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Acta Polytechnica Vol. 52 No. 1/2012

Prediction of Lower Extremity Movement by Cyclograms

P. Kutilek, S. Viteckova

Abstract

Human gait is nowadays undergoing extensive analysis. Predictions of leg movements can be used for orthosis and
prosthesis programming, and also for rehabilitation. Our work focuses on predicting human gait with the use of angle-
angle diagrams, also called cyclograms. In conjunctionwith artificial intelligence, cyclograms offer awide area ofmedical
applications. We have identified cyclogram characteristics such as the slope and the area of the cyclogram for a neural
network learning algorithm. Neural networks learnedbycyclograms offerwide applications inprosthesis control systems.

Keywords: gait, artificial intelligence, cyclogram, artificial neural networks.

1 Introduction

In medical practice, there is no appropriate widely-
used application of a system based on artificial in-
telligence (AI) for identifying defects in the move-
ment of human legs, or for controlling the actuators
of prosthesis or rehabilitation facilities. Above all, it
is difficult to evaluate the quality of walking. Medical
decisions are often based only on the subjective views
of physicians, and appropriate accurate methods and
are not widely used in clinical practice.

Several methods can be used in medical prac-
tice and in physiotherapeutic research for identifying
defects in the movement of the human body. The
most widely-used method for studying gait behav-
ior in clinical practice is gait phase analysis by gait
time phase cycles [1, 2]. The time phase cycles of gait
behavior have been used to analyze gait with the ap-
plication of artificial intelligence methods [3–9], but
the findings have not subsequently been applied in
medical practice. Very intensive research is now be-
ing done on predicting leg movements by artificial
intelligence and EMG signal measurements [10–12],
mainly with a view to their possible application in
myoelectrical prosthesis control systems.

For a study of gait we have used new methods
based on an analysis of gait angles using cyclograms
(also called angle-angle diagrams or cyclokinograms)
and artificial intelligence to predict human motion
of legs/prostheses. The concept of angle-angle di-
agrams, although known to the biomechanics com-
munity, has not been mentioned much in the recent
literature. The first mention of the cyclogram [13]
argued that a cyclic process such as walking is bet-
ter understood if studied with a cyclic plot, e.g. an
angle-angle diagram. The creation of cyclograms is
based on gait angles that are objective, reliable and
well suited for statistical study [14]. This technique is
strongly rooted in geometry, and the quantities are

intuitively understandable [15]. Depending on the
cyclicity of the gait, cyclograms are closed trajec-
tories generated by simultaneously plotting two (or
more) joint quantities. In gait studies, most atten-
tion has traditionally been given to easily identifiable
planar knee-hip cyclograms . In order to quantify the
symmetry of human walking, we have obtained and
studied a cyclogram for the same joint and two sides
of the body [16].

Applications of cyclograms in conjunction with
artificial intelligence can offer a wide range of medi-
cal applications, but this approach has not yet been
studied or applied in practice.

2 Methods

To create and study angle-angle diagrams, we use
a model of the human body created in MatLab
Simulink and SimMechanic software. The movement
of the model of a body is controlled by data measured
by the motion capture system, which identifies the
position of points/markers in the Cartesian coordi-
nate system. We can generally use several methods
for measuring movements in two/three dimensional
space, e.g. an infrared (IR) camera with active mark-
ers, or a web camera, which is cheaper. For our ap-
plication, we used an IR medical camera with active
markers (Lukotronic AS 200 system) and placed LED
diode markers on the following points on the per-
son being measured: fifth metatarsophalangeal joint,
malleolus lateralis, epicondylus lateralis, trochanter
major, spina iliaca anterior superior, etc., Figure 1.

