elliptic 3 body problem

Stationkeeping on UnstableOrbits: Generalization to the

Elliptic Restricted Three-BodyProblem

Pini Gurfil1 and Dani Meltzer2

Abstract

We develop new methods for generating periodic orbits about the collinear librationpoints and for stabilizing motion on libration orbits using the general formalism of theelliptic restricted three-body problem (ER3BP). Calculation of periodic orbits is accom-plished by formulating the ER3BP as a control problem. This approach yields halo-likeorbits that do not exist without applying active control, having arbitrarily small amplitudes.Linearization about the libration points in pulsating coordinates yields an unstable linearparameter-varying (LPV) system with periodic coefficients. We introduce a continuousacceleration control term into the state-space dynamics and use an LPV-generalized versionof the pole-assignment technique to find linear periodic reference trajectories. The nonlin-ear terms of the equations of motion are then treated as periodic disturbances. A disturbance-accommodating control is used to track the libration-point reference orbit in the presence ofnonlinear periodic disturbances. Simulation experiments show that stationkeeping is robustto propulsive dispersions.

Introduction

Recent years have seen a rising interest in launching spacecraft into libration-point orbits for scientific missions. Past missions such as ISEE-3, SOHO and Gene-sis successfully utilized quasi-periodic orbits about the and collinear librationpoints of the Sun-Earth system. With the embarkation of future NASA and ESAmissions such as Darwin, TPF, and SAFIR [1], [2] to be launched into librationpoint orbits, there is an opportunity to design and simulate novel trajectory plan-ning and control schemes.

In practical analysis and design of missions around libration points, the circularrestricted three-body problem (CR3BP) model is usually adopted [3]–[6]. Although

L2L1

The Journal of the Astronautical Sciences, Vol. 54, No. 1, January–March 2006, pp. 000–000

1

1Senior Lecturer, Faculty of Aerospace Engineering, Technion— Israel Institute of Technology, Haifa 32000,Israel. Email: [email protected] Student, Israel Institute of Technology, Haifa 32000, Israel. Email: [email protected].

this model has proven fruitful, it possesses an inherent approximation, assumingthat the orbit of the secondary is circular. However, both the motion of the Eartharound the Sun and the motion of Moon around the Earth are eccentric. Incorporat-ing the eccentricity term into the equations of motion renders a more general model,known as the elliptic restricted three-body problem (ER3BP). The ER3BP has sig-nificant topological differences compared to the CR3BP: the position of the libra-tion points in the Earth-Moon system is not constant, but rather pulsating with respectto Earth, and moreover, the Jacobi integral is time- (true anomaly-) dependent.

Several works have addressed the problem of finding natural periodic orbits inthe planar ER3BP based on specialized regularizations [7], [8]. Derivation of suchorbits through numerical searches was accomplished in references [9] and [10].These methods provided initial conditions resulting in periodic or quasi-periodicballistic trajectories. However, with the persistent improvement in electric propulsion,it is now possible to design libration point trajectories, which are not necessarilycenter-manifold solutions of the unperturbed dynamics, but are rather approximatetrajectories, the stationkeeping on which is performed by continuous low-thrustpropulsion

The method of reduction to a center-manifold along with other methods give aset of initial conditions that result in families of ballistic libration-point orbits(depending on the Hamiltonian values) [22], such as the Lyapunov family of planarorbits or the North/South halo orbits. While the minimal amplitude of ballistic or-bits is limited to a few thousands of kilometers [23], the approach developed hereinpermits designing periodic orbits with much smaller amplitudes. As a result, the tra-jectory generation process is dictated by the requirement, or, alternatively, by themaximum allowable excursion from the libration point, yielding arbitrarily small“halo” orbits. These orbits are therefore more flexible and adaptable than ballisticorbits, rendering them suitable for diverse science missions; For example, small“halo” orbits may be used for future space-based radio telescopes if the spacecraftis in a region completely shaded by the Moon.

