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20  Приборы и методы измерений, № 2 (3), 2011 
УДК 621.023.6 
INSPECTION ROBOTS WITH PIEZO ACTUATORS: 
MODELING, SIMULATION AND PROTOTYPES 
 
Becker F.
1
, Zimmermann R.
1
, Minchenya V.
2
, Lysenko V.
2
, Chigarev A.
2
 
 
1
Ilmenau University of Technology, Ilmenau, Germany  
2
Belarusian National Technical University, Minsk, Belarus  
 
 
Models, simulations and experimental setups of resonant inspection robots are presented. The goal 
is to show ways to cope with the new requirements and to use the given chances to create novel mo-
bile robots. For the creation of a directed motion the vibration behavior of simple beams and plates 
is used. It is possible to design robots for 2-dimesnional locomotion which are characterized by a 
light weight, small size, relative simple design and the ability to create controllable motion using on-
ly one actuator. Different types of actuators for micro robots are presented and compared. Further-
more the dynamical behavior of a piezoelectric bending actuator under elastic boundary conditions 
is investigated and a model for the motion of the locomotion-generating limbs is presented. The 
comparison with experiments and prototypes shows that the results of the analytical and computa-
tional models agree (E-mail : Klaus.Zimmermann@tu_ilmenau.de) 
 
Key Words: resonant robots, piezo-driven systems, vibrations of continua.  
 
Introduction 
 
Inspection mobile robots are dominated by 
rigid multibody systems with constant mass distri-
bution. The bodies are coupled by kinematic pairs. 
The modification of relative positions is provided 
mostly by actuators in the joints. Systems with legs 
and wheels to perform the locomotion are well 
known and used in many fields of application.  
Actual developments, especially in medicine, 
biology and inspection technology, require new 
mobile robots, which are characterized on the one 
hand by relative small size and weight and one the 
other hand by small costs for production and appli-
cation. In this context established robots are not 
optimal for the new challenges. Joints and typical 
actuators cannot be minimized arbitrary because of 
the scaling effects of the used physical principles. 
Also the costs for manufacturing and assembling 
would increase disproportionally. To provide mul-
ti-dimensional movement several actuators are 
needed, at least one for every degree of freedom. 
The energy consumption and effort for controlling 
for the established systems are relative high. 
The objective of this scientific project is to 
break through these disadvantages and to develop 
systems which are customized for the new re-
quirements on mobile systems. Using the theory of 
forced vibration of continua, especially bending 
vibration of beams and plates, it is possible to cre-
ate systems which are small, light and cheap with a 
relative simple design. Following the principle «in-
telligence in the mechanics» [2] robots are in the 
design process (figure 1), which are controllable by 
only one actuator to perform motion in three de-
grees of freedom. 
In the following figures four mobile robots for 
2-dimensional locomotion on a flat surface are pre-
sented. 
 
 
 
Figure 1 – Beetle-Robot [1] – autonomous, pro-
grammable and remote controlled robot driven by 
a single piezo actuator using resonance characteris-
tics of elastic continua 
 
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Приборы и методы измерений, № 2 (3), 2011  21 
Design issues 
 
In choosing the right actuator the most critical 
issue to consider is the speed to excite the reso-
nance characteristics of small mechanical devices. 
Ideally would be a light and compact actuator with 
a high speed, low power consumption, large output 
force and strain. A high potential for the use in 
small micro robots have actuators made form pie-
zoelectric materials, shape memory alloy (SMA), 
ionic polymer-metal composite (IPMC) and DC 
motors [4]. The main disadvantages of SMA and 
IPMC are the slow speed, the small force and the 
relative high power consumption. By consuming a 
large amount of energy DC motors have the possi-
bility to create a very large displacement with a 
high speed and a moderate force. A problem is the 
heavy weight and the limits in miniaturization be-
cause of the scaling of the electromagnetic force in 
relation to the friction force in the sliding bearings. 
Piezo actuators are characterized by light weight, 
high speed, large force and high energy efficiency. 
It is possible to convert more than 90 % of the 
electrical energy into mechanical energy. The main 
disadvantage is the small strain and the tempera-
ture behavior in the dynamic mode.  
For a small weight of a robot the piezo ele-
ment can be used as actuator and base body. This 
vibratory drive creates the needed ultrasonic ex-
citement. Assuming that the average value of the 
excitation vibrations equals zero over one period of 
time an asymmetry in the systems characteristics is 
needed to create a directed locomotion. The cha-
racter of the system in the first half period of the 
vibration needs to be another than in the second. 
For this the vibration behavior of simple mechani-
cal structures like beams or plates can be used.  
In the exemplary robot presented in figure 1 
bended metal wires are used as vibration transdu-
cers which are excited by circular piezo unimorph 
drives.  
 
