Acta Astronautica 63 (2008) 1283–1298www.elsevier.com/locate/actaastro

Optimal reconfigurationmaneuvers for spacecraft imaging arrays inmulti-body regimes

Lindsay D. Millard∗, Kathleen C. HowellSchool of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA

Received 5 December 2007; received in revised form 7 April 2008; accepted 6 May 2008Available online 18 July 2008

Abstract

Upcoming National Aeronautics and Space Administration (NASA) mission concepts include satellite arrays to facilitateimaging and identification of distant planets. These mission scenarios are diverse, including designs such as the terrestrial planetfinder-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 16 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 controlmethods for satellite imaging array reconfiguration in multi-body systems. Specifically, optimal nonlinear control and geometriccontrol methods are derived and compared to the more traditional linear quadratic regulators, as well as input state feedbacklinearization. These control strategies are implemented and evaluated for the TPF-O mission concept.© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Over the next two decades, one focus within NationalAeronautics and Space Administration’s (NASAs)Origins program is the development of sophisticatedtelescopes and technologies to begin a search for ex-traterrestrial life and to monitor the birth of neighboringgalaxies. Part of the program’s mission is (1) to locateand image Earth-like planets orbiting other stars and(2) to determine the probability that life has existed orcould develop on the surface [1].

Traditional Earth-based telescopes must continuouslyexpand in size to resolve increasingly distant and faintobjects. Unfortunately, as the size of a telescope in-creases, it becomes too heavy and too expensive to

∗Corresponding author.E-mail addresses: lmillard@purdue.edu (L.D. Millard),

howell@purdue.edu (K.C. Howell).

0094-5765/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.actaastro.2008.05.016

launch, thus motivating alternative designs for deep-space observers. One such example is a free-flying arrayor constellation of smaller satellites that work togetherto form a single optical platform.

Upcoming NASA mission concepts include satelliteformations to facilitate imaging and identification ofdistant planets. These scenarios are diverse, includ-ing designs such as the terrestrial planet finder-occulter (TPF-O) [2], where a monolithic telescope isaided by a single occulter spacecraft, and the micro-arcsecond X-ray interferometry mission (MAXIM)[3], where as many as 16 spacecraft move to-gether to form a space interferometer. Each design,however, requires precise reconfiguration and startracking in potentially complex dynamic regimes.Numerous inter-spacecraft positioning and systemgeometry constraints must be satisfied to assuresuccessful imaging. Spacecraft separated over largedistances must constantly be re-oriented and pointed

1284 L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298

at different stars, while the distance between the vehi-cles is expanding and contracting. All must be accom-plished with minimum fuel expenditure, and for manymissions, autonomously.

Extensive progress in spacecraft formation controlhas been achieved. Previously, the focus for much ofthe analysis was on satellites moving under the gravi-tational influence of the Earth, or simply in free-space[4,5]. Recently, however, more potential applications in-volve satellite arrays near the collinear libration pointsand, consequently, generate considerable interest in for-mation control strategies in multi-body regimes [6–14].

Howell and Marchand [15,16] successfully applyboth discrete and continuous control methods to de-ploy, enforce, and reconfigure formations near thevicinity of libration points in the Sun–Earth/Moon sys-tems. Specifically, these methods include input-statefeedback linearization (IFL) and input–output feedbacklinearization, linear quadratic regulator theory, and im-pulsive maneuvers derived via a differential correctionsscheme. In their examples, spacecraft follow both natu-ral and non-natural trajectories in the circular restricted3-body Problem (CR3BP). This analysis is also transi-tioned to an ephemeris model, including solar radiationpressure.

Marchand and Howell [17] also indicate that the re-quired discrete velocity changes for individual satelliteswithin a formation are often impossibly small, consid-ering current thruster capabilities. Carlson et al. [18,19]expand upon this result via the development of a methodfor identification of formation size and control toler-ances for which impulsive maneuvers become a prac-tical option. This research focuses on the analysis ofdiscrete maneuvering techniques using differential cor-rections in the CR3BP.

Very recently, control of interferometric satellite ar-rays has been pursued. Because of the complexity of theproblem, much of the available research assumes satel-lites within the formation to be moving through freespace [20–25]. However, Millard and Howell [26] de-velop two control strategies that are advantageous forinterferometric imaging arrays in multi-body regimes.A continuous geometric control algorithm is based onthe dynamical characteristics of the phase space nearperiodic orbits. By adjusting parameters in the controlalgorithm appropriately, satellites in the formation fol-low trajectories that aid image reconstruction. Secondly,a nonlinear, locally optimal control law is developedthat minimizes fuel while optimizing image resolutionin the CR3BP [27].

The goal in the present investigation is a comparisonof control methods for array re-configuration in multi-

body systems. Specifically, optimal nonlinear control aswell as geometric control methods are derived and com-pared to more traditional linear quadratic regulators andfeedback linearization. The control strategies are imple-mented and evaluated for the TPF-O mission concept.

