Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs
The purpose of the article is to develop a technique for detecting singular states in GMC based on three collinear pairs. Results. The analysis was carried out and the singular states of the GMC with three collinear pairs were revealed.
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України
2019
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nasplib_isofts_kiev_ua-123456789-1617482025-02-23T17:55:37Z Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs Топологічний аналіз області допустимих значень кінетичного моменту силових гіроскопічних комплексів кратних схем Топологический анализ области допустимых значений кинетического момента силовых гироскопических комплексов кратных схем Yefymenko, M.V. Kudermetov, R.K. Intelligent Control and Systems The purpose of the article is to develop a technique for detecting singular states in GMC based on three collinear pairs. Results. The analysis was carried out and the singular states of the GMC with three collinear pairs were revealed. Мета роботи — розроблення методики виявлення сингулярних поверхонь в СГК кратних схем. Результат — проведено аналіз та виявлено сингулярні стани схеми СГК, що містить три Колінеарні пари. Цель работы — разработка методики выявления сингулярных поверхностей в СГК кратных схем. Результат — проведен анализ и выявлены сингулярные состояния схемы СГК, содержащей три коллинеарные пары. 2019 Article Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs / M.V. Yefymenko, R.K. Kudermetov // Cybernetics and computer engineering. — 2019. — № 2 (196). — С. 43-58. — Бібліогр.: 17 назв. — англ. 2663-2578 DOI: https:// 10.15407/kvt196.02.043 https://nasplib.isofts.kiev.ua/handle/123456789/161748 550:531; 681.51 en Кибернетика и вычислительная техника application/pdf Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Intelligent Control and Systems Intelligent Control and Systems |
| spellingShingle |
Intelligent Control and Systems Intelligent Control and Systems Yefymenko, M.V. Kudermetov, R.K. Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs Кибернетика и вычислительная техника |
| description |
The purpose of the article is to develop a technique for detecting singular states in GMC based on three collinear pairs. Results. The analysis was carried out and the singular states of the GMC with three collinear pairs were revealed. |
| format |
Article |
| author |
Yefymenko, M.V. Kudermetov, R.K. |
| author_facet |
Yefymenko, M.V. Kudermetov, R.K. |
| author_sort |
Yefymenko, M.V. |
| title |
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs |
| title_short |
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs |
| title_full |
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs |
| title_fullStr |
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs |
| title_full_unstemmed |
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs |
| title_sort |
topological analysis of angular momentum range values of the gyro moment clusters based on collinear gyrodines pairs |
| publisher |
Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
| publishDate |
2019 |
| topic_facet |
Intelligent Control and Systems |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/161748 |
| citation_txt |
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters Based on Collinear Gyrodines Pairs / M.V. Yefymenko, R.K. Kudermetov // Cybernetics and computer engineering. — 2019. — № 2 (196). — С. 43-58. — Бібліогр.: 17 назв. — англ. |
| series |
Кибернетика и вычислительная техника |
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ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196)
Intelligent Control and Systems
DOI: https:// 10.15407/kvt196.02.043
UDC 550:531; 681.51
YEFYMENKO M.V.1, PhD (Engineering),
associate professor of Zaporizhzhia National Technical University,
Chief Designer of SMC
e-mail: nefimenko@gmail.com
KUDERMETOV R.K.2, PhD (Engineering),
associate professor of Zaporizhzhia National Technical University,
Head of Computer Systems and Networks Department
1 "HARTRON-UCOM LTD.",
166, Soborniy av., Zaporizhzhia, 69035, Ukraine
2 Networks Department,
64, Zhukovsky str. Zaporizhzhia, 69063, Ukraine
TOPOLOGICAL ANALYSIS OF ANGULAR MOMENTUM
RANGE VALUES OF THE GYRO MOMENT CLUSTERS
BASED ON COLLINEAR GYRODINES PAIRS
Introduction. To ensure the high dynamic characteristics of Earth remote sensing satellites
in their orientation systems, the gyro moment clusters (GMCs) based on excessive number
(more than three) two-gimbals control moment gyrodines (GDs) can be used as actuators.
