Study on Stationkeeping for Halo Orbits at EL {1 ...

Study on Stationkeeping for Halo Orbits at EL1: Dynamics Modeling

and Controller Designing

By Ming XU and Shijie XU

Department of Aerospace Engineering, School of Astronautics, Beihang University, Beijing, China

(Received March 7th, 2011)

This paper deals with the stationkeeping control for halo orbits at EL1 in the Sun-Earth/Moon system, and proposes

an effective adaptive robust controller for the unknown spacecraft mass and perturbation boundaries. The controller has

to deal with two divergence sources: one is the instability of the halo orbit, and the other is the perturbation imposed by

the natural model onto the nominal model. The former source is displayed by the Floquet multiplier from the Poincare

mapping. However, the latter is revealed by the difference of Hamiltonian functions between the nominal reference

model, the circular restricted three-body problem (CR3BP) and the natural simulation model, the spatial bicircular

model (SBCM). Firstly, the algorithm of backstepping control theory is employed to generate the initial controller in

the nominal reference model of CR3BP. Some improvements are then implemented for the estimations of the unknown

parameters as the perturbation boundaries and the spacecraft mass, which may cause the failure of the initial unimproved

controller in stationkeeping. The controller proves to be effective in terms of adaptive robust estimation and asymptotic

stability from Lyapunov’s stability theory. Furthermore, further improvements of the triggers for the on/off schedule are

proposed to remedy the weakness in the capability of estimating for excessively long (infinite) time required to converge.

Finally, the controller developed in this paper is implemented in the natural simulation model of SBCM to evaluate its

performance. In the numerical simulation, the mass and perturbation boundaries will converge only after approximately

three iterations. The deviation of the estimating mass is 1 kg from its true mass, but 55 kg for the unimproved controller.

The total velocity increment over five years is only 126m/s, which is equivalent to the fuel consumption of 3.8 kg for the

Hall thrust engine carried by SMART-1.

Key Words: Libration Point, Halo Orbit, Stationkeeping, Robust Adaptive Strategy, Backstepping Design

1. Introduction

Since the success of ISEE-3 in 1978, libration points and

halo orbits have drawn much attention from the astronauti-

cal society all over the world.1) So far there have been seven

missions involving in the libration point or halo orbits for

solar observation, early warning systems for solar wind or

deep-space telescopes, such as ISEE-3 (1978), Wind

(1994), SOHO (1995), ACE (1997), Genesis (2001), MAP

(2001) and Planck (2009). Of these, the last two missions

were at EL2 and the others at EL1.

It is well known that halo orbits near the collinear libra-

tion points are unstable,2) thus the stationkeeping control

is necessary to keep the spacecraft near the nominal trajec-

tory. The method of achieving this has been a common focus

of research since the first mission for the libration point was

implemented. Up to now, some control strategies have been

proposed by employing different control theories, and they

can be classified into two modes: the target and Floquet

modes.1) The former keeps the spacecraft flying on the nom-

inal trajectory with frequent controls. The latter, however,

locates the spacecraft on the invariant manifolds of the halo

orbit with the controller compensating for the unstable

manifolds in order to avoid divergence. The Floquet mode

was developed by Simo et al.3) from the view of dynamical

system theory. It is more complicated in terms of controller

design but results in less fuel consumption than the target

mode. Despite this higher fuel consumption, the target mode

has more potential applications from modern control theo-

ries. Farquhar2) and Breakwell et al.4) are pioneers in

employing the feedback control in this opening problem.

Howell and Pernicka5) designed the linear controller for

the stationkeeping on halo orbits from the balanced view

of control precision and fuel assumption. Giamberardino

and Monaco6) designed a nonlinear controller to track the

halo orbit asymptotically based on a linear model. Cielaszyk

and Wie7) used a numerical method developed from LQR to

generate and maintain the halo orbit. Kulkarni and Camp-

bell8) applied the H1 theory to the stationkeeping for halo

orbits, and then applied the development strategy to main-

tain the formation of several spacecrafts flying around halo

orbits. Wong and Kaplia9) developed the adaptive control

strategy for this question and proved the stability of the con-

troller designed. Xu and Xu10) achieved the linear periodic

strategy for stationkeeping control based on the periodicity

of halo orbits.

