
with the displacement x. Fig. 5 shows the right side of the
stator when the displacement increases. Parameters at x0
are expressed using the prime symbol (0) to distinguish from
JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 4
ORIGINAL INSERTED
r
1
¨r
P
P
r
2
b
h
P
b
Front
Side
L
¨r
(a)
ds
ORIGINAL
ds
INSERTED
ds
y
¨ds
1
2
h
(b)
Fig. 3. Geometric relationship between the coil spring and the stator. (a) A coil spring with a slightly larger diameter than the stator hole diameter is inserted
to the stator hole. The coil spring shrunk to the stator hole generates pre-pressure in the radial direction. (b) The detail of the deformation of a coil spring
element. The cross section either lengthen or shorten, creating the strain.
x
L
k
R
l
ini
Stator
l
all
x
R
m
R
c
R
x
F
c
L
k
L
m
L
Fig. 4. A generalized model of the flexible ultrasonic motor, which is
expressed as three-degrees of freedom system.
those of x0. The mass and the spring coefficient become
a function of x. The mass mR
0is expressed as
mR
0=m
lall
(lini +x)(9)
where mand lall are the mass and length of the whole coil
spring, respectively, lini is the initial length between the stator
center and the coil spring end, defined in Fig. 4. The spring
coefficient kR
0is
kR
0= 2lall
k
lini +x(10)
where kis the spring constant of the whole coil spring. There
is a damping coefficient, but it can be regarded as constant
because its change is small. When the stator generates a force
F, the displacement xoccurs. Regarding that the behavior
of the displacement xis independent of the motion of the
masses, the displacement xcan be simply estimated from
the axial velocity of the elliptical motion. The relationship
between displacements xand xR
0is expressed by the equation
of motion with the variable mass and spring coefficient as
follow.
mR
0¨xR+cR( ˙x−˙xR) + kR
0(x−xR)=0 (11)
The motion at the left side in the model can be estimated by
replacing index Rwith Lin (9) to (11).
The natural angular frequency of the coil spring depends
on length of the coil at both the sides of the stator. Seeing
k
R
c
R
m
R
x = 0
x = X
T
t= 0 [s] x
R
t= T[s]
m
R
’
c
R
k
R
’
Fig. 5. A model at the right side of the coil spring. The mass and spring
coefficient become variables of the displacement.
the right side from the stator, the natural angular frequency is
described as
ω=skR
0
mR0(12)
This equation shows that the natural angular frequency de-
creases at larger displacements; that is, an end of the coil
spring vibrates slowly as it moves away from the stator.
III. PERFORMANCE EVALUATI ON
A. Prototype of the Stator
We describe how to build the flexible ultrasonic motor. The
stator consists of a metallic cube and four piezoelectric plates
on its four sides. The cube, made of phosphor bronze, has a
side length of 14 mm and a hole of 10 mm. Nickel plating is
coated inside the hole to reduce wear. In the neighborhood of
a corner of the stator, an internal thread of 1 mm in diameter
is opened to connect to a ground line. Each piezoelectric
plate with a length of 14 mm, a width of 10 mm, and a
thickness of 0.5 mm, has two silver electrodes on one side. The
four piezoelectric plates are bonded using an epoxy adhesive
(TB2280E, ThreeBond, Japan) at 120 ◦C for 2 hours. Although
the driving principle is explained by two piezoelectric plates in
section II, more piezoelectric plates are used to supply larger
electric power and to enhance motor output.
The resonances of the stator can be found by analyzing
the frequency characteristics of admittance. To move the coil
spring slider linearly, both the T1 and T2 modes should
JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 5
(a)
0.0001
0.001
0.01
0.1
70 75 80 85 90
Admittance [S]
Frequency [kHz]
T1 mode
T2 mode
(b)
-60
-30
0
30
60
90
120
70 75 80 85 90
Phase [deg]
Frequency [kHz]
T1 mode
T2 mode
(c)
Fig. 6. Frequency response of the T1 and T2 modes. (a) Connection of the
piezoelectric plates to the impedance analyzer. Frequency characteristic of (b)
admittance and (c) phase. The result shows that T1 and T2 modes occur at
the same resonant frequency.
be simultaneously excited at the same driving frequency.
