1

Stationkeeping on Libration Point Orbits in the Elliptic

Restricted Three-Body Problem

Pini Gurfil* and Dani Meltzer†

Faculty of Aerospace Engineering, Technion – Israel Institute of Technology,

Haifa 32000, Israel

Abstract

We develop new methods for generating periodic orbits about the collinear

libration points and for stabilizing motion on libration orbits using the general

formalism of the elliptic restricted three-body problem (ER3BP). Calculation

of periodic orbits was accomplished by formulating the ER3BP as a control

problem. Linearization about the libration points in pulsating coordinates

yields an unstable linear parameter-varying (LPV) system with periodic

coefficients. We introduce a continuous acceleration control term into the

state-space dynamics and use an LPV-generalized version of the pole-

assignment technique to find linear periodic reference trajectories. The

nonlinear terms of the equations of motion are then treated as periodic

disturbances. A disturbance accommodating control is used to track the

libration-point reference trajectory under the nonlinear periodic disturbances.

Simulation experiments show that small amounts of propellant are required.

* Senior Lecturer. email: pgurfil@technion.ac.il

† Graduate Student. email: meltzer@aerodyne.technion.ac.il

AIAA/AAS Astrodynamics Specialist Conference and Exhibit21 - 24 August 2006, Keystone, Colorado

AIAA 2006-6035

Copyright © 2006 by The Authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2

1. Introduction

Recent years saw a rising interest in launching spacecraft into libration-point

orbits for scientific missions. Past missions such as ISEE-3, SOHO and

Genesis successfully utilized quasi-periodic orbits about the 1L and 2L

collinear libration points of the Sun-Earth system. With the embarkation of

future NASA and ESA missions such as Darwin, TPF, and SAFIR [1], [2], to

be launched into libration point orbits, there is an opportunity to design and

simulate novel trajectory planning and control schemes.

In practical analysis and design of missions around libration points, the

circular restricted three-body problem (CR3BP) model is usually adopted [3]-

[6]. Although this model has proven fruitful, it possesses an inherent

approximation, assuming that the orbit of the secondary is circular. However,

both the motion of the Earth around the Sun and the motion of Moon around

the Earth are eccentric. Incorporating 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 significant topological differences

compared to the CR3BP: The position of the libration points in the Earth-

Moon system is not constant, but rather pulsating with respect to Earth, and

moreover, the Jacobi integral is time- (true anomaly-) dependent.

Several works have addressed the problem of finding natural periodic orbits in

the planar ER3BP based on specialized regularizations [7], [8]. Derivation of

such orbits through numerical searches was accomplished in Refs. [9] and [10].

These methods provided initial conditions resulting in periodic or quasi-

periodic ballistic trajectories. However, with the persistent improvement in

electric propulsion, it is now possible to design libration point trajectories,

3

which are not necessarily center-manifold solutions of the unperturbed

dynamics, but are rather approximate trajectories, the stationkeeping on

which is performed by continuous low-thrust propulsion. These orbits are

more flexible and adaptable than ballistic trajectories.

In this work, we use the ER3BP model to calculate periodic reference

trajectories around collinear libration points by applying the following steps.

First, we derive a simple form of the equations of motions by normalizing into

dimensionless coordinates and using the true anomaly as the independent

variable [7], [11]. Second, we calculate the position of the libration points in

these coordinates and derive a first-order approximation to the equations of

motion around the libration points, resulting in a periodic linear parameter-

varying (LPV) system. Finally, we introduce a 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 by

Scheeres et al. [20] for the CR3BP. However, we use here a generalized pole-

placement technique based on monodromy matrix calculations for the general

setup of the ER3BP without resorting to the CR3BP or Hill’s CR3BP

approximations. Application of such a generalized pole-placement technique

for periodic systems constitutes a novel approach to the problem of stabilizing

motion on unstable orbits.

To find our reference trajectories, we temporarily neglect the nonlinear

dynamical constituent. Nevertheless, the nonlinearity is later introduced back

into our model as a persistent disturbance [18]. This disturbance is modeled as

a second-order dynamical system driven by a nonlinear dynamical term.

Thus, while Howell and Pernicka [14] utilized an impulsive linear quadratic

regulator (LQR) and Cielaszyk and Wei [18] adopted an infinite-horizon

4

continuous LQR, in this work we utilize a finite-horizon LQR scheme to track

reference trajectories while rejecting persistent disturbances using the realistic

ER3BP model.

