Download Natural Regions Near the Collinear Libration Points Ideal for Space

J of Astronaut Sci
DOI 10.1007/s40295-014-0027-8
Natural Regions Near the Collinear Libration Points
Ideal for Space Observations with Large Formations
Aurélie Héritier · Kathleen C. Howell
© American Astronautical Society 2014
Abstract This investigation explores regions near libration points that might prove
suitable for space observations with large formations. Recent analyses have considered occulters located at relatively large distances from the telescope near the
L2 Sun-Earth libration point for detection of exoplanets. During the science mode,
the telescope-occulter distance, as well as the pointing direction toward the star,
are typically fixed. Quasi-periodic Lissajous trajectories are employed as a tool to
determine regions near the telescope orbit where the large formation can be maintained. By placing the occulter in these locations, the control required to maintain the
line-of-sight is reduced.
Introduction
In the last decade, hundreds of planets orbiting other stars, called extrasolar planets
or exoplanets, have been detected. Thus far, all of them are gas giants like Jupiter, but
improvements in technology are moving the detection limits to planets with smaller
masses. Using multiple spacecraft to create a large observation platform enables the
detection of smaller and smaller planets. Additional investigations on formation flying in multi-body regimes have emerged to support space astronomy missions. Some
new concepts using large formations can detect not only Earth-like planets, but can
A. Héritier ()
Advanced Concepts Team, ESA/ESTEC, PPC-PF,
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
e-mail: [email protected]
K. C. Howell
School of Aeronautics and Astronautics, Purdue University,
West Lafayette, IN 47907, USA
e-mail: [email protected]
J of Astronaut Sci
also characterize them via spectroscopy, providing information such as atmospheric
conditions, internal structure, mass estimates, as well as life signs. For example, the
original New Worlds Observer (NWO) design concept employed an external occulter placed at a relatively large distance (∼80,000 km) from its telescope for the
detection and characterization of extrasolar planets [1]. In the science mode, the
occulter is maintained along the line-of-sight from the telescope to the target star to
block out the starlight. It suppresses the starlight by many orders of magnitude, to
enable the observation of habitable terrestrial planets.
From a control perspective, the telescope-occulter architecture concept can be
decomposed into two mission phases.
•
•
The observation phase: the occulter must be maintained precisely along the
telescope line-of-sight to some inertially fixed target stars.
The reconfiguration phase: the occulter is realigned between each observation
from one target star line-of-sight to the next.
During the observation mode, the two spacecraft must be aligned within a few meters
along the line-of-sight. This is most easily accomplished if these spacecraft are in
a low-acceleration environment such as the vicinity of the Sun-Earth L2 libration
point, the future home of astrophysical observatories. Ideally, regions with zero relative velocity and zero relative radial acceleration maintain the formation, and for
a small distance between the spacecraft, these regions can be determined via linear
analysis. However, for a large telescope-occulter distance, up to tens of thousands of
kilometers, linearization relative to the telescope orbit may no longer be an acceptable
planning option.
The Sun-Earth L2 libration point region, providing a low-acceleration environment that is ideal for astronomical instruments, has been a popular destination for
satellite imaging formations. Such imaging scenarios were the original motivation for
this work. Barden and Howell investigate the natural behavior on the center manifold
near the libration points and compute some natural six-spacecraft formations, which
demonstrate that quasi-periodic trajectories could be useful for formation flying [2].
Later, Marchand and Howell extend this study and use some control strategies, continuous and discrete, to maintain non-natural formations near the libration points [3].
Most of the formation flying missions have been considering spacecraft at a relatively
small distance from the reference orbit, however. Space-based observatory and interferometry missions, such as the Terrestrial Planet Finder, have been the motivation
for the analysis of many control strategies. Gómez et al. investigate discrete control methods to maintain such a formation [4]. Howell and Marchand consider linear
optimal control, as applied to nonlinear time-varying systems, as well as nonlinear
control techniques, including input and output feedback linearization [5]. Recently,
Gómez et al. derive regions around a halo orbit with zero relative velocity and zero
relative radial acceleration that ideally maintain the mutual distances between spacecraft [6]. Their analysis, based on linearization relative to the reference orbit, assume
small formations of spacecraft. Lo examines the Terrestrial Planet Finder architecture
with an external occulter at a much larger relative distance, and considers different
mission scenarios with the formation placed on a halo orbit around the Sun-Earth L2
libration point as well as a formation on a Earth leading heliocentric orbit [7]. Millard
J of Astronaut Sci
and Howell evaluate some control strategies for the Terrestrial Planet Finder-Occulter
mission for the two phases of the mission scenario [8]. They compare different
control methods for satellite imaging arrays in multi-body systems such as optimal
nonlinear control, geometric control methods, a linear quadratic regulator and input
state feedback linearization. Kolemen and Kasdin focus on the realignment problem
and investigate optimal control strategies for the reconfiguration phase, to enable
the imaging of the largest possible number of planetary systems with the minimum
mass requirement [9]. Their analysis does not exploit the natural dynamics specific
to the region near the libration points. Most recently, Héritier and Howell investigate
the natural dynamics in the collinear libration point region for the control of large
formations [10]. In the current work, this analysis is extended.
