New Developments in Pure and Applied Mathematics
MODELLING OF EXOSKELETON
MOVEMENT IN VERTICALIZATION
PROCESS
Sergey Jatsun, Doctor of science, Professor, Head of the department of mechanics, mechatronics and
robotics, Southwest State University
Sergei Savin, Candidate of science, Junior research fellow of the department of mechanics,
mechatronics and robotics, Southwest State University
Petr Bezmen, Candidate of science, Associate professor of the department of mechanics, mechatronics
and robotics, Southwest State University
The paper is devoted to the analytical construction of
exoskeleton motion trajectories by means of the
experimental data characterizing the movement of a
person in the getting up process. The solution of this
problem will allow adapting the dynamic characteristics
exoskeleton to the motion of person.
Abstract—The present paper focuses on comparison
and analysis of different techniques of data processing as
related to the problem of acquiring experimental data of
human getting up process in such form that it can be used
as an input to the control system of an exoskeleton. Use
of approximation by trigonometric series, polynomials
and spline functions is discussed.
Keywords—Experimental data processing, exoskeleton
II.
control inputs.
I.
The purpose of this paper is the comparative analysis
of data processing methods to process information about
the person motions during the process of getting up. It is
possible to synthesize the control actions for the
exoskeleton control system on the basis of these methods.
The methods of obtaining the experimental data may be
different and are not described here. We assume that the
source data is written in the form of numerical sequences
which define a person position at each time interval. A
person position is determined by the generalized
coordinates. In this paper, we consider the case when the
system of three generalized coordinates defines the
orientation of a shin, a hip and a trunk. These parts of a
person body execute the plane-parallel motions but a foot
remains stationary. Hence it appears that the person
movement can be described as the four-link mechanism
motion with one fixed link (figure 1).
INTRODUCTION
Nowadays, various robotics facilities are systems
composed of two main elements – a man and a machine.
These systems include objects called the exoskeleton and
used to extend the functionality of a person. The
interaction of these two components determines the
quality of the system as a whole.
Recently, the new generation of exoskeletons came
into service and it allows a person to move in space, even
in case of damage of the lower extremities. In addition, it
becomes possible to significantly expand human
capabilities when a person performs tasks that impose
high demands on endurance and physical strength of a
man.
Obviously, the creation of such devices is possible with
a well-developed theory of the functioning of “manmachine” systems, and the particular emphasis should be
given to the control. The main aim demands the
consideration of interaction between man and machine.
The common problems in the theory of walking
mechanisms have been developed in [1-4]. The
mathematical modelling of elements motion is one of the
most important tools in the study of behavior of the
systems like an exoskeleton. At the same time it is
necessary to process a large number of experimentally
obtained movement curves, solve the problem of
approximation and obtain the analytical dependences
which reflect the change in generalized coordinates
describing the position of the mechanism links.
ISBN: 978-1-61804-287-3
STATEMENT OF THE PROBLEM
Figure 1 A person in the process of getting up and the four-link
mechanism
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New Developments in Pure and Applied Mathematics
Moreover, the smoothing shifts the boundaries of
transitions between different stages of the movement.
A shin orientation is determined by the angle ϕ1 , a hip
orientation – by the angle ϕ 2 , and a trunk orientation –
by the angle ϕ 3 . The figure 2 shows the human rising
process data derived from experiments in the laboratory
of the department of mechanics, mechatronics and
robotics, Southwest State University. The corresponding
numerical dependences are shown in figure 2 (a).
III.
THE APPROXIMATION BY
TRIGONOMETRIC SERIES
An approximation is a method of data processing that
allows to selectively retain the required information about
the motion by eliminating unwanted high-frequency
components. We consider an approximation of the
original dependences by the trigonometric Fourier series:
1
f i (t ) = ai ,0 +
2
where:
fi
∑ (a
n
i ,k
cos(k ⋅ t ) + bi ,k cos(k ⋅ t )) (2)
k =1
– the function, that approximates the
experimental dependence ϕ i , ai ,k , bi ,k – the Fourier
a)
series coefficients, i = 1, 2, 3 .
The figure 3 shows the results of approximation at n =
30. The series coefficients for the i-th generalized
coordinate are chosen by minimizing of the following
positive definite function:
Ei =

