Celest Mech Dyn Astr (2010) 107:471–485
DOI 10.1007/s10569-010-9285-3
ORIGINAL ARTICLE
The use of invariant manifolds for transfers between
unstable periodic orbits of different energies
Kathryn E. Davis · Rodney L. Anderson ·
Daniel J. Scheeres · George H. Born
Received: 16 November 2009 / Revised: 13 April 2010 / Accepted: 3 May 2010 /
Published online: 5 June 2010
© Springer Science+Business Media B.V. 2010
Abstract Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used
to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on
a trajectory contained within the initial orbit’s unstable manifold and later asymptotically
approaches the final orbit on a trajectory contained within the stable manifold of the final
orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Twobody parameters dictate the selection of the individual manifold trajectories used to construct
efficient transfers. A bounding sphere centered on the secondary, with a radius less than the
sphere of influence of the secondary, is used to study the manifold trajectories. A two-body
parameter, κ, is computed within the bounding sphere, where the gravitational effects of the
secondary dominate. The parameter κ is defined as the sum of two quantities: the difference
in the normalized angular momentum vectors and eccentricity vectors between a point on
the unstable manifold and a point on the stable manifold. It is numerically demonstrated that
as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary
results indicate that this method of constructing transfers produces a significant cost savings
over methods that do not employ the use of invariant manifolds.
Keywords Circular restricted three-body problem · Unstable periodic orbits ·
Invariant manifolds · Dynamical systems theory · Transfer trajectories ·
Libration point orbits
1 Introduction
Libration point orbits (LPOs), a subset of unstable periodic orbits in the three-body problem,
have enjoyed a growing prominence in mission design. The first spacecraft to fly on an LPO
K. E. Davis (B) · R. L. Anderson · D. J. Scheeres · G. H. Born
Colorado Center for Astrodynamics Research, University of Colorado at Boulder,
429 UCB, Boulder, CO 80309, USA
e-mail: [email protected]
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K. E. Davis et al.
was ISEE-3, which launched in 1978. Over the last 30 years, several more missions have
flown on these three-body trajectories, and many are still in orbit collecting data, including ACE, SOHO, WMAP, and Herschel/Planck (Dunham and Farquhar 2003; Hechler and
Cobos 2003). There are missions in the planning phases to send spacecraft, or constellations
of spacecraft, to LPOs in the Sun–Earth and Earth–Moon systems. LPOs in the Sun–Earth
system are planned as the science orbits for the upcoming James Webb Space Telescope
and Gaia missions, and in 2010, the two spacecraft of the Artemis mission will be the first
to navigate and perform station-keeping operations about the libration points in the Earth–
Moon system (Gardner 2003; Broschart et al. 2009). An orbit about Sun–Earth L2 is slated
as the nominal science orbit for New Worlds Observer, a proposed space-based observatory
that aims to detect and analyze terrestrial extrasolar planets (Folta and Lowe 2008). Additionally, various LPOs have been proposed for lunar navigation and communication relay
constellations (Farquhar and Kamel 1973; Grebow et al. 2006; Hill et al. 2006; Hamera et al.
2008).
As interest in three-body orbits continues to increase and more missions utilize these
orbits, an inexpensive method of constructing transfers between unstable periodic orbits will
be a powerful mission design tool. The ability to transfer spacecraft between orbits will add
a great deal of flexibility to a mission. Such flexibility would allow mission designers to create transfers for spacecraft to travel between different orbits to achieve mission objectives,
insert into a new orbit for a follow-on mission, or, in the case of a lunar communication
relay, change orbits to improve coverage characteristics for certain regions of the lunar surface. The success of the Genesis mission and the design of the Artemis transfer trajectory
illustrate the importance and benefits of incorporating techniques from dynamical systems
theory into mission design (Lo et al. 2001). Invariant manifolds were used by Koon et al.
(2002) and Gómez et al. (2004) to construct a Petit Grand Tour of the Jovian system. Later,
Russell (2006) located families of unstable periodic orbits around Europa. As the final leg
of a Jovian Moon tour, a spacecraft could transfer from an LPO to an unstable periodic orbit
about Europa. Additionally, invariant manifolds can be used to model Earth–Mars transfers
(Pergola et al. 2009) or transfers from the Earth to the Moon (see e.g., Parker and Born 2008;
Mingotti et al. 2009).