We confirm that the two vibration modes exist at the same
frequency by observing the admittance curve. The admittance
and phase of the stator are measured by an impedance analyzer
(IM3570, Hioki E. E. Corp., Nagano, Japan). Changing the
connection between the stator and the impedance analyzer can
clarify the existence of both T1 and T2 modes. The left of Fig.
6(a) shows how to connect the piezoelectric plate electrodes
to the analyzer. To excite T1 mode, the input voltage wire is
connected to all eight electrodes of the piezoelectric plates,
and the ground wire is connected to the metallic cube of the
stator. When the voltage Vin is applied, all the piezoelectric
plates repeat extension and contraction and generate T1 mode.
On the other hand, to excite T2 mode, the input voltage is
connected to the four electrodes at backward and the ground
wire is connected to the other four electrodes at forward as
shown in the right of Fig. 6(a). In this case, when the backward
extends, the forward contracts, or vice versa. The repetition of
these extension and contraction generates T2 mode.
Fig. 6(b) and (c) show the frequency response of the
admittance and phase, respectively, in which the solid lines
and dashed lines show T1 and T2 modes, respectively. The
resonance of T1 and T2 modes is observed as a steep change at
almost the same frequency at around 82.0 kHz. These figures
show that the T1 and T2 modes can be excited at the same
driving frequency. When two voltages described in (1) and (2)
are applied at 82.0 kHz, the excitation of T1 and T2 generate
a translation as shown in Fig. 2 (c).
B. Relation between the Output and the Pre-pressure
The important characteristic of the proposed motor is how
the motor output changes with respect to the pre-pressure. We
experimentally clarify the relation of the pre-pressure with the
thrust force and velocity generated by the motor. The coil
spring model are given as a length L= 94.2 mm, width b= 3
mm, height (thickness) h= 0.15 mm, and Young’s modulus E
= 196 GPa. To change the pre-pressure, we insert several coil
springs with different diameters ranging from 10 to 11 mm into
the stator hole with a diameter of 10 mm. i.e., the coil spring
with a larger diameter generates a large pre-pressure because
the diameter of the stator hole is constant. The pre-pressure can
be estimated from (4) after the strain energy is computed using
(5). For example, when the coil spring diameter is 10 mm, the
pre-pressure becomes zero. When a coil spring with a diameter
of 11 mm is inserted into the stator hole, it generates a pre-
pressure of P=0.036 N/mm2and stores a strain energy of
U=2.8 mJ. (The strain energy is evaluated by the experiment
shown in the Appendix).
In the experiments, the force is measured by a force gauge
(ZP-20N, Imada Co., Japan) attached to the end of the coil
spring. The velocity is measured by a laser displacement sen-
sor (ZX2-LD50, OMRON Corp., Kyoto, Japan) placed in the
travelling direction of the coil spring. In general, the velocity
is calculated from the differentiation of the displacement,
but estimating the motor velocity has large noise because
the coil spring vibrates. We define the velocity from the
displacement of the coil end after vibration and the period
that the voltages apply. In this measurement, the transient
time is ignored because the motor velocity peaks within a
few milliseconds—the mass of the coil spring is much smaller
than the output or brake force. During the experiments, the
amplitude of the voltages is constant at 120 Vp−p, and the
frequency is adjusted to about 82 kHz to maximize the force
and velocity. The optimum frequency has a slightly different
value by coil springs because it depends on the pre-pressure
value. For example, the optimum frequency at a coil diameter
of 10.8 mm is 81.6 kHz, 0.2 kHz higher than the natural
frequency at that of 10.15 mm.
Fig. 7 shows the behavior of the force and velocity when the
pre-pressure varies. The force increases with the pre-pressure
and peaks at 0.02 N/mm2(a coil diameter of 10.5 mm). A too
large pre-pressure over 0.03 N/mm2(a coil diameter of 10.8
mm) decreases the force. On the other hand, the velocity is
200 mm/s at maximum, and decreases at higher pre-pressures.