2. 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 massive gravitational bodies – the primaries. The most popular

coordinate system used to model the ER3BP has its origin set at the

barycenter of the large primary, M1 and the small primary, M2. The x-axis is

positive in the direction of M2, the z-axis is perpendicular to the plane of

rotation and is positive upwards, and the y-axis completes the setup to yield a

Cartesian, rectangular, dextral reference frame, as shown in Figure 1. The

small primary is orbiting the large primary on an elliptic orbit with

eccentricity e . This orbit complies with the two-body Keplerian motion; the

distance between the primaries, ρ , depends upon the true anomaly, f, through

the conic equation

1 cos

pe f

ρ =+

, (1)

where p is the semi-latus rectum. The rate of change of the true anomaly, f ,

satisfies 2/f h ρ= , where h is the magnitude of the angular momentum, given

by ( )21 2h G M M p= + . Here G is the universal gravitational constant, M1

denotes the mass of the first primary, and M2 is the mass of second primary.

The position vector, R , of the spacecraft in the rotating barycentric frame

shown in Fig. 1 is

ˆˆ ˆXi Yj Zk= + +R . (2)

5

The coordinates system rotates at the rate f about k , so the angular velocity vector satisfies ω ˆfk= and the velocity vector,V , can be written as ( ) ( ) ( )ω ˆˆ ˆX fY i Y fX j Z k= + × = + + + +V R R . (3)

By defining 2 1 2( )M M Mµ + , M1 is located at (-µ,0,0) and M2 is located at

(1-µ,0,0). The position of the spacecraft with respect to the primaries (cf.

Figure 1) can be expressed by

( )1 , 1T T

X Y Z X Y Zµρ µ ρ⎡ ⎤ ⎡ ⎤= + = + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦2r r (4)

Figure 1 - Definition of the coordinate system

Denoting the kinetic energy by K , the potential energy by U , and the

Lagrangian byL , we have

( )

1 2

11, ,

2

µ µ−= ⋅ = − − = −V V

r rK U L K U . (5)

(X, Y, Z )

X

(X2, 0, 0)

S/C

C

ω

Z

r2

r1

R

Y

( X 1 , 0 ,0 )

M2

M 1

6

Writing the Euler-Lagrange equations with the components of the position

vector as the generalized coordinates,

0ddt

⎛ ⎞∂ ∂⎟⎜ − =⎟⎜ ⎟⎜⎝ ⎠ ∂∂ RRL L , (6)

yields the equations of motion

( )( )

( )

( )( )

( ) ( )

( ) ( ) ( )

( ) ( )

23 3

2 22 22 2 2 2

23 3

2 22 22 2 2 2

3 32 22 22 2 2 2

112

1

12

1

1

1

XXX f X fY fY

X Y Z X Y Z

Y YY f Y fX fX

X Y Z X Y Z

Z ZZ

X Y Z X Y Z

µ µ ρµ ρ

µρ µ ρ

µ µ

µρ µ ρ

µ µ

µρ µ ρ

+ −− +− − − = − −

⎡ ⎤ ⎡ ⎤+ + + + − + +⎢ ⎥ ⎣ ⎦⎣ ⎦−

− − − = − −⎡ ⎤ ⎡ ⎤+ + + + − + +⎢ ⎥ ⎣ ⎦⎣ ⎦

−= − −

⎡ ⎤ ⎡ ⎤+ + + + − + +⎢ ⎥ ⎣ ⎦⎣ ⎦ (7)

In order to simplify Eqs. (7), a transformation to rotating-pulsating

coordinates is required [11]. This transformation consists of normalizing time

by 31 1( )/G M M ρ+ , normalizing position by the instantaneous distance,

, ,X Y Zρξ ρη ρζ= = = , (8)

and then transforming time derivatives into derivatives with respect to true

anomaly:

( ) ( )

( )* * 2

cos' , '

(1 cos )d d df pe f

fdt df dt e f

ρ= = =+

, (9)

7

where *t is the dimensional time. The relationships between the dimensional,

time-dependent and the dimensionless, true anomaly-dependent velocities and

accelerations is as follows:

( )

( ) ( )[ ]* ' ' sin 1 cos 'd h

X f e f e fdt p

ρξρ ξ ρξ ξ ξ= = + = + + (10)

( ) ( )[ ]( )2

322

2 ' 1 cos " cos 1 cosh

X f f e f e f e fp

ρξ ρ ρ ξ ξ ξ ξ= + + + = + + + (11)

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

( )2 2 21 1cos

1 cos 2e f

e fξ η ζ

⎡ ⎤Ω = + − −⎢ ⎥

⎢ ⎥+ ⎣ ⎦U . (12)

