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Steven Cooley Rich Luquette Greg Marr Scott Starin

Micro-Arcsecond X-ray Imaging Mission, Pathfinder (MAXIM-PF). Flight Dynamics. Steven Cooley Rich Luquette Greg Marr Scott Starin. May 13-17, 2002. 5cm control 15 m m Knowledge. 200 km +/- 5 m. Requirements & Assumptions (1 of 2). Phase 1. Detector S/C. Optics Hub S/C. Phase 2.

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Steven Cooley Rich Luquette Greg Marr Scott Starin

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  1. Micro-Arcsecond X-ray Imaging Mission, Pathfinder (MAXIM-PF) Flight Dynamics Steven Cooley Rich Luquette Greg Marr Scott Starin May 13-17, 2002

  2. 5cm control 15 mm Knowledge 200 km +/- 5 m Requirements & Assumptions (1 of 2) • Phase 1 Detector S/C Optics Hub S/C • Phase 2 Optics Hub S/C 5cm control 15 mm Knowledge 20,000 km +/- 5 m Detector S/C FreeFlyer S/C 100-500 m separation Control to ~10 microns MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  3. Requirements & Assumptions (2 of 2) • Mission Orbit • L2 Lissajous • Heliocentric “Drift-Away” • Variations on Drift Away (e.g., DROs stay closer to Earth) • Orbit Control and Knowledge Requirements • Orders of Magnitude above Current Operational Missions • Not Addressed Here • V and Acceleration Magnitude Values • Very Coarse Approximations • No Noise • CRTBP or Free Space Model • No Perturbations (Moon, Jupiter, etc.) • No Navigation Errors • Further Analysis Required MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  4. Lissajous Orbit Option • Orbit Characteristics • Quasi Orbit Period of ~6 months • Can Choose small or Large Amplitude Lissajous • No Earth Eclipses • MAXIM Adds Requirement of No Lunar Shadows (MAP) • Advantages • Spacecraft do not Drift too Far from Earth • Communications (High Data Rate Missions) • Spacecraft can be More Easily Replaced/Repaired • Important for Long Missions • Small Launch Vehicle C3 (-2.6 for Phasing Loops, -0.7 for Direct) • Disadvantages • Unstable Complicated Dynamics • Can Lose Spacecraft (e.g., Propulsion Failure) • All s/c in formation require propulsion (Operational Complexity) • Formation Keeping Costs May be Greater (Further Analysis Needed) • May Have increased variation in Formation Keeping Control Acceleration Magnitude (Harder to size thrusters) • 6 Month Transfer Time • High Thrust Propulsion System Likely Needed (Need to Correct LV Errors QUICKLY) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  5. Heliocentric Orbit Option • Orbit Characteristics • Drift Away Orbit (0.1 AU/year) • Advantages • Stable Dynamics • Simpler Operations • Potentially No Orbit Overhead Costs • Optics Hub may Not need propulsion • Relatively Short Transfer Times • May Require Less Formation Keeping Costs (?) • May be Able to Eliminate Need for High Thrust Propulsion System • Disadvantages • Higher Launch Vehicle C3 (0.4) • Drift Away Concerns MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  6. Lissajous Orbit Option Phase 1 MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  7. Lissajous Orbit Description(Phase 1) • Desired Characteristics • Two S/C in formation, 200km apart • Maintain inertial orientation of SC-to-SC line for 1 week observation • Optics Hub follows a ‘Ballistic’ lissajous orbit during Observation (the “Leader”) • Detector SC (the “Follower”) follows a shifted trajectory • For Given Observation, Position differs by a constant baseline vector • Driving Requirements • Time allocated for reorienting the SC-to-SC line • SC-to-SC line remains inertially fixed during observation MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  8. Formation Initialization • Direct Transfer (One LV with a C3 of -0.7 km2/s2) • Large ‘Halo’ Orbit • No Lunar Shadows • Max L2-Earth-Vehicle Angle  30 • Orbit Does Not “Collapse” • Detector SC is maneuvered to the shifted orbit 200 km away • Consider Initialization V as 6 Formation Re-Orientations • FreeFlyers Stay Attached to Optics Hub MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  9. Formation KeepingSolar Radiation Pressure (SRP) • Acceleration Magnitude (1 AU)  4.5 x 10-6 (1+r) A/M (m/s2) • A = Cross Sectional Area exposed to Sun (m2) • M = Mass of Spacecraft (kg) • r = Reflection Factor. (r  [0,1]) • Approximate Result for all Mission Orbits Considered • SMAD (3rd Edition, not 2nd edition) • SRP Acceleration Magnitude Differential Between 2 Spacecraft •  4.5 x 10-6 | (1+r1) (A1/m1) – (1+r2) (A2/m2)| • Assumed Dominant Term for 200 km Baseline (CRTBP model) • Assume Control Acceleration Magnitude  10-6 m/s2 Needed MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  10. Optics Hub 10 200 km 200 km Detector at Obs 2 Detector at Obs 1 Distance Formation Re-OrientationFree Space Analysis (1/4) • Preliminary “Drift-Away” Orbit Results • For “small” reorientation times (< 1 week), solar gravity has “small” effect on V costs. • Free space analysis (ie, gravity free) is a reasonable approximation for small reorientation times in a “Drift Away” • Further Study Needed (Especially for Applicability to Lissajous Orbits) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  11. Formation Re-OrientationFree Space Analysis (2/4) • Impulsive Burn Analysis • One burn after obs 1 initiates translation of detector to the obs 2 location • Magnitude: VImpulse = distance / reorientation time • Equal but opposite burn stops translation when obs 2 location is reached • Total V = 2* VImpulse • Continuous Thrust Analysis • Acceleration is constant toward obs2 location for first half of the time • Acceleration is of the same magnitude, but reversed for the remaining time • Total V (m/s) = 4*Vimpulse • Acceleration = 4*Vimpulse / reorientation time = 4*distance/ (reorientation time)2 MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  12. Formation Re-OrientationFree Space Analysis (3/4) • V Costs (both Continuous and Impulsive) • Linear Relationship with Distance • Inverse Linear Relationship with Re-Orientation Time • Control Acceleration Magnitude (Continuous) • Linear Relationship with Distance • Inverse Square Relationship with Re-Orientation Time MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  13. Formation Re-OrientationFree Space Analysis (4/4) Notes: (1) 200 km baseline, (2) 10 re-orientation of Detector SC MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  14. Continuous Low Thrust SummaryDetector - Phase 1 Notes: (1) Formation Keeping Costs Highly Dependent on SRP and thus the relative A/M ratios for the spacecraft. (2) The Formation Re-Orientation Costs are based on Free Space Calculations. This number should be multiplied by a “CorrectionFactor” > 1 to account for the L2 orbit. Low Thrust Software Needed for Future Refinements. (3) The Formation Reorientation values are considered a “delta” above the baseline Formation Keeping costs. (4) All Numbers are Coarse approximations. MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  15. Lissajous Orbit Option Phase 2 MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  16. Optics Hub S/C 20,000 km 200 km Detector S/C (Phase 1) Detector S/C (Phase 2) 500 m FreeFlyer S/C Formation Initialization(Phase 2, 20000 km Baseline) • Possible Configuration • Optics Hub has Minimal or no Propulsion • Detector SC moves to a distance of 20,000 km from Optics Hub • FreeFlyer SC Separates from Optics hub to a maximum separation of 500 m • New Baseline May Require New Class of Continuous Thrusters for Detector SC MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  17. Formation Keeping(Phase 2) • Same 10-6 m/s2 from SRP Differential Assumed • Larger Baseline  Dynamics Plays a Greater Role • Control Acceleration Magnitude Depends on • Position of SC in its Orbit • Choice of Target • Sample Mission Orbit (Calculation Purposes Only) • Optics Hub at L2 • Detector SC moves in a Circle about L2 • 20,000 km Radius • In Ecliptic Plane • Clockwise Motion (360/yr) • Circular Restricted Three Body Problem • No Other Forces modeled • Control Acceleration  10-5 m/s2 • Combined Accel Mag  1.1 x 10-5 m/s2 MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  18. Formation Re-OrientationFree Space Analysis (Detector, Phase 2) Notes: (1) 20000 km baseline, (2) 10 re-orientation MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  19. Continuous