In this work, we use the ER3BP model to calculate periodic reference trajecto-ries around collinear libration points by applying the following steps. First, we de-rive a simple form of the equations of motions by normalizing into dimensionlesscoordinates and using the true anomaly as the independent variable [7], [11]. Sec-ond, we calculate the position of the libration points in these coordinates and derivea first-order approximation to the equations of motion around the libration points,resulting in a periodic linear parameter-varying (LPV) system. Finally, we introducea control term into the linearized system and use generalized pole-placement [12],[13] to cancel the diverging modes.

Cancellation of unstable modes by pole-placement has been studied before byScheeres et al. [20] for the CR3BP. However, we use here a generalized pole-placement technique based on monodromy matrix calculations for the generalsetup of the ER3BP without resorting to the CR3BP or Hill’s CR3BP approxi-mations. Application of such a generalized pole-placement technique for periodicsystems constitutes a novel approach to the problem of stabilizing motion on un-stable orbits.

To find our reference trajectories, we temporarily neglect the nonlinear dynami-cal constituent. The nonlinearity is later introduced back into our model as a persist-ent disturbance [18]. Wie [21] proposed a similar approach based on approximatingthe ER3BP dynamics using eccentricity power series. In the current paper, we do

�V

2 Gurfil and Meltzer

not use any approximation, but rather treat the full nonlinear time-varying problemin pulsating coordinates.

Disturbances are subsequently modeled as a second-order dynamical systemsdriven by nonlinear dynamical terms. Thus, while Howell and Pernicka [14] utilizedan impulsive linear quadratic regulator (LQR) and Cielaszyk and Wei [18] adoptedan infinite-horizon continuous LQR, in this work we utilize a finite-horizon LQRscheme to track reference trajectories while rejecting persistent disturbances usingthe realistic ER3BP model.

In order to evaluate the applicability of the developed control scheme, the station-keeping algorithm is tested for robustness under propulsive dispersions using a Monte-Carlo simulation that includes a realistic model of an electric propulsion system.

Equations of Motion

The ER3BP is a dynamical model that describes the motion of an infinitesimal-mass body—a space vehicle—under the gravitational influence of two massivegravitational bodies— the primaries. The most popular coordinate system used tomodel the ER3BP has its origin set at the barycenter of the large primary, and thesmall primary, The x-axis is positive in the direction of the z-axis is perpen-dicular to the plane of rotation and is positive upwards, and the y-axis completes thesetup to yield a Cartesian, rectangular, dextral reference frame, as shown in Fig. 1.The small primary is orbiting the large primary on an elliptic orbit with eccentric-ity e. This orbit complies with the two-body Keplerian motion; the distance betweenthe primaries, depends upon the true anomaly, f, through the conic equation

(1)

where p is the semi-latus rectum. The rate of change of the true anomaly, satis-fies where h is the magnitude of the angular momentum, given by

Here G is the universal gravitational constant, denotes the massof the first primary, and is the mass of second primary.M2

M1G�M1 � M2�p.h2 �f � h��2,

f ,

� �p

1 � e cos f

�,

M2,M2.M1

Stationkeeping: Generalization to the Elliptic Restricted Three-Body Problem 3

FIG. 1. Definition of the Coordinate System.

The position vector, R, of the spacecraft in the rotating barycentric frame shownin Fig. 1 is

(2)

The coordinates system rotates at the rate about so the angular velocity vectorsatisfies and the velocity vector, V, can be written as

(3)

By defining is located at and is located at The position of the spacecraft with respect to the primaries (cf.Fig. 1) can be expressed by

(4)

Denoting the kinetic energy by the potential energy by and the Lagrangianby we have

(5)

Writing the Euler-Lagrange equations with the components of the position vector asthe generalized coordinates

(6)

yields the equations of motion

(7)

In order to simplify equations (7), a transformation to rotating-pulsating co-ordinates is required [11]. This transformation consists of normalizing time by

normalizing position by the instantaneous distance

(8)

and then transforming time derivatives into derivatives with respect to trueanomaly