2. Dynamic model: CR3BP

The motion of a spacecraft in the CR3BP is describedin terms of rotating coordinates relative to the barycen-ter of the system primaries. For this analysis, emphasisis placed on formations near the libration points in theSun–Earth/Moon system. Thus, the rotating x-axis isdirected from the Sun toward the Earth/Moon barycen-ter. The z-axis is normal to the plane of motion of theprimaries, and the y-axis completes the right handedtriad. The system is represented in Fig. 1.

Let X = [x y z x y z]T, then the general non-dimensional equations of motion of a spacecraft, rela-tive to the system barycenter, are

x − 2 y − x = − (1 − �)(x + �)

r31− �(x − (x − �))

r32(1)

y + 2x − y = − (1 − �)y

r31− �y

r32(2)

z = − (1 − �)z

r31− �z

r32(3)

where

r1 = ((x + �)2 + y2 + z2)1/2

r2 = ((x − (1 − �))2 + y2 + z2)

The quantity � is a non-dimensional mass parameterassociated with the system. For the Sun–Earth/Moonsystem, � ≈ 3.0404 × 10−6.

yx

yintertial^

xintertial^

L2

P2

P1

r2

r1

θ

Fig. 1. Circular restricted three-body problem.

L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298 1285z

[km

] 10

-1

5

0

-51.502

1.506

1.51

x [km]

x108y [km]

x10 5

x105

L2

Fig. 2. Nominal Lissajous orbit for TPF-O mission.

A more compact notation incorporates a pseudo-potential function, U , such that

U = (1 − �)

r1+ �

r2+ 1

2(x2 + y2) (4)

Then, the scalar equations of motion are rewritten in theform

x =Ux + 2 y = fx (x, x, y, y, z, z) (5)

y =Uy − 2x = fy(x, x, y, y, z, z) (6)

z =Uz = fz(x, x, y, y, z, z) (7)

where the symbolUj denotes the partial derivative ofUwith respect to j . Eqs. (1)–(3) or Eqs. (5)–(7) comprisethe dynamic model in the CR3BP.

Given a solution to the nonlinear differential equa-tions, linear variational equations of motion in theCR3BP can be derived in matrix form as

� ˙X(t) = A(t)�X(t) (8)

where �X = [�x �y �z �x �y �z]T represents varia-tions with respect to a reference trajectory. Generally,the reference solution of interest is not constant. In thisanalysis, the reference trajectory is a Lissajous orbit,or a quasi-periodic orbit, near L2. This particular orbitwas identified by NASA Goddard Space Flight Center(GSFC) [28] as a possibly advantageous orbit for theTPF-O mission, and appears in Fig. 2.

For the nominal Lissajous orbit, A(t) is time-varyingand of the form

A(t) =

⎡⎢⎢⎢⎢⎢⎣

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

Uxx Uxy Uxz 0 2 0Uyx Uyy Uyz −2 0 0Uzx Uzy Uzz 0 0 0

⎤⎥⎥⎥⎥⎥⎦ (9)

The partials Ukl are also functions of time, although thefunctional dependence on time, t , has been dropped.

The general form of the solution to the system inEq. (8) is

�X(t) = �(t, t0)�X(t0) (10)

where �(t, t0) is the state transition matrix (STM). TheSTM is a linear map from the initial state at the initialtime, t0, to a state at some later time, t , and thus offersan approximation for the impact of the variations in theinitial state on the evolution of the trajectory. The STMmust satisfy the matrix differential equation

�(t, t0) = A(t)�(t, t0) (11)

given the initial condition,

�(t, t0) = I6×6 (12)

where I6×6 is the 6 × 6 identity matrix.The STM at time t can be numerically determined

by integrating Eq. (10) from the initial value. Sincethe STM is a 6 × 6 matrix, propagation of Eq. (10)requires the integration of 36 first-order, scalar differ-ential equations. Because the elements of A(t) dependon the reference trajectory, Eq. (10) must be integratedsimultaneously with the nonlinear equations of motionto generate reference states. This requirement necessi-tates the integration of an additional 6 first-order equa-tions, for a total of 42 differential equations.

3. Description of control methods

Optimal nonlinear control and geometric control arecompared to more traditional linear quadratic regula-tors and feedback linearization. The control strategiesare implemented and evaluated for the TPF-O missionconcept.

3.1. Time-varying linear quadratic regulator

Let the motion of a spacecraft moving in the CR3BPbe described by

x = fx (x, x, y, y, z, z) + ux (13)

y = fy(x, x, y, y, z, z) + uy (14)

z = fz(x, x, y, y, z, z) + uz (15)

where fx , fy , and fz are defined in Eqs. (5)–(7) andu(t) = [ux uy uz]T is an applied control (acceleration)vector. Let Xo(t) represent some reference motion that

1286 L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298

satisfies the system Eqs. (13)–(15) and, then, uo(t) de-notes the control effort required to maintain Xo(t). Lin-earization about this reference solution yields a systemof the form

� ˙X(t) = A(t)�X + B(t)�u(t) (16)

where �X(t) and �u(t) represent perturbations relativeto Xo(t) and uo(t), respectively. The matrix A(t) is de-fined in Eq. (9) and B(t)=B=[03×3 I3×3]T. The matrixB(t) is selected such that the control input can instan-taneously change only the velocity of the satellite (notposition).

