Optimal Reconfiguration Maneuvers for Spacecraft

IACIAC----07070707----C1.7.03C1.7.03

OOOPTPTPTIMALPT IMAL RRRECONFIGURATION MMMANEUVERS FFFORFOR SSSPACECRAFT IIIMAGING AAARRAYS IN MMMULTI ---B-BBBODY RRREGIMES

Lindsay D. Millard Graduate Student School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana [email protected]

Kathleen C. Howell Hsu Lo Professor of Aeronautical and Astronautical Engineering School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana [email protected]

AAABSTRACT

Upcoming National Aeronautics and Space Administration (NASA) mission concepts include satellite arrays to facilitate imaging and identification of distant planets. These mission scenarios are diverse, including designs such as the Terrestrial Planet Finder-Occulter (TPF-O), where a monolithic telescope is aided by a single occulter spacecraft, and the Micro-Arcsecond X-ray Imaging Mission (MAXIM), where as many as sixteen spacecraft move together to form a space interferometer. Each design, however, requires precise reconfiguration and star-tracking in potentially complex dynamic regimes. This paper explores control methods for satellite imaging array reconfiguration in multi-body systems. Specifically, optimal nonlinear control and geometric control methods are derived and compared to the more traditional linear quadratic regulators, as well as to input state feedback linearization. These control strategies are implemented and evaluated for the TPF-O mission concept.

IIINTRODUCTION

Over the next two decades, one focus of NASA‘s Upcoming National Aeronautics and Space Origins program is the development of sophisticated Administration (NASA) mission concepts include telescopes and technologies to begin a search for satellite formations to facilitate imaging and extraterrestrial life and to monitor the birth of identification of distant planets. These scenarios are neighboring galaxies. Part of the program‘s mission is diverse, including designs such as the Terrestrial to (1) locate and image Earth-like planets orbiting Planet Finder-Occulter (TPF-O),2 where a monolithic other stars; and (2) determine the probability that life telescope is aided by a single occulter spacecraft, and has existed or could develop on the surface. 1 the Micro-Arcsecond X-ray Interferometry Mission (MAXIM),3 where as many as sixteen spacecraft move Traditional Earth-based telescopes must continuously together to form a space interferometer. Each design, expand in size to resolve increasingly distant and faint however, requires precise reconfiguration and star- objects. Unfortunately, as the size of a telescope tracking in potentially complex dynamic regimes. increases, it becomes too heavy and too expensive to Numerous inter-spacecraft positioning and system launch, thus motivating alternative designs for deep- geometry constraints must be satisfied to assure space observers. One such example is a free-flying successful imaging. Spacecraft separated over large array or constellation of smaller satellites that work distances must constantly be re-oriented and pointed together to form a single optical platform. at different stars, while the distance between the vehicles is expanding and contracting. All must be

1 accomplished with minimum fuel expenditure, and The goal in the present investigation is a comparison for many missions, autonomously. of control methods for array re-configuration in multi-body systems. Specifically, optimal nonlinear Extensive progress in spacecraft formation control has control as well as geometric control methods are been achieved. Previously, the focus for much of the derived and compared to more traditional linear analysis was on satellites moving under the quadratic regulators and feedback linearization. The gravitational influence of the Earth, or simply in free- control strategies are implemented and evaluated for space. 4,5 Recently, however, potential also applications the TPF-O mission concept. involve satellite arrays near the collinear libration points and, consequently, generate considerable DDDYNAMIC MMMODEL ::: CR3BP interest in formation control strategies in multi-body regimes. 6-14 The motion of a spacecraft in the CR3BP is described in terms of rotating coordinates relative to the Howell and Marchand successfully apply both discrete barycenter of the system primaries. For this analysis, and continuous control methods to deploy, enforce, emphasis is placed on formations near the libration and reconfigure formations near the vicinity of points in the Sun-Earth/Moon system. Thus, the libration points in the Sun-Earth/Moon system. 15,16 rotating x-axis is directed from the Sun toward the Specifically, these methods include input-state and Earth/Moon barycenter. The z-axis is normal to the input-output feedback linearization, linear quadratic plane of motion of the primaries, and the y-axis regulator theory, and impulsive maneuvers derived completes the right-handed triad. The system is via a differential corrections scheme. In their represented in Figure 1. examples, spacecraft follow both natural and non- natural trajectories in the Circular Restricted 3-Body xÙ yÙ inertial Problem (CR3PB). This analysis is also transitioned to yÙ P2 an ephemeris model, including solar radiation L2 pressure. L1

r2 Marchand and Howell further indicate that the required discrete velocity changes for individual G satellites within a formation are often impossibly xÙ 17 P1 inertial small, considering current thruster capabilities. r1 Carlson, Pernicka, and Balakrishnan expand upon this result via the development of a method for identification of formation size and control tolerances for which impulsive maneuvers become a practical Figure 1: Circular Restricted Three-Body Problem. option. 18,19 This research focuses on the analysis of discrete maneuvering techniques using differential Let Χ = [ x y z x& y& z& ] T , then the general non- corrections in the CR3BP. dimensional equations of motion of a spacecraft, relative to the system barycenter, are Very recently, control of interferometric satellite arrays has been pursued. Because of the complexity of 1 − x + x − x − x&& − 2y& − x = − ( µ)( µ) − µ ( ( µ)) [1] 3 3 the problem, much of the available research assumes r1 r2 satellites within the formation are moving through 1 − y y y&& + 2x& − y = − ()µ − µ 20-25 3 3 free space. However, Millard and Howell develop r1 r2 [2 ] two control strategies that are advantageous for 1 − z z z&& = − ()µ − µ 3 3 interferometric imaging arrays in multi-body regimes. r1 r2 [3 ] A continuous geometric control algorithm is based on the dynamical characteristics of the phase space near where periodic orbits. 26 By adjusting parameters in the 2/1 control algorithm appropriately, satellites in the r = x + 2 + y 2 + z 2 1 (( µ) ) formation follow trajectories that aid image r = x − 1− 2 + y 2 + z 2 2 ()()µ reconstruction. Secondly, a nonlinear, locally-optimal () control law is developed that minimizes fuel while optimizing image resolution in the CR3BP. 27