This yields the compact equations

" 2 '

" 2 '

"

ξ ηξ

η ξη

ζζ

∂Ω− =

∂∂Ω

+ =∂∂Ω

=∂

(13)

The location of the libration points in rotating-pulsating coordinates is

constant and identical to their location in the CR3BP setup. It is therefore

straightforward to perform linearization of the equations of motion about the

libration points. To that end, we define a state vector as

1 2 3 4 5 6' ' 'T T

x x x x x xδξ δη δζ δξ δη δζ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦x (14)

8

For an initial true anomaly If , the linearized equations of motion assume a

periodic linear parameter-varying (LPV) state-space representation of the

form

( ) ( ) ( )

( )

( ) ( )

'

I I

f f f

f

f f T

=

=

= +

r r

r r

x A x

x x

A A

(15)

with a period 2T π= .

The matrix A satisfies

( )

( )

41

21 52

63

0 2 00 01

, 0 0 , 2 0 01 cos

0 0 00 0 cos

a

ae f

a e f

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= = = −⎢ ⎥⎢ ⎥ ⎢ ⎥+ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

3X3 3X3

2221 22

0 IA A AA A

(16)

It is well-known [11] that the linear approximation about the collinear points

in the CR3BP is unstable. However, the linear CR3BP is autonomous,

whereas the linear ER3BP is parameter (time) varying. Thus, in order to

study the stability of the linearized 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

systems can be deduced from analyzing the stability of an equivalent, time-

invariant, system. Letting ( )Φ , If f denote the state transition matrix,

Floquet’s equivalent system is obtained from the transformation

( ) ( ) ( ) ( ) ( )Φ 1, ,fIf f f f e f f−= = Jz P x P . (17)

9

The equivalent time-invariant system is then simply

( ) ( )' f f=z Jz . (18)

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

( )Ψ Φ ,I If f T f= = + as follows:

Ψ1ln

T=J . (19)

A well-known theorem [16] states that a periodic LPV system is exponentially

stable if and only if all the characteristic exponents (eigenvalues of the

Floquet matrix) have negative real parts:

( )[ ]Re 0iλ <J . (20)

The monodromy matrix can be calculated in several ways, the most common

being a direct numerical integration of the state transition matrix through the

matrix differential equations

( ) ( ) ( )

( )

Φ Φ

Φ

, ,

, .

I I

I I

df f f f f

df

f f

=

=

A

I (21)

In this paper, we calculate the state transition and monodromy matrices

numerically, and verify the computations using a recently developed semi-

analytical method based on orthogonal Chebyshev polynomials [17]. A brief

description of this approach is given in Appendix A.

10

We shall subsequently utilize the ER3BP dynamical model presented in this

section for designing a feedback stabilizer for the motion on libration point

orbits. We shall accomplish this goal in two steps: First, we will design

controlled periodic reference trajectories about the collinear points using a

pole-placement technique generalized to periodic LPV systems; second, we

shall introduce the inherent nonlinearity of the ER3BP back into the

equations of motion using an LPV-generalized version of disturbance-

accommodating control.

3. Controlled Periodic Libration Point Orbits

We start by generating reference trajectories about the libration points. In

order to stabilize the unstable modes of the LPV system (15), we introduce a

control vector m∈u :

( ) ( ) ( ) ( )

( )

( ) ( )

'

I I

f f f f

f

f T f

= +

=

+ =

r r

r r

x A x Bu

x x

A A

(22)

We assume that (22) is controllable and observable, and use pole-placement

to render a periodic reference trajectory. Since we deal with a periodic LPV

system, a generalized version of the pole-placement technique must be

adopted [12], [13]. In particular, we shall adapt the static output feedback

method proposed in Ref. [12], which relies on using a sampled output at the

beginning of each period and calculating a generalized hold function ( )fK as

follows:

( ) ( )

( ) ( )

Φ , , 0f T f t T

f T f

− ⎡ ⎤= ∈ ⎢ ⎥⎣ ⎦+ =

T T 1K B W L

K K (23)

11

In Eq. (23), ( )Φ ,T f is the state transition matrix (calculated either

numerically or semi-analytically, as explained in Appendix A),W is the

controllability Gramian and L is a pole-placement gain matrix. The static

output feedback controller for some output q∈y is given by

( ) ( ) ( ) ( ), [ 1 ), 0,1,f f kT f kT k T k= ∈ + =u K y … (24)

Controller (24) does not necessary result in a continuous signal. Appendix B

develops a methodology for deriving a smooth controller; however, smoothness

is not required to yield a smooth reference orbit.