The goal in the present investigation is the further exploration of the natural
dynamics in the collinear libration point region to aid in the control of large formations. Examining the dynamical environment, and better understanding the flow in
the vicinity of the telescope orbit, assists in the design of mission scenarios. Quasiperiodic Lissajous trajectories are employed as a new tool to determine regions near
the telescope orbit where the large formation can be relatively easily maintained.
Studying these trajectories yields some insight into the behavior of a formation of
spacecraft in this region. First, extensive computations of quasi-periodic Lissajous
trajectories near the telescope orbit are completed. Arcs along the Lissajous trajectories are analyzed and viewing spheres at various points along the telescope orbit are
developed. These space spheres are used as a tool to categorize regions along the orbit
with less natural drift when the distance between two vehicles is large. Locating the
occulter in these zones leads to a smaller variation in the telescope-occulter vector,
both in magnitude and direction. The effectiveness of the low natural drift regions is
then tested for the observation of an inertially fixed target star at different times along
the telescope orbit. A linear quadratic regulator is used to maintain the occulter along
the telescope line-of-sight to the inertially fixed star. If the observation of the target star begins when the orientation of the formation lies in a low natural drift zone,
then the control effort to maintain the formation is reduced. Finally, given a set of
inertial target stars, a star sequence design process is proposed with both observation
and reconfiguration phases. Impulsive maneuvers are applied for realignment during the reconfiguration phase. As the observations of target stars which possess long
observation intervals increase the total cost of observation, the observation phase is
designed such that the occulter is located in a low drift zone during the long observations. The remaining target stars are then placed consistent with the reconfiguration
phase, such that the overall cost for the mission is reduced.
Dynamical Model
The motion of the spacecraft is described within the context of the circular restricted
three-body problem (CR3BP). In this model, it is assumed that the Sun and the Earth
move in a circular orbit around their barycenter. The mass of the spacecraft (P3 ) is
negligible compared to the masses of the two primaries. The equations of motion are
described in a rotating coordinate frame where P3 is located relative to the barycenter
J of Astronaut Sci
B of the primaries, with the x-axis directed from the Sun to the Earth. Let X define a
general vector in the rotating frame from B to P3 , i.e.
X = [x
y
z
ẋ
ẏ
ż]T
(1)
where superscript ‘T ’ implies transpose. The equations of motion are expressed in
terms of a pseudo potential function
μ
1
(1 − μ)
+
+ (x 2 + y 2 )
(2)
r1
r2
2
with r1 = (x + μ)2 + y 2 + z2 and r2 = (x − (1 − μ))2 + y 2 + z2 , and where
μ is the mass parameter associated with the Sun-Earth system. The scalar nonlinear
equations of motion in their non-dimensional forms are then written as
U=
ẍ − 2ẏ = Ux
(3)
ÿ + 2ẋ = Uy
(4)
z̈ = Uz
(5)
where Uj represents the partials
for j = x, y, z. One advantage of the CR3BP as
the framework for this analysis lies in its dynamical properties that may be exploited
for mission design. The equations of motion in the CR3BP possess five equilibrium
solutions, or libration points. These consist of the three collinear points (L1 , L2 , and
L3 ) and the two equilateral points (L4 and L5 ). In the vicinity of each equilibrium
point, some structure can be identified: periodic and quasi-periodic orbits (center
manifold), as well as unstable and stable manifolds. Knowledge of this natural flow
is very useful for trajectory design.
The design process generally relies on variations relative to a reference arc. Given
a solution to the nonlinear differential equations, linear variational equations of
motion are derived in matrix form as
∂U
∂j
δ Ẋ = A(t)δX
(6)
where δX = [δx δy δz δ ẋ δ ẏ δ ż]T represents variations about a reference
trajectory. The A(t) matrix is time-varying of the form
03×3 I3×3
A(t) =
(7)
F
J
where the F matrix and J matrix are defined as
⎤
⎡
Uxx Uxy Uxz
F = ⎣ Uyx Uyy Uyz ⎦
Uzx Uzy Uzz
⎡
⎤
0 20
J = ⎣ −2 0 0 ⎦
0 00
(8)
(9)
2
∂U
The partials Uj k represent ∂j
∂k for j, k = x, y, z and are evaluated along the
reference. The general form of the solution to the vector Eq 6 is
δX(t) = (t, t0 )δX(t0 )
(10)
J of Astronaut Sci
where (t, t0 ) is the state transition matrix (STM), which is essentially a linear
mapping that approximates the impact of the initial variations on the variations
downstream. The STM satisfies the matrix differential equation
˙ t0 ) = A(t)(t, t0 )
(t,
(11)
(t0 , t0 ) = I6,6
(12)
where I6,6 is the identity matrix. Equation 11 is integrated simultaneously with the
nonlinear equations of motion to generate reference states and updates to the trajectory. Since the STM is a 6 × 6 matrix, it requires the integration of 36 first-order,
scalar, differential equations, plus the additional 6 first-order state equations, hence
a total of 42 differential equations.
Space Spheres
The first objective in this study is a better understanding of the dynamical
environment at a relatively large distance from the telescope orbit. The CR3BP model
possesses dynamical properties that may be exploited for mission design. In the current scenario, the telescope path evolves along a halo orbit in the vicinity of the
Sun-Earth L2 libration point as illustrated in Fig. 1. A fixed distance between the
telescope and the occulter of 50,000 km is selected to develop the mission concept.
