J of Astronaut SciDOI 10.1007/s40295-014-0027-8

Natural Regions Near the Collinear Libration PointsIdeal for Space Observations with Large Formations

Aurelie Heritier · Kathleen C. Howell

© American Astronautical Society 2014

Abstract This investigation explores regions near libration points that might provesuitable for space observations with large formations. Recent analyses have con-sidered occulters located at relatively large distances from the telescope near theL2 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 todetermine regions near the telescope orbit where the large formation can be main-tained. By placing the occulter in these locations, the control required to maintain theline-of-sight is reduced.

Introduction

In the last decade, hundreds of planets orbiting other stars, called extrasolar planetsor exoplanets, have been detected. Thus far, all of them are gas giants like Jupiter, butimprovements in technology are moving the detection limits to planets with smallermasses. Using multiple spacecraft to create a large observation platform enables thedetection of smaller and smaller planets. Additional investigations on formation fly-ing in multi-body regimes have emerged to support space astronomy missions. Somenew concepts using large formations can detect not only Earth-like planets, but can

A. Heritier (�)Advanced Concepts Team, ESA/ESTEC, PPC-PF,Keplerlaan 1, 2201 AZ Noordwijk, The Netherlandse-mail: relieheritier@hotmail.fr

K. C. HowellSchool of Aeronautics and Astronautics, Purdue University,West Lafayette, IN 47907, USAe-mail: howell@purdue.edu

J of Astronaut Sci

also characterize them via spectroscopy, providing information such as atmosphericconditions, internal structure, mass estimates, as well as life signs. For example, theoriginal New Worlds Observer (NWO) design concept employed an external occul-ter placed at a relatively large distance (∼80,000 km) from its telescope for thedetection and characterization of extrasolar planets [1]. In the science mode, theocculter is maintained along the line-of-sight from the telescope to the target star toblock out the starlight. It suppresses the starlight by many orders of magnitude, toenable the observation of habitable terrestrial planets.

From a control perspective, the telescope-occulter architecture concept can bedecomposed into two mission phases.

• The observation phase: the occulter must be maintained precisely along thetelescope line-of-sight to some inertially fixed target stars.

• The reconfiguration phase: the occulter is realigned between each observationfrom one target star line-of-sight to the next.

During the observation mode, the two spacecraft must be aligned within a few metersalong the line-of-sight. This is most easily accomplished if these spacecraft are ina low-acceleration environment such as the vicinity of the Sun-Earth L2 librationpoint, the future home of astrophysical observatories. Ideally, regions with zero rel-ative velocity and zero relative radial acceleration maintain the formation, and fora small distance between the spacecraft, these regions can be determined via linearanalysis. However, for a large telescope-occulter distance, up to tens of thousands ofkilometers, linearization relative to the telescope orbit may no longer be an acceptableplanning option.

The Sun-Earth L2 libration point region, providing a low-acceleration environ-ment that is ideal for astronomical instruments, has been a popular destination forsatellite imaging formations. Such imaging scenarios were the original motivation forthis work. Barden and Howell investigate the natural behavior on the center manifoldnear the libration points and compute some natural six-spacecraft formations, whichdemonstrate that quasi-periodic trajectories could be useful for formation flying [2].Later, Marchand and Howell extend this study and use some control strategies, con-tinuous and discrete, to maintain non-natural formations near the libration points [3].Most of the formation flying missions have been considering spacecraft at a relativelysmall distance from the reference orbit, however. Space-based observatory and inter-ferometry missions, such as the Terrestrial Planet Finder, have been the motivationfor the analysis of many control strategies. Gomez et al. investigate discrete con-trol methods to maintain such a formation [4]. Howell and Marchand consider linearoptimal control, as applied to nonlinear time-varying systems, as well as nonlinearcontrol techniques, including input and output feedback linearization [5]. Recently,Gomez et al. derive regions around a halo orbit with zero relative velocity and zerorelative radial acceleration that ideally maintain the mutual distances between space-craft [6]. Their analysis, based on linearization relative to the reference orbit, assumesmall formations of spacecraft. Lo examines the Terrestrial Planet Finder architecturewith an external occulter at a much larger relative distance, and considers differentmission scenarios with the formation placed on a halo orbit around the Sun-Earth L2libration 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-Occultermission for the two phases of the mission scenario [8]. They compare differentcontrol methods for satellite imaging arrays in multi-body systems such as optimalnonlinear control, geometric control methods, a linear quadratic regulator and inputstate feedback linearization. Kolemen and Kasdin focus on the realignment problemand investigate optimal control strategies for the reconfiguration phase, to enablethe imaging of the largest possible number of planetary systems with the minimummass requirement [9]. Their analysis does not exploit the natural dynamics specificto the region near the libration points. Most recently, Heritier and Howell investigatethe natural dynamics in the collinear libration point region for the control of largeformations [10]. In the current work, this analysis is extended.

