РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ
DOI: https://doi.org/10.15407/itm2026.02.023 The focus of the study is a compact orbital tether consisting of two bodies that must be deployed from a spacecraft so that after deployment it rotates at a constant angular speed and attains a prescribed length. The end masses are taken as equal, and the...
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Technical Mechanics| _version_ | 1870650009034162176 |
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| author | WANG, Ch. ZAKRZHEVSKYI, O. E. |
| author_facet | WANG, Ch. ZAKRZHEVSKYI, O. E. |
| author_institution_txt_mv | [
{
"author": "Ch. WANG",
"institution": "https:\/\/orcid.org\/0000-0002-1358-7731 Northwestern Polytechnical University 1.127 YouyiXilu, Xi'an 710072 Shaanxi, P. R. China; e-mail: wangcq@nwpu.edu.cn"
},
{
"author": "O. E. ZAKRZHEVSKYI",
"institution": "https:\/\/orcid.org\/0000-0003-2106-2086 Space Research Institute of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine Kiev 03680, Ukraine; e-mail: alex.e.zakr@gmail.com"
}
] |
| author_sort | WANG, Ch. |
| baseUrl_str | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/oai |
| collection | OJS |
| datestamp_date | 2026-07-13T20:32:47Z |
| description | DOI: https://doi.org/10.15407/itm2026.02.023
The focus of the study is a compact orbital tether consisting of two bodies that must be deployed from a spacecraft so that after deployment it rotates at a constant angular speed and attains a prescribed length. The end masses are taken as equal, and the connecting cable is assumed massless. The aim of the research is to develop a feedforward control for the tether length, taking into account the gravitational torque exerted by the central Newtonian field of forces. Initial rotation of the tether about the orbit’s binormal is employed for deployment. The research consists of two stages. The development of the control law is based on the tether length in the first stage, which provides the performance of the desired deployment. The tether motion equations, expressed in spherical polar coordinates for a particular case of tether motion in the orbital plane, are used at this stage. In the second investigative phase, a numerical simulation of the deployment dynamics of a tether under the effect of the developed feedforward control law is performed using the tether length. As a mathematical model of the tether, the full set of motion equations for a variable‑length tether in spherical polar coordinates is employed, describing the spatial motion of bodies. The tensile force of the cable entering these equations is described by the control law governing length variation and its first two time derivatives. The novelty of this study lies in devising a feedforward control law that treats the tether as an underactuated mechanical system. Analytical mechanics, numerical techniques, and authors‑developed methods were employed in this research. The results enable determination of the permissible ranges of deployment parameters, facilitating this type of deployment. Practically, the findings allow small tethers to be deployed in orbit and brought to uniform rotation at a specified length via tether length control.
REFERENCES
1. Alpatov A. P., Beletsky V. V., Dranovskii V. I., Khoroshilov V. S. et al. Dynamics of Tethered Space Systems. Advances in Engineering. Boca Raton: CRC Press, 2010. 223 pp.
2. Alpatov A. P., Zakrzhevskii A. E. Passive deployment of a tether between two bodies in orbit. International Applied Mechanics. 1999. V. 35. No.10. Pp. 1053-1058. https://doi.org/10.1007/BF02682318
3. Beletsky V. V. Motion of an artificial satellite about its center of mass. Jerusalem: Israel Program for Scientific Translations. 1966.
4. Beletsky V. V., Levin E. M. Dynamics of Space Tether Systems. V. 83. Advances in the Astronautical Sciences, San Diego: Univelt, 1993. 499 pp.
5. Huang P., Zhang F., Chen L. et al. A review of space tether in new applications. Nonlinear Dynam. 2018. V. 94. Pp. 1-19.
6. Kang J., Zhu Z., Santaguida L. F. Analytical and experimental investigation of stabilizing rotating uncooperative target by tethered space tug. IEEE Trans. Aero. Electron. Syst. 2021. V. 57. No. 4. Pp. 2426-2437. https://doi.org/10.1109/TAES.2021.3061798
7. Li Z., Meng Z., Huang P. Spin-up control of tethered space station for artificial gravity task. IEEE International Conference on Robotics and Biomimetics (ROBIO). 2019. Pp. 2502-2508.https://doi.org/10.1109/ROBIO49542.2019.8961375
8. Rajkumar A., Bannova O. A three-body spacecraft as a testbed for artificially-induced gravity research in low earth orbit. 2020. ARC AIAA. 2020-4110. https://doi.org/10.2514/6.2020-4110
9. Trushlyakov V.I., Yudintsev V.V. Rotary space tether system for active debris removal. J. Guid. Control Dynam. 2020. V. 43. No. 2. Pp. 354-364. https://doi.org/10.2514/1.G004615
10. Wang Ch., Zakrzhevskiy O. E. Deployment of the space tether in the centrifugal force field with alignment to the local vertical. Teh. Meh. 2024. No.1. Pp. 26-39. https://doi.org/10.15407/itm2024.01.026
11. Zakrzhevskii A. E. Method of deployment of a space tethered system aligned to the local vertical. J. of Astronaut. Sci. 2016. V. 63. Pp. 221-236. https://doi.org/10.1007/s40295-016-0087-z |
| first_indexed | 2026-07-03T01:01:11Z |
| format | Article |
| fulltext |
23
UDK 629.5 https://doi.org/10.15407/itm2026.01.023
Ch. WANG1, https://orcid.org/0000-0002-1358-7731
O. E. ZAKRZHEVSKYI2, https://orcid.org/0000-0003-2106-2086
DEPLOYMENT OF THE ROTATING TWO BODIES TETHER IN AN ORBIT
1Northwestern Polytechnical University 1.127 YouyiXilu, Xi'an 710072 Shaanxi, P. R. China;
e-mail: wangcq@nwpu.edu.cn
2Space Research Institute of National academy of sciences and State space agency, Kiev, 03680, Ukraine;
e-mail: alex.e.zakr@gmail.com
Предметом дослідження є невелика орбітальна зв’язка, що складається з двох тіл, які мають бути
розгорнуті з космічного апарата таким чином, щоб після завершення розгортання вона оберталася з
постійною кутовою швидкістю та мала задану довжину. Маса кінцевих тіл вважається однаковою, а трос,
який з'єднує ці тіла, розглядається як безмасовий. Метою дослідження є розробка програми управління
довжиною троса з урахуванням впливу на його динаміку гравітаційного моменту центрального
ньютонівського силового поля. Для розгортання використовується попереднє обертання зв’язки навколо
бінормалі орбіти. Дослідження складається з двох етапів. На першому етапі розробляється закон
управління довжиною троса для досягнення бажаного розгортання. На цьому етапі використовуються
рівняння руху зв’язки, записані в сферичних полярних координатах для спеціального випадку руху зв’язки
в площині орбіти. На другому етапі проводиться числове моделювання динаміки розгортання троса під
дією розробленого програмного закону управління, зосереджуючи увагу на зміні довжини зв’язки.