To minimize some combination of the error stateand required control, Bryson and Ho [29] employ aquadratic cost function

min J = 1

2

∫ t f

t0(�X(t)TQ�X(t) + �u(t)TR�u(t))dt (17)

subject to the governing state equations in Eq. (16) withinitial conditions �X(t0) = �X0. The matrices Q andR represent weighting factors on the state error andcontrol effort, respectively. Both are positive definite.Because the individual state errors are decoupled fromthe required control, Q and R are diagonal matrices.

Application of the Euler–Lagrange theorem to thesystem in Eqs. (16) and (17) yields the following opti-mality requirement on the control variable:

�u(t) = −R−1BP(t)�X(t) (18)

where P(t) is a solution of the matrix differential Riccatiequation

P(t) = − A(t)TP(t) − P(t)A(t)

+ P(t)BR−1BTP(t) − Q (19)

subject to P(t f ) = [06×6].Eqs. (18) and (19) represent a two-point boundary

value problem with 42 first-order, ordinary differentialequations. However, for application to the CR3BP, thiscomplexity can be avoided by exploiting the time invari-ance properties associated with the dynamical model.Marchand [30] develops a time transformation,

P(t f − t) = G−1S(t)G (20)

where G j j =(−1) j−1, that enables simple forward inte-gration to numerically solve a transformed Riccati ma-trix differential equation

S(t) = − A(t)TS(t) − S(t)A(t) − S(t)BR−1BTS(t)

+ G−1QG (21)

Thus, the resulting optimal gain is defined as

K (t f − t) = −R−1BTS(t)G (22)

and can be implemented at each numerical time step.The nonlinear system of equations in Eq. (16) and thematrix differential equation in Eq. (21) are numericallyintegrated forward in time, subject to X(t0) = Xo(t0)and S(t0) = 06×6, respectively. Simultaneously, an op-timal gain matrix is calculated at each integration timestep and control is applied to the perturbed spacecraftmotion.

It is notable that nearly the same optimal controlgains can be computed by solving the algebraic Riccatiequation at each integration step, rather than numeri-cally integrating the differential Riccati equation. Thismethod, although arguably less rigorous, requires farless computation time and, therefore, is very useful forpreliminary analysis. Also, if the reference trajectory isperiodic or well known a priori, the gain matrix can becomputed and stored as a function of time for later im-plementation. If the integration time steps are definedsuch that no interpolation of K (t) is necessary, compu-tation time can be greatly reduced.

3.2. Input state feedback linearization

One alternative to LQR control is IFL. By implement-ing IFL, it is possible to specify desired characteristicsof the state response.

To develop an IFL scheme, the motion of a spacecraftmoving in CR3BP is separated into linear and nonlinearcomponents, such that

x = hxnonlinear + hxlinear + ux (23)

y = hynonlinear + hylinear + uy (24)

z = hznonlinear + hzlinear + uz (25)

where the h functions are defined as the linear and non-linear terms in Eqs. (5)–(7), that is,

hxnonlinear = − (1 − �)(x + �)

r31− �(x − (x − �))

r32(26)

hxlinear = 2 y + x (27)

hynonlinear = − (1 − �)y

r31− �y

r32(28)

hylinear = −2x + y (29)

hznonlinear = − (1 − �)z

r31− �z

r32(30)

hzlinear = 0 (31)

L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298 1287

There are many ways to implement feedback lineariza-tion. One option is to select the control, u(t), such that

u(t) =[uxuy

uz

]=

[−hxnonlinear−hynonlinear−hznonlinear

]+ u(t) (32)

By first removing the nonlinear dynamics from the sys-tem, the control acceleration vector u(t) is determinedto produce the desired response characteristics of thelinear, time-invariant dynamics. Linearization about areference solution yields a subsystem of the form

� ˙X(t)linear = A�X(t)linear + B�u(t) (33)

where B is defined above, and A is determined fromthe linear dynamics of the spacecraft motion such that

A =

⎡⎢⎢⎢⎢⎢⎣

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 2 00 0 0 −2 0 00 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎦ (34)

Because the pair ( A, B) is completely controllable, con-trol effort is defined to place the poles of the linear sys-tem anywhere on the complex plane.

To minimize the state error with minimum control ef-fort, �u(t) is computed via LQR analysis. In this analy-sis, full-state feedback is assumed. The system is lineartime-invariant, thus, the optimal control effort is definedby

�u(t) = −R−1BP�X(t) (35)

where the algebraic Riccati equation yields P:

− ATP − P A + PBR−1BTP − Q = 0 (36)

Feedback linearization offers both advantages and dis-advantages over LQR control. This approach does notrely on a priori knowledge of the controller gain matrixsince it is determined directly from the nonlinear sys-tem rather than the linear variational equations. It alsoallows a designer to specify desired response charac-teristics of the state. Thus, the stability of the systemresponse can be more easily analyzed. However, in gen-eral, feedback linearization is implemented assumingfull-state feedback, which may or may not be available.