The goal in the present investigation is the further exploration of the naturaldynamics in the collinear libration point region to aid in the control of large forma-tions. Examining the dynamical environment, and better understanding the flow inthe vicinity of the telescope orbit, assists in the design of mission scenarios. Quasi-periodic Lissajous trajectories are employed as a new tool to determine regions nearthe telescope orbit where the large formation can be relatively easily maintained.Studying these trajectories yields some insight into the behavior of a formation ofspacecraft in this region. First, extensive computations of quasi-periodic Lissajoustrajectories near the telescope orbit are completed. Arcs along the Lissajous trajecto-ries are analyzed and viewing spheres at various points along the telescope orbit aredeveloped. These space spheres are used as a tool to categorize regions along the orbitwith less natural drift when the distance between two vehicles is large. Locating theocculter 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 isthen tested for the observation of an inertially fixed target star at different times alongthe telescope orbit. A linear quadratic regulator is used to maintain the occulter alongthe telescope line-of-sight to the inertially fixed star. If the observation of the tar-get 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 ofinertial target stars, a star sequence design process is proposed with both observationand reconfiguration phases. Impulsive maneuvers are applied for realignment dur-ing the reconfiguration phase. As the observations of target stars which possess longobservation intervals increase the total cost of observation, the observation phase isdesigned such that the occulter is located in a low drift zone during the long observa-tions. The remaining target stars are then placed consistent with the reconfigurationphase, 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 restrictedthree-body problem (CR3BP). In this model, it is assumed that the Sun and the Earthmove in a circular orbit around their barycenter. The mass of the spacecraft (P3) isnegligible compared to the masses of the two primaries. The equations of motion aredescribed 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 ageneral vector in the rotating frame from B to P3, i.e.

X = [x y z x y z]T (1)

where superscript ‘T ’ implies transpose. The equations of motion are expressed interms of a pseudo potential function

U = (1 − μ)

r1+ μ

r2+ 1

2(x2 + y2) (2)

with r1 = √(x + μ)2 + y2 + z2 and r2 = √

(x − (1 − μ))2 + y2 + z2, and whereμ is the mass parameter associated with the Sun-Earth system. The scalar nonlinearequations of motion in their non-dimensional forms are then written as

x − 2y = Ux (3)

y + 2x = Uy (4)

z = Uz (5)

where Uj represents the partials ∂U∂j

for j = x, y, z. One advantage of the CR3BP asthe framework for this analysis lies in its dynamical properties that may be exploitedfor mission design. The equations of motion in the CR3BP possess five equilibriumsolutions, or libration points. These consist of the three collinear points (L1, L2, andL3) and the two equilateral points (L4 and L5). In the vicinity of each equilibriumpoint, some structure can be identified: periodic and quasi-periodic orbits (centermanifold), as well as unstable and stable manifolds. Knowledge of this natural flowis very useful for trajectory design.

The design process generally relies on variations relative to a reference arc. Givena solution to the nonlinear differential equations, linear variational equations ofmotion are derived in matrix form as

δX = A(t)δX (6)

where δX = [δx δy δz δx δy δz]T represents variations about a referencetrajectory. The A(t) matrix is time-varying of the form

A(t) =[

03×3 I3×3F J

](7)

where the F matrix and J matrix are defined as

F =⎡

⎣Uxx Uxy Uxz

Uyx Uyy Uyz

Uzx Uzy Uzz

⎤

⎦ (8)

J =⎡

⎣0 2 0

−2 0 00 0 0

⎤

⎦ (9)

The partials Ujk represent ∂U2

∂j∂kfor 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 linearmapping that approximates the impact of the initial variations on the variationsdownstream. The STM satisfies the matrix differential equation

�(t, t0) = A(t)�(t, t0) (11)

�(t0, t0) = I6,6 (12)

where I6,6 is the identity matrix. Equation 11 is integrated simultaneously with thenonlinear equations of motion to generate reference states and updates to the trajec-tory. 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, hencea total of 42 differential equations.

