final version steven cooley rich luquette greg marr scott starin flight dynamics may 13-17, 2002...

Final Version

Steven CooleyRich Luquette

Greg MarrScott Starin

Flight Dynamics

May 13-17, 2002

Micro-Arcsecond X-ray Imaging Mission, Pathfinder (MAXIM-PF)

Flight Dynamics

Final Version

MAXIM-PF, May 13-17, 2002Goddard Space Flight Center

Requirements & Assumptions (1 of 2)

Phase 1

200 km +/- 5 m

5cm control

15 m Knowledge

Phase 2

5cm control

15 m Knowledge

Optics Hub S/C

Detector S/C

20,000 km +/- 5 m

FreeFlyer S/C 100-500 m separationControl to ~10 microns

Detector S/C

Optics Hub S/C

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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

Requirements & Assumptions (2 of 2)

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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)

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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

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Lissajous Orbit Option Phase 1

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Desired Characteristics

Two S/C in formation, 200 km 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

Lissajous Orbit Description(Phase 1)

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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

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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

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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)

Optics Hub

200 km

200 km

10

DistanceDetector at Obs 1

Detector at Obs 2

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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

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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

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1 dayImpulsi

ve

1 dayContinuo

us

1 WeekImpulsive

1 WeekContinuous

Total V (m/s)

0.8 1.61 0.12 0.23

Acceleration (m/s2)

N/A 1.9 e-5 N/A 3.81 e–7

Notes: (1) 200 km baseline, (2) 10 re-orientation of Detector SC

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Continuous Low Thrust Summary

Detector - Phase 1

Formation Keeping1 day1

Formation Reorientati

on1 day2,3

Formation Reorientation

(Delta)7 days2,3

Total V (m/s)

0.0864 1.61 0.23

Acceleration (m/s2)

1e-6 1.9 e-5 3.81 e-7

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.

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Lissajous Orbit Option

Phase 2

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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

Formation Initialization (Phase 2, 20000 km Baseline)

Detector S/C(Phase 2)

Optics Hub S/C 20,000 km

FreeFlyer S/C

500 m

200 km

Detector S/C (Phase 1)

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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

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Formation Re-Orientation

Free Space Analysis (Detector, Phase 2)

1 dayImpulsi

ve

1 dayContinuo

us

1 WeekImpulsive

1 WeekContinuous

Total V (m/s)

0.8 e2 1.61 e2 1.2 e1 2.31 e1

Acceleration (m/s2)

N/A 1.9 e-3 N/A 3.81 e-5

Notes: (1) 20000 km baseline, (2) 10 re-orientation

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Detector - Phase 2


Formation Reorientatio

n1 day2,3


(Delta)7 days2,3

Total V (m/s)

0.95 1.61 e2 2.31 e1

Acceleration (m/s2)

1.1 e-5 1.9 e-3 3.8 e-5

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.

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FreeFlyer - Phase 2


Formation Reorientatio

n 1 day2,3


(Delta)7 days2,3

Total V (m/s)

0.0864 4.1 e-3 6 e-4

Acceleration (m/s2)

1e-6 4.7 e-8 1 e-9

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.

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DeltaV Analysis (All Phases)

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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

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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)

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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

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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

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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)

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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

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SupplementaryMaterial

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Miscellany

Satellite Operators Should employ strategies to balance the fuel usage amongst all the SC in the Formation

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SupplementaryMaterial – Lissajous Orbit

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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

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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”

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Sample Lissajous Orbit Delta-V Budgets

Direct TransferC3 = -0.677 km2/kg2

Lissajous 800(y-amplitude ~ 800K

km)

Direct TransferC3 = -0.677 km2/kg2


km)

Transfer with Phasing Loops

andLunar FlybyC3 = -2.6 km2/kg2


km)

Correct Delta Inaccuracy

50 m/s 50 m/s 20 m/s

Phasing Loops n/a n/a 50 m/s

Final Perigee Correction

n/a n/a 15 m/s

Midcourse Corrections 5 m/s 5 m/s 5 m/s

Lissajous Insertion 2 m/s 108 m/s 5 m/s

Lunar Shadow Avoidance

N/A 10 m/s per yr 10 m/s per yr

Trajectory Maintenance

4 m/s per yr 4 m/s per yr 4 m/s per yr

Total, 5 years 77 m/s 233 m/s 165 m/s

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

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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

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Libration Point Trajectory Manifolds

L1 L2L3

L5

L4

Y

zeclipticnorthpole

xview from the

ecliptic north pole

~1.5 x106 km

Earth/Moon

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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)

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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)

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Small Lissajous / Direct Transfer




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Small Lissajous / Lunar Gravity Assist