Previous research has been successful in developing techniques to transfer between LPOs
of the same energy using invariant manifolds (Koon et al. 2000; Gómez and Masdemont
2000; Gómez et al. 2004). These are theoretically zero-cost transfers, as a particle asymptotically departs the first orbit on its unstable manifold and asymptotically approaches the
second orbit on its corresponding stable manifold. Studies have also explored strategies to
transfer between orbits of different energy without the use of invariant manifolds. Howell
and Hiday-Johnston (1994) developed a method in which they selected departure and arrival
states on two halo orbits and connected them using a portion of a Lissajous trajectory. They
employed the use of primer vector theory and extended it to the elliptic restricted three-body
problem to establish optimal transfers. Gómez et al. (1998) used a combination of manifold
theory and Floquet theory to construct transfers between halo orbits of different energies.
They constructed two-maneuver transfers where the first maneuver is performed in the direction tangent to the family containing the halo orbits, and the second maneuver is performed
in the direction of the stable manifold.
The method presented in this paper is different in nature from previously developed strategies. The transfers constructed in this method involve connecting a trajectory from within
the unstable manifold of the initial orbit to a trajectory contained within the stable manifold
of the final orbit. Specifically, this paper proposes a method in which two-body dynamics
are employed to determine specific trajectories within the stable and unstable manifolds that
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The use of invariant manifolds
473
produce small transfer costs. A subsequent paper will detail the methods used to create locally
optimal trajectories.
2 The Circular Restricted Three-Body Problem
Trajectories in this research have been modeled by the equations of motion of the Circular
Restricted Three-Body Problem (CRTBP). The CRTBP models the motion of a spacecraft
acting under the influence of two massive bodies. The two bodies are assumed to be in circular
orbits about the barycenter of the system. The reference frame, centered on the barycenter,
rotates at the same rate as the orbital motion of the two bodies. Henceforth, the larger of the
masses will be referred to as the primary, and the smaller of the two will be called the secondary. The x-axis extends from the barycenter through the secondary, the z-axis extends in the
direction of the angular momentum of the system, and the y-axis completes the right-hand
coordinate frame. The equations describing the motion of the third body may be written as
x +µ
x −1+µ
−µ
R13
R23
y
y
ÿ = −2 ẋ + y − (1 − µ) 3 − µ 3
R1
R2
z
z
z̈ = −(1 − µ) 3 − µ 3 ,
R1
R2
ẍ = 2 ẏ + x − (1 − µ)
(1)
where µ is the mass parameter used to nondimensionalize the system and R1 and R2 are equal
to the distance from the third body to the primary and secondary, respectively. The reader
is directed to Szebehely (1967) for a derivation of the equations of motion. The dynamics
of the CRTBP allow an integral of motion to exist in the rotating frame. The equations of
motion given by Eq. 1 can be multiplied by 2 ẋ, 2 ẏ, and 2ż, respectively, summed together,
and integrated to obtain an integral of motion known as the Jacobi constant,
C = 2Ω − V 2 ,
(2)
where
" 1−µ
1! 2
µ
x + y2 +
+
,
2
R1
R2
V 2 = ẋ 2 + ẏ 2 + ż 2 .
Ω=
(3)
(4)
The Jacobi constant of a particle in the CRTBP cannot change unless it is perturbed by something other than the two primaries. In this way, the Jacobi constant is analogous to energy
in the two-body problem. Note that it is only a function of the nondimensional position and
velocity magnitude expressed in the rotating frame.
2.1 Periodic orbits
The equations of motion of the CRTBP allow the existence of five equilibrium points known
as the libration points, denoted Li . Conventionally, L1 lies between the primary and the
secondary, L2 lies on the far side of the secondary in the positive x-direction, L3 lies on the
far side of the primary in the negative x-direction, and the points L4 and L5 form equilateral
triangles with the two massive bodies.