Low Thrust SummaryDetector - Phase 2 Notes: (1) Formation Keeping Costs Highly Dependent on SRP and thus the relative A/M ratios for the spacecraft. (2) The Formation Re-Orientation Costs are based on Free Space Calculations. This number should be multiplied by a “Correction Factor” > 1 to account for the L2 orbit. Low Thrust Software Needed for Future Refinements. (3) The Formation Reorientation values are considered a “delta” above the baseline Formation Keeping costs. (4) All Numbers are Coarse approximations. MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  20. Continuous Low Thrust SummaryFreeFlyer - Phase 2 Notes: (1) Formation Keeping Costs Highly Dependent on SRP and thus the relative A/M ratios for the spacecraft. (2) The Formation Re-Orientation Costs are based on Free Space Calculations. This number should be multiplied by a “CorrectionFactor” > 1 to account for the L2 orbit. Low Thrust Software Needed for Future Refinements. (3) The Formation Reorientation values are considered a “delta” above the baseline Formation Keeping costs. (4) 500 m baseline (5) All Numbers are Coarse approximations. MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  21. DeltaV Analysis (All Phases) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  22. DeltaV Summary (1 of 3) • L2 Propulsion Insertion Module • Carries All SC in Formation • Launch Vehicle Correction • Contingency • Mid-Course Correction (MCC) • Lissajous Orbit Insertion (LOI) • 200 m/s – High Thrust MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  23. DeltaV Summary (2 of 3) • Detector SC • 125 m/s High Thrust for Lissajous Stabilization and Contingencies • 25 m/s * 5 years • 32 m/s Continuous Low Thrust for Formation Keeping in Phase 1 • 1e-6 m/s2 * 1 yr • 117 m/s Continuous Low Thrust for Re-Orientation (1 day) in Phase 1 • (45 targets) * (1e-6 + 1.9 e-5) m/s2 * (1 day to reorient) * (Correction Factor of 1.5) • 1389 m/s Continuous Low Thrust for Formation Keeping in Phase 2 • 1.1 e-5 m/s2 * 4 yr • 2042 m/s Continuous Low Thrust for Re-Orientation (7 day) in Phase 2 • (45 targets) * (1.1 e-5 + 3.8 e-5) m/s2 * (7 day to reorient) * (Correction Factor of 1.5) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  24. DeltaV Summary ( 3 of 3) • Optics Hub • 125 m/s High Thrust for Lissajous Stabilization and Contingencies • 25 m/s * 5 years • FreeFlyer SC (per SC) • 100 m/s High Thrust for Lissajous Stabilization and Contingencies • 25 m/s * 4 years • 380 m/s Continuous Low Thrust for Formation Keeping (Phase 2) • 1e-6 m/s2 * 4 yr * (Correction Factor of 3) • 13 m/s Continuous Low Thrust for Re-Orientation in 1 day (Phase 2) • (45 targets) * (1 e-6 + 4.7 e-8) m/s2 * (1 day to reorient) * (Correction Factor of 3) Notes: (1) In Phase 2, the Detector SC re-orients in 1 week while the FreeFlyers re-orient in 1 day. (2) All V values for all SC do not include engineering penalties, ACS Penalties, and cant angles. (3) Formation Re-Orientation (10) values include the necessary Formation Keeping contribution. (4) Double Counting of Formation Keeping costs during a Re-Orientation used to account for formation Acquisition Costs. (5) Formation Initialization Costs not explicitly listed here MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  25. Flight DynamicsTechnologies Required • Control Law algorithm development • Improved Control Performance • Collision Avoidance • Re-Acquisition of Formation after Re-Orientation • Simulation • Continuous Thrust model • High Fidelity Force model • Relative Navigation needed • Current Ground based Orbit Determination : 5 km position knowledge MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  26. Flight DynamicsAdditional Trades to Consider • Continuous Low Thrust Transfer to L2 • Feasibility of Using Low Thrust for Lissajous Stabilization • Consider Surface Coatings on SC or Other Methods to minimize SRP Differentials • Formation Keeping Costs are a function of Both Position in Orbit and Choice of Target. By judicious choice of target sequence, Some V Optimization can be Realized. • Detailed Trajectory Design Study to Include Lissajous vs. Heliocentric Trade • Heliocentric Orbits