Математична модель зв’язки враховує повний набір рівнянь руху зв’язки зі змінною довжиною в
сферичних полярних координатах, що описують її тривимірний рух. Сила натягу тросу в цих рівняннях
виражена через запрограмований закон зміни довжини та його перші дві часові похідні. Новизна
дослідження полягає в формулюванні програмного закону управління, де трос розглядається як
малоприводна (underactuated) механічна система. У дослідженні використовувалися методи аналітичної
механіки, числові методи та методики, розроблені самими авторами. Отримані результати дозволяють
встановити діапазони значень параметрів розгортання, які забезпечують виконання такого типу операцій.
Практична значимість результатів полягає у можливості розгортання невеликих зв’язок на орбіті та
приведення їх до стану рівномірного обертання із заданою довжиною за допомогою контролю довжини.
Ключові слова: космічний трос, розгортання, обертання, динаміка, управління.
The focus of the study is a compact orbital tether consisting of two bodies that must be deployed from a
spacecraft so that after deployment it rotates at a constant angular speed and attains a prescribed length. The end
masses are taken as equal, and the connecting cable is assumed massless. The aim of the research is to develop a
feedforward control for the tether length, taking into account the gravitational torque exerted by the central
Newtonian field of forces. Initial rotation of the tether about the orbit’s binormal is employed for deployment. The
research consists of two stages. The development of the control law is based on the tether length in the first stage,
which provides the performance of the desired deployment. The tether motion equations, expressed in spherical
polar coordinates for a particular case of tether motion in the orbital plane, are used at this stage. In the second
investigative phase, a numerical simulation of the deployment dynamics of a tether under the effect of the
developed feedforward control law is performed using the tether length. As a mathematical model of the tether, the
full set of motion equations for a variable‑length tether in spherical polar coordinates is employed, describing the
spatial motion of bodies. The tensile force of the cable entering these equations is described by the control law
governing length variation and its first two time derivatives. The novelty of this study lies in devising a
feedforward control law that treats the tether as an underactuated mechanical system. Analytical mechanics,
numerical techniques, and authors‑developed methods were employed in this research. The results enable
determination of the permissible ranges of deployment parameters, facilitating this type of deployment.
Practically, the findings allow small tethers to be deployed in orbit and brought to uniform rotation at a specified
length via tether length control.
Keywords: space tether, deployment, rotation, dynamics, control.
Introduction. The use of tethers represents one of the promising directions in
the current development of space technologies. Cable systems are regarded as
attractive for solving numerous tasks in space [1 – 11]. Particular interest is drawn
to centrifugal systems, whose deployment and shape maintenance are achieved
through the centrifugal forces generated when a structure rotates about its center of
mass. In recent years many projects have employed centrifugal forces to preserve
the shape of space antennas. Since there are no drag forces in space, maintaining a
© Ch. Wang, О. E. Zakrzhevskyi, 2026
The article is an open access article distributed underthe terms and conditions of the Creative
Commons Attributions ( CC BY) license (https/creativecommons.org/licenses/by/4.0/)
ISSN 1561-9184 (Print) ISSN 2616-6380 (Online) Технічна механіка. 2026. № 2.
https://doi.org/10.15407/itm2025.01.00
24
constant rotational speed of bodies is relatively straightforward. Although
theoretical studies examine the contours of extremely large tethers, researchers
continue to focus mainly on small tethers, typically supporting systems ranging
from a few meters to several kilometers in length and having masses from a few
kilograms up to a few hundred kilograms [4].
Such systems are of practical relevance because they are suitable both for
solving independent scientific tasks and for implementing full‑scale experiments
and checks of a number of hypotheses concerning tether deployment dynamics and
methods of creating their steady motions. For instance, a compact rotating tether
can serve as a length standard for calibrating onboard optical and radar servo-
mechanisms, as a built-in sensor of the planetary force field, and so on.
One possible implementation of a small-size space tether (a few dozen meters
long) is a system of two bodies linked by an elastic cable. Typically, the masses of
the end bodies determine the total mass of the tether (Fig. 1); the mass of the cable
connecting the ends usually accounts for no more than a percent of the total tether
mass. As a model, one may choose two equal point masses joined by a flexible,
massless cable. The dimensions of the end bodies can be neglected when studying
motions in which the cable cannot be wound onto the end bodies.
The simplest and most economical approach, both in construction and in
mathematical description of deployment in orbit, is to rely on centrifugal forces.
These forces can arise from an initial rotation of the system in the plane that passes
through the centers of mass of its end bodies. Two actuators (for example, springs)
must be installed on the spacecraft for this purpose; on command from Earth or
from the spacecraft’s onboard computer, they push the system away. As it
separates from the spacecraft, it begins to rotate in the preset plane relative to the
spacecraft–normally the orbital plane.
The initial angular velocity must be high enough to ensure, on the one hand,
deployment of the system by centrifugal forces and, on the other hand that the resulting
angular velocity and tether length correspond to the intended operational duty.
