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 L 2 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: relieheritier@hotmail.fr K. C. Howell School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA e-mail: howell@purdue.edu
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 L 2 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 L 2 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 L 2 libration point as well as a formation on a Earth leading heliocentric orbit [7]. Millard
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 (P 3 )is negligible compared to the masses of the two primaries. The equations of motion are described in a rotating coordinate frame where P 3 is located relative to the barycenter
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 P 3,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 (1 μ) U = + μ + 1 r 1 r 2 2 (x2 + y 2 ) (2) with r 1 = (x + μ) 2 + y 2 + z 2 and r 2 = (x (1 μ)) 2 + y 2 + z 2,andwhere 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 ẍ 2ẏ = U x (3) ÿ + 2ẋ = U y (4) z = U z (5) where U j represents the partials U j 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 (L 1, L 2,and L 3 ) and the two equilateral points (L 4 and L 5 ). 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 δẋ = A(t)δX (6) where δx =[δx δy δz δẋ δẏ δż] T represents variations about a reference trajectory. The A(t) matrix is time-varying of the form [ ] 03 3 I A(t) = 3 3 (7) F J where the F matrix and J matrix are defined as U xx U xy U xz F = U yx U yy U yz (8) U zx U zy U zz 0 2 0 J = 2 00 (9) 0 0 0 U 2 The partials U jk 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, t 0 )δx(t 0 ) (10)
where (t, t 0 ) 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 (t, t 0 ) = A(t) (t, t 0 ) (11) (t 0,t 0 ) = I 6,6 (12) where I 6,6 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 L 2 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 z rotating (km) x 10 5 2 0 2 5 0 x 10 5 y rotating (km) 5 t = 0 days L 2 1.505 1.51 A x ~ 200 960 km A y ~ 686 200 km A z ~ 148 900 km x rotating(km) 1.515 x 10 8 Fig. 1 Reference orbit for the telescope (period 180 days)
V ref t 0 Telescope-occulter vector direction ~ 50,000 km 1. Difference in velocity norm t1 t 0 Δ V = V liss V ref V liss 2. Angle between velocity vectors Natural drift V ref V liss t 1 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
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
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. TheF matrix possesses three orthogonal eigenvectors defined by V 1, V 2 and V 3.Twoofthem,V 1 and V 2, are plotted in blue and red, respectively. The third one, V 3, 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 = δ r [ 03 3 I 3 3 F J ][ ] δr δṙ 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 (13) δ r = Fδr (14) x 10 5 x 10 5 6 3 4 2.8 y rotating (km) 2 0 2 y rotating (km) 2.6 2.4 2.2 V 2 4 6 1.502 1.504 1.506 1.508 1.51 1.512 1.514 1.516 1.518 1.52 x rotating (km) x 10 8 (a) 2 1.8 1.5078 1.508 1.5082 1.5084 1.5086 1.5088 1.509 x rotating (km) x 10 8 (b) V 1 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
The normoftherelative acceleration is computed fromthe dotproductofδ r with δ r. δ r 2 = δ r δ r = Fδr Fδr (15) The positions such that the relative acceleration is equal to zero, are then the set of points that satisfy the equation δr T F T Fδr = 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 F = P P T = [ ] λ 1 0 0 V 1 V 2 V 3 0 λ 2 0 [ ] T V 1 V 2 V 3 (17) 0 0 λ 3 is a real diagonal matrix with λ 1,λ 2 and λ 3, the eigenvalues of F, on its diagonal. P is an orthogonal matrix whose columns V 1, V 2 and V 3 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 = [y 1 y 2 y 3 ] T such that y = P T δr. Equation 16 is then rewritten in the following canonical form y T P T F T FPy = y T 2 y = λ 2 1 y2 1 + λ2 2 y2 2 + λ2 3 y2 3 = 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 x 10 5 x 10 5 3 y rotating (km) 6 4 2 0 2 y rotating (km) 2.8 2.6 2.4 2.2 V 2 4 6 1.502 1.504 1.506 1.508 1.51 1.512 1.514 1.516 1.518 1.52 x rotating (km) x 10 8 (a) 2 1.8 1.5078 1.508 1.5082 1.5084 1.5086 1.5088 x rotating (km) x 10 8 (b) V 1 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
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-planeangleβ, oriented with respect to the (x,y) plane. These angles are represented in Fig. 8. AnyV vectors as illustrated in Fig. 8 can be defined as cos(θ) cos(β) q = q sin(θ) cos(β) (19) 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
calculated at each time along the telescope orbit as represented by the skematic in Fig. 9.Table1 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