2 The quantity is a non-dimensional mass parameter For the nominal Lissajous orbit, A(t) is time-varying associated with the system. For the Sun-Earth/Moon and of the form system, ≈ 3.0404 þ 10 -6.  0 0 0 1 0 0 A more compact notation incorporates a pseudo-    0 0 0 0 1 0 potential function, U, such that  0 0 0 0 0 1 A()t =   [9] U xx U xy U xz 0 2 0 1− 1 U = ( µ) + µ + x2 + y 2 [4] U U U − 2 0 0 r r 2 ()  yx yy yz  1 2 U U U 0 0 0  zx zy zz  Then, the scalar equations of motion are rewritten in the form The partials Ukl are also functions of time, although the functional dependence on time, t, has been x&& = U + 2y& = f x, x&, y, y&, z, z& [5 ] dropped. x x ( )

y&& = U − 2x& = f x, x&, y, y&, z, z& [6 ] y y () The general form of the solution to the system in z&& = U = f x, x&, y, y&, z, z& [7 ] z z () equation [8] is where the symbol Uj denotes the partial derivative of δΧ t = Φ ,tt δΧ t [10] ( ) ( 0 ) ( 0 ) U with respect to j. Equations [1]-[3] or equations [5]- [7] comprise the dynamic model in the circular where (t,t 0) is the state transition matrix (STM). The restricted three-body problem. STM is a linear map from the initial state at the initial

time, t0, to a state at some later time, t, and thus offers Given a solution to the nonlinear differential an approximation for the impact of the variations in equations, linear variational equations of motion in the initial state on the evolution of the trajectory. the CR3BP can be derived in matrix form as The STM must satisfy the matrix differential equation

& Χ = Χ & δ t A t δ t [8] Φ ,tt = A t Φ ,tt ()() () ( 0 ) ( ) ( 0 ) [11]

δΧ = δ δ δ δ& δ& δ& T where [ x y z x y z ] represents given the initial condition, variations with respect to a reference trajectory. Generally, the reference solution of interest is not Φ ,tt = I ( 0 ) 6 x6 [12] constant. In this analysis, the reference trajectory is a

Lissajous orbit, or a quasi-periodic orbit, near L 2. This where I is the 6 þ 6 identity matrix. particular orbit was identified by NASA Goddard 6x6

Space Flight Center (GSFC) as a possibly advantageous The STM at time t can be numerically determined by orbit for the TPF-O mission, and appears in Figure 2. integrating equation [10] from the initial value. Since

the STM is a 6 þ 6 matrix, propagation of equation [10]

requires the integration of 36 first-order, scalar 5 x 10 differential equations. Because the elements of A(t)

depend on the reference trajectory, equation [10]

1 must be integrated simultaneously with the nonlinear [km]

z 0 equations of motion to generate reference states. This

-1 requirement necessitates the integration of an

additional 6 first-order equations, for a total of 42 5 differential equations.

5 L2 x 10 0 1.51 y [km] 8 DDDESCRIPTION OF CCCONTROL MMMETHODS 1.506 x 10 [km] -5 x 1.502 Optimal nonlinear control and geometric control

methods are compared to more traditional linear Figure 2: Nominal Lissajous Orbit for TPF-O Mission. quadratic regulators and feedback linearization. The

control strategies are implemented and evaluated for the TPF-O mission concept.

3 Time-Time -VaryiVaryingng Linear Quadratic Regulator subject to P t = 0 . Time --Varyi ng Linear Quadratic Regulator ( f ) [ 6 x6 ] Let the motion of a spacecraft moving in the CR3BP be described by Equation [18] and [19] represent a two-point boundary value problem with 42 first-order, ordinary x&& = f x, x&, y, y&, z, z& + u [13 ] x ( ) x differential equations. However, for application to y&& = f x, x&, y, y&, z, z& + u [14 ] y () y the CR3BP, this complexity can be avoided by && = & & & + [15 ] exploiting the time invariance properties associated z f z x, x, y, y, z, z u z () 28 with the dynamical model. Marchand develops a where f , f , and f are defined in equations [5]-[7] time transformation, x y z = and u t u u u is an applied control −1 ( ) [ x y z ] P t − t = G S t G [20] ( f ) ( ) Χ o t (acceleration) vector. Let ( ) represent some reference motion that satisfies the system equations j−1 where G = −1 , that enables simple forward o jj ( ) [13]-[15] and, then, u t denotes the control effort ( ) integration to numerically solve a transformed Riccati required to maintain Χ o t . Linearization about this ( ) matrix differential equation, reference solution yields a system of the form & = − T − − −1 T + −1 & S(t) A(t) S(t) S(t)A(t) S(t)BR B S(t) G QG δΧ = δΧ + δ ()()t A t B()t u ()t [16] [21] where δΧ(t) and δu(t) represent perturbations Thus, the resulting optimal gain is defined as relative to Χ o (t) and u o (t) , respectively. The matrix K t − t = −R −1 BT S t G [22] A(t) is defined in equation [9] ( f ) ( ) B t = B = 0 I T B(t) and ( ) [ 3 x3 3 x3 ] . The matrix is selected such that the control input can instantaneously and can be implemented at each numerical time step. change only the velocity of the satellite (not position). The nonlinear system of equations in [13]-[16] and the matrix differential equation in [21] are To minimize some combination of the error state and numerically integrated forward in time, subject o 29 Χ t = Χ t S t = 0 required control, Bryson and Ho employ a quadratic to ( 0 ) ( 0 ) and ( 0 ) , respectively. cost function Simultaneously, an optimal gain matrix is calculated at each integration time step and control is applied to t f the perturbed spacecraft motion. = 1 δΧ T δΧ + δ T δ min J ∫ ()()t Q ()t u()t R u()t dt [17] 2 t 0 It is notable that nearly the same optimal control

gains can be computed by solving the algebraic Riccati subject to the governing state equations in [16] with equation at each integration step, rather than initial conditions δΧ t = δΧ . The matrices Q and R ( 0 ) 0 numerically integrating the differential Riccati represent weighting factors on the state error and equation. This method, although arguably less control effort, respectively. Both are positive definite. rigorous, requires far less computation time and, Because the individual state errors are decoupled from therefore, is very useful for preliminary analysis. the required control, Q and R are diagonal matrices. Also, if the reference trajectory is periodic or well known a priori, the gain matrix can be computed and Application of the Euler-Lagrange theorem to the stored as a function of time for later implementation. system in equations [16] and [17] yields the following If the integration time steps are defined such that no optimality requirement on the control variable, interpolation of K(t) is necessary, computation efficiency can be greatly improved. − δu (t) = −R 1 BP(t)δΧ(t) [18] Input State Feedback Linearization where P(t) is a solution of the matrix differential One alternative to LQR control is input state feedback Riccati equation, linearization (IFL). By implementing IFL, it is possible to specify desired characteristics of the state & T −1 T P(t) = −A(t) P(t)− P(t)A(t)+ P(t)BR B P(t)− Q [19] response.

