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, , · · (TOF) for human missions to the vicinity of the EM ·libration points is strictly constrained by the life support systems aboard the spacecraft. The Orion spacecraft, for example, has an operational 5 maximum round~tripduration of 21 days. Thus, the trajectory solutions must be both low-cost, in terms of required propellant, and of reasonable duration. Preliminary transfer design is con ducted within the context of the circular restricted three-body (CR3B) problem to utilize elements of dynamical systems theory including Poincare mapping and invariant manifolds when applicable. Other known solutions in the CR3B problem, such as three-body free return trajectories, are also exploited. Flexible multiple shooting algorithms are employed to blend multiple segments into con tinuous trajectories while constraining the location and/or magnitude of any Ll V maneuvers. Both direct transfers to the regions near the EM £1 and £2 points as well as transfer options incorporat ing a close lunar passage are investigated in terms of the CR3B as well as higher-fidelity ephemeris models that include the effects of lunar eccentricity and solar gravity.
BACKGROUND Circular Restricted Three-Body Model
For preliminary analysis, the CR3B model6 is assumed as the force model. The focus of this investigation is on solutions in the Earth-Moon (EM) system. Then, the motion of a spacecraft, assumed massless, is determined by the gravitational forces of the Earth and the Moon, each repre sented as a point mass. The orbits of the Earth and Moon are assumed to be circular relative to the system barycenter. A barycentric rotating frame is defined such that the rotating x-axis is directed from the Earth to the Moon, the z-axis is parallel to the direction of the angular velocity of the primary system, and the y-axis completes the right-handed, orthonormal triad. The position of the spacecraft relative to the EM barycenter is defined in terms of rotating coordinates as r = [x, y, z], where bold symbols denote vector quantities. The mass parameter is defined as p, ___!!!L_+, where = m1 m2 m1 and m2 correspond to the mass of the Earth and Moon, respectively. The first-order, nondimen- simial, vector equation of motion is X """f(x), (1) where the vector field, f (x), is defined
f (x) = [x,y,z,2ny + Ux, -2nx + Uy, Uz], (2) noting that the nondimensional mean motion of the primary system is n = 1. In Equation (2), 2 2 2 U (x, y, z, n) = l-;t+~+!n(x + y ) is the pseudo-potential function, with the non-dimensional Earth-spacecraft and Moon-spacecraft distances written as d and r, respectively, and the quantities Ux, Uy, Uz represent partial derivatives of U with respect to rotating position coordinates_ In this analysis, Dormand-Prince 8(9) and Adams-Bashforth-Moulton numerical integration schemes are utilized to propagate the first-order equations of motion. The only known integral of the motion is 2 2 2 2 1 2 the Jacobi constant, evaluated as C = 2U- v , where v = (x + y + z ) / ; this quantity is a constant of the motion in the rotating frame.
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. .
part part
towards towards
-
Example Example
1
corrector corrector
the the
the the
to to
time time
. .
not not
!:l.\12 !:l.\12
z-amplitudes z-amplitudes
orbit orbit
very very
of of
1 1
EM-L1 EM-L1
240
247.14 247.14
transfer transfer
130.17 130.17
188.10 188.10
the the
80.35 80.35
80.17 80.17
79.12 79.12
80.22 80.22
82.90 82.90
96
the the
computation computation
perilune perilune
the the
including including
of of
.
(rnls) (rnls)
61 61
x x .
mission mission
similar similar
EM EM
initial initial
63 63
two two
flight flight
initial initial
Earth-L
11 11
(x (x
and and
transfer transfer
£
trajectories trajectories
design design
2
altitude altitude
10
EM EM
continuation continuation
. . is is
!:l.\12 !:l.\12
to to
the the
guess guess
design. design.
range range
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2 2
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the the of of
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km) km)
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Earth Earth
.