Space Spheres

The first objective in this study is a better understanding of the dynamicalenvironment at a relatively large distance from the telescope orbit. The CR3BP modelpossesses dynamical properties that may be exploited for mission design. In the cur-rent scenario, the telescope path evolves along a halo orbit in the vicinity of theSun-Earth L2 libration point as illustrated in Fig. 1. A fixed distance between thetelescope 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 thetelescope orbit is not recommended. Instead, natural quasi-periodic Lissajous trajec-tories are employed as a tool to determine regions in the vicinity of the telescope orbitwhere a large formation yields small relative motion. Such trajectories yield someinsight into the behavior of a large formation of spacecraft in this region.

First, extensive computations of Lissajous trajectories near the telescope orbit arecompleted. Then, arcs along the Lissajous trajectories that correspond to the appro-priate telescope-occulter distances are analyzed. At each instant of time, the velocityvector along the telescope orbit and the corresponding Lissajous velocity vector arecompared as illustrated in Fig. 2. The state on the Lissajous arc, that corresponds tothe state of the occulter, is located at a distance of 50,000 km from the state alongthe telescope orbit. Two parameters are derived: the difference in the norm of the twovelocities, i.e., between the velocity vector along the telescope orbit and the velocity

1.5051.51

1.515

x 108

−50

5

x 105

−2

0

2

x 105

x rotating(km)y rotating (km)

z rot

atin

g (k

m)

L2

t = 0 daysA

x ~ 200 960 km

Ay ~ 686 200 km

Az ~ 148 900 km

Fig. 1 Reference orbit for the telescope (period ∼180 days)

J of Astronaut Sci

1. Difference in velocity norm

2. Angle between velocity vectors

refV

lissV

Telescope-occulter

vector direction

~ 50,000 km

Natural drift

0t

0t

1t refliss VVV −=Δ

refVV

Natural drift

1t

lissV

φ

Fig. 2 Schematic of the velocity vector comparison

vector on the Lissajous arc, as well as the angle between the two velocity vectors ateach location around the telescope orbit. The natural drift of the telescope-occulterline-of-sight from its initial orientation, as defined in red in Fig. 2, depends on thesize of these two parameters: difference in velocity norm and angle between veloc-ity vectors. Figure 3 illustrates the variation of the telescope-occulter line-of-sightvector after a specific time interval with respect to these two parameters. The timeinterval for star observation may vary between 1 day to 40 days depending on the starlocation and the method employed (detection or characterization). The time intervalselected for this investigation is 3 days. The initial state considered is selected at timet = 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 isderived as illustrated in Fig. 4. A vector originating at the center of the sphere (pointon the reference orbit) and terminating at a point on the sphere defines a line-of-sightdirection. On the surface of each sphere, different zones are defined that identifytelescope-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 tothe 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 cal-culated after a three-day time interval. Hence, each color represents some maximumvalues for the two initial velocity parameters: the difference in the initial velocitynorm and the initial angle between the velocity vectors. One space sphere at an iso-lated time appears in Fig. 5. The blue zones represent regions with less natural driftthan the red zones. Effectively, if the line-of-sight to a star is directed along a linefrom the origin of the sphere to a blue dot, an occulter along this line experiences lessnatural drift than a star line-of-sight through a red dot. Locating the occulter in theregions with small natural drift leads to a smaller variation in the telescope-occultervector and, therefore, to a potential reduction in the control effort to maintain theformation 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 differ-ent 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 thetelescope orbit are represented in Fig. 6. These dark blue rings reflect a differencein the initial velocity norm that is less than 5 m/s and an initial angle between thevelocity vectors of less than 5 deg, that corresponds to the dark blue regions from thenatural drift surface in Fig. 3. In addition to the low natural drift zones, the eigen-structure of the F matrix defined in Eq. 8 is also illustrated in Fig. 6. The F matrixpossesses three orthogonal eigenvectors defined by V1, V2 and V3. Two of them, V1and V2, are plotted in blue and red, respectively. The third one, V3, is mainly directedalong the z-axis. These eigenvectors represent the principal directions of the low driftregions. This is demonstrated by studying the variational equations defined in Eq. 6and by deriving some regions suitable to maintain a small formation of spacecraft.