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(a)
K. E. Davis et al.
(b)
Fig. 1 Planar views of the invariant manifolds of an Earth–Moon LPO. The arrows indicate the direction of
motion. (a) Unstable invariant manifold (b) Stable invariant manifold
Families of periodic orbits and quasi-periodic orbits exist about the libration points and
have been studied extensively by a multitude of researchers, prominently among them (Darwin 1897; Moulton 1920; Strömgren 1935; Broucke 1968; Bray and Goudas 1967; Breakwell
and Brown 1979; Gómez et al. 2001; Hénon 2003. There are three common classifications
of LPOs: Lissajous, halo, and Lyapunov orbits. A Lissajous orbit is a quasi-periodic orbital
trajectory that winds around a torus, but never closes in on itself. Halo orbits are a special
case of periodic Lissajous orbits, as the in plane and out of plane frequencies are equal.
Halo orbits are three-dimensional while Lyapunov orbits are planar and periodic. Two common methods for computing LPOs are a Richardson-Cary expansion and a Single-Shooting
Algorithm (Richardson and Cary 1975; Howell 1984).
2.2 Invariant manifolds
Libration point orbits are classified as either unstable or neutrally stable (Strogatz 1994). As
such, these unstable orbits have an associated set of invariant manifolds. Invariant manifolds
!
"
are trajectories that asymptotically depart or approach an orbit. The unstable manifold WU
includes the set of all possible trajectories that a particle on a nominal orbit could traverse if
it was perturbed in the direction of the orbit’s unstable eigenvector. The unstable manifold
contains all of the trajectories! that" exponentially depart the nominal orbit as time moves
forward. The stable manifold W S includes the set of all possible trajectories that a particle
could take to arrive onto the nominal orbit along the orbit’s stable eigenvector. Converse to
the unstable manifold, the stable manifold contains all of the trajectories the particle could
take to exponentially depart the nominal orbit as time moves backwards (Parker and Chua
1989). Each orbit has two associated stable and unstable manifold sets: one corresponding
to a positive perturbation, and one corresponding to a negative perturbation. Planar views of
the unstable and stable manifolds of an Earth–Moon LPO are shown in Fig. 1.
3 Using invariant manifolds for transfers between orbits
The trajectories within the unstable and stable manifolds of an unstable periodic orbit often
traverse a wide range of locations within the three-body system. The idea to use manifolds for
transfers between orbits within the restricted three-body problem has been explored. Conley
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The use of invariant manifolds
475
(1968) and McGehee (1969) proved the existence of homoclinic orbits, orbits that are both
forward and backward asymptotic to an unstable periodic orbit. A homoclinic orbit lies in
the intersection of the stable manifold and the unstable manifold of an equilibrium point or
periodic orbit. In contrast, a heteroclinic connection or heteroclinic orbit joins two different
equilibrium points or periodic orbits. A heteroclinic orbit lies at the intersection of the unstable manifold of one periodic orbit and the stable manifold of another periodic orbit. Koon
et al. (2000) numerically demonstrated the existence of heteroclinic connections between
pairs of periodic orbits. In the three-body problem, heteroclinic connections can only occur
between orbits with the same energy, or Jacobi constant.
Invariant manifolds, homoclinic orbits, and heteroclinic transfers have been shown to
play an important role in the distribution and transport of material within the solar system
(Belbruno and Marsden 1997; Lo and Ross 1998; Koon et al. 2000; Gómez et al. 2004;
Wilczak and Zgliczyński 2005). Specifically, Lo and Ross found the orbits of the comets
Oterma and Gehrels 3 appeared to shadow the invariant manifolds of the L1 and L2 libration
points in the Sun–Jupiter frame. Koon et al. showed that the path of the comet Oterma closely
followed a homoclinic-heteroclinic chain and explored the numerical construction of orbits
with prescribed itineraries to describe the resonant transitions exhibited by Oterma. Lo and
Parker (2005) investigated the use of invariant manifolds to chain together periodic threebody orbits. Many of the techniques employed in previous research have been used here.