A similar problem was examined in the scientific literature [1] without
considering the gravitational field. Deployment of a three-body tether was
performed in a centrifugal-force field with tension control. That control was
proposed to be realized by generating viscous and Coulomb friction forces that
resist pulling the cable from the middle body, depending on deployment length and
speed. Here we consider the deployment problem under orbital flight conditions for
a two-body tether with control over the length of the connecting cable.
We introduce an orbital reference frame Cx y zor or or with its origin at the
tether’s center of mass C , where the Cxor
axis points along the local vertical,
Cyor aligns with the orbital velocity vector, and Czor
is normal to the orbital plane
(Fig. 1). This frame rotates about the Czor
axis at a constant angular velocity or
.
25
Fig. 1
The mathematical description of the process of ejecting a system as a rigid
body from the spacecraft is rather simple, and we will not dwell on it. If we
consider the motion of the system since the instant of its separation from the
actuator, its mechanical model represents a free system of two bodies tied by a
cable, undergoing relative motion under the influence of centrifugal and
gravitational forces. We will assume that the ejection system will not cause motion
of the bodies about their centers of mass at the moment of separation from the
actuators. In that case the terminal bodies can be treated as point masses. Let us
further assume that this condition is satisfied. The cable linking the bodies will be
regarded as elastic because at the initial stage of rotation the tension in the cable
can be quite large, and therefore it is essential to account for its elastic
deformations when controlling the tether length. We will assume deployment of the
tether from one of the bodies so that the masses of the bodies are equal at the end
of deployment.
Since the dimensions of the considered tether are substantially smaller than
the size of the orbit, the problem statement can be “limited” [3]; the equations of
motion of the tether’s center of mass are decoupled from the equations concerning
the spacecraft’s center of mass in this case. Consequently, we will focus solely on
studying the system’s attitude. When the tether is ejected from the spacecraft by
impulses of varying magnitude, its center of mass acquires an initial translational
velocity. As a result, the tether’s center of mass moves in inertial space relative to
the spacecraft.
At the same time, the motion of the deploying system occurs in a region of
space that must be free of obstacles. Therefore, when simulating the
tether-deployment dynamics, it is helpful to define the limits of such a region. This
issue is trivial and of little scientific interest. Hence, we will concentrate further
only on the rotational motion of the tether in the orbital frame of reference.
Assume the deployment of the tether takes place in the orbital plane. Let the
distance between the centers of mass of the tether’s end bodies be its length.
Denote the tether length at the initial instant by
0L and its final length by
FL .
When the end masses are equal, it is sufficient to examine the motion of a single
end body. In planar motion the state of the point mass can be described by its
distance r from the tether’s center of mass and the pitch angle , measured in the
orbital plane from the local vertical.
Typically a single control input associated with varying the tether length or its
tension is used in models of real tethers with a massless cable whose motion lies in
the plane of a circular orbit, yet the system possesses two degrees of freedom. The
second degree of freedom is associated with the pitch rotational motion of the
26
tether. In this formulation the tether can be regarded as an under-actuated
mechanical system, where the number of degrees of freedom exceeds the number
of control inputs. We cannot directly influence the pitch angle, so it is necessary to
devise an appropriate control law for the tether length that produces the desired
pitch-angle dynamics.
Our goal is to find a control law ( )L t that achieves deployment of the tether
in the orbital plane to the desired length and a prescribed rotational condition while
avoiding longitudinal oscillations.
To derive the needed control laws, one may employ the angular-momentum
conservation theorem for the tether. A simpler approach is to employ the motion
equation for the pitch angle of an inextensible tether, whose length can vary via
unwinding. Owing to the monotonic nature of the deployment process in this
model, it is straightforward to introduce corrections for the tether length due to
elastic deformation. Following [4] (pp. 65), the equation for the planar motion of a
tether in a circular orbit can be written as follows:
22( ) / 3( ) sin cos 0 + + + =or orr r (1)
here or
is the orbital angular velocity; ( ) ( ) / 2=r t L t is half of the tether length;
is the pitch angle.
Although we assume that the tether during deployment will be displaced
relative to the spacecraft, the speed of its motion is negligibly small compared with
the spacecraft’s velocity in the inertial frame. Consequently, the angular velocity of
the tether’s motion in orbit can be taken as identical to that of the spacecraft.
To develop the feedforward controlling tether deployment to a specified
length while maintaining uniform rotation at the desired angular velocity, we will
employ the method described in [0, 11].
After elementary manipulations of equation (1), one can obtain a first-order
ordinary differential equation together with the appropriate initial condition.
( )
23( ) sin 2 2
4
+
= −
+
or
or
L L ,
0(0) =L L . (2)
Its solution can be written down as the following:
( )
2
0
0
3( ) sin(2 ( )) 2 ( )
( ) exp
4 ( )
+
= −
+
t or
or
L t L d . (3)
It follows from equation (3) that ( )L t is a function of the pitch angle ( ) t , its
first two time derivatives, and the duration of the maneuver =F Ft T . It is clear that
the variation law for the pitch angle and its time derivatives must satisfy certain
conditions for the deployment mode under consideration, both at the start of
deployment and at its conclusion. We now examine how to select such boundary
conditions on physical grounds.
We choose a variation law for the angle ( ) t that yields a smooth deployment
of the tether to a prescribed constant length. Numerous possible laws ( )L t for the
tether-length change can be generated for different values of the maneuver duration
27
FT . From this family of solutions, one can be selected that achieves the desired
final tether length, avoids loss of tension during deployment, and satisfies the
additional requirements that will be stated below.
Next, we consider the constraints that must be imposed on the pitch-angle
variation law ( ) t in order to define the time-dependent tether-length law ( )L t
that permits solving the problem via the Cauchy-problem solution (2).
As mechanisms for generating the initial rotation of the tether, two spring
pushers acting in opposite directions with different impulses may be employed.
