The document analyzes the chaotic motions that can occur for tethered satellite systems with low thrust. It describes the system and assumptions, presents the motion equations, and identifies stationary solutions. Orbital eccentricity and out-of-plane oscillations are shown to induce chaos if they cause an unstable equilibrium condition. The choice of thrust level, satellite masses, and tether length must satisfy conditions to ensure regular in-plane motion even in an elliptic orbit.

1. CHAOTIC MOTIONS OF
TETHERED SATELLITES WITH
LOW THRUST
Vladimir S. Aslanov1, Arun K. Misra2, Vadim V. Yudintsev1
1Samara National Research University, Samara, Russia
2McGill University, Montreal, QC, Canada
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 1

2. OUTLINE
1. System description and assumptions
2. Motion equations
3. Stationary solutions
4. Chaotic motion due to eccentricity of the orbit
5. Chaotic motion due to out-of-plane oscillations
6. Conclusion
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 2

3. SYSTEM DESCRIPTION
• Chaotic behaviour of the system
(Space Tug + Tether + Debris)
is considered
• The following disturbances are
assumed to affect the in-plane
motion of the tether
• Orbital eccentricity
• Out-of-plane motion of the tether
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 3

4. ASSUMPTIONS
• The tug and debris are point
masses
• Tug’s thrust is low
• 𝑀𝑐 𝑷 is comparable to 𝑀𝑔𝑟𝑎𝑣
• The orbit is not changed
• |R|= const
• e = const
• Tether length is constant
• 𝐶1 𝐶2 = 𝑙 = const
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 4

5. MOTION EQUATIONS
• In-plane motion
cos2 𝛾 𝛼 + 𝜗 − 2 𝛾 𝛼 + 𝜗 tan 𝛾 +
3
2
𝜇
𝑅3
sin 2𝛼 =
𝑄 𝛼
𝑙2 𝑚12
• Out-of-plane motion
𝛾 + 𝛼 + 𝜗
2
+ 3
𝜇
𝑅3
cos2 𝛼 sin 𝛾 cos 𝛾 =
𝑄 𝛾
𝑙2 𝑚12
• Generalized forces
𝑄 𝛼 =
𝑙 𝑚2 cos 𝛾 cos 𝛼
𝑚12
𝑃, 𝑄 𝛾 = −
𝑙𝑚2 sin 𝛾 sin 𝛼
𝑚12
𝑃
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 5

6. DIMENSIONLESS EQUATIONS
• In-plane motion
cos2 𝛾 𝛼′′ − 2𝛾′ tan 𝛾 + 𝐾 𝛼′ + 1 +
3
2
𝐺 sin 2𝛼 = 𝑄 𝛼
• Out of plane motion
𝛾′′ − 𝐾𝛾′ + 𝛼′ + 1 2 + 3𝐺cos2 𝛼 sin 𝛾 cos 𝛾 = 𝑄 𝛾
where
𝛼′′ =
𝑑2 𝛼
𝜕𝜗2
, 𝛾′′ =
𝑑2 𝛾
𝜕𝜗2
, 𝐺 =
1
1 + 𝑒 cos 𝜗
, 𝐾 =
2 𝑒 sin ϑ
1 + 𝑒 cos 𝜗
• Generalized forces
𝑄 𝛼 =
𝑄 𝛼
𝑙2 𝑚12
⋅
𝐺4 𝑝3
𝜇
, 𝑄 𝛾 =
𝑄 𝛾
𝑙2 𝑚12
⋅
𝐺4 𝑝3
𝜇
, 𝑚12 =
𝑚1 𝑚2
𝑚1 + 𝑚2
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 6

7. STATIONARY MOTIONS
In-plane motion in a circular orbit
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 7

8. STATIONARY MOTION
• For in-plane motion in a circular orbit
(e=0, 𝛾 =0)
𝛼′′ =
𝑃
𝑚1 𝑙0 𝜔2
cos 𝛼 −
3
2
sin 2𝛼 = 𝑚 𝛼
• First integral
𝛼′ 2
2
+ 𝑊 𝛼 = 𝐸
Torque 𝑚 𝛼
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 8

9. POTENTIAL ENERGY
• First Integral
𝛼′ 2
2
+ 𝑊 𝛼 = 𝐸
• Potential energy
𝑊 𝛼 = −𝑎 sin 𝛼 − 𝑏 cos2
𝛼
where
𝑎 =
𝑃
𝑚1 𝑙0 𝜔2
, 𝑏 =
3
2
, 𝜔 =
𝜇
𝑅3
Potential energy 𝑊 𝛼
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 9
a=2

10. POTENTIAL ENERGY
• First Integral
𝛼′ 2
2
+ 𝑊 𝛼 = 𝐸
• Potential energy
𝑊 𝛼 = −𝑎 sin 𝛼 − 𝑏 cos2
𝛼
where
𝑎 =
𝑃
𝑚1 𝑙0 𝜔2
, 𝑏 =
3
2
, 𝜔 =
𝜇
𝑅3
Potential energy 𝑊 𝛼
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 10
a=4