Using this method we can record the movement
in a three-dimensional space, though we primarily
study the movement in a two-dimensional sagittal
plane. Human gait data commonly consists of the
recorded positions of markers on the skin/dress at
the extremities of the limb segments (the thigh, the
shank, etc.) of a subject. If we have information on

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Acta Polytechnica Vol. 52 No. 1/2012

Fig. 1: Location of IR markers and angles measured during the examination

the movement of points/markers in space, and the
points characterize parts of the human body, we can
use these points to define the vectors of the positions
of the body parts. The difference of the coordinates
of two points in space defines a vector. The angles
between each two segments are calculated by assum-
ing the segments to be idealized rigid bodies. For the
computing angles, we use the formula:

β = arccos
u · v

|u| · |v|
, 0 ≤ ϕ ≤ π, [rad], (1)

for the two-dimensional system, because we assume
that this knee or ankle joint has only one rotational
degree of freedom in the case of simplified assump-
tions [17]. u and v are vectors of body segments
(thigh, shin, foot, etc.) represented by at least two
points, i.e. markers. The calculation is performed in
an equivalent way for three-dimensional space. If we
are interested in the angle between a body segment
and a physical horizontal we determine the angle be-
tween the horizontal vector (1, 0) and vector that is
defined by the coordinates of the points on the body
segments evaluated in the Cartesian coordinate sys-
tem. The markers, i.e. the points, move in space
together with the body segments, and the individual
segments of the body move by the translational or
angular movement per unit time. We therefore use
numerical derivations to determine the translational
and rotational speed and the acceleration of the in-
dividual segments, i.e. markers

The aim of this study is not to evaluate the cen-
ter of rotation of a joint, because even in practice, in
the case of the motor control system of a prosthetic

device, we do not expect complete conformity with
the anatomical characteristics, and the deviations are
negligible for our purpose, which is to provide veri-
fication of movement prediction methods. For the
calculation we therefore used vectors based on the
coordinates of the markers, and vector u and v in
the equation does not necessarily correspond to the
directional unit vectors of the body parts.

During the measurement, the measured data is
grouped according to the age groups of the subjects,
and could also be grouped according to the diagnoses
of the subjects. We create cyclograms of the an-
gles, angular velocities and angular acceleration: left
knee – left hip, right knee – right hip, left hip – right
hip, left knee – right knee, right ankle – right hip, left
ankle – left hip, etc. We were also able to study the
following characteristics of cyclograms: length of the
trajectory, frequency of the loops, slope of the loops,
maximum range, average speed, total circumscribed
area of the loops.

The most important part of our work is on de-
signing methods for applying cyclograms in practice
in order to identify movement defects, and to apply
cyclograms in the control system of the actuators of
prosthesis or rehabilitation facilities. For this pur-
pose, we use artificial intelligence methods which are
implemented in MatLab toolboxes [18, 19]. For ex-
ample, we can use Artificial Neural Networks (ANN)
to predict the joint angles, i.e. for predicting cyclo-
grams.

Artificial neural networks are based on the neu-
ral structure of the brain [20, 21]. They process
records one at a time, and “learn” by comparing

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Acta Polytechnica Vol. 52 No. 1/2012

their prediction of the record with the known actual
record [22, 23]. The input to the first layer consists of
the values in a data record. The final layer is the out-
put layer, where there is one node for each physical
quantity. The prediction of time series using a neu-
ral network consists of teaching the net the history of
the variable in a selected limited time and applying
the taught information to the future. Data from the
past is provided to the inputs of the neural network,
and we expect future data from the outputs of the
network.

Table 1: Learning process of artificial neural network ac-
cording to the calculated joint angles

Input data Target data

Joint angles Joint angle

x1 x2 x3 x4 x5 x6

x2 x3 x4 x5 x6 x7

x3 x4 x5 x6 x7 x8

x4 x5 x6 x7 x8 x9
...

...
...

...
...

...

xn−4 xn−3 xn−2 xn−1 xn xn+1

Our learning method is based on the premise of
the proposal of a table with m + 1 columns of states.
We assume that five columns of previous states plus
one column for the prediction will be sufficient, Ta-
ble 1. The first row of the table records the first five
angles computed from the measured data, and the
sixth column indicates the next calculated value of
the angle, as a target to which the artificial neural
network learns by example. The second row records
the second to sixth calculated value of the angle, and
in the sixth column in the same row we insert the
seventh value of the angle, as a target. This cascad-
ing method fills a table of n-4 rows, where n + 1 is

the number of known values of the joint angles that
we decided to use for the learning process, Table 1.
Thus the method is generally based on information
about a man walking. Walking is described by the
cyclogram, and the cyclogram is used to learn neural
networks. We chose the calculated values of the joint
angles as an approach for selecting data for artificial
neural network learning. For neural network learning
we use part of the curve of the cyclogram that rep-
resents a set of states for learning. These states are
divided into past states and prospective states. With
each presentation, the output of the neural network
was compared to the desired output, and the error
was computed. This error was then fed back (back-
propagated) to the neural network, and was used to
adjust the weights in such a way that the error de-
creases with each iteration and the neural model gets
closer to producing the desired output.