3.3. Impulsive control via differential corrections

The use of a differential corrector is arguably thesimplest method for computation of the required impul-sive velocity change to reach a specified future position.

An impulsive velocity change is also, potentially, theeasiest to implement using available chemical thrusters.While LQR, IFL, and nonlinear methods may revealadvantageous trajectories that require less fuel, theseapproaches may be significantly more complicated toexecute. It may be necessary to discretize and optimizethe control results in order to use common propulsionsystems.

Consider the general form of the discretized solutionto the linear system in Eq. (8)

�Xk+1 = �(tk+1, tk)�Xk (37)

where �(tk+1, tk) denotes the STM for the kth segmentassociated with the nominal Lissajous orbit. The statecan be partitioned into position and velocity compo-nents, �r and �v, respectively, such that[

�rk+1�v−

k+1

]= �(tk+1, tk)

[�rk�v+

k

]

=[�rr �rv

�vr �vv

] [�rk

�v−k + �Vk

](38)

where the superscript “+” or “−” signifies the be-ginning or end of a segment originating at time tk ,respectively, and �Vk represents an impulsive maneuverapplied at tk . To accomplish the desired change inposition, �rk+1, the required impulsive change invelocity can be expressed as

�Vk = B−1k (�rk+1 − �rr�rk) − �v−

k (39)

Of course, in the nonlinear system, this linearly com-puted velocity change is inadequate. Differential correc-tions are incorporated in an iterative scheme to identifythe precise maneuver that meets the end-state constraintto within the specified tolerance levels, as discussed inHowell and Barden [31].

3.4. Nonlinear optimal control

To determine minimum fuel maneuvers for a space-craft moving in the full nonlinear CR3BP, a traditionalnonlinear optimal control problem is formulated. Let acost function be defined as

min J =∫ t f

0�xu

2x + �yu

2y + �zu

2z dt (40)

where the vector �=[�x �y �z]T represents the weights

on the corresponding control vector components, ux ,uy , and uz , defined in Eq. (32). The Hamiltonian is thenrepresented by

H = �xu2x + �yu

2y + �zu

2z + �1 x

+ �2 y + �3 z + �4 x + �5 y + �6 z (41)

1288 L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298

where �(t)=[�1 �2 �3 �4 �5 �6] is a vector of co-statesand X(t) = [x y z x y z] is the vector of spacecraftstates. The Euler equations are derived via the partialderivative of the Hamiltonian with respect to the state,as follows:

�1 = −�H

�x= −�4

�

�x(x) − �5

�

�x(y) − �6

�

�x(z) (42)

�2 = −�H

�y= −�4

�

�y(x) − �5

�

�y(y) − �6

�

�y(z) (43)

�3 = −�H

�z= −�4

�

�z(x) − �5

�

�z(y) − �6

�

�z(z) (44)

�4 = −�H

�x= −�1 + 2�5 (45)

�5 = −�H

�y= −�2 − 2�4 (46)

�6 = −�H

�z= −�3 (47)

Then, the conditions for optimality become

�H

�ux= 2�xux + �4 = 0 (48)

�H

�uy= 2�yuy + �5 = 0 (49)

�H

�uz= 2�zuz + �6 = 0 (50)

This control strategy is implemented to maneuver aspacecraft from a specified initial state (both positionand velocity) to a specified final position. Therefore,the boundary conditions include three “free” final con-ditions on both the state and co-state variables

X(0) = [x0 y0 z0 x0 y0 z0] (51)

X(t f ) = [x f y f z f free free free] (52)

�(t f ) = [free free free 0 0 0] (53)

The two-point boundary value problem is solved usinga function (bvp4c) available in the Matlab OptimizationToolbox�. Final conditions for the “free” state and co-state elements are determined based upon knowledge ofthe optimal LQR solution for the same maneuver.

3.5. Geometric “floquet” control

Floquet control is an impulsive control strategy thattakes advantage of the phase space near a nominal or-bit. At discrete intervals, a maneuver is performed in

the direction of the center or stable manifolds. The mainresult of Floquet is that, for a time-varying periodic, lin-ear system, the STM can be factored into two matrices,E and J , in the form

�(t, 0) = E(t)eJ t E(0) (54)

The matrix J is a constant matrix, usually in the Jordannormal form. Its diagonal entries, �i , are the Poincaréexponents, that is, the analog of the eigenvalues forconstant-coefficient systems.