Modeling, simulation and experiments connec-
ted to the actuator dynamics 
 
Analytical description of the actuator dynamics  
 
The actuation system is modeled as a thin 
elastic plate, which can be described with the help 
of the hypothesis of plates and laminates. A cha-
racteristic parameter of a plate is the bending stiff-
ness N . For the introduction of this parameter and 
the use of the Kirchhoff hypothesis a neutral area 
is needed. In a homogenous plate this strain- and 
stress-free area is situated exactly in the middle. 
The position changes in a laminated plate. This is 
illustrated in figure 2, where ze  and re  are the unit 
vectors, 
1E  and 2E  the Young’s modulus of the 
materials, 1h  and 2h  the thickness of the plates and 
nh  the distance between the adherend and the neu-
tral area which can be calculated using equations 
(1) and (2). It is assumed that there is no motion 
between the adherend surfaces. Furthermore the 
distributions of strain )(е z  and stress )(у z  are pre-
sented.  
 
1E
2E
1h
2h
nh
ze
re
Adherend
Neutral area
ε( )z σ( )z
 
 
Figure 2 – The distribution of strain and stress in a 
laminate of two layers 
 
In equation (3) and (4) the calculation of the 
bending stiffness of a homogenous plate as well as a 
laminate is shown, where   is Poisson’s ratio [3]: 
 
1
1
2
2
1



ea
eah
hn
, (1) 
 
1
2
h
h
a  , 
1
2
E
E
e  , (2) 
 
)1(12 2
3
v
Eh
Nhomogenuos

 , (3) 
 
)1(
)1()1(4
)1(12
222
2
3
1 1
ae
eaaae
v
hE
NLaminate




 . (4) 
 
The equations for the bending vibration of a 
circular plate and the general solution are written 
in (5) and (6), where ρ is the density, ω the natural 
angular frequency, λ the eigenvalue, J and I the 
Bessel function and the modified Bessel function 
of first kind. The problem is considered to be 
rotational symmetric so that the function, which is 
 Средства  измерений 
 
22  Приборы и методы измерений, № 2 (3), 2011 
describing the bending of the plate w , depends 
only on the radius r and the time t. 
 
0),(
с
),( 2  trw
N
h
trw tt
, (5) 
)]л()л([)]щsin()щcos([),( 040321 rJcrJctctctrw  . (6) 
 
The influence of the elastic robot legs to the 
actuator plate is modeled as linear springs (figure 
3). We assume that the stiffness of the plate along 
the circumference is relatively high so that the 
springs can be modeled as evenly distributed over 
the circumference of the plate (figure 4).  
 
 
 
Figure 3 – Boundary conditions of robots' actuator 
 
ic
 
 
Figure 4 – Boundary conditions of the analytical 
model 
 
The boundary conditions are presented in (7), 
where rQ  is the shear force, rM  the bending mo-
ment and c  a characteristic stiffness, which can 
be calculated using (8).  
 
,,,г0 t)w(Rct)(RQ 
 ,,г0 t)(RM
 
 (7) 
 
.
р2 R
c
c i
i
  
 
(8) 
 
The characteristic equation, to calculate the 
eigenvalues and natural frequencies of a such plate, 
is formulated in (9) where 
0F , 1F  and 2F  are 
functions of the material and geometrical 
properties, as well as of the modified Bessel 
function of the first kind and different orders. The 
parameter c  could be calculated as іc cR N  and 
is the relation between characteristic stiffness 
parameters. For different boundary conditions, on 
the circumference of the plate, i.e. fixed or flexible 
support, the eigenvalues could be found in the 
literature. They are used to verify the analytical 
model as well as the FEM model. Some results of 
the numerical analysis of equation (9) are 
presented in table 1.  
 
