Invariant Manifolds, Material Transport and Space Mission Design

Transcription

1 C A L T E C H Control & Dynamical Systems Invariant Manifolds, Material Transport and Space Mission Design Shane D. Ross Control and Dynamical Systems, Caltech Candidacy Exam, July 27, 2001

2 Acknowledgements W. Koon, M. Lo, J. Marsden H. Poincaré, J. Moser C. Conley, R. McGehee C. Simó, J. Llibre, R. Martinez E. Belbruno, B. Marsden, J. Miller G. Gómez, J. Masdemont K. Howell, B. Barden, R. Wilson L. Petzold, S. Radu 2

3 Outline Theme Using dynamical systems theory for understanding solar system dynamics and identifying useful orbits for space missions. 3

4 Outline Theme Using dynamical systems theory for understanding solar system dynamics and identifying useful orbits for space missions. Current research importance development of some NASA mission trajectories, such as the Genesis Discovery Mission to be launched Monday 3

5 Outline Theme Using dynamical systems theory for understanding solar system dynamics and identifying useful orbits for space missions. Current research importance development of some NASA mission trajectories, such as the Genesis Discovery Mission to be launched Monday of current astrophysical interest for understanding the transport of solar system material (eg, how ejecta from Mars gets to Earth, etc.) 3

6 Genesis Discovery Mission Genesis will collect solar wind samples at the Sun- Earth L1 and return them to Earth. It was the first mission designed start to finish using dynamical systems theory. Moon's Orbit Return Trajectory to Earth Transfer to Halo Earth z (km) 0 1E+06 Halo Orbit Sun y (km) E+06 0 x (km) Y Z X 4

7 Previous Work Topics: solar system dynamics (eg, dynamics of comets) 5

8 Previous Work Topics: solar system dynamics (eg, dynamics of comets) the role of the three and four body problems 5

9 Previous Work Topics: solar system dynamics (eg, dynamics of comets) the role of the three and four body problems space mission trajectory design 5

10 Previous Work Topics: solar system dynamics (eg, dynamics of comets) the role of the three and four body problems space mission trajectory design Petit Grand Tour of Jupiter s moons 5

11 Previous Work Topics: solar system dynamics (eg, dynamics of comets) the role of the three and four body problems space mission trajectory design Petit Grand Tour of Jupiter s moons low energy transfer from Earth to Moon 5

12 Previous Work Topics: solar system dynamics (eg, dynamics of comets) the role of the three and four body problems space mission trajectory design Petit Grand Tour of Jupiter s moons low energy transfer from Earth to Moon Earth-Moon L1 Gateway station 5

13 Previous Work Topics: solar system dynamics (eg, dynamics of comets) the role of the three and four body problems space mission trajectory design Petit Grand Tour of Jupiter s moons low energy transfer from Earth to Moon Earth-Moon L1 Gateway station optimal control 5

14 Previous Work Topics: solar system dynamics (eg, dynamics of comets) the role of the three and four body problems space mission trajectory design Petit Grand Tour of Jupiter s moons low energy transfer from Earth to Moon Earth-Moon L1 Gateway station optimal control trajectory correction maneuvers for Genesis 5

15 Jupiter Comets We consider the historical record of the comet Oterma from 1910 to 1980 first in an inertial frame then in a rotating frame a special case of pattern evocation similar pictures exist for many other comets 6

16 Jupiter Comets Rapid transition: outside to inside Jupiter s orbit. Captured temporarily by Jupiter during transition. Exterior (2:3 resonance) to interior (3:2 resonance). Oterma s orbit 2:3 resonance Second encounter Jupiter y (inertial frame) START Sun 3:2 resonance END First encounter Jupiter s orbit x (inertial frame) 7

17 Viewed in Rotating Frame Oterma s orbit in rotating frame with some invariant manifolds of the 3-body problem superimposed. Homoclinic Orbits Oterma's Trajectory Oterma y (rotating) 1980 Sun L 1 Jupiter L 2 y (rotating) 1910 Jupiter Heteroclinic Connection x (rotating) Boundary of Energy Surface x (rotating) 8

18 Viewed in Rotating Frame oterma-rot.qt 9

19 Three-Body Problem Circular restricted problem the two primary bodies move in circles; the much smaller third body moves in the gravitational field of the primaries, without affecting them 10

20 Three-Body Problem Circular restricted problem the two primary bodies move in circles; the much smaller third body moves in the gravitational field of the primaries, without affecting them the two primaries could be the Sun and Earth, or Earth and Moon, or the Sun and Jupiter, etc. the smaller body could be a spacecraft, comet, or asteroid 10

21 Three-Body Problem Circular restricted problem the two primary bodies move in circles; the much smaller third body moves in the gravitational field of the primaries, without affecting them the two primaries could be the Sun and Earth, or Earth and Moon, or the Sun and Jupiter, etc. the smaller body could be a spacecraft, comet, or asteroid we consider the planar and spatial problems there are five equilibrium points in the rotating frame, places of balance which generate interesting dynamics 10