Because the nominal Lissajous orbit is not perfectlyperiodic, a similar halo orbit is successfully employedto approximate the surrounding phase space. The matrixE(t) corresponding to the halo orbit is periodic withperiod, T . Solving a Floquet problem for a periodic haloorbit over all time, thus, requires the determination ofthe constant matrix J and the periodic matrix E(t) overone period. Since the matrix E is periodic, E(0)=E(T )and at t = T , Eq. (54) becomes

�(T, 0) = E(0)eJT E(0) (55)

Thus, E(0) is the matrix of eigenvectors of the mon-odromy matrix, �(t, t0). The eigenvalues of the mon-odromy matrix, �i , are related to the Poincaré exponents�i , i.e.,

�i = e�i T (56)

After numerically integrating Eq. (11) over one period,the Poincaré exponents and the eigenvector matrix att = 0 are obtained. Since E(t) is periodic and there-fore bounded, the stability of the system is governedby �i (or �i ) alone. Both the halo orbit and nominalLissajous orbit of interest can be modeled as inherentlyunstable. In fact, the six-dimensional eigenstructure ischaracterized by one unstable eigenvalue (�1), one sta-ble eigenvalue (�2), and four eigenvalues in the centersubspace. Two of these neutrally stable eigenvalues arelocated on or near the unit circle (�3 and �4) and theremaining two are approximately equal to one (�5 and�6). At any point in time, the perturbation �X(t) can beexpressed in terms of any six-dimensional basis. TheFloquet modes (e j ), defined by the columns of E(t),form a non-orthogonal six-dimensional basis. Hence,

�X(t) =6∑j=1

�X j (t) =6∑j=1

c j (t)e j (t) (57)

where �X j (t) denotes the component of �X(t) alongthe j th mode, e j (t), and the coefficients c j (t) are theelements of the vector c(t) defined by

c(t) = E(t)−1�X(t) (58)

L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298 1289

This Floquet structure is implemented by Gómez et al.[11] and, later, by Howell and Keeter [32] as the ba-sis of a station keeping strategy for a single spacecraftevolving along a halo orbit. Both investigations deter-mine the impulsive maneuver scheme that is required toremove the component of error in the direction of theunstable mode, �X1(t) (as defined in Eq. (57)), at dis-crete intervals. Because impulsive maneuvers, are cal-culated based on the phase space associated with a haloorbit, yet implemented on a spacecraft moving near aLissajous orbit, additional error is incurred. This impliesslightly larger impulsive maneuvers may be required tomove a spacecraft state to the center subspace. How-ever, in general, the error incurred is extremely smallwhen compared to the magnitude of the required impul-sive maneuver. Marchand and Howell, for instance, let

�Xdesired (t) =6∑j=2

(1 + � j (t))�X j (t) (59)

denote the desired perturbation relative to the referenceorbit, where �(t) is some, yet to be determined, coeffi-cient vector. Note that the limits of the summation rangefrom 2 to 6 which implies that the perturbation in thedirection of the unstable mode, e1 has been removed.The linearized control problem then reduces to a com-putation of the impulsive maneuver, �v(t) such that

6∑j=2

(1 + � j (t))�X j (t) =6∑

k=1

�Xk(t) +[

03×1�v(t)3×1

](60)

After some reduction, Eq. (60) can be rewritten in ma-trix form as[

�X2(t) �X3(t) �X4(t) �X5(t) �X6(t)03×3I3×3

]×

[�(t)5x1

�v(t)3×1

]= �X1(t) (61)

where � represents a 5 × 1 vector formed by the � j

components in Eq. (59). Therefore, the maneuver,�v3×1, that is required to remove the unstable mode,e1, from the perturbation at any given time can beapproximated from Eqs. (61). Howell and Keeter [31]identify this required �v3×1 via a minimum norm so-lution. If the maneuver is constrained to the Sun–Earthline (x-direction), �v1×1 is actually a scalar, and thesystem possesses an exact solution.

If exactly three modes are removed and control ispossible in three directions, the system also possessesan exact solution. For example, Marchand and Howell[17] demonstrate the impulsive �v3×1 that is requiredto remove the portion of the perturbation along the

unstable and short period modes (e1, e3, and e4). It canbe determined exactly from[

�(t)3×1�v(t)3×1

]=

[�X2(t) �X5(t) �X6(t)

03×3−I3×3

]−1

× (�X1(t) + �X3(t) + �X4(t)) (62)

Similarly, the �v3×1 vector that is required to removethe unstable and long period modes (e1, e5, and e6) isexactly determined from[

�(t)3×1�v(t)3×1

]=

[�X2(t) �X3(t) �X4(t)

03×3−I3×3

]−1

× (�X1(t) + �X5(t) + �X6(t)) (63)

As formulated in Eqs. (62) and (63), �v3×1 is the im-pulsive control necessary to remove perturbations inthe direction of one unstable mode and two semi-stablemodes in the linearized system represented in Eq. (8)at a given time.

4. Results from TPF-O mission analysis

4.1. TPF-O mission concept

The NASA TPF-O mission is an optical formationthat will search for Earth-like planets in other solar sys-tems. Two spacecraft are included in the formation: onetelescope spacecraft and one occulter spacecraft. Thetelescope spacecraft moves near a nominal L2 Lissajousorbit. During an observation period, the telescope andthe occulter align along the direction to a star of inter-est in some inertial direction. The two spacecraft arerequired to maintain a fixed relative distance during

TransitPeriod

ObservationPeriod

Fixeddistance

t1

t2

t3 Orbitnear L2

Fig. 3. Schematic of TPF-O mission sequence.

