Dnamical Sstems and Space Mission Design Wang Koon, Martin Lo, Jerrold Marsden and Shane Ross Wang Sang Koon Control and Dnamical Sstems, Caltech koon@cds.caltech.edu
Acknowledgements H. Poincaré, J. Moser C. Conle, R. McGehee C. Simó, J. Llibre, R. Martinez E. Belbruno, B. Marsden, J. Miller G. Gómez, J. Masdemont K. Howell and the Purdue group
Introduction: Hohmann Transfer Traditional transfer from to Moon is b Hohmann transfer. 2 bod Keplerian ellipse from to Moon. Need 2 Vs. Transfer Ellipse V2 V1 Moon's
Sun-Perturbed E-M Transfer with Ballistic Capture In 1991, Muses-A did not have enough propellant to reach Moon b Hohmann transfer. Belbruno/Miller designed a Sun-perturbed -to-moon transfer with ballistic capture at Moon. Similar techniques used b Japanese team to save mission. (from Belbruno and Miller [1993])
Sun-Perturbed E-M Transfer with Ballistic Capture We provide a theoretical basis and a numerical procedure for constructing such ballistic capture transfer. B considering Sun--Moon-SC 4-bod sstem as 2 coupled 3-bod sstems. Better seen in Sun- rotating frame. Inertial Frame Sun- Rotating Frame V V Moon's x Ballistic Capture L 1 Sun Moon's x Ballistic Capture
Introduction: Coupled 3-Bod Sstems Find position/velocit for spacecraft integrating forward, SC guided b -Moon manifold and get ballisticall captured at Moon; integrating backward, SChugsSun- manifolds with a twist and return to. Based on Koon, Lo, Marsden and Ross [2000]. Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon's orbit x
Planar Circular Restricted Three-Bod Problem Sun and total mass normalized to 1; rotate about center of mass, with angular velocit equal to 1. 3rd bod has infinitesimal mass. Rotating coordinate sstem with origin at center of mass, SandEfixedat( µ, 0) and (1 µ, 0). Has 3 unstable equilibrium points on S-E line. 1 L 4 spacecraft 0.5 (rotating frame) 0-0.5 L 3 S L 1 E m S = 1 - µ m E = µ -1 L 5 s orbit -1-0.5 0 0.5 1
PCR3BP: The Flow near L 1 and For energ value just above that of, Hill s region contains a neck about L 1 &. SC canmaketransition through these equilibrium regions. 4 tpes of orbits: periodic, asmptotic. transit & nontransit. 1 Forbidden Region Exterior Region (rotating frame) 0.5 0-0.5 Interior Region S L 1 E (rotating frame) -1-1 -0.5 0 0.5 1 x (rotating frame) Capture Region x (rotating frame)
PCR3BP: Invariant Manifold as Separatrix Invariant manifold tubes act as separatrices for the flow in equilibrium regions. Those inside the tubes are transit orbits. Those outside the tubes are non-transit orbits. Forbidden Region Stable Manifold Unstable Manifold Sun (rotating frame) Stable Manifold Forbidden Region Periodic Unstable Manifold x (rotating frame)
Schematic of Shoot the Moon Trajector, 2 portions of the trajector in Sun- rotating frame: Sun- libration point portion. Lunar ballistic capture portion. Maneuver ( V) at Patch Point Sun--S/C Sstem -Moon-S/C Sstem Sun Moon x
Lunar Ballistic Capture Portion Stable manifold tube provides temporar capture mechanism b second primar. Stable manifold tube of periodic orbit around guides spacecraft towards ballistic capture b Moon. -0.25-0.3 Transit s Ballistic Capture Tube of Transit s -0.35 Ballistic Capture Initial Condition Poincare Section. r -0.4 Poincare Cut of Stable Manifold orbit -0.45-0.5 2.2 2.3 2.4 2.5 2.6 2.7 2.8 r Forbidden Region x Moon
Lunar Ballistic Capture Portion B saving (on-board) fuel for lunar ballistic capture portion, this design uses less fuel than -to-moon Hohmann transfer. Trajector Begins Inside Tube (rotating frame) Moon Ends in Ballistic Capture Passes through Equilibrium Region x (rotating frame)
Sun- Libration Point Portion Pick initial condition outside Poincaré cut, backward integrate to produce a trajector: hugs unstable manifold back to region with a twist, hugs stable manifold back towards. Poincare Section Backward Targeting Portion Using "Twisting" Near Tubes (Sun- rotating frame). 0.01 0.01 0.02 0.03 0.04 0.05 0.06 0 Moon's Sun- Unstable Manifold Cut Initial Condition Sun Moon's 0 0.001 0.002 0.003 0.004 0.005 0.006 (Sun- rotating frame) x
Sun- Libration Point Portion Amount of twist depends sensitivel on distance from manifold, can change dramaticall with small V. With small V, can target back to (200 km) parking orbit. Poincare Section Targeting Using "Twisting" P -1 (q 2 ) P -1 (q 1 ) q 1 q 2 -velocit Pre-Image of Strip S Stable Manifold Strip S Unstable Manifold -position Sun orbit -position x-position
Connecting Two Portions Recall: Sun--Moon-SC 4-bod sstem as 2 coupled 3-bod sstems. Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon's orbit x
Connecting Two Portions Var phase of Moon until -Moon manifold cut intersects Sun- manifold curve. (Sun- rotating frame). 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Poincare Section in Sun- Rotating Frame -Moon Stable Manifold Cut with Moon at Different Phases B Sun- Unstable Manifold Cut 0 0.001 0.002 0.003 0.004 0.005 0.006 (Sun- rotating frame) A Stable Manifold Tube in -Moon Rotating Frame Stable Manifold Tube A x B orbit Moon
Connecting Two Portions Pick initial condition in region in interior of green curve but in exterior of red curves. Sun- Unstable Manifold Cut Initial Condition Backward Targeting Portion Using "Twisting" (Sun- rotating frame). 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Initial Condition -Moon Stable Manifold Cut Lunar Capture Portion Sun Moon's orbit 0 0.001 0.002 0.003 0.004 0.005 0.006 (Sun- rotating frame) Poincare Section x
Connecting Two Portions With slight modification ( a 34 m/s V at patch point), this produces a solution in bicircular 4-bod problem. Since capture at Moon is natural (zero V ), amount of on-board fuel needed is lowered (b about 20%). Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon s x
x Arguments for Coupled 3-Bod Model Outside Moon s sphere of influence (20,000 km), can neglectmoon s perturbation on S-E-SC 3-bod sstem, can use Sun- invariant manifold structure. Midcourse V is performed as SC is entering s sphere of influence (900,000 km), can neglect Sun s perturbation on E-M-SC 3-bod sstem, can use -Moon manifold structuere for capture. Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon's orbit
Future Work Using differential correction, can utilize this trajector as initial guess to find3-dimensional Shoot the Moon trajector, with full solar sstem model. Optimize trajector b appling optimal control (e.g., COOPT), with continuous (low) thrust. Develop procedure for coupling multiple 3-bod sstems, which will aid in design of innovative space missions in understanding subtle non-keplerian transport throughout solar sstem.
References and Other Informations Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, vol. 10, (2000) pp. 427-469. http://www.cds.caltech.edu/ marsden/ Clip on current issue of http://ojps.aip.org/chaos/ Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross Shoot the Moon. Email: koon@cds.caltech.edu