C C Dynamical A L T E C S H Cylindrical Manifolds and Tube Dynamics in the Restricted Three-Body Problem Shane D. Ross Control and Dynamical Systems, Caltech www.cds.caltech.edu/ shane/pub/thesis/ April 7, 2004
Acknowledgements Committee: M. Lo, J. Marsden, R. Murray, D. Scheeres W. Koon G. Gómez, J. Masdemont CDS staff & students JPL s Navigation & Mission Design section 2
Motivation Low energy spacecraft trajectories Genesis has collected solar wind samples at the Sun- Earth L1 and will return them to Earth this September. First mission designed using dynamical systems theory. Genesis Spacecraft Where Genesis Is Today 3
Motivation Low energy transfer to the Moon 4
Outline of Talk Introduction and Background Planar circular restricted three-body problem Motion near the collinear equilibria My Contribution Construction of trajectories with prescribed itineraries Trajectories in the four-body problem patching two three-body trajectories e.g., low energy transfer to the Moon Current and Ongoing Work Summary and Conclusions 5
Three-Body Problem Planar circular restricted three-body problem P in field of two bodies, m 1 and m 2 x-y frame rotates w.r.t. X-Y inertial frame y Y P x m 2 t X m 1 6
Three-Body Problem Equations of motion describe P moving in an effective potential plus a coriolis force y _ U(x,y) L 3 P (x,y) L 5 L 1 L 4 m 1 m 2 ( µ,0) (1 µ,0) Position Space x L 2 Effective Potential 7
Hamiltonian System Hamiltonian function H(x, y, p x, p y ) = 1 2 ((p x + y) 2 + (p y x) 2 ) + where p x and p y are the conjugate momenta, and p x = ẋ y = v x y, p y = ẏ + x = v y + x, Ū(x, y), Ū(x, y) = 1 2 (x2 + y 2 ) 1 µ µ r 1 r 2 where r 1 and r 2 are the distances of P from m 1 and m 2 and m 2 µ = (0, 0.5]. m 1 + m 2 8
Equations of Motion Point in phase space: q = (x, y, v x, v y ) R 4 Equations of motion, q = f(q), can be written as ẋ = v x, ẏ = v y, v x = 2v y Ū x, v y = 2v x Ū y, conserving an energy integral, E(x, y, ẋ, ẏ) = 1 2 (ẋ2 + ẏ 2 ) + Ū(x, y). 9
Motion in Energy Surface Fix parameter µ Energy surface for energy e is M(µ, e) = {(x, y, ẋ, ẏ) E(x, y, ẋ, ẏ) = e}. For a fixed µ and energy e, one can consider the surface M(µ, e) as a three-dimensional surface embedded in the four-dimensional phase space. Projection of M(µ, e) onto position space, M(µ, e) = {(x, y) Ū(x, y; µ) e}, is the region of possible motion (Hill s region). Boundary of M(µ, e) places bounds on particle motion. 10
Realms of Possible Motion For fixed µ, e gives the connectivity of three realms P m 1 m 2 m 1 m 2 m 1 m 2 P P Case 1 : E<E 1 Case 2 : E 1 <E<E 2 Case 3 : E 2 <E<E 3 P m 1 m 2 P m 1 m 2 Case 4 : E 3 <E<E 4 Case 5 : E>E 4 11
Realms of Possible Motion Neck regions related to collinear unstable equilibria, x s 1 L 4 0.5 y 0 L 3 m L 1 1 m L 2 2-0.5-1 L 5-1 -0.5 0 0.5 1 The location of all the equilibria for µ = 0.3 x 12
Realms of Possible Motion Energy Case 3: For m 1 = Sun, m 2 = Jupiter, we divide the Hill s region into five sets; three realms, S, J, X, and two equilibrium neck regions, R 1, R 2 X S J R 1 R 2 13
Equilibrium Points Find q = ( x, ȳ, v x, v y ) s.t. q = f( q) = 0 Have form ( x, ȳ, 0, 0) where ( x, ȳ) are critical points of Ū(x, y), i.e., Ūx = Ūy = 0, where Ūa Ū a _ U(x,y) L 3 L 4 L 5 L 1 L 2 Critical Points of Ū(x, y) 14