Due to the relatively large telescope-occulter distance, linearization relative to the
telescope orbit is not recommended. Instead, natural quasi-periodic Lissajous trajectories are employed as a tool to determine regions in the vicinity of the telescope orbit
where a large formation yields small relative motion. Such trajectories yield some
insight into the behavior of a large formation of spacecraft in this region.
First, extensive computations of Lissajous trajectories near the telescope orbit are
completed. Then, arcs along the Lissajous trajectories that correspond to the appropriate telescope-occulter distances are analyzed. At each instant of time, the velocity
vector along the telescope orbit and the corresponding Lissajous velocity vector are
compared as illustrated in Fig. 2. The state on the Lissajous arc, that corresponds to
the state of the occulter, is located at a distance of 50,000 km from the state along
the telescope orbit. Two parameters are derived: the difference in the norm of the two
velocities, i.e., between the velocity vector along the telescope orbit and the velocity
5
t = 0 days
z rotating (km)
x 10
A ~ 200 960 km
x
Ay ~ 686 200 km
Az ~ 148 900 km
2
L2
0
−2
1.515
5
5
1.51
0
−5
x 10
1.505
y rotating (km)
Fig. 1 Reference orbit for the telescope (period ∼180 days)
x rotating(km)
8
x 10
J of Astronaut Sci
t0
Telescope-occulter
vector direction
~ 50,000 km
Vref
t0
t1
Vliss
1.
Difference in velocity norm
Δ V = Vliss − Vref
2.
Angle between velocity vectors
Vref
Vliss
Natural drift
t1
φ
Fig. 2 Schematic of the velocity vector comparison
vector on the Lissajous arc, as well as the angle between the two velocity vectors at
each location around the telescope orbit. The natural drift of the telescope-occulter
line-of-sight from its initial orientation, as defined in red in Fig. 2, depends on the
size of these two parameters: difference in velocity norm and angle between velocity vectors. Figure 3 illustrates the variation of the telescope-occulter line-of-sight
vector after a specific time interval with respect to these two parameters. The time
interval for star observation may vary between 1 day to 40 days depending on the star
location and the method employed (detection or characterization). The time interval
selected for this investigation is 3 days. The initial state considered is selected at time
t = 0 days, that is represented in Fig. 1. At each location around the telescope orbit,
a sphere of points with radius equal to the reference telescope-occulter distance is
derived as illustrated in Fig. 4. A vector originating at the center of the sphere (point
on the reference orbit) and terminating at a point on the sphere defines a line-of-sight
direction. On the surface of each sphere, different zones are defined that identify
telescope-occulter directions (that is, line-of-sight orientations). Each point on the
Fig. 3 Variation of the telescope-occulter line-of-sight vector after a 3-day time interval with respect to
the difference in the norm of the velocities and to the angle between the two velocity vectors
J of Astronaut Sci
Fig. 4 Space spheres of 50,000 km radius at different times along the telescope orbit
sphere is colored to reflect the value of the natural drift at that location that is calculated after a three-day time interval. Hence, each color represents some maximum
values for the two initial velocity parameters: the difference in the initial velocity
norm and the initial angle between the velocity vectors. One space sphere at an isolated time appears in Fig. 5. The blue zones represent regions with less natural drift
than the red zones. Effectively, if the line-of-sight to a star is directed along a line
from the origin of the sphere to a blue dot, an occulter along this line experiences less
natural drift than a star line-of-sight through a red dot. Locating the occulter in the
regions with small natural drift leads to a smaller variation in the telescope-occulter
vector and, therefore, to a potential reduction in the control effort to maintain the
formation during an observation.
Characteristics of the Low Drift Regions on the Space Spheres
Given the projections of the 3D space spheres onto the (x, y) rotating plane at different instants of times, the variations in different regions on the space spheres through
Fig. 5 Sample space sphere at time t = 127 days along the telescope orbit
J of Astronaut Sci
time are observed. The regions with small natural drift at different times along the
telescope orbit are represented in Fig. 6. These dark blue rings reflect a difference
in the initial velocity norm that is less than 5 m/s and an initial angle between the
velocity vectors of less than 5 deg, that corresponds to the dark blue regions from the
natural drift surface in Fig. 3. In addition to the low natural drift zones, the eigenstructure of the F matrix defined in Eq. 8 is also illustrated in Fig. 6. The F matrix
possesses three orthogonal eigenvectors defined by V1 , V2 and V3 . Two of them, V1
and V2 , are plotted in blue and red, respectively. The third one, V3 , is mainly directed
along the z-axis. These eigenvectors represent the principal directions of the low drift
regions. This is demonstrated by studying the variational equations defined in Eq. 6
and by deriving some regions suitable to maintain a small formation of spacecraft.