Assuming that the separation between the spacecraft is “small” compared to theradius of the reference orbit, i.e, no greater than a few kilometers maximum, regionsof low drift are computed via the first-order variational equations with respect to thereference orbit, and an analytical expression for them is determined. Equation 6 isrewritten in terms of relative position and relative velocity vectors as

[δrδr

]=

[03×3 I3×3F J

] [δrδr

](13)

where F and J are defined in Eqs. 8 and 9, respectively. As the spacecraft are placedat a small distance from each other, the relative velocity is assumed to be equal tozero, i.e., δr = 0. Notice that this assumption is only valid for a small formationof spacecraft. At large distances, the relative velocity may not be negligible. FromEq. 13, the relative acceleration is then

δr = Fδr (14)

1.502 1.504 1.506 1.508 1.51 1.512 1.514 1.516 1.518 1.52

x 108

−6

−4

−2

0

2

4

6

x 105

x rotating (km)

y ro

tatin

g (k

m)

(a)

1.5078 1.508 1.5082 1.5084 1.5086 1.5088 1.509

x 108

1.8

2

2.2

2.4

2.6

2.8

3

x 105

x rotating (km)

y ro

tatin

g (k

m)

V1

V2

(b)

Fig. 6 a Low drift regions on the space spheres at different times along the telescope orbit; b One lowdrift region at time t = 100 days

J of Astronaut Sci

The norm of the relative acceleration is computed from the dot product of δr with δr.

‖δr‖2 = δr · δr = Fδr · Fδr (15)

The positions such that the relative acceleration is equal to zero, are then the set ofpoints that satisfy the equation

δrT F T Fδr = 0 (16)

Equation 16 represents quadrics with zero relative velocity and zero relative acceler-ation. These quadrics are plotted in Fig. 7. Using a change in coordinate, Eq 16 canbe transformed into its canonical form. The matrix F is a real symmetric matrix, i.e.,F = FT and therefore can be diagonalized. The singular value decomposition of F

takes the form

F = P�P T = [V1 V2 V3

]⎡

⎣λ1 0 00 λ2 00 0 λ3

⎤

⎦[V1 V2 V3

]T (17)

� 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 eigenvec-tors of F previously defined. These eigenvectors are orthogonal to each other andrepresent 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 followingcanonical form

yT P T FT FP y = yT �2y = λ21y

21 + λ2

2y22 + λ2

3y23 = 0 (18)

Although these quadrics describe the dynamics close to the reference orbit, it isinteresting to compare the similarity with the low drift regions as plotted in Fig. 6 at

1.502 1.504 1.506 1.508 1.51 1.512 1.514 1.516 1.518 1.52

x 108

−6

−4

−2

0

2

4

6

x 105

x rotating (km)

y ro

tatin

g (k

m)

(a)

1.5078 1.508 1.5082 1.5084 1.5086 1.5088

x 108

1.8

2

2.2

2.4

2.6

2.8

3

x 105

x rotating (km)

y ro

tatin

g (k

m)

V1

V2

(b)

Fig. 7 a Quadrics with zero relative velocity and zero relative acceleration determined via linear analysisat 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 suit-able to maintain a formation of spacecraft, either large or small. They are derived viasignificantly different methods, either from a linear analysis or numerically by propa-gating in the nonlinear dynamical model. The surface of the small quadrics possessesno relative velocity and no relative acceleration whereas some relative velocity andrelative acceleration still remain in the low drift regions (the difference in the veloc-ity norm is less than 5 m/s and the angle between the velocity vectors is less than5 deg). But, interestingly, they both are defined with the same orientation in spaceas represented in Fig. 7. This orientation, derived from the eigenspace of the linearsystem, seems to persist as the distance between the spacecraft increases to at least50,000 km.

In contrast to a planar projection, an alternative approach to visualizing the lowdrift 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 tothe x-axis in the rotating frame, and an out-of-plane angle β, oriented with respect tothe (x, y) plane. These angles are represented in Fig. 8. Any V vectors as illustratedin Fig. 8 can be defined as

q = q

⎡

⎣cos(θ) cos(β)

sin(θ) cos(β)

sin(β)

⎤

⎦ (19)