 ϕi t j − 1 ai ,0 −

2
j =1 
∑ ()
mi


i ,k cos k ⋅ t j + bi ,k cos k ⋅ t j 

∑ (a
30
k =1
( )
( ))
2
, (3)
where mi – the total number of the dependence ϕ i
points, i = 1, 2, 3 .
b)
1, 2 and 3 – The time dependences ϕ1 (t ) , ϕ 2 (t ) and ϕ3 (t ) ,
respectively
Figure 2 The experimental data: a) – before smoothing process,
b) – after smoothing process
The resulting graphs (figure 2 (a)) are largely nonlinear, which is due to the inaccuracy of measurement.
High-frequency components of the signal give a stepped
appearance and do not carry useful information and
should be removed before the signal can be used as the
input action for the exoskeleton control system. The
smoothing method is the simplest way to process these
signals. The figure 2 (b) shows the dependences received
after data processing by the sliding window method that
uses the following expression:
ϕ~i (t j ) =
1
n
∑ϕ (t ) ,
a)
n
i
j+k
i = 1, 2, 3 ,
(1)
k =1
( )
where: ϕ i t j – the generalized coordinate ϕ i value at the
time t before data processing, ϕ~i t j – the generalized
( )
b)
1, 2 and 3 – the approximation of the time dependences ϕ1 (t ) ,
coordinate ϕ i value at the time t after data processing, n –
the width of the sliding window.
The use of these dependences has several
disadvantages. Smoothed graphs and their derivatives
retain high-frequency components described above,
(although their amplitude decreased significantly) that
can impair the performance of the control system.
ISBN: 978-1-61804-287-3
ϕ 2 (t ) and ϕ3 (t ) , respectively
Figure 3 The approximation of the initial data by trigonometric
series:a) with using of the formula (3), b) with using of the
formula (4)
It is possible to pay attention to the occurrence of
oscillations on the “direct” sections of the graphs (where
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New Developments in Pure and Applied Mathematics
the first order time-derivative is equal to null, figure 3
(a)). The approximation by piecewise-defined function


ϕ i (t1 ) if t ∈ [t i1 , ti 2 ]

ρ i (t ) = ϕ i (t3 ) if t ∈ [ti 3 , ti 4 ]

n
1 a +
(a cos(k ⋅ t ) + bi, k cos(k ⋅ t )) otherwise,
 2 i , 0 k =1 i , k
can avoid these oscillations. We consider the case when
the graph has two straight sections:
(4)
IV.
THE APPROXIMATION BY POLYNOMIAL
FUNCTIONS AND SPIELS
∑
The n-th order polynomial can be written in
form:
where: ρi (t ) – the piecewise-defined function used for
n
the approximation of original dependence ϕ i , [ti1 , ti 2 ] and
[ti 3 , ti 4 ] – the first and the second sections, respectively,
Pi (t ) =
∑c
i,k
tk ,
i = 1, 2, 3 ,
(5)
k =0
where: Pi (t ) – the polynomial functions that are used for
where ϕ i has the first order derivative that is equal to
null. The figure 3 (b) presents the experimental data
approximation results of the piecewise-defined function.
We note that in both cases, the dependences have
significant vibrational state, and it is especially
conspicuous in the graphs of the time-derivatives. The
figure 4 shows the first order time-derivatives plots of the
functions f k in modulo.
the approximation of the original dependence ϕ i , ci , k –
the polynomial coefficients.
The polynomial coefficients for the i-th generalized
coordinate are chosen in the course of minimizing of the
following function:
(6)
i = 1, 2, 3 ,
 ,


