new trends in astrodynamics and applications princeton, nj. june 3-4 2005 hamilton-jacobi modelling...

New Trends in Astrodynamics and ApplicationsPrinceton, NJ. June 3-4 2005

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Hamilton-Jacobi Modelling of Hamilton-Jacobi Modelling of Relative Motion for Formation FlyingRelative Motion for Formation Flying

byby

Egemen KolemenEgemen Kolemen11, N. Jeremy Kasdin, N. Jeremy Kasdin11 & Pini Gurfil & Pini Gurfil22

1.1. Princeton University, Mechanical & Aerospace EngineeringPrinceton University, Mechanical & Aerospace Engineering

2.2. Technion Israel Institute of Technology, Aerospace EngineeringTechnion Israel Institute of Technology, Aerospace Engineering

http://facilities.princeton.edu/conference/sports_camps.htm

2/28“Hamilton-Jacobi Modelling of Relative Motion for Formation Flying”

E. Kolemen, J. Kasdin, P. Gurfil

Content

2. Hamilton Jacobi Analysis

3. Results - Eccentric Orbit

5. Conclusions

4. Results - J2, J3, J4 Perturbation

1. Literature




Formation Flying

Co-observingStereo Imaging

Interferometry

Large ConstellationsIn situ observations

Aerobots

Tethered Interferometry

Space Station“Aircraft Carrier” to Fleets of

Distributed Spacecraft

Sensor Webs

Multi-point observing




Dynamical Modeling of Relative Motion

• Control is expensive

• Use natural forces to control

• Aim:

1. Initial conditions

• Bounded orbits

2. Relative Trajectory

• Position, velocity

3. 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)




• 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:

Clohessy-Wiltshire (C-W) Equations

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.

)1()0( HHH




Relative Motion Around Eccentric Orbits

• Relative velocity:

• Find Hamiltonian, expand in eccentricity:

• 1st order perturbing Hamiltonian:

tp

txpyp

zyxxe

H

x

yx

sin2

cos2124

33642

2

222

)1(

Relative motion around an eccentric orbit

)0()1( HHH onperturbati




Relative Motion Around Eccentric Orbits

• Variation of epicyclic elements:

• Solution:

• No drift condition for 1st order:




Eccentricity/Nonlinearity Perturbation Results

e = 0.001m/orbit

e = 0.012mm/orbit

e = 0.027cm/orbit

• Third order no drift condition.

• Approximation good up to e ~ 0.03

• Code: http://www.princeton.edu/ ~ekolemen/eccentricity.m




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 conditionAngular 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

NORTHERN HALO Orbit

LISSAJOUS Orbits

NORTHERN QUASIHALO

Orbits

VERTICAL LYAPUNOV Orbit

SOUTHER Orbits

HORIZONTAL LYAPUNOV Orbit

(?)

Univ. Barcelona: Jorba, Gomez, Simo Univ. Catalunya: Masdemont

(?)




Reference Frames and Relative Motion

Bounded Orbit 2

Bounded Orbit 1

Mean Rotating Frame

Osculating Rotating Frame

Real Relative Motion




J2 and J22 Periodic Solutions (Two Spacecrafts)

Short Term: 20 Orbits Long Term: 200 Orbits

Error: 30cm/orbit




Tumbling Effect

Tumbling of the Relative Orbit

Examples:

Figure: S. A. Schweighart, “Development and Analysis of a High Fidelity Linearized J2 Model for Satellite Formation Flying”




• 5 boundedness conditions,

• Constraining all the parameters except the u offset.

One osculating orbit for one mean orbit.

Higher Order Zonal Perturbation: J2, J22, J3, J4

Relative motion around the rotating frame Relative motion around osculating orbit

Error: 30cm/orbit




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, 155-195

• Gomez, Masdemont, Simo, “Quasi-Periodic Orbits Associated with the Libration Points”, JAS, 1998(2), 46, 135-176

• 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. Terminal guidance system for satellite rendezvous. J. Astronaut. Sci. 27(9): 653-678.

• Carter, T.E. & M. Humi. 1987. Fuel-optimal rendezvous near a point in general Keplerian orbit. J. Guid. Control Dynam. 10(6): 567-573.

• Inalhan, G., M. Tillerson & J.P. How. 2002. Relative dynamics and control of spacecraft formations in eccentric orbits. J. Guid. Control Dynam. 25(1): 48-60.

• Gim, D.W. & K.T. Alfriend. 2001. 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 01-222.