4 To develop an IFL scheme, the motion of a spacecraft To minimize the state error with minimum control moving in the CR3BP is separated into linear and effort, δu~(t) is computed via LQR analysis. In this nonlinear components, such that analysis, full-state feedback is assumed. The system is linear time-invariant, thus, the optimal control effort && = + + x hxnonlinear hxlinear u x [23 ] is defined by && = + + y hynonlinear hylinear u y [24 ] δ~ = − −1 δΧ && = + + u t R BP t linear [35] z hznonlinear hzlinear u z [25 ] ( ) ( ) where the h functions are defined as the linear and where the algebraic Riccati equation yields P, nonlinear terms in the equations [5]-[7], that is, ~ ~ − AT P − PA + PBR −1 B T P − Q = 0 [36] 1− x + x − x − = − ( µ)( µ) − µ( ( µ)) [26 ] hxnonlinear 3 3 r1 r2 Feedback linearization offers both advantages and = & + hxlinear 2y x [27 ] disadvantages over LQR control. This approach does 1− y y not rely on a priori knowledge of the controller gain hy = − ()µ − µ [28 ] nonlinear 3 3 matrix since it is determined directly from the r1 r2 = − & + nonlinear system rather than the linear variational hylinear 2x y [29 ] − equations. It also allows a designer to specify desired = − ()1 µ z − µz hznonlinear [30 ] response characteristics of the state. Thus, the r 3 r 3 1 2 stability of the system response can be more easily hz = 0 [31 ] linear analyzed. However, in general, feedback linearization is implemented assuming full state feedback, which There are many ways to implement feedback may or may not be available. linearization. One option is to select the control, u(t), such that Impulsive Control via Differential CorrectionCorrectionssss The use of a differential corrector is arguably the

u x  − hxnonlinear  simplest method for computation of the required     u t = u = − hy + u~ t impulsive velocity change to reach a specified future ()() y   nonlinear  [32]   −  position. An impulsive velocity change is also, u z   hznonlinear  potentially, the easiest to physically implement using By first removing the nonlinear dynamics from the available chemical thrusters. While LQR, IFL, and system, the control acceleration vector u~(t) is nonlinear methods may reveal advantageous trajectories that require less fuel, these approaches determined to produce the desired response may be significantly more complicated to execute. It characteristics of the linear, time-invariant dynamics. may be necessary to discretize and optimize the Linearization about a reference solution yields a control results in order to use common propulsion subsystem of the form systems. & ~ δΧ t = A δΧ t + B δu~ t [33] ()linear ()()linear Consider the general form of the discretized solution

~ to the linear system in equation [8], where B is defined above, and A is determined from the linear dynamics of the spacecraft motion such that δΧ = Φ t ,t δΧ k +1 ( k +1 k ) k [37]

0 0 0 1 0 0 Φ where t + ,t denotes the state transition matrix for   ( k 1 k ) 0 0 0 0 1 0   the kth segment associated with the nominal Lissajous ~ 0 0 0 0 0 1 A =   [34] orbit. The state can be partitioned into position and 0 0 0 0 2 0 velocity components, δr and δv , respectively, such 0 0 0 − 2 0 0 that   0 0 0 0 0 0 δr  δr  Φ Φ   δr  k +1 k rr rv k  −  = Φ tk +1 ,tk  +  =    −  [38] ~ δv () δv Φ Φ δv + ∆V Because the pair (A, B) is completely controllable,  k +1   k   vr vv   k k  control effort is defined to place the poles of the linear system anywhere on the complex plane.

5 ∂ where the superscript —+“ or —-“ signifies the λ& = − H = −λ 6 3 beginning or end of a segment originating at time t k, ∂z& ∆ respectively, and Vk represents an impulsive Then, the conditions for optimality become, maneuver applied at tk. To accomplish the desired δ ∂H change in position, r + , the required impulsive [48 ] k 1 = 2u + λ = 0 ∂u x 4 change in velocity (in the linear system) can be x ∂H expressed as = 2u + λ = 0 [49 ] ∂ y 5 u y ∆ = −1 δ − Φ δ −δ − V B r + r v [39] ∂H k k ( k 1 rr k ) k = 2u + λ = 0 [50 ] ∂ z 6 u z Of course, in the nonlinear system, this linearly computed velocity change is inadequate. Differential This control strategy is implemented to maneuver a corrections are incorporated in an iterative scheme to spacecraft from a specified initial state (both position identify the precise maneuver that meets the end- and velocity) to a specified final position. Therefore, state constraint to within the specified tolerance the boundary conditions include three —free“ final levels, as discussed in Howell and Barden. 30 conditions on both the state and co-state variables,

Nonlinear Optimal Control Χ 0 = x y z x& y& z& [51 ] ( ) [ 0 0 0 0 0 0 ] To determine minimum fuel maneuvers for a Χ t = x y z free free free [52 ] spacecraft moving in the full nonlinear CR3BP, a ( f ) [ f f f ] λ t = free free free 0 0 0 [53 ] traditional nonlinear optimal control problem is ( f ) [ ] formulated. Let a cost function be defined as The two-point boundary value problem is solved t f using a function ( bvp4c ) available in the Matlab min J = u 2 + u 2 + u 2 dt [40] ∫ β x x β y y β z z ® 0 Optimization Toolbox . Final conditions for the —free“ state and co-state elements are determined = based upon knowledge of the optimal LQR solution where the vector β β x β y β z is comprised of [ ] for the same maneuver. the weights on the corresponding control vector components, ux, uy, and uz, defined in [32]. The Geometric —Floquet“ Control Hamiltonian is then represented by Floquet control is an impulsive control strategy that exploits the phase space near a nominal orbit. At H = u 2 + u 2 + u 2 + x& + y& + z& + x&& + y&& + z&& β x x β y y β z z λ1 λ2 λ3 λ4 λ5 λ6 discrete intervals, a maneuver is performed in the [41] direction of the center or stable manifolds. The main result of Floquet theory is that, for a time-varying λ t = λ λ λ λ λ λ periodic, linear system, the STM can be factored into where ( ) [ 1 2 3 4 5 6 ] is a vector of two matrices, E and J, in the form co-states and Χ t = x y z x& y& z& is the vector ( ) [ ] of spacecraft states. The Euler equations are derived Jt Φ t 0, = E t e E 0 [54] via the partial derivative of the Hamiltonian with ( ) ( ) ( ) respect to the state, as follows, The matrix J is a constant matrix, usually in the