).л()л()л(0 221100 RJFRJFRJF   (9) 
 
FEM modeling and simulation 
 
To analyze to vibration behavior of the actua-
tor, under the described boundary conditions, a 
FEM model is formulated. The plate is modeled to 
be homogenous. With the help of a modal analysis, 
the natural frequencies and normal modes are si-
mulated and compared with the results from the 
analytical calculations. It could be noticed that the 
results of both modeling methods agree. With the 
full rotationally symmetric mathematical modeling 
can be calculated only the natural frequencies of 
such normal modes. With FEM simulation also the 
non-symmetric modes are determined. Some ex-
amples are given in figures 5 and 6. 
 
Table 1 – Natural frequencies for the rotation-
ally symmetric normal modes of a plate under line-
ar elastic boundary conditions for one set of pa-
rameters – analytical and FEM calculations 
 
No. Analytical [Hz] FEM [Hz] 
1 179 216 
2 798 782 
3 2062 2012 
4 4495 4430 
5 7957 7903 
6 12415 12432 
 
  
 
a 
 
b 
 
Figure 5 – rotational symmetric normal modes: 
a – 3rd with 2012 Hz; b – 4th with 4430 Hz 
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Приборы и методы измерений, № 2 (3), 2011  23 
  
  
 
a 
 
b 
 
Figure 6 – normal modes: a – 9th with 1206 Hz;   
b – 15th with 1862 Hz 
 
Experiment 
 
The natural modes of a circular piezo uni-
morph actuator are investigated and presented in 
figure 7. In agreement with the analytical and 
computational models, it is possible to establish 
different normal modes. The boundary conditions 
of this plate are different than the presented mo-
dels. Also the soldered dots have an influence to 
the vibration behavior.  
 
  
  
  
 
Figure 7 – Natural modes of a robots' actuator 
 
Analytical modeling of the legs motion 
 
The geometry of the robot determines the fre-
quency spectrum of its oscillations. Resonant exci-
tation of the robot body, which is realized by the 
described actuator, leads to a transfer of the vibra-
tion energy to the limbs. It allows the transforming 
of the periodic motion into a forward one, at a cer-
tain synchronization of their oscillations. The main 
role in the transformation of these motions plays 
the geometry of the legs, which consist of three 
links. The length of the links and the angles be-
tween the links affect significantly the nature of the 
transformation of the vibrations. To make first as-
sertions about the movement of the endpoints of 
the legs, which are the contact points between the 
robot and the flat surface, an analytical model is 
presented. 
The reference point of the robot leg B  per-
forms vibrations in a spatial coordinate system. 
The origin lies in point A  which is connected to 
the body of the robot (figure 8).  
Characteristic is the so called radius vector 
( )R t  connecting the points A  and B . Then its 
projections on the reference axes 
xe , ye , ze  are 
1xR l , 2yR l  and 3zR l  . The angels between 
and the unit vectors of the axes are φ1(t), φ2(t) 
and φ1(t). φ1(0), φ2(0) and φ3(0) define the ge-
ometry of the robot leg. The relations (10) and 
(11) are given. 
 
cos
2
 φ1+ cos
2
 φ2+ cos
2
 φ3  = 1, (10) 
 
  
.223
2
2
2
1 Rlll   (11) 
 
1l
A
2l
3lR
B
1
2
3
ye
ze
xe
 
 
 
Figure 8 – Estimated model of the robot leg 
 
Using a spherical coordinate system to de-
scribe the radius vector ( )R t  three variables can be 
used: the length ( )l t  of the vector and the angles 
( )t  and )(t  between ( )R t  and the unit vectors 
xe  and ze  (figure 9). 
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24  Приборы и методы измерений, № 2 (3), 2011 
ze
ye
xe
A
B


R
 
 
Figure 9 – Model of robot leg in a spherical coor-
dinate system 
 
In a first approximation it is considered that 
the motion of the reference point B  occurs on a 
sphere of a constant radius 0( )l t l const  . The 
equations of motion for   and   are presented in 
(12) and (13), where the mass m  of the leg is mo-
deled to be concentrated in B . The motion of the 
reference point is considered to be small. The line-
arized equations are (14) and (15). 
 