1290 L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298

Table 1Features of DRM4 and DRM5 sequences

Star sequence Mission duration(days)

Number ofobservations

Distance to occulter(km)

Average observation(days)

Average transit(days)

DRM4 1832 134 72 000 2.67 10.99DRM5 1840 102 37 800 3.24 14.80

this phase. During the transit, or maneuver, period thereis no constraint on the geometry of the occulter andtelescope. The “transit period” refers to the time periodbetween t1 and t2, while the “observation period” de-notes the time interval between t2 and t3, as illustratedin Fig. 3.

The path of the occulter is dictated by a star sequence.This star sequence represents a list of stars that are inthe same spectral class as the Sun, indicating a higherpotential for the existence of nearby, orbiting Earth-like planets. The required observation time at each starand the specific length of time allotted for maneuver-ing between each observation is also included in thestar sequence. In this analysis, two star sequences areused for comparison: design reference mission (DRM)4 and DRM 5. These sequences are specified as op-tions for TPF-O. The general characteristics of the starsequences are compared in Table 1. Placement of thestars, in a given sequence on a celestial sphere, is notedin Fig. 4.

4.2. Control analysis results

The various control strategies are implementedin differing combinations for DRM 4 and DRM 5.For example, in one scenario, a nonlinear optimalcontroller is used for transit between t1 and t2 (inFig. 3) and an LQR controller is used to track thestar during an observation period. Alternatively, animpulsive maneuver can support the transit betweent1 and t2, as well as t2 and t3; this approach impliesthat no star tracking occurred during the observa-tion period. A summary of the different scenariosand the resulting required control effort is listed inTables 2 and 3.

The first column of these tables, labeled “Total �V ”is an estimate of the control effort required for theentire mission. Because each mission contains a dif-ferent number of star targets, it may be helpful to de-termine the average control effort used for each starin the sequence. This is shown in the second columnlabeled, “Ave.�V per star.” In columns 3 and 4, the totalcontrol effort is divided into mission segments: transit

Fig. 4. Star sequences for DRM 4 (top) and DRM 5 (bottom).

and observation periods, respectively. Finally, the lastcolumn is a measure of only the control effort neededto track the star during the observation period. This

L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298 1291

Table 2Required control effort for DRM 4

Control method Total �V(km/s)

Ave. �V per star(m/s)

Transit �V(km/s)

Observation �V(km/s)

Star track(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 30.074Impulsive transit + IFL star track 5.966 44.522 2.704 3.263 30.166LQR transit + LQR star track 5.998 44.761 2.745 3.253 30.074IFL transit + IFL star track 6.067 45.275 2.804 3.263 30.166NL 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 30.074NL optimal transit + IFL star track 6.111 45.604 2.848 3.263 30.166Geometric 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 30.074Geometric transit + IFL star track 6.811 50.828 3.548 3.263 30.166

Table 3Required control effort for DRM 5

Control method Total �V(km/s)

Ave. �V per star(m/s)

Transit �V(km/s)

Observation �V(km/s)

Star track(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 13.555Impulsive transit + IFL star track 3.013 29.539 1.451 1.561 13.581LQR transit + LQR star track 3.032 29.725 1.474 1.558 13.555IFL transit + IFL star track 3.111 30.500 1.549 1.561 13.581NL 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 13.555NL optimal transit + IFL star track 3.104 30.431 1.543 1.561 13.581Geometric 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 13.555Geometric transit + IFL star track 3.302 32.373 1.741 1.561 13.581

number is defined as the amount of control necessary tomaintain the occulter within a specified tolerance of thedesired state, after it has made a maneuver to reach thatstate.

A few observations are noted in the results from thecontrol analysis. Tables 2 and 3 both indicate that theimpulsive method, used during the transit period, re-quires the least amount of control effort. The geomet-ric “Floquet” maneuvers require the most control effortduring the transit period.

DRM 4 has a significantly higher average con-trol effort per star than DRM 5, as indicated bycolumn 2 of Tables 2 and 3. Upon further investi-gation, it was determined that this is a result of thelarger distance between the telescope and the occulter:72000km for DRM 4 as opposed to 38700km forDRM 5. As the occulter is placed further and furtheraway from the nominal telescope orbit, the control

cost tends to increase. This may be due, in part, tothe longer distance the occulter must move in orderto create the desired angle during each observationperiod.

The required control at each star can also be repre-sented graphically, as plotted in Figs. 5 and 6. Fig. 5illustrates a notional thrust profile for DRM 4. This pro-file corresponds to row 3 in Table 2: impulsive controlduring transit periods and LQR control during observa-tion periods. The horizontal axis represents each star inthe DRM 4 sequence. The vertical axis is the amountof control required at a particular star in the sequence.The red line represents the total required control effortfor the DRM 4 mission. This control effort is dividedinto the control required for the observation period andthat required for the transit period, that is, the greenand the cyan line, respectively. The dark blue linecorresponds to the amount of control required if

1292 L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298

150

125

100

75

50

25

00 20 40 60 80 100 120

Star Sequence

ΔV [

m/s

]

Transit Impulsive ΔV: 2.7022km/sTotal ΔV without Dynamics: 10.1837km/sObservation LQR Control: 3.2534km/sTotal ΔV: 5.955km/s

Fig. 5. Notional thrust profile for DRM 4, impulsive control during transit periods and LQR control during observation periods.