Assuming that the separation between the spacecraft is “small” compared to the
radius of the reference orbit, i.e, no greater than a few kilometers maximum, regions
of low drift are computed via the first-order variational equations with respect to the
reference orbit, and an analytical expression for them is determined. Equation 6 is
rewritten in terms of relative position and relative velocity vectors as
δṙ
δr̈
=
03×3 I3×3
F
J
δr
δṙ
(13)
where F and J are defined in Eqs. 8 and 9, respectively. As the spacecraft are placed
at a small distance from each other, the relative velocity is assumed to be equal to
zero, i.e., δṙ = 0. Notice that this assumption is only valid for a small formation
of spacecraft. At large distances, the relative velocity may not be negligible. From
Eq. 13, the relative acceleration is then
δr̈ = F δr
(14)
5
x 10
6
3
4
2.8
y rotating (km)
y rotating (km)
x 10
2
0
−2
5
2.6
2.4
V2
2.2
−4
2
V
−6
1
1.8
1.502 1.504 1.506 1.508 1.51 1.512 1.514 1.516 1.518 1.52
8
x rotating (km)
x 10
(a)
1.5078 1.508 1.5082 1.5084 1.5086 1.5088 1.509
8
x rotating (km)
x 10
(b)
Fig. 6 a Low drift regions on the space spheres at different times along the telescope orbit; b One low
drift region at time t = 100 days
J of Astronaut Sci
The norm of the relative acceleration is computed from the dot product of δr̈ with δr̈.
δr̈2 = δr̈ · δr̈ = F δr · F δr
(15)
The positions such that the relative acceleration is equal to zero, are then the set of
points that satisfy the equation
δrT F T F δr = 0
(16)
Equation 16 represents quadrics with zero relative velocity and zero relative acceleration. These quadrics are plotted in Fig. 7. Using a change in coordinate, Eq 16 can
be transformed into its canonical form. The matrix F is a real symmetric matrix, i.e.,
F = F T and therefore can be diagonalized. The singular value decomposition of F
takes the form
⎤
⎡
λ1 0 0 T
(17)
F = P P T = V1 V2 V3 ⎣ 0 λ2 0 ⎦ V1 V2 V3
0 0 λ3
is a real diagonal matrix with λ1 , λ2 and λ3 , the eigenvalues of F , on its diagonal.
P is an orthogonal matrix whose columns V1 , V2 and V3 represent the eigenvectors of F previously defined. These eigenvectors are orthogonal to each other and
represent the principal directions of the quadrics. Let define a new vector y =
[y1 y2 y3 ]T such that y = P T δr. Equation 16 is then rewritten in the following
canonical form
yT P T F T F P y = yT 2 y = λ21 y12 + λ22 y22 + λ23 y32 = 0
(18)
Although these quadrics describe the dynamics close to the reference orbit, it is
interesting to compare the similarity with the low drift regions as plotted in Fig. 6 at
5
x 10
5
x 10
3
6
2.8
y rotating (km)
y rotating (km)
4
2
0
2.6
2.4
−2
2.2
−4
2
V2
V
−6
1
1.8
1.502 1.504 1.506 1.508 1.51 1.512 1.514 1.516 1.518 1.52
8
x rotating (km)
x 10
(a)
1.5078 1.508 1.5082 1.5084 1.5086 1.5088
8
x rotating (km)
x 10
(b)
Fig. 7 a Quadrics with zero relative velocity and zero relative acceleration determined via linear analysis
at different times along the telescope orbit (enlarged size for illustration); b One quadric at time t =
100 days
J of Astronaut Sci
50,000 km. Both the low drift zones and the small quadrics represent regions suitable to maintain a formation of spacecraft, either large or small. They are derived via
significantly different methods, either from a linear analysis or numerically by propagating in the nonlinear dynamical model. The surface of the small quadrics possesses
no relative velocity and no relative acceleration whereas some relative velocity and
relative acceleration still remain in the low drift regions (the difference in the velocity norm is less than 5 m/s and the angle between the velocity vectors is less than
5 deg). But, interestingly, they both are defined with the same orientation in space
as represented in Fig. 7. This orientation, derived from the eigenspace of the linear
system, seems to persist as the distance between the spacecraft increases to at least
50,000 km.
In contrast to a planar projection, an alternative approach to visualizing the low
drift regions on each space sphere is the consideration of two orientation angles.
These ring-shaped regions are defined by an in-plane angle θ , measured relative to
the x-axis in the rotating frame, and an out-of-plane angle β, oriented with respect to
the (x, y) plane. These angles are represented in Fig. 8. Any V vectors as illustrated
in Fig. 8 can be defined as
⎡
⎤
cos(θ ) cos(β)
(19)
q = q ⎣ sin(θ ) cos(β) ⎦
sin(β)
Some characteristics of these low drift regions are apparent by examining the in-plane
angle θ . For most of the low drift directions identified by blue points, θ approaches
either 90 or −90 degrees as illustrated in Fig. 6a, and this feature is reproduced
at each time along one revolution of the orbit. The mean value of the angle θ is
Fig. 8 Low drift zone on a space sphere described by in-plane angle θ and out-of-plane angle β
J of Astronaut Sci
calculated at each time along the telescope orbit as represented by the skematic in
Fig. 9. Table 1 gives the mean value of the angle θ and the corresponding time along
the reference orbit. These angles describe the orientation of the low drift regions
along one revolution of the telescope orbit (∼180 days). For reference, the initial
state on the telescope orbit at time t = 0 days is selected as
X(0) = [1.011222412261165 0
0.0009 0
− 0.009036145801608 0]T
(20)
Space Observation of Inertial Target Stars Using the Space Spheres
The effectiveness of the low natural drift regions is evaluated for the observation of
an inertially fixed target star at different times along the telescope orbit. Although
inertially fixed target stars describe paths which are not fixed in the rotating frame,
the natural motion of a fixed star relative to the rotating frame is still smaller than the
natural drift present at 50,000 km in the low drift regions. A linear quadratic regulator
is used to maintain the occulter along the telescope line-of-sight to the inertially
fixed star. If the observation of the target star originates when the orientation of the
formation lies in a low natural drift zone, then the control effort to maintain the
formation is reduced. The control cost is considerably higher if the direction to be
maintained lies in the high drift regions in contrast to those with low drift.
