jrealm

C A L T E C H
C
Dynamical S 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 SunEarth 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, m1 and m2
– x-y frame rotates w.r.t. X-Y inertial frame
Y
y
x
P
m2
t
X
m1
6
Three-Body Problem
Equations of motion describe P moving in an effective
potential plus a coriolis force
_
U(x,y)
y
L3
L4
P
(x,y)
L5
m1
(−µ,0)
m2
(1−µ,0)
Position Space
x
L1
L2
Effective Potential
7
Hamiltonian System
Hamiltonian function
1
H(x, y, px, py ) = ((px + y)2 + (py − x)2) + Ū (x, y),
2
where px and py are the conjugate momenta,
px = ẋ − y = vx − y,
py = ẏ + x = vy + x,
and
1−µ µ
1 2
2
−
Ū (x, y) = − (x + y ) −
2
r1
r2
where r1 and r2 are the distances of P from m1 and m2
and
m2
µ=
∈ (0, 0.5].
m1 + m2
8
Equations of Motion
Point in phase space: q = (x, y, vx, vy ) ∈ R4
Equations of motion, q̇ = f (q), can be written as
ẋ = vx,
ẏ = vy ,
∂ Ū
,
v˙x = 2vy −
∂x
∂ Ū
v˙y = −2vx −
,
∂y
conserving an energy integral,
1 2
E(x, y, ẋ, ẏ) = (ẋ + ẏ 2) + Ū (x, y).
2
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
m1
m1
P
m1
m2
m2
Case 1 : E<E1
m2
Case 2 : E1<E<E2
P
Case 3 : E2<E<E3
P
P
m1
m2
Case 4 : E3<E<E4
m1
m2
Case 5 : E>E4
11
Realms of Possible Motion
Neck regions related to collinear unstable equilibria, x’s
1
L4
0.5
y
L3
L1
m1
0
m2
L2
-0.5
L5
-1
-1
-0.5
0
0.5
1
x
The location of all the equilibria for µ = 0.3
12
Realms of Possible Motion
Energy Case 3: For m1 = Sun, m2 = Jupiter, we divide
the Hill’s region into five sets; three realms, S, J, X,
and two equilibrium neck regions, R1, R2
X
S
R1
J
R2
13
Equilibrium Points
Find q̄ = (x̄, ȳ, v̄x, v̄y ) s.t. q̄˙ = f (q̄) = 0
Have form (x̄, ȳ, 0, 0) where (x̄, ȳ) are critical points of
Ū (x, y), i.e., Ūx = Ūy = 0, where Ūa ≡ ∂∂aŪ
_
U(x,y)
L3
L4
L5
L1
L2
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
L3
-1.5
L1
L2
-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 q̄ = L (either L1 or L2)
Eigenvalues of linearized equations about L are ±λ and
±iν with corresponding eigenvectors u1, u2, w1, w2
Equilibrium region has a saddle × center geometry
16
Equilibrium Regions
Eigenvectors Define Coordinate Frame
Let the eigenvectors u1, u2, w1, w2 be the coordinate
axes with corresponding new coordinates (ξ, η, ζ1, ζ2).
The differential equations assume the simple form
ξ˙ = λξ,
η̇ = −λη,
ζ̇1 = νζ2,
ζ̇2 = −νζ1,
and the energy function becomes
ν 2
2
El = λξη + ζ1 + ζ2 .
2
Two additional integrals: ξη and ρ ≡ |ζ|2 = ζ12 + ζ22,
where ζ = ζ1 + iζ2
17
Equilibrium Regions
For positive ε and c, the region R (either R1 or R2), is
determined by
El = 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
El = ε determines the two-sphere
λ
λ
ν 2
2
2
(η + ξ) + ζ1 + ζ2 = ε + (η − ξ)2,
4
2
4
in the variables ((η + ξ), ζ1, ζ2).
18
Bounding Spheres of R
n1, the left side (η − ξ = −c)
n2, the right side (η − ξ = c)
η
2 =0
|ζ|
η−ξ=−c
η−ξ=+c
η−ξ=0
ξ
η+ξ=0
n2
2
∗
=ρ
2 =0
|
ζ
|
|ζ|
|ζ| 2
=ρ
∗
n1
The projection of the flow onto the η-ξ plane
19
Transit & Non-transit Orbits
There are transit orbits, T12, T21 and non-transit orbits, T11, T22, separated by asymptotic sets to a p.o.