The attitude control by GD actuators task is the most difficult among the tasks of spacecraft
(SC) reorientation control. The central issue in solving this task is the synthesis of the control
laws for precession angles of individual GDs when there are excessive. Success in solving the
control problem is substantially determined by the choice of the GMC structure, it means the
number of GDs used and their mutual positions of the precession axes. From this choice
depends on the possibility of forming by GMC the necessary control momentum, the existence
and number of special GMC states, the complexity of the control laws for the precession
angles of the individual GDs included in the GMC. This is because in order to maintain the
desired SC orientation for a long time and to perform its turns with the required angular
rate, the GMC must have a sufficient margin of angular momentum. The allowable values of
the total angular momentum created by the GDs form a certain area that is bounded by a
closed surface of complex shape in a coordinate system rigidly attached to main SC body.
Inside this area there are particular surfaces on which the control of the GDs is complicated
or unfeasible. These surfaces are called singular. In this regard, for SC attitude control in
addition to control the precession rate of individual GDs it is also necessary to control the
mutual orientation of the angular momenta of the GDs in GMC. That is why one of the most
important problems of the control laws synthesis with the use of GMC is the identifying sin-
gular surfaces (topological analysis) in the area allowable angular momentum of the GMC.
© YEFYMENKO M.V., KUDERMETOV R.K., 2019
43
Yefymenko M.V., Kudermetov R.K.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 44
The purpose of the article is to develop a technique for detecting singular states in
GMC based on three collinear pairs.
Results. The analysis was carried out and the singular states of the GMC with three col-
linear pairs were revealed.
Conclusion. An original technique of a topological analysis of GMC based on collinear GD’s
pairs is proposed. This technique may be useful to developers of SC attitude control systems.
Keywords: spacecraft, gyrodine, singular vector, singular surface.
INTRODUCTION
The task of a SC attitude control is the control by angular motion around CS
mass center. Nowadays this task is very actual due to the ever-increasing re-
quirements for the dynamic characteristics of CS angular maneuvers. The turns
must run from any current position to any given position. Wherein the attitude
accuracy after turning should be angular minutes, and the angular rates can attain
the values 2–3 degrees per second. For example, the French SC Spot 7 for shoot-
ing the Earth’s surface with high-resolution launched on June 30, 2014, provides
the following dynamic characteristics of spatial maneuvers:
− angular orientation accuracy — 1,7 arcmin;
− the maximum angular rate — 2,1 deg/s.
The assurance of such high dynamic characteristics is complicated by the
fact that the mass increase trend is observed for high resolution Earth remote
sensing satellites. If the mass of the earlier satellites Ikonos, OrbView-3 was 720
and 304 kg, respectively, then the mass of the subsequent satellites QuickBird-2,
WorldView-1, Geoeye-1, WorldView-2 exceeds 2000 kg. It is known that for
SC with a considerable mass the most effective actuators of attitude control sys-
tems are the GMCs based on excessive number (more than three) two-gimbals
control moment GDs [1, 2]. The main advantage of GMC is that they have the
best “created control moment /own mass” ratio among other actuator types and
at the same time allow for complex rotational movements of the SC necessary
for solving many practically important attitude control tasks. The tasks of con-
trolling angular motion with the help of GMC are one of the most difficult tasks
among the SC attitude control tasks. The general approaches to the solution of
these tasks and some fundamental results were first presented in the domestic
publications by papers of Ye. N. Tokar in the 1970s – 1980s. A significant con-
tribution to the development of this subject matter was also made by the results
presented in [3, 4, 5, 6]. The modern approaches to the development SC attitude
control algorithms using GDs are considerd in [7, 8].
In order to maintain the SC given orientation for a long time and to turn
with the required angular rate the GMC must have a sufficient angular momen-
tum store. Possible summary angular momentum generated by the GDs form a
certain area HS in SC body fixed frame. This area is bounded by a closed sur-
face of complex shape. Inside area HS there are particular surfaces on which the
control of the GDs is complicated or unfeasible. These surfaces are called singu-
lar. There are two types of singular surfaces: passable and impassable. Passable
are called surfaces that can be passed by changing the mutual configuration of
GDs angular momenta without changing the summary angular momentum of
GMC. If this is not possible, then the surface is called impassable. In the class of
GDs systems, the most rational are the GMC on the base of collinear pairs. The
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 45
aggregating of GDs into so-called collinear groups, in which the precession axes
are arranged in parallel, gives one very important advantage: if there are six or
more GDs, all the singular surfaces of such schemes are strictly passable [9].