Locations Along the Telescope Orbit for Star Observation
The direction of an inertial target star is usually specified in terms of two angles, the
right ascension and the declination, defined in the equatorial frame. The direction is
Fig. 9 Low drift regions
described by in-plane angle θ
t = 0 days
y
mean of θ
x
J of Astronaut Sci
Table 1 Orientation of the low drift regions along the reference orbit
Time along the reference orbit (days)
In-plane angle θ of the low drift regions (degrees)
0
90.0000
10
84.5521
20
79.5085
30
75.1828
40
71.8661
50
69.9950
60
70.2755
70
73.5659
80
80.3493
90
89.8369
100
80.6235
110
73.7303
120
70.3306
130
69.9661
140
71.7800
150
75.0578
160
79.3556
170
84.3804
180
89.7661
time dependent as viewed from the rotating frame of the CR3BP and, therefore, a
coordinate transformation is necessary to locate the star, i.e., the target direction, at
each instant of time. The coordinate transformation from the equatorial frame to the
rotating frame is derived as
⎡
⎤⎡ ⎤
⎡ ⎤
cos α cos sin α sin sin α
x
x
⎣y ⎦
= ⎣ − sin α cos cos α sin cos α ⎦ ⎣ y ⎦
(21)
0
− sin cos z Equatorial
z CR3BP
with = 23.0075 degrees and α = ω + α0 where α0 represents the initial angle
between the inertial and rotating frames (α0 is assumed to be equal to zero in this
analysis) and the angular rate ω is equal to one in nondimensional coordinates. The
in-plane angle of an inertial target star measured relative to the x-axis in the rotating
frame is defined as γ . This in-plane angle γ varies with time in the (x, y) rotating
plane, approximatively 1 degree/day, and therefore, the direction of the target star
might eventually lie in a low drift zone at a specific time along the telescope orbit.
For most of the low drift directions identified by blue points, the in-plane angle
θ approaches either 90 or −90 degrees, and this feature is reproduced at each time
along one revolution of the orbit as illustrated in Fig. 6a. Hence, at each time, the
mean value of the angle θ is calculated and compared to the in-plane angle γ of
the inertial target star direction. The low drift regions are ring-shaped objects, and
therefore, matching the in-plane angles also assures the out-of-plane angle of the star
J of Astronaut Sci
Fig. 10 Schematic of the
method used to determine the
starting times for observation
Star 1 at t1
at t1
Star 1 at t2
Telescope
Occulter
at t2
to match an out-of-plane angle of the blue regions. The closest value between the inplane angle of the star and the mean value of the angle θ occurs at the “best” location
along the telescope orbit, and this location assures that the star direction lies in a low
natural drift region during the observation phase. The schematic in Fig. 10 illustrates
this process. Two space spheres are represented at time t1 and time t2 . The low drift
regions and high drift regions are represented in blue and red, respectively, on each
sphere. Given an inertial star direction for observation, (Star 1 in the schematic), if the
observation is initiated at time t1 , the occulter lies in a high drift zone and the control
to maintain it along the line-of-sight is relatively high. However, if the observation
of the same star is delayed to begin at time t2 , the occulter lies in a low drift region
at this time, and therefore the cost of the control is reduced. An example with one
inertial target star with a right ascension of 80 degrees and a declination of 10 degrees
appears in Fig. 11. The in-plane angle γ of the inertial target star represented in green
in Fig. 11b is computed at each time along the telescope orbit. As noted in the figure,
γ shifts approximately 1 degree/day. The best location to begin the observation of
150
y
Star 1
mean of θ
γ
x
θ = in-plane angles of the low drift region
γ = in-plane angle of the inertial star direction
(a)
In−plane angles (deg)
100
mean of θ > 0
50
0
−50
mean of θ < 0
γ
−100
−150
0
50
100
150
200
Time (days)
(b)
Fig. 11 Best locations along the telescope orbit to begin observation: example with one inertial star
direction. The ‘best’ location to initiate the observation is approximately 160 days (red square)
J of Astronaut Sci
this specific target star is near the time t = 160 days, which is the time when the
value of γ is close to the mean value of the angle θ .
Control Strategy for the Observation Phase Using the Low Natural Drift Regions
Even in the low drift zones, drift between the two spacecraft (telescope and occulter)
still occurs and the design of a controller to maintain the formation is necessary, but
these low drift zones can be used as a tool from a control perspective. If the specific
orientation of the formation lies in a zone of low natural drift, then the control to
maintain the orientation is reduced. A linear quadratic regulator (LQR) is applied to
demonstrate this result.