For the initial condition of the tether deployment, we assume it is aligned with
the local vertical, possessing a prescribed angular velocity (both magnitude and
direction) and an initial translational velocity of the tether’s mass center in the
orbital reference frame. In this case the following initial conditions for the pitch
angle and its angular velocity can be written:
(0) 0 = ,
0(0) = , (4)
and an initial condition for the relative velocity of the tether mass center in the
orbital frame of reference:
0=Cv v . (5)
As in an initial instant the contacts of bodies of a tether with the device of
creation of its initial motion stop, at this instant both the angular and linear
accelerations turn into zero, i.e.,
(0) 0 = , (6)
0=Cv . (7)
The following conditions have to be satisfied at the initial instant of
deployment and at the time of its termination;
0(0) =L L , ( ) =F FL T L . (8)
The first of these conditions can be regarded as satisfied; the second condition
can be used further to define the a priori unknown time at which the maneuver
ends. This value will enable the selection of the appropriate solution of the problem
from the set of solutions obtained for various values.
Two additional conditions follow from the requirement of constancy of the
tether length at the initial and final instants:
(0) 0=L , ( ) 0=FL T . (9)
The emergence of jumps in the tension of the cable is unacceptable at the
initial and final instants of the maneuver and throughout the entire maneuver,
because they can lead to a loss of tension that would render the adopted mechanical
model inadequate. The absence of tension jumps at the initial and final instants is
ensured when the following conditions are satisfied:
(0) 0=L , ( ) 0=FL T . (9)
To obtain the restriction on the law governing the change of the pitch angle
from these conditions, we differentiate equation (2) with respect to time:
28
( ) ( )
( )
2
2
2 2
1 / [4 ]{ 2 [3( ) sin 2 / 2 ]
[3( ) sin 2 2 ] 2 [3( ) cos2 ]}.
= + − + + +
+ − + +
or or or
or or or
L L
L L
(11)
At the initial instant, using conditions (4, 6, 8 – 10), expression (11) becomes
( )2
0 0(0) [6( ) 2 (0)] / 4 / = − + +or orL L . (12)
Considering condition (10) yields an additional requirement that must be
satisfied when the pitch angle varies with time:
2
0(0) 3( ) = − or . (13)
Next, we examine further constraints that must be imposed on the law ( ) t at
the final instant = Ft T of tether deployment in the designated state. At the final
state, when the deployment process ends, it is convenient to assume that the tether
has rotated an integer number of turns relative to its initial position:
( ) 2 =F rotT n . (14)
Hence, the tether at = Ft T will be aligned with the local vertical. Since the
gravitational torque vanishes at this moment and no other external forces act on the
tether in the present formulation, we can introduce the additional condition
( ) 0 =FT . (15)
Rather than constraining the final tether length, we prescribe its final angular
velocity.
( ) =F FT . (16)
The final tether length can be selected, if required, by adjusting the value F
.
Finally, using condition (15) and by analogy with (13), we obtain another condition
2( ) 3( ) = − or
FF . (17)
Consequently, the law ( ) t must satisfy eight conditions (4), (6), and (13 –
17). The required law ( ) t can be expressed as any suitable series of functions,
with its coefficients determined from the eight constraints. Here we choose a
seventh-order power series for the law ( ) t .
7
0
( )
=
=
i
i
i F
t
t c
T
. (18)
At satisfaction of eight specified boundary conditions for the law ( ) t the
expression for coefficients of the power series (18) take the following form:
29
0
1 0
2
2
3 0
2 3 2 3 4
4 0 0
0,
,
0,
( ) / 2,
[ 140 40 30 4( ) ( ) ] / 2 / .
=
=
=
= −
= − − + + − −
or
or or
rot F F F F F F F
c
c
c
c
c n T T T T T
2 3 2 3 5
5 0 0
2 3 2 3 6
6 0 0
2 3 2 3 7
7 0 0
3[112 30 26 2( ) ( ) ] / 2 / .
[ 280 72 68 4( ) 3( ) ] / 2 / .
[80 20 20 ( ) ( ) ] / 2 / .
= − − − + +
= − − + + − −
= − − − + +
or or
rot F F F F F F F
or or
rot F F F F F F F
or or
rot F F F F F F F
c n T T T T T
c n T T T T T
c n T T T T T
(19)
Now, having set parameters of the problem and having substituted their values
in the coefficients (19), we will obtain the law of change of pitch angle in time
corresponding to the required mode of the motion of the tether at its deployment
from an initial state in the state of rotation with the set angular velocity. The law of
change of the tether length in time, which implements the required mode of the
tether deployment, can be obtained from the solution of the Cauchy problem (3),
having substituted laws of change of the pitch angle and its time derivatives there.
Conditions (5) and (7) define only the translatory motion of the tether mass center
in the orbital frame of reference and have no influence on its attitude.
Numerical research. Let us consider further implementation of the stated
method of creation of the feedforward control of the space tether deployment in the
condition of uniform rotation in the orbit plane. Let us choose the following basic
values of the tether parameters:
The masses of the end bodies are 10 kg.
The duration of the deployment process is 50 s.
The initial length of the tether
0L is 4 m.
The initial angular velocity of the tether
0 = 15 rad / s.
The final length of the tether
FL is 15 m.
The longitudinal rigidity of the cable EF is 26104 N.
The velocity of leaving of the center of mass of the tether
0v is 0.2 m/s,
The basic value of the number of turns
rotn is 70.
The radius of the orbit of the tether is 7000 km.
Let us show further how to find the law, which solves the problem.
Apparently, from expressions (2) and (19), the required control law is defined
by values of such parameters as the number of turns of the tether during
deployment
rotn , the initial value of the pitch angle of the tether
0 , the initial
value of the angular velocity
0 of rotation of the tether in the orbital frame of
reference, the final value of the angular velocity F
of the tether rotation in the
orbital frame of reference, and the deployment process duration
FT . These
parameters define the law of change in time of the tether pitch angle ( ) t
according to expression (18). The initial value of the semi-length of the tether or is
included directly in the control law (3) of program change of the tether length.