Quite a lot of work has been devoted to the problem of analyzing singular sur-
faces. As an example, one can cite the works [10, 11], in which singular passable
and impassable surfaces were classified based on analytical criteria for GMC
arbitrary schemes, and a technique for the type surface determining was pro-
posed. The works [12–16] deserve attention, where the attitude control problems
using GMC are studied in detail. Despite the fact that a lot of work has been
devoted to the problem of GMC topological analysis, interest to this problem
continues unabated today.
The main goal of this paper is to develop an analytical technique for identi-
fying singular surfaces for GMC based on collinear GD’s pairs and to carry out a
topological analysis of such GMC built on six GDs.
FORMULATION OF THE PROBLEM
The SC will be considered as a rigid body containing arbitrarily mounted collin-
ear pairs of GDs. Let us introduce the right orthogonal coordinate system B
rigidly attached to main SC body or Body Frame (BF) and the right orthogonal
coordinate system iG rigidly attached to mounting plains of the GDs in the thi
pair. Let us determine the position of the basis iG relative to the basis B by the
transition matrix iBGC . We assume that all the GDs have the same proper angu-
lar momenta const=0h .
For the GDs angular momenta vectors 12 −ih , ih2 , Ni , ,2 ,1 L= in the thi
collinear pair, the following relationships are valid:
.
,
22
1212
iGiB
iGiB
iBGi
iBGi
hCh
hCh
=
= −−
(1)
Hereinafter, the subscript in the vector designation indicates what basis it is
projected.
Introduce the vector ( )T21 , , Nαα=α K and Jacobi gradient matrix
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
α∂
∂
⋅⋅⋅
α∂
∂
=α
N
NBB hhL
2
2
1
1)( . (2)
To ensure SC angular motion complete controllability it is necessary that
the rank of the Jacobi gradient matrix )(αL , which is determined by the dimen-
sion of its column space, is three. Consider the following problem: to find re-
strictions on the vector of the GDs precession angles α in the scalar equations
form (equations of special surfaces)
Kkfk K2,1,0)( ==α ,
where K is the number of singular surfaces in GMC at which the rank of the
Jacobi gradient matrix )(αL is less than three
Yefymenko M.V., Kudermetov R.K.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 46
3)(rang <αL (3)
The solution of the problem. If the condition (3) is satisfied, then the col-
umns of the matrix )(αL lie in the same plane. In this case GMC cannot gener-
ate control torque components along unit vector Bλ of the normal to this plane.
Such an unit vector ( )T321 , , BBBB λλλ=λ is called the GMC singular vector
and is determined from the condition
0)( =αλ LT
B . (4)
This task can be reformulated as follows: to find all the restrictions on the
GDs precession angles vector, for which the homogeneous overdetermined sys-
tem of equations (4) has nonzero solutions. Equation (4) taking into account
expressions (1) and (2) can be represented as follows:
0
22
11
11
11
T
2
2
T
12
12
T
2
2
T
1
1
=λ
⎟⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
α∂
∂
α∂
∂
α∂
∂
α∂
∂
−−−
−
B
BGN
N
BGN
N
BG
BG
NNG
NNG
G
G
Ch
Ch
Ch
Ch
M . (5)
Denote by
0h
h
g i
i = the angular momenta of the GDs in relative units. Then
equation (5) can be represented as:
0
22
11
11
11
T
2
2
T
12
12
T
2
2
T
1
1
=λ
⎟⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
α∂
∂
α∂
∂
α∂
∂
α∂
∂
−−−
−
B
BGN
N
BGN
N
BG
BG
NNG
NNG
G
G
Cg
Cg
Cg
Cg
M . (6)
Suppose iR is the coordinate system associated with thi GD rotor. The axis
Rx of this system coincides with the precession axis of the thi GD, the axis Ry
coincides with the rotor rotation axis of thi GD and the Rz axis complements
the system to a right-hand one. Further assume that the mutual orientation of the
bases iG and iR corresponds to Fig. 1.