This completes the formalism required for designing libration-point reference

orbits using the linearized LPV model. In the next section, we shall introduce

the nonlinearities back into the model through a disturbance-accommodating

control law.

4. Accommodating the Nonlinearities

To render the treatment general, the nonlinear dynamical effects must be

incorporated into the model. In Ref. [18], Cielaszyk and Wie modeled the

nonlinear gravitational terms in the CR3BP as periodic disturbances. The

main idea was to model the nonlinear dynamical effects as an output of a

linear second-order system driven by a nonlinear input. An LQR scheme was

subsequently used to stationkeep on the reference trajectory while rejecting

the disturbance. In this section, we extend the Cielaszyk-Wie formalism to the

non-autonomous ER3BP model.

Recall the ERTBP equations of motion in pulsating coordinates (13). Let a

reference trajectory of a spacecraft flying near a collinear libration point be

given by ( ), ,r r rξ η ζ , and let

12

2

2

.

r rr

r rr

rr

u

u

u

ξ

η

ζ

ξ ηξ

η ξη

ζζ

∂Ω′′ ′= − −∂∂Ω′′ ′= + −∂

∂Ω′′= −∂

(25)

Explicitly, (25) can be written as

( )( )( ) ( )

( )( )

( )( ) ( )

3 31 2

3 31 2

3 31 2

12 1 1

1 cos

12 1

1 cos

1cos 1

1 cos

r r r r r

r r r r r

r r r r

u r re f

u r re f

u e f r re f

ξ

η

ζ

ξ η ξ µ ξ µ ξ µ

η ξ η µ η µη

ζ ζ µ ζ µζ

− −

− −

− −

⎡ ⎤′′ ′= − − − − + − + −⎣ ⎦+

⎡ ⎤′′ ′= + − − − −⎣ ⎦+

⎡ ⎤′′= − − − − −⎣ ⎦+

(26)

where

( ) ( )2 22 2 2 21 2, 1r r r r r rr rξ µ η ζ ξ µ η ζ= + + + = + − + + . (27)

Next, , ,u u uξ η ζ are numerically reconstructed by evaluating the power

spectral density (PSD). This calculation is performed by transforming the

signals , ,u u uξ η ζ into the frequency domain using a discrete Fourier

transform, also known as a fast Fourier transform (FFT):

( )1

1 2 /0

0

NN ink N

n k k kkk

F f n f e π−

− −=

=

⎡ ⎤= =⎢ ⎥⎣ ⎦ ∑F (28)

The resulting vector is multiplied by its conjugate transpose to obtain the

intensity as a function of frequency. Then, the intensity-dominant frequency

of each disturbing signal is found, yielding three frequencies, , ,ξ η ζω ω ω , for

the three signals , ,u u uξ η ζ , respectively. We note that the nonlinear term

13

constituting , ,u u uξ η ζ are non-dimensional; therefore, the frequencies

, ,ξ η ζω ω ω are non-dimensional as well.

In order to model the nonlinear disturbances as an output of a second-order

system, we feed the nonlinear term into a disturbance filter of the following

form:

2

2

2 .

u

u

u

ξ ξ

η η

ζ ξ

α ω α

β ω β

γ ω γ

+ =

+ =

+ =

(29)

Denoting [ ]Td α β γ α β γ= , , , , ,x , the disturbance filter dynamics are given by

d d d d nl= +x A x B u (30)

where the system matrix of the disturbance filter is

( )

( ) ( )3 3 32 2 2

3 3

, diag , ,d dd

ξ η ζω ω ω×

×

⎡ ⎤⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

2121

0 IA A

A 0, (31)

the input matrix satisfies

3 3

3d

×⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

0B

I, (32)

and the control vector nlu is to be designed so as to reject the periodic

disturbances. The linearized ERTBP (15) is augmented with the disturbance

filter states, resulting in the augmented system

14

( ) 6 6

6 6nl

f ×

×

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ dd dd

x x BA 0uxx B0 A

(33)

Thus, now the control term nlu is set to track a reference trajectory rx while rejecting

periodic disturbances:

( )nl f−⎡ ⎤

⎢ ⎥= −⎢ ⎥⎣ ⎦

r

d

x xu P x (34)

Eq. (33) constitutes a periodic LPV system, with ( ) ( )f f T= +A A . The

controller can be determined by a standard finite-horizon LQR using the cost

functional

( ) ( ) ( ) ( ) ( ) ( )0

ff

w w f fJ f f f f df f f= + +∫ T T Tnl nl fx Q x u R u x P x , (35)

where , ,w w fQ R P are constant weight matrices. The parameter ff is the final true

anomaly.

