Libration Orbit Mission Design: Applications of Numerical and Dynamical Methods

Libration Orbit Mission Design: Applications Of Numerical And Dynamical Methods

David Folta and Mark Beckman NASA - Goddard Space Flight Center

Libration Point Orbits and Applications June 10-14, 2002, Girona, Spain Agenda • NASA Enterprises • Challenges • Historical and future missions • Libration mission design via numerical and dynamical methods • Applications ¾Direct transfer ¾Low thrust ¾Servicing ¾Formations: Constellation-X and Stellar Imager ¾Conceptual studies • Improved tools • Conclusions NASA Themes and Libration Orbits NASA Enterprises of Space Sciences (SSE) and Earth Sciences (ESE) are a combination of several programs and themes SSE ESE SEU

SEC Origins

• Recent SEC missions include ACE, SOHO, and the L1/L2 WIND mission. The Living With a Star (LWS) portion of SEC may require libration orbits at the L1 and L3 Sun-Earth libration points. • Structure and Evolution of the Universe (SEU) currently has MAP and the future Micro Arc-second X-ray Imaging Mission (MAXIM) and Constellation-X missions. •Space Sciences’ Origins libration missions are the Next Generation Space Telescope (NGST) and The Terrestrial Planet Finder (TPF). • The Triana mission is the lone ESE mission not orbiting the Earth. • A major challenge is formation flying components of Constellation-X, MAXIM, TPF, and Stellar Imager. Future Libration Mission SSE and ESE Challenges

¾ Orbit ¾Biased orbits when using large sun shades ¾Shadow restrictions ¾Very small amplitudes ¾Reorientation to different planes and Lissajous classes ¾Rendezvous and formation flying ¾Low thrust transfers ¾Quasi-stationary orbits ¾Earth-moon libration orbits

¾Equilateral libration orbits: L4 & L5 ¾ L3 orbits ¾Other ¾Servicing of resources in libration orbits ¾Minimal fuel ¾Constrained communications ¾Limited ∆V directions ¾Solar sail applications ¾Continuous control to reference trajectories ¾Tethered missions ¾Human exploration Historical and Future Missions in Libration Orbits Mission Location / Amplitudes Launch Total ∆V Transfer Type (Ax, Ay, Az) Year Allocation Type (m/s) ISEE-3 L1Halo/L2/Comet 175000, 660670, 120000 1978 430 Direct 1st mission WIND+ L1 – Lissajous 10000, 350000, 250000 1994 685 Multiple Lunar Gravity Assist SOHO L1 – Lissajous 206448, 666672, 120000 1995 275 Direct

ACE L1 – Lissajous 81775, 264071, 157406 1997 590 Direct 1st small amplitude (Constrained) MAP L2-Lissajous n/a, 264000, 264000 2001 127 Single Lunar 1st L2 Mission Gravity Assist Genesis L1-Lissajous 250000, 800000, 250000 2001 540 Direct

Triana L1-Lissajous 81000, 264000, 148000 # 620 Direct Launch Constrained NGST* L2-Quasi-Periodic 290000, 800000, 131000 # 150 Direct Lissajous SPECs L2-Lissajous 290000, 800000, 131000 # Tbd Direct Tethered Formation MAXIM L1 – Lissajous Large Lissajous # # Direct Formation Constellation-X L2 – Lissajous Large Lissjaous 2010 150-250 Single Lunar Loose Formation Gravity Assist Darwin L1-Lissajous 300000, 800000, 350000 2014 # # Large Lissajous Stellar Imager L2 – Lissajous Large Lissajous 2015 # Direct ~30 S/C Formation TPF L2 – Lissajous Lissajous # # # Formation? + WIND was originally design as an L1 orbiter, * NGST Concept only, # is tbd Numerical Targeting •High Fidelity Perturbation Theory Modeling •Intuitive Numerical Targeting Via DC & Optimization •Spacecraft Modeling for Attitude and Maneuvers •Currently Single Trajectory Design •Limited Set of Initial Conditions •Operational Software •Can perform Monte Carlo and parametric scan studies GSFC’s Swingby Program Astrogator • Astrogator is a COTS mission design tool designed as an add- on module to Satellite Tool Kit by Analytical Graphics Inc. • Astrogator was originally designed based upon Swingby mathematical specifications • Astrogator combines the trajectory design capabilities of Swingby with the visualization and computational capabilities of STK • MAP was the first LPO mission to use Astrogator operationally Example of a Forward Shooting Method of a Libration Point Orbit Direct Transfer • Initial escape trajectory towards a libration point with the Moon at the appropriate geometry • Target the anti-Sun right ascension and declination at the appropriate launch epoch • Target velocity of the Sun- Earth rotating coordinate x-z plane crossing condition to achieve a quasi-libration orbit, L2 x-axis velocity ~ 0 • Target a second x-z plane crossing velocity which yields a subsequent x-z plane crossing, then target to a one period revolution • Vary the launch injection C3 and parking orbital parameters

