Chaotic Motions of Tethered Satellites with Low Thrust

Abstract

The chaotic motion of the tethered tug-debris system with the low thrust tug is considered in the context of active debris removal (ADR) missions.Stable and unstable stationary solutions are presented for the in-plane motion of the system in a circular orbit, which depend on the tug's thrust and orbital angular rate.The unstable solutions give rise to the chaotic motion of the system in the presence of additional disturbances. The chaotic motion is caused by the effects of the eccentricity of the orbit of the system's center of mass and the out-of-plane roll oscillations of the tether. The system is investigated using Poincaré sections and Lyapunov exponents. It is shown that the chaotic motions of the system depend on the tug's thrust.The results obtained are useful in determining the parametersof the tethered tug-debris system for ADR missions.

Full text

67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016.
Copyright ©2016 by the International Astronautical Federation (IAF). All rights reserved.
IAC-16,C1,8,11,x32993
Chaotic Motions of Tethered Satellites with Low Thrust
Vladimir Aslsnova*, Aruk K. Misrab, Vadim V. Yudintsevc
a Theoretical Mechanics Department, Russia, Samara National Research University, 34, Moskovskoye shosse,
Samara, 443086, Russia, aslanov_vs@mail.ru
b Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, PQ, Canada
H3A 2K6, arun.misra@mcgill.ca
c Theoretical Mechanics Department, Russia, Samara National Research University, 34, Moskovskoye shosse,
Samara, 443086, Russia, yudintsev@classmech.ru
* Corresponding Author
Abstract
The chaotic motion of the tethered tug-debris system with the low thrust tug is considered in the context of active
debris removal (ADR) missions. Stable and unstable stationary solutions are presented for the in-plane motion of the
system in a circular orbit, which depend on the tug’s thrust and orbital angular rate. The unstable solutions give rise
to the chaotic motion of the system in the presence of additional disturbances. The chaotic motion is caused by the
effects of the eccentricity of the orbit of the system’s center of mass and the out-of-plane roll oscillations of the
tether. The system is investigated using Poincaré sections and Lyapunov exponents. It is shown that the chaotic
motions of the system depend on the tug’s thrust. The results obtained are useful in determining the parameters of the
tethered tug-debris system for ADR missions.
Keywords: space debris, tether, chaos, orbit, eccentricity
1. Introduction
Space tethers have been studied extensively [1–13].
Space tethers can be used for space debris removal.
Understanding the dynamical behavior of the tethered
tug-debris system is essential for the success of active
debris removal missions. Active debris removal using
tethered tug-debris system is a relatively new topic.
Although there have been several recent studies [14-24],
many aspects of this problem still remain unexplored.
For example, the chaotic behavior of the system, which
includes a space tug, an elastic tether, and space debris,
remains unexplored. The chaotic motion of a tethered
system was reported in a paper by Misra et al. [7] and
by Aslanov [8]. In [6,7], Misra et al. show that chaos
could occur only for elliptic orbits or in the presence of
both pitch and roll. In [8], it is shown that a chaotic
planar motion exists due to the flexibility of the tether
and the presence of the low thrust.
This paper focuses on the attitude motion of a tether
connecting two bodies (a passive space debris and an
active space tug). This paper is an extension of papers
[6], [7] and [8]. In this paper we consider the chaotic
motion of the system due to the eccentricity of the orbit
of the center of mass and due to the out-of-plane
oscillations of the tether.
In the second section of the paper, the equations of
the spatial motion of the system are written for the
general case. Stationary solutions of the motion in a
circular orbit are found in the third section of the paper.
In the fourth section, chaotic motions of the system are
investigated using Poincaré sections and Lyapunov
exponents.