A feedback controller that minimizes the cost function (35) while tracking a

reference trajectory has the following form [19]:

[ ]*( ) min ( ) ( ) ( )T

uf J f f f−= = − −1

nlu R B P x g (36)

The vector-valued function ( )fg satisfies

( ) ( )σ

σ

( ) ( ) ( )

( ) ,

w w

f f f

f f f f f

f

−⎡ ⎤= − − −⎣ ⎦=

T1 T T

T

g A BR B P g C Q

g C P (37)

15

where σ f is the final value of the augmented reference state, ( )σ f , designed

to converge to the reference trajectory for the dynamical states and set to

zero for the disturbance states:

( )σ 1 6

T

f ff ×⎡ ⎤= ⎢ ⎥⎣ ⎦rx 0 (38)

We assume that the reference signal, ( )σ f , depends linearly on the true

anomaly:

( )σ σ If

f I

f ff

f f−

=−

. (39)

The gain matrix ( )fP satisfies the matrix Riccati differential equation

( ) ( )( ) ( ) ( ) ( ) ( )

( ) .

w

f f

f f f f f f f

f

−= − − + −

=

T 1 T T

T

P P A A P P BR B P C Q C

P C P C (40)

Eqs. (37) and (40) are integrated backwards for the initial values ( )ifg and

( )ifP is found. These systems are then augmented with Eqs. (33) and

integrated forward in time, with the overall model

( ) ( )

( ) 1 6

( ) ( ) ( )

( )T

I i

f f f f f

f f

− −

×

⎡ ⎤= − +⎣ ⎦⎡ ⎤= ⎢ ⎥⎣ ⎦

1 T 1 T

r

x A BR B P x BR B g

x x 0 (41)

16

Illustrative Example

In his section, we shall illustrate the newly-developed formalism for the Earth-

Moon system, for which e=0.0549 and µ 0.01215= . The collinear equilibria

for this system are given by

1 2 3=-1.005062402, =0.8569180073, =1.155679913L L L .

The entries ija appearing in (16) depend upon the location of the three

collinear points, as elaborated by Table 1:

63a 52a 41a

-1.010691642 -0.01069164180 3.021383283 1L

-5.147609411 -4.147609411 11.29521865 2L

-3.190417208 -2.190417208 7.380834356 3L

Table 1: Numerical values for the coefficients of the linearized system

The characteristic exponents, iλ - the eigenvalues of the matrix J - are given

by Table 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.

Eq. (20)).

5,6λ 3,4λ 1,2λ

0.16i 0.1605i 0.028407± 1L

0.36136i 0.37184i 0.46703± 2L

0.28444i 0.29661i 0.3439± 3L

Table 2: Characteristic exponents

17

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

( ) ( )5 6 5 6

1 11 1 ln lni T T

ω ν λ ν λ λ λ⎡ ⎤

= −⎢ ⎥⎢ ⎥⎣ ⎦

(42)

where ν is a free parameter and iλ is the characteristic exponent. The

resulting reference trajectory around 2L for a 1-year simulation, calculated

with 2ν = , is depicted by Figure 2. This figure shows the xy, yz and zx

projections as well as the three-dimensional orbit. The objective of the pole-

placement control was to cancel out the diverging modes of the periodical

dynamical system resulting from linearization of ERTBP equations. In this

example, the closed-loop characteristic exponents were set to yield a "figure

eight"-shaped orbit. The amplitude of this orbit is dictated by the initial

conditions chosen to lie in close proximity to the libration point (200 - 600 m

away).

−0.6 −0.4 −0.2 0 0.2 0.4−0.5

0

0.5

X [km]

Y [k

m]

−0.5 0 0.5−0.4

−0.2

0

0.2

0.4

Y [km]

Z [k

m]

−0.3 −0.2 −0.1 0 0.1−0.2

−0.1

0

0.1

X [km]

Z [k

m]

−0.5

0

0.5

−1

−0.5

0

0.5

1−0.3

−0.2

−0.1

0

0.1

0.2

0.3

X [km]Y [km]

Z [k

m]

Figure 2: Reference orbit around 2L

18

Figure 3 depicts the state variables as a function of time for each of the

libration points, and Figure 4 depicts the required control acceleration. We

notice that under the applied control the states converge in a relatively short

time. The required control effort expressed as V∆ varies between 1.08 m/s

per year for the orbit about L1 and 1.29m/s per year for the orbit about L2

and L3 points.