The table of input and target data can be ex-
tended to other parameters that are also very impor-
tant for predicting the movement of the lower limbs.
Appropriate parameters are, for example, the angular
acceleration (i.e. four states of acceleration), and also
the subject’s weight in kilograms and age in years,
Table 2. We tested several modifications to ANNs.
The first designed and tested ANNs were for predict-
ing the angle in only one joint of the left leg (hip, knee
and ankle). We also designed ANNs for predicting
the complete knee-hip cyclogram or the ankle-knee
cyclogram, i.e. for predicting two angles. An ANN
structure that we designed for predicting the angles
of the hip-knee curve of a cyclogram is presented in
Figure 2.

For learning the NN, we can also add the height
of the subject in meters, or other additional param-
eters, e.g. a predefined code for an illness, operation
of the musculoskeletal apparatus, etc. The backprop-
agation algorithm is used for training the neural net-
works. With backpropagation, the input data was
repeatedly presented to the neural network.

Table 2: Learning process of artificial neural networks according to the angles and angular accelerations calculated in a
joint, subject’s weight and subject’s age

Input data Target data

Joint angles Angular accelerations
Patient’s
weight

Patient’s
age

First
joint
angle

Second
joint
angle

x1 x2 x3 x4 x5 ε1 ε2 ε3 ε4 m N x6 y6

x2 x3 x4 x5 x6 ε2 ε3 ε4 ε5 m N x7 y7

x3 x4 x5 x6 x7 ε3 ε4 ε5 ε6 m N x8 y8

x4 x5 x6 x7 x8 ε4 ε5 ε6 ε7 m N x9 y9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xn−4 xn−3 xn−2 xn−1 xn εn−4 εn−3 εn−2 εn−1 m N xn+1 yn+1

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Acta Polytechnica Vol. 52 No. 1/2012

Fig. 2: Artificial neural network 11-7-2 for predicting the movement of the lower limb

The above-described methods for predicting mo-
tion using learned neural networks reflect only a short
part of the curve of the angle-angle diagram of a gait
cycle. The values by which the neural network is
learned reflect information on only a limited number
of states within a short period of time. This method
may not always adequately describe the stereotypes
of human walking. The main problem is that the
method does not describe the whole gait cycle and
the transition from the current gait cycle into a new
gait cycle. It is theoretically possible to design a
number of input neurons of a neural network to de-
scribe the whole gait cycle, but the structure of the
neural network will then be very complex, and the
calculation will therefore be difficult and slow. For
this reason, we proposed new ways to describe the
gait cycle by characteristics of cyclogram.

The proposed method takes the distribution of
the values less into account, but to a greater extent
reflects the geometrical shape of the area A of one cy-
cle. We can use several methods and we decided to
use area moment of inertia to describe the property
of a two-dimensional plane shape of the cyclogram.
The reason for using second moment is that we can
very easily describe the distribution of the circum-
scribed area of a cyclogram and we can determine
the inclination angle θ of diagram. The value of an-
gle θ, which is given a product moment of zero, is

equal to:

tan 2θ =
2Ixy

Ixx − Iyy
, (2)

Ixx is the second moment of the area about the x-
axis, Ixx is the second moment of the area about the
y-axis and Ixy is the product moment of area. The θ
is the inclination angle between axes of the original
coordinate system of the diagram and the principal
axes of the area. The values of the inclination angle
or the moments of the area can be used for learn-
ing the neural network. The neural network will be
extended for neurons with regard to other input val-
ues, e.g. the inclination angle of one cycle before the
actual value of the joint angles. The neural network
will also be extended for new output neurons, so that
the neural network learns by target values, e.g. the
inclination angle of a subsequent one cycle after the
actual value of the joint angles, Table 3.