A A
strategies strategies
8 8
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311.81 311.81
459.68 459.68 modified modified
308.58 308.58
516.01 516.01 314.20 314.20
325.35 325.35
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512
three-bum three-bum
360.27 360.27
397.07 397.07
V V one one
so so
days days
from from
is is
southern southern
!:l. !:l.
are are
costs costs
.
constrained constrained
that that
V3 V3
43 43
parking parking
transfer transfer
discussed discussed
which which
5000 5000
computed computed
( (
procedure, procedure,
mls) mls)
the the
3 4 4 3
initial initial
is is
halo halo
Earth-£2 Earth-£2
that, that,
spacecraft spacecraft
km km
is is
orbit orbit
well well
TOF TOF
guess guess
orbit orbit
previou
to to
(Orbit (Orbit
to to
for for
departure departure
7
7
7
4 the the
7.67 7.67
7
6
5.80 5.80
5.42 5.42
5.09 5.09
200 200
reach reach
within within
.
.
.
.
.
.
(days) (days) 50 50
61 61 transfer transfer
40 40
27 27
81 81
87 87
of of
select select
scheme scheme
intermediate intermediate
will will
km km
s
1) 1)
interest interest
ly ly
L
to to
the the
pass pass
for for
for' for'
2
members members
maneuver maneuver
, a a ,
incorporat
76,000 76,000
is is
stipulated stipulated
Earth-
purposes purposes
cislunar cislunar
appears appears
close close
demon
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km km
£
of of
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to to
- 1 1 2~~----~----~----~--~-----,
0 "'~ x Earth ..._.. 0 ~ /" L1
-IL-~----~----~---•-m~e_nn_e_d_ia~te~A_r_c_~~ 0 1 2 3 4 5 x (x 105 km)
Figure 12. Three-bum EM L 2 transfer initial guess strategy
2 ,..-... 1.5 ] 1 0 "'~ 0.5 ..._,X ~ 0 -0.5 0 1 2 3 4 5 x (x 105 km)
Figure 13. Example three-bum Earth-L 2 transfer
cost is reduced to a value of D.V2 + D.113 = 202 m/s corresponding to a time of flight of 14.70 days. This Earth-L2 transfer approach sharply contrasts the two-burn direct Earth-L2 transfers presented in the previous section in that this transfer possesses. a significantly lower .6.V ·as well as a much longer TOF. The trajectory in Figure 13 is used as an initial guess to compute a transfer to a second EM L2 southern halo orbit and the process is repeated for the remaining orbits of interest. The cost of associated with each of the three-bum EM-L2 transfer trajectories are presented in Table 2. It
Table 2. EM-L2 transfer A V costs
Orbit No. ..6.v2 (m/s) ..6.V2(rnls) ..6.V2 +D. v3(m/s) TOF(days) 1 190.50 11.95 202.45 14.70 2 192.64 28.04 220.68 14.14 3 187.71 51.74 239.45 13.83 4 183.78 69.94 253.72 13.90 5 179.93 87.35 267.28 13.97 6 177.27 120.28 297.55 14.23 7 175.20 138.42 313.62 14.36 8 173.18 130.58 303.76 14.76
12
the the
the the
in in
sitioned sitioned
parture parture
The The
ephemeris ephemeris
a a
enforce enforce
The The
may may
s
halo halo
with with
TRANSFER TRANSFER
Earth-bound Earth-bound
depicted depicted
21-day 21-day
human human
possible possible
two-week two-week algorithm algorithm
is is
is is olution
velocity velocity
As As
While While
Figure Figure
Despite Despite
significant significant
transfer transfer
associated associated
effect effect
CR3B CR3B
be be
orbit orbit
a a
a a
large large
inclination inclination
preliminary preliminary
transitioned transitioned
trip trip
full
time time
from from
the the
to to
in in
15. 15.
to to
of of
model model discontinuity discontinuity
time time
to to
integrated integrated
the the
trajectories trajectories
model model
-
Figure Figure
compute compute
duration duration
state state
portion portion
return return
transfers transfers
varying varying
of of
compute compute
return return
to to
the the
RESULTS RESULTS
In In
CR3B CR3B
relatively relatively
of of
flight flight
note note
this this
can can
CR3B CR3B
continuity continuity
flight flight
on on
solutions solutions
to to
14. 14.
demonstration demonstration
of of
leg leg
into into
the the
that, that,
trajectory trajectory
constraint. constraint.
round round
influences influences
example, example,
Earth-£1/£2 Earth-£1/£2
Earth
within within
"' "'
-
...-.. ...-..