For long flights in deep space far from the Earth, space-

craft should have the capabilities to deal with uncertainties

such as mismodeled accelerations in the N-body gravita-

tional fields (with unknown upper-boundary) and unknown

parameters of the spacecraft (such as mass and moment of� 2012 The Japan Society for Aeronautical and Space Sciences

Trans. Japan Soc. Aero. Space Sci.

Vol. 55, No. 5, pp. 274–285, 2012

inertia). There are several studies on estimating the inertia

moment associated with the attitude control.11–13) However,

there are no papers discussing the stationkeeping control for

halo orbits with the unknown parameters or perturbations.

Therefore, we will deal with the stationkeeping for the halo

orbit at the libration point EL1 in the Sun-Earth/Moon sys-

tem, and design a robustly adaptive controller associated

with estimating the spacecraft mass and the upper-boundary

of the disturbances.

To understand the stability of nominal trajectory gener-

ated from the reference model of the circular restricted three

body problem (CR3BP), the Floquet multiplier is calculated

from the Poincare mapping to indicate the instability of halo

orbits, which needs to be compensated by the controller. To

understand the unknown perturbations between the nominal

reference model and natural model, the difference of Ham-

iltonian functions is derived to check whether the lunar

gravitational perturbation is small enough that the reference

model can be considered as a reasonable approximation of

the natural simulation model from the view of the Hamilto-

nian dynamical system. So the controller developed in this

paper has to deal with two divergence sources: one is the

perturbation between nominal and natural models, and the

other is the instability of the halo orbit caused by its large

Floquet multiplier.

The controller is designed based on the nominal reference

model of the CR3BP, and is then implemented in the natural

simulation model of the spatial bicircular model (SBCM) to

evaluate its performance. The algorithm of backstepping

control is employed to generate the initial controller. How-

ever, the controller requires some improvements for the

estimations of the unknown parameters such as the bound-

ary of perturbations from the nominal and natural models

and the mass of spacecraft, which may cause the failure of

the unimproved controller. The improvements prove to be

effective in terms of adaptive robust estimation and asymp-

totic stability from Lyapunov’s stability theory. However,

the controller is still weak in dealing with the estimation re-

quired to converge for excessively long (infinite) time. So

further improvements of the on/off schedule for the trigger

are proposed.

2. Dynamical Modeling

2.1. Unit normalization and coordinate system defini-

tion

The equations derived in this paper can be normalized by

means of the characteristic length, time and mass, as below:

½L� ¼ RS-E/M; average distance between the Sun and the barycenter of the Earth and Moon

½M� ¼ mS þ mE/M; total mass in the Sun-Earth/Moon system

½T� ¼ R3S-E/M=GðmS þ mE/MÞ

� �1=28><>:

where mS and mE/M are the mass of the Sun and Earth/

Moon, respectively, and G is the universal gravitation con-

stant.

Three different coordinates are referred to in this paper as:

the syzygy frame in the Sun-Earth/Moon system (SS-E/M),

the inertial frame in the Sun-Earth/Moon system (IS-E/M),

and the syzygy frame in the Earth/Moon system (SE/M).

The reason for introducing the syzygy frames, such as

SS-E/M and SE/M, is so that they can be used to reduce the

computational work in Kepler circular motions in the Sun-

Earth/Moon and Earth/Moon systems, and to greatly im-

prove the computing efficiency for long simulations.

The IS-E/M frame with its components of ðX; Y ; ZÞ has thefollowing features: the origin O is fixed at the barycenter of

the Sun-Earth/Moon system; the Z axis is perpendicular to

the ecliptic plane and along the revolution axis of the Earth/

Moon system; the X axis, which follows an inertial direction

in the system, is along the intersection of the ecliptic and the

lunar plane; and the Y axis is determined by the right-hand-

side rule.