Jordan normal form. Its diagonal entries, ) , are the & ∂H ∂ ∂ ∂ i λ = − = −λ x&& − λ y&& − λ z&& [42 ] Poincaré exponents, that is, the analog of the 1 ∂x 4 ∂x () 5 ∂x () 6 ∂x () eigenvalues for constant-coefficient systems. & ∂H ∂ ∂ ∂ λ = − = −λ x&& − λ y&& − λ z&& 2 ∂ 4 ∂ () 5 ∂ () 6 ∂ () [43 ] y y y y Because the nominal Lissajous orbit is not perfectly & ∂H ∂ ∂ ∂ λ = − = −λ x&& − λ y&& − λ z&& periodic, a similar halo orbit is successfully employed 3 4 () 5 () 6 () [44 ] ∂z ∂z ∂z ∂z to approximate the surrounding phase space. The matrix E(t) corresponding to the halo orbit is periodic & ∂H with period, T. Solving a Floquet problem for a λ = − = −λ + 2λ [45 ] 4 ∂x& 1 5 periodic halo orbit over all time, thus, requires the & ∂H determination of the constant matrix J and the λ = − = −λ − 2λ [46 ] 5 ∂y& 2 4 periodic matrix E(t) over one period. Since the matrix [47 ]

6 E is periodic, E(0)=E(T) and, at t=T , equation [54] spacecraft state to the center subspace. However, in becomes general, the error incurred is extremely small when Φ(T 0, ) = E(0) e JT E(0) [55] compared to the magnitude of the required impulsive maneuver. Marchand and Howell, for instance, let Thus, E(0) is the matrix of eigenvectors of the 6 monodromy matrix, Φ ,tt . The eigenvalues of the ( 0 ) δΧ t = 1+α t δΧ t [59] desired ()()∑()j j () j=2 monodromy matrix, +i, are related to the Poincaré exponents )i, i.e., denote the desired perturbation relative to the ω T α λ = i reference orbit, where t is some, yet to be i e [56] ( ) determined, coefficient vector. Note that the limits of After numerically integrating equation [11] over one the summation range from 2 through 6 which implies period, the Poincaré exponents and the eigenvector that the perturbation in the direction of the unstable mode, e has been removed. The linearized control matrix at t=0 are obtained. Since E(t) is periodic and 1 therefore bounded, the stability of the system is problem then reduces to a computation of the ∆v t governed by )i (or +i) alone. Both the halo orbit and impulsive maneuver, ( ), such that nominal Lissajous orbit of interest can be modeled as inherently unstable. In fact, the six-dimensional 6 6  0  1+α t δΧ t = δΧ t + 3 x1 [60] ∑()j () j () ∑ k ()   eigenstructure is characterized by one unstable = = ∆v t j 2 k 1  ()3 x1  eigenvalue ( + ), one stable eigenvalue ( + ), and four 1 2 eigenvalues in the center subspace. Two of these After some reduction, equation [60] can be rewritten neutrally stable eigenvalues are located on or near the in matrix form as unit circle ( + and + ) and, given a perfectly periodic 3 4 orbit, the remaining two are equal to one ( + and + ). 5 6  0  At any point in time, the perturbation δΧ t can be δΧ t δΧ t δΧ t δΧ t δΧ t 3 x3 ( )  2 () 3 () 4 () 5 () 6 ()   I 3 x3  expressed in terms of any six-dimensional basis. The [61] α e E(t)  t 5 x1  Floquet modes ( j ), defined by the columns of , () δ ×   = Χ1 t ∆v t () form a non-orthogonal six-dimensional basis. Hence,  ()3 x1 

6 6 where α represents a 5 x 1 vector formed by the 3j δΧ t = δΧ t = c t e t [57] () ∑j ()() ∑ j j () j=1 j =1 components in equation [59]. Therefore, the ∆v maneuver, 3 x1 , that is required to remove the

δΧ t δΧ t unstable mode, e1 , from the perturbation at any given where j ( ) denotes the component of ( ) along jth e t c t time can be approximated from equations [61]. the mode, j ( ), and the coefficients j ( ) are Howell and Keeter 31 identify this required ∆v via a determined as the elements of the vector c(t) defined 3 x1 minimum norm solution. If the maneuver is by constrained to the Sun-Earth line ( xÙ -direction), ∆v 1x1 − c(t) = E(t) 1 δΧ(t) [58] is actually a scalar, and the system possesses an exact solution.

This Floquet structure is implemented by Gómez et If exactly three modes are removed and control is al. 11 and, later, by Howell and Keeter 31 as the basis of a possible in three directions, the system also possesses station keeping strategy for a single spacecraft an exact solution. For example, Marchand and evolving along a halo orbit. Both investigations Howell 17 demonstrate the impulsive ∆v that is determine the impulsive maneuver scheme that is 3 x1 required to remove the component of error in the required to remove the portion of the perturbation e e δΧ t along the unstable and short period modes ( 1 , 3 , and direction of the unstable mode, 1 ( ) (as defined in e ). It can be determined exactly from [57]), at discrete intervals. Because impulsive 4 maneuvers are calculated based on the phase space −1 associated with a halo orbit, yet implemented on a  α t   0  ( )3x1 δ δ δ 3x3   =  Χ 2 t Χ 5 t Χ 6 t  spacecraft moving near a Lissajous orbit, additional ∆v t () () () − I  ()3x1   3 x3  [62] error is incurred. This implies slightly larger × δΧ t +δΧ t +δΧ t impulsive maneuvers may be required to move a ()1 () 3 () 4 ()

7 ∆ Similarly, the v3 x1 vector that is required to remove The path of the occulter is dictated by a star sequence. This star sequence represents a list of stars that are in the unstable and long period modes ( e1 , e5 , and e6 ) is exactly determined from the same spectral class as the Sun, indicating a higher potential for the existence of nearby, orbiting Earth- −1 like planets. The required observation time at each  α t   0  ( )3x1 δ δ δ 3x3 star and the specific length of time allotted for   =  Χ 2 t Χ 3 t Χ 4 t  ∆v t () () () − I [63]  ()3x1   3x3  maneuvering between each observation is also × δΧ t +δΧ t +δΧ t included in the star sequence. In this analysis, two ()1 () 5 () 6 () star sequences are used for comparison: Design ∆ Reference Mission (DRM) 4 and DRM 5. These As formulated in equations [62] and [63], v3 x1 is the sequences are specified as options for TPF-O. The impulsive control necessary to remove perturbations general characteristics of the star sequences are in the direction of one unstable mode and two semi- compared in Table 1. Placement of the stars, in a stable modes in the linearized system represented in given sequence on a celestial sphere, is noted in Figure equation [8] at a given time. 4.