0 100 200 300 400−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

time [days]

x [k

m]

0 100 200 300 400−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time [days]

y [k

m]

100 200 300−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

time [days]

z [k

m]

0 100 200 300 400−0.5

0

0.5

1

1.5

2

2.5x 10

−6

time [days]

xd [k

m/s

]

0 100 200 300 400−5

0

5x 10

−6

time [days]

yd [k

m/s

0 100 200 300 400−6

−4

−2

0

2

4

6x 10

−6

time [days]

zd [k

m/s

]

L1

L2

L3

L1

L2

L3

L1

L2

L3

Run #1 Run #2 Run #3

L1

L2

L3

L1

L2

L3

Figure 3: State variables comparison for three libration points

19

0 50 100 150 200 250 300 350 400−5

0

5x 10

−8

time [days]

ux [m

/s2 ]

0 50 100 150 200 250 300 350 400−4

−3

−2

−1

0

1

2

3

4x 10

−8

time [days]

uy [m

/s2 ]

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1x 10

−7

time [days]

uz [m

/s2 ]

L1

L2

L3

L1

L2

L3

L1

L2

L3

Figure 4: Control acceleration required for stationkeeping about each of the

libration points

20

To illustrate the stationkeeping algorithm including the nonlinear terms, we

chose the following values for the weight matrices:

( )7

12

10 diag 1 1 1 0 0 0 1 1 1 0 0 0w

w

f

= ⋅

=

=

Q

R I

P 0

The simulation was performed around each one of the collinear libration

points for a period of 1 year. Under the nonlinear disturbances, the reference

trajectory shown previously results in a bounded trajectory as depicted by

Figure 5. The time history of the state ( )tx around each of the collinear

libration points is depicted by Figure 6. The required control accelerations are

shown in Figure 7. In this example, we see that the closed-loop states track

the given reference trajectory while accommodating the periodic disturbances

originating from the nonlinearity of the equations of motion. The weight

matrices are set to penalize deviation from the reference trajectory while

allowing the nonlinear terms to affect the states. Thus, we result in high-

bandwidth tracking in expense of reasonable control effort. The control effort

required to track the reference trajectory while accommodating the nonlinear

disturbances varies between -31.28 10 ⋅ m/s per year about L1, -31.68 10⋅ m/s

about L2 and -31.56 10⋅ m/s per year about L3. Similar to the reference

trajectory tracking control components, the closed-loop states converge to

yield a bounded trajectory in relatively short time. The control term nlu is a

continuous signal that is much more efficient, in the sense of control effort,

compared to the reference-trajectory-generating control u .

21

−0.4 −0.2 0 0.2 0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

X [km]

Y [k

m]

−0.2 0 0.2−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Y [km]

Z [k

m]

−0.2 −0.1 0−0.15

−0.1

−0.05

0

0.05

0.1

X [km]

Z [k

m]

−0.2−0.1

0−0.2

0

0.2

−0.1

0

0.1

X [km]

Y [km]

Z [k

m]

Figure 5: Bounded trajectory resulting from tracking a reference trajectory

under nonlinear disturbances around L2

22

0 100 200 300 400−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

time [days]

x [k

m]

0 100 200 300 400−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [days]

y [k

m]

0 100 200 300 400−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

time [days]

z [k

m]

0 100 200 300 400−6

−4

−2

0

2

4

6

8

10x 10

−6

time [days]

xd [k

m/s

]

0 100 200 300 400−3

−2

−1

0

1

2

3

4

5x 10

−6

time [days]

yd [k

m/s

100 200 300

−4

−2

0

2

4

x 10−6

time [days]

zd [k

m/s

]

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

Figure 6: State vector comparison for three libration points under nonlinear

disturbances

23

50 100 150 200 250 300 350 400

−2.5

−2

−1.5

−1

−0.5

0

0.5

x 10−10

time [days]

ux [m

/s2 ]

0 50 100 150 200 250 300 350 400−2

−1

0

1

2x 10

−10

time [days]

uy [m

/s2 ]

0 50 100 150 200 250 300 350 400−6

−4

−2

0

2

4x 10

−11

time [days]

uz [m

/s2 ]

L1

L2

L3

L1

L2

L3

L1

L2

L3

Figure 7: Control acceleration required for stationkeeping about each of the

libration points under nonlinear disturbances

24

6. Conclusions

In this work, we developed a closed-loop control scheme for derivation of

periodic reference trajectories based on the elliptic restricted three-body

problem (ER3BP) model. The nonlinear dynamics of the ERTBP about the

libration points was controlled by a method previously used for the circular

restricted three-body problem only. An LQR technique developed for tracking

a reference trajectory while rejecting nonlinear disturbances was used. This

resulted in a bounded trajectory close to the libration points for a reasonable

control effort.