In the beginning we will estimate the influence of the parameter
rotn on the
nature of the required law of feedforward control, the tether length, and the
behavior of other characteristic values. In Fig. 2 the change of shape of the control
law ( )L t depending on values of the parameter
rotn at basic values of other
parameters is shown.
30
Fig. 2
Here the number of whole revolutions of the tether during the deployment
rot
n changes from 45 to 90. In all cases the length of the tether deployment is
equal; nevertheless, the nature of its change considerably depends on the accepted
number of turns. In the range from 45 to 60 turns at approach to the end of
deployment, the tether length becomes more than the final length. For the
realization of such a mode in the device of deployment, it will be required to carry
out not only the controlled deployment of the cable but also its controlled retrieval.
Therefore, for practical implementation, the range of values 60rotn is
represented the most appropriate. The straight dashed line on it and other figures
corresponds to the time of the deployment completion 50=t s.
Fig. 3 shows the program law of change in time of the tether length in the case
of rigid cable.
Fig. 3
31
In this figure the parts of curves where the velocity of exit of the cable has to
be negative are good visible [45,60)rotn . Most strongly this effect is expressed
at 45=rotn . At the same time, such parts of the curves corresponding to the
condition 80rotn are expressed less considerably.
The behavior of the pitch angle of the tether in time depending on value
rot
n is
shown in Fig. 4.
Fig. 4
Here the direct dependence between the number of turns and the change of the
angle of rotation of the tether during deployment is visible, and essential effects are
not noticeable. But in Fig. 5, in which the dependence of the relative angular
velocity of the tether in orbital basis on the number of program turns is represented,
it is visible that in the range (45,60)rotn and in the range 80rotn in the law
of program change of the tether length, there are parts where it is necessary to
increase the angular velocity of the tether, i.e., to reduce its length. As it was
already noted, it will lead to complication of the mechanism of the system of
deployment. Note that the range 80rotn , where cable length reduction is
required, is also present in Fig. 2. However, due to the chosen viewing angle, the
decrease in the coordinates L (representing length vs. time) on this interval is
nearly indistinguishable from adjacent ranges.
32
Fig. 5
Program change in time of the tether angular acceleration ( ) t depending on
the number of its turns during deployment
rotn is shown in Fig. 6
.
Fig. 6
In this figure it is visible that the number of turns in the program motion is
less, and the maximum delay of rotation is closer to the beginning of deployment.
Here it is also possible to note that in the range of the modes of the program motion
acceptable for implementation ( (60,80)rotn ) the range of change of the angular
acceleration is narrower than at the edges of the considered area of values of the
parameter
rotn .
Having constructed the program law of change of the tether length in time, it
is possible to construct the program law of change of the tether tension
tethT in
time. For this purpose we use one of the equations of motion of a tether of variable
length in spherical coordinates [4] on the condition of the motion of the tether in
the orbit plane. As a result, it is possible to write down:
33
𝑇𝑡𝑒𝑡ℎ = 𝑚/2{−𝐿̈ + 𝐿[(𝜔𝑜𝑟 + 𝜗̇)
2
𝑐𝑜𝑠 𝜑 + 𝜑̇2 + 3(𝜔𝑜𝑟)2 𝑐𝑜𝑠2 𝜗 𝑐𝑜𝑠2𝜑 −
(𝜔𝑜𝑟)2 = 𝑚/2{−𝐿̈ + 𝐿[(𝜔𝑜𝑟 + 𝜗̇)
2
+ 3(𝜔𝑜𝑟)2 𝑐𝑜𝑠2 𝜗 − (𝜔𝑜𝑟)2 . (20)
In Figure 7, the change in time of program values of the tether tension is
shown in the mode of deployment depending on the chosen number of turns of the
tether.
Fig. 7
Both in the initial and in the final instants of the deployment, the force of the
tether tension does not depend on the total number of turns. Nevertheless, the
nature of the change of the tension during deployment depends on value
rotn
essentially. In the range of values (80,90)rotn time intervals are visible on
which the maximum tension force considerably exceeds the initial tension. It
occurs where, according to Fig. 5, at the same values
rotn there is a significant
increase in angular velocity of the tether. Respectively, significant decrease in the
force of tension on an interval of time in the neighborhood of a point 35=t s at
(45,60)rotn happens where, in Fig. 5, the considerable decrease in angular
velocity in comparison with its value at the end of deployment is visible.
Lengthening of the tether during deployment is shown in Fig. 8.
Fig. 8
34
The surface shape
rot( , )L n t at first sight reminds the surface shape
rot( , )tethT n t , nevertheless, here the similarity is rather mirror-like. This surface is
much closer in shape to the surface
rot( , ) n t .
Having analyzed the graphic information connected with studying the
influence of the parameter
rotn on the nature of the feedforward control law for the
tether length during its deployment, we will note that the choice of value of this
parameter does not allow us to obtain the set size of the final length of the tether. It
follows from physical reasons that the final length of the rotating tether depends on
its initial angular momentum and the specified final angular velocity of the tether
in the orbital frame of reference. The initial relative tether angular momentum in
the orbital frame of reference is
2
0 0{0,0,2 }=CK mr , (21)
i.e., with constant end masses, the parameters
0r and
0 equally influence the
value of the angular moment and the size of the final length of the tether.
Further, we will turn to consideration of the influence of the parameter
Fr on
the control law and behavior of the tether. The nature of the change of the tether
length at change of size
0r in the range
0 [2,2.9]r is shown in Fig. 9. Other
parameters at the same time have their basic values.