100

90

80

70

60

50

40

30

20

10

00 10 20 30 40 50 60 70 80 90 100

Star Sequence

ΔV [

m/s

]

Continuous Maneuver Control Effort: 1.543km/sRequired Control without Dynamics: 5.2174km/sObservation ISFL Control: 1.561km/sTotal Required ΔV: 3.104km/s

Fig. 6. Notional thrust profile for DRM 5, nonlinear optimal control during transit periods and input-state feedback linearization control duringobservation periods.

analysis were performed without CR3BP dynamics. Inother words, spacecraft were assumed to be moving infree space. It is notable that control effort required forspacecraft moving in free space is significantly higherthan that required for spacecraft moving in the CR3BP.

Thus, in this mission scenario, the natural dynamicsin this regime reduce the control cost of spacecraftmaneuvers.

Similarly, Fig. 6 illustrates a notional thrust pro-file for DRM 5. This profile corresponds to row 9 in

L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298 1293

z

y

^X

x

h1 h2

CR3B

Radial-Transverse

Frame

Fig. 7. Coordinate transformation from CR3BP to radial-transverseframe.

Table 3: nonlinear optimal control during transit pe-riods and IFL during observation periods. Again, therequired control for a spacecraft formation moving infree-space is significantly higher than for a formationmoving in the CR3BP.

4.3. LQR control for TPF-O

For the TPF-O mission, the occulter position tol-erances are provided in terms of a radial and trans-verse component. The radial direction is defined as thedirection from the telescope to the target star, as illus-trated in Fig. 7. The transverse directions lay anywherein a plane perpendicular to this radial direction. Theocculter is constrained to remain within ±100km of thedesired path in the radial direction and ±10m from thedesired path in the transverse direction. Because theLQR problem is formulated in the CR3BP, it is difficultto determine the effect of the weighting matrices, Q andR, on the precision in the defined radial and transversedirections. Thus, the LQR problem is reformulated interms of radial and transverse coordinates. Let the di-rection to the star target from the telescope be definedas r = r1 x + r2 y + r3 z and the direction of the velocityof the telescope near the Lissajous orbit be defined asv = v1 x + v2 y + v3 z.

Then, a vector perpendicular to both r and v is

h1 = r × v (64)

The direction h1 is defined as the first transverse direc-tion. This choice is arbitrary, since the specific direc-tion of the transverse coordinate frame does not effectthe problem formulation. The new coordinate frame iscompleted by h2 such that

h2 = r × h1 (65)

To transform the original LQR problem into the newcoordinate frame, a transformation matrix,C , is derived.

3.8

3.7

3.6

3.5

3.4

3.3

3.2

3.1

Dis

tanc

e fr

om N

omin

al O

rbit

[km

]3.8

3.7

3.6

3.5

3.4

3.3

3.2

3.1

Dis

tanc

e fr

om N

omin

al O

rbit

[km

]

0 2 4 6 8 10 12 14 16 18

Time [days]

0 2 4 6 8 10 12 14 16 18

Time [days]

x104

x104

Final desiredlocation

Path without maneuverPath after maneuver

Fig. 8. Geometric control method, transit period.

This matrix transforms the perturbations �X to the newcoordinate system �Z (t)= [r h1 h2 r h1 hT2 ] such that

�X(t) = C�Z (t) (66)

� ˙X(t) = C� ˙Z (t) + C�Z (t) (67)

Substituting Eqs. (66) and (67) into Eq. (16) yields

�Z (t) = [C(t)−1A(t)C(t) − C(t)−1C(t)]�Z (t)

+ C(t)−1Bu(t) (68)

and, thus, the new cost function for the LQR formulationis

min J = 1

2

∫ t f

t0(�Z (t)TQZ�Z (t)

+ �u(t)TRZ�u(t))dt (69)

1294 L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298

Δ

Δ

Fig. 9. Required control and state error response in the transverse direction over observation period: ±10m.

where the diagonal entries of QZ and RZ now representthe weighting on the radial and transverse error compo-nents. In this analysis weighting matrices

QZ = I6×6 (70)

RZ = diag([10−5 10−10 10−10]) (71)

are sufficient in almost all cases to maintain the desiredtolerances on the state. However, recall that only therelative magnitudes of QZ and RZ are important, notthe exact numbers used.

4.4. Floquet control for TPF-O

During a transit period, several small maneuversmaintain the spacecraft state near the center subspaceof the reference Lissajous orbit. Because the Lissajousorbit is not perfectly periodic, a similar halo orbit isused to approximate the surrounding phase space whilecalculating the required velocity change.

Because the spacecraft should eventually reach aspecified observation state (which may or may not bein the center subspace), an impulsive targeting ma-neuver is required at some point during the transitperiod. This maneuver is added at the time when thesmallest total amount of thrust is required to move tothe desired observation location. A schematic of the

geometric control method, as applied to the TPF-Omission, appears in Fig. 8.