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 x y z CR3BP cos α = cos ɛ sin α sin ɛ sin α x y z sin α cos ɛ cos α sin ɛ cos α 0 sin ɛ cos ɛ Equatorial with ɛ = 23.0075 degrees and α = ω + α 0 where α 0 represents the initial angle between the inertial and rotating frames (α 0 isassumedtobeequaltozerointhis 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 (21)
Fig. 10 Schematic of the method used to determine the starting times for observation Star 1 at t 1 at t 1 Star 1 at t 2 Telescope Occulter at t 2 to match an out-of-plane angle of the blue regions. The closest value between the inplaneangleofthe starand themean valueoftheangleθ 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 t 1 and time t 2. 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 t 1, 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 t 2, 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-planeangle 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 y 150 100 mean of θ > 0 mean of θ γ Star 1 In plane angles (deg) 50 0 50 x 100 mean of θ < 0 γ θ = in-plane angles of the low drift region γ = in-plane angle of the inertial star direction (a) 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)
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 wheref x,f y,f z are defined from Eqs. 3 5 as ẍ = f x (x, ẋ,y,ẏ,z,ż) + u x (22) ÿ = f y (x, ẋ,y,ẏ,z,ż) + u y (23) z = f z (x, ẋ,y,ẏ,z,ż) + u z (24) f x = U x + 2ẏ (25) f y = U y 2ẋ (26) f z = U z (27) and u(t) =[u x u y u z ] T is the control vector. Let X 0 be some reference motion and ū 0 the respective control effort to maintain X 0. 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 δẋ = A(t)δX + B(t)δu (28) where δx and δu are the variations relative to the reference arc X 0 and its respective control u 0. The time-varying matrix A(t) is defined in Eq. 7 and B(t) =
[0 3,3 I 3,3 ] T. The LQR controller minimizes some combination of the state error and the required control, which is represented as the following quadratic cost function min J = 1 2 tf t 0 (δx T QδX + δu T Rδu)dt (29) 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 = I 6 6 (30) R = diag([10 5 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 appearsincyaninfig.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. z rotating (km) x 10 5 2 0 t = 120 days 2 5 0 x 10 5 y rotating (km) 5 t = 0 days t = 7 days t = 80 days 1.515 1.51 1.505 x 10 8 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
45 40 DV for LQR controller (m/s) 35 30 25 20 15 10 0 20 40 60 80 100 120 140 160 180 t = 7 days Low drift region t = 80 days High drift region 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 δx k+1 = (t k+1,t k )δx k (32) The state is partitioned into position and velocity vector components, δr and δv, respectively, such that
[ δrk+1 [ δrk+1 δv k+1 δv k+1 ] [ ] δrk = (t k+1,t k ) δu + k ] [ ][ ] rr = rv δr k vr vv δv k + V k (33) (34) where the superscripts (plus or minus) imply the beginning or the end of a segment originating at time t k, respectively, and V k represents an impulsive maneuver applied at t k. To accomplish the desired change in position, δr k+1, the required impulsive change in velocity is expressed as V k = 1 rv (δr k+1 rr δr k ) δv k (35) This approach to the computation of a maneuver V k 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
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
x 10 5 z rotating (km) 2 0 2 5 0 x 10 5 y rotating (km) 5 1.505 1.51 1.515 x rotating (km) x 10 8 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 80 DV (m/s) 60 40 20 0 0 2 4 6 8 10 Star Sequence 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
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. Cost of observation (m/s) 200 190 180 170 160 150 140 130 120 110 Cost of reconfiguration (m/s) 700 650 600 550 500 450 400 350 100 0 20 40 60 80 100 100 different sequences of the same set of target stars (a) Total cost (observation and reconfiguration) (m/s) 900 850 800 750 700 650 600 550 500 0 20 40 60 80 100 100 different sequences of the same set of target stars 300 0 20 40 60 80 100 100 different sequences of the same set of target stars Fig. 17 Cost comparison with 100 random sequences generated with the same target stars (c) (b)
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 L 1 and L 2 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 L 2 for the TPF mission. In: AAS Astrodynamics Specialist Conference, Quebec City, Canada. Paper No. AAS 01-305 (2001)
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