2. Mathematical model
The system considered in this paper consists of two
point end-masses
m1
(space tug) and
m2
(debris) connected by a massless tether of free length
l0
. The position of the center of mass of the system
is described by the radial distance
R
and true
anomaly
ϑ
. Orientation of the tether relative to the
orbital frame
Cxyz
is described by angles
α
and
γ
(Fig. 1). The configuration of the system is
described by 5 generalized coordinates
R ,ϑ, α , γ , l
. The motion of the system is
represented by the equations [9]
m´
R−mR ´
ϑ2+μm
R2−3μ m12 l2
2R4×
(
1−3 cos2αcos γ
)
=QR
(1)
2mR ´
R´
ϑ+m R2´
ϑ+2m12 l
(
´
ϑ+´
α
)
´
lcos2γ+m12l2
[
(
´
ϑ+´
α
)
cos2γ−
(
´
ϑ+´
α
)
´
γsin2 γ
]
=Qϑ
(2)
2m12l
(
´
ϑ+ ´α
)
´
lcos2γ+m12 l2
[
(
´
ϑ+ ´α
)
cos2γ−2
(
´
ϑ+ ´α
)
´γcos γsin γ
]
+3μ m12 l2
2R3sin 2 αcos2γ=Qα
(3)
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m12 l2´γ−2m12 l´
l´γ+m12 l2
(
´
ϑ+ ´α
)
2sin γcosγ+3μ m12 l2
R3cos γsin γcos2α=Qγ
(4)
m12´
l+
(
μ m12
(
1−3 cos2αcos2γ
)
R3−m12
(
(
´
ϑ+ ´α
)
2cos2γ+ ´γ2
)
)
l=Ql
(5)
Fig. 1. Geometry of the system
where
m=m1+m2, m12=m1m2/m
,
QR,
Qϑ,
Qα,
Qγ,
Ql
are the generalized
forces corresponding to the tether elasticity and
damping and the tug’s thrust
P
and are given by
QR=0,
Qϑ=−(m R−l m2cos γcos ϑ)
mP
(6)
Qα=lm2cos γcos α
mP ,
Qγ=−l m2sin γsin α
mP
(7)
Ql=m2cos γsin α
mP−k(l−l0)−c´
l
(8)
where
k
is the tether stiffness and
c
is the
damping coefficient of the tether. In these expressions
for the generalized forces we suppose that the tug’s
thrust
P
acts in a direction opposite to the axis
Cy
(Fig. 1), so the coordinates of the vector
P
in
Cxyz
are
¿
[
0,−P , 0
]
T
.
Let us simplify equations (1)-(5) after making some
assumptions about the motion of the center of mass of
the system. Suppose that the acceleration due to the low
thrust tug is small
aτ=P
m1+m2
≪g=μ
R2
(9)
Based on this simplification, the attitude motion of the
system can be studied assuming that the geometry of the
orbit is preserved
p=const , e =const
We also do not consider the perturbations of the system
induced by the longitudinal oscillations of the tether [8],
so in this paper we leave out the corresponding
equation. Taking into account these accepted
assumptions, the equations of the tethered system can be
written as [6], [9]
cos2γ
[
(
´
α+´
ϑ
)
−2´γ
(
´
α+´
ϑ
)
tan γ+3
2
μ
R3sin 2 α
]
=Qα
l2m12
(10)
´γ+
[
(
´
α+´
ϑ
)
2+3μ
R3cos2α
]
sin γcos γ=Qγ
l2m12
(11)
For the convenience of analysis, the independent
variable can be changed from time to true anomaly. In
this case the equations get the form [6]
cos2γ
[
α' '−
(
2γ'tan γ+K
)(
α'+1
)
]
+3
2Gsin 2 α=´
Qα
(12)
γ''−K γ'+
[
(
α'+1
)
2+3Gcos2α
]
sin γcos γ=´
Qγ
(13)
where prime denotes differentiation with respect to true
anomaly
ϑ.