XY Projection Lissajous Orbit XZ Projection

Lunar Orbit

YZ Projection

Solar Rotating - Ecliptic Plane Projection Example of a Forward Shooting Method of a Libration Point Orbit Lunar Gravity Assist

• Target the moon at the appropriate encounter epoch to achieve an anti-sun outgoing asymptote vector • Target the lunar b-plane condition to achieve a gravity assist and a perpendicular sun- earth rotating coordinate x-z plane crossing • Target x-z plane crossing velocities which yields a second x-z plane crossing and target to a one period revolution at L2 • Re-target lunar b-plane conditions to achieve the correct orientation of the lissajous pattern with respect to the ecliptic plane

Lissajous Orbit Target goals may include; Lunar Orbit •Time • B-plane conditions. SUN Three Phasing Loops L2 • Libration sun-earth line crossing conditions • Mathematical computation (eigenvectors) Lunar Swingby • Targets may be single event string, nested, Or branched to allow repeatable targeting • Maneuvers can be inserted were appropriate Solar Rotating - Ecliptic Plane Projection Numerical and Dynamical Targeting using Generator Sample Generator Menus • Qualitative Assessments • Global Solutions • Time Saver / Trust Results •Robust • Helps in choosing numerical shooting methods

Utility Input Output Phase (Generic Orbit) User Data Control Angles For Lissajous Lissajous Universe And User Data Patch Point And Lissajous Orbit Monodromy (Periodic Orbit) Universe And Lissajous Output Fixed Points And Stable And Unstable Manifold Approximations Manifold Universe And Monodromy Output 1-Dimensional Manifold Transfer Universe, User Selected Patch Transfer Trajectory From Earth Points, Manifold Output To L1 Or L2

NGST Mission Applications

•Mission: NGST Is Part of Origins Program Artist Conception •Designed to Be the Successor to the Hubble Space Telescope •Launch: ~2010, Direct Transfer •Lissajous Orbit about L2:

400,000km > Ay < 800,000km •Spacecraft: Three Axis Stabilized, ‘Star’ Pointing •Notable: Observations in the Infrared Part of the Spectrum •Important That the Telescope Be Kept at Low Temperatures (~30K): Large Solar Shade/Solar Sail Dynamical System Approach to Generate the Lissajous • Input desired libration orbit parameters, e.g. 800,000 km y-amplitude liss orbit • Class I configuration, epoch • Provides invariant manifolds • Select the manifold that best meets mission and parking orbit conditions, shadows, and minimum insertion ∆V

5 LIS S AJ OUS -RLP F RAME 5 x 10 x 10

5 5 0 0 Projections of All Invariant Manifolds z [km]

y [km] -5

-5 for Time Interval 5 5 5 x 10 0 0 5 -5 -5 x 10 -5 0 5 y [km] x [km] x [km] 5 5 x 10 5 x 10 x 10

5 5

0 0 z [km] z [km]

-5 -5 S EV ANGLE 40

30 -5 0 upper 5 -5 0 5 ] g e 5 5 +x-axis d[ 20 x [km] y [km] x 10 x 10 V E S 10 lowe r 0 0 200 400 600 800 1000 1200 1400 1600 1800 TOF [da ys ] 40

30 upper +y-axis ] g e d 20 [ V E S 10 lowe r 0 -4 -3 -2 -1 0 1 2 3 x [km] 5 x 10 Application of Dynamical System Approach Results • Input Into Numerical Targeting Scheme With Full Perturbation Modeling • Direct Shooting Method to Meet Trajectory Goals Via DC • ∆V Corrections of 1 cm/s at Injection, 0.3 cm/s at LOI NGST Transfer to Libration Orbit Via Low-Thrust •Low thrust trajectory solutions exist for the collinear libration points •No lunar gravity assist is used •The trajectory generally consists of spiraling out to lunar orbit with periods of thrusting and coasting and targeting the post-lunar leg to insert into the periodic orbit using coast times •The thrust can be along the velocity vector or at an angle to it to achieve maximum efficiency •Results show possible extensive time-of-flight amplified by the mass •A conceptual nuclear powered electric propulsion provides 1.2 N of thrust at an Isp of 4800 sec •Transfer trajectory takes 510 days of continuous thrust followed by an 85-day coast Servicing of Libration Missions •Return to LEO to be serviced at the ISS. •Unstable manifolds that pass near the Earth are used as initial estimates. •The transfer is targeted to meet inclination and dynamic pressure constraints •Drag Mechanism would be used to aerocapture at the Earth. •Retarget perigee to the original 107 km altitude at first apogee. •After several perigees, altitude remains constant and the spacecraft is aerocaptured within 4 days. Reflective Outgoing and Incoming Transfers Aerobrake Design using a 1000 m2 drag area Outgoing Transfer

Solar Rotating View GCI View Constellation - X •Challenge of placing 4 spacecraft into a formation in libration orbit

•Launch two spacecraft at a time

•Achieve a small Lissajous via a lunar gravity assist

•Targeted using DC with phasing loops and deterministic ∆Vs to align both spacecraft in mission orbit Constellation-x Lunar Gravity Assist Trajectory