E = ∑
 ϕ (t ) − ∑ c t 
mi
i
j =1
2
n

i
i,k
j
k =0
k
j

The figure 5 shows the graphs obtained due to the
approximation of the functions ϕ i by the sixth order
polynomials.
a)
a)
b)
b)
c)
Figure 4 The time dependences a) f1 , b) f2 , c) f3 in a
logarithmic scale
c)
The figures 3 and 4 allow us to conclude that the use of
the approximation by trigonometric series make it
possible to reliably reproduce the original dependences,
but this method leads to additional high-frequency
components in the signal spectrum. The use of functions
obtained by this way as the input action for the
exoskeleton control system can negatively affect the
control process.
1 – the polynomial functions Pi (t ) , 2 – the time dependences
ϕi
Figure 5 The approximation of the experimental data by the
sixth order polynomials for the generalized coordinate: a) ϕ1 b)
ϕ 2 , c) ϕ3
As well as in the case of approximation by
trigonometric series, the use of polynomial functions
ISBN: 978-1-61804-287-3
85
New Developments in Pure and Applied Mathematics
leads to errors in straight segment of the dependences ϕ i .
To obtain the better results, we use the approximation by
spline functions. For this purpose we divide the
dependences ϕ1 (t ) and ϕ 2 (t ) into three portions, and the
dependence ϕ3 (t ) into the four sections. Let this
partitioning occur at the points corresponding to the time
instants: t11 = 3.48 sec, t12 = 4.93 sec for the dependence
ϕ1 (t ) , and t 21 = 2.69 sec, t22 = 4.4 sec for the dependence
ϕ 2 (t ) , and t31 = 1.38 sec, t32 = 2.56 sec, t33 = 4.03 sec
for the dependence ϕ3 (t ) . The first section and the last
c)
functions S1 (t ), S2 (t ), S3 (t ) , 7,8,9 – the functions
S1 (t ), S2 (t ), S3 (t )
section of each spline are specified by the zero order
polynomial, and the rest is defined by the seventh order
polynomials.
To calculate the polynomial coefficients, we can write
the following conditions:
S1 (t11 ) = ϕ1 (t11 )
,

S1 (t12 ) = ϕ1 (t12 )



S1 (t11 ) = S1 (t11 ) = S1 (t11 ) = 0
S (t ) = S (t ) = S (t ) = 0
1 12
1 12
 1 12
1, 2, 3 – the functions S1 (t ), S 2 (t ), S3 (t ) , 4, 5, 6 – the
Figure 6 The graphs: a) the spline functions, b) the first order
time-derivatives of spline functions, c) the second order timederivatives of spline functions
S 2 (t 21 ) = ϕ 2 (t 21 )

,
S 2 (t 22 ) = ϕ 2 (t 22 )

 (t ) = S (t ) = 0
(
)
S
t
S
=
2 21
2 21
 2 21
S (t ) = S (t ) = S (t ) = 0
2 22
2 22
 2 22
Because of the seven order spline functions, it is
possible to achieve absence of function discontinuities in
the graphs of the time dependences of the generalized
velocities and accelerations (figure 6 (b) and (c)). Also,
the spline functions eliminate the high-frequency
oscillations.
S3 (t31 ) = ϕ3 (t31 )

S3 (t32 ) = ϕ3 (t32 )
S3 (t33 ) = ϕ3 (t33 )
, (7)


 (t ) = S (t ) = 0
(
)
=
S
t
S
3
31
3
31
3
31

S (t ) = S (t ) = S (t ) = 0
3 32
3 32
 3 32
S3 (t33 ) = S3 (t33 ) = S3 (t33 ) = 0
V.
THE DETERMINATION OF MOMENTS
NEEDED TO IMPLEMENT THE OBTAINED
MOVEMENT TRAJECTORIES OF THE MECHANISM
where S1 (t ), S 2 (t ), S3 (t ) are the spline functions used for
approximation of the time dependences ϕ1 (t ) , ϕ 2 (t ) и
Different approaches with some accuracy allow
obtaining the mechanism movement that is determined by
certain changes of the generalized coordinates. For
example, it is possible to build the automatic control
system using negative feedback to control the generalized
coordinates. We consider another approach: the moments
sequence realizes the desired movement and can be
determined by solving the inverse problem of dynamics.
The equations of the flat three-link mechanism
dynamics can be found in a number of papers, including
[5], we do not give them in the paper. The flat three-link
mechanism is a series of connected links by joints. In
general terms, the equation of the mechanism dynamics
can be written as follows:
      