Homoclinic orbits and heteroclinic connections lay the foundations for a more general case
of transfers between unstable periodic orbits in the CRTBP. These maneuver-free transfers
are theoretical and only exist between orbits with the same energy. In practical mission implementation, small maneuvers are required to transition from the initial orbit onto the unstable
manifold and to transition from the stable manifold onto the second orbit. Additionally, a
trajectory with a long duration may also require small correction maneuvers along its path
due to the compilation of perturbations, such as solar radiation pressure or the gravitational
forces of other bodies not modeled in the three-body problem. Obviously, if a transfer is
desired between orbits that possess different energies, at least one maneuver will be required
to complete the transfer.
3.1 Poincaré maps
Poincaré maps have been successfully used to locate heteroclinic connections between orbits
of the same energy in the CRTBP. A Poincaré map replaces the flow of an nth order system with a discrete-time system with the order (n − 1). For a given n-dimensional system
ẋ = f (x), an (n − 1)-dimensional surface of section # may be placed transverse to the flow.
A Poincaré map is created by intersecting a trajectory in Rn with the Poincaré section, #. A
Poincaré mapping, P, may be described as a function that maps the state of a trajectory at
the kth intersection, xk , to the next, xk+1 ,
xk+1 = P(xk ).
(5)
3.2 Constructing a transfer using a Poincaré map
Figure 2 illustrates how a Poincaré map can be used to locate heteroclinic connections between
unstable periodic orbits. The initial orbit is an Earth–Moon Lyapunov orbit about L1 and the
final orbit is an Earth–Moon Lyapunov orbit about L2 . Both orbits have the same Jacobi
constant. Figure 2a shows the unstable manifold of the first orbit and the stable manifold of
the second orbit integrated to the first intersection with the Poincaré section, located at the
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K. E. Davis et al.
(b)
Fig. 2 An illustration of the process of using a Poincaré map to identify heteroclinic connections between two
Lyapunov orbits in the Earth–Moon system. a The manifolds of two Lyapunov orbits integrated to the surface
of section #, and b A Poincaré map showing the stable and unstable manifold crossings. The intersections of
the manifold curves, denoted by the circles, correspond to two heteroclinic connections
x-coordinate of the Moon (x = 379, 729 km from the barycenter). As Lyapunov orbits are
planar, the corresponding invariant manifolds will also be planar and may be characterized
by a four-dimensional state [x, y, ẋ, ẏ]. The Poincaré section, #, is placed in R4 and the
resulting intersection is a surface in R3 . Because the two orbits have the same Jacobi constant,
all points along the manifolds of both orbits will have the same Jacobi constant, and the phase
space of the problem is reduced to R2 . Therefore, intersections in the Poincaré map denote
heteroclinic connections between the initial and final orbits.
The positions and velocities of both manifolds in the y-direction at the Poincaré section
are plotted in Fig. 2b and appear as curves in the two-dimensional Poincaré map. The two
intersections correspond to two heteroclinic connections, which are shown in the upper half
of Fig. 2b. Additional heteroclinic connections between these two orbits exist and may be
found by plotting multiple intersections of the manifolds with the surface of section in the
Poincaré map.
More complicated Poincaré maps are required to locate transfers between three-dimensional orbits. For examples, the reader is directed to the work of Gómez and Masdemont
(2000) and Gómez et al. (2004). If the initial and final orbits are three-dimensional and do
not have the same energy, Poincaré maps will have a limited value. However, invariant manifolds may still be used to construct transfers, although at least one maneuver will be required
to complete a transfer connecting the unstable manifold of the first orbit to the stable manifold
of the second orbit. There are an infinite number of trajectories within an orbit’s invariant
manifold, and a method is necessary to determine the individual manifold trajectories used
to construct a low cost transfer.
4 Bounding sphere
The concept of a bounding sphere is introduced to take the place of the planar Poincaré
section. A sphere of
R is placed around the center of mass
! radius
"
! of" the secondary. The
unstable manifold WU of the first orbit and the stable manifold W S of the second orbit
are integrated in time, and the states are stored each time a trajectory pierces the sphere.
Figure 3 presents a planar projection of the unstable manifold of a halo orbit about L1 , the
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The use of invariant manifolds
477
Fig. 3 A bounding sphere of
radius R is placed around the
center of mass of the secondary.