ξ
T21
η
T22
n1
n2
T11
T12
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
0 2
1
2λ(η )
T =
− ln(ρ∗ − ρ) ,
ln
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), vx(t), vy (t)),
= α1eλtu1 + α2e−λtu2 + 2Re(βeiνtw1),
where α1, α2 are real and β = β1 + iβ2 is complex.
Four categories of orbits, depending on the signs of α1
and α2.
By a theorem of Moser [1958], all the qualitative results
carry over to the nonlinear system.
22
Orbits in Position Space
α1 < 0
α2 > 0
α1α2<0
y
α1α2>0
α1α2>0
α1 > 0
α2 < 0
L
α1α2=0
α1α2<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
R1
R2
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
S Realm
y
Sun
Forbidden
Realm
J Realm
x
28
Construction of Trajectories
seek area on 2D Poincaré section corresponding to
(X, J, S) itinerary region; an “itinerarea”
X Realm
U4
Forbidden
Realm
U3
U1
R1
R2
S Realm
U2
J Realm
The location of four Poincaré sections Ui
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[X],J
T[J],S
TX,[J]
U3
L2 p.o.
L1 p.o.
TJ,[S]
U3
U4
U1
U2
T[S],J
TS,[J]
U2
T[J],X
TJ,[X]
How the tubes connect the Ui
30
Construction of Trajectories
0.6
([J],S)
U
= (T[J],S U3)
0.4
0.2
y
0
-0.2
(X,[J])
U
= (TX,[J] U3)
-0.4
-0.6
0
0.02
0.04
0.06
0.08
y
Poincare Section U3
{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)
T
Denote the intersection (X, [J]) ([J], S) by (X, [J], S)
0.6
([J],S)
0.4
0.2
y
(X,[J],S)
U
= (X,[J]) ([J],S)
0
-0.2
(X,[J])
-0.4
-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
S Realm
y
Sun
Forbidden
Realm
J Realm
x
33
Construction of Trajectories
Trajectories with longer itineraries can be produced
– e.g., (X, J, S, J, X)
0.08
Jupiter
1
0.04
y
0
0
Sun
-0.04
-1
Initial Condition
-0.08
-1
0
1
0.92
0.96
1
1.04
1.08
x
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. 3body problem solutions
37
Low Energy to the Moon
(1) Sun-Earth-S/C
(2) Earth-Moon-S/C
Sun-Earth-S/C
System
Maneuver (∆V)
at Patch Point
Earth-Moon-S/C
System
Moon
L2
y
Sun
Earth
x
38
Low Energy to the Moon
Consider the intersection of tubes in these two systems
(if any exists) on a Poincaré section
Sun-Earth L2 Portion
Using "Twisting"
Maneuver (∆V)
at Patch Point
Lunar Capture
Portion
Moon's
Orbit
L2
y
Sun
Earth
L2 orbit
x
39
Low Energy to the Moon
Earth-Moon-S/C – Ballistic capture
Find boundary of tube of lunar capture orbits
Ballistic
Capture
Orbit
Sun-Earth Rotating Frame
Initial
Condition
L2
y
Moon's Orbit
.
Initial
Condition
Earth
y (Sun-Earth rotating frame)
Tube Containing Lunar
Capture Orbits
Earth-Moon L2
Orbit Stable
Manifold Cut
Sun
Moon
Earth
x
y (Sun-Earth rotating frame)
Poincare Section
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
Earth Targeting
Using "Twisting"
P-1(q2)
P-1(q1)
Earth
Poincare Section
q2
Strip S
Pre-Image
of Strip S
Stable
Manifold
Unstable
Manifold
y-position
y-velocity
q1
L2
Earth
Sun
L2 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 L2 Orbit
Portion Using "Twisting"
Initial
Condition
0.01
0
Lunar Capture
Portion
Initial
Condition
0.01
0.02
Sun
L2
y
y (Sun-Earth rotating frame)
Sun-Earth L2 Orbit
Unstable Manifold Cut
0.03
Earth-Moon L2
Orbit Stable
Manifold Cut
0.04
0.05
Moon's
Orbit
Earth
L2 orbit
0.06
0
0.001
0.002
0.003
0.004
0.005
0.006
y (Sun-Earth rotating frame)
Poincare Section
x
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
Inertial Frame
Sun-Earth Rotating Frame
∆V
Earth
y
L1
Earth
L2
y
∆V
Moon's
Orbit
Sun
Moon's
Orbit
Ballistic
Capture
Ballistic
Capture
x
x
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