ESULTS FROM ISSION NALYSIS RRR TPFTPF----OOOO MMM AAA Star Mission Number of Distance Average Average Sequence Duration Observations to Occulter Observation Transit TPFTPF----OO Mission CoCoConceptCo ncept DRM4 1832 days 134 72000 km 2.67 days 10.99 days The NASA Terrestrial Planet Finder-Occulter mission DRM5 1840 days 102 37800 km 3.24 days 14.80 days is an optical formation that will search for Earth-like planets in other solar systems. Two spacecraft are Table 1: Features of DRM4 and DRM5 Sequences included in the formation: one telescope spacecraft and one occulter spacecraft. The telescope spacecraft moves near a nominal L 2 Lissajous orbit. During an observation period, the telescope and the occulter align along the direction to a star of interest in some inertial direction. The two spacecraft are required to maintain a fixed relative distance during this phase. Unit During the transit, or maneuver, period there is no Sphere constraint on the geometry of the occulter and Rotating (Ecliptic) telescope. The —transit period“ refers to the time - Z period between t 1 and t 2, while the —observation period“ denotes the time interval between t 2 and t 3, as illustrated in Figure 3. Y-Rotating

X-Rotating

(Sun-Earth Line)

Transit Observation Fixed Period Period Unit distance Sphere

Rotating (Ecliptic) - Z t2 Y-Rotating t3 t1 Orbit X-Rotating (Sun-Earth Line) near L 2

Figure 3: Schematic of TPF-O Mission Sequence. Figure 4: Star Sequences for DRM 4 (top) and DRM 5 (bottom).

8 Control AnalysiAnalysiss Results The first column in these tables, labeled —Total UV“ is The various control strategies are implemented in an estimate of the control effort required for the differing combinations for DRM 4 and DRM 5. For entire mission. Because each mission contains a example, in one scenario, a nonlinear optimal different number of star targets, it may be helpful to controller is used for transit between t 1 and t 2 (in determine the average control effort used for each star Figure 3) and an LQR controller is used to track the in the sequence. This appears in the second column star during an observation period. Alternatively, an labeled, —Ave. UV per star.“ In columns 3 and 4, the impulsive maneuver can support the transit between total control effort is divided into mission segments: t1 and t 2, as well as t 2 and t 3; this approach implies that transit and observation periods, respectively. Finally, no star tracking occurred during the observation the last column is a measure of only the control effort period. A summary of the different scenarios and the needed to track the star during the observation period. resulting required control effort is listed in Tables 2 This number is defined as the amount of control and 3. necessary to maintain the occulter within a specified

Total UV Ave. UV Transit Observation Star Track Control Method [km/s] per star [m/s] UV [km/s] UV [km/s] [m/s]

Impulsive Transit without Star Track 4.728 35.284 4.728 Impulsive Transit + Impulsive Star Track 5.783 43.157 2.795 2.947 Impulsive Transit + LQR Star Track 5.955 44.440 2.702 3.253 30.074 Impulsive Transit + IFL Star Track 5.966 44.522 2.704 3.263 30.166

LQR Transit + LQR Star Track 5.998 44.761 2.745 3.253 30.074 IFL Transit + IFL Star Track 6.067 45.275 2.804 3.263 30.166

NL Optimal Transit + Impulsive Star Track 5.802 43.299 2.855 2.947

NL Optimal Transit + LQR Star Track 6.103 45.545 2.850 3.253 30.074 NL Optimal Transit + IFL Star Track 6.111 45.604 2.848 3.263 30.166

Geometric Transit + Impulsive Star Track 6.792 50.687 3.845 2.947 Geometric Transit + LQR Star Track 6.804 50.776 3.551 3.253 30.074 Geometric Transit + IFL Star Track 6.811 50.828 3.548 3.263 30.166

Table 2: Required Control Effort for DRM 4.

Total UV Ave. UV Transit Observation Star Track Control Method [km/s] per star [m/s] UV [km/s] UV [km/s] [m/s]

Impulsive Transit without Star Track 2.721 26.676 2.721

Impulsive Transit + Impulsive Star Track 2.899 28.422 1.506 1.392

Impulsive Transit + LQR Star Track 3.009 29.500 1.452 1.558 13.555 Impulsive Transit + IFL Star Track 3.013 29.539 1.451 1.561 13.581

LQR Transit + LQR Star Track 3.032 29.725 1.474 1.558 13.555 IFL Transit + IFL Star Track 3.111 30.500 1.549 1.561 13.581

NL Optimal Transit + Impulsive Star Track 2.995 29.363 1.603 1.392

NL Optimal Transit + LQR Star Track 3.101 30.402 1.543 1.558 13.555

NL Optimal Transit + IFL Star Track 3.104 30.431 1.543 1.561 13.581

Geometric Transit + Impulsive Star Track 3.213 31.500 1.821 1.392 Geometric Transit + LQR Star Track 3.298 32.333 1.740 1.558 13.555 Geometric Transit + IFL Star Track 3.302 32.373 1.741 1.561 13.581

Table 3: Required Control Effort for DRM 5.