The patterns in Table 4 are used for learning the
neural network using second moments and for pre-
dicting the one joint angle. For learning the ANNs,
including the variables mentioned above, we can the-
oretically also use the area of one angle-angle dia-
gram A, the area of the ellipse of inertia, the center
of the area, etc. Description of the cyclogram by area,
area moment and inclination angle has great advan-
tage because calculated characteristics of the cyclo-

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Acta Polytechnica Vol. 52 No. 1/2012

Table 3: Learning process of artificial neural networks according to the joint angles and inclination angles of the angle-
angle diagram

Input data Target data

First joint angle Second joint angle

Inclination
of the

previous
gait cycle

1st joint
angle

2nd joint
angle

Inclination
of the

subsequent
gait cycle

x1 x2 x3 x4 y1 y2 y3 y4 θ1 x5 y5 θ
′
1

x2 x3 x4 x5 y2 y3 y4 y5 θ2 x6 y6 θ
′
2

x3 x4 x5 x6 y3 y4 y5 y6 θ3 x7 y7 θ
′
3

x4 x5 x6 x7 y4 y5 y6 y7 θ4 x8 y8 θ
′
4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xn−3 xn−2 xn−1 xn yn−3 yn−4 yn−2 yn θn xn+1 yn+1 θ
′
n

Table 4: Learning process of artificial neural networks according to the joint angle, the moments of the area and the
inclination angles of the one angle-angle diagram

Input data Target data

Joint angles
2nd moment

of area:
x-axis

2nd moment
of area:
y-axis

Inclination
of the

diagram

Joint
angle

2nd moment
of area:
x-axis

2nd moment
of area:
y-axis

Inclination
of the

subsequent
diagram

x1 x2 x3 x4 Ixx1 Iyy1 θ1 x5 I
′
xx1 I

′
yy1 θ

′
1

x2 x3 x4 x5 Ixx2 Iyy2 θ2 x6 I
′
xx2 I

′
yy2 θ2

x3 x4 x5 x6 Ixx3 Iyy3 θ3 x7 I
′
xx3 I

′
yy3 θ

′
3

x4 x5 x6 x7 Ixx4 Iyy4 θ4 x8 I
′
xx4 I

′
yy4 θ

′
4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xn−3 xn−2 xn−1 xn Ixxn Iyyn θn xn+1 I
′
xxn I

′
yyn θ

′
n

gram are not negatively affected if the data without
a normal distribution. For this reason, alternatively
we can also use linear regression to calculate slope
of angle-angle diagram. Simple linear regression fits
a straight line (axis) through the set of points i.e.
states. The polynomial equation of the regression
principal axis is

y = a0 + a1 · x (3)

the a0 and a1 are parameters identified by estimator.
When y is the dependent variable of the first angle,
e.g. knee angle, and the x is the independent vari-
able of the second angle, e.g. hip angle, the direction
(slope) of the principal axis, i.e. inclination angle θ of
angle-angle diagram, is obtained from the parameter
a1 as follows

tan θ = a1. (4)

The trained neural networks prefer the typical
changes from the previous cycle to the subsequent
cycle, and avoid the use of atypical changes. In ad-
dition, we allow NN to estimate the expected gait
cycle based on a specified slope of the angle-angle di-
agram of a gait cycle. Method based on the moment
of inertia is proposed for training neural networks on
the basis of individual data for a particular person.
However, the method can be modified to take into ac-
count anthropometric data such as the weight, height
or age of the subject, and the neural network can be
expanded to include the appropriate number of input
neurons, and can thus become universal.

The main object of our study was to predict the
trajectory of cyclogram curves of on the basis of the
current state of the lower extremities, and to make
the prediction with the use of artificial intelligence.
The set of data for training artificial neural networks

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Acta Polytechnica Vol. 52 No. 1/2012

was measured on 10 volunteers recruited from stu-
dents of the Czech Technical University in Prague.
The subjects were asked to walk properly on a tread-
mill at a variable walking speed. The main human
walking speed was 1.5 km/h for studying and adjust-
ing the proposed method.