The unstable manifold of Orbit 1
and the stable manifold of Orbit 2
are propagated, and the states are
stored each time a trajectory
pierces the bounding sphere (not
sized to scale)
(a)
(b)
Fig. 4 a The manifold trajectories that pass through the sphere are highlighted. b Each trajectory is integrated
such that successive points are spaced approximately equal in position
stable manifold of a halo orbit about L2 , and a bounding sphere centered on the secondary.
Once the manifolds have been propagated and the trajectory intersections with the bounding
sphere have been stored, each trajectory piercing is integrated inside the bounding sphere as
shown in Fig. 4. Rather than integrating such that states are spaced equally in time, the trajectories are integrated such that the position differences of successive integration points are
nearly constant. This ensures that trajectories passing through the sphere at higher velocities
will have an equal distribution
as trajectories traveling with! slower
! of points
"
" velocities. An
S
unstable manifold trajectory WU
T and a stable manifold trajectory WT will be selected
and then connected by the execution of maneuvers which will form a bridging trajectory
between the manifolds.
The final transfer trajectory will have a minimum distance to the center of mass of the secondary which is less than the radius of the bounding sphere, R. This is why the primary is not
selected for the origin of the bounding sphere. In any Sun-planet system, transfers that occur
near the Sun are not feasible due to extreme environmental conditions. The same problem
may also exist for planet-moon systems, such as the Jupiter–Europa system. Additionally,
following the invariant manifolds from an orbit at L1 or L2 to the vicinity of the primary may
take a considerable amount of time-longer than feasible for mission design purposes. Thus,
for practical reasons, the bounding sphere will be centered on the secondary.
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K. E. Davis et al.
4.1 Two-body dynamics within the bounding sphere
Trajectories within the bounding sphere will be mainly influenced by the secondary if the
radius of the bounding sphere is less than the radius of the sphere of influence of the secondary. If this case exists, techniques from two-body dynamics can be used to help determine
manifold trajectories that produce small transfer costs. The bounding sphere radius is selected
to be approximately one-third the radius of the sphere of influence of the secondary in order
to ensure that the motion of the third body is dominated by the secondary and to limit the
number of manifold trajectories for analysis.
Information concerning the shape and orientation of a trajectory can be captured through
analysis of two parameters associated with the two-body problem, the normalized angular
momentum vector, hnor m , and the eccentricity vector, e. The vectors hnor m and e are computed
from the dimensional position and velocity of the third body with respect to the secondary,
Rsec and Vsec , and gravitational parameter of the secondary, µsec (not to be confused with
µ, the parameter used to nondimensionalize the three-body system),
Rsec × Vsec
hnor m = √
µsec · a
Vsec × (Rsec × Vsec )
Rsec
.
e=
−
|Rsec |
µsec
(6)
(7)
The vectors hnor m and e are independent of the periapsis epoch and semimajor axis, a, (only
used for normalization), but are functions of the other four orbital elements (e, i, ω, Ω).
Each point along a manifold trajectory within the bounding sphere may be expressed in
terms of these two vectors. In a two-body dynamics sense, minimizing the differences in the
vectors hnor m and e between two points minimizes the differences in the shape and orientation of the respective orbits that contain the two points. Thus, these vectors will be used to
select unstable and stable manifold trajectories that closely match in shape and orientation.
!
"
Let us define the differences of the vectors between a point on the unstable manifold W PU
! S"
and a point on the stable manifold W P as
#
#
(8)
∆hnor m = #hnor m,S − hnor m,U #
∆e = |e S − eU | ,
(9)
where the subscripts U and S denote the unstable and stable manifolds, respectively. Let κ
be defined as the sum of the differences of the hnor m and e vectors:
κ = ∆hnor m + ∆e.
(10)
The parameter κ quantifies the sum of the differences of the selected two-body parameters
between a point on an unstable manifold trajectory and a point on a stable manifold trajectory. We will numerically demonstrate that within the bounding sphere, smaller values of κ
correspond to smaller total transfer costs.
4.2 Constructing a transfer between two orbits
In order to test the relationship between the κ parameter and the total cost (∆V) of the transfer, a method must be developed to construct a transfer between the two orbits. This section
provides a brief description of a preliminary approach to constructing a transfer between two
orbits.