22 Three-Body Problem 3 unstable points on line joining two main bodies L 1,L 2,L 3 2stablepointsat±60 along the circular orbit L 4,L 5 1 L 4 spacecraft 0.5 y (rotating frame) L 3 S L 1 E m S = 1 - µ m E = µ L 2-1 L 5 Earth s orbit x (rotating frame) Equilibrium points 11

23 Three-Body Problem orbits exist around L 1 and L 2 ; both periodic and quasiperiodic Lyapunov, halo and Lissajous orbits one can draw the invariant manifolds assoicated to L 1 (and L 2 ) and the orbits surrounding them these invariant manifolds play a key role in what follows 12

24 Three-Body Problem Equations of motion: ẍ 2ẏ = U eff x, ÿ +2ẋ= U eff y where U eff = (x2 + y 2 ) 1 µ µ. 2 r 1 r 2 Have a first integral, the Hamiltonian energy, given by E(x, y, ẋ, ẏ) = 1 2 (ẋ2 +ẏ 2 )+U eff (x, y). 13

25 Three-Body Problem Equations of motion: ẍ 2ẏ = U eff x, ÿ +2ẋ= U eff y where U eff = (x2 + y 2 ) 1 µ µ. 2 r 1 r 2 Have a first integral, the Hamiltonian energy, given by E(x, y, ẋ, ẏ) = 1 2 (ẋ2 +ẏ 2 )+U eff (x, y). Energy manifolds are 3-dimensional surfaces foliating the 4-dimensional phase space. This is for the planar problem, but the spatial problem is similar. 13

26 Regions of Possible Motion Effective potential In a rotating frame, the equations of motion describe a particle moving in an effective potential plus a magnetic field (goes back to work of Jacobi, Hill, etc). U eff Forbidden Region Exterior Region (X) S L 1 J L 2 Interior (Sun) Region (S) Effective potential Jupiter Region (J) Level set shows accessible regions 14

27 Transport Between Regions Invariant manifolds of L 1 /L 2 orbits red = unstable, green =stable z x y 15

28 Transport Between Regions These manifold tubes play an important role in what passes by Jupiter (transit orbits) and what bounces back (non-transit orbits) transit possible for objects inside the tube, otherwise no transit this is important for transport issues 16

29 Transport Between Regions These manifold tubes play an important role in what passes by Jupiter (transit orbits) and what bounces back (non-transit orbits) transit possible for objects inside the tube, otherwise no transit this is important for transport issues earlier work in this direction by Conley and McGehee in the 1960 s was extended by Koon, Lo, Marsden, and Ross [2000] discovery of heteroclinic connection between L 1 and L 2 orbits was key 16

30 Transport Between Regions Theorem of global orbit structure says we can construct an orbit with any itinerary, eg (...,J,X,J,S,J,S,...), where X, J and S denote the different regions (symbolic dynamics) X U 3 U 4 U 1 S U 3 J L 1 L 2 J U 2 U 2 17

31 Construction of Trajectories One can systematically construct new trajectories, which use little fuel. by linking stable and unstable manifold tubes in the right order and using Poincaré sections to find trajectories inside the tubes One can construct trajectories involving multiple 3-body systems. 18

32 Tour of Jupiter s Moons Tours of planetary satellite systems. Example 1: Europa Io Jupiter 19

33 Tour of Jupiter s Moons Example 2: Ganymede Europa injection into Europa orbit Maneuver performed Jupiter Ganymede's orbit Europa's orbit z Close approach to Ganymede y (a) x Injection into high inclination orbit around Europa z z y x x y (b) (c) 20

34 Tour of Jupiter s Moons pgt-3d-orbit-eu.qt 21

35 Earth to Moon Transfer In 1991, a Japanese mission was saved by using a more fuel efficient way to the Moon (Miller and Belbruno) now a deeper understanding of this is emerging 22

36 Earth to Moon Transfer In 1991, a Japanese mission was saved by using a more fuel efficient way to the Moon (Miller and Belbruno) now a deeper understanding of this is emerging we approach this problem by systematically implementing the view that the Sun-Earth-Moonspacecraft 4-body system can be regarded as two coupled 3- body systems and using invariant manifold ideas 22

37 Earth to Moon Transfer In 1991, a Japanese mission was saved by using a more fuel efficient way to the Moon (Miller and Belbruno) now a deeper understanding of this is emerging we approach this problem by systematically implementing the view that the Sun-Earth-Moonspacecraft 4-body system can be regarded as two coupled 3- body systems and using invariant manifold ideas we transfer from the Sun-Earth-spacecraft system to the Earth-Moon-spacecraft system 22

38 Earth to Moon Transfer 20% more fuel efficient than Apollo-like transfer but takes longer; a few months compared to a few days Inertial Frame Sun-Earth Rotating Frame V Earth V y Moon's Orbit x Ballistic Capture y L 1 Earth Sun Moon's Orbit x Ballistic Capture L 2 23

39 Earth to Moon Transfer shootthemoon-rotating.qt 24

40 L1 Gateway Station The Earth-Moon L 1 point is of interest as a permanent manned site. could operate as a transportation node for going to the moon, asteroids and planets could provide servicing for telescopes at Sun-Earth L 2 point Efficient transfers can be created using the 3-body and invariant manifold techniques discussed 25