These results show that low-thrust continuous control schemes can be

adopted for missions targeted to stay at the vicinity of the libration points.

Miniscule amounts of propellant are required to that end. Our results show

that while dynamical systems theory provides much benefit and insight into

libration point orbits, similar results may be obtained by re-casting the

astrodynamical problem as a control problem. Recent advances in electric

propulsion may therefore facilitate future exploitation of libration points

without the need to employ high-end dynamical systems analyses.

Appendix A: Semi-Analytical Calculation of the Monodromy Matrix

The state transition and monodromy matrices may be accurately computed

using Chebyshev polynomials. This step is important for verifying the

numerical evaluation of these matrices using direct integration [17].

Chebyshev polynomials of the first kind, ( )V φ , and the second kind, ( )U φ

are defined respectively by the identities

25

( ) ( )

( ) ( )

cos ,

sin[ 1 ]/ sin , 0 , 0,1,2,

r

r

V rf

U r f f f r

φ

φ π

=

= + ≤ ≤ = … (43)

where ( )cos 1, 1fφ ⎡ ⎤= ∈ −⎢ ⎥⎣ ⎦ . To approximate the entries of the monodromy

matrix, we must perform the dilation-translation transformation

( )* 1 /2φ φ= + , (44)

so that in the modified coordinates, * 0, 1φ ⎡ ⎤∈ ⎢ ⎥⎣ ⎦ . The shifted Chebyshev polynomials

of the first kind, written in recursive form, are given by

( )

*0

* *1

* * * * *1 1

1

2 1

2 2 1 , 0, 1 ,r r r

V

V

V V V

φ

φ φ+ −

=

= −

⎡ ⎤= − − ∈ ⎢ ⎥⎣ ⎦

(45)

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

( )( )

*0

* *1

* * * * *1 1

1

2 2 1

2 2 1 , 0, 1 .r r r

U

U

U U U

φ

φ φ+ −

=

= −

⎡ ⎤= − − ∈ ⎢ ⎥⎣ ⎦

(46)

Chebyshev polynomials of both kinds are orthogonal; the shifted polynomials

of the fist kind are orthogonal with respect to the weight function

( ) ( ) 1/2* * *2Vw φ φ φ

−= − , (47)

so that

26

( ) ( ) ( )1

* * * * * *

0

0,

/2, 0

, 0,

r k V

r k

V V w d r k

r k

φ φ φ φ π

π

⎧⎪ ≠⎪⎪⎪⎪= = ≠⎨⎪⎪⎪ = =⎪⎪⎩

∫ (48)

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

weight function

( ) ( )1/2* * *2Uw φ φ φ= − , (49)

satisfying

( ) ( ) ( )1

* * * * * *

0

0,

/ 8, .r k U

r kU U w d

r kφ φ φ φ

π

⎧ ≠⎪⎪⎪= ⎨⎪ =⎪⎪⎩∫ (50)

To derive semi-analytical expressions for the monodromy matrix, we write

( )* 1 cos /2fφ = + (cf. (44)) and expand each true anomaly-dependent entry, *( )ija φ , of the submatrix 21A (cf. (16)) into Chebyshev polynomials. We

repeat the same procedure for each component , 1, ,6ix i = … of the state of

system (15), and then use the special properties of Chebyshev polynomials to

obtain a semi-analytical approximation to the transition and monodromy

matrices.

The m-1-order expansion of any function ( ), [0,1]g τ τ ∈ into Chebyshev

polynomials can be written as

( ) ( )1

*

0

mi

r rr

g Sτ κ τ−

=

= ∑ (51)

where rκ is determined by the quadrature

27

( )

1*

0

1( ) ( ) , 0,1,2,...r rw g S d rκ τ τ τ τ

δ= =∫ (52)

In Eqs. (51), (52), Vw w= and * *r rS V= for polynomials of the first kind,

and Uw w= and * *r rS U= for polynomials of the second kind. The constant

δ satisfies

* *

* *

/2, 0for

, 0

/8, 0,1,2,... for .

r r

r r

rS V

r

r S U

πδ

π

δ π

⎧ ≠⎪⎪= =⎨⎪ =⎪⎩= = =

(53)