The telescope is assumed to move along the reference orbit. Given an inertial target
star direction with an observation time of 20 days, the occulter must be maintained
precisely along the line-of-sight from the telescope to the star during the observation
interval. Particularly, the occulter is constrained to remain within ±100 km of the
baseline path in the radial direction (line-of-sight direction) and within a few meters
from the baseline position in the transverse direction (orthogonal to the line-of-sight)
[11]. Various reference arcs are selected as potential baseline paths for the occulter
along the telescope line-of-sight to the inertially fixed star. Each reference path is
an arc along a suitable Lissajous trajectory. These reference arcs are computed for
different starting locations along the telescope orbit. The LQR controller is formulated to track these reference arcs during the observation period. The motion of the
occulter in the CR3BP is described as
ẍ = fx (x, ẋ, y, ẏ, z, ż) + ux
(22)
ÿ = fy (x, ẋ, y, ẏ, z, ż) + uy
(23)
z̈ = fz (x, ẋ, y, ẏ, z, ż) + uz
(24)
wherefx , fy , fz are defined from Eqs. 3–5 as
fx = Ux + 2ẏ
(25)
fy = Uy − 2ẋ
(26)
f z = Uz
(27)
and u(t) = [ux uy uz ]T is the control vector. Let X̄0 be some reference motion
and ū0 the respective control effort to maintain X̄0 . The selected reference arcs represent the baseline paths of the inertial star direction in the rotating frame. These paths
are computed at the distance of 50,000 km from the telescope, and the occulter is
controlled to track these arcs using the LQR controller. Linearization relative to these
reference solutions yields a linear system of the form
δ Ẋ = A(t)δX + B(t)δu
(28)
where δX and δu are the variations relative to the reference arc X0 and its respective control u0 . The time-varying matrix A(t) is defined in Eq. 7 and B(t) =
J of Astronaut Sci
[03,3 I3,3 ]T . The LQR controller minimizes some combination of the state error and
the required control, which is represented as the following quadratic cost function
1
min J =
2
tf
(δXT Q δX + δuT R δu)dt
(29)
t0
The matrices Q and R are positive definite matrices and represent weighting factors
on the state error and control effort, respectively. In this analysis, weighting matrices,
Q = I6×6
R = diag([10−5
(30)
10−10
10−10 ])
(31)
are selected such that the error requirements in the transverse and radial directions
are satisfied [8]. The observation of the given target star is computed at three different starting times around the telescope orbit. The telescope in its reference orbit
appears in cyan in Fig. 12. Then, the path of the occulter controlled via the LQR
controller is plotted in blue for the three different starting times. For a given starting time, the LQR controller tracks the reference path, which corresponds to the
path of the inertial target star in the rotating frame at 50,000 km from the telescope
orbit. With knowledge of the low drift zones, the best correlation between locations
along the telescope orbit and the occulter drift is identified to ensure that the star
direction lies in a blue region. The strategic time of observation for this particular
star was determined as time t = 7 days using the low drift regions. For comparison, two other starting times, t = 80 days and t = 120 days are also plotted in
Fig. 12. The direction of the star at each of these starting times is represented by
a red star. The observation phase lasts for 20 days in this example. The total cost
associated with each of these observations is illustrated in Fig. 13. The cost of observation is reduced if the observation begins around time t = 7 days, the time that
corresponds to the low drift direction. Therefore, observing the stars at the strategic times via the low drift zones reduces the control effort during the observation
phase.
t = 0 days
5
z rotating (km)
x 10
t = 7 days
2
0
t = 120 days
−2
1.515
t = 80 days
5
5
1.51
0
−5
x 10
y rotating (km)
8
x 10
1.505
x rotating (km)
Fig. 12 Path of the occulter for the observation of the same target star originating at different locations
around the telescope orbit
J of Astronaut Sci
45
DV for LQR controller (m/s)
40
35
30
25
20
15
10
0
20
t = 7 days
Low drift region
40
60
80
t = 80 days
High drift region
100
120
140
160
180
t = 120 days
Medium drift region
Starting times along telescope orbit (days)
Fig. 13 Cost associated with the observation of the selected target star at the different starting times
Design with Reconfiguration and Observation Phases
Given a set of inertial target stars, a star sequence design process is proposed for both
the observation and the reconfiguration phases. As the observation of target stars
which possess long observation intervals increases the total cost of observation, the
observation phase is designed such that the occulter is located in a low drift zone
at the time of the long observations. The remaining target stars are then placed as
determined for the reconfiguration phase, such that the overall cost remains as low
as possible without regard for any science requirements. A linear quadratic regulator
maintains the occulter during the observation phase along the telescope line-of-sight
to some inertially fixed star directions, and impulsive maneuvers are applied for
realignment during the reconfiguration phase.
Reconfiguration Phase
The reconfiguration phase consists of realigning the occulter between each observation from one target star line-of-sight to the next. As an impulsive change
in velocity is potentially the easiest control strategy to implement using available chemical thrusters, impulsive maneuvers are applied for the reconfiguration
phase. Consider the general form of the discretized solution to the linear system in
Eq. 6
δXk+1 = (tk+1 , tk )δXk
(32)
The state is partitioned into position and velocity vector components, δr and δv,
respectively, such that
J of Astronaut Sci
δrk+1
δv−
k+1
δrk+1
δv−
k+1
=
= (tk+1 , tk )
rr rv
vr vv
δrk
δu+
k
δrk
δv−
k + Vk
(33)
(34)
where the superscripts (plus or minus) imply the beginning or the end of a segment originating at time tk , respectively, and Vk represents an impulsive maneuver
applied at tk . To accomplish the desired change in position, δrk+1 , the required
impulsive change in velocity is expressed as
−
Vk = −1
rv (δrk+1 − rr δrk ) − δvk
(35)
This approach to the computation of a maneuver Vk represents, of course, a simple
targeter. Kolemen and Kasdin focus on the reconfiguration phase for the telescopeocculter mission and investigate trajectory optimization of the occulter path between
each observation phase with a different approach and employ optimization [9]. In
particular, given a fixed transfer time of two weeks, the optimal cost is derived with
respect to two parameters, the distance between the spacecraft as well as the line-ofsight angle between consecutive star directions. As these two parameters get larger,
the optimal cost increases. Similarly, in this analysis, Fig. 14 illustrates the cost
of reconfiguration determined using the two impulsive maneuvers with respect to
the time of reconfiguration in days and the line-of-sight angle between consecutive
star directions in degrees for the telescope-occulter distance of 50,000 km. No optimization is incorporated. The results obtained are comparable to the optimal costs
computed from Kolemen and Kasdin given the same fixed transfer time and same
distance between the spacecraft. Notice that for a given transfer time, the cost of
reconfiguration is significantly reduced if the line-of-sight angle between consecutive stars is small. Hence, for the design of the star sequence, the line-of-sight angle
between the set of stars is computed to assure that the cost of reconfiguration stays
relatively small.