41 L1 Gateway Station Below is a near-optimal transfer between the L 1 Gateway station and a Sun-Earth L 2 orbit Moon L1 to Earth L2 Transfer: Earth-Moon Rotating Frame Moon L1 to Earth L2 Transfer: Earth-Sun Rotating Frame V = 14 m/s Transfer Trajectory Moon Transfer Trajectory (38 days) y 0 y Earth Moon L1 orbit x Earth Moon's Orbit Sun Earth L2 orbit x 26

42 Optimal Control Halo Orbit Insertion After launch, the Genesis Discovery Mission will get onto the stable manifold of its eventual periodic orbit around L 1 Launch velocity errors necessitate corrective maneuvers The software coopt has been used to determine the necessary corrections (burn sizes and timing) systematically for a variety of launch conditions It gets one onto the orbit at the right time, while minimizing fuel consumption 27

43 Optimal Control A very nice mixture of dynamical systems (providing guidance and first guesses) and optimal control Delay in first maneuver (days) Perturbation in initial velocity (m/s) see Serban, Koon, Lo, Marsden, Petzold, Ross, and Wilson [2001] Pert. in initial vel. (m/s) Delay in first maneuver (days) Optimal cost C 2 (m/s) Optimal cost C 2 (m/s)

44 Proposed Research Topics: extension of work to 3D symbolic dynamics in 3D, another theorem? 29

45 Proposed Research Topics: extension of work to 3D symbolic dynamics in 3D, another theorem? trajectory design rendezvous problems (to/from Moon L1 Gateway) optimal control using NTG, COOPT, etc. 29

46 Proposed Research Topics: extension of work to 3D symbolic dynamics in 3D, another theorem? trajectory design rendezvous problems (to/from Moon L1 Gateway) optimal control using NTG, COOPT, etc. continuous (low) thrust 29

47 Proposed Research solar system dynamics probabilities of transition, capture, collision comets between planets / Kuiper Belt Shoemaker-Levy 9 type collisions (Chodas, et al.) Earth collision, eg. KT impactor (Muller, et al.) impact ejecta between planets (Burns, Levison, et al.) 30

48 Proposed Research solar system dynamics probabilities of transition, capture, collision comets between planets / Kuiper Belt Shoemaker-Levy 9 type collisions (Chodas, et al.) Earth collision, eg. KT impactor (Muller, et al.) impact ejecta between planets (Burns, Levison, et al.) large scale structure dust clouds around stellar systems (for TPF) 30

49 Transport Between Planets Comets transfer between the giant planets eg, jumping between tubes of Saturn and Jupiter Semimajor Axis of Comet Smirnova-Chernykh (AD ) Inertial Plot of Comet Smirnova-Chernykh 9 Saturn 10 Saturn Semimajor Axis (AU) Smirnova-Chernykh Jupiter y (AU, inertial frame) Sun Jupiter Year -10 Smirnova-Chernykh x (AU, inertial frame) 31

50 Minor Body Statistics Computation of long term statistics is possible Compare manifold computation (green) with comet data Shortperiod comet orbits (~800 yrs, black) and SunJupiter interior L1 manifold (~5,000,000 yrs, green) relative number of orbits (period of orbit)/(jupiter s period) 32

51 Impact Trajectories Deeper understanding of low velocity impacts eg, Shoemaker-Levy 9 and Earth crossers Example Collision Trajectory x Collision! Comes from Interior Region 33

52 Circumstellar Dust Clouds 1 y (AU, Sun-Earth Rotating Frame) (A Sun Earth x (AU, Sun-Earth Rotating Frame) Earth's Orbit Relative Particle Density 34

53 Proposed Research Tools to use and develop use knowledge of phase space geometry and ideas from transport theory (MacKay, Meiss, Wiggins, Rom- Kedar, Jaffé, Uzer, et al.) 35

54 Proposed Research Tools to use and develop use knowledge of phase space geometry and ideas from transport theory (MacKay, Meiss, Wiggins, Rom- Kedar, Jaffé, Uzer, et al.) use graph theoretic methods (Dellnitz, et al.) 35

55 Proposed Research Tools to use and develop use knowledge of phase space geometry and ideas from transport theory (MacKay, Meiss, Wiggins, Rom- Kedar, Jaffé, Uzer, et al.) use graph theoretic methods (Dellnitz, et al.) use symplectic integrators (Wisdom, Marsden, et al.) combine the above methods 35

56 References Gómez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross [2001] Invariant manifolds, the spatial three-body problem and space mission design. AAS/AIAA Astrodynamics Specialist Conference. Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2001] Resonance and capture of Jupiter comets. Celestial Mechanics and Dynamical Astronomy, to appear. Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2001] Low energy transfer to the Moon. Celestial Mechanics and Dynamical Astronomy. toappear. Serban, R., Koon, W.S., M.W. Lo, J.E. Marsden, L.R. Petzold, S.D. Ross, and R.S. Wilson [2001] Halo orbit mission correction maneuvers using optimal control. Automatica, toappear. Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2000] Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10(2), The End 36