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 47
Fig. 1. Mutual orientation of the bases Gi and Ri
In this case, for the thi collinear pair GDs angular momenta in the relative
units, the following expressions are valid:
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
=
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
α
α=
j
j
j
jG
z
yg j
0
sin
cos
0
, Nj 2 , ,2,1 K= . (7)
It follows from (7) that
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
−=
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
α
α−=
α∂
∂
j
j
j
j
j
G
y
z
g j
0
cos
sin
0
. (8)
By substituting (8) into (6), we obtain N subsystems
0
0
0
22
1212 =λ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
− −−
iG
ii
ii
yz
yz
, Ni , ,2,1 K= , (9)
where
BBGG ii C λ=λ . (10)
Consider the matrices
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
= −−
−
ii
ii
ii yz
yz
А
22
1212
2,12 . (11)
Denote by iiD 2,12 − the determinant of the matrix iiA 2,12 − . Finding this de-
terminant, we obtain
Yefymenko M.V., Kudermetov R.K.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 48
=αα−αα=−= −−−−− 122122212122212 iiiiiiiii,i sincoscossinyzyzD
)sin( ii 122 −α−α= (12)
The solution of the systems (9) depends on the values of the determinants
and is determined by the expression:
⎪
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎪
⎨
⎧
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε±
ε±
ε
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε
≠
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛±
=λ=λ
−
−
−
−
;0,
cos
cos
sin
cos
cos
sin
;0 ,
0
0
1
2,12
12
12
2
2
2,12
ii
ii
ii
i
ii
ii
i
ii
GBGB
D
z
y
z
y
D
C ii , (13)
where iε is an arbitrary angle. The geometric interpretation of the solution (13) is as
follows. According to equation (9), the vector iGλ is a singular vector of the thi col-
linear pair. If the angular momenta of the included in the collinear pair GDs do not lie
on one straight line, then the singular vector iGλ is perpendicular to the plane passing
through the vectors 12 −iGg , iGg 2 and parallel to the GDs of precession axes, which,
according to Fig. 1, is the axis ix . If the angular momenta of the included into collin-
ear pair GDs lie on one straight line, then they form a straight line to which an infinite
number of perpendiculars can be drawn, which position in the basis iG is determined
by the angle iε . Since, if the condition 02,12 =− iiD is satisfied, the angular momenta
in the thi collinear pair are related by the relationship ii GG gg 212 ±=
−
, hereafter only
vectors iGg 2 are considered. Then, taking into account expressions (7) and (10), for a
singular vector Bλ it is true the relationship
⎪⎩
⎪
⎨
⎧
=++
≠±
==
−
−
;0,coscossin
;0,
2,1223221
2,121
iiii
BG
ii
BG
i
BG
ii
BG
GBGB Dzcycc
Dc
C
iii
i
ii εεε
λλ , (14)
where 3 ,2 ,1, =kc iBG
k are the columns of matrix iBGC . N relationships (14)
define the same vector Bλ , therefore, if Bλ is not a zero solution to system (4),
then for any ji ≠ , Nji , ,2 ,1, K= , the equalities should be satisfied
jjii GBGGBG CC λ=λ .
(15)
Thus, if there are the GDs precession angles vector α and angles iε for
which all 2/)1( −NN equalities (15) hold, then there exist nonzero solutions of
system (4) and there is a singular state. If there are no such angles, then 0=λB
and there is no singular state.
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 49
TOPOLOGICAL ANALYSIS OF THE 3-SPE SCHEME
As an example of the application of the proposed technique, consider the GMC
containing three collinear pairs. In the original work of J.W.Crenshaw [17], the
excessive multiple scheme based on three collinear GDs pairs was named as
3-Scissored Pair Ensemble (3-SPE).
We will assume that the precession axes of the GDs of the first group coincide
with the axis Bz , the precession axes of the second group coincide with the axis By
and the precession axes of the third group coincide with the axis Bx (Fig. 2). For such
a GMC scheme the transition matrixes iBGC , 3 ,2 ,1=i have the forms:
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛ −
=
001
010
100
1BGC ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛ −
=
100
001
010
2BGC ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
100
010
001
3BGC .