•Y-Amp ~ 200k

•Z-Amp ~ 300k




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Triana (L1 Lissajous Orbit) DSN/USN Support Requirements

(Example from Triana Peer Review)

Mission Phase Tracking RequirementsTTI => TTI + 6 hrs All DSN 26-m stations with view of Triana will be scheduled

for continuous support;USN station may be prime for first contact, depending on

TTI longitude

TTI + 6 hrs => TTI + 72 hrs DSN prime, continuous support from 26-m and 34-m sites;USN as backup

TTI + 72 hrs => TTI + 144 hrs At least 4 hrs per day of range and range ratedata from USN sites in alternating hemispheres;At least 2 hrs per day of range and range rate

data from DSN sites in alternating hemispheres

TTI + 144 hrs => LOI + 6 weeks At least 4 to 6 hrs per day of range and range ratedata from USN sites in alternating hemispheres;

DSN as backup

After LOI + 6 weeks 16 hrs of range rate and 20 minutes of range data per dayfrom USN sites, alternating between hemispheres

Note: Since USN had planned Dedicated Triana Support, Some of these Requirements may be Overkill. Data Courtesy Greg Marr.

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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

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SupplementaryMaterial – Heliocentric Orbit

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MAXIM-PF Range From Earth(Heliocentric Orbit Option)

Reference: August 99 MAXIM IMDC Study

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MAXIM-PF Trajectory in Solar Rotating Coordinates(Heliocentric Orbit Option)

Reference: August 99 MAXIM IMDC Study

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Heliocentric Orbit Option

Phase 1

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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

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Desired Characteristics

Two S/C in formation, 200 km 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 200 km from Sun

Driving Requirements

Time allocated for reorienting the SC-to-SC line

SC-to-SC line remains inertially fixed during observation

Heliocentric Orbit Description(Phase 1)

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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

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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

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SupplementaryMaterial – General

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Formation Re-OrientationFree Space Analysis Revisited

Impulsive Analysis One Burn at Observation 1 (Magnitude V1) & One Burn (Same Magnitude,

Opposite Direction) at Observation 2 V1 (m/s) = Distance (m) / t0 (s) Total V (m/s) = 2 V1 = 2 * Distance (m) / t0(s) (t0 is time to re-orient)

Continuous Thrust Analysis Acceleration is a positive constant (magnitude A) from t = 0 to t = t0/2 Acceleration is a negative constant (same magnitude) from time t0 /2 to

time, t0 At time, t=0 & t = t0, Velocity is 0 At time, t= t0/2, Velocity reaches a maximum of V2 = 2 V1 = 2 * Distance

/t0 Total V (m/s) is Twice that of Impulsive Case: 4 * Distance / t0 A = Distance / (t0/2)2 = 2 V2 / t0 = 4 V1 / t0

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References (1 of 3)

Marr, Cooley, Franz, Roberts, Triana Trajectory Design Peer Review, 2001.

Cuevas, Newman, Mesarch, Woodard, An Overview of Trajectory Design Operations for the MAP Mission, AIAA 2002-4425, AIAA Astrodynamics Specialist Conference, August 2002.

Mesarch, Andrews, The Maneuver Planning Process for the MAP Mission, AIAA 2002-4427, AIAA Astrodynamics Specialist Conference, August 2002.

Mesarch, Contingency Planning for the MAP Mission, AIAA 2002-4426, AIAA Astrodynamics Specialist Conference, August 2002.

L. Newman, Constellation-X Reference Mission Description Document, Govind Gadwal, ed., 2002.

Mesarch, Vaughn, Concha, Flight Dynamics IMDC Study for the MAXIM Mission, August 1999.

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References (2 of 3)

Cooley, Marr, Starin, Petruzzo, Flight Dynamics IMDC Study for the Fresnel Lens Gamma Ray Telescope, January 2002.

Cooley, Marr, Starin, Petruzzo, Flight Dynamics IMDC Study for the Fresnel Lens Gamma Ray PathFinderTelescope, January 2002.

Concha, Cooley, Folta, Hamilton, Flight Dynamics IMDC Study for the Stellar Imager, July 2001.

Markley, Maxim Mission White Paper, January 31, 2002. Grady, MAXIM Pathfinder Mission Concept Design

Powerpoint Presentation, MPF Mission Definition Team Meeting, September 18, 2000.

Wertz, ed., Spacecraft Mission Analysis and Design, 3rd Edition, Microcosm, 1999.

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References (3 of 3)

Luquette, Sanner, A nonlinear approach to spacecraft formation control in the vicinity of a collinear libration point, AAS001-330, 2001.

final version steven cooley rich luquette greg marr scott starin flight dynamics may 13-17, 2002...

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