Equilibrium Points Consider x-axis solutions; the collinear equilibria Ūx = Ūy = 0 polynomial in x depends on parameter µ -1.5 L 3 L 1 L 2-2.0 _ U(x,0) -2.5-3.0-3.5-4.0-4.5-2.0-1.5-1.0-0.5 0 0.5 1.0 1.5 2.0 x The graph of Ū(x, 0) for µ = 0.1 15
Equilibrium Regions Phase space near equilibrium points Consider the equilibrium q = L (either L 1 or L 2 ) Eigenvalues of linearized equations about L are ±λ and ±iν with corresponding eigenvectors u 1, u 2, w 1, w 2 Equilibrium region has a saddle center geometry 16
Equilibrium Regions Eigenvectors Define Coordinate Frame Let the eigenvectors u 1, u 2, w 1, w 2 be the coordinate axes with corresponding new coordinates (ξ, η, ζ 1, ζ 2 ). The differential equations assume the simple form ξ = λξ, η = λη, ζ 1 = νζ 2, ζ2 = νζ 1, and the energy function becomes E l = λξη + ν ( ζ 2 2 1 + ζ2 2 ). Two additional integrals: ξη and ρ ζ 2 = ζ1 2 + ζ2, 2 where ζ = ζ 1 + iζ 2 17
Equilibrium Regions For positive ε and c, the region R (either R 1 or R 2 ), is determined by E l = ε, and η ξ c, is homeomorphic to S 2 I; namely, for each fixed value of (η ξ) on the interval I = [ c, c], the equation E l = ε determines the two-sphere λ 4 (η + ξ)2 + ν ( ζ 2 2 1 + ζ2) 2 λ = ε + 4 (η ξ)2, in the variables ((η + ξ), ζ 1, ζ 2 ). 18
Bounding Spheres of R n 1, the left side (η ξ = c) n 2, the right side (η ξ = c) ξ η ξ=0 ζ 2 =0 η η ξ= c η ξ=+c η+ξ=0 n 1 n 2 ζ 2 =ρ ζ 2 =0 ζ 2 =ρ The projection of the flow onto the η-ξ plane 19
Transit & Non-transit Orbits There are transit orbits, T 12, T 21 and non-transit orbits, T 11, T 22, separated by asymptotic sets to a p.o. ξ T 21 η T 22 n 1 n 2 T 11 T 12 Transit, non-transit, and asymptotic orbits projected onto the η-ξ plane 20
Twisting of Orbits We compute that d arg ζ = ν, dt i.e., orbits twist while in R in proportion to the time T spent in R, where T = 1 (ln 2λ(η0 ) 2 ) ln(ρ ρ), λ ν where η 0 is the initial condition on the bounding sphere and ρ = ρ = 2ε/ν only for the asymptotic orbits. Amount of twisting depends sensitively on how close an orbit comes to the cylinders of asymptotic orbits, i.e., depends on (ρ ρ) > 0. 21
Orbits in Position Space Appearance of orbits in position space The general (real) solution has the form u(t) = (x(t), y(t), v x (t), v y (t)), = α 1 e λt u 1 + α 2 e λt u 2 + 2Re(βe iνt w 1 ), where α 1, α 2 are real and β = β 1 + iβ 2 is complex. Four categories of orbits, depending on the signs of α 1 and α 2. By a theorem of Moser [1958], all the qualitative results carry over to the nonlinear system. 22
Orbits in Position Space α 1 α 2 <0 y L α 2 < 0 α 1 > 0 α 1 < 0 α 2 > 0 α 1 α 2 >0 α 1 α 2 >0 α 1 α 2 =0 α 1 α 2 <0 x 23
Equilibrium Region: Summary The Flow in the Equilibrium Region In summary, the phase space in the equilibrium region can be partitioned into four categories of distinctly different kinds of motion: (1) periodic orbits, a.k.a., Lyapunov orbits (2) asymptotic orbits, i.e., invariant stable and unstable cylindrical manifolds (henceforth called tubes) (3) transit orbits, moving from one realm to another (4) non-transit orbits, returning to their original realm These categories help us understand the connectivity of the global phase space 24
Tube Dynamics All motion between realms connected by equilibrium neck regions R must occur through the interior of the cylindrical stable and unstable manifold tubes R 25
