1. Egemen Kolemen1, N. Jeremy Kasdin1 & Pini Gurfil2by
Egemen Kolemen1, N. Jeremy Kasdin1 & Pini Gurfil2
Princeton University, Mechanical & Aerospace Engineering
Technion Israel Institute of Technology, Aerospace Engineering

2. Content 1. Literature 2. Hamilton Jacobi Analysis
3. Results - Eccentric Orbit
4. Results - J2, J3, J4 Perturbation
5. Conclusions

3. Formation Flying Sensor Webs Interferometry Co-observing
Stereo Imaging
Aerobots
Multi-point observing
Large Constellations
In situ observations
Tethered Interferometry
Space Station
“Aircraft Carrier” to Fleets of Distributed Spacecraft

4. Dynamical Modeling of Relative Motion
Control is expensive
Use natural forces to control
Aim:
Initial conditions
Bounded orbits
Relative Trajectory
Position, velocity
Local measurement / Local Control suitability
Artist’s impressions of Orion/Emerald Spacecrafts

5. Background
Linearized relative motion in a Cartesian rotating frame. (C-W, `60)
Described linear relative motion using orbital elements (Schaub, Vadali, Alfriend, `00)
Applied symmetry and reduction technique to J2 perturbation (Koon, Marsden, Murray `01)
Used approximate generating function for the relative Hamiltonian (Guibout, Scheeres, `04)
Provided detailed description of our method and J2 solution (Kasdin, Gurfil, & Kolemen, `05)

6. Clohessy-Wiltshire (C-W) Equations
Aim: To solve the rendezvous problem
Two-body equation:
where r is the position of the satellite,
where  is the relative position,
Linearizing the equation around =0 and keeping only the 1st order terms, we get:
Relative motion in rotating Euler-Hill reference frame

7. Clohessy-Wiltshire (C-W) Equations
Periodic motion in z-direction.
A stationary mode, an oscillatory mode, and a drifting mode in the x-y plane.
The non-drifting condition:
2 by 1 ellipse x-y plane.

8. How to extend this solution?
What is the effect of perturbations on periodic orbits?
How do we get into these periodic orbits?
Not only finding Initial Conditions but finding Periodic Orbits.
Approach:
Solve Hamilton-Jacobi Equations.
Express the periodic relative orbits by canonical elements.
Perform perturbation analysis on these elements.

9. Canonical Analysis of Relative Motion
Velocity in the rotating frame:
Kinetic Energy:
Potential Energy:
Relative motion in rotating frame

10. The Hamiltonian
Separate low and higher order Hamiltonian:
The unperturbed Hamiltonian:
Corresponds to C-W equations:
Solve Hamilton-Jacobi Equations for the 6 constants of motion.

11. Hamilton-Jacobi (H-J) Solution
Transform Cartesian elements to new constants using H-J Theory.
Termed Epicyclic Elements
The new Hamiltonian:
Cartesian coordinates:

12. Physical meaning of the Epicyclic Elements
Osculating 2:1 elliptic relative orbit
1  size
3  drift in y-axis
Q3  center position in y-axis
Q1  phase
Osculating elliptic relative orbit

13. Modified Epicyclic Elements
For ease of calculations, symplectic transformation to amplitude only variable:
The new Hamiltonian:
Cartesian coordinates:

14. Perturbations to Relative Motion
Perturbation Analysis:
Perturbing Hamiltonian:
Hamilton’s Equations:
Transform the perturbation on the relative orbit to perturbations on the canonical elements.

15. Relative Motion Around Eccentric Orbits
Relative velocity:
Find Hamiltonian, expand in eccentricity:
1st order perturbing Hamiltonian:
Relative motion around an eccentric orbit

16. Relative Motion Around Eccentric Orbits
Variation of epicyclic elements:
Solution:
No drift condition for 1st order:

17. Eccentricity/Nonlinearity Perturbation Results
Third order no drift condition.
Approximation good up to e ~ 0.03
Code:
~ekolemen/eccentricity.m
e = 0.001
m/orbit
e = 0.01
2mm/orbit
e = 0.02
7cm/orbit

18. Placing Multiple Satellites on Relative Orbit
Specify:
Size in x-y plane and z-plane.
Phase of each satellite
Place a group of satellites in one relative orbit.