We used the very accurate Lukotronic AS200 mo-
tion capture system to record the data. The IR
camera with active markers was manufactured by
Lutz Mechatronic Technology e.U. According to the
CE-certification for medical products, EU-Directive
93/42/EEC, the manufacturer declares that the
Lukotronic motion analysis system can be used for
patient care in hospitals and rehabilitation centers.
One camera system mounted in front of or behind
the subject moving on the treadmill recorded the 3D
motion of the lower limbs. Markers were placed in ac-
cordance with the manufacturer’s recommendations
for gait analysis by GaitLab software. The recom-
mended marker set model is the same as the set de-
fined by the Helen Hayes Hospital model [26] for Vi-
con Clinical Manager and the sagittal plane. Physi-
cians defined the location of the markers by the Helen
Hayes marker set model. The position of the mark-
ers on the foot was chosen mainly according to the
requirements for a good record, because the move-
ment of the feet is not usually measured in practice,
and the manufacturer does not mention placing more
than one marker on the foot.

After obtaining the measured data, we created
the necessary cyclograms in MATLAB software. We
obtained graphs of the changes of angles in all the
main points in the lower part of human body and
the sagittal plane. This was important for subse-
quent computation of the curve of the cyclograms
and for predicting the motion. We used cyclograms
as models/patterns for training the artificial neural
networks. After training the neural networks we used
them to predict the future states of the angles in the
joints of the lower limbs.

3 Results

Our measured data indicates that the angle in the
knee usually changes from 5◦ (stretched leg) to 70◦

(shrugged leg). In the hip, the angle is usually from
5◦ to 40◦, Figure 3, and in the ankle the angle is
usually from 60◦ to 100◦, Figure 6. Under the as-
sumption of neglecting inaccurate placement of the
markers, and other simplifying assumptions, we can
state that the angles correspond to the angle between
the femur and the tibia and between the tibia and the
metatarsus. The cyclogram in Figure 3 shows that
the swing phase typically starts at a thigh extension
angle of 0◦ and knee flexion of about 80 % of the
maximum. The subject weighed 65 kilograms and
was 23 years old.

Fig. 3: Knee-hip cyclogram (treadmill, walking speed
1.5 km/h)

Fig. 4: Predicted knee-hip cyclogram (NN training, not
taking into account the inclination angles of the dia-
grams): predicted values of angles — circular symbols;
measured known values — cross symbols

After training the neural networks, we used parts
of the cyclograms for predicting the future states of
the joint angles. The result was that by using short
sections of the cyclogram curve, which was loaded
into a neural network, the trained neural network
predicted the subsequent behavior of the gait by the
predicted cyclogram curve. The two results of our
proposed method for gait prediction are shown by
the predicted knee-hip cyclogram (Figure 4) and the
predicted ankle-knee cyclogram (Figure 7). For the
prediction, we used NN learning without taking into
account the inclination angles of the diagrams. The
two-dimensional knee-hip and ankle-knee cyclogram
shows the prediction cyclogram curves based only on
measurements of past states, i.e. the remaining part
of the curve is predicted by assuming knowledge of a
short curve segment. We can see that the predicted
curves correspond only partially to the usual form
of cyclograms [13–16]. The 11-7-2 artificial neural

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Acta Polytechnica Vol. 52 No. 1/2012

network, Figure 2, was used for predicting the cyclo-
grams (Figure 4 and Figure 7). The results show that
the prediction is inaccurate, especially in the ankle-
knee cyclogram, because of its complicated shape.
The predicted knee-hip cyclogram is a relatively ac-
curate, and is similar to the pattern, Figure 3.

We found that predictions of movement based
only on an evaluation of a small section of the cyclo-
gram are not suitable for predicting complex move-
ments. In the case of gait, which often changes, the
prediction is not always appropriate and accurate.

Fig. 5: Predicted knee-hip cyclogram (NN training, tak-
ing into account the inclination angles of the diagrams):
predicted values of the angles — circular symbols; mea-
sured known values — cross symbols

Fig. 6: Ankle-knee cyclogram (treadmill, walking speed
1.5 km/h)

Improved predictions are achieved by increasing
the monitored section of the curve of the cyclogram,
but the complexity of the neural network and the
computing time also increase. These features are un-
desirable, and we had to modify the neural networks
and the methods of learning. For ANN training and
for the prediction, we used the variables identified by

the moment of inertia method. The predicted knee-
hip cyclogram (Figure 5) and the predicted ankle-
knee cyclogram (Figure 8) show that the prediction
is very accurate. We identified slight variability in the
prediction of the angles, but the variability was neg-
ligible, because there are small variations in angles
even in typical gait. This way of making predictions
is suitable for situations where the movements of-
ten use stereotypes and typical gait changes between
these stereotypes of human walking.