Fig. 9
In this figure it is visible that during the uniform increase of the initial size
0r
from 2.0 to 2.9 m, the radius of the deployed tether
Fr evenly increases from 15.48
m up to 22.45 m. Here it should be noted that the choice of a way of increasing the
initial angular momentum for an increase in the tether's final length, taking into
account the tensile strength of the cable, will unambiguously lead to an increase in
the initial radius of the tether, but not its initial angular velocity. It is easy to show,
for example, that a fourfold increase in the module of the angular momentum due
to an increase
Fr twice, centrifugal force
2
0 0( )=CF mr will increase twice. To
reach the same value of the angular momentum due to an increase in the initial
angular velocity of rotation, it needs to be increased four times. At the same time,
35
centrifugal force will increase eight times. It will demand an increase in tensile
strength and of the cable mass.
The described result can be used for obtaining the specified final length of the
tether, applying the method of linear interpolation.
Fig. 10
In Figure 10 the velocity of change of the tether length during deployment is
shown. The shown curves have smooth character. The less initial size of the radius
0r , the less maximum value of velocity of the tether deployment.
Change of the initial value
0r at constant value of the parameter
rotn does not
influence laws of change of the pitch angle and all its time derivatives. In general,
at correctly chosen value of the parameter
rotn the modes of deployment of the
tether can be theoretically realized at any value
rotn .
Let's consider what impact the gravitational field has on the studied process.
The plot of the gravitational torque acting on the tether during deployment at basic
values of all parameters is provided in Fig. 11.
Fig. 11
Here it is visible that the value of the torque changes periodically, being
synchronized with the frequency of rotation of the tether and with its length. In the
beginning, the amplitude of the torque grows slowly, as well as the tether length in
36
Fig. 9, and quickly increases at the end, reaching a constant value at the instant
= Ft T . In general, the maximum value of the amplitude of the gravitational torque
is insignificant because of the small sizes of the tether. Therefore, this torque
cannot have a significant effect on the nature of the deployment process of the
tether. Nevertheless, in Fig. 12 the influence of this torque on the angular velocity
of the deployed tether is shown upon termination of deployment.
Fig. 12
Here it is visible that the angular velocity of the deployed tether rotating
around the orbit binormal periodically changes with a frequency twice greater than
the frequency of the tether rotation. Such perturbations of the angular velocity from
the practical point of view do not matter. Naturally, the same perturbations may
also be shown in the change of the tension force of the tether. The amplitude of
fluctuations of the tether tension
tethT is about 0.0002 N for the considered case
that taking into account the accepted rigidity of the cable on stretching has the
negligible effect from the practical point of view.
Let us carry out a numerical simulation of the use of the constructed
feedforward control law ( )L t for deployment of the tether to the condition of
rotation with the specified length of 16 m and angular velocity of rotation in orbital
basis 1 rad/s. Linear interpolation of the results of the numerical experiment
represented in Fig. 9 shows that for obtaining the tether's final length =FL 16 m
the initial length of the tether
0L has to be equal to 4.1304 m. For numerical
simulation of the deployment, we will use the full system of the equation of motion
of the tether of variable length in a circular orbit in spherical coordinates [0] which
describes the spatial motion of a tether. We will adopt the initial position of the
tether in the orbital plane.
In Figure 13, the trajectory of the end body, upper at the initial instant is
shown during deployment. The trajectory of the lower body is shifted in a phase on
the angle . As well as it was specified, the final length of the tether is 16 m.
Now it is necessary to turn to the elastic properties of the cable. Apparently in
Fig. 8, at the initial instant when the tether tension in the majority of the considered
cases is as much as possible, lengthening of the cable L can reach several
centimeters. Therefore, the mechanism of deployment has to have an opportunity
to consider adjustment for lengthening of the cable at deployment of the cable from
one of the bodies according to the obtained the length control law.
37
Fig. 13
Conclusion. Summarizing the results, it can be noted that a new approach is
presented here to solve the problem of deploying a two body space tether under
uniform rotation in a circular orbit, theoretically sound and numerically validated
for length control. This approach relies on programmatic control of tether length,
acknowledging the impossibility of directly controlling the tether pitch angle. The
tether length control law is derived by imposing constraints on the motion law of
the tether concerning pitch motion. These constraints arise from physical reasons
and do not contradict the nature of the studied process. Consequently, the number
of degrees of freedom decreases by one, and the problem obtains a simple solution.
In control theory terms, the method for creating the programmed length control is
developed, allowing deployment of the two tethered bodies under uniform rotation
with a specified size and rotation frequency without triggering longitudinal
fluctuations. The resulting control law is very smooth in time and lacks features
that could cause such fluctuations.
1. Alpatov A. P., Beletsky V. V., Dranovskii V. I., Khoroshilov V. S, et al. Dynamics of tethered space systems.
Advances in engineering. CRC Press, Boca Raton, 2010. 223 p.
2. Alpatov A. P., Zakrzhevskii A. E. Passive deployment of a tether between two bodies in orbit. International
Applied Mechanics. 1999. Vol.35, 10. P. 1053–1058. https://doi.org/10.1007/BF02682318
3. Beletsky V. V. Motion of an artificial satellite about its center of mass. Jerusalem: Israel Program for Scientific
Translations. 1966.
4. Beletsky V. V., Levin E. M. Dynamics of space tether systems. Volume 83. Advances in the astronautical
sciences, San Diego: Univelt., 1993. 499 p.
5. Huang P., Zhang F., Chen L., et al. A review of space tether in new applications. Nonlinear Dynam. 2018.
Vol. 94. P. 1–19.