•Each spacecraft independently controlled Lunar orbit

3.5 Phasing Loops, Final mission ~1 week period each lissajous orbit •Total Mission ∆V Earth Sun-Earth line L2 S/C #1 = 157m/s, S/C#2 = 151m/s

Cruise phase •B-plane Targets Lunar Swingby, S/C #1: B.T = 13581km, B.R = 20894km ~1 month after launch S/C #2: B.T = 13838km, B.R = 20910km Constellation - X

Spacecraft separation distances

Launch though libration orbit

Constellation X L2 Mission SC1/SC2 Position Difference Epoch: 2000/12/14 08:00:36 at Perigee 1 (P1) in Phasing Loop SC1 and SC2 Separate at P1 Launch though gravity assist 22500 ) 20000 Constellation X L2 Mission SC1/SC2 Position Difference 17500 Epoch: 2000/12/14 08:00:36 at Perigee 1 in Phasing Loop Phasing Loop Period with Maneuvers 15000 22500 12500 ) 20000 10000 P3 DV 17500 7500 15000 P2 DV 5000 12500

Position Magnitude Difference (km 2500 10000 0 0 200 400 600 800 1000 1200 7500 Elapsed Time from Epoch (Days) 5000 P1 DV Swingby

PositionMagnitude Difference (km 2500

0 0 5 10 15 20 25 30 Elapsed Time from Epoch (Days) Deeptail Conceptual Study

Trajectories derived from a standard libration orbit

Notes: Launch C3 = -0.59 Total mission duration = 375 days Total mission ∆V = 333 m/s for mother ship

Red represents the trajectory in the Geotail region ( 0 to 20 deg)

∆V increments of 10m/s Stellar Imager • SI is a concept for a space-based, uv-optical interferometer. The leading concept for SI is a 500-meter diameter fizeau-type interferometer composed of 30 small drone satellites that reflect incoming light to a hub satellite. • Focal lengths of both 0.5 km and 4 km are being considered. This would make the radius of the sphere either 1 km or 8 km. • Perturbation models generate lissajous mission orbit monodromy matrix for accurate modeling and STM for propagation

SI Lissajous Orbit About L2

6 5 x 10reference for hub satellite x 10 SI Geometry 1 1

0.5 0.5

0 0 z y

-0.5 -0.5

-1 -1 0 0.5 1 1.5 2 0 0.5 1 1.5 2 x 6 x 6 5 x 10 5 x 10 x 10 x 10 1

1 0.5

0 0 z z

-0.5 -1 1 2 6 0 x 10 1 6 -1 x 10 -1 -0.5 0 0.5 1 y -1 0 x y 6 x 10 Stellar Imager • Three different scenarios make up the SI formation control problem; maintaining the Lissajous orbit, slewing the formation, and reconfiguring • Using a LQR with position updates, the hub maintains an orbit while drones maintain a geometric formation

The magenta circles represent drones at the beginning of the simulation, and the red circles represent drones at the end of the simulation. The hub is the black asterisk at the origin. Formation 0.5 SI Slewing Geometry0.5 Formation DV Cost per slewing maneuver 0 0 y (km) z (km) Focal Slew Hub Drone 2 Drone 31 Length Angle ∆V (m/s) ∆V (m/s) ∆V (m/s) (km) (deg) -0.5 -0.5 -0.5 0 0.5 -0.5 0 0.5 0.5 30 1.0705 0.8271 0.8307 Dronesx (km) at beginning x (km) 0.5 90 1.1355 0.9395 0.9587 4 30 1.2688 1.1189 1.1315 0.5 4 90 1.8570 2.1907 2.1932 0.5

0 0 z (km) z (km) -0.5 0.5 0.5 0 -0.5 0 -0.5 0 0.5 y (km) -0.5 -0.5 x (km) y (km) Improved Tools

• Utilize capabilities of numerical and dynamical methods. The idea being to merge the best of current targeting, optimizing, and control applications. • Include search methods and optimization to numerical and dynamical approaches • Additionally, these tools must be able to interface with one another

Intelligent / Auto Intelligent systems Systems Reference and Formation Optimization Tools Controllers Generator & Dynamical Approaches Optimization Capabilities LTOOL Current ----- Future Basic Numerical Design Capabilities Swingby & Astrogator Conclusions

•Trajectory design in support libration missions is increasingly challenging as more constrained mission orbits are envisioned in the next few decades.

•Software tools for trajectory design in this regime must be further developed to incorporate better understanding of the solution space, improving the efficiency, and expand the capabilities of current approaches.

•Improved numerical and dynamical systems must offer new insights into the natural dynamics associated with the multi-body problem and provide to methods to use this information in trajectory design.

•The goal of this effort is the blending of analysis from dynamical systems theory with the well-established NASA Goddard software programs such as Swingby to enhance and expand the capabilities for mission design and to make trajectories more operationally efficient.