123
The use of invariant manifolds
(a)
479
(b)
Fig. 5 The bridging trajectory connecting an unstable manifold trajectory of the first orbit to a stable manifold
trajectory of the second orbit. ∆V1 , executed on the unstable manifold trajectory, is denoted by the diamond
and targets a state on the stable manifold trajectory. ∆V2 , denoted by the square, corrects the velocity at the
end of the bridging trajectory to match the velocity on the stable manifold trajectory
The unstable and stable manifolds have different energies, hence, at least one maneuver
S
will be necessary to connect them. If a WU
T has a position intersection with a WT , the transfer
can be accomplished by one maneuver at the point of the intersection to correct the velocity.
However, it is highly unlikely that the manifolds will intersect. Furthermore, if the manifolds do intersect, the cost to change the velocities at that particular point may be quite high.
Therefore, two maneuvers will be used in this preliminary approach to construct a trajectory.
The two maneuvers create a bridging trajectory between the manifolds.
A reference bridging trajectory is constructed as follows: First, unstable manifold points
of the first orbit and stable manifold points of the second orbit are computed within the
S
bounding sphere, as depicted in Figs. 3 and 4. For a given WU
T and WT , the values of κ are
U
S
U
S
computed for each W P /W P combination. The W P /W P combination with the smallest value
of κ is located. The points on the manifolds that produce the smallest κ values are denoted
S
WU
κ,min and Wκ,min .
U
The first maneuver, ∆V1 , will be performed on the WU
T at a time ∆t1 from Wκ,min . The
S
first maneuver will target the position on the WTS at a time ∆t2 from Wκ,min
. The magnitude
and direction of ∆V1 are computed based on the equations of motion of the CRTBP using
Level 1 of a differential corrector (see Wilson 2003). Once this maneuver is computed, the
bridging trajectory is propagated forward in time after ∆V1 to the intersection with the WTS .
Here, the second maneuver, ∆V2 , is executed to match the bridging trajectory’s velocity to
the velocity on the WTS . The trajectory computation process is illustrated in Fig. 5. Figure 5a
S
shows the first maneuver on the WU
T , denoted by the diamond, which targets a state on the WT .
The second maneuver, denoted by the square, corrects the velocity at the end of the bridging
trajectory to match the velocity on the WTS . In Fig. 5b, the final trajectory connecting a WU
T
of the first orbit to a WTS of the second orbit is shown. The trajectory is continuous in position
and requires two impulsive maneuvers. The total duration of the bridging trajectory is some
∆t, which can be determined based on the maneuver locations.
4.3 Optimizing the ∆V between two points
It is possible to determine the time of flight between the maneuvers, ∆t, that will minimize
the total transfer ∆V required, given the two states where the maneuvers are performed. A
cost function, J , is defined in terms of the two maneuvers, such that
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K. E. Davis et al.
J = |∆V1 | + |∆V2 | =
$
∆V1 · ∆V1 +
$
∆V2 · ∆V2 .
(11)
Take the partials of the cost function, J , with respect to ∆t to obtain
∂|∆V1 | ∂∆V1
∂|∆V2 | ∂∆V2
∂J
=
·
+
·
.
∂∆t
∂∆V1
∂∆t
∂∆V2
∂∆t
(12)
1
∆Vi
1
∂|∆Vi |
= √
· (2∆Vi ) =
,
∂∆Vi
2 ∆Vi · ∆Vi
|∆Vi |
(13)
∂J
∆V1 ∂∆V1
∆V2 ∂∆V2
=
·
+
·
.
∂∆t
|∆V1 |
∂t
|∆V2 |
∂t
(14)
R S = ϕ R (tU + ∆t, RU , VU + ∆V1 , tU ) ,
(15)
Given that
rewrite Eq. 12 as
The value of ∆t that minimizes J is the time of flight that will minimize the sum of the
maneuvers that are required to complete the transfer. Note also that for J to be minimized,
∂2 J
> 0. The unknown terms in Eq. 14 will now be computed.
