Volume 04 Issue 11-2024
101
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
11
Pages:
101-105
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
This article explores the methods of controlling objects whose motion is described by a system of linear differential-
difference equations. It provides an analysis that serves as a basis for developing an approach to eliminate existing
shortcomings.
KEYWORDS
Control function parameters, optimal solution, kinematic chain, system of differential-difference equations.
INTRODUCTION
The shortcomings of controllability in linear stationary
systems are described in [1], and these issues are also
relevant for systems of differential-difference
equations. The primary criterion for checking the full
controllability of such systems is Kalman's criterion,
which applies universally to controllable systems.
The general form of a system of linear differential-
difference equations is as follows::
( )
(
)
( )
( )
( )
x t
Ax t
h
Bu t
y t
Hx t
=
−
+
=
(1)
where:
( )
x
x t
n
=
− −
a vector representing the state of the system at time
t
;
Research Article
ON THE METHOD OF CONTROLLING OBJECTS WHOSE MOTION IS
DESCRIBED BY A SYSTEM OF LINEAR DIFFERENTIAL-DIFFERENCE
EQUATIONS
Submission Date:
November 20, 2024,
Accepted Date:
November 25, 2024,
Published Date:
November 30, 2024
Crossref doi:
https://doi.org/10.37547/ajast/Volume04Issue11-15
Dilshod Davronovich Aroev
PhD, Associate Professor, Kokand State Pedagogical Institute, Uzbekistan
Orchid: - https://orcid.org/0009-0004-0392-7597
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 11-2024
102
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
11
Pages:
101-105
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
A
−
constant matrices of specific dimensions;
(
)
n n
−
ўлчовли ўзгармас матрица;
(
)
B
n m
−
−
A constant matrix with non-zero elements;;
h
−
an infinitesimally small quantity;;
( )
(1
)
u t
m
−
−
a control function vector;
0
0,
h
t
t
T
;
( )
y t
n
− −
a vector of specific dimensions depending on the state vector
(1
)
n
−
;
(
)
H
r n
− −
A constant matrix of specified dimensions.
The first equation in (1) represents the differential-
difference equation, while the second equation
describes the phase state of the first one.
Main Section.
Objects whose motion is described by a
system
of
differential-difference
equations,
particularly for industrial robots, have the following
advantages and disadvantages for the system in (1):
1.
Motion system:
Deterministic.
2.
Control system:
Deterministic.
3.
Form of the motion model:
Linear
differential-difference
equation-
( )
(
)
( )
x t
Ax t
h
Bu t
=
− +
.
4.
Advantage of the model:
The form of the
model is simple for systematic analysis and
convenient for verifying full controllability
using Kalman's criterion.
5.
Disadvantage of the model:
It is known that if
2
1
[ ,
,
,...,
]
n
rank B AB A B
A
B
n
−
=
the system (1) is fully controllable according to
Kalman's criterion. In the system of equations (1), the
matrix
B
has dimensions of
(
)
n m
−
, particularly
for the motion of industrial robots,
m n
=
.
Thus, the matrix,
B
(
)
n n
−
is extended to the
required dimensions. The dimension of the
controllable subspace of the system (1) becomes equal
to
𝑛
n. As a result, the obtained system satisfies the
property of full controllability in
n
R
-dimensional space
but does not satisfy the property of observability. To
ensure the property of full controllability, the system
(1) is considered as two autonomous systems as
follows [2]:
1
1
1
11 1
11 1
1
1
2
22
2
1
1
1 1
2
2
( )
(
)
( )
( )
(
)
( )
( )
( )
x t
A x t
h
B u t
x t
A x t
h
y t
H x t
H x t
=
−
+
=
−
=
+
(2)
The second part of the system (2) is discarded because
it does not satisfy the property of observability. In
practice, particularly in the complex phase motion of
industrial robots, the second part of system (2)
complements the properties of the first part [1].
Volume 04 Issue 11-2024
103
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
11
Pages:
101-105
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
From this, it becomes evident that additional scientific
research is required to develop control methods for
linear differential-difference systems.