with Better Communication • Some can be Achieved Via Only Launch Vehicle Considerations • Distant Retrograde Orbits (~200 m/s) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  27. Flight DynamicsIssues and Concerns • Continuous Thrusting will make OD more Difficult • Very Difficult to Choose Class (acceleration magnitudes) of Propulsion Systems Needed • Very Coarse Estimates of Control Acceleration Magnitudes • Different Phases of Mission • New Technology: Thrusters with Greater Range of Thrust Modulation? • Relative Orbit Position Control & Knowledge Requirements Orders of Magnitude above Current Operational Capability • Collision Avoidance • Further extensive analysis required • High fidelity simulation w/ all force perturbations and sensor/actuator noise and error • Control Law Evaluation • Continuous Low Thrust Simulations • Continuous Low Thrust Trajectory Optimization Software Needed MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  28. Supplementary Material MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  29. Miscellany • Satellite Operators Should employ strategies to balance the fuel usage amongst all the SC in the Formation MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  30. Supplementary Material – Lissajous Orbit MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  31. Sample Impulsive Re-Orientation (1 of 2)(20,000 km baseline, 10 in 7 days) • Force model • Full Ephemeris • Sun/Earth/Moon/Jupiter Point Mass • SRP • Both SC Have same A/M Ratio (Cr A/M = 0.013) • Initial Optics Hub State (ECI MJ2000) • UTC Gregorian Date: 23 Jan 2003 05:02:45.56 UTC Julian Date: 2452662.71024955 • X: -993733.7803065266900000 km Vx: -0.3149752411661734 km/sec • Y: 913746.5347422765300000 km Vy: -0.2540742769815505 km/sec • Z: 396534.8804631549300000 km Vz: -0.0421253073023613 km/sec • Initial Detector State • Offset Position by b1 = 20000*(1, 0, 0) • Identical Velocity • Final Optics Hub State • UTC Gregorian Date: 30 Jan 2003 05:02:45.56 • X: -1.1621310681357966e+006 km Vx: -0.2462221518003097 km/sec • Y: 757799.1103539797500000 km Vy: -0.2561905904275567 km/sec • Z: 369320.0722157274700000 km Vz: -0.0457073285913774 km/sec • Final Detector State • Offset Position by b2 = 20000*(cos(10),sin(10), 0) • Identical Velocity MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  32. Sample Impulsive Re-Orientation (2 of 2) • Astrogator Simulation • First Maneuver Magnitude of 7.3 m/s • Second Maneuver Magnitude of 4.4 m/s • Total Maneuver Magnitude of 11.7 m/s • Free Space Approximations (Impulsive) • Two Equal Impulsive Maneuvers of 6 m/s • Total V of 12 m/s • Comparison of Astrogator vs. Free Space • Fairly Good Agreement for this Sample Case • Small Re-Orientation Times • Astrogator’s Unequal Maneuver Size  Need for Previously Discussed “Correction Factor” MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  33. Sample Lissajous Orbit Delta-V Budgets Notes: (1) Total does not include engineering penalties,ACS Penalties, finite burn losses, cant angle, contingencies. Low Thrust not Considered here. (2) No Corresponding Chart for Heliocentric Orbit Option MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  34. Direct vs. Phasing Loop Transfer(Lissajous Orbit Option) • Phasing Loops with Lunar Swingby • More Robust • Operationally Complex • 10 Launch Days per Month (MAP 3 & 5 loop option) • Reduced C3 Costs (Not really a factor here) • Direct Transfer • Higher Risk (Little Time to React to Unforeseen Contingencies) • Simpler Operationally • 22 Launch Days per Month Constellation-X Example. Courtesy Lauri Newman MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  35. Libration Point Trajectory Manifolds L4 Earth/Moon L3 L1 L2 Y L5 x ~1.5 x106 km view from the ecliptic north pole z ecliptic north pole MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  36. Selected Lissajous Orbit Option Issues • Define Lissajous Orbit Parameters • Phasing Loop vs. Direct Transfer • Define Maximum L2-Earth-Spacecraft Angle for Communication Purposes (MAP was 10.5 degrees) • Define how sensitive Spacecraft is to Shadow in Phasing Loops • Review Lessons Learned from Other Libration