Tube Dynamics: Itineraries We can find/construct an orbit with any itinerary, e.g., (..., J, X, J, S, J,...), where X, J and S denote the different realms (symbolic dynamics) X Realm Forbidden Realm R 1 R 2 S Realm J Realm 26
Construction of Trajectories Systematic construction of trajectories with desired itineraries trajectories which use no fuel. by linking tubes in the right order tube hopping 27
Construction of Trajectories Ex. Trajectory with Itinerary (X, J, S) search for an initial condition with this itinerary X Realm y S Realm Sun Forbidden Realm x J Realm 28
Construction of Trajectories seek area on 2D Poincaré section corresponding to (X, J, S) itinerary region; an itinerarea X Realm U 4 Forbidden Realm U 1 U 3 R 1 R 2 S Realm U 2 J Realm The location of four Poincaré sections U i 29
Construction of Trajectories T [X],J is the solid tube of trajectories currently in the X realm and heading toward the J realm Let s seek itinerarea (X, [J], S) T J,[S] T [X],J L 1 p.o. T [J],S U 3 T X,[J] L 2 p.o. U 4 U 3 U 1 T [S],J U 2 T S,[J] U 2 T [J],X T J,[X] How the tubes connect the U i 30
Construction of Trajectories 0.6 0.4 0.2 U ([J],S) = (T [J],S U 3 ) y 0-0.2-0.4 U (X,[J]) = (T X,[J] U 3 ) -0.6 0 0.02 0.04 0.06 0.08 y Poincare Section U 3 {x = 1 µ, y > 0, x < 0} T [J],S TX,[J] J realm 31
Construction of Trajectories An itinerarea with label (X,[J],S) Denote the intersection (X, [J]) ([J], S) by (X, [J], S) 0.6 0.4 ([J],S) y 0.2 0 (X,[J],S) = (X,[J]) ([J],S) U -0.2-0.4 (X,[J]) -0.6 0 0.02 0.04 0.06 0.08 y 32
Construction of Trajectories Forward and backward numerical integration of any initial condition within the itinerarea yields a trajectory with the desired itinerary X Realm y S Realm Sun Forbidden Realm x J Realm 33
Construction of Trajectories Trajectories with longer itineraries can be produced e.g., (X, J, S, J, X) 1 Jupiter 0.08 0.04 y 0 Sun 0-1 Initial Condition -0.04-0.08-1 0 1 x 0.92 0.96 1 1.04 1.08 34
Restricted 4-Body Problem Solutions to the restricted 4-body problem can be built up from solutions to the rest. 3-body problem One system of particular interest is a spacecraft in the Earth-Moon vicinity, with the Sun s perturbation Example mission: low energy transfer to the Moon 35
Low Energy to the Moon Motivation: systematic construction of trajectories like the 1991 Hiten trajectory. This trajectory uses significantly less on-board fuel than an Apollo-like transfer using third body effects. The key is ballistic, or unpropelled, capture by the Moon Originally found via a trial-and-error approach, before tube dynamics in the system was known (Belbruno and Miller [1993]) 36
Low Energy to the Moon Patched three-body approximation: we assume the S/C s trajectory can be divided into two portions of rest. 3- body problem solutions 37
Low Energy to the Moon (1) Sun-Earth-S/C (2) Earth-Moon-S/C Maneuver ( V) at Patch Point Sun-Earth-S/C System Earth-Moon-S/C System Sun y L 2 Moon Earth x 38
Low Energy to the Moon Consider the intersection of tubes in these two systems (if any exists) on a Poincaré section Maneuver ( V) at Patch Point Sun-Earth L 2 Portion Using "Twisting" Lunar Capture Portion Sun y L 2 Moon's Orbit Earth x L 2 orbit 39
Low Energy to the Moon Earth-Moon-S/C Ballistic capture Find boundary of tube of lunar capture orbits Tube Containing Lunar Capture Orbits Sun-Earth Rotating Frame y (Sun-Earth rotating frame). Earth Moon's Orbit Initial Condition Earth-Moon L 2 Orbit Stable Manifold Cut Ballistic Capture Orbit Sun Moon y Earth Initial Condition L 2 y (Sun-Earth rotating frame) Poincare Section x 40