0 0
] ]
in in
...... constraint constraint
Earth-£
X X
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the the
presented presented
fluctuates fluctuates
using using
between between
-2.5 -2.5
model model
-1.5 -1.5 inclination inclination
-0.5 -0.5
a a
the the
long long
IN IN
unlike unlike
0.5 0.5
1.5 1.5
-2 -2
-1 -1
cost cost
higher-fidelity higher-fidelity
. .
Figure Figure
total total
2 2
trip trip
I I
are are
previous previous
the the
AN AN
along along
times times
2 2
a a
0 0
attributed attributed
the the
-
and and
the the
Earth-£
steps steps
/:::,. /:::,.
circumlunar circumlunar
sampled sampled
ephemeris ephemeris
is is
the the
EPHEMERIS EPHEMERIS
As As
the the in in
950 950
by by
14. 14.
V V
transfer transfer
of of
the the
imposed imposed
/:::,. /:::,.
characteristics characteristics
this this
reconverged reconverged
at at
of of
Earth-£1 Earth-£1
an an
cost-
less less
transfer transfer
V V
the the
through through
section section
Example Example
m/s
LEO LEO
transfer transfer
flight flight
1 1
example, example,
2
investigation investigation
cost cost
-Earth -Earth
impact impact
than than
to to
at at
ephemeris ephemeris
. .
trajectories, trajectories,
not not
model. model.
Recall Recall
is is
a a
a a
to to
for for
free free
associated associated
965 965
arc arc
are are
x x
the the
selected selected
transfers, transfers,
one one
also also
including including
produce produce
Earth-L:r
13 13
arcs arcs
(
transfers transfers
MODEL MODEL
the the
2 3 3 2
of of
in in
x x
and and
computed computed
EM EM
a a
return return
m/s m/s
of of
day day
explored
that, that,
10
A A
a a
lunar lunar
14.3
model model
ephemeris ephemeris
and and
both both
are are
5 5
higher-fidelity higher-fidelity
the the
£
multiple-shooting multiple-shooting
bum bum
for for
time time
a a
km) km)
2 2
the the
in in
the the
-
with with
trajectory trajectory
·
direct, direct,
day day
generally generally
along along
/:::,. /:::,.
Earth Earth
that that
southern southern
orbit orbit
eccentricity, eccentricity,
this this class
the the
the the
V1 V1
via via
intermediate intermediate
required required . .
resulting resulting
in in
interval
Earth-£
these these
transfer transfer
do do
-is -is
CR3B CR3B
the the
differential differential
is is
transfer
trajectories trajectories
two-bum two-bum
the the
not not
permitted. permitted.
4 4
/:::,. /:::,.
halo halo
CR3B CR3B
nearly nearly segment segment
of of Earth-~ Earth-~
libration libration
, ,
to to
ephemeris ephemeris
V2 V2
violate violate
and and
2 2
round round
model, model,
and and Earth-~ Earth-~
solar solar
depart depart
transfer transfer
. .
family. family.
+ +
/:::,. /:::,.
algorithm algorithm
Earth-£
problem, problem,
the the
planar. planar.
the the
/:::,. /:::,.
V V
correct
5 5
i
trip trip
gravity, gravity,
as as
V3 V3
s s
the the
point point
the the
costs costs
The The
trajectories, trajectories,
resulting resulting
libration libration
the the
fixed fixed
an an
The The + +
model model
transfers, transfers,
is is
transfer transfer
21-day 21-day
outbound outbound
2 2
The The
EM EM
i
transition transition
/:::,. /:::,.
initial initial
connected connected ons ons
transfer transfer
increases increases
orbit, orbit,
is is
these these
to to
V4 V4
approximately approximately
and and
employed employed
trajectory trajectory
£
procedures. procedures.
be be
as as
point point
= =
discretized discretized
2 2
maximum maximum
Earth Earth
trajectory trajectory
guess. guess.
solutions solutions
although although
1468 1468
however
depicted depicted
southern southern
equal equal
it it
is is
legs legs
to to
is is
orbit. orbit.