Some characteristics of these low drift regions are apparent by examining the in-planeangle θ . For most of the low drift directions identified by blue points, θ approacheseither 90 or −90 degrees as illustrated in Fig. 6a, and this feature is reproducedat 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 inFig. 9. Table 1 gives the mean value of the angle θ and the corresponding time alongthe reference orbit. These angles describe the orientation of the low drift regionsalong one revolution of the telescope orbit (∼180 days). For reference, the initialstate 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 ofan inertially fixed target star at different times along the telescope orbit. Althoughinertially 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 thenatural drift present at 50,000 km in the low drift regions. A linear quadratic regulatoris used to maintain the occulter along the telescope line-of-sight to the inertiallyfixed star. If the observation of the target star originates when the orientation of theformation lies in a low natural drift zone, then the control effort to maintain theformation is reduced. The control cost is considerably higher if the direction to bemaintained 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, theright ascension and the declination, defined in the equatorial frame. The direction is

Fig. 9 Low drift regionsdescribed by in-plane angle θ

t = 0 days

mean of θ

y

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, acoordinate transformation is necessary to locate the star, i.e., the target direction, ateach instant of time. The coordinate transformation from the equatorial frame to therotating frame is derived as

⎡

⎣x

y

z

⎤

⎦

CR3BP

=⎡

⎣cos α cos ε sin α sin ε sin α

− sin α cos ε cos α sin ε cos α

0 − sin ε cos ε

⎤

⎦

⎡

⎣x

y

z

⎤

⎦

Equatorial

(21)

with ε = 23.0075 degrees and α = ω + α0 where α0 represents the initial anglebetween the inertial and rotating frames (α0 is assumed to be equal to zero in thisanalysis) and the angular rate ω is equal to one in nondimensional coordinates. Thein-plane angle of an inertial target star measured relative to the x-axis in the rotatingframe is defined as γ . This in-plane angle γ varies with time in the (x, y) rotatingplane, approximatively 1 degree/day, and therefore, the direction of the target starmight 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 angleθ approaches either 90 or −90 degrees, and this feature is reproduced at each timealong one revolution of the orbit as illustrated in Fig. 6a. Hence, at each time, themean value of the angle θ is calculated and compared to the in-plane angle γ ofthe inertial target star direction. The low drift regions are ring-shaped objects, andtherefore, matching the in-plane angles also assures the out-of-plane angle of the star

J of Astronaut Sci

Fig. 10 Schematic of themethod used to determine thestarting times for observation

Star 1 at t1

Star 1 at t2

at t1

at t2Telescope

Occulter

to match an out-of-plane angle of the blue regions. The closest value between the in-plane angle of the star and the mean value of the angle θ occurs at the “best” locationalong the telescope orbit, and this location assures that the star direction lies in a lownatural drift region during the observation phase. The schematic in Fig. 10 illustratesthis process. Two space spheres are represented at time t1 and time t2. The low driftregions and high drift regions are represented in blue and red, respectively, on eachsphere. Given an inertial star direction for observation, (Star 1 in the schematic), if theobservation is initiated at time t1, the occulter lies in a high drift zone and the controlto maintain it along the line-of-sight is relatively high. However, if the observationof the same star is delayed to begin at time t2, the occulter lies in a low drift regionat this time, and therefore the cost of the control is reduced. An example with oneinertial target star with a right ascension of 80 degrees and a declination of 10 degreesappears in Fig. 11. The in-plane angle γ of the inertial target star represented in greenin 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

Star 1

y

mean of θ

x

γ

θ = in-plane angles of the low drift region

γ = in-plane angle of the inertial star direction

(a)

0 50 100 150 200−150

−100

−50

0

50

100

150

Time (days)

In−

pla

ne

angle

s (d

eg)

mean of θ > 0

mean of θ < 0

γ

(b)

Fig. 11 Best locations along the telescope orbit to begin observation: example with one inertial stardirection. 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 thevalue 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, butthese low drift zones can be used as a tool from a control perspective. If the specificorientation of the formation lies in a zone of low natural drift, then the control tomaintain the orientation is reduced. A linear quadratic regulator (LQR) is applied todemonstrate this result.