,0)sin( 220 ml
dt
d  (12) 
 
,0sincossin
0
2 
l
g
  (13) 
 
,0   (14) 
 
.0cos
0
2 
l
g
  (15) 
 
From equation (14) we get: 
 
0  or .0  (16) 
 
In the first case of (16) the equation (15) has 
the form of (17) with the solution (18), where g  is 
the gravitational constant, 0v  the initial velocity, t  
the time and 
0
щ
l
g  the angular frequency of the 
vibration. In the case point B  performs an har-
monic vibration in the plane with φ(t) = φ0  = const, 
. 
,0
0

l
g  (17) 
 
.0щtsin
щ
щcos)( 00 

tt  (18) 
 
In the second case ( 0 ) equation (15) has 
the form of (19) with the solution (20). B  per-
forms a circular movement in the xy plane with 
constt  0)( . 
 
2
0
g
l
  , (19) 
0
( )
g
t t
l
   . (20) 
 
In general, the motion of the reference point 
B  is described by three coordinates ( )l t , φ(t) and 
)(t  with ( ) ( )R t l t . In that case the equations for 
φ and   has the form (21) and (22), where α and β 
are arbitrary dimensionless constants of the legs’ 
mass geometry. These are the well-known equa-
tions of motion of s spherical pendulum. 
 
,
sin)(
бщ
22
0
2
0


tml
ml
  (21) 
 
.вщ
2
1
sin)(
бщ
2
1
cos)()(
2
1
2
0
2
22
22
0
4
0
22
ml
tl
ml
tmgltml




 
(22) 
 
Regarding equations (21) and (22) three cases 
can be considered.  
In the first case when α = 0 and φ(t) = φ0  = 
= const  the plane motion of the pendulum is 
given, described by the laws ( )l t  and (t) . 
For the second case the constant α has a value 
0 < α2 < f(β) f(β) is presented in equation (23). 
Then 
12   and the motion occurs on a 
sphere with the variable radius ( )l t  between the 
parallels 2z  and 1z  (figure 10). 1z  describes the 
lifting phase of robots’ platform (the leg is in con-
tact to the ground). 2z  gives the phase of lowering, 
where the leg loses the contact to the ground. 
 
3
2 32
1
(в) (в в) 36в-в .
54
f
 
   
 
 (23) 
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Приборы и методы измерений, № 2 (3), 2011  25 
1 1z l cos
2 2z l cos
ze
( )R t
 
Figure 10 – Scheme of the motion of the robot 
platform in the second case 
 
In the third case (α2 = f(β)) the radius vector 
( )R t  moves on a conical surface and 
2
р
)( 0  t
. 
The reference point B  moves in a circle of the ra-
dius 
0sinl  in the horizontal plane 00 cos lz  
(figure 11). 
A
0
( )R t
( )t
B
ye
ze
xe
 
 
Figure 11 – Scheme of movement in the third case 
 
Prototypes and Experiments 
 
Follow the mentioned goals and using the 
described models for creating small mobile robots 
the two prototypes presented in Figure 1 are 
developed. Structural data are given in Table 2. 
The actuator is controlled through a sinusoidal 
electrical signal with amplitude of 20 V. 
To analyze the motion of the contact point of 
the leg and the surface (reference point B ) exper-
iments using a scanning electron microscope 
(SEM) are made. 
 
Table 2 – Data of prototypes 
 
Plate-Robot 
Length × Width × 
Height 
58 × 42 × 10 mm³ 
Mass 3,5 g 
Max. velocity on glass 150 mm/s 
Excitation frequency 10–60 kHz 
Beetle-Robot 
Length × Width × 
Height 
69 × 80 × 30 mm³ 
Mass 31,7 g 
Max. velocity on glass 20 mm/s 
Excitation frequency 12–70 kHz 
 
As experimental setup Plate-Robot is used. In 
figure 12, the overlay of two photos is presented. 
The solid scheme represents the endpoint of a leg 
in a static state. 
 
