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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)

IAC-16-16,C1,8,11,x32993 Page 1 of 8

67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016.

Copyright ©2016 by the International Astronautical Federation (IAF). All rights reserved.

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)

IAC-16-16,C1,8,11,x32993 Page 2 of 8

67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016.

Copyright ©2016 by the International Astronautical Federation (IAF). All rights reserved.

´

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

IAC-16-16,C1,8,11,x32993 Page 3 of 8

67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016.

Copyright ©2016 by the International Astronautical Federation (IAF). All rights reserved.

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.

IAC-16-16,C1,8,11,x32993 Page 4 of 8

67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016.

Copyright ©2016 by the International Astronautical Federation (IAF). All rights reserved.

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.

IAC-16-16,C1,8,11,x32993 Page 5 of 8

67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016.

Copyright ©2016 by the International Astronautical Federation (IAF). All rights reserved.

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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Copyright ©2016 by the International Astronautical Federation (IAF). All rights reserved.

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