as as
tran
to to
de
m/s m/s
still still
the the
the the
of of
in in
to to
to to
an an
A A . . ]' 0.5 ..., ~ 0
~-0.5
2 1 0 105 y (x 105 km) x (x km)
Figure 15. Example two bum Earth-L2 ephemeris transfer
Figure 15 is computed with an epoch in mid-December 2017 when the lunar orbital plane is inclined approximately 20 degrees with respect to Earth's equatorial plane. Thus, to transition the CR3B tra jectories to higher-fidelity transfers with an Earth departure inclination of 28.5 degrees (consistent with a launch from Kennedy Space Center), the trajectories in this analysis are first reconverged with a departure inclination of 20 degrees and a continuation procedure is implemented to increase the inclination to the desired angle. An identical procedure is used compute three-bum Earth-L1 transfers in a higher-fidelity Earth Moon-Sun ephemeris model as well. A continuation procedure is again incorporated to raise the Earth departure inclination from 20 degrees to 28.5 degrees. To demonstrate the effects of Earth departure inclination on this particular collection of three-bum transfer trajectories, the 6.V costs for each trajectory in the continuation process are summarized in Table 3. To isolate, the effect of inclination, the time of flight and perilune altitude are fixed at 5 days and 400 km for each trajectory. The cost, .6.V 1 associated with departing Earth from a 200-km altitude parking orbit is also included
Table 3. Three-bum EM-L1 ephemeris transfer a V costs
Inclination (deg) 6.V1 (m/s) 6.V2 (m/s) 6.V3 (m/s) .6.V2 + 6. v3(m/s) 20 3151.40 235.63 364.37 600.00 22 3151.80 235.03 367.97 603.00 24 3151.13 239.02 367.48 606.50 26 3151.39 247.34 364.66 612.00 28.5 3151.96 259.26 358.74 618.00 in the table to illustrate that, in this instance, changing the inclination does not significantly alter the Earth departure maneuver. The final converged three-bum Earth-Lt ephemeris transfer at an inclination of 28.5 degrees appears in Figure 16. The intermediate 6.V cost of .6.Vi + 6. V2= 618 m/s represents only a 3% increase in cost compared to the initial transfer with an Earth departure inclination of 20 degrees and illustrates the inclination does not appear to have a significant effect on libration point orbit transfer cost for the cases examined in this investigation.
STATIONKEEPING RESULTS Given the unstable nature of many EM Lt and L2 libration point orbits relevant to human space exploration, the stationkeeping method associated with maintaining these orbits can be as critical to the mission design process as the construction of the transfer trajectories themselves. Using the
14
from from
trajectories' trajectories'
Lyapunov Lyapunov
the the
at at
appear appear
from from
generally generally
EM EM
the the
maintaining maintaining
halo halo
costs costs maintaining maintaining
which which
average average
keeping keeping
500-trial 500-trial
is is
potential potential
cuted cuted
Randomly-distributed Randomly-distributed
Stationkeeping Stationkeeping behavior behavior
reference reference
the the
neuver neuver
long-term long-term
revolutions revolutions
Annual Annual
extrapolated extrapolated
over over
Stationkeeping Stationkeeping
largest largest
impact impact
average average
L2 L2
orbit orbit
a a
2.6 2.6
can can
twice twice
is is
communications communications
in in
execution execution
southern southern
39 39
total total
operations. operations.
to to
relevance relevance
approximately approximately
Monte Monte
reasonably reasonably
as as
Figure Figure
less less
orbit orbit libration libration
stationkeeping stationkeeping
examined examined
stationkeeping stationkeeping
m/s
of of
amplitude amplitude
18.5 18.5
effectively effectively
total total
various various
the the
line line
per per
the the
stationkeeping stationkeeping
libration libration
costly costly
. .
requires requires
to to
largest largest
orbit orbit
Costs Costs
m/s m/s
original original
Carlo Carlo
halo halo
of of
Note Note
18. 18.