9 tolerance of the desired state, after it has made a The required control at each star can also be maneuver to reach that state. represented graphically, as plotted in Figures 5 and 6. Figure 5 illustrates a notional thrust profile for DRM A few observations are noted in the results from the 4. This profile corresponds to row 3 in Table 2: control analysis. Tables 2 and 3 both indicate that the impulsive control during transit periods and LQR impulsive method, used during the transit period, control during observation periods. The horizontal requires the least amount of control effort. The axis represents each star in the DRM 4 sequence. The geometric —Floquet“ maneuvers require the most vertical axis is the amount of control required at a control effort during the transit period. particular star in the sequence. The red line represents the total required control effort for the DRM 4 has a significantly higher average control DRM 4 mission. This control effort is divided into the effort per star than DRM 5, as indicated by column 2 control required for the observation period and that of Tables 2 and 3. Upon further investigation, it was required for the transit period, that is, the green and determined that this is a result of the larger distance the cyan line, respectively. The dark blue line between the telescope and the occulter: 72000km for corresponds to the amount of control required if the DRM 4 as opposed to 38700km for DRM 5. As the analysis is performed without CR3BP dynamics, in occulter is placed further and further away from the other words, if spacecraft is assumed to be moving in nominal telescope orbit, the control cost tends to free space. It is notable that control effort required increase. This may be due, in part, to the longer for spacecraft moving in free space is significantly distance traveled by the occulter to create the desired higher than that required for spacecraft moving in the angle during each observation period. CR3BP. Thus, in this mission scenario, the natural

150 ∆ Transit Impulsive V: 2.7022km/s Total ∆V without Dyanmics: 10.1837km/s

Observation LQR Control: 3.2534km/s

Total ∆V: 5.9555km/s 125

100

75 V[m/s]

∆

0 0 20 40 60 80 100 120 Star Sequence

Figure 5: Notional Thrust Profile for DRM 4, Impulsive Control during Transit Periods and LQR Control during Observation Periods.

10 100 Continuous Maneuver Control Effort: 1.543km/s Required Control without Dyanmics: 5.2174km/s 90 Observation ISFL Control: 1.561km/s Total Required ∆V: 3.104km/s 80

V [m/s] ∆ 40

0 0 10 20 30 40 50 60 70 80 90 100 Star Sequence

Figure 6: Notional Thrust Profile for DRM 5, Nonlinear Optimal Control during Transit Periods and Input-State Feedback Linearization Control during Observation Periods. dynamics in this regime reduce the control cost of precision in the defined radial and transverse spacecraft maneuvers. directions. Thus, the LQR problem is reformulated in terms of radial and transverse coordinates. Let the Similarly, Figure 6 illustrates a notional thrust profile direction to the star target from the telescope be = + + for DRM 5. This profile corresponds to row 9 in Table defined as rÙ 1xr Ù r2 yÙ 3 zr Ù and the direction of the 3: nonlinear optimal control during transit periods velocity of the telescope near the Lissajous orbit be and input-state feedback linearization during = + + defined as vÙ v1xÙ v2 yÙ v3 zÙ . observation periods. Again, the required control for a spacecraft formation moving in free-space is significantly higher than for a formation moving in zÙ the CR3BP.

LQR Control for TPFTPF----OOOO CR3B rÙ

For the TPF-O mission, the occulter position yÙ tolerances are provided in terms of a radial and transverse component. The radial direction is defined Radial- xÙ Χ as the direction from the telescope to the target star, Transverse as illustrated in Figure 7. The transverse directions Frame lay anywhere in a plane perpendicular to this radial direction. The occulter is constrained to remain within ±100km of the desired path in the radial hÙ hÙ 1 2 direction and ±10m from the desired path in the transverse direction. Because the LQR problem is Figure 7: Coordinate Transformation from CR3BP to formulated in the CR3BP, it is difficult to determine Radial-Transverse Frame. the effect of the weighting matrices, Q and R, on the

11 Then, a vector perpendicular to both rÙ and vÙ is of the reference Lissajous orbit. Because the Lissajous orbit is not perfectly periodic, a similar halo orbit is Ù = × used to approximate the surrounding phase space h1 rÙ vÙ [64] while calculating the required velocity change.

The direction hÙ is defined as the first transverse 1 Because the spacecraft should eventually reach a direction. This choice is arbitrary, since the specific specified observation state (which may or may not be direction of the transverse coordinate frame does not in the center subspace,) an impulsive targeting effect the problem formulation. The new coordinate maneuver is required at some point during the transit hÙ frame is completed by 2 such that, period. This maneuver is added at the time when the smallest total amount of thrust is required to move to Ù = × Ù the desired observation location. A schematic of the h2 rÙ h1 [65] geometric control method, as applied to the TPF-O

mission, appears in Figure 8. To transform the original LQR problem into the new coordinate frame, a transformation matrix, C, is The blue line in the top plot in Figure 8 indicates the derived. This matrix transforms the perturbations difference between the path of the spacecraft and the δΧ to the new coordinate system & & nominal Lissajous orbit in the rotating x-direction δZ t = r h h r& h h such that ( ) [ 1 2 1 2 ] during one transit period (approximately 16 days). A small maneuver is applied at each blue star to keep δΧ = δ t C Z t [67] 4 ( ) ( ) x 10 & & & δΧ t = CδZ t + CδZ t [68] () () () 3.8

Substituting equations [67] and [68] into equation [16] 3.7 yields,

3.6

& −1 −1 −1 δZ t = C t A t C t − C t C t δZ t + C t Bu t ( ) [ ( ) ( ) ( ) ( ) ( )] ( ) ( ) ( ) 3.5 [69] 3.4 and, thus, the new cost function for the LQR 3.3 formulation is Distance Nominalfrom Orbit [km] 3.2 Path without maneuver min Path after maneuver t f 3.1 1 T T J = δZ t Q δZ t + δu t R δu t dt [70] 0 2 4 6 8 10 12 14 16 18 2 ∫()() Z () () Z () Time [days] t0 4 x 10 where the diagonal entries of QZ and RZ now 3.8 represent the weighting on the radial and transverse error components. In this analysis, the weighting 3.7 matrices, 3.6 = QZ I 6 x6 [71] Final desired 3.5 R = diag 10−5 10−10 10−10 [72] location Z ([ ]) 3.4 are sufficient in almost all cases to maintain the 3.3 desired tolerances on the state. However, recall that Distance from Nominal Orbit [km] only the relative magnitude of QZ and RZ are 3.2 important, not the exact numbers used. 3.1 0 2 4 6 8 10 12 14 16 18 Time [days] Floquet Control for TPFTPF----OOOO During a transit period, several small maneuvers maintain the spacecraft state near the center subspace Figure 8: Geometric Control Method, Transit Period.