Fig. 2. Precession axes arrangement of GDs in 3-SPE
scheme
Yefymenko M.V., Kudermetov R.K.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 50
In this case, for the projections of the relative angular momenta along the
axes of B basis, the following expressions are valid:
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
000
cos
sin
1
1
1
1
1
1
1 B
B
B y
x
y
z
б
б
g ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
000
cos
sin
2
2
2
2
2
2
2 B
B
B y
x
y
z
б
б
g ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
α
−
=
B
B
B
z
x
z
yб
g
3
3
3
3
3
3
00
sin
0
cos
3 ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
α
−
=
B
B
B
z
x
z
yб
g
4
4
4
4
4
4
00
sin
0
cos
4 ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
α
α=
B
BB
z
y
z
yg
5
5
5
5
5
5
00
sin
cos
0
5 ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
α
α=
B
BB
z
y
z
yg
6
6
6
6
6
6
00
sin
cos
0
6 .
(16)
To determine the conditions under which the homogeneous system (4) has a
nonzero solution, we analyze the dependence of the solution of this system on
the values of the determinants 2,1D , 4,3D , 6,5D . According to (12) these deter-
minants are defined by the formulae
( )122,1 sin α−α=D , ( )344,3 sin α−α=D , ( )566,5 sin α−α=D .
Depending on the numerical values of the determinants 2,1D , 4,3D , 6,5D ,
the following different options are possible:
1) 02,1 ≠D , 04,3 ≠D , 06,5 ≠D .
In this case, according to (15), the condition for the existence of a nonzero
solution of system (4) is
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛±
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
±=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
± 0
0
1
0
1
0
1
0
0
.
But obviously equalities are impossible. Therefore, 0=λВ and no singular state.
2) 02,1 =D , 04,3 ≠D , 06,5 ≠D .
From relationships (14) and (15) it follows that for the existence of a non-
zero solution the equalities
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛±
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
±=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε−
0
0
1
0
1
0
sin
cos
cos
1
12
12
y
z
must be satisfied, which is impossible and there is no singular state.
3) 02,1 ≠D , 04,3 =D , 06,5 ≠D .
In this case, for the existence of a nonzero solution the equalities
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 51
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛±
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
± 0
0
1
cos
sin
cos
1
0
0
24
2
24
z
y
.
must be satisfied, which is impossible and there is no singular state.
4) 02,1 ≠D , 04,3 ≠D , 06,5 =D .
In this case, for the existence of a nonzero solution the equalities
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
±=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
± 36
36
3
cos
cos
sin
0
1
0
1
0
0
z
y
must be satisfied. These equalities are not satisfied, 0=λВ , therefore there is no
single state.
5) 02,1 ≠D , 04,3 =D , 06,5 =D .
A necessary condition for the existence of a nonzero solution for this case is
the validity following equalities:
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
± 36
36
3
24
2
24
cos
cos
sin
cos
sin
cos
1
0
0
z
y
z
y
.
These equalities are possible when π=ε ,02 ; π=ε ,03 ; 04 =y , 06 =y ,
14 ±=z , 16 ±=z . Therefore 0≠λB and there is a singular state. Let us find the
equation of the surface in the basis B on which this state arises. From the equal-
ities of the determinants 4,3D and 6,5D to zero and expressions (16) it follows
that 04343 ==−=±= BB xxyy m , 06565 =±==±= BB yyyy . Then
02
6
2
5
2
4
2
3 =+++ BBBB yyxx . (17)
The constraint (17) defines a surface in the basis B in the form of a unit circle ly-
ing in a plane BBYOX , to which the singular vector ( )T1 ,0 ,0 ±=λB corresponds.
6) 02,1 =D , 04,3 ≠D , 06,5 =D .
In this case, the condition for the zero solution existence is
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
±=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε−
36
36
3
1
12
12
cos
cos
sin
0
1
0
sin
cos
cos
z
yy
z
.
It is satisfied if π=ε ,01 ; π=ε ,03 ; 12 ±=y , 16 ±=y , 02 =z , 06 =z . From
the conditions 02,1 =D and 06,5 =D , the equalities follow:
02121 ==−=±= BB xxyy m , 06565 =±==±= BB zzzz . A nonzero solution will
occur if there is a constraint
Yefymenko M.V., Kudermetov R.K.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 52
02
6
2
5
2
2
2
1 =+++ BBBB zzxx . (18)
To the constraint (18) corresponds a circle lying in a plane BB ZOX of the
basis B and a singular vector ( )T0 ,1 ,0 ±=λB .