6. Kang J., Zhu Z., Santaguida L. F. Analytical and experimental investigation of stabilizing rotating
uncooperative target by tethered space tug. IEEE Trans. Aero. Electron. Syst. 2021. Vol. 57, 4. P. 2426–
2437. https://doi.org/10.1109/TAES.2021.3061798
7. Li Z., Meng Z., Huang P. Spin-up control of tethered space station for artificial gravity task, IEEE International
Conference on Robotics and Biomimetics (ROBIO). 2019. P. 2502–2508.
https://doi.org/10.1109/ROBIO49542.2019.8961375
8. Rajkumar A., Bannova O. A three-body spacecraft as a testbed for artificially-induced gravity research in low
earth orbit. 2020. ARC AIAA. 2020-4110. https://doi.org/10.2514/6.2020-4110
9. Trushlyakov V.I., Yudintsev V.V. Rotary space tether system for active debris removal. J. Guid. Control Dynam.
2020. Vol. 43, 2. P. 354–364. https://doi.org/10.2514/1.G004615
10. Wang Ch, Zakrzhevskiy O. E. Deployment of the space tether in the centrifugal force field with alignment to
the local vertical. Technical mechanics. 2024. N.1. P. 26–39. https://doi.org/10.15407/itm2024.01.026
11. Zakrzhevskii A. E. Method of deployment of a space tethered system aligned to the local vertical. J. of
Astronaut Sci. 2016. Vol. 63. P. 221–236. https://doi.org/10.1007/s40295-016-0087-z
Received on January 2, 2026;
approved on June 29, 2026;
published on July 2, 2026.
https://doi.org/10.1109/TAES.2021.3061798
https://doi.org/10.1109/ROBIO49542.2019.8961375
https://doi.org/10.15407/itm2024.01.026
https://doi.org/10.2514/1.G004615
https://doi.org/10.15407/itm2024.01.026
https://doi.org/10.1007/s40295-016-0087-z
|
| id | oai:ojs2.journal-itm.dp.ua:article-188 |
| institution | Technical Mechanics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-14T01:01:08Z |
| publishDate | 2026 |
| publisher | текст 3 |
| record_format | ojs |
| resource_txt_mv | journal-itmdpua/27/b432f1e52cb72186d6fe09e30739d527.pdf |
| spelling | oai:ojs2.journal-itm.dp.ua:article-1882026-07-13T20:32:47Z DEPLOYMENT OF THE ROTATING TWO BODIES TETHER IN AN ORBIT РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ WANG, Ch. ZAKRZHEVSKYI, O. E. space tether, deployment, rotation, dynamics, control. космічний трос, розгортання, обертання, динаміка, управління. DOI: https://doi.org/10.15407/itm2026.02.023 The focus of the study is a compact orbital tether consisting of two bodies that must be deployed from a spacecraft so that after deployment it rotates at a constant angular speed and attains a prescribed length. The end masses are taken as equal, and the connecting cable is assumed massless. The aim of the research is to develop a feedforward control for the tether length, taking into account the gravitational torque exerted by the central Newtonian field of forces. Initial rotation of the tether about the orbit’s binormal is employed for deployment. The research consists of two stages. The development of the control law is based on the tether length in the first stage, which provides the performance of the desired deployment. The tether motion equations, expressed in spherical polar coordinates for a particular case of tether motion in the orbital plane, are used at this stage. In the second investigative phase, a numerical simulation of the deployment dynamics of a tether under the effect of the developed feedforward control law is performed using the tether length. As a mathematical model of the tether, the full set of motion equations for a variable‑length tether in spherical polar coordinates is employed, describing the spatial motion of bodies. The tensile force of the cable entering these equations is described by the control law governing length variation and its first two time derivatives. The novelty of this study lies in devising a feedforward control law that treats the tether as an underactuated mechanical system. Analytical mechanics, numerical techniques, and authors‑developed methods were employed in this research. The results enable determination of the permissible ranges of deployment parameters, facilitating this type of deployment. Practically, the findings allow small tethers to be deployed in orbit and brought to uniform rotation at a specified length via tether length control. REFERENCES 1. Alpatov A. P., Beletsky V. V., Dranovskii V. I., Khoroshilov V. S. et al. Dynamics of Tethered Space Systems. Advances in Engineering. Boca Raton: CRC Press, 2010. 223 pp. 2. Alpatov A. P., Zakrzhevskii A. E. Passive deployment of a tether between two bodies in orbit. International Applied Mechanics. 1999. V. 35. No.10. Pp. 1053-1058. https://doi.org/10.1007/BF02682318 3. Beletsky V. V. Motion of an artificial satellite about its center of mass. Jerusalem: Israel Program for Scientific Translations. 1966. 4. Beletsky V. V., Levin E. M. Dynamics of Space Tether Systems. V. 83. Advances in the Astronautical Sciences, San Diego: Univelt, 1993. 499 pp. 5. Huang P., Zhang F., Chen L. et al. A review of space tether in new applications. Nonlinear Dynam. 2018. V. 94. Pp. 1-19. 6. Kang J., Zhu Z., Santaguida L. F. Analytical and experimental investigation of stabilizing rotating uncooperative target by tethered space tug. IEEE Trans. Aero. Electron. Syst. 2021. V. 57. No. 4. Pp. 2426-2437. https://doi.org/10.1109/TAES.2021.3061798 7. Li Z., Meng Z., Huang P. Spin-up control of tethered space station for artificial gravity task. IEEE International Conference on Robotics and Biomimetics (ROBIO). 2019. Pp. 2502-2508.https://doi.org/10.1109/ROBIO49542.2019.8961375 8. Rajkumar A., Bannova O. A three-body spacecraft as a testbed for artificially-induced gravity research in low earth orbit. 2020. ARC AIAA. 2020-4110. https://doi.org/10.2514/6.2020-4110 9. Trushlyakov V.I., Yudintsev V.V. Rotary space tether system for active debris removal. J. Guid. Control Dynam. 