11. STABLE POINT
• If
𝑃
𝑚1 𝑙0 𝜔2
> 3
• Stable position only
𝑎 𝑠 =
𝜋
2
Bifurcation diagram
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 11
𝑎 𝑠

12. UNSTABLE POINTS
• If
𝑃
𝑚1 𝑙0 𝜔2
< 3
• Stable positions
𝑎 𝑠1 = asin
𝑎
2𝑏
, 𝑎 𝑠2 = 𝜋 − asin
𝑎
2𝑏
• Unstable position
𝑎 𝑢 = asin
𝑎
2𝑏
Bifurcation diagram
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 12
𝑎 𝑠1
𝑎 𝑠2
𝑎 𝑢

13. ORBIT ECCENTRICITY
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 13

14. CHAOTIC MOTION
• Parameters
𝑎 = 7371 km, 𝑒 = 0.05, 𝑙0 = 100 m
𝑚1 = 500 kg, 𝑚2 = 3000 kg
• Tug’s thrust
𝑃 = 0.1 N → 𝑎 =
𝑃
𝑚1 𝑙0 𝜔 𝑚𝑖𝑛
2 < 3
• Orbital rate
𝜔 𝑚𝑖𝑛 = 𝜇𝑝−3 1 − 𝑒 ≈ 0.0009 𝑠−1
𝜔 𝑚𝑎𝑥 = 𝜇𝑝−3 1 + 𝑒 ≈ 0.0011 𝑠−1
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 14

15. REGULAR MOTION
• Parameters
𝑎 = 7371 km, 𝑒 = 0.05, 𝑙0 = 100 m
𝑚1 = 500 kg, 𝑚2 = 3000 kg
• Tug’s thrust
𝑃 = 0.2 N → 𝑎 =
𝑃
𝑚1 𝑙0 𝜔 𝑚𝑖𝑛
2 > 3
• Orbital rate
𝜔 𝑚𝑖𝑛 = 𝜇𝑝−3 1 − 𝑒 ≈ 0.0009 𝑠−1
𝜔 𝑚𝑎𝑥 = 𝜇𝑝−3 1 + 𝑒 ≈ 0.0011 𝑠−1
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 15
Poincare section

16. LARGEST LYAPUNOV EXPONENT
• Lyapunov exponents for trajectory
• 𝛼0 = 𝜋/2, 𝛼0 = 0
• 𝛾0 = 0.00, 𝛾0 = 0
• 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
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 16

17. OUT-OF-PLANE OSCILLATIONS
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 17

18. CHAOTIC MOTION
• Parameters
𝑎 = 7371 km, 𝑒 = 0, 𝑙0 = 100 m
𝑚1 = 500 kg, 𝑚2 = 3000 kg, 𝛾0 = 0.1
• Tug’s thrust
𝑃 = 0.1 N → 𝑎 =
𝑃
𝑚1 𝑙0 𝜔2
< 3
• Orbital rate
𝜔 = 𝜇𝑝−3 ≈ 0.0009 𝑠−1
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 18

19. REGULAR MOTION
• Parameters
𝑎 = 7371 km, 𝑒 = 0.05, 𝑙0 = 100 m
𝑚1 = 500 kg, 𝑚2 = 3000 kg, 𝛾0 = 0.1
• Tug’s thrust
𝑃 = 0.2 N → 𝑎 =
𝑃
𝑚1 𝑙0 𝜔 𝑚𝑖𝑛
2 > 3
• Orbital rate
𝜔 = 𝜇𝑝−3 ≈ 0.0009 𝑠−1
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 19

20. LARGEST LYAPUNOV EXPONENT
• Lyapunov exponents for trajectory
• 𝛼0 = 𝜋/2, 𝛼0 = 0
• 𝛾0 = 0.10, 𝛾0 = 0
• 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
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 20

21. CONCLUSION
• The pitch motion of the system is perturbed by the
• out-of-plane roll motion of the tether and
• change in the gravity gradient due to motion of the center of mass of the
system in an elliptic orbit
• The orbital motion of the center of mass of the system and out-of-plane
oscillation of the tether can cause chaos if there is an unstable equilibrium.
• The choice of the thrust and mass of the space tug, as well as the tether
length, should be such as to satisfy the condition for in-plane motion in an
elliptic orbit
𝑃
𝑚1 𝜇 𝑙0 1 − 𝑒 4
𝑝3 > 3
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 21

22. Thank you!
Vladimir S. Aslanov (aslanov_vs@mail.ru)
Arun K. Misra (arun.misra@mcgill.ca)
Vadim V. Yudintsev (yudintsev@classmech.ru)
67th International Astronautical Congress (IAC), Guadalajara, Mexico, 26-30 September 2016 22