The SS-E/M frame with its components of ðx; y; zÞ has thefollowing features: the origin O is fixed at the barycenter of

the Sun-Earth/Moon system; the z axis is perpendicular to

the ecliptic orbital plane; the x axis points from the Sun to

OE/M; and the y axis is determined by the right-hand-side

rule. The mass ratio of the Earth and Moon subject to the

total mass of the Sun-Earth/Moon system can be defined

as �S ¼ mE/M=ðmS þ mE/MÞ. Then the coordinate compo-

nents of the Sun and OE/M in the syzygy frame can then

be solved from the normalization as ½��S 0 0�T and

½1� �S 0 0�T, respectively, to implement the reduction

in computing the Kepler circular motion introduced by the

definition of the syzygy frame.

The SE/M frame with its components ð�; �; �Þ has the fol-lowing features: the origin O is set at the barycenter of the

Earth/Moon system; the � axis is perpendicular to the lunar

plane and along the revolution axis of the Moon; the � axis

points from the Earth to the Moon; and the � axis is deter-

mined by the right-hand-side rule. Thus, the position vectors

rE and rM of the Earth and Moon in SE/M can be expressed

simply as

rE ¼RE-M

RS-E/M

� �mM

mE þ mM

0 0

� �Tand

rM ¼RE-M

RS-E/M

�mE

mE þ mM

0 0

� �T;

where mE and mM are the mass of Earth and Moon, respec-

tively, and the relationship exists whereby mE þ mM ¼ �S,

and RE-M is the averaging distance from Earth to Moon. Fur-

thermore, the identical equation can be achieved from the

definition of the SE/M frame and Newton’s gravitational

equation as mE � rE þ mM � rM ¼ 0.

Sep. 2012 M. XU and S. XU: Study on Stationkeeping for Halo Orbits at EL1 275

2.2. CR3BP

2.2.1. CR3BP assumption

The astrodynamics in the gravitational field of the Sun-

Earth/Moon system has a famous simplified form in celes-

tial mechanics referred as the circular restricted three body

problem (CR3BP), which is an open unsolved problem first

proposed by Newton.14) The assumptions for the simplified

model in the Sun-Earth/Moon system are summarized as:

a) The Earth and Moon act as one whole gravitational

point with its position located in their centroid OE/M;

b) The gravitational point stays circumsolar in the

ecliptic plane with relatively small eccentricity (e � 0:01)

ignored.

2.2.2. Dynamics in CR3BP

The CR3BP has different dynamics equations in the dif-

ferent frames referred in section 2.1. The dynamics in

SS-E/M, however, have a more simplified form than in the

others and reduces the computing work in the Kepler circu-

lar motion introduced by the definition of the syzygy frame.

The position vector of the spacecraft in the syzygy frame

is denoted as R ¼ ½x y z�T, and the generalized momen-

tum P ¼ ½px py pz�T is defined as

px ¼ _xx� y

py ¼ _yyþ x

pz ¼ _zz:

8<: ð1Þ

Then, the Hamiltonian function is

H0 ¼1

2p2x þ p2y þ p2z

� �� xpy þ ypx �

1� �S

rS�

�S

rE-Mð2Þ

where

rS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxþ �SÞ2 þ y2 þ z2

q

is the distance between the spacecraft and the Sun, and

rE-M ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxþ �S � 1Þ2 þ y2 þ z2

qis the distance between the spacecraft and OE/M. The

Hamiltonian dynamics in the model of CR3BP can be ex-

pressed as:

_RR ¼@H0

@P

_PP ¼ �@H0

@R:

8>><>>: ð3Þ

Expanding the above equation yields the Hamiltonian

dynamics in their components as

€xx

€yy

€zz

264

375 ¼ �2

� _yy

_xx

0

264

375þ

x

y

0

264

375�

1� �S

r3S½Rþ �Se1�

��S

r3E-M½R� ð1� �SÞe1� ð4Þ

where e1 ¼ ½1 0 0�T is the unit vector along the x axis.