By applying Eqs. (51), (52) on each component of the state vector, ix , we get

1* *

0

0 1 1

* * * *0 1 1

, 1, ,6

, , ,

, , ,

m Ti ii r r

r

Ti i i i

m

Ti i i

m

x b S i

b b b

S S S

−

=

−

−

= = =

⎡ ⎤= ⎢ ⎥⎣ ⎦

=

∑ S b

b

S

…

(54)

where ib is a yet unknown Chebyshev polynomial coefficient vector. In a

similar manner, we can calculate a Chebyshev polynomial coefficient vector

for each (true-anomaly dependent) entry, ija , of 21A , wherefrom

1* *

0

0 1 1

, , 1, ,6

, , , ,

mTij ij

ij r rr

Tij ij ij ij

a d S i j

d d d

−

=

−

= = =

⎡ ⎤= ⎢ ⎥⎣ ⎦

∑

m

S d

d

… (55)

where ijd are polynomial coefficients calculated according to the quadrature

(52).

28

We can now utilize the unique properties of Chebyshev polynomials, that is,

express the product of any two polynomials using the product operational

matrix, and the quadrature of shifted Chebyshev polynomials using the

integration operational matrix. This formalism transforms the problem of

solving the vector differential equation (15) into the following system of

algebraic equations:

( ) ( ) ( ) ( )ˆ ˆ ˆ ˆT T T T

It t t t⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− = +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦S S X S P S RB B B (56)

In (56), ( ) ( )1 6, ,T T⎡ ⎤= ⎢ ⎥⎣ ⎦

b b…B is a 6m -dimensional column vector of unknown

polynomial coefficients (cf. (54)), and

( ) ( ) *6

ˆ T Tt t⎡ ⎤ = ⊗⎢ ⎥⎣ ⎦S I S , (57)

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

3 3

3 3

ˆ

ˆ

m m

m m

×

×

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0 GP

0 C (58)

where

3 22ˆ ˆ,T T= ⊗ = ⊗G I G C A G , (59)

and TG is the 3 3m m× integration operational matrix. The matrix R

assumes the form

3 3 3 3

* *ˆ ˆ

m m m m× ×⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

0 0R

R C (60)

with the sub-matrices

29

* *21 22

ˆ ˆ ˆ ˆ,= =R GQ C GQ (61)

where 21Q and 22Q are the 3 3m m× product operational matrices, whose

entries depend upon ijd (cf. Eq. (55)). Finally, we choose

6 1 11,T

I m× −⎡ ⎤= ⊗ ⎢ ⎥⎣ ⎦X I 0 (62)

and calculate the monodromy matrix through

( )ˆ TT⎡ ⎤Ψ = ⎢ ⎥⎣ ⎦S B (63)

where 1 6, ,⎡ ⎤= ⎢ ⎥⎣ ⎦B B B .

Appendix B: Generating a Smooth Controller

In order to avoid discontinuities we require that

( ) ( ),t t t kT− += ∀ =u u . (64)

This requirement leads to a slightly modified control law,

( ) ( ) ( ) ( ), [ 1 ), 0,1,f f kT f kT k T k= ∈ + =u K y … (65)

where K is a generalized hold function of the form

( ) ( ) ( ) ( )1 01 2 02 ,f f f f m qα α= + + ≥K K K K (66)

30

and 1 2,α α are constants, to be determined shortly. ( ) ( )01 02,f fK K are given

by

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

101 1 1

102 2 2

, , [0, )

, , [0, )

T T

T T

f f f T f t T

f f f T f t T

−

−

= − Φ ∈

= − Φ ∈

K K B W L

K K B W L (67)

where ( ) ( )1 2,f fK K are arbitrary continuous matrix functions. We chose the

following functions, which are continuous in ( )0,T :

( )

( ) ( )1

2 1

f f

f f

=

= −

K I

K I (68)

The matrices 1 2,L L of (67) are controllability maps of ( ) ( )1 2,f fK K ,

respectively:

( ) ( ) ( )

( ) ( ) ( )

1 10

2 20

,

,

T

T

T d

T d

τ τ τ τ

τ τ τ τ

= Φ

= Φ

∫

∫

L B K

L B K (69)

The continuity requirement (64) gives rise to the constraint

( ) 0, 0,1,...kT k= ∀ =K , (70)

which, in turn, determines the constants 1 2,α α of Eq. (66) through the

following system of linear equations:

( ) ( )

( ) ( )

( )( )

01 01

1 2

02 02

0 0

0

T

TTα α

+ − +

−+ −

⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥⎡ ⎤ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ −⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

K K K

KK K, (71)

31

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