The blue line in the top plot in Fig. 8 indicates thedifference between the path of the spacecraft and thenominal Lissajous orbit in the rotating x-direction dur-ing one transit period (approximately 16 days). A smallmaneuver is applied at each blue star to keep the space-craft state near the center manifold during the transitperiod. Over longer periods of time, this trajectory isquasi-periodic. The red line in the top plot representsthe path of the spacecraft if no maneuver is imple-mented. In the bottom plot, a point along the centermanifold has been identified which minimizes the to-tal required control effort to reach the specified obser-vation point. This point appears in the black box. Atthis point, a targeting maneuver enables the vehicle toreach the final desired location. Again, the blue line in-dicates the trajectory of the spacecraft relative to thenominal orbit after the maneuvers have been performed.The red line indicates the path of the spacecraft withoutmaneuvers.

4.5. Effectiveness of control methods on mission goals

During the transit period, no requirements ex-ist on the position and velocity of the spacecraft

L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298 1295

Δ

Fig. 10. Required control and state error response in the radial direction over observation period: ±100km.

formation to facilitate imaging. During the observa-tion, though, the occulter must remain within 100kmof the specified path in the radial direction and within10m of the specified path in the transverse direc-tion. When impulsive control is employed during theobservation period, generally, these mission require-ments are not met. However, when either IFL or LQRcontrol are applied, weighting matrices can be se-lected such that position errors are within requiredtolerances.

An example of the required control effort and stateerror response in the transverse direction, during oneobservation period of the TPF-O mission, appears inFig. 9. The weighting matrices, QZ and RZ , are de-fined such that the state error is just within the definedmission tolerances. In the first column of Fig. 9, therequired control in the transverse direction is plottedas a function of time for three different control meth-ods. In the second column of the plot, the state error inthe transverse direction for the corresponding control

method is displayed. This specific observation periodcorresponds to the first star of interest in the DRM 5mission.

The typical required control effort and the state er-ror response in the corresponding radial direction arerepresented in Fig. 10. Because the position tolerancein the radial direction is much more lax, the state errorresponse is much higher in magnitude. This, again, re-sults in part from the selection of QZ and RZ . In thefirst column of the figure, the required control in theradial direction appears as a function of time for threedifferent control methods applied to star tracking. Thesecond column in the plot displays the state error in theradial direction for the corresponding control method.

Finally, an example of the path of both the occulterand telescope spacecraft can be seen in Fig. 11. Thetrajectory of the telescope appears in blue, following thenominal Lissajous orbit. The occulter path is both redand cyan, indicating the transit and observation periods,respectively.

1296 L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298

Fig. 11. Occulter and telescope spacecraft path, impulsive control during transit and lqr control during observation period.

5. Concluding remarks and future work

This work represents a preliminary comparison ofvarious control strategies for the TPF-O mission con-cept. Error inclusion as well as estimation filters canbe incorporated in future analyses. Also, the continuouscontrol methods described may be discretized and op-timized to provide competing control solutions for cur-rent chemical propulsion systems in a full ephemerismodel.

The TPF-O DRM 4 and DRM 5 mission conceptsare feasible, i.e., required mission tolerances can bemaintained with limited error inclusion. However, notall control strategies are capable of meeting missionrequirements on the state of the occulter spacecraft.The control strategy that meets all mission require-ments while using the smallest amount of control isa combination of LQR and impulsive control meth-ods. The LQR control method is implemented dur-ing the observation period and the impulsive methodis used during the transit period. Using this controlstrategy, the total mission �v is approximately 6km/sfor DRM 4 and 3km/s for DRM 5, as presentedin Tables 2 and 3.

For the two TPF-O design reference missions an-alyzed, it is observed that inclusion of the naturaldynamics in the model reduces the control cost formaneuvers. Such a result highlights the importanceof including dynamics in mission control analysis inmulti-body regimes. If all forces acting on the space-craft were to be approximated as simply the force dueto the spacecraft thrusters (moving in “deep space”),significantly higher control requirements are predicted.

A reduction in overall mission control cost may beaccomplished in two ways. First, control savings canbe found by designing a mission in which the occulteris closer, rather than farther, from the telescope space-craft. As the distance between the occulter and tele-scope spacecraft is increased, the required control formaneuvers increases. This may be due, in part, to theadvantageous natural dynamics near the nominal Lis-sajous orbit; the dynamic regime changes as the oc-culter moves farther from the nominal orbit. Second,the order and location of stars in a sequence could af-fect the necessary control effort. If these stars couldbe chosen based, in part, on the amount of controlrequired to maneuver, the cost could be reduced. How-ever, the current mission design stipulates a specific

L.D. Millard, K.C. Howell / Acta Astronautica 63 (2008) 1283–1298 1297

star sequence to be followed to maximize scientificresults.

Acknowledgments

The authors would like to thank Dave Folta, Con-rad Schiff, and the TPF-O working group at NASAGoddard Space Flight Center for providing informationand guidance for the current study. This work was per-formed at Purdue University with support from NASAGoddard Space Flight Center under contract numbersNCC50-727 and NNG04GP69G. Support was also pro-vided from Purdue University via the Bilsland Disser-tation Fellowship.

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