Though the Hamiltonian dynamics in the model of

CR3BP are non-integrable, Eq. (4) has five special solu-

tions, named as libration points (or Lagrange points, denoted

as Li, i ¼ 1; � � � ; 5), of which L1, L2 and L3 are collinear

points (with the second and third components of yL ¼ 0,

zL ¼ 0) and L4 and L5 are triangular points (with the third

components of zL ¼ 0). The locations of the libration

points can be achieved by solving the algebraic equations

generated by setting the right sides of Eq. (4) to zero. This

paper concentrates more on the point of EL1 with its first

component as

xL ¼ ð1� �SÞ

��S

3

1=3

1þ1

3

��S

3

1=3

�1

9

��S

3

2=3

þ � � �

" #

� 0:98999093: ð5Þ

2.3. SBCM

2.3.1. SBCM assumption

The SBCM originates from the planar BCM (PBCM)

introduced by Andreu15) and Koon et al.16) in the planar

analysis on lunar low-energy transfer, which is a reasonable

approximation of the CR3BP. However, the SBCM shows

significant improvements in the inclination between the

ecliptic and lunar planes, which has an important application

in spatial analysis on astrodynamics.17,18) Compared with

the CR3BP, the SBCM has the following assumptions:

a) The Earth and Moon act as different simple gravita-

tional points, and move round their centroid in Kepler circu-

lar motions with their eccentricity ignored;

b) The centroid of the Earth/Moon system stays cir-

cumsolar in the ecliptic plane with the eccentricity ignored;

c) The inclination of the lunar plane relative to the

ecliptic plane is considered as the average angle of 5�90.

The SBCM shows significant improvements in terms of

the extra gravitational force implemented by the Moon

and the inclination between the ecliptic and lunar planes,

which are neglected by the CR3BP. The difference of the

two models’ Hamiltonian functions will be checked in sec-

tion 2.3.4, and whether lunar gravity is so small that the

CR3BP can be considered as a reasonable approximation

of the SBCM from the view of the Hamiltonian dynamical

system will be stated.

SunEarth

MoonAss

C

Rr

Spacecraft

Lunarplane:N

: EclipticplaneM

θ ϕ

Fig. 1. SBCM.

276 Trans. Japan Soc. Aero. Space Sci. Vol. 55, No. 5

2.3.2. Kinematics in the SBCM

RI ¼ ½X Y Z�T is defined as the position vector from

the barycenter of the Sun-Earth/Moon system to the space-

craft in the frame IS-E/M, and r ¼ ½� � ��T is defined as

the position vector from the barycenter of the Earth-Moon

system to the spacecraft in the frame SE-M. �S and ’ are

the phasic angles of the Sun and Moon (shown in Fig. 1),

and i is the inclination between the ecliptic and lunar planes.

AS is defined as the position vector from the Sun to the cent-

roid OE/M in the frame IS-EM.

According to definitions of coordinate systems and

SBCM assumptions, the required relationships are listed as:

R ¼ Rzð�SÞRI ð6Þ

AS ¼ ð1� �SÞ½cos �S sin �S 0�T ð7Þ

RI ¼ Rxð�iÞRzð�’ÞrþAS ð8Þ

where Rzð�Þ and Rxð�Þ are, respectively, the elementary

transformation matrixes around the Z (or z) and X (or x)

axes. The position vector R is another form of RI in the

SS-E/M frame.

