1. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Dynamics and Control of Formation Flying Satellites by S. R. Vadali NASA Lunch & Learn Talk August 19, 2003

2. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Outline lFormation vs. Constellation lIntroduction to Orbital Mechanics lPerturbations and Mean Orbital Elements lHill’s Equations lInitial Conditions lA Fuel Balancing Control Concept lFormation Establishment and Maintenance lHigh-Eccentricity Orbits lWork in Progress lConcluding Remarks

3. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Global Positioning System (GPS) Constellation No Inclination Difference

4. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Formation Flying: Relative Orbits

5. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Distributed Space Systems- Enabling New Earth & Space Science (NASA) Co-observation Multi-point observation Interferometry Tethered Interferometry Large Interferometric Space Antennas Large Interferometric Space Antennas

6. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 32 optics (300  10 cm) held in phase with 600 m baseline to give 0.3 micro arc-sec 34 Formation Flying Spacecraft 1 km Optics 10 km Combiner Spacecraft 500 km Detector Spacecraft Black hole image! System is adjustable on orbit to achieve larger baselines The Black Hole Imager: Micro Arcsecond X-ray Imaging Mission (MAXIM) Observatory Concept

7. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Optics Detector Spacecraft Landsat7/EO-1 Formation Flying 450 km in-track and 50m Radial Separation. Differential Drag and Thrust Used for Formation Maintenance

8. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Motivation for Research lAir Force: Sparse Aperture Radar. lNASA and ESA: Terrestrial Planet Finder (TPF) Stellar Imager (SI) LISA, MMS, Maxim LISA, MMS, Maxim lSwarms of small satellites flying in precise formations will cooperate to form distributed aperture systems. lDetermine Fuel efficient relative orbits. Do not fight Kepler!!! lEffect of J 2 ? lHow to establish and reconfigure a formation? lBalance the fuel consumption for each satellite and minimize the total fuel.

9. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Introduction to Orbital Mechanics-1 lFormation Flying: Satellites close to each other but not necessarily in the same plane. Radial (up), x Along-Track, (y) Out-of-plane (z) Deputy ChiefDeputy Dynamics Communications Navigation Control

10. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Introduction to Orbital Mechanics-2 lOrbital Elements: Five of the six elements remain constant for the 2-Body Problem. lVariations exist in the definition of the elements. lMean anomaly:

11. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-3 lWays to setup a formation: Inclination difference. Node difference. Combination of the two. Inclination DifferenceNode Difference

12. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-4 lJ 2 Perturbation: Gravitational Potential: J 2 is a source of a major perturbation on Low-Earth satellites. Equatorial BulgePotential of an Aspherical body

13. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-5 lJ 2 induces short and long periodic oscillations and secular Drifts in some of the orbital elements l Secular Drift Rates Node: Perigee: Mean anomaly: Drift rates depend on mean a, e, and i

14. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-6 lAnalytical theories exist for obtaining Osculating elements from the Mean elements. lBrouwer (1959) lIf two satellites are to stay close, their periods must be the same (2-Body). lUnder J 2 the drift rates must match. lRequirements:

15. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-7 lFor small differences in a, e, and i lExcept for trivial cases, all the three equations above cannot be satisfied with non-zero a, e, and i elemental differences. lNeed to relax one or more of the requirements.

16. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-8 lJ 2 -invariant Relative Orbits (Schaub and Alfriend, 2001). lThis condition can sometimes lead to large relative orbits (For Polar Reference Orbits) or orbits that may not be desirable.

17. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-9 lJ 2 -invariant Relative Orbits (No Thrust Required)

18. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Orbital Mechanics-10 lGeometric Solution in terms of small orbital element differences l For small eccentricity lA condition for No Along-track Drift (Rate- Matching) is:

19. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Remarks: Our Approach lThe and constraints result in a large relative orbit for small eccentricity and high inclination of the Chief’s orbit. (J 2 -Invariant Orbits) lEven if the inclination is small, the shape of the relative orbit may not be desirable. lUse the no along-track drift condition (Rate-Matching) only. lSetup the desired initial conditions and use as little fuel as possible to fight the perturbations. End of Phase-1

20. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Hill-Clohessey-Wiltshire Equations-1  Eccentric reference orbit relative motion dynamics (2-Body) :  Assume zero-eccentricity and linearize the equations:

21. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Hill-Clohessey-Wiltshire Equations-2  HCW Equations:  Bounded Along-Track Motion Condition Velocity vector Along orbit normal z Chief y Deputy

22. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Bounded HCW Solutions lProjected Circular Re. Orbit. lGeneral Circular Re. Orbit.

23. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 PCO and GCO Relative Orbits Projected Circular Orbit (PCO)General Circular Orbit (GCO)

24. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Initial Conditions in terms of Mean Element Differences : General Circular Relative Orbit. Eccentricity Difference Inclination Difference Node Difference Mean Anomaly Difference Perigee Difference Semi-major axis Difference

25. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Simulation Model  Equations of motion for one satellite Rotating frame coordinates Inertial Relative DisplacementInertial Relative Velocity  Initial conditions: Convert Mean elements to Osculating elements and then find position and velocity.

26. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Hill’s Initial Conditions with Rate-Matching lChief’s orbit is eccentric: e=0.005 lFormation established using inclination difference only. Relative Orbits in the y-z plane, (2 orbits shown) 150 Relative Orbits

27. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Drift Patterns for Various Initial Conditions The above pattern is for a deputy with no inclination difference, only node difference. End of Phase-2

28. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Fuel Requirements for a Circular Projection Relative Orbit Formation  Sat #1and 4 have max and zero  Sat #3 and 6 have max but zero  1 and 4 will spend max fuel; 3 and 6 will spend min fuel to fight J 2. y z Snapshot when the chief is at the equator. Pattern repeats every orbit of the Chief 1 3 4 2 5 6

29. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Fuel Balancing Control Concept  Balance the fuel consumption over a certain period by rotating all the deputies by an additional rate y z Snapshot when the chief is at the equator. 1 3 4 2 5 6

30. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Modified Hill’s Equations to Account for J 2  Analytical solution  The near-resonance in the z-axis is detuned by Assume no in-track drift condition satisfied.

31. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Balanced Formation Control Saves Fuel  Ideal Trajectory  Ideal Control for perfect cancellation of the disturbance and for  Optimize over time and an infinite number of satellites

32. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Analytical Results Benefits of Rotation (Circular Projection Orbit) Fuel Balanced in 90 days

33. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Nonlinear Simulation Results Cost (m 2 /sec 3 ) Equivalent to 28 m/sec/yr/sat Equivalent to 52 m/sec/yr Formation cost equivalent to 32 m/sec/yr/sat Cost (m 2 /sec 3 ) Benefits of Rotation (Circular Projection Orbit)

34. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Nonlinear Simulation Results lOrbit Radii over one year(8 Satellites)

35. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Disturbance Accommodation lDo not cancel J 2 and Eccentricity induced periodic disturbances above the orbit rate. lUtilize Filter States lNo y-bias filter lLQR Design lTransform control to ECI and propagate orbits in ECI frame. lThe Chief is not controlled. End of Phase-3

36. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Formation Establishment and Reconfiguration lChanging the Size and Shape of the Relative Orbit. lCan be Achieved by a 2-Impulse Transfer. lAnalytical solutions match numerically optimized Results. lGauss’ Equations Utilized for Determining Impulse magnitudes, directions, and application times. lAssumption: The out-of-plane cost dominates the in-plane cost. Node change best done at the poles and inclination at the equator crossings.

37. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Formation Establishment 1 km GCO Established with 1 km PCO Established with

38. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Formation Reconfiguration 1 km, PCO to 2 km, PCO 1 km, PCO to 2 km, PCO Chief is at the Asc. Node at the Beginning.

39. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Reconfiguration Cost Cost vs. Final Phase Angle This plot helps in solving the slot assignment problem. The initial and final phase angles should be the same for fuel optimality for any initial phase angle.

40. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Optimal Assignment Objectives: (i) Minimize Overall Fuel Consumption (ii) Homogenize Individual Fuel Consumption End of Phase-4

41. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Relative Motion on a Unit Sphere Chief Deputy Unit Sphere x y ECI Relative Position Vector

42. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Relative Motion Solution on the Unit Sphere  Valid for Large Angles

43. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Analytical Solution using Mean Orbital Elements-1 n Mean rates are constant.

44. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Analytical Solution using Mean Orbital Elements-2 n Actual Relative Motion.

45. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 High Eccentricity Reference Orbits  Eccentricity expansions do not converge for high e.  Use true anomaly as the independent variable and not time.  Need to solve Kepler’s equation for the Deputy at each data output point.

46. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Formation Reconfiguration for High-Eccentricity Reference Orbits

47. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 High Eccentricity Reconfiguration Cost Cost vs. Final Phase Angle Impulses are applied close to the apogee. No symmetry is observed with respect to phase angle.

48. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Research in Progress  Higher order nonlinear theory and period matching conditions for large relative orbits.  Continuous control Reconfiguration (Lyapunov Functions).  Nonsingular Elements (To handle very small eccentricity)  Earth-moon and sun-Earth Libration point Formation Flying.

49. Texas A&M University - Dept. of Aerospace Engineering August 19, 2003 Concluding Remarks  Discussed Issues of Near-Earth Formation Flying and methods for formation design and maintenance.  Spacecraft that have similar Ballistic coefficients will not see differential drag perturbations.  Differential drag is important for dissimilar spacecraft (ISS and Inspection Vehicle).  Design of Large Near-Earth Formations in high- eccentricity orbits pose many analytical challenges.  Thanks for the opportunity and hope you enjoyed your lunch!!