(9)�� pe cos f

�1 � e cos f �2

d� �dt* �

d� �df

df

dt* � � ��f ,

Z � ��Y � ��,X � ��,

�G�M1 � M1��3,

Z � �1 �Z

��X � ��2 � Y 2 � Z2�3/2 Z

��X � � 1��2 � Y 2 � Z23/2

Y

��X � � 1��2 � Y2 � Z23/2

Y f 2Y 2 f X f X � �1 �Y

��X � ��2 � Y 2 � Z2�3/2

�X � � 1��

��X � � 1��2 � Y 2 � Z23/2

X f 2X 2 f Y f Y � �1 � �X � ��

��X � ��2 � Y 2 � Z2�3/2

d

dt�L�R�

�L�R

� 0

L � K UU � �1 �

�r1�

�r2�,K �

1

2V � V,

L,U,K,

r2 � �X � � 1�� Y Z�Tr1 � �X � � Y Z�T,

�1 , 0, 0�.M2�, 0, 0�M1 �� M2��M1 � M2�,

V � R � � R � �X � f Y�i � �Y � f X�j � �Z�k

� � f kk,f

R � X i � Yj � Zk


where is the dimensional time. The relationships between the dimensional, time-dependent and the dimensionless, true anomaly-dependent velocities and accelera-tions are

(10)

(11)

The final step is to define a pseudo-potential function as

(12)

This yields the compact equations

(13)

The location of the libration points in rotating-pulsating coordinates is constant andidentical to their location in the CR3BP setup. It is therefore straightforward to per-form linearization of the equations of motion about the libration points. To that end,we define a state vector as

(14)

For an initial true anomaly the linearized equations of motion assume a periodiclinear parameter-varying (LPV) state-space representation of the form

(15)

with a period The matrix A satisfies

(16)

It is well-known [11] that the linear approximation about the collinear points in theCR3BP is unstable. However, the linear CR3BP is autonomous, whereas the linear

A22 � 0

2

0

2

0

0

0

0

0�

A21 �1

1 � e cos� f � a41

0

0

0

a52

0

0

0

a63 e cos� f ��,A � O3 3

A21

I3 3

A22�,

T � 2�.

A� f � � A� f � T�xr� fI� � xrI

xr�� f � � A� f �xr� f �

fI,

x � �� T � �x1 x2 x3 x4 x5 x6�T

��

��

�� 2��

��

�� 2��

��

1

2��2 � �2 e�2 cos f � U��

1

1 � e cos f

�h2

p2 ��1 � e cos f �� e� cos f � �1 � e cos f �3

X � �� 2� � � f �� f 2�

X �d��

dt* � f �� h

p�e sin f� � �1 � e cos f ��

t*


ER3BP is parameter (time) varying. Thus, in order to study the stability of the lin-earized ER3BP, we recall Floquet’s theory [15], dedicated to periodic LPV systems.

The main result of Floquet’s theory shows that the stability of periodic LPV sys-tems can be deduced from analyzing the stability of an equivalent, time-invariant,system. Letting denote the state transition matrix, Floquet’s equivalent sys-tem is obtained from the transformation

(17)

The equivalent time-invariant system is then simply

(18)

The Floquet matrix, J, is calculated from the monodromy matrix as

(19)

A well-known theorem [16] states that a periodic LPV system is exponentiallystable if and only if all the characteristic exponents (eigenvalues of the Floquetmatrix) have negative real parts

(20)

The monodromy matrix can be calculated in several ways, the most common beinga direct numerical integration of the state transition matrix through the matrix dif-ferential equations

(21)

In this paper, we calculate the state transition and monodromy matrices numericallyand verify the computations using a recently developed semi-analytical methodbased on orthogonal Chebyshev polynomials [17]. A brief description of this ap-proach is given in Appendix A.

We shall subsequently utilize the ER3BP dynamical model presented in this sec-tion for designing a feedback stabilizer for the motion on libration point orbits. Weshall accomplish this goal in two steps: First, we will design controlled periodicreference trajectories about the collinear points using a pole-placement techniquegeneralized to periodic LPV systems; second, we shall introduce the inherent non-linearity of the ER3BP back into the equations of motion using an LPV-generalizedversion of disturbance-accommodating control.