Low Energy to the Moon Sun-Earth-S/C Twisting of orbits Amount of twist depends sensitively on distance from tube boundary; use this to target Earth parking orbit P -1 (q 2 ) P -1 (q 1 ) Poincare Section Earth q 1 Earth Targeting Using "Twisting" y-velocity Pre-Image of Strip S Stable Manifold q 2 Strip S Unstable Manifold y-position Earth Sun L 2 L 2 orbit y-position x-position 41
Low Energy to the Moon Integrate initial conditions forward and backward to generate desired trajectory, allowing for velocity discontinuity (maneuver of size V to tube hop ) Sun-Earth L 2 Orbit Unstable Manifold Cut Initial Condition Sun-Earth L 2 Orbit Portion Using "Twisting" y (Sun-Earth rotating frame). 0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Initial Condition Earth-Moon L 2 Orbit Stable Manifold Cut 0 0.001 0.002 0.003 0.004 0.005 0.006 y (Sun-Earth rotating frame) Lunar Capture Portion Sun Moon's Orbit y Poincare Section Earth x L 2 L 2 orbit 42
Low Energy to the Moon Verification: use these initial conditions as an initial guess in restricted 4-body model, the bicircular model Small velocity discontinuity at patch point: V = 34 m/s Uses 20% less on-board fuel than an Apollo-like transfer the trade-off is a longer flight time 43
Low Energy to the Moon Sun-Earth Rotating Frame Inertial Frame V V Earth L 1 Earth L 2 y y Sun Moon's Orbit Moon's Orbit x Ballistic Capture x Ballistic Capture 44
Current and Ongoing Work Multi-moon orbiter, V = 22 m/s (!!!) JIMO Low Energy Tour of Jupiter s Moons Seen in Jovicentric Inertial Frame Callisto Ganymede Europa Jupiter 45
Current and Ongoing Work Ongoing challenges Make an automated algorithm for trajectory generation Consider model uncertainty, unmodeled dynamics, noise Trajectory correction when errors occur Re-targeting of original (nominal) trajectory vs. regeneration of nominal trajectory Trajectory correction work for Genesis is a first step 46
Current and Ongoing Work Getting Genesis onto the destination orbit at the right time, while minimizing fuel consumption from Serban, Koon, Lo, Marsden, Petzold, Ross, and Wilson [2002] 47
Current and Ongoing Work Incorporation of low-thrust Spiral out from Europa Europa to Io transfer 48
Current and Ongoing Work Coordination with goals/constraints of real missions e.g., time at each moon, radiation dose, max. flight time Decrease flight time: evidence suggests large decrease in time for small increase in V 49
Current and Ongoing Work Spin-off: Results also apply to mathematically similar problems in chemistry and astrophysics phase space transport Applications chemical reaction rates asteroid collision prediction 50
Summary and Conclusions For certain energies of the planar circular rest. 3-body problem, the phase space can be divided into sets; three large realms and equilibrium regions connecting them We consider stable and unstable manifolds of p.o. s in the equil. regions The manifolds have a cylindrical geometry and the physical property that all motion from one realm to another must pass through their interior The study of the cylindrical manifolds, tube dynamics, can be used to design spacecraft trajectories Tube dynamics applicable in other physical problems too 51