Point Missions such as MAP & Triana • Insure that Thrusters are sized large enough to produce Desired DeltaV in a Reasonable time (For Transfer Trajectory) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  37. Large Lissajous / Direct Transfer projection onto ecliptic plane(ie, top view) projection onto xz plane (ie, side view) projection onto yz plane (ie, view from earth) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  38. Small Lissajous / Direct Transfer projection onto ecliptic plane(ie, top view) projection onto xz plane (ie, side view) projection onto yz plane (ie, view from earth) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  39. Small Lissajous / Lunar Gravity Assist • Y-Amp ~ 200k • Z-Amp ~ 300k projection onto ecliptic plane(ie, top view) projection onto xz plane (ie, side view) projection onto yz plane (ie, view from earth) MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  40. Triana (L1 Lissajous Orbit) DSN/USN Support Requirements(Example from Triana Peer Review) Note: Since USN had planned Dedicated Triana Support, Some of these Requirements may be Overkill. Data Courtesy Greg Marr. MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  41. MAP Lunar Shadows (L2 Lissajous Mission Orbit) • Sample Worst Cases • MAP is a small amplitude Lissajous • Moon Farther from L2 • 8 Hour Shadow with Maximum Depth of 4.5% • Moon Closer to L2 • 6 Hour Shadow with Maximum Depth of 13% Note: Data courtesy Mike Mesarch MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  42. Supplementary Material – Heliocentric Orbit MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  43. MAXIM-PF Range From Earth(Heliocentric Orbit Option) Reference: August 99 MAXIM IMDC Study MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  44. MAXIM-PF Trajectory in Solar Rotating Coordinates(Heliocentric Orbit Option) Reference: August 99 MAXIM IMDC Study MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  45. Heliocentric Orbit Option Phase 1 MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  46. Formation Initialization • One LV with a C3 of 0.4 km2/s2 • Needed to put the trajectories beyond Earth’s sphere of influence (SOI is ~106 km) Relatively Quickly • One SC is maneuvered to the shifted orbit 200 km away from the other’s origin MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  47. Heliocentric Orbit Description(Phase 1) • Desired Characteristics • Two S/C in formation, 200km apart • Maintain inertial orientation of SC-to-SC line for 1 week observation • One SC follows a circular, heliocentric orbit • Other SC follows a shifted, circular, heliocentric trajectory with orbit plane parallel to the plane of the first SC • Center of shifted trajectory lies on the Sun-target line 200km from Sun • Driving Requirements • Time allocated for reorienting the SC-to-SC line • SC-to-SC line remains inertially fixed during observation MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  48. Formation Keeping (1/2)(Heliocentric Orbit, Phase 1, 200 km Baseline) • Apply control accelerations continuously to maintain the inertial orientation of the SC-to-SC line • ~0.01 m/s per week • Only Solar Gravity modeled • Circular Earth Orbit about Sun • SRP Differential Acceleration not considered here (Very Important Term) • Maximum control accelerations are needed when the trajectories are coplanar (it’s counter-intuitive) • 0.8 x 10-8 to 1.6 x 10-8 m/s2 • 8 to 16 micro-newton thrust for a 1000 kg SC Control acceleration magnitude -vs- time since station-keeping starts MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  49. Formation Keeping (2/2)(Heliocentric Orbit, Phase 1, 200 km Baseline) • Control Acceleration Magnitude Depends on • Position of SC in its Orbit • Choice of Target • Control Acceleration Magnitude Varies (Approximately) Linearly with Baseline Assuming: • For Our Range of Baselines • Ecliptic Target with RA=DEC=0 • Only Solar Gravity modeled • Circular Orbit MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

  50. Supplementary Material – General MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

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