The position vector r in the SE-M frame can be obtained

from Eq. (8), as

r ¼ Rzð’ÞRxðiÞRzð��SÞ½R� ð1� �SÞe1�: ð9Þ

2.3.3. Dynamics in the SBCM

The Hamiltonian function for the SBCM in SS-E/M is18)

H1 ¼1

2p2x þ p2y þ p2z

� �� xpy þ ypx

�1� �S

rS�

mE

kr� rEk�

mM

kr� rMkð10Þ

where rS is the distance between the Sun and the spacecraft,

i.e., rS ¼ kR� ½��S 0 0�Tk. Then, the dynamics in

Hamiltonian form are

_RR ¼@H1

@P

_PP ¼ �@H1

@R¼ �

@

@R

�1

2p2x þ p2y þ p2z

� �� xpy þ ypx �

1� �S

rS

�þ

@r

@R�@

@r

�mE

kr� rEkþ

mM

kr� rMk

�8>><>>: ð11Þ

where

@r

@R¼ Rzð�SÞRxð�iÞRzð�’Þ;

and the vector r can be calculated from Eq. (9).

Expand the above dynamical equations as

€xx

€yy

€zz

264

375 ¼ �2

� _yy

_xx

0

264

375þ

x

y

0

264

375� ð1� �SÞ

Rþ �Se1

kRþ �Se1k3� Rzð�SÞRxð�iÞRzð�’Þ mE

r� rE

kr� rEk3þ mM

r� rM

kr� rMk3

� �: ð12Þ

2.3.4. Alternating gravitation caused by the moon

The SBCM considers the Earth and Moon as two different

gravitational points, so that the gravitational force from the

Moon will impose the periodic perturbation on the CR3BP,

which can be measured by the difference of Hamiltonian

functions as �H ¼ H1 � H0.19) When the spacecraft stays

near the halo orbit at EL1, there exists the relationship krk �4 � k �RRE-Mk (where �RRE-M is 384,400 km, the average distance

between the Earth and Moon). Then we can attain the fol-

lowing proposition:

Proposition 1. For spacecraft flying on the halo orbit

near EL1 (or EL2), the periodic alternating gravitation is on-

ly small perturbation.

Proof. The location of OE/M in the SE/M frame is

½0 0 0�T, and the fact of rE-M ¼ krk ensures the expan-

sion of �H as

�H ¼ H1 � H0 ¼�S

rE-M�

mE

kr� rEk�

mM

kr� rMk

¼�S

krk1�

mE

�S

�r

r� rE

� mM

�S

�r

r� rM

� �:

We can transfer the mathematical expression as kak=kaþ bk as follows:

kakkaþ bk

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaTa

ðaþ bÞTðaþ bÞ

s

¼�1þ 2

aTb

aTaþ

bTb

aTa

�12

¼ 1�aTb

kak2�

1

2

kbk2

kak2þ

3

2

ðaTbÞ2

kak4þ O

�kbkkak

3

:

ð13ÞTherefore, with the help of the above transformation tech-

nique and the following identical equation:

mErE þ mMrM ¼ 0 ð14Þ

we can deduce the simplified form of �H as


�H ¼�S

krk1�

mE

�S

� 1þrTrE

krk2�

1

2

krEk2

krk2þ

3

2

ðrTrEÞ2

krk4þ o

�krEkkrk

2 !" #

��S

krkmM

�S

� 1þrTrE

krk2�

1

2

krMk2

krk2þ

3

2

ðrTrMÞ2

krk4þ o

�krMkkrk

2 !" #

:

The mass ratios of the Earth and Moon in the Sun-Earth/Moon system are as follows:

�M :¼mM

�S

� 0:01215; �E :¼mE

�S

with the identical relationship of �E þ �M ¼ 1. Thus, �H can be expressed as:

�H ¼ ��S

krk�

�EkrEk2 þ �MkrMk2

2krk2þ

3

2

rTrTð�ErErE þ �MrMrMÞkrk4

þ o

�RE-M

krk

2" #

¼ ��S

krk�

R2E-M

2krk2ð�E�

2M þ �M�

2EÞ þ

3

2

R2E-Mx

2

krk4ð�E�

2M þ �M�

2EÞ þ o

�RE-M

krk

2" #

¼ ��S

krkð�E�

2M þ �M�

2EÞ

R2E-M

krk2

��

1

2þ

3

2

x2

krk2þ oð1Þ

�

where rErE and rMrM represent the dyadic matrixes gener-

ated from the position vectors rE and rM of the Earth and

Moon in SE/M, respectively.