7) 02,1 =D , 04,3 =D , 06,5 ≠D .
Making the transformations similar to those performed in cases 6) and 7),
we obtain the following equation for the singular surface:
02
4
2
3
2
2
2
1 =+++ BBBB zzyy . (19)
The constraint (19) defines a circle lying in a plane BB ZOY of the basis B
and a singular vector ( )T0 ,0 ,1±=λB .
The spatial positions of singular planes for cases 5) – 7) are shown in Fig. 3, [14].
8) 012 =D , 013 =D , 023 =D .
The condition for the existence of a nonzero solution in this case is
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε
ε
ε−
=
36
36
3
24
2
24
1
12
12
cos
cos
sin
cos
sin
cos
sin
cos
cos
z
y
z
y
y
z
лB . (20)
Considering that the sines of angles 3,2,1 , =ε kk are the coordinates of the
vector Bλ :
31 sin ε=λ B , 22 sin ε=λ B , 13 sin ε=λ B (21)
and Bλ is the normalized vector, then from the equality of its norm to one fol-
lows the condition for the existence of a nonzero solution
Fig. 3. The set of special GMC surfaces when
two pairs GDs are in singular state
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters
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1sinsinsin 3
2
2
2
1
2 =ε+ε+ε . (22)
To satisfy equalities (20), it is necessary that the following relationships
hold for the coordinates of vectors ig2 :
1
3
2
1
2
2 cos
sin,
cos
sin
ε
ε
−=
ε
ε
= zy ;
2
1
4
2
3
4 cos
sin,
cos
sin
ε
ε
=
ε
ε
−= zy ;
3
1
6
3
2
6 cos
sin,
cos
sin
ε
ε
=
ε
ε
−= zy .
(23)
Consider a few special cases:
a) 01 =ε , 02 ≠ε , 03 ≠ε ;
b) 01 ≠ε , 02 =ε , 03 ≠ε ;
c) 01 ≠ε , 02 ≠ε , 03 =ε ;
d) 01 =ε , 02 =ε ,
23
π
±=ε .
In the case a) from conditions (22–23) we have the following relationships:
1sinsinsinsinsin 3
2
2
2
3
2
2
2
1
2 =ε+ε=ε+ε+ε ,
3
2
3
2
2
2 cossin1sin ε=ε−=ε , 32 cossin ε±=ε , 32 sincos ε±=ε ,
3
1
3
23
1
2
2 sin
cos
sin,cos
cos
sin
ε−=
ε
ε
−=ε±=
ε
ε
= zy ,
0
cos
sin,1
cos
sin
2
1
4
2
3
4 =
ε
ε
=±=
ε
ε
−= zy ,
0
cos
sin,1
cos
sin
3
1
6
3
2
6 =
ε
ε
=±=
ε
ε
−= zy ,
2121 , zzyy ±=±= ,
4343 , zzyy ±=±= ,
6565 , zzyy ±=±= .
By substituting these relationships into (16) we find
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε±
ε
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
±
±
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
0
cos
sin
00
3
3
2
2
1
1
1
m
y
z
y
x
g B
B
B ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
ε±
ε
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
0
cos
sin
00
3
3
2
2
2
2
2 y
z
y
x
g B
B
B ,
Yefymenko M.V., Kudermetov R.K.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 54
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛±
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
0
0
1
00
3
3
3
3
3
z
y
z
x
g
B
B
B ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛±
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
0
0
1
00
4
4
4
4
4
z
y
z
x
g
B
B
B ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
±=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
0
1
000
5
5
5
55
z
y
z
yg
B
BB ,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
±=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
0
1
000
6
6
6
66
z
y
z
yg
B
BB .
In this case, the singular surface is a unit circle lying in a plane BBYOX .
Having done the similar calculations for cases b) and c), we obtain that in
case b) the singular surface is a unit circle in the plane BBZOX , and in case c) it
is a unit circle lying in the plane BBZOY .