Dimensionless functions
G
,
K
and dimensionless generalized forces
´
Qα
,
´
Qγ
are the following [6]:
G=1
1+ecosϑ, K =2esin ϑ
1+ecos ϑ
(14)
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´
Qα=Qα
l2m12
⋅G4p3
μ,
´
Qγ=Qγ
l2m12
⋅G4p3
μ
(15)
3. Stationary motion for a circular orbit
Let us consider the stationary motion of the system
in the orbital plane under the action of the tug’s thrust
force. In-plane motion of the system is described by the
following equation (
γ=0, γ'=0
)
α' ' +3
2sin 2 α=P
m1l0ω2cos α
(16)
where
ω=
√
μ R−3
is the orbital angular rate.
Equation (16) can be rewritten as
α'' =mα(α)
(17)
were
mα
(
α
)
is a dimensionless torque
mα
(
α
)
=acosα−bsin 2 α
(18)
and
a=P
m1l0ω2>0∧b=3
2
(19)
The first integral of equation (17) is
α'2
2+W
(
α
)
=E
(20)
where
W(α)
is the potential energy
W
(
α
)
=∫
(
−acos α+bsin 2 α
)
dα=−asin α−bcos2α
(21)
The potential energy (21) has two minima
corresponding to the two stationary states and one
maxima corresponding to the unstable configuration
(Fig. 2).
Fig. 2. Potential energy
W
(
α
)
To get the stationary solutions, let us equate Eq. 18
to zero. That leads to two stationary solutions
α¿=π
2+kπ∧¿
α¿=
(
−1
)
kasin
(
a
2b
)
+kπ , k ∈Z
(22)
If
a<2b
(23)
or according to (19) if
P
m1l0ω2<3
(24)
then there are stable equilibrium positions
αs1=asin
(
a
2b
)
, a s2=π−asin
(
a
2b
)
(25)
and the unstable equilibrium position
αu=π
2
(26)
The unstable equilibrium could also correspond to
αu=π/2+πk
, however, it is not realizable for a
flexible tether. If the condition (24) is not satisfied, then
the stable position is
αs=π/2
and the unstable
position is absent (Fig. 4).
Fig. 3. Separatrices for
a=0
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Fig. 4. Bifurcation diagram
4. Chaotic motion
In this section the motions of the system in the
presence of the following disturbing factors: eccentricity
of the orbit and the out of plane oscillation of the tether.
For the unstable equilibrium position given by (26),
small perturbations caused by these factors can lead to
chaos.
4.1 Orbit eccentricity
Let us consider the motion of the system in a
Keplerian orbit with semi-major axis 7371 km and
eccentricity
e=0.05
. These parameters correspond
to the orbit with
ha=1369 km
and
hp=631km
. The space tug of mass
m1=500
kg and the debris of mass
m2=3000
kg are
connected by a tether of length
l0=l=100
m.
When
P=0.1
N, condition (24)
amin=P
m1l0ωmax
2<P
m1l0ωmin
2=amax<3
is satisfied for minimum and maximum orbital rates
ωmin=
√
μ
p3(1−e)
2
, ωmax=
√
μ
p3(1+e)
2
where
amin=P
m1l0ωmax
2≈1.6, amax=P
m1l0ωmin
2≈2.5
So there are two equilibrium positions that are described
by Eqs. (25) and (26). Fig. 5 shows Poincaré maps for
this case for four trajectories with the following initial
conditions
a.
a0=0
,
b.
a0=0.05
,
c.
a0=0.4,
d.
a0=π/2,
´α0=γ0= ´γ0=0
for all cases. We can see a large
area of chaotic motion depicted by a diffused set of
points.
Fig. 5. Poincaré map for P=0.1 N and e=0.05
When
P=0.2
N
amin=P
m1l0ωmax
2≈3.3>3
Fig. 6 shows Poincaré map for this case. Due to the
orbital eccentricity, there are no stable points in this case
but the chaotic motion of the system is eliminated. Fig.
6 demonstrates quasi-periodic orbits only in the (
α , ´α
) phase space.
Fig. 6. Poincaré map for P=0.2 N and e=0.05
Fig. 7 plots the largest Lyapunov exponents for
trajectories with
α0=π/2
,
´α0=0
and for
P=0.1 N , 0.2 N
over 500 orbits. The largest
Lyapunov exponent tends to zero if
P=0.2
N. For
P=0.1
N it tends to a positive value, about 0.08.