• Alfriend, K.T. & H. Schaub. 2000. Dynamics and control of spacecraft formations: challenges and some solutions. J. Astronaut. Sci. 48(2): 249-267.

• Hill, G.W. 1878. Researches in the lunar theory. Am. J. Math. 1: 5-26. • Schaub, H., S.R. Vadali & K.T. Alfriend. 2000. Spacecraft formation flying control using mean orbital elements. J.

Astronaut. Sci. 48(1): 69-87. • Namouni, F. 1999. Secular interactions of coorbiting objects. Icarus 137: 293-314.• Gurfil, P. & N.J. Kasdin. 2003. 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. 2001. Second-order relative motion equations. Proceedings of the AAS/AIAA

Astrodynamics Conference, Quebec City, Quebec, July. AAS 01-464. • Alfriend, K.T., H. Yan & S.R. Vadali. 2002. Nonlinear considerations in satellite formation flying. Proceedings of the

2002 AIAA/AAS Astrodynamics Specialist Conference, Monterey, CA, August. AIAA 2002-4741. • Koon, W.S., J.E. Marsden & R.M. Murray. 2001. J2 dynamics and formation flight. Proceedings of the 2001 AIAA

Guidance, Navigation, and Control Conference, Montreal, Canada, August. AIAA 2001-4090. • Broucke, R.A. 1999. Motion near the unit circle in the three-body problem. Celest. Mech. Dynam. Astron. 73(1): 281-

290.• Goldstein, H. 1980. Classical Mechanics. Addison-Wesley. • Battin, R.H. 1999. An Introduction to the Mathematics and Methods of Astrodynamics. AIAA.




Clohessy, W.H. & R.S. Wiltshire. 1960. Terminal guidance system for satellite rendezvous. J. Astronaut. Sci. 27(9): 653-678.

Carter, T.E. & M. Humi. 1987. Fuel-optimal rendezvous near a point in general Keplerian orbit. J. Guid. Control Dynam. 10(6): 567-573.

Inalhan, G., M. Tillerson & J.P. How. 2002. Relative dynamics and control of spacecraft formations in eccentric orbits. J. Guid. Control Dynam. 25(1): 48-60.

Gim, D.W. & K.T. Alfriend. 2001. 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 01-222. Alfriend, K.T. & H. Schaub. 2000. Dynamics and control of spacecraft formations: challenges

and some solutions. J. Astronaut. Sci. 48(2): 249-267. Hill, G.W. 1878. Researches in the lunar theory. Am. J. Math. 1: 5-26.

Schaub, H., S.R. Vadali & K.T. Alfriend. 2000. Spacecraft formation flying control using mean orbital elements. J. Astronaut. Sci. 48(1): 69-87.

Namouni, F. 1999. Secular interactions of coorbiting objects. Icarus 137: 293-314.[CrossRef] Gurfil, P. & N.J. Kasdin. 2003. 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. 2001. Second-order relative motion equations. Proceedings of the AAS/AIAA Astrodynamics Conference, Quebec City, Quebec, July. AAS 01-464.

Alfriend, K.T., H. Yan & S.R. Vadali. 2002. Nonlinear considerations in satellite formation flying. Proceedings of the 2002 AIAA/AAS Astrodynamics Specialist Conference, Monterey, CA,

August. AIAA 2002-4741. Koon, W.S., J.E. Marsden & R.M. Murray. 2001. J2 dynamics and formation flight. Proceedings of the 2001 AIAA Guidance, Navigation, and Control Conference, Montreal, Canada, August.

AIAA 2001-4090. Broucke, R.A. 1999. Motion near the unit circle in the three-body problem. Celest. Mech. Dynam.

Astron. 73(1): 281-290.[CrossRef] Goldstein, H. 1980. Classical Mechanics. Addison-Wesley.

Battin, R.H. 1999. 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

TAMUH. Schaub, S. R. Vadali, K. T. Alfriend, “Spacecraft Formation Flying Control Using Mean Orbital Elements”, 2000

MITG. 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

CaltechW. S. Koon, J. E. Marsden and R. M. Murray, “J2 DYNAMICS AND FORMATION FLIGHT”, AIAA 2001-4090

UMichV.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): 693-704.

Univ. SurreyY. Hashida, P. Palmer , “Epicyclic Motion of Satellites Under Rotating Potential”, Journal of Guidance, Control and

Dynamics, Vol. 25, No. 3, pp. 571-581, 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


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