For the case d) a nonzero solution will exist if the following equalities hold
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛±
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛−
=
0
0
1
0
0 4
4
2
2
z
y
y
z
лB ,
from which it follows that 022 == Byy , 122 ±=−= Bxz , 144 ±=−= Bxy ,
044 == Bzz and the singular surface is a straight line coinciding with the axis
BOX . Similarly, it can be shown that when
21
π
±=ε , 02 =ε , 03 =ε the singular
surface is transformed into a straight line that coincides with the axis BOZ , and
as 01 =ε ,
22
π
±=ε , 03 =ε the singular surface is transformed into a straight line
that coincides with the axis BOY .
Fig. 4. The set of special GMC surfaces when three pairs GDs are in
singular state
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 55
Thus, when one of the angles ,kε 3 ,2 ,1=k tends to zero or to π , the sin-
gular surface is transformed into a circle lying in one of the coordinate planes,
and when one of the angles kε tends to
2
π or to
2
3π the singular surface is trans-
formed into a straight line coinciding with one of the BF axes. In the general
case, when 012 =D , 013 =D , 023 =D , the singular surface has a quite complex
shape. The boundary of this shape has six cavity (“craters”), the axes of which
coincide with the BF axes. The set of possible native singular states of the 3-SPE
scheme is shown in Fig. 4.
CONCLUSION
An original technique of a topological analysis of GMC based on collinear GD’s
pairs is proposed. Using this technique, the analysis was carried out and the
singular states of the GMC containing three collinear pairs were revealed. The
proposed technique may be useful to developers of SC attitude control systems.
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Отримано 15.02.2019
Topological Analysis of Angular Momentum Range Values of the Gyro Moment Clusters
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 57
Єфименко М.В.1, канд.техн.наук, доцент, головний конструктор,
доцент Запорізького національного технічного університету,
e-mail: nefimenko@gmail.com
Кудерметов Р.К.2, канд.техн.наук, доцент,
завідувач кафедри комп’ютерних систем та мереж,
e-mail: kudermetov@gmail.com
1 НВП «ХАРТРОН-ЮКОМ»,
пр.Соборний,166, Запоріжжя, Україна, 69035
2 Запорізькій національний технічний університет,
вул. Жуковського, 64, Запоріжжя, Україна, 69063
ТОПОЛОГІЧНИЙ АНАЛІЗ ОБЛАСТІ ДОПУСТИМИХ ЗНАЧЕНЬ КІНЕТИЧНОГО
МОМЕНТУ СИЛОВИХ ГІРОСКОПІЧНИХ КОМПЛЕКСІВ КРАТНИХ СХЕМ
Вступ. Для забезпечення високих динамічних характеристик супутників дистанційного
зондування Землі в їхніх системах орієнтації в якості виконавчих органів можуть викорис-
товуватися силові гіроскопічні комплекси (СГК), які є надлишковою (більше 3) системою
двоступеневих силових гіроскопів (гіродінов). Завдання гіросилового керування кутовим
рухом є одними з найскладніших серед завдань керування переорієнтацією КА. Централь-
ним питанням під час вирішення цих завдань є питання синтезу законів керування кутами
прецесії окремих гіродінов за їх надлишковості. Успіх у розв’язанні завдання керування
багато в чому визначається вибором структури СГК, під якою розуміється кількість вико-
ристовуваних гіродінов і взаємне розташування їхніх осей прецесії. Від такого вибору
залежить можливість формування СГК необхідного керуючого моменту, наявність і кіль-
кість особливих станів СГК, складність законів керування кутами прецесії окремих гіроді-
нов входять в СГК. Це зумовлено тим, що для тривалого підтримання постійної орієнтації
апарата і виконання ним розворотів з необхідною кутовою швидкістю СГК повинен мати
достатній запас кінетичного моменту (КМ). Допустимі значення сумарного КМ, створюва-
ного гіродінами, утворюють в системі координат, жорстко пов'язаної з КА, деяку область,
яка обмежена замкнутою поверхнею складної форми. Усередині цієї області розташову-
ються особливі поверхні, на яких керування гіродінами ускладнено або взагалі неможливе.