Fig. 14 Cost of reconfiguration with respect to the time of reconfiguration and to the line-of-sight angle
between consecutive star directions
J of Astronaut Sci
Design of a Star Sequence with Both Observation and Reconfiguration phases
Given a set of inertial target stars with their respective intervals of observation, the
design of a star sequence is proposed that includes both observation and reconfiguration phases. This design does not incorporate any science objectives. Some
stars denoted ‘long observation’ are assumed to possess significantly longer times of
observation, and the remaining ones, denoted ‘short observation’ are assumed to possess short intervals of observation in comparison. The different steps for the design
of the star sequence are described below:
•
•
•
Timing the long observations to coincide with the low natural drift regions. The
long observations are the largest contributors to the total cost during the observation phase, especially if the occulter is located in a high natural drift zone at the
time of the observation. Therefore, with knowledge of the low drift zones, the
best correlation between locations along the telescope orbit and the occulter drift
is identified to ensure that a star direction, i.e., an inertially fixed direction, with
long observation time lies in a low drift zone. Observing the stars at these different times reduces the cost for the control during the observation phase. For each
long observation, the best time of observation is determined along one period of
the telescope orbit. If some starting times are determined to be too close to allow
the required length of observation, the observation is postponed for one period
of the telescope orbit.
Determining the number of short observations to place between each long
observation. The number of short observations to assign between each long
observation interval is based on their respective interval of observation and the
initial reconfiguration time. If the required number is higher than the actual number of short observations, the time of reconfiguration is increased to assure the
correct number of short observations.
Selecting the short observations based upon their line-of-sight angle. The reconfiguration phase detailed previously demonstrates that the cost of reconfiguration
is reduced if the angle between each consecutive star is small. Hence, each short
observation is selected such that the angle between the consecutive star is relatively small. Notice that the inertial target star has a direction that changes in
time as observed in the rotating frame and, therefore, the angle also changes in
time depending on the starting time of the observation.
Simulation of the Star Sequence and Cost Comparison with Random Star Sequences
The effectiveness of this design process is now examined for a mission scenario with
a set of 10 inertially fixed target stars with their respective intervals of observation.
Half of the observation phases are long, i.e, the time of observation is 30 days, and
the other half, the short observations, all require a short time of observation of 2 days.
The two phases of the mission are controlled via different control schemes:
•
An LQR controller tracks a reference trajectory during the observation phase.
This reference arc represents the path of each inertial target star, and the occulter
J of Astronaut Sci
5
z rotating (km)
x 10
2
0
−2
5
5
x 10
0
−5
1.515
1.51
1.505
8
x 10
x rotating (km)
y rotating (km)
Reconfiguration phases
Observation phases
Fig. 15 Simulation of the observation/reconfiguration phases for 10 inertial target stars
•
must follow this reference accurately to maintain its position along the line-ofsight from the telescope to the target star. This control strategy is similar to the
one described in the previous section.
Impulsive maneuvers are assumed for the reconfiguration phase. The occulter is
realigned between each observation from one target star line-of-sight to the next
using two impulsive maneuvers.
The star sequence is designed and the results are illustrated in Fig. 15. Each blue
arc represents the path of the occulter during each observation phase, and each red
arc reflects the path of the occulter during each reconfiguration phase. The associated costs for each target star are plotted in Fig. 16. The cost of observation appears
in blue, the cost of reconfiguration is plotted in red and pink for respectively, the
initial and final impulsive maneuvers and the total cost (reconfiguration and observation) is indicated in black. For this particular set of stars, the total cost of the
100
DV (m/s)
80
60
40
20
0
0
2
4
6
Star Sequence
8
10
Cost of observation
Cost of reconfiguration (DV initial)
Cost of reconfiguration (DV final)
Total cost (reconfiguration + observation)
Fig. 16 Cost associated with the different phases of the mission for each of the 10 stars
J of Astronaut Sci
sequence computed is approximately 590 m/s, that is 120 m/s for the observation
phase and 470 m/s for the reconfiguration phase with an average time of reconfiguration of about 25 days. The total duration of the sequence is equal to 480
days.