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Fig. 7. Largest Lyapunov exponents for P=0.1 N,
P=0.2 N and e=0.05
Figures 5-7 illustrate that the system exhibits chaotic
dynamic behavior if
a=P
m1l0ω2<3
4.2 Out of plane motion
Chaotic motion of the considered system can also be
caused by the out-of-plane oscillations of the tether. The
spatial motion of the system in a circular orbit is given
by the following equations
cos2γ
[
α' '−2γ'
(
α'+1
)
tan γ
]
+3
2sin 2 α=´
Qα
(27)
γ' ' +
[
(
α'+1
)
2+3 cos2α
]
sin γcos γ=´
Qγ
(28)
The generalized forces are
´
Qα=p3cos γcos α
m1μ l P ,
´
Qγ=−p3sin γsin α
m1μl P
(29)
We consider the motion of the system in a circular
orbit with
R=7371 km
= const. As in the previous
case, the space tug of mass 500 kg and the debris of
mass 3000 kg are connected by a tether of length 100 m.
Fig. 8 gives Poincaré map for
P=0.1
N and for
four trajectories with the following initial conditions
a.
a0=0.1
,
b.
a0=π/2−0.3
,
c.
a0=π/2+0.3,
d.
a0=π/2+0.05,
and
γ0=0.1
rad,
´α0=´γ=0
for all cases. Fig.
8 illustrates that there is a chaotic region near the
separatrix of the undisturbed map (Fig. 3).
Fig. 8. Poincaré map for P=0.1 N and
γ0
=0.1
Fig. 9 gives Poincaré map for
P=0.2
N. In this
case
a=P
m1l0ω2≈4>3
There are no stable points in this case and chaotic
motion is absent. Poincare map shows quasi-periodic
trajectories. Fig. 10 plots the largest Lyapunov
exponents for trajectories with
α0=π/2
,
´α0=0
,
γ0=0.1
rad,
´γ0=0
and for
P=0.1 N , 0.2 N
over 500 orbits. The largest
Lyapunov exponent tends to zero if
P=0.2
N. For
P=0.1
N it tends to a positive value, about 0.1.
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Fig. 9. Poincaré map for P=0.2 N and
γ0
=0.1
Fig. 10. Largest Lyapunov exponents for P=0.1 N,
P=0.2 N and
γ0
=0.1
5. Conclusions
The pitch motion of the two-body tethered system in
circular and elliptic orbits under the action of the tug’s
thrust has been studied. The pitch motion is perturbed
by the out-of-plane roll motion and change in the
gravity gradient due to motion of the center of mass of
the system in an elliptic orbit. The simulation results
show that the ADR using tethered low thrust space tug
can lead to chaotic motion of the system if there is an
unstable equilibrium of the undisturbed system. The
stationary solutions obtained for the undisturbed system
allowed us to estimate the lowest tug’s thrust below
which the chaotic motion can occur in the presence of
disturbances. To avoid chaotic motion of the system, the
choice of the thrust and mass of the space tug, as well as
the tether length, should be such as to satisfy the
following condition:
a=P p3
m1μ l0
(
1−e
)
4>3
In the case of out-of-plane motion of the tether even for
a large tug’s thrust
P
the initial pitch angle of the
system should be closer to
π/2
to avoid chaotic
motion of the system. This estimate is approximate,
because it ignores the motion in an elliptical orbit and
out-of-plane oscillation of the system, so the possibility
of the chaos should be investigated in-depth if tug’s
thrust P is close to
3m1μ l0
(
1−e
)
4/p3
.
The obtained results can be applied to select the
possible physical properties of the space tug and the
tether for the space debris removal system. Further
studies may be aimed at analysing the phase space of
the system for different values of the tug’s thrust to
investigate transitions from chaos to order in this
system.
Acknowledgements
This work was supported by Ministry of Education
and Science of Russia (contract no. 9.540.2014/K).
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