Ці поверхні прийнято називати сингулярними. У зв'язку з цим, у разі керуванні орієнтацією
КА за допомоги СГК, крім керування швидкістю прецесії окремих гіродінов, необхідно
керувати і взаємною орієнтацією кінетичних моментів гіродінов, що входять в СГК. Вод-
ночас одним з найважливіших завдань синтезу законів керування з використанням СГК є
завдання виявлення сингулярних поверхонь (топологічного аналізу) області допустимого
кінетичного моменту СГК.
Мета роботи — розроблення методики виявлення сингулярних поверхонь в СГК
кратних схем.
Результат — проведено аналіз та виявлено сингулярні стани схеми СГК, що міс-
тить три Колінеарні пари.
Висновки. Запропоновано оригінальну методику проведення топологічного аналізу
СГК кратних схеми. Методика може бути корисна розробникам систем орієнтації КА.
Ключові слова: космічний апарат, гіродін, сингулярний вектор, сингулярна поверхня.
Yefymenko M.V., Kudermetov R.K.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 2 (196) 58
Ефименко Н.В.1, канд. тех. наук,
доцент Запорожского национального технического университета,
главный конструктор
e-mail: nefimenko@gmail.com
Кудерметов Р.К.2, канд. техн. наук, доцент,
заведующий кафедрой компьютерных систем и сетей
e-mail: kudermetov@gmail.com
1 Научно-производственное предприятие «Хартрон-ЮКОМ»,
пр. Соборный, 166, г. Запорожье, 69035, Украина
2 Запорожский национальный технический университет
ул. Жуковского, 64, г. Запорожье, 69063, Украина
ТОПОЛОГИЧЕСКИЙ АНАЛИЗ ОБЛАСТИ ДОПУСТИМЫХ ЗНАЧЕНИЙ
КИНЕТИЧЕСКОГО МОМЕНТА СИЛОВЫХ
ГИРОСКОПИЧЕСКИХ КОМПЛЕКСОВ КРАТНЫХ СХЕМ
Введение. Для обеспечения высоких динамических характеристик спутников дистан-
ционного зондирования Земли в их системах ориентации в качестве исполнительных
органов могут использоваться силовые гироскопические комплексы (СГК), представ-
ляющие собой избыточную (более 3) систему двухстепенных силовых гироскопов
(гиродинов). Задачи гиросилового управления угловым движением являются одними
из наиболее сложных среди задач управления переориентацией КА. Центральным
вопросом при решении этих задач является вопрос синтеза законов управления углами
прецессии отдельных гиродинов при их избыточности. Успех в решении задачи управ-
ления во многом определяется выбором структуры СГК, под которой понимается ко-
личество используемых гиродинов и взаимное расположение их осей прецессии. От
такого выбора зависит возможность формирования СГК требуемого управляющего
момента, наличие и количество особых состояний СГК, сложность законов управления
углами прецессии отдельных гиродинов, входящих в СГК. Обусловлено это тем, что
для продолжительного поддержания заданной ориентации аппарата и выполнения им
разворотов с требуемой угловой скоростью, СГК должен обладать достаточным запа-
сом кинетического момента (КМ). Допустимые значения суммарного КМ, создаваемо-
го гиродинами, образуют в системе координат, жестко связанной с КА, некоторую
область, которая ограничена замкнутой поверхностью сложной формы. Внутри этой
области располагаются особые поверхности, на которых управление гиродинами
усложнено или вообще неосуществимо. Эти поверхности принято называть сингуляр-
ными. В связи с этим, при управлении ориентацией КА с помощью СГК, кроме управ-
ления скоростью прецессии отдельных гиродинов, необходимо управлять и взаимной
ориентацией кинетических моментов гиродинов, входящих в СГК. При этом одной из
важнейших задач синтеза законов управления с использованием СГК является задача
выявления сингулярных поверхностей (топологического анализа) области допустимого
кинетического момента СГК.
Цель работы — разработка методики выявления сингулярных поверхностей в
СГК кратных схем.
Результат — проведен анализ и выявлены сингулярные состояния схемы СГК,
содержащей три коллинеарные пары.
Выводы. Предложена оригинальная методика проведения топологического ана-
лиза СГК кратных схем. Методика может быть полезна разработчикам систем ориен-
тации КА.
Ключевые слова: космический аппарат, гиродин, сингулярный вектор, сингулярная
поверхность.
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
/NOR <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>
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/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|