A(ϕ ) ⋅ ϕ + b ϕ ,ϕ + g (ϕ ) = T ⋅ τ , (8)
  
where A(ϕ ) – the matrix of kinetic energy, ϕ ,ϕ ,ϕ – the
vectors of the generalized coordinates, generalized
velocities, and generalized accelerations, respectively,
  
b (ϕ , ϕ ) – the vector bound with the compound centrifugal
 
forces, g (ϕ ) – the vector of the generalized potential

forces, τ – the vector consists of some elements – the
moments which are generated by electrical drives, T –
the transition matrix.
The initial data for solving of the inverse
problem of dynamics is the law of the generalized
coordinates alteration and their first and second order
time-derivatives.
As
a
law,
we
use
the
functions S1 (t ), S 2 (t ), S3 (t ) , described in the previous
section. In the solving of the inverse problem of
ϕ3 (t ) , respectively.
Due to using of the criterion (7), we can find the
desired coefficients to plot splines (figure 6).
( )
a)
b)
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New Developments in Pure and Applied Mathematics
dynamics, we obtained the results which depend on the
moments of electrical drives (figure 7).
1, 2, 3 – the moments of electrical drives τ 1 (t ) , τ 2 (t ) и
τ 3 (t ) mounted in the ankle joint, the knee joint and the
coxofemoral joint of exoskeleton, respectively
Figure 7 The time dependence of the moments generated by the
exoskeleton drives
Two graphs τ 1 (t ) and τ 2 (t ) have function
discontinuities at t = 2.69 sec. (figure 7). Before this time
moment t = 2.69 sec a hip and a shin were in static
equilibrium under the influence of reaction at supports
(i.e. a chair and a floor). Thus, the time moment t = 2.69
sec is the power up time of first and second electrical
drives.
VI.
CONCLUSION
This paper discusses various processing methods of
experimental data which describe the motion of a person
in the getting up process. It is shown that the
approximation of initial relationships by trigonometric
series provides a sufficient accuracy to reproduce the
shape of the original dependences, but adds the highfrequency harmonics in the spectrum of the signal. These
harmonics can adversely affect the quality of the control
process. The paper demonstrates that this problem can be
avoided by using the spline approximation. The solution
results of the inverse problem of dynamics are presented.
These results were gotten by means of the approximating
spline functions and their derivatives. The spline
functions make it possible to reduce the peak magnitude
of the second order time-derivatives with respect to the
original dependences and other types of approximating
functions.
REFERENCES
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A.
M.
Peremeshcheniye
antropomorfnykh
mekhanizmov. M.: Nauka, 1982.
[2]. Beletskiy V. V., Berbyuk V. Ye. Nelineynaya model dvunogogo
shagayushchego apparata, snabzhennogo upravlyayemymi
stopami. M.: Nauka, 1982.
[3]. Beletskiy V. V. Dvunogaya khodba: Model'nyye zadachi dinamiki
i upravleniya. M.: Nauka, 1984.
[4]. Vukobratovich M. K. Shagayushchiye roboty i antropomorfnyye
mekhanizmy. M.: Mir, 1976.
[5]. Vorochaeva L. Yu. Simulation of Motion of a Three Link Robot
with Controlled Friction Forces on a Horizontal Rough Surface /
L. Yu. Vorochaeva, G. S. Naumov, S. F. Yatsun // Journal of
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1, pp. 151–164.
ISBN: 978-1-61804-287-3
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