A A
errors errors
costs costs
produce produce
to to
point point
V. V.
sight sight
EM EM
for for
Figure Figure
It) It)
in in
1 1
Monte Monte
'-" '-"
be be
~ ~
to to
~ ~ X X
Similar Similar
human human
trajectory trajectory
family, family,
-
navigation/modeling navigation/modeling
that that
Across Across
point point
act act
simulations simulations
Figure Figure
halo halo
are are
However, However,
-0
the the
extrapolated extrapolated
are are
standpoint. standpoint.
maintain maintain
approximately approximately
the the
L1 L1
3.5 3.5
0.
y y
orbits orbits
reference reference
to to
costs costs
strategy strategy
.
0 0
as as
5 5
5 5
first first
an an
(
&.\1 &.\1
largest largest
16. 16.
and and
included included
Carlo Carlo
selected selected
the the
x x
to to
orbit orbit
orbit orbit
cost cost
exploration exploration
the the
1 1
an an
trends trends
approximate approximate
Families Families
10
l7(a) l7(a)
for for
7.5 7.5
explored explored
Example Example
to to
Earth Earth
~ ~
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requires requires
"
0.5 0.5
5 5
larger larger
anchor
than than
type, type,
it it
of of
tested tested
outlined outlined
simulation simulation
maintain maintain
km
annual annual
selected selected
months months
southern southern
for for
solution
vertical vertical
is is
EM EM
with with
are are
over over
for for
to to
) )
is is
important important
their their
each each
once once
size, size,
0 0
simulate simulate
the the
of of
" "
required required
longer longer
regularly regularly
observed observed
for for
two two
activities
L
the the
-0
an an
to to
maintenance maintenance
35 35
2 2
Periodic Periodic
previously, previously,
for for
amplitude amplitude
members members
. .
annual annual
orbits orbits
1-CT 1-CT
.
the the
several several
L1 L1
halo halo
southern southern
orbit orbit
and and
5 5
per per
ensure ensure
orbital orbital
bum bum
lowest lowest
mls mls
is is
In In
the the
durations. durations.
implemented implemented
counterparts counterparts
desired desired
15 15
errors errors
the the
periodicity periodicity
week, week,
this this
orbits orbits
to to
the the
is is
. .
of of
blocked blocked
Earth-L
in in
are are
trajectories trajectories
stationkeeping stationkeeping
Each Each
periodic periodic
observe observe
that that
period period
uncertainty uncertainty
L
associated associated
average average
both both
interest interest
of of
lowest lowest
halo halo
analysis, analysis,
of of
1
this this
not not
of of
!::J.. !::J..
/L
are are
in in
the the
orbit orbit
the the
the the
trajectory trajectory
V V
1 1
2 2
Stationkeeping Stationkeeping
the the
general general
1 1
orbits. orbits.
considered considered
investigation investigation
by by
presented presented
of of
km km
EM EM
of of
Orbits Orbits
ephemeris ephemeris
on on
EM EM
EM EM
spacecraft spacecraft
Ll Ll
in in
total total
with with
and and
for for
x x
to to
approximately approximately
Lyapunov Lyapunov
the the
any any
V V
considered considered
Figure Figure
stationkeeping stationkeeping
stationkeeping stationkeeping
with with
and and
(
L
associated associated
make make
L2 L2
L
x x
of of
12 12
the the
1 1
Moon Moon
average average
Ll Ll
cost
of of
-
1 1
is is
10
Lyapunov Lyapunov
approximately approximately
and and
1 1
southern southern
V. V.
at at
revolutions
the the
trajectory trajectory
first first
maintained maintained
5 5
12-revolution 12-revolution
cmls cmls
in in
a a
. .
transfer transfer
17(b) 17(b)
maintains maintains
km) km)
crossings crossings
incorporates incorporates
L2 L2
preliminary preliminary
The The
The The
smallest smallest
which which
and and
this this
in in
costs costs
here, here,
total total
with with
as as
libration libration
Figure Figure
that that
4 4
costs costs
smallest smallest
vertical vertical
halo halo
12 12
analysis analysis
and and
cost
well well
maneuvers maneuvers
in in
are are
can can
. .
but but
real-world real-world
Ll Ll
days
for for
of of
the the
the the
this this
EM EM
The The
V V
vertical vertical
orbits, orbits,
. .
estimated estimated
3 3
associated associated
17
as as
be be
average average
stationkeeping stationkeeping
the the
point point
12 12
assessment assessment
mls
costs costs
. .