The telescope is assumed to move along the reference orbit. Given an inertial targetstar direction with an observation time of 20 days, the occulter must be maintainedprecisely along the line-of-sight from the telescope to the star during the observationinterval. Particularly, the occulter is constrained to remain within ±100 km of thebaseline path in the radial direction (line-of-sight direction) and within a few metersfrom 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 occulteralong the telescope line-of-sight to the inertially fixed star. Each reference path isan arc along a suitable Lissajous trajectory. These reference arcs are computed fordifferent starting locations along the telescope orbit. The LQR controller is formu-lated to track these reference arcs during the observation period. The motion of theocculter in the CR3BP is described as

x = fx(x, x, y, y, z, z) + ux (22)

y = fy(x, x, y, y, z, z) + uy (23)

z = fz(x, x, y, y, z, z) + uz (24)

wherefx, fy, fz are defined from Eqs. 3–5 as

fx = Ux + 2y (25)

fy = Uy − 2x (26)

fz = Uz (27)

and u(t) = [ux uy uz]T is the control vector. Let X0 be some reference motionand u0 the respective control effort to maintain X0. The selected reference arcs repre-sent the baseline paths of the inertial star direction in the rotating frame. These pathsare computed at the distance of 50,000 km from the telescope, and the occulter iscontrolled to track these arcs using the LQR controller. Linearization relative to thesereference solutions yields a linear system of the form

δX = A(t)δX + B(t)δu (28)

where δX and δu are the variations relative to the reference arc X0 and its respec-tive 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 andthe required control, which is represented as the following quadratic cost function

min J = 1

2

∫ tf

t0

(δXT Q δX + δuT R δu)dt (29)

The matrices Q and R are positive definite matrices and represent weighting factorson the state error and control effort, respectively. In this analysis, weighting matrices,

Q = I6×6 (30)

R = diag([10−5 10−10 10−10]) (31)

are selected such that the error requirements in the transverse and radial directionsare satisfied [8]. The observation of the given target star is computed at three dif-ferent starting times around the telescope orbit. The telescope in its reference orbitappears in cyan in Fig. 12. Then, the path of the occulter controlled via the LQRcontroller is plotted in blue for the three different starting times. For a given start-ing time, the LQR controller tracks the reference path, which corresponds to thepath of the inertial target star in the rotating frame at 50,000 km from the telescopeorbit. With knowledge of the low drift zones, the best correlation between locationsalong the telescope orbit and the occulter drift is identified to ensure that the stardirection lies in a blue region. The strategic time of observation for this particularstar was determined as time t = 7 days using the low drift regions. For compar-ison, two other starting times, t = 80 days and t = 120 days are also plotted inFig. 12. The direction of the star at each of these starting times is represented bya red star. The observation phase lasts for 20 days in this example. The total costassociated with each of these observations is illustrated in Fig. 13. The cost of obser-vation is reduced if the observation begins around time t = 7 days, the time thatcorresponds to the low drift direction. Therefore, observing the stars at the strate-gic times via the low drift zones reduces the control effort during the observationphase.

1.5051.51

1.515

x 108

−50

5

x 105

−2

0

2

x 105

x rotating (km)y rotating (km)

z rot

atin

g (k

m) t = 7 days

t = 80 days

t = 120 days

t = 0 days

Fig. 12 Path of the occulter for the observation of the same target star originating at different locationsaround the telescope orbit

J of Astronaut Sci

0 20 40 60 80 100 120 140 160 18010

15

20

25

30

35

40

45

Starting times along telescope orbit (days)

DV

for

LQ

R c

ontr

oll

er (

m/s

)

t = 7 daysLow drift region

t = 80 daysHigh drift region

t = 120 daysMedium drift region

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 boththe observation and the reconfiguration phases. As the observation of target starswhich possess long observation intervals increases the total cost of observation, theobservation phase is designed such that the occulter is located in a low drift zoneat the time of the long observations. The remaining target stars are then placed asdetermined for the reconfiguration phase, such that the overall cost remains as lowas possible without regard for any science requirements. A linear quadratic regulatormaintains the occulter during the observation phase along the telescope line-of-sightto some inertially fixed star directions, and impulsive maneuvers are applied forrealignment during the reconfiguration phase.

Reconfiguration Phase

The reconfiguration phase consists of realigning the occulter between each obser-vation from one target star line-of-sight to the next. As an impulsive changein velocity is potentially the easiest control strategy to implement using avail-able chemical thrusters, impulsive maneuvers are applied for the reconfigurationphase. Consider the general form of the discretized solution to the linear system inEq. 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

]= �(tk+1, tk)

[δrk

δu+k

](33)

[δrk+1

δv−k+1

]=

[�rr �rv

�vr �vv

] [δrk

δv−k + �Vk

](34)

where the superscripts (plus or minus) imply the beginning or the end of a seg-ment originating at time tk , respectively, and �Vk represents an impulsive maneuverapplied at tk . To accomplish the desired change in position, δrk+1, the requiredimpulsive change in velocity is expressed as