This is because a higher pre-pressure increases the friction at
the stator-slider interface. Such a relation between the motor
output and the pre-pressure has been seen in the preload
JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 6
0
0.1
0.2
0.3
0.4
0.5
0
50
100
150
200
250
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Force [ N]
Velocity [mm/s]
Pre-press ure [N/mm
2
]
Velocity
Force
10
Coil spring diameter [mm]
10.15 10.3 10.5 10.8 10.95
Fig. 7. Relation of the velocity and the force to pre-pressure. The pre-pressure
is changed by using several coil springs with different diameters.
characteristic of ultrasonic motors with a friction drive [26],
[27].
C. Load Characteristic
The relation between the force and velocity is a fundamental
characteristic of linear actuators. It can be seen by measuring
the motion of a coil spring that lifts load attached to its end.
Fig. 8 shows an experimental setup to examine the force–
velocity curve. The coil spring that generates an optimal pre-
pressure of 0.02 N/mm2is placed in the setup vertically.
External weights are connected to the coil spring as load.
Because the weight of the coil spring is approximately 6 g,
the sum of the coil spring and the additional weights is the
force generated by the motor. The velocity is measured by the
laser displacement sensor while the coil spring moves upward
with the weights. Fig. 9 shows the force-velocity curve when
the weights change from 0 to 50 g. The velocity decreases as
the load increases, and the motor cannot generate a motion
over a load of 50 g. This behavior is roughly linear as with
the load characteristic of other ultrasonic motors [24].
D. Relation to the Bending Radius of the Coil Spring
Another important characteristic of the proposed motor is
its flexibility. Evaluating the flexibility should be to examine
the motion of the coil spring curved by constraints and/or
external forces. The top of Fig. 10 shows the experimental
setup to clarify how the velocity of the coil spring behaves
under constraints. The coil spring end is fixed to a rotary
constraint component that transfers the linear motion of the
coil spring into a motion around an arc trajectory with a
radius R. As shown in the bottom of Fig. 10, when the motor
generates a linear motion, the coil spring moves to the left side
of the stator and bends by the rotary constraint. The distance
between the rotary constraint center and the coil spring end
is equal to the bending radius Rof the rotary constraint. The
bending radius can be changed in the experimental setup. The
laser displacement sensor installed at the right side measures
another end of the coil spring that moves away from the sensor
linearly.
Fig. 11 shows the relation between the bending radius and
the velocity when the bending radius varies from 55 to 15
Fig. 8. Experimental setup for measuring the velocity when the flexible
ultrasonic motor lifts a load. The coil spring slider moves upward in this
experiment.
0
30
60
90
120
150
0 10 20 30 40 50
Velocity [mm/s]
load [g]
Fig. 9. Load characteristic of the flexible ultrasonic motor. It means a force-
velocity curve.
mm. The result shows that the velocity is constant regardless
of the bending radius—a smaller bending radius travels a
shorter distance at less travelling time, and vice versa. This is
because the velocity of the coil spring slider is determined by
the steady-state vibration velocity of the stator in the friction
drive [25]. Even at the smallest bending radius of 15 mm,
the motor can generate an average translation velocity. The
bending radius of 15 mm is close to the limit of bending
because smaller bending radii are over the range of elastic
deformation.
The flexibility of the coil spring slider can be designed by
the material and dimensions of the coil spring. When the coil
spring slider is very flexible by the design, it is difficult to
transfer a force to the slider end, and buckling might occur.
The coil spring slider with high stiffness can transfer a force,
but it makes the pre-pressure unstable at smaller radii.
IV. DYNA MI C EVALUATI ON
A. Step Response
We measure the step response of the flexible ultrasonic
motor to show the vibration of the coil spring and verify the
dynamic model by experiments. The experimental setup is the
same as that without the rotary constraint component shown
JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 7
Fig. 10. Experimental setup to examine the relationship between the velocity
and bending radius. The bending radius can be changed by the mechanical
constraints.