Controlled Periodic Libration Point Orbits

We start by generating reference trajectories about the libration points. In orderto stabilize the unstable modes of the LPV system (15), we introduce a control vec-tor

(22)

We assume that (22) is controllable and observable, and use pole-placement torender a periodic reference trajectory. Since we deal with a periodic LPV system, a

A� f � T� � A� f �xr� fI� � xrI

xr�� f � � A� f �xr� f � � Bu� f �u � �m

�� fI, fI� � I

�� f, fI� � A� f �� f, fI�d

df

Re��i�J�� 0

J �1

Tln �

�� f � fI � T, fI��

z�� f � � Jz� f �

P� f � � eJf�1� f, fI�z� f � � P� f �x� f �,

�� f, fI�


generalized version of the pole-placement technique must be adopted [12], [13].In particular, we shall adapt the static output feedback method proposed in refer-ence [12], which relies on using a sampled output at the beginning of each periodand calculating a generalized hold function as follows:

(23)

In equation (23), is the state transition matrix (calculated either numericallyor semi-analytically, as explained in Appendix A), W is the controllability Gramianand L is a pole-placement gain matrix. The static output feedback controller forsome output is given by

(24)

Controller (24) does not necessary result in a continuous signal. Appendix B de-velops a methodology for deriving a smooth controller; however, smoothness is notrequired to yield a smooth reference orbit.

This completes the formalism required for designing libration-point reference or-bits using the linearized LPV model. In the next section, we introduce the nonlin-earities back into the model through a disturbance-accommodating control law.

Accommodating the Nonlinearities

To render the treatment general, the nonlinear dynamical effects must be incor-porated into the model. In reference [18], Cielaszyk and Wie modeled the nonlin-ear gravitational terms in the CR3BP as periodic disturbances. The main idea wasto model the nonlinear dynamical effects as an output of a linear second-order sys-tem driven by a nonlinear input. An LQR scheme was subsequently used to sta-tionkeep on the reference trajectory while rejecting the disturbance. In this section,we extend the Cielaszyk-Wie formalism to the nonautonomous ER3BP model.

Recall the ERTBP equations of motion in pulsating coordinates (13). Let a ref-erence trajectory of a spacecraft flying near a collinear libration point be givenby and let

(25)

Explicitly, (25) can be written as

(26)

u� � �r� 1

1 � e cos� f ��e cos� f ��r �1 ��rr1

3 �rr23�

u� � �r� � 2�r� 1

1 � e cos� f ��r �1 ��rr1

3 �rr23�

u� � �r� 2�r� 1

1 � e cos� f ��r �1 � ��r � �r1

3 ��r � 1�r23�

u� � �r� ��

��r.

u� � �r� � 2�r� ��

��r

u� � �r� 2�r� ��

��r

��r, �r, �r�,

k � 0, 1, . . .f � �kT �k � 1�T�,u� f � � K� f �y�kT�,

y � �q

��T, f �

K� f � T� � K� f �t � �0 T�K� f � � BT�T�T, f �W1L,

K� f �


where

(27)

Next, are numerically reconstructed by evaluating the power spectraldensity (PSD). This calculation is performed by transforming the signals into the frequency domain using a discrete Fourier transform, also known as a fastFourier transform (FFT)

(28)

The resulting vector is multiplied by its conjugate transpose to obtain the intensityas a function of frequency. Then, the intensity-dominant frequency of each disturb-ing signal is found, yielding three frequencies, for the three signals

respectively. We note that the nonlinear term constituting arenondimensional; therefore, the frequencies are nondimensional as well.

In order to model the nonlinear disturbances as an output of a second-order sys-tem, we feed the nonlinear term into a disturbance filter of the form

(29)

Denoting the disturbance filter dynamics are given by

(30)

where the system matrix of the disturbance filter is

(31)

the input matrix satisfies

(32)

and the control vector is to be designed so as to reject the periodic disturbances.The linearized ERTBP (15) is augmented with the disturbance filter states, result-ing in the augmented system

(33)

Thus, now the control term is set to track a reference trajectory while reject-ing periodic disturbances

(34)

Equation (33) constitutes a periodic LPV system, with Thecontroller can be determined by a standard finite-horizon LQR using the costfunctional

(35)J � �ff

0

x� f �TQwx� f � � unlT � f �Rwunl� f �df � xT� ff�Pfx� ff�

A� f � � A� f � T�.