These results are compared by generating 100 random sequences of the same
given stars with their respective times of observation. The times of reconfiguration
are selected to be the same as the ones determined from the previous design method
for comparison. In Fig. 17 each dot represents the total cost for each random star
sequence that is generated. The cost of observation, reconfiguration and the total
(observation + reconfiguration) are represented by blue dots, pink dots, and black
dots respectively, in Fig. 17. The big red dot in each plot represents the result obtained
from the design process previously described using the low drift regions. By applying
this design process, the cost of observation is always small compared to the random
star sequences. The cost of reconfiguration is usually smaller than the mean value of
the cost for these random sequences, although the design method does not guarantee
such a result.
200
700
190
650
Cost of reconfiguration (m/s)
Cost of observation (m/s)
180
170
160
150
140
130
120
550
500
450
400
350
110
100
0
600
20
40
60
80
100 different sequences of the same set of target stars
100
300
0
20
40
60
80
100 different sequences of the same set of target stars
(a)
(b)
Total cost (observation and reconfiguration) (m/s)
900
850
800
750
700
650
600
550
500
0
20
40
60
80
100 different sequences of the same set of target stars
100
(c)
Fig. 17 Cost comparison with 100 random sequences generated with the same target stars
100
J of Astronaut Sci
Summary and Concluding Remarks
In this present study, quasi-periodic Lissajous trajectories in the collinear libration
point region are employed as a tool to determine regions near the telescope orbit
where a large formation of spacecraft can be relatively easily maintained. The low
natural drift regions derived from the computation of these orbits, aid in the control
of large formations. By placing the occulter in these locations, the control required
to maintain the telescope-occulter distance, as well as the pointing direction toward a
star is demonstrated to be reduced. Finally, given a set of inertial target stars, an automatic star sequence design process is proposed with observation and reconfiguration
phases using the low natural drift regions. This design is demonstrated to create star
sequences that lead to a relatively small overall cost for the mission.
Optimal star sequences can be produced via classical optimization techniques.
However, most of these approaches do not incorporate the natural dynamics in the
multi-body regime. Understanding the existing environment can lead to the development of techniques and aid in the design of mission scenarios, while exploiting
the natural structure of the phase space. Although this work was motivated by
the telescope-occulter mission design, it also provides some insight on the relative
motion between vehicles separated by a long distance and this knowledge is also
useful for other applications.
Future work includes analyzing the evolution of the low drift regions as parameters
change. As the telescope-occulter distance varies, the low drift regions vary and their
evolution can be analyzed for smaller and larger telescope-occulter distances. If the
radius of the space sphere decreases, the low drift zones could potentially expand
and may exist for a wider range of observations. Also, the telescope orbit in this
investigation is a Lissajous orbit with relative amplitudes very close to a halo orbit
and, therefore, the space spheres derived on it are approximatively the same for each
Lissajous revolution. A more quasi-periodic trajectory may yield new space spheres
with features that vary from one revolution to the next.
Acknowledgments The authors wish to thank Purdue University and the School of Aeronautics and
Astronautics for providing financial support. The computational capabilities available in the Rune and
Barbara Eliasen Aerospace Visualization Laboratory are also appreciated.
References
1. Cash, W., the New Worlds Team: The new worlds observer: the astrophysics strategic mission concept
study. In: EPJ Web of Conferences, vol. 16 (2011). doi:10.1051/epjconf/20111607004
2. Barden, B.T., Howell, K.C.: Fundamental motions near collinear libration points and their transitions.
J. Astronaut. Sci. 46, 361–378 (1998)
3. Howell, K.C., Marchand, B.G.: Natural and non-natural spacecraft formations near the L1 and L2
libration points in the Sun-Earth/Moon ephemeris system. Dyn Syst: an International Journal, Special
Issue: Dynamical Systems in Dynamical Astronomy and Space Mission Design 20, 149–173 (2005)
4. Gómez, G., Lo, M., Masdemont, J., Museth, K. Simulation of formation flight near L2 for the TPF
mission. In: AAS Astrodynamics Specialist Conference, Quebec City, Canada. Paper No. AAS 01-305
(2001)
J of Astronaut Sci
5. Marchand, B.G., Howell, K.C.: Control strategies for formation flight in the vicinity of the libration
points. J. Guid. Control Dyn. 28, 1210–1219 (2005)
6. Gómez, G., Marcote, M., Masdemont, J., Mondelo, J.: Zero relative radial acceleration cones and
controlled motions suitable for formation flying. J. Astronaut. Sci. 53(4), 413–431 (2005)
7. Lo, M.: External occulter trajectory study. NASA JPL (2006)
8. Millard, L., Howell, K.C.: Optimal reconfiguration maneuvers for spacecraft imaging arrays in multibody regimes. Acta Astronaut. 63, 1283–1298 (2008). doi:10.1016/j.actaastro.2008.05
9. Kolemen, E., Kasdin, J.: Optimal trajectory control of an occulter-based planet-finding telescope. Adv.
Astronaut. Sci. 128, 215–233 (2007). Paper No. AAS 07-037
10. Héritier, A., Howell, K.C.: Natural regions near the sun-earth libration points suitable for space observations with large formations. In: AAS Astrodynamics Specialist Conference. Girdwood, Alaska.
Paper No. AAS 11-493 (2011)
11. Luquette, R., Sanner, R.: Spacecraft formation control: managing line-of-sight drift based on the
dynamics of relative motion. In: 3rd International Symposium on Formation Flying, Missions, and
Technologies, Noordwijk, The Netherlands (2008)