∂∆t 2
S
Recall that the first maneuver is executed on the WU
T to target some state on the WT . Let
S
the position on the WT be denoted R S , which can be expressed as
where R denotes position, V denotes velocity, and the subscripts U and S denote the unstable
and stable manifolds, respectively. Let the velocity after the execution of first maneuver be
V0 where
V0 = VU + ∆V1 .
(16)
Take the partials of Eq. 15 with respect to ∆t to obtain the following
#
∂R S
∂ϕ R ∂V0
∂ϕ R ##
+0+
·
=
.
∂∆t
∂∆t #tU +∆t
∂V0 ∂∆t
(17)
The position on the WTS and the velocity on the WU
T prior to ∆V1 remain constant, despite
variations to ∆t. After simplifying and rearranging Eq. 17, the variations of ∆V1 with respect
to changes in the transfer time are found,
0 = ϕV (tU + ∆t) + Φ RV ·
∂∆V1
∂∆t
∂∆V1
= −Φ −1
RV ϕV (tU + ∆t) ,
∂∆t
(18)
where Φ is the state transition matrix, integrated forward ∆t from the state after the execution
of ∆V1 . If the 6 × 6 Φ matrix is partitioned into 4 submatrices, Φ RV is upper right 3 × 3
submatrix:
%
&
Φ R R Φ RV
Φ=
.
ΦV R ΦV V
The quantity ϕV (tU + ∆t) is the velocity of the state at time tU + ∆t, i.e., the velocity on
the bridging trajectory immediately before the execution of ∆V2 .
The second maneuver, ∆V2 , may be expressed as
∆V2 = V S − ϕV (tU + ∆t, RU , VU + ∆V1 , tU ) .
123
(19)
The use of invariant manifolds
Take the partial derivative of Eq. 19 with respect to ∆t,
'
(
#
∂V S
∂ϕV ##
∂ϕV ∂V0
∂∆V2
=
−
+0+
·
.
∂∆t
∂∆t
∂∆t #tU +∆t
∂V0 ∂∆t
481
(20)
Simplify and rearrange to obtain an expression for the variations of ∆V2 with respect to
changes in the transfer time,
'
(
#
∂ϕV ##
∂∆V1
∂∆V2
= 0−
+ ΦV V ·
∂∆t
∂∆t #tU +∆t
∂∆t
&
%
∂∆V2
∂∆V1
.
(21)
= − ϕ A (tU + ∆t) + Φ V V ·
∂∆t
∂∆t
The term Φ V V is the lower right submatrix of the partitioned state transition matrix and the
subscript A denotes acceleration. The partials of ∆V1 and ∆V2 with respect to transfer time,
given by Eqs. 18 and 21, can be substituted back into Eq. 14. Then, the transfer time that
minimizes Eq. 14 can be quickly computed by an iterative secant method process.
5 Two-Body parameters and their correlation to total transfer cost
The total ∆V was computed to complete a transfer between every combination of WU
T and
WTS that passed within the bounding sphere in order to determine the relationship between
the two-body parameter κ and the total transfer cost.
5.1 κ vs. ∆V: Sun–Earth system
The two-body parameters and total transfer ∆V were computed for six different pairs of
orbits about L1 in the Sun–Earth system. The z-amplitude of the initial halo orbit, Az 1 , varied from 160,000 to 260,000 km, and the z-amplitude of the final halo orbit, Az 2 , was held
constant at 110,000 km. The results are shown in Fig. 6.
The plots in Fig. 6 show an approximate linear trend between decreasing values of κ and
decreasing total transfer ∆V. As the two-body parameters of a WU
T more closely match the
two-body parameters of a WTS , in general, the total ∆V required to complete the transfer
S
decreases. Thus, within the bounding sphere, WU
T /WT combinations with small values of
κ should produce small ∆V costs. All of the plots in Fig. 6 are shown on the same scale.
It should be noted that as the initial z-amplitude difference increases (i.e., increasing differences in the energies between the initial and final orbits), the total ∆V required for the
transfer increases.