�Vk = �−1rv (δrk+1 − �rrδrk) − δv−

k (35)

This approach to the computation of a maneuver �Vk represents, of course, a simpletargeter. Kolemen and Kasdin focus on the reconfiguration phase for the telescope-occulter mission and investigate trajectory optimization of the occulter path betweeneach observation phase with a different approach and employ optimization [9]. Inparticular, given a fixed transfer time of two weeks, the optimal cost is derived withrespect to two parameters, the distance between the spacecraft as well as the line-of-sight angle between consecutive star directions. As these two parameters get larger,the optimal cost increases. Similarly, in this analysis, Fig. 14 illustrates the costof reconfiguration determined using the two impulsive maneuvers with respect tothe time of reconfiguration in days and the line-of-sight angle between consecutivestar directions in degrees for the telescope-occulter distance of 50,000 km. No opti-mization is incorporated. The results obtained are comparable to the optimal costscomputed from Kolemen and Kasdin given the same fixed transfer time and samedistance between the spacecraft. Notice that for a given transfer time, the cost ofreconfiguration is significantly reduced if the line-of-sight angle between consecu-tive stars is small. Hence, for the design of the star sequence, the line-of-sight anglebetween the set of stars is computed to assure that the cost of reconfiguration staysrelatively small.

Fig. 14 Cost of reconfiguration with respect to the time of reconfiguration and to the line-of-sight anglebetween 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, thedesign of a star sequence is proposed that includes both observation and recon-figuration phases. This design does not incorporate any science objectives. Somestars denoted ‘long observation’ are assumed to possess significantly longer times ofobservation, and the remaining ones, denoted ‘short observation’ are assumed to pos-sess short intervals of observation in comparison. The different steps for the designof the star sequence are described below:

• Timing the long observations to coincide with the low natural drift regions. Thelong observations are the largest contributors to the total cost during the observa-tion phase, especially if the occulter is located in a high natural drift zone at thetime of the observation. Therefore, with knowledge of the low drift zones, thebest correlation between locations along the telescope orbit and the occulter driftis identified to ensure that a star direction, i.e., an inertially fixed direction, withlong observation time lies in a low drift zone. Observing the stars at these differ-ent times reduces the cost for the control during the observation phase. For eachlong observation, the best time of observation is determined along one period ofthe telescope orbit. If some starting times are determined to be too close to allowthe required length of observation, the observation is postponed for one periodof the telescope orbit.

• Determining the number of short observations to place between each longobservation. The number of short observations to assign between each longobservation interval is based on their respective interval of observation and theinitial reconfiguration time. If the required number is higher than the actual num-ber of short observations, the time of reconfiguration is increased to assure thecorrect number of short observations.

• Selecting the short observations based upon their line-of-sight angle. The recon-figuration phase detailed previously demonstrates that the cost of reconfigurationis reduced if the angle between each consecutive star is small. Hence, each shortobservation is selected such that the angle between the consecutive star is rel-atively small. Notice that the inertial target star has a direction that changes intime as observed in the rotating frame and, therefore, the angle also changes intime 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 witha 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, andthe 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

1.5051.51

1.515

x 108

−50

5

x 105

−2

0

2

x 105

x rotating (km)y rotating (km)

z ro

tatin

g (k

m)

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-of-sight from the telescope to the target star. This control strategy is similar to theone described in the previous section.

• Impulsive maneuvers are assumed for the reconfiguration phase. The occulter isrealigned between each observation from one target star line-of-sight to the nextusing two impulsive maneuvers.

The star sequence is designed and the results are illustrated in Fig. 15. Each bluearc represents the path of the occulter during each observation phase, and each redarc reflects the path of the occulter during each reconfiguration phase. The associ-ated costs for each target star are plotted in Fig. 16. The cost of observation appearsin blue, the cost of reconfiguration is plotted in red and pink for respectively, theinitial and final impulsive maneuvers and the total cost (reconfiguration and obser-vation) is indicated in black. For this particular set of stars, the total cost of the

0 2 4 6 8 100

20

40

60

80

100

Star Sequence

DV

(m

/s)

Total cost (reconfiguration + observation)Cost of reconfiguration (DV final)

Cost of reconfiguration (DV initial)Cost of 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 observationphase and 470 m/s for the reconfiguration phase with an average time of recon-figuration of about 25 days. The total duration of the sequence is equal to 480days.