For linear differential-difference systems, the following
problem is posed:
Is it possible to ensure full controllability of the system
(1) by optimizing the number of parameters of the
control function during its management?
The problem was posed in a similar form for
autonomous control systems in [1]. However, although
the system was given in an autonomous linear form,
the problem concerning the differential-difference
nature of the system was not resolved., pecifically, in
mathematical
terms
1
:
(
n
k
n
f R
R
R
R
f
→
−
−
continuous
function) does there exist a control function
( )
k
n
v t
R
R
−
?
The problem condition is defined as follows:
The system is in the form of a linear differential-
difference equation.
The system is finite-dimensional.
For robotic systems, the problem is posed as follows:
It is known that the stages of industrial robot motion
can be divided into three phases: strategic, tactical,
and execution. In the strategic phase, design issues are
addressed; in the tactical phase, modeling and
planning are carried out; and in the execution phase,
control issues are solved. This is schematically
represented as shown in Figure 1:
In this case, MADS
–
the methods of automated design
systems
; М
M- methods of modeling; MP- Methods of
planning.; MC- Methods of control.
Figure 1. Control diagram of an industrial robot
.
Volume 04 Issue 11-2024
104
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
11
Pages:
101-105
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
CONCLUSION
The issues of designing, modeling, and controlling
industrial robot motion have been thoroughly studied,
investigated, and the necessary results have been
obtained. An analysis of the literature on the research
topic revealed the following shortcomings and gaps
related to industrial robot motion modeling and
control:
It is known that the dynamic motion of an industrial
robot is expressed using Lagrange's second law,
Newton-Euler, D'Alembert, Gauss, Appel, and Kane
equations, and there are several methods for solving
these equations [3, 4]. When solving these equations,
the number of equations increases twofold in relation
to the number of unknowns [4]. This indicates that in
the case of the robot's complex phase motion, the
optimal trajectory is not unique. The decision-maker is
unable to reach a consensus in choosing the
appropriate trajectory. As a result, the process stops
due to the collision of the industrial robot with objects
in the external environment [5]. The SR (robot arm)
changes the structure of the working hand. Such cases
lead to excessive dependencies in the kinematic chain
of the industrial robot, which requires the acceptance
of alternative solutions and further research [1].
2. For the optimal control of an industrial robot, errors
in calculating the coefficients of the system of
equations representing its motion in trapezoidal form,
as well as obstacles in communication channels, can
lead to delays in the movement of the robot's links and
an increase in the overall time required for
technological operations. Furthermore, this can cause
a decrease in the stability of the robot's motion and
deteriorate its performance.
From these shortcomings in the mathematical models
of the industrial robot's motion and their solving
methods, it is evident that the correct specification of
robot motion and the execution of complex phase
operations,
based
on
technical
and
design
characteristics, require the development of an optimal
solution. Effective use of control function parameters
in selecting the appropriate solution enhances the
robot's operational efficiency. This, in turn, highlights
the necessity to improve the methods of modeling and
controlling industrial robot motion.
REFERENCES
1.
Onorboev B.O., Abdullaev A.Q., Siddiqov R.Y.
"Issues of improving positional accuracy and
sensitivity in the control of robotic systems" //
Current state and development trends of
information technologies
—
Republic Scientific
and Technical Conference, Tashkent, September
23-25, 2008, pp. 209-212.
2.
Li E.B., Markus L.M. Fundamentals of Optimal
Control Theory
—
Moscow: Nauka Publishing,
1972.
—
p. 576.
Volume 04 Issue 11-2024
105
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
11
Pages:
101-105
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
3.
Belousov I.R. Formation of robot manipulator
dynamics equations. Preprint. IPM M.V. Keldysh
RAN, 2002.
—
36 p.
4.
Fu K., Gonzalez R., Lee K. Robotics
—
Moscow: Mir
Publishing, 1989.
—
p. 624.
5.
Khonboboev Kh.I., Siddiqov R.Y., Aroev D.D. On
industrial robots in unknown environments //
Uzbek Journal of Informatics and Energy Problems
—
Tashkent, 2011, No. 2, pp. 31-34.