According to the method described here, the cy-
clograms inform us about the previous position of the
lower limbs, and we can infer the expected conditions,
i.e. the states during future predicted walking. This
information is important in rehabilitation medicine,
and also as an expected value of the angles used in
control algorithms for lower limb prostheses.

Fig. 7: Predicted ankle-knee cyclogram (NN training,
not taking into account the inclination angles of the di-
agrams): predicted values of the angles — circular sym-
bols; measured known values — cross symbols

Fig. 8: Predicted ankle-knee cyclogram (NN training,
taking into account the inclination angles of the dia-
grams): predicted values of the angles — circular sym-
bols; measured known values — cross symbols

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Acta Polytechnica Vol. 52 No. 1/2012

4 Discussion

Cyclograms in conjunction with artificial intelligence
are broadly applicable in medicine. We have de-
scribed a method for predicting the motion of the
lower extremities, and this predicted data can be used
for evaluating human gait in physiotherapeutic prac-
tice, based on a study of angle-angle diagrams. Pre-
dictions of the joint angles of the leg are based on
the principles of artificial intelligence. In our study
we have used artificial backpropagation neural net-
works. By analogy with the analytical method based
on NNs for a study of two-dimensional cyclograms,
we designed and tested an methods based on NNs
for a study of three-dimensional knee-hip-ankle cy-
clograms. The new methods can be applied in clin-
ical practice for studies of disorders or characteris-
tics in the motion function of the human body [27],
and the method can be used in advanced control
systems for controlled prostheses of the lower ex-
tremities [28]. In the past, it was almost impos-
sible to use complex algorithms based on artificial
intelligence in the slow control systems of a con-
trolled prosthesis, but today we can consider apply-
ing the methods described here in the algorithms for
new prosthetic knee control systems [29, 30]. There
is an obvious opportunity to continue in this re-
search, and to use these methods to study and de-
sign a new hydraulic or pneumatic knee prosthe-
sis.

A new robotic orthosis offers a second very im-
portant possibility of applications in rehabilitation
medicine. We can use the proposed methods in algo-
rithms for a driven robotic gait orthosis for the pur-
poses of locomotion therapy [31, 32]. The therapy is
based on simulating the movement of the lower limbs
of healthy people by sophisticated robotic devices.
An example of such a system is the Hocoma Lokomat
system, which supports the rehabilitation of patients
suffering from neurological diseases (multiple sclero-
sis, post-stroke), patients after spinal cord injury or
after a traumatic brain injury resulting in partial loss
of the ability to walk. The Lokomat has been on
the market since 2001, and has crucially improved
the art and science of locomotion therapy at the Mo-
tol University Hospital in Prague. We assume that
our method will be applied in the control systems of
such a device, because artificial intelligence applied in
control systems could extend the possibilities of reha-
bilitation i.e. training possibilities. In addition, the
identified and predicted gait pattern could be individ-
ually adjusted to the patient’s needs [33, 34], because
our movement prediction method also takes into ac-
count the patient’s weight and age. Moreover, the
proposed new methods for identifying the technique
of human walking, which is used for training neural
networks, can be modified and used in other areas

of artificial intelligence, such as reinforcement learn-
ing [35, 36].

This work has not attempted to describe all po-
tential ways of applying cyclograms in conjunction
with artificial intelligence. We have shown new meth-
ods that have subsequently been proved by a number
of simulations in MATLAB software. These methods
based on cyclograms and ANNs could be suitable for
a broad range of applications.

Acknowledgement

This work was done at the Joint Department of
Biomedical Engineering CTU and Charles Univer-
sity in Prague in the framework of research program
No. VG20102015002 (2010–2015, MV0/VG), spon-
sored by the Ministry of the Interior of the Czech
Republic.

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Patrik Kutilek
Phone: +420 312 608 302, +420 224 358 490
E-mail: kutilek@fbmi.cvut.cz
Faculty of Biomedical Engineering
Czech Technical University in Prague
Sitna Sq. 3105, Kladno, Czech Republic

Slavka Viteckova
Phone: +420 224 358 401
E-mail: slavka.viteckova@fbmi.cvut.cz
Faculty of Biomedical Engineering
Czech Technical University in Prague
Sitna Sq. 3105, Kladno, Czech Republic

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