0
50
100
150
200
10 20 30 40 50 60
Velocity [mm/s]
Bending radius [mm]
Fig. 11. The relationship between the velocity and bending radius (error bars
indicate SD from 6 tests of one bending radius). The result shows that the
velocity stays constant even if the coil spring is bent.
in Fig. 10. When the voltages are applied, the coil spring
starts to move linearly. The displacement xwith vibration is
measured using the laser sensor. The vibration shown in the
displacement xdepends on the length of the coil spring, and
the step responses are measured at the initial lengths lini of 40,
80, and 120 mm. The step responses are compared with the
simulation. Assuming that the displacement xis independent
of the vibration of the coil spring, the displacement xcan
be estimated from the coil spring’s mass and the vibration
velocity of the stator, as expressed in (7). The coil spring
slider is supported by the experimental setup and is guided
to move linearly, but a friction occurs by contact with the
setup’s base. A friction term Ff=µmR
0gbetween the coil
spring slider and the experimental setup is added in the left
hand side of (11). When xis determined in (7), the motion of
the coil spring end xR
0can be obtained in (11).
Fig. 12 shows the step responses when the control signal is
on from time t= 0 to 50 ms. While the control signal is on,
the voltages are applied to the motor from an external power
0
2
4
6
8
10
0 0.05 0.1 0.15 0.2 0.25 0.3
Position [mm]
Time [s]
Model 120mm
0
2
4
6
8
10
0 0.05 0.1 0.15 0.2 0.25 0.3
Position [mm]
Time [s]
Model 80mm
0
2
4
6
8
10
0 0.05 0.1 0.15 0.2 0.25 0.3
Position [mm]
Time [s]
Model 40mm
OFF
ON
0 0.05 0.1 0.15 0.2 0.25 0.3
Control P ulse
Time [s]
Fig. 12. Step response of the flexible ultrasonic motor when changing the
initial length lini. The dashed and solid lines show the predicted result and
the measured result, respectively.
TABLE I
MOD EL PROP ERT IES O F TH E FLEXIBLE ULTRASONIC MOTOR
Symbol Quantity Value
mSlider mass 0.006 kg
kSpring constant 4.4 N/m
cDamping coefficient 3.7 N·s/m
cRDamping coefficient 0.008 N·s/m
µDynamic friction coefficient 0.24
lall Coil spring length 210 mm
FMotor output 0.45 N
source. In all responses, when t= 50 ms, the driving force
generated in the stator stops, but the coil spring still has an
elastic energy. After the input signal is off, the vibration of
the coil spring remains for 0.1–0.2 seconds. The experimental
step response is compared with the simulations. The model
parameters for simulation are given in Table I. The mass m,
spring constant kand spring length lall are determined from
the design of the coil spring, and the damping coefficients c,
cR, and the friction coefficient µare empirical. The motion
of the coil spring is in agreement with the simulation. As
the initial length lini shortens, the natural angular frequency
increases as estimated in (11). This is because, at the short
initial length, the mass mR
0reduces and spring constant kR
0
enlarges. The natural angular frequencies of approximately 69,
100, and 201 rad/s, at the initial length lini of 120, 80, and 40
mm, respectively, are in agreement with the estimation.
V. CONCLUSION
In this paper, we demonstrated the first flexible ultrasonic
motor using an elastic elongated coil spring. The proposed
idea is the simplest way that provides a flexibility and a
pre-pressure because there is no additional mechanism. The
experiments showed the sufficient flexibility under a me-
chanical constraint and an accordance between the model
and experiments. Although only one example of the flexible
JOURNAL OF TRANSACTIONS ON MECHATRONICS, VOL. XX, NO. X, AUGUST 2018 8
ultrasonic motor is shown in this paper, the design strategy
can be extended to the other designs for soft and flexible
actuation technologies. In addition to the flexibility, this idea
should be valuable as a simple pre-pressure mechanism for a
rotary or linear motor with a rigid output shaft. Taking it into
account that a typical advantage of ultrasonic motors is a high
energy density, the pre-pressure mechanism has a potential to
be miniaturized for narrow spaces, such as the inside of camera
lenses and cell phones.