19. Earth Oblateness Perturbation
Earth’s Oblateness ( J2 zonal harmonic) is often the largest perturbation.
Axially symmetric gravitational potential zonal harmonics:
where  is the spacecraft’s latitude angle:
Z is the distance from the Equatorial Plane and the Jk's:

20. Effect of the J2 Perturbation on a single spacecraft
The long term variation of the mean orbital elements:
where a, e, i,  , u orbital element and n mean motion. And,
Note: For circular orbits, argument of perigee,  does not make sense.
Regression of the ascending node under the J2 perturbation

21. The Average J2 Drifting Frame
Rotating with the average J2 drift:
The velocity of the follower:
First order no drift condition
Angular velocity of the average J2 drifting frame

22. Periodic Orbits around J2 and J22 drifting frame
3 different types of periodic orbits. Comparison with differential orbital elements
 : Invariance under .
i, e, a: Solutions for
 u: Invariance under u. 
i, e, a u

23. Families of Periodic Orbit around L2
(?)
HORIZONTAL LYAPUNOV Orbit
NORTHERN QUASIHALO Orbits
(?)
SOUTHER Orbits
NORTHERN HALO Orbit
VERTICAL LYAPUNOV Orbit
LISSAJOUS Orbits
Univ. Barcelona: Jorba, Gomez, Simo
Univ. Catalunya: Masdemont

24. Reference Frames and Relative Motion
Osculating Rotating Frame
Real Relative Motion
Bounded Orbit 2
Mean Rotating Frame
Bounded Orbit 1

25. J2 and J22 Periodic Solutions (Two Spacecrafts)
Short Term: 20 Orbits
Long Term: 200 Orbits
Error: 30cm/orbit

26. Tumbling of the Relative Orbit
Tumbling Effect
Tumbling of the Relative Orbit
Figure: S. A. Schweighart, “Development and Analysis of a High Fidelity Linearized J2 Model for Satellite Formation Flying”
Examples:

27. Higher Order Zonal Perturbation: J2, J22, J3, J4
5 boundedness conditions,
Constraining all the parameters except the u offset.
One osculating orbit for one mean orbit.
Relative motion around the rotating frame
Relative motion around osculating orbit
Error: 30cm/orbit

28. Relative Motion for J2, J3, J4 perturbations
Conclusion
A Hamiltonian approach to solve for the relative motion is applied.
Bounded relative motion for nonlinearity and eccentricity perturbation is solved.
Effect of zonal harmonics, J2, J3, and J4 on the relative bounded orbit is investigated.
Outlook: Combine eccentricity and zonal harmonics perturbations.
Relative Motion for J2, J3, J4 perturbations

29. References
Koon, Lo, Marsden and Ross, “Dynamical Systems, the Three-Body Problem and Space Mission Design”, To be published
Jorba, Masdemont, “Dynamics in the center manifold of the collinear points of the restricted three body problem”, Physica D 132 (1999) 189–213
Jorba, “A Methodology for the Numerical Computation of Normal Forms, Centre Manifolds and First Integrals of Hamiltonian Systems”, Experimental Mathematics 8,
Gomez, Masdemont, Simo, “Quasi-Periodic Orbits Associated with the Libration Points”, JAS, 1998(2), 46,
S. A. Schweighart, “Development and Analysis of a High Fidelity Linearized J2 Model for Satellite Formation Flying” Master of Science, MIT, June 2001
T. A. Lovella, S. G. Tragesser, “Near-Optimal Reconfiguration and Maintenance of Close Spacecraft Formations” Ann. N.Y. Acad. Sci. 1017: 158–176 (2004).

30. Clohessy, W. H. & R. S. Wiltshire. 1960
Clohessy, W.H. & R.S. Wiltshire Terminal guidance system for satellite rendezvous. J. Astronaut. Sci. 27(9):
Carter, T.E. & M. Humi Fuel-optimal rendezvous near a point in general Keplerian orbit. J. Guid. Control Dynam. 10(6):
Inalhan, G., M. Tillerson & J.P. How Relative dynamics and control of spacecraft formations in eccentric orbits. J. Guid. Control Dynam. 25(1):
Gim, D.W. & K.T. Alfriend The state transition matrix of relative motion for the perturbed non-circular reference orbit. Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA, February. AAS
Alfriend, K.T. & H. Schaub Dynamics and control of spacecraft formations: challenges and some solutions. J. Astronaut. Sci. 48(2):
Hill, G.W Researches in the lunar theory. Am. J. Math. 1: 5-26.
Schaub, H., S.R. Vadali & K.T. Alfriend Spacecraft formation flying control using mean orbital elements. J. Astronaut. Sci. 48(1):
Namouni, F Secular interactions of coorbiting objects. Icarus 137:
Gurfil, P. & N.J. Kasdin Nonlinear modeling and control of spacecraft relative motion in the configuration space. Proceedings of the AAS/AIAA Spacecflight Mechanics Meeting, Puerto Rico, February.
Karlgaard, C.D. & F.H. Lutze Second-order relative motion equations. Proceedings of the AAS/AIAA Astrodynamics Conference, Quebec City, Quebec, July. AAS
Alfriend, K.T., H. Yan & S.R. Vadali Nonlinear considerations in satellite formation flying. Proceedings of the 2002 AIAA/AAS Astrodynamics Specialist Conference, Monterey, CA, August. AIAA
Koon, W.S., J.E. Marsden & R.M. Murray J2 dynamics and formation flight. Proceedings of the 2001 AIAA Guidance, Navigation, and Control Conference, Montreal, Canada, August. AIAA
Broucke, R.A Motion near the unit circle in the three-body problem. Celest. Mech. Dynam. Astron. 73(1):
Goldstein, H Classical Mechanics. Addison-Wesley.
Battin, R.H An Introduction to the Mathematics and Methods of Astrodynamics. AIAA.