unl � P� f � xr xxd

�xrunl

xxd� � A� f �

O6 6

O6 6

Ad� x

xd� � B

Bd�unl

unl

Bd � O3 3

I3�

�Ad�21 � diag��2, ��

2, ��2�Ad � O3 3

�Ad�21

I3

O3 3�,

xd � Adxd � Bdunl

xd � ��, �, �, �, �, ��T,

� � ��2� � u�

� � ��2� � u�

� � ��2� � u�

��,��,u�u�,u�,u�,u�,

u�,��,��,��,

Fn � Fk�� fkk�0N1� �n� � �N1

k�0fke2�ink/N

u�u�,u�,u�u�,u�,

r2 � ��r � 1�2 � �r2 � � r

2r1 � ��r � �2 � �r2 � � r

2,


where are constant weight matrices. The parameter is the final trueanomaly.

A feedback controller that minimizes the cost function (35) while tracking a ref-erence trajectory has the form [19]

(36)

The vector-valued function satisfies

(37)

where is the final value of the augmented reference state, designed to con-verge to the reference trajectory for the dynamical states and set to zero for the dis-turbance states

(38)

We assume that the reference signal, depends linearly on the true anomalythrough

(39)

The gain matrix satisfies the matrix Riccati differential equation

(40)

Equations (37) and (40) are integrated backwards for the initial values andis found. These systems are then augmented with equation (33) and integrated

forward in time, with the overall model

(41)

Illustrative Example

In his section, we shall illustrate the newly-developed formalism for the Earth-Moon system, for which and Our example is inspiredby future libration point missions aimed at space-borne radio astronomy. For efficientradio-telescope observations with no Earth-induced interference, a spacecraft has tobe in a region completely shaded by the Moon. Such a region exists a few hundredkilometers away from the libration point [7]. Due to geometric constraints, themaximum amplitude of the libration point orbit must be limited to a few hundreds ofkilometers. We subsequently demonstrate how to design a small-amplitude figure-eight controlled “halo” orbit that cannot be obtained using common ballistic meth-ods, and implement the developed control strategy for stationkeeping on this orbit.

Recall that in our coordinate system, is behind the Earth, is the interior li-bration point and lies behind the Moon. Thus, the collinear equilibria for thissystem are given by

The entries appearing in (16) depend upon the location of the three collinearpoints, as elaborated in Table 1.

aij

L3 � 1.155679913.L2 � 0.8569180073,L1 � 1.005062402,

L3

L2L1

L3

� 0.01215.e � 0.0549

x� fI� � �xr� fi� O1 6�T

x� f � � �A� f � BR1BTP� f ��x� f � � BR1BTg� f �

P� fi�g� fi�

P� ff� � CTPfC

P� f � � P� f �A� f � AT� f �P� f � � P� f �BR1BTP� f � CTQwC

P� f �

�� f � � �ff fI

ff fI

�� f �,

�f � �xr� ff� O1 6�T

�� f �,�f

g� ff� � CTPf�f,

g� f � � �A� f � BRw1BTP� f ��Tg� f � CTQw�� f �

g� f �

unl* � f � � min

uJ � R1BT�P� f �x� f � g� f ��

ffPtRw,Qw,


The characteristic exponents, — the eigenvalues of the matrix J—are given byTable 2. As expected, we find that all three collinear libration points are unstable,since due to the existence of eigenvalues with positive real parts (cf. equation (20)).

We require that the monodromy matrix eigenvalues (for the closed-loop system)be at the center manifold, to obtain a periodic trajectory. An example set is

(42)

where is a free parameter and is the characteristic exponent. The resulting ref-erence trajectory around for a one-year simulation, calculated with is de-picted by Figure 2. This figure shows the xy, yz and zx projections as well as thethree-dimensional orbit. The objective of the pole-placement control was to cancelthe diverging modes of the periodic dynamical system resulting from linearizationof ERTBP equations. In this example, the closed-loop characteristic exponentswere set to yield a “figure eight”-shaped orbit. The amplitude of this orbit is dic-tated by the initial conditions chosen to lie in close proximity to the libration point(200–600 m away).

Figure 3 depicts the state variables as a function of time for each of the librationpoints, and Fig. 4 depicts the required control acceleration. We notice that under theapplied control the states converge in a relatively short time. The required controleffort expressed as varies between 1.08 m�s per year for the orbit about and1.29 m�s per year for the orbit about and points.