These results are compared by generating 100 random sequences of the samegiven stars with their respective times of observation. The times of reconfigurationare selected to be the same as the ones determined from the previous design methodfor comparison. In Fig. 17 each dot represents the total cost for each random starsequence that is generated. The cost of observation, reconfiguration and the total(observation + reconfiguration) are represented by blue dots, pink dots, and blackdots respectively, in Fig. 17. The big red dot in each plot represents the result obtainedfrom the design process previously described using the low drift regions. By applyingthis design process, the cost of observation is always small compared to the randomstar sequences. The cost of reconfiguration is usually smaller than the mean value ofthe cost for these random sequences, although the design method does not guaranteesuch a result.

0 20 40 60 80 100100

110

120

130

140

150

160

170

180

190

200

100 different sequences of the same set of target stars

Cost

of

obse

rvat

ion (

m/s

)

(a)

0 20 40 60 80 100300

350

400

450

500

550

600

650

700

100 different sequences of the same set of target stars

Cost

of

reco

nfi

gura

tion (

m/s

)

(b)

0 20 40 60 80 100500

550

600

650

700

750

800

850

900

100 different sequences of the same set of target stars

To

tal

cost

(o

bse

rvat

ion

an

d r

eco

nfi

gu

rati

on

) (m

/s)

(c)

Fig. 17 Cost comparison with 100 random sequences generated with the same target stars

J of Astronaut Sci

Summary and Concluding Remarks

In this present study, quasi-periodic Lissajous trajectories in the collinear librationpoint region are employed as a tool to determine regions near the telescope orbitwhere a large formation of spacecraft can be relatively easily maintained. The lownatural drift regions derived from the computation of these orbits, aid in the controlof large formations. By placing the occulter in these locations, the control requiredto maintain the telescope-occulter distance, as well as the pointing direction toward astar is demonstrated to be reduced. Finally, given a set of inertial target stars, an auto-matic star sequence design process is proposed with observation and reconfigurationphases using the low natural drift regions. This design is demonstrated to create starsequences 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 themulti-body regime. Understanding the existing environment can lead to the devel-opment of techniques and aid in the design of mission scenarios, while exploitingthe natural structure of the phase space. Although this work was motivated bythe telescope-occulter mission design, it also provides some insight on the relativemotion between vehicles separated by a long distance and this knowledge is alsouseful for other applications.

Future work includes analyzing the evolution of the low drift regions as parameterschange. As the telescope-occulter distance varies, the low drift regions vary and theirevolution can be analyzed for smaller and larger telescope-occulter distances. If theradius of the space sphere decreases, the low drift zones could potentially expandand may exist for a wider range of observations. Also, the telescope orbit in thisinvestigation is a Lissajous orbit with relative amplitudes very close to a halo orbitand, therefore, the space spheres derived on it are approximatively the same for eachLissajous revolution. A more quasi-periodic trajectory may yield new space sphereswith features that vary from one revolution to the next.

Acknowledgments The authors wish to thank Purdue University and the School of Aeronautics andAstronautics for providing financial support. The computational capabilities available in the Rune andBarbara Eliasen Aerospace Visualization Laboratory are also appreciated.

References

1. Cash, W., the New Worlds Team: The new worlds observer: the astrophysics strategic mission conceptstudy. 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 L2libration points in the Sun-Earth/Moon ephemeris system. Dyn Syst: an International Journal, SpecialIssue: Dynamical Systems in Dynamical Astronomy and Space Mission Design 20, 149–173 (2005)

4. Gomez, G., Lo, M., Masdemont, J., Museth, K. Simulation of formation flight near L2 for the TPFmission. 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 librationpoints. J. Guid. Control Dyn. 28, 1210–1219 (2005)

6. Gomez, G., Marcote, M., Masdemont, J., Mondelo, J.: Zero relative radial acceleration cones andcontrolled 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 multi-

body regimes. Acta Astronaut. 63, 1283–1298 (2008). doi:10.1016/j.actaastro.2008.059. Kolemen, E., Kasdin, J.: Optimal trajectory control of an occulter-based planet-finding telescope. Adv.

Astronaut. Sci. 128, 215–233 (2007). Paper No. AAS 07-03710. Heritier, A., Howell, K.C.: Natural regions near the sun-earth libration points suitable for space obser-

vations 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 thedynamics of relative motion. In: 3rd International Symposium on Formation Flying, Missions, andTechnologies, Noordwijk, The Netherlands (2008)