Our next step is the use of two or more flexible ultrasonic
motors as a flexible and elongated continuum robot. Further
investigation about contact problems at the stator-slider inter-
face is important to generate a stable motion for controlling
multiple flexible ultrasonic motors. Because the stator can
generate a rotary motion [20], the combination of rotation and
translation might be more attractive as a robotic application.
The position sensors are required for control. We have started
the study on the position sensor using the coils spring as a
variable resistance. In this idea, when the coil spring moves,
the position can be measured by reading the value of the
variable resistance. It can provide a flexible sensor system into
the motor without additional components. Another study about
a further miniaturization of the flexible ultrasonic motor might
be required for smaller diameter continuum robots. We have
achieved an ultrasonic motor that can generate both rotary and
linear motions by the stator with a cube of 3.5 mm [28]. This
miniaturization technology might be applied to build a medical
continuum robot with smaller diameter.
APPENDIX A
EXP ER IM EN TAL VER IFI CATION OF TH E STRAIN ENERGY
In (5), we estimated the strain energy of the coil spring by
the model of the bending deformation of the Euler–Bernoulli
beam. There is another computational method for the strain
energy using the bending moment [23]. The advantage of
using the bending moment is that shows the strain energy
from the angular displacement by a simple experiment. In
this section, we derive the strain energy using the bending
moment and verify it experimentally. We consider the case
that the coil spring inserted to the stator hole generates the
pre-pressure Pbetween the stator and coil (Fig. 3(a)). At this
time, the bending moment Macts both the ends of the coil
spring and makes the beam planes either lengthen or shorten,
thereby creating strains. The bending moment is expressed by
integrating the strains:
M=ZA
EεydA (A1)
where dA is the differential element of the beam area. In the
Euler–Bernoulli beam, the bending moment Mcan be solved
from a given angular displacement φ. This is expressed as
φ=ML
EI (A2)
φcan be geometrically determined by the parameters of the
coil spring and the hole diameter. Using the relation between
φand M, the strain energy Usis rewritten to
Us=M2L
2EI (A3)
(a)
0
1
2
3
4
0 50 100 150 200
Energy [mJ]
Twist angle [deg]
Actual
Predicted
(b)
Fig. A1. Strain energy stored by the twist of the coil spring. The strain energy
estimated is accorded to the experimental result obtained from the change in
the potential energy.
The energy Usequals to the strain energy derived in (4).
Let us confirm the relation between the strain energy and
the angular displacement experimentally. Fig. A1(a) shows the
experimental setup. One end of the coil spring is fixed and
the other free end is attached to a pulley. The coil spring
has a diameter of 11 mm and a wire total length of 310
mm. A weight is loaded to the string connected to the pulley
that fixes the free end. When a weight is applied, it moves
downwards due to gravity and twists the coil spring with
an angular displacement φcircumferentially. In this case, the
strain energy stored by the twist is equal to the work done by
the displacement of the weight. It is described as a half of the
potential energy of the weight mand the change in height h:
Up=1
2mgh (A4)
This value should take the same value as the strain energies
in (A3).
Fig. A1(b) shows the experimental behavior of Usand Up
when φand hare determined. The plots obtained in the change
in the displacement of the mass are in good agreement with
the curve of the strain energy. This fact means that the strain
energy computed by (4) and (A3) is correct.
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Ayato Kanada received the B.S. and M.S. degrees
in mechanical engineering from Toyohashi Univer-
sity of Technology, Japan, in 2015 and 2017, re-
spectively. He is currently working toward the Ph.D.
degree at Toyohashi University of Technology.
His research interests are soft actuators and bio
mimetic robotics.
Tomoaki Mashimo (M’06) received the Ph.D. de-
gree in mechanical engineering from the Tokyo
University of Agriculture and Technology, Fuchu,
Japan, in 2008. He was a Robotics Researcher at
the Robotics Institute, Carnegie Mellon University,
Pittsburgh, PA, USA, from 2008 to 2010. After
being an Assistant Professor (tenure-track) with Toy-
ohashi University of Technology, Toyohashi, Japan,
in 2011, he became an Associate Professor in 2016.
His research interests include piezoelectric actuators
and the robotic applications.