Transfers between the initial and final orbits given in Fig. 6a, c, and e have been previously
examined by Howell and Hiday-Johnston (1994) sans invariant manifolds. The transfer costs
for cases (a), (c), and (e) given by Howell and Hiday-Johnston were 26.36, 44.2, and 64.9
m/s, respectively. The minimum total transfer costs shown in Fig. 6a, c, and e are 11.2, 14.6,
and 39.8 m/s, respectively, a significant cost reduction. These costs can be further decreased
if the trajectories are optimized, as the transfers costs computed here are based on a reference
transfer trajectory. The computed costs shown in Fig. 6 are a baseline to gauge how the total
cost correlates to the two-body parameter κ.
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(a)
(b)
(c)
(d)
(e)
(f)
Fig. 6 Variations in κ versus total transfer ∆V for transfers between L1 halo orbits in the Sun–Earth system
5.2 κ vs. ∆V: Earth–Moon system
The correlation between κ and total ∆V was also explored in the Earth–Moon system to
verify that the method is applicable in another system with a vastly different three-body
parameter. Two cases were investigated and the results are shown in Fig. 7. Figure 7a shows
the κ vs. ∆V relationship for transfers constructed between two Southern halo orbits about
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(a)
483
(b)
Fig. 7 Variations in κ versus total transfer ∆V for halo-to-halo orbit transfers in the Earth–Moon system. (a)
Southern L2 halo (C = 3.0327) to Southern L2 halo (C = 3.0597) (b) Northern L2 halo orbit (C = 3.0319) to
Southern L2 halo orbit (C = 3.0492)
L2 . The initial orbit is a very small halo orbit with a period of 7.5 days and a Jacobi constant
C = 3.0327. The final halo orbit is much larger and has a period of 13.6 days and a Jacobi
constant C = 3.0597. The two orbits used to create the plot in Fig. 7b were a Northern L2
halo orbit with a period of 12.6 days and Jacobi constant C = 3.0319 and a Southern L2 halo
orbit with a period of 13.3 days and a Jacobi constant C = 3.0492. Again, an approximate
linear trend is observed: as the κ parameter decreases, the total ∆V decreases.
6 Conclusions
This paper has demonstrated a method of using invariant manifolds for constructing transfer
trajectories between unstable periodic orbits. An unstable manifold trajectory of the initial
orbit can be connected to a stable manifold trajectory of the final orbit by impulsive maneuvers to form a bridging trajectory. Additionally, a method has been proposed for locating
the individual unstable and stable manifold trajectories that should be used to construct the
transfer. A two-body parameter, κ, was computed within the bounding sphere, a region dominated by the gravitational force of the secondary. The parameter κ is defined as the sum of
two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It
was numerically demonstrated that as the κ parameter decreases, the total ∆V to complete
the transfer decreases. In other words, as the two-body parameters of an unstable manifold
trajectory more closely match the two-body parameters of a stable manifold trajectory, the
cost to complete the transfer will decrease. Preliminary results indicate a substantial fuel
savings when transfer trajectories are constructed using invariant manifolds. For example,
the transfer cost for case (c) in the Sun–Earth system represents a 70% improvement over a
method that does not employ invariant manifolds. It is also important to note that the costs
of the transfers shown in Figs. 6 and 7 do not represent the minimum cost, as the transfers
have not been optimized. A future publication will focus on optimizing the reference transfer
trajectories. Primer vector theory will be used to alter the maneuver locations and/or include
additional interior impulses to render an optimal trajectory. Transfers will be constructed in
multiple three-body systems to demonstrate the applicability of the method.
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Subsequent research will also seek to implement this method for practical missions. The
Petit Grand Tour of the Jovian system, designed by Gomez et al., ended with a capture into
a highly inclined two-body orbit about Europa. Russell (2006) recently located families of
periodic orbits that exist around Europa in the Jupiter–Europa three-body system. Rather
than inserting into a two-body orbit, the methods presented here will be used to locate a
transfer trajectory between a Jupiter–Europa L2 halo orbit and a periodic orbit about Europa
by connecting their respective invariant manifolds.
Acknowledgments This research was funded from a National Science Foundation Graduate Research Fellowship, a Zonta International Amelia Earhart Fellowship, and funds from the Alliance for Graduate Education
and the Professoriate.
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