To illustrate the stationkeeping algorithm including the nonlinear terms, we chosethe following values for the weight matrices:

Pf � O

Rw � I12

Qw � 107 � diag�1 1 1 0 0 0 1 1 1 0 0 0�

L3L2

L1�V

� � 2,L2

�i�

�i � 1 11

Tln��5

1

Tln��6 �5 �6�

�i


TABLE 1. Numerical Values for the Coefficients of the Linearized System

0 3.021383283 0.01069164180 1.010691642

0 11.29521865 4.147609411 5.147609411

0 7.380834356 2.190417208 3.190417208L3

L2

L1

a63a52a41

TABLE 2. Characteristic Exponents

0 �0.028407 0.1605i 0.16i

0 �0.46703 0.37184i 0.36136i

0 �0.3439 0.29661i 0.28444iL3

L2

L1

�5,6�3,4�1,2

The simulation was performed around each one of the collinear libration points fora period of one year. Under the nonlinear disturbances, the reference trajectoryshown previously results in a bounded trajectory as depicted by Fig. 5. The timehistory of the state around each of the collinear libration points is depicted byFig. 6. The required control accelerations are shown in Fig. 7. In this example, wesee that the closed-loop states track the given reference trajectory while accommo-dating the periodic disturbances originating from the nonlinearity of the equationsof motion. The weight matrices are set to penalize deviation from the reference tra-jectory while allowing the nonlinear terms to affect the states. Thus, the result ishigh-bandwidth tracking at the expense of reasonable control effort. The control ef-fort required to track the reference trajectory while accommodating the nonlineardisturbances varies between per year about about and per year about Similar to the reference trajec-tory tracking control components, the closed-loop states converge to yield abounded trajectory in relatively short time. The control term is a continuous sig-nal that is much more efficient, in the sense of control effort, compared to the ref-erence-trajectory-generating control u.

unl

L3.1.56 103 m�sL2,1.68 103 m�sL1,1.28 103 m�s

x�t�


FIG. 2. Reference Orbit Around L2.

The developed control law is designed for implementation using electric propul-sion. As other types of actuators, electric thrusters are prone to dispersions anduncertainties, meaning that the actual acceleration does not equal the commandedacceleration. For modeling propulsive dispersions, we adopt the linear model


FIG. 3. State Variables Comparison for Three Libration Points.

(43)

where SF is a scale factor, b is a bias vector for each direction and w is vector ofzero-mean Gaussian white noise with a 3-sigma value of 10% of the maximum

ua � SF � uc � b � w


FIG. 4. Control Acceleration Required for Stationkeeping About each of the Libration Points.

nominal thrust magnitude, (for and for ). The actual control accelerations, are fed into the

closed-loop dynamics instead of the commanded accelerations, A Monte-Carlosimulation of the closed-loop dynamics was then performed. In every run, theparameter values were randomly generated using a Gaussian distribution with meanand standard deviation that assume the values

In each Monte-Carlo run, 100 simulations are performed for a period of one year.The resulting performance envelopes are depicted by Figs. 8 and 9. Figure 8 shows

the mean, minimum, and maximum values of the position error components, denotedby (relative to the nominal case) for motion about Figure 9 shows thecorresponding errors in the velocity components, denoted by Similar quali-tative results are obtained for motion about and are hence not shown.

These figures show that following a short transient, the envelopes converge intodeviations of up to 1 meter relative to the nominal orbit amplitude (a few hundreds ofmeters). This observation implies that the stationkeeping control is robust to propul-sive dispersions.

L3,ez,ey,ex,

L2.ez,ey,ex,

w N�0, 0.033 � umax�b N�0, 0.033 � umax�SF � N�1, 0.1�3�

uc.ua,1.5 1010 m�s2umax �

L3,umax � 1.25 1010 m�s2,L2,umax


FIG. 5. Bounded Trajectory Resulting from Tracking a Reference Trajectory Under NonlinearDisturbances Around L2.