12 the spacecraft state near the center manifold during the occulter must remain within 100km of the the transit period. Over longer periods of time, this specified path in the radial direction and within 10m trajectory is quasi-periodic. The red line in the top of the specified path in the transverse direction. plot represents the spacecraft path if no maneuver is When impulsive control is employed during the implemented. In the bottom plot, a point along the observation period, generally, these mission center manifold has been identified which minimizes requirements are not met. However, when either the total required control effort to reach the specified input-state feedback linearization or LQR control are observation point. This point appears in the black applied, weighting matrices can be selected such that box. At this point, a targeting maneuver enables the position errors are within required tolerances. vehicle to reach the final desired location. Again, the blue line indicates the trajectory of the spacecraft An example of the required control effort and state relative to the nominal orbit after the maneuvers have error response in the transverse direction, during one been performed. The red line indicates the path of observation period of the TPF-O mission, appears in the spacecraft without maneuvers. Figure 9. The weighting matrices, QZ and RZ, are defined such that the state error is just within the Effectiveness of Control Methods on Mission Goals defined mission tolerances. In the first column of During the transit period, no requirements exist on Figure 9, the required control in the transverse the position and velocity of the spacecraft formation direction is plotted as a function of time for three to facilitate imaging. During the observation, though, different control methods. In the second column

20 40 ∆ ∆ Impulsive V ImpulsiveImpulsive V State Error 15 30

10 20

V[m/s] ∆ 5 10 StateError [km]

0 0 16 16.5 17 17.5 18 16 16.5 17 17.5 18

0.4 15 ]

2 LQR Control LQR State Error 0.3 10

0.2

5 0.1 StateError [m] ControlEffort [m/s 0 0 16 16.5 17 17.5 18 16 16.5 17 17.5 18

0.4 15

] 2 IFL Control IFL State Error 0.3 10 0.2 5 0.1 State Error [m]

Control Effort[m/s 0 0 16 16.5 17 17.5 18 16 16.5 17 17.5 18 Time [days] Time [days]

Figure 9: Required Control and State Error Response in the Transverse Direction over Observation Period: ± 10m .

13 of the plot, the state error in the transverse direction Finally, an example of the path of both the occulter for the corresponding control method is displayed. and telescope spacecraft can be seen in Figure 11. The This specific observation period corresponds to the trajectory of the telescope appears in blue, following first star of interest in the DRM 5 mission. the nominal Lissajous orbit. The occulter path is both red and cyan, indicating the transit and observation The typical required control effort and the state error periods, respectively. response in the corresponding radial direction are represented in Figure 10. Because the position CCCONCLUDING RRREMARKS AND FFFUTURE WORK tolerance in the radial direction is relatively lenient, the state error response is higher in magnitude. This, This work represents a preliminary comparison of

again, results in part from the selection of QZ and RZ. various control strategies for the TPF-O mission In the first column of the figure, the required control concept. Error inclusion as well as estimation filters in the radial direction appears as a function of time for can be incorporated in future analyses. Also, the three different control methods applied to star continuous control methods described may be tracking. The second column in the plot displays the discretized and optimized to provide competing state error in the radial direction for the control solutions for current chemical propulsion corresponding control method. systems in a full ephemeris model.

Impulsive ∆V Impulsive ∆V 20 40 Impulsive UV Impulsive State Error 15 30

10 20

V [m/s] ∆ 5 10 State[km] Error 0 0 16.5 17 17.5 16.5 17 17.5 Time [days] Time [days] -3 x 10 LQR Control LQR 4 100 ] 2 LQR Control LQR State Error

2 50

State Error State[km] Error

[m/s Control Effort 0 0 16 16.5 17 17.5 18 16 16.5 17 17.5 18 Time [days] Time [days] -3 x 10 IFL Control IFL 3 150 ]

2 IFL Control IFL State Error

2 100

1 50

[km] State Error Control Effort [m/s Control Effort 0 0 16 16.5 17 17.5 18 16 16.5 17 17.5 18 Time [days] Time [days]

Figure 10: Required Control and State Error Response in t he Radial Direction over Observation Period: ± 100km

14 Occulter Path for TPF-O Mission Concept #4: Impuslive ∆V for Transit and LQR for Star-Tracking 5 x 10 Nominal Halo Orbit Maneuver Coast Period Observation Period (LQR Control) 1 Star Target Implusive Maneuver 0

Z-Rotating [km] -1

2 5 x 10

-2

-4 1.512 1.511 Y-Rotating [km] 1.51 8 -6 1.509 x 10

X-Rotating [km]

Figure 11: Occulter and Telescope Spacecraft Path, Impulsive Control during Transit and LQR Control during Observation Period.

For these two star sequences, it is observed that under contract numbers NNG04GP69G and inclusion of the natural dynamics near the nominal NNX06AC22G. Support was also provided from Lissajous orbit in the model reduces the control cost Purdue University via the Bilsland Dissertation for maneuvers. Such a result highlights the Fellowship. importance of including dynamics in mission control analyses in multi-body regimes. RRREFERENCES

Control costs can be reduced whenever it is possible 1. Origins Science Subcommittee Members, —Origins Roadmap,“ National Aeronautics and Space to use a shorter baseline between the occulter and Administration, © 2002. telescope spacecraft. The optimal relative distance 2. Traub, W., S. Shaklan, and P. Lawson, —Prospects for between the occulter and telescope spacecraft, based Terrestrial Planet Finder (TPF-C, TPF-I, and TPF-O),“ on control cots, can be determined from further Proceedings of the Conference In the Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar investigation. Disks in the 21st Century, University of California, Berkeley, California, June 4-8, 2007. Finally, and not surprisingly, the order and specific 3. Cash, W., N. White, and M. Joy, —The Maxim Pathfinder Mission: X-Ray Imaging at 100 Micro-Arcseconds,“ stars in a sequence affect the necessary control effort. Proceedings of the Soc. Photo-Opt. Instr. Eng., Vol. 4012, However, the current mission design requirements 2000, p. 258-269 specify stipulates a unique star sequence to be 4. Scharf, D. P., F. Y. Hadaegh, and S. R. Ploen, —A Survey followed to maximize scientific results. of Spacecraft Formation Flying Guidance and Control (Part I): Guidance,“ American Control Conference, Denver, Colorado, June 4-6, 2003. AAACKNOWLEDGEMENTS 5. Scharf, D. P., F. Y. Hadaegh, and S. R. Ploen, —A Survey of Spacecraft Formation Flying Guidance and Control (Part II): Control,“ American Control Conference, The authors would like to thank Dave Folta, Conrad Boston, Massachusetts, June 30-July 2, 2004. Schiff, and the TPF-O working group at NASA 6. Gurfil, P., and N. J. Kasdin, —Dynamics and Control of Goddard Space Flight Center for providing Spacecraft Formation Flying in Three-Body Trajectories,“ information and guidance for the current study. This AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, Canada, August 6-9, 2001. work was performed at Purdue University with 7. Gurfil, P., M. Idan, and N. J. Kasdin, —Adaptive Neural support from NASA Goddard Space Flight Center Control of Deep-Space Formation Flying,“ American