16-revolution 16-revolution
same same
L2 L2
orbits orbits
. .
investigation investigation
L
1% 1%
y-amplitude y-amplitude
four four
because because
Conversely, Conversely,
The The
undesirable undesirable
revolutions revolutions
1 1
x-z x-z
. .
orbits orbits
the the
southern southern
Like Like
orbits orbits
are are
families families
1-CT 1-CT
ranging ranging
station
general general
highest highest
unused unused
in in
plane. plane.
Ll Ll
lower lower
using using
with with
exe
Vtot Vtot
that that
ma
the the
are are
the the
of of of of 4 8 6 2 4 j 0 :§:] 2 25 :§: 8 "" '-"8 s -2 --;~ 0 , .. 20 ~ X ~ X ..__, 8 10 35 6 30 4 5 j 2 25:§: ]' 25:§: 8 '--"s"" 0 '-" ""~ 0 0 20 ~ ~ X ~X '-" -2 ~ 15 Figure 18. SK costs for the EM L 1 Lyapunov and vertical families Additional insight into the orbit maintenanc~costs depicted in Figures 17 and 18 is gained by exploring the correlation between the orbit stability index, v, and annual stationkeeping ~ V costs. 16 with with models, models, of of eccentricity eccentricity because because keeping keeping results results system, system, Quasi-periodic Quasi-periodic family family that that stationkeeping stationkeeping lated lated First, First, orbit orbit periodic periodic annual annual where where The The In In a a libration libration the the maintaining maintaining orbit orbit addition addition and and while while note note _ results results earlier, earlier, Amax Amax stationkeeping stationkeeping it it orbits orbits libration libration (1), (1), costs costs orbit. it it is is stability stability that that is is and and a a mission mission also also point, point, represents represents apparent apparent decrease decrease in in stable stable to to become become 11 11 it it !::. !::. for for solar solar Orbit Orbit higher higher 00 00 ..0 ..0 ...... "'0 "'0 1-1 1-1 ...... ...... V V EM EM can can the the ~ ~ ~ ~ beneficial beneficial point point ~ ~ ~ ~ ~ ~ ~ ~ Fundamentally, Fundamentally, index index each each as as Figure Figure goes goes 1 1200 1200 1 periodic periodic requirements requirements analyzing analyzing 200 200 400 400 L be be gravity. gravity. 600 600 400 400 800 800 000 000 opposed opposed Stationkeeping Stationkeeping costs costs that, that, less less as as orbits orbits transfer transfer 1 1 the the 00 00 family, family, said said is is and and decreases. decreases. the the 19. 19. computed computed largest largest unstable. unstable. as as to to for for ~ --<.>--- - -&---- family family orbit orbit ~ L2 L2 that, that, Figures Figures will will periodic periodic explore explore Effect Effect to to -Ll -Ll - costs, costs, the the each each the the southern southern having having L2 L2 L L1 L1 a a eigenvalue eigenvalue inherently inherently corresponds corresponds for for may may ., 1 1 · · stability stability costs costs larger larger grows. grows. Lastly, Lastly, Vertical Vertical Halo Halo Halo Halo Lyapunov Lyapunov . . of of of of Orbit Orbit as as but but 20(a) 20(a) 10 10 the the stationkeeping stationkeeping orbits orbits Costs Costs stability stability the the only only to to lower lower required required quasi-halo quasi-halo EM EM stability stability rigidly rigidly four four maintenance maintenance Conversely, Conversely, and and combining combining index index require require within within be be of of ~V ~V £ to to 17 17 index, index, stationkeeping stationkeeping periodic periodic 20(b) 20(b) the the quasi 1 1 decreases decreases adhere adhere v v southern southern to to index index 20 20 monodromy monodromy the the (m/s) (m/s) = = a a trajectories, trajectories, maintain maintain - costs costs illustrate illustrate v, v, periodic periodic spacecraft spacecraft 1 the the family family . . stepping stepping families families on on cost cost indicates indicates The The to to stationkeeping stationkeeping for for stationkeeping stationkeeping halo halo a a as as and and periodic periodic stability stability become become !::. !::. the the the the associated associated various various due due matrix matrix through through V V 30 30 orbits, orbits, explored explored to to