# Egemen Kolemen1, N. Jeremy Kasdin1 & Pini Gurfil2

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

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

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

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

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)

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

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.

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.

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

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.

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

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

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

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

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

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

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

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.

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:

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

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

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

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

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

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

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:

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

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

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).

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.

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.

Clohessy-Wiltshire Equations

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.

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

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