2020. V. 43. No. 2. Pp. 354-364. https://doi.org/10.2514/1.G004615 10. Wang Ch., Zakrzhevskiy O. E. Deployment of the space tether in the centrifugal force field with alignment to the local vertical. Teh. Meh. 2024. No.1. Pp. 26-39. https://doi.org/10.15407/itm2024.01.026 11. Zakrzhevskii A. E. Method of deployment of a space tethered system aligned to the local vertical. J. of Astronaut. Sci. 2016. V. 63. Pp. 221-236. https://doi.org/10.1007/s40295-016-0087-z DOI: https://doi.org/10.15407/itm2026.02.023 Предметом дослідження є невелика орбітальна зв’язка, що складається з двох тіл, які мають бути розгорнуті з космічного апарата таким чином, щоб після завершення розгортання вона оберталася з постійною кутовою швидкістю та мала задану довжину. Маса кінцевих тіл вважається однаковою, а трос, який з'єднує ці тіла, розглядається як безмасовий. Метою дослідження є розробка програми управління довжиною троса з урахуванням впливу на його динаміку гравітаційного моменту центрального ньютонівського силового поля. Для розгортання використовується попереднє обертання зв’язки навколо бінормалі орбіти. Дослідження складається з двох етапів. На першому етапі розробляється закон управління довжиною троса для досягнення бажаного розгортання. На цьому етапі використовуються рівняння руху зв’язки, записані в сферичних полярних координатах для спеціального випадку руху зв’язки в площині орбіти. На другому етапі проводиться числове моделювання динаміки розгортання троса під дією розробленого програмного закону управління, зосереджуючи увагу на зміні довжини зв’язки. Математична модель зв’язки враховує повний набір рівнянь руху зв’язки зі змінною довжиною в сферичних полярних координатах, що описують її тривимірний рух. Сила натягу тросу в цих рівняннях виражена через запрограмований закон зміни довжини та його перші дві часові похідні. Новизна дослідження полягає в формулюванні програмного закону управління, де трос розглядається як малоприводна (underactuated) механічна система. У дослідженні використовувалися методи аналітичної механіки, числові методи та методики, розроблені самими авторами. Отримані результати дозволяють встановити діапазони значень параметрів розгортання, які забезпечують виконання такого типу операцій. Практична значимість результатів полягає у можливості розгортання невеликих зв’язок на орбіті та приведення їх до стану рівномірного обертання із заданою довжиною за допомогою контролю довжини. ПОСИЛАННЯ 1. Alpatov A. P., Beletsky V. V., Dranovskii V. I., Khoroshilov V. S, et al. Dynamics of tethered space systems. Advances in engineering. CRC Press, Boca Raton, 2010. 223 p. 2. Alpatov A. P., Zakrzhevskii A. E. Passive deployment of a tether between two bodies in orbit. International Applied Mechanics. 1999. Vol.35, 10. P. 1053–1058. https://doi.org/10.1007/BF02682318 3. Beletsky V. V. Motion of an artificial satellite about its center of mass. Jerusalem: Israel Program for Scientific Translations. 1966. 4. Beletsky V. V., Levin E. M. Dynamics of space tether systems. Volume 83. Advances in the astronautical sciences, San Diego: Univelt., 1993. 499 p. 5. Huang P., Zhang F., Chen L., et al. A review of space tether in new applications. Nonlinear Dynam. 2018. Vol. 94. P. 1–19. 6. Kang J., Zhu Z., Santaguida L. F. Analytical and experimental investigation of stabilizing rotating uncooperative target by tethered space tug. IEEE Trans. Aero. Electron. Syst. 2021. Vol. 57, 4. P. 2426–2437. https://doi.org/10.1109/TAES.2021.3061798 7. Li Z., Meng Z., Huang P. Spin-up control of tethered space station for artificial gravity task, IEEE International Conference on Robotics and Biomimetics (ROBIO). 2019. P. 2502–2508. https://doi.org/10.1109/ROBIO49542.2019.8961375 8. Rajkumar A., Bannova O. A three-body spacecraft as a testbed for artificially-induced gravity research in low earth orbit. 2020. ARC AIAA. 2020-4110. https://doi.org/10.2514/6.2020-4110 9. Trushlyakov V.I., Yudintsev V.V. Rotary space tether system for active debris removal. J. Guid. Control Dynam. 2020. Vol. 43, 2. P. 354–364. https://doi.org/10.2514/1.G004615 10. Ван Чанцін, Закржевський О. Е. Розгортання космічної зв'язки в полі відцентрових сил із приведенням до місцевої вертикалі. The. Meh.2024. N.1. P. 26–39. https://doi.org/10.15407/itm2024.01.026 11. Zakrzhevskii A. E. Method of deployment of a space tethered system aligned to the local vertical. J. of Astronaut Sci. 2016. Vol. 63. P. 221–236. https://doi.org/10.1007/s40295-016-0087-z текст 3 2026-07-02 Article Article application/pdf https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/188 Technical Mechanics; No. 2 (2026): Technical Mechanics; 23-37 Институт технической механики Национальной академии наук Украины и Государственного космического агентства Украины; № 2 (2026): Technical Mechanics; 23-37 ТЕХНІЧНА МЕХАНІКА; № 2 (2026): ТЕХНІЧНА МЕХАНІКА; 23-37 en https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/188/83 Copyright (c) 2026 Technical Mechanics |
| spellingShingle | космічний трос розгортання обертання динаміка управління. WANG, Ch. ZAKRZHEVSKYI, O. E. РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ |
| title | РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ |
| title_alt | DEPLOYMENT OF THE ROTATING TWO BODIES TETHER IN AN ORBIT |
| title_full | РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ |
| title_fullStr | РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ |
| title_full_unstemmed | РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ |
| title_short | РОЗГОРТАННЯ ЗВ’ЯЗКИ ДВОХ ТІЛ, ЩО ОБЕРТАЄТЬСЯ, НА ОРБІТІ |
| title_sort | розгортання зв’язки двох тіл, що обертається, на орбіті |
| topic | космічний трос розгортання обертання динаміка управління. |
| topic_facet | space tether deployment rotation dynamics control. космічний трос розгортання обертання динаміка управління. |
| url | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/188 |
| work_keys_str_mv | AT wangch deploymentoftherotatingtwobodiestetherinanorbit AT zakrzhevskyioe deploymentoftherotatingtwobodiestetherinanorbit AT wangch rozgortannâzvâzkidvohtílŝoobertaêtʹsânaorbítí AT zakrzhevskyioe rozgortannâzvâzkidvohtílŝoobertaêtʹsânaorbítí |