respectively. respectively. stability stability family family stationkeeping stationkeeping greater greater .requirements. .requirements. to to remain remain orbit orbit perturbations perturbations less less index index associated associated periodic periodic both both the the goes goes results results costs costs quasi-periodic quasi-periodic previously previously index index instability instability unstable, unstable, in in and and EM EM is is transfer transfer the the 40 40 larger larger For For plotted plotted (2), (2), orbits orbits are are lrJ lrJ with with costs costs with with general general the the in in such such clearly clearly their their southern southern which which in in in in and and full-fidelity full-fidelity the the each each a a in in against against associated associated Figure Figure a a particular particular behavior behavior required required as as the the vicinity vicinity periodic periodic transfer transfer station of of corre means means lunar lunar halo halo EM EM the the the the 19 (3) (3) , , 4 4 2 2 j 0 ,, ..., Moon 0 ~ -2 X -4 10 -6 -6 5 -sL_~~~--~--~--~ -~SL---~6----4~---~2--~0~0 2 4 6 8 x (x 104 Ian) x (x 104 km) (a) EM L1 southern quasi-halo family (b) EM £2 southern quasi-halo family Figure 20. SK costs for the EM L 1/ L 2 southern quasi-halo families quasi-periodic trajectories simulated, the observed stationkeeping costs are very similar to the costs required to maintain the associated periodic orbits. This is not a completely unexpected result, but, nevertheless, it is important to confirm that periodic orbit maintenance costs can provide reasonable predictions for the !:J.V required to maintain nearby quasi-periodic trajectories. CONCLUSIONS This research represents a preliminary investigation into transfer trajectory design options be tween the Earth and Earth-Moon £ 1 and £ 2 libration point orbits that may be relevant to future human space exploration activities. Both direct transfers and those incorporating close lunar pas sage are examined. Direct transfers provide relatively fast transport to orbits in the vicinity of £ 1 and £ 2, however, total !:J.V costs vary greatly with orbit type. Generally, cost to transfer may be decreased by inserting onto nearly a planar revolution along 'large' quasi-periodic (quasi-halo and Lissajous) orbits. Increasing the energy level (decreasing C) additionally decreases the required !:J.V to transfer. To further decrease !:J.V costs for cases where time-of-flight is less important, it is useful to incorporate stable manifolds as a segment of the transfer. Three-bum, lunar assisted transfers are developed to both EM £ 1 and £ 2 that leverage existing lunar free return trajectories solutions as part of an initial guess architecture. For the classes of three burn transfer trajectories considered in this investigation, EM £1 is reached more quickly than EM £2, but at a higher l:J.V cost. Examples of round trip Earth-Lt/£2-Earth trajectories are presented to demonstrate that either EM L 1 or £ 2 can be accessed within the 21-day total time of flight limit. Both Earth-Lt and Earth-£2 transfers are reconverged in a higher-fidelity model to demonstrate that these orbit architectures are valid in real-world mission applications and that Earth departure inclination appears to have minimal effect on transfer !:J.V costs. Optimal periodic and quasi-periodic orbit stationkeeping costs are assessed using a previously 18 [11] [11] (10] (10] REFERENCES REFERENCES GrantNo. GrantNo. ACKNOWLEDGMENTS ACKNOWLEDGMENTS NNX12AC57G NNX12AC57G dynamical dynamical made made analysis analysis stationkeeping stationkeeping transfer transfer lower lower established established [9] [9] [8] [8] [7) [7) [6] [6] [5] [5] [4] [4] [3] [3] [2] [2] [1] [1] TOF TOF Perilune Perilune SK SK !:lV !:lV z-amp z-amp The The ference ference cations cations University, University, D. D. Advanced Advanced E. E. Body Body July July of of D. D. R. R. Mechanics, Mechanics, York, York, 63rd 63rd V. V. K. K. M. M. Report, Report, Exploration%20Report_508_6-4-12 2011. 2011. 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