The effect of propulsive dispersions on the required for stationkeeping aboutand is depicted by Fig. 10 and in Fig. 11, respectively. It can be seen that

the control effort is normally distributed around the mean values of per year for motion about about and per year�V � 1.56 103 m�sL2103 m�s

�V � 1.68 L3L2

�V


FIG. 6. State Vector Comparison for Three Libration Points Under Nonlinear Disturbances.

for motion about (these values are those required by the stationkeepingcontroller only). The standard deviations are per year for motionabout and per year for motion about L3.106 m�sL2

1.2 106 m�s�VL3


FIG. 7. Control Acceleration Required for Stationkeeping About each of the Libration Points UnderNonlinear Disturbances.


FIG. 8. Envelope of Position Errors Under Propulsive Dispersions for Stationkeeping Control:Maximal, Minimal and Mean Values.

FIG. 9. Envelope of Velocity Errors Under Propulsive Dispersion for Stationkeeping Control:Maximal, Minimal and Mean Values.


FIG. 10. Distribution of the Total Control Effort per Year of the Stationkeeping Control Under PropulsiveDispersions: Motion About L2.

FIG. 11. Distribution of the Total Control Effort per Year of the Stationkeeping Control Under PropulsiveDispersions: Motion About L3.

Conclusions

In this work, we developed a closed-loop control scheme for derivation of peri-odic reference orbits based on the elliptic restricted three-body problem (ER3BP)model. The nonlinear dynamics of the ERTBP about the libration points was con-trolled by a method previously used for the circular restricted three-body problemonly. An LQR technique developed for tracking a reference trajectory while reject-ing nonlinear disturbances was used. This resulted in a bounded trajectory close tothe libration points for a reasonable control effort. The developed trajectory-track-ing control exhibits robustness to propulsive dispersions.

These results show that low-thrust continuous control schemes can be adoptedfor missions targeted to fly on small halo-like orbits that do not exist without activecontrol. Miniscule amounts of propellant are required to that end. Our results showthat while dynamical systems theory provides much benefit and insight into libra-tion point orbits, similar results may be obtained by re-casting the astrodynamicalproblem as a control problem. Recent advances in electric propulsion may there-fore facilitate future exploitation of libration points without the need to employhigh-end dynamical systems analyses.

Appendix A: Semi-Analytical Calculation of the Monodromy Matrix

The state transition and monodromy matrices may be accurately computed usingChebyshev polynomials. This step is important for verifying the numerical evalua-tion of these matrices using direct integration [17].

Chebyshev polynomials of the first kind, and the second kind, are de-fined respectively by the identities

(44)

where To approximate the entries of the monodromy ma-trix, we must perform the dilation-translation transformation

(45)

so that in the modified coordinates, The shifted Chebyshev polynomi-als of the first kind, written in recursive form, are given by

(46)

whereas the shifted Chebyshev polynomials of the second kind are written as

(47)

Chebyshev polynomials of both kinds are orthogonal; the shifted polynomials ofthe fist kind are orthogonal with respect to the weight function

(48)wV��*� � ��* �*2�1/2

�* � �0, 1�U *r�1 � 2�2�* 1�U*

r U *r1,

U 1* � 2�2�* 1�

U 0* � 1

�* � �0, 1�V *r�1 � 2�2�* 1�V *

r V *r1,

V 1* � 2�* 1

V0* � 1

�* � �0, 1�.

�* � �� 1��2

� � cos� f � � �1, 1�.

r � 1, 2, 3, . . .0 � f � �,Ur�� sin��r � 1�f ��sin f,

Vr�� cos�rf �

U��V��,


so that

(49)

whereas the polynomials of the second kind are orthogonal with respect to theweight function

(50)satisfying

(51)

To derive semi-analytical expressions for the monodromy matrix, we write (cf. (45)) and expand each true anomaly-dependent entry,

of the submatrix (cf. (16)) into Chebyshev polynomials. We repeat the same pro-cedure for each component of the state of system (15), and then usethe special properties of Chebyshev polynomials to obtain a semi-analytical approxi-mation to the transition and monodromy matrices.