 
 
 
Figure 12 – SEM picture of a vibrating leg 
 
The moveable part shows the systems under exci-
tation. It should be noticed, that with the used mi-
croscope, it is possible to take one picture every 
three seconds. The vibration frequency of the leg is 
much higher, which means that the marked ampli-
tudes do not represent necessarily the maximum 
va-lue. The leg is excited by the bending vibrations 
of the actuator plate (figure 5 to 7). According to 
the frequency different vibration forms are pro-
duced. The endpoint of the leg performs longitudi-
nal and transversal vibrations. The trajectory corre-
sponds to the relation between the amplitude of the 
longitudinal and transversal vibrations (figure 10). 
This behavior can be described using the third case 
α2 = f(β) of chapter 4 (figure 11). The motion of 
the robot depends on this vibration behavior. Dur-
ing the movement the friction forces in the contact  
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26  Приборы и методы измерений, № 2 (3), 2011 
points between robot and environment are 
changed periodically.The motion direction could 
be controlled using the resonance shift between 
the legs. This resonance shift is caused by the 
asymmetric system properties. Furthermore the 
robot legs lose the contact to the surface during the 
vibration, described by the second case 0 < α2 < 
f(β) (figure 10). The motion is influenced by the 
resulting shock effects. 
 
Conclution  
 
Actuators for small mobile robots are 
compared. The dynamical behavior of a 
piezoelectric unimorph actuator was studied using 
analytical and computational methods as well as 
an experimental setup. An analytical model for the 
description of the motion of robot legs is presented 
and analyzed. Following this ideas experimental 
investigations and prototypes are shown.  
Further investigations will be connected with 
robots legs. Including the boundary conditions, 
given by the contact properties between robot and 
environment, will be determined the resonance 
characteristics. The objective of the future work is 
to find the qualitative and quantitative relations 
between the excitation frequency and the locomo-
tion properties. Models with a lower grad of ab-
straction are needed. Further micro robots, which 
are in the design process, will use the described 
principle of motion. The concentration on the mo-
tion of the mobile robots in the different environ-
ments will arise in the further work. 
 
Acknowledgments 
 
The work has been supported by the German 
Research Foundation (DFG) under grant Zi 
540/11-1 as well as by the Free State of Thuringia 
via graduation scholarship. 
 
References 
 
1. Becker, F. Single Piezo Actuator Driven Micro 
Robots for 2-dimensional Locomotion / F. 
Becker. – Aachen : Electro. Proceedings of 
Workshop on Microactuators and Micromecha-
nisms, 2010. 
2. Blickhan, R. Intelligence by Mechanics. / R. 
Blickhan. – London : Philosophical Transactions 
of the Royal Society 365, 2007. 
3. Pfeifer, G. Piezoelektrische lineare Stellantriebe. 
/ G. Pfeifer. – Karl-Marx-Stadt : Wissenschaftli-
che Schriftenreihe der Technischen Hochschule, 
1982.   
4. Song, S.Y. Surface-Tension-Driven Biologically 
Inspired Water Strider Robot: Theory and Ex-
periments / S.Y. Song, M. Sitti // IEEE Robotics 
and Automation Society: Transactions on Ro-
botics 23. – No. 3. –  2007.  
 
 
 
Беккер Ф., Циммерманн К., Минченя В., Лысенко В., Чигарев А. 
 
Инспекционные роботы с пьезоприводом: моделирование, симуляция, опытные образцы 
 
Приведены конструкции, представляющие собой симуляции и экспериментальные модели инспекци-
онных микророботов, работающих на основе резонансных колебаний. Использование разработанных по-
движных систем позволит создать микророботы, соответствующие новым требованиям разработчиков. 
Для создания направленного движения инспекционных микророботов используются вибрационные коле-
бания балки на опорах и пластин. Показана возможность создания резонансных инспекционных роботов 
для двумерного движения и контроля, которые будут иметь небольшой вес, маленькие размеры и относи-
тельно несложный дизайн. Измерение параметров резонанса позволяет получить информацию о харак-
терных особенностях опорной исследуемой поверхности Управляемое движение и контроль осуществля-
ется за счет активации только одного элемента микроробота. Сравниваются различные типы механизмов 
для микророботов; представлены результаты исследования динамических свойства пьезоэлектрического 
механизма изгиба пластин в границах упругой зоны. Представлена модель движения, порождаемого нож-
ками робота. (E-mail : Klaus/Zimmermann@tu_ilmenau.de) 
 
Ключевые слова: резонансные роботы, системы пьезопривода, вибрационные колебания. 
 
Поступила в редакцию 19.09.2011.