31. Hill, G.W. 1878. Researches in the lunar theory. Am. J. Math. 1: 5-26.
Clohessy, W.H. & R.S. Wiltshire Terminal guidance system for satellite rendezvous. J. Astronaut. Sci. 27(9):
Carter, T.E. & M. Humi Fuel-optimal rendezvous near a point in general Keplerian orbit. J. Guid. Control Dynam. 10(6):
Inalhan, G., M. Tillerson & J.P. How Relative dynamics and control of spacecraft formations in eccentric orbits. J. Guid. Control Dynam. 25(1):
Gim, D.W. & K.T. Alfriend The state transition matrix of relative motion for the perturbed non-circular reference orbit. Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA, February. AAS
Alfriend, K.T. & H. Schaub Dynamics and control of spacecraft formations: challenges and some solutions. J. Astronaut. Sci. 48(2):
Hill, G.W Researches in the lunar theory. Am. J. Math. 1: 5-26.
Schaub, H., S.R. Vadali & K.T. Alfriend Spacecraft formation flying control using mean orbital elements. J. Astronaut. Sci. 48(1):
Namouni, F Secular interactions of coorbiting objects. Icarus 137: [CrossRef]
Gurfil, P. & N.J. Kasdin Nonlinear modeling and control of spacecraft relative motion in the configuration space. Proceedings of the AAS/AIAA Spacecflight Mechanics Meeting, Puerto Rico, February.
Karlgaard, C.D. & F.H. Lutze Second-order relative motion equations. Proceedings of the AAS/AIAA Astrodynamics Conference, Quebec City, Quebec, July. AAS
Alfriend, K.T., H. Yan & S.R. Vadali Nonlinear considerations in satellite formation flying. Proceedings of the 2002 AIAA/AAS Astrodynamics Specialist Conference, Monterey, CA, August. AIAA
Koon, W.S., J.E. Marsden & R.M. Murray J2 dynamics and formation flight. Proceedings of the 2001 AIAA Guidance, Navigation, and Control Conference, Montreal, Canada, August. AIAA
Broucke, R.A Motion near the unit circle in the three-body problem. Celest. Mech. Dynam. Astron. 73(1): [CrossRef]
Goldstein, H Classical Mechanics. Addison-Wesley.
Battin, R.H An Introduction to the Mathematics and Methods of Astrodynamics. AIAA.

32. Clohessy-Wiltshire Equations

33. Background
W.H. Clohessy, R. S. Wiltshire, “Terminal Guidance for Satellite Rendezvous”, 1960
TAMU
H. Schaub, S. R. Vadali, K. T. Alfriend, “Spacecraft Formation Flying Control Using Mean Orbital Elements”, 2000
MIT
G. Inalhan, M. Tillerson, J. P. How, “Relative Dynamics and Control of Spacecraft Formation in Eccentric Orbits”, 2002
Princeton/Technion
N. J. Kasdin and P. Gurfil, “Canonical Modelling of Relative Spacecraft Motion via Epicyclic Orbital Elements”, In publish
Caltech
W. S. Koon, J. E. Marsden and R. M. Murray, “J2 DYNAMICS AND FORMATION FLIGHT”, AIAA
UMich
V.M. Guibout, D.J. Scheeres, “Solving relative two-point boundary value problems: Application to spacecraft formation flight transfer” Journal of Guidance, Control, and Dynamics 27(4):
Univ. Surrey
Y. Hashida, P. Palmer , “Epicyclic Motion of Satellites Under Rotating Potential”, Journal of Guidance, Control and Dynamics, Vol. 25, No. 3, pp , 2002.

34. Perturbations Question:
H^{(1)}, perturbing Hamiltonian, How do we get into these periodic orbits?
How will be the form of the new periodic orbits?
Answer:
Hamilton’s

35. Analytical Solution for the Base Orbit
Kozai  Sgp
Brower  Sgp4
Hoots  HANDE  Von Ziepel
1st order in J2 short term (not precise)
Coffey and Deprit
Vinti  Lie Method
Too many terms