15 Control Conference Proceedings, Anchorage, Alaska, 20. Hyland, D. C., —Interferometric Imaging Concepts with Vol. 4, May 8-10, 2002, p. 2842-2847. Reduced Formation-Keeping Constraints,“ AIAA Space 8. Luquette, R. J., and R. M. Sanner, —A Nonlinear 2001 Conference, Albuquerque, New Mexico, August Approach to Spacecraft Formation Control in the Vicinity 2001. of a Collinear Libration Point,“ AAS/AIAA 21. Hussein, I. I., D. J. Scheeres, and D. C. Hyland, Astrodynamics Conference Proceedings, Quebec, July 30- —Formation Path Planning for Optimal Fuel and Image August 2, 2001, p. 437-445. Quality for a Class for Interferometric Imaging Missions,“ 9. Hamilton, N. H., —Formation Flying Satellite Control February, 2003. Around the L2 Sun-Earth Libration Point,“ M.S. Thesis, 22. Hussein, I. I., D. J. Scheeres, and D. C. Hyland, —Control George Washington University, Washington, DC, of Satellite Formation for Imaging Applications,“ December 2001. Proceedigns fo the American Control Conference, Copper 10. Folta, D., J. R. Carpenter, and C. Wagner, —Formation Mountain, Colorado, January 23-27, 2005. Flying with Decentralized Control in Libration Point 23. Hussein, I. I., D. J. Scheeres, and D. C. Hyland, —Optimal Orbits,“ International Symposium: Spaceflight Dynamics, Formation Control for Imaging and Fuel Usage with Biarritz, France, June 2000. Terminal Imaging Constraints,“ IEEE Conference on 11. Gómez, G., K. C. Howell, J. Masdemont, and C. Simó, Control Applications, 2005. —Station-keeping Strategies for Translunar Libration Point 24. Hussein, I. I., and A. M. Bloch, —Dynamic Coverage Orbits,“ Advances in Astronautical Sciences, Vol. 99, Pt. Optimal Control for Interferometric Imaging Spacecraft 2, 1998, p. 949-967. Formations,“ 43 rd IEEE Conference on Decision and 12. Gómez, G., M. Lo, J. Masdemont, and K. Museth, Control, Paradise Island, Bahamas, December 2004, p. —Simulation of Formation Flight Near Lagrange Points for 1812-1817. the TPF Mission,“ AAS/AIAA Astrodynamics 25. Hussein, I. I., and A. M. Bloch, —Dynamic Coverage Conference, Quebec, Canada, July 30-August 2, 2001. Optimal Control for Interferometric Imaging Spacecraft 13. Vadali, S. R., H. W. Bae, and K. T. Alfriend, —Design and Formations (Part II): The Nonlinear Case,“ Proceedings Control of Libration Point Satellite Formations,“ 2004. fo the American Control Conference, 2005, p. 2391-2396. 14. Li, H. M., and T. Williams, —Formation Keeping for Sun- 26. Howell, K. C., and L. D. Millard, —Control of Satellite Earth/Moon Libration Point Formation by Using Solar Imaging Formations in Multi-Body Regimes,“ IAF 57 th Radiation Pressure,“ AAS/AIAA 15 th Space Flight International Astronautical Congress, Valencia, Spain, Mechanics Meeting, Cooper Mountain, Colorado, January September 28-October 2, 2006. 23-27, 2005. 27. Millard, L. D., and K. C. Howell, —Control of 15. Howell, K. C., and B. G. Marchand, —Control Strategies Interferometric Spacecraft Arrays for ( u, v) Plane for Formation Flight in the Vicinity of the Libration Coverage in Multi-Body Regimes,“ AAS/AIAA Points,“ AIAA/AAS Space Flight Mechanics Conference, Spaceflight Mechanics Conference, Sedona, Arizona, Ponce, Puerto Rico, February 9-13, 2003. January 28-February 1, 2007. 16. Marchand, B. G., and K. C. Howell, —Formation Flight 28. Marchand, B., —Spacecraft Formation Keeping Near the Near L1 and L2 in the Sun-Earth/Moon Ephemeris Libration Points of the Sun-Earth/Moon System,“ System Including Solar Radiation Pressure,“ AAS/AIAA PhD Thesis, Purdue University, August, 2004. Astrodynamics Specialist Conference, Big Sky, Montana, 29. Bryson, A. E., and Y. Ho, Applied Optimal Control: August 3-8, 2003. Optimization, Estimation, and Control , Hemisphere 17. Marchand, B. G. and K. C. Howell, —Spherical Formation Publishing Corporation, New York, 1975. Near the Libration Points in the Sun-Earth/Moon 30. Barden, B. T., and K. C. Howell, —Formation Flying in Ephemeris System,“ AAS/AIAA Spaceflight Mechanics the Vicinity of Libration Point Orbits,“ Advances in Conference, Maui, Hawaii, February 8-11, 2004. Astronautical Sciences , Vol. 99, 1998, p. 969-988. 18. Carlson, B. A., H. J. Pernicka, and S. N. Balakrishnan, 31. Howell, K.C., and T. M Keeter, —Station-Keeping —Spacecraft Formation Flight about Libration Points,“ Strategies for Libration Point Orbits: Target Point and AIAA/AAS Astrodynamics Specialist Conference and Floquet Mode Approaches,“ AAS/AIAA Spaceflight Exhibit, Providence, Rhode Island, August 16-19, 2004. Mechanics Meeting, Albuquerque, New Mexico, 19. Pernicka, H. J., B. A. Carlson, and S. N. Balakrishnan, February 13-16, 1995. —Spacecraft Formation Flight About Libration Points Using Impulsive Maneuvering,“ Journal of Guidance, Control, and Dyanmics, Vol. 29, No. 5, September- October 2006, p. 1122-1130.