The m-1-order expansion of any function into Chebyshev poly-nomials can be written as

(52)

where is determined by the quadrature

(53)

In equations (52), (53), and for polynomials of the first kind, andand for polynomials of the second kind. The constant satisfies

for (54)

for

By applying equations (52), (53) on each component of the state vector, we get

(55)

where is a yet unknown Chebyshev polynomial coefficient vector. In a similarmanner, we can calculate a Chebyshev polynomial coefficient vector for each (true-anomaly dependent) entry, of wherefrom

(56)

dij � �d0ij, d1

ij, � � � , dm1ij �T

i, j � 1, . . . , 6aij � �m1

r�0dr

ijS*r � �S*Tdij,

A21,aij,

bi

�S* � �S*0i, S*

1i, � � � , S*

m1i T

bi � �b0i , b1

i , � � � , bm1i �T

i � 1, . . . , 6xi � �m1

r�0br

iS*r � �S*Tbi,

xi,

S*r � U*

rr � 0, 1, 2, . . .� � ��8,

S*r � V *

r� � ��2,

�,

r � 0

r � 0

�S*r � U*

rw � wU

S*r � V *

rw � wV

r � 0, 1, 2, . . .�r �1

��1

0w��g��S*

r��d�,

�r

g�� m1

r�0�r

i S*r��

� � �0, 1�g��,

i � 1, . . . , 6xi,A21

aij��*�,�1 � cos f ��2�* �

�1

0U *

r��*�Uk��*�wU��*�d�* � � 0,

��8,

r � k

r � k

wU��*� � ��* �*2�1/2

�1

0V *

r��*�V *k��*�wV��*�d�* � � 0,

��2,

�,

r � k

r � k � 0

r � k � 0


where are polynomial coefficients calculated according to the quadrature (53).We can now utilize the unique properties of Chebyshev polynomials, that is, ex-

press the product of any two polynomials using the product operational matrix, andthe quadrature of shifted Chebyshev polynomials using the integration operationalmatrix. This formalism transforms the problem of solving the vector differentialequation (15) into the system of algebraic equations

(57)

In (57), is a 6m-dimensional column vector of unknown poly-nomial coefficients (cf. (55)), and

(58)

where denotes the Kronecker product. The matrix P is given by

(59)

where

(60)

and is the integration operational matrix. The matrix R assumes theform

(61)

with the submatrices

(62)

where and are the product operational matrices, whose entriesdepend upon (cf. equation (56)). Finally, we choose

(63)

and calculate the monodromy matrix through

(64)

where

Appendix B: Generating a Smooth Controller

In order to avoid discontinuities we require that

(65)

This requirement leads to a slightly modified control law

(66)

where is a generalized hold function of the form

(67)

and are constants, to be determined shortly. are given byK02� f �K01� f �,�2�1,

m qK� f � � K� f � � �1K01� f � � �2K02� f �,

K

k � 0, 1, . . .f � �kT �k � 1�T�,u� f � � K� f �y�kT�

! t � kTu�t� � u�t��,

B � �B1, � � � , B6�.

� � �S�T��TB

XI � I6 � �1, O1 m1�T

dij3m 3mQ22Q21

C* � GQ22R* � GQ21,

R � O3m 3m

R*

O3m 3m

C* �3m 3mGT

C � A22 � GTG � I3 � GT,

P � O3m 3m

O3m 3m

G

C��

�S�t��T � I6 � �S*�t�T

B � ��b1�T, � � � , �b6�T�

�S�t��TB �S�t��TXI � �S�t��TPB � �S�t��TRB

dij


(68)

where are arbitrary continuous matrix functions. We chose the fol-lowing functions, which are continuous in

I(69)

The matrices of (68) are controllability maps of respectively:

(70)

The continuity requirement (65) gives rise to the constraint

(71)

which, in turn, determines the constants of equation (67) through the systemof linear equations

(72)

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��1 �2� K01�0��K02�0��

K01�T�K02�T�� K�0��

K�T��2�1,

! k � 0, 1, . . .K�kT� � 0,

L2 � �T

0��T, ��B��K2��d�

L1 � �T

0��T, ��B��K1��d�

K2� f �,K1� f �,L2L1,

�IK2� f � � �1 �f

K1� f � � �f

�0, T�:K2� f �K1� f �,

t � �0, T�K02� f � � K2� f � BT� f ��T�T, f �W1L2,

t � �0, T�K01� f � � K1� f � BT� f ��T�T, f �W1L1,


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