For spacecraft flying in the vicinity of EL1, the distance

from the Earth to the spacecraft is about four times larger

than the distance between the Earth and Moon. So we can

obtain an interesting consequence as

�H � 7:5� 10�4 ��S

krk¼ 7:5� 10�4 � UE-M: ð15Þ

According to the analysis on the energy of the halo orbit per-

formed in section 3.1.3, we can attain

H0

UE-M

� 1:5: ð16Þ

That is,H0 andUE-M are in the same order of magnitude. As-

sociated with �H � 7:4� 10�4 � UE-M, �H=H0 is of 10�3

order of magnitude, which can be considered a small pertur-

bation onto the Hamiltonian system H0.

3. Nominal Trajectory for Stationkeeping

The halo orbit, which bifurcates from the planar Lyapu-

nov orbit,8) is another solution of the CR3BP (quite different

from the fixed libration points), and is symmetrical with

respect to the x–z plane in the SS-E/M frame (Fig. 2 illus-

trates the three-dimensional (3D) representation of the halo

orbit family at EL1, and the normalized length is the mean

distance between the Sun and the Earth/Moon). The states

of some points on the halo orbit (with the period of THalo)

depend solely on the two variables: magnitude Ax and phase

#, such that

P ¼ � ðAx; #Þ ¼ HðAx; tÞ; # 2 R=2�Z; t 2 R=THaloZ

ð17Þ

where P is the state variable, and P ¼ ½x y z _xx _yy _zz�T.

There is no analytic solution to the CR3BP due to the

non-integrability. The equivalence relation � can be de-

fined as

P1 � P2 , 9� 2 R=THaloZ; P2 ¼ ’�ðP1Þ ð18Þ

where ’� is the flow of the dynamics. So any point P1 can

represent the whole flow.

3.1. Calculation of the halo orbit

3.1.1. Richardson expansion

The Lindstedt-Poincare technique in the analytic approx-

imation method to solve the dynamical equation can be used

to improve the three-order analytic solution to the halo or-

bits, which is called the Richardson expansion, named after

its inventor.8)

The Richardson expansion has the form:

x ¼ a0 þX3i¼1

ai cosði � !t þ �0Þ

y ¼X3i¼1

bi sinði � !t þ �0Þ

z ¼ c0 þX3i¼1

ci cosði � !t þ �0Þ

8>>>>>>>>>><>>>>>>>>>>:

ð19Þ

where ai, bi, ci and ! are the functions of Ax, and the initial

value for �0 can be set as 0. The parameters of the halo orbit

selected in this paper, ai, bi, ci and !, are listed in the Ap-

pendix.

3.1.2. Differential correction

Put P1 on the symmetry plane of the halo orbit, and then

the numerical algorithm for it is:

Step 1: use the Richardson expansion to obtain the ap-

proximate value of P1;

Step 2: take P1 ¼ ½x0 0 z0 0 _yy0 0�T as the initial

state to integrate the differential equations describing the

CR3BP until the flow arrives at the symmetry plane x–z again

with the final state denoted as P f ¼ ½xf 0 zf _xxf _yyf _zzf �T;


Step 3: iterate x0 and _yy0 to adjust _xxf and _zzf closed to zero

after four to five iterations.

The correction of x0 and _yy0 are:

x0

_yy0

� �¼

�41 ��21

€xxf

_yyf�45 ��25

€xxf

_yyf

�61 ��21

€zzf

_yyf�65 ��25

€zzf

_yyf

26664

37775

�1

�_xxf

_zzf

� �ð20Þ

where � is the monodromy symplectic matrix defined in

section 3.2, and �ij is the element of the matrix in the ith

row and jth column.

3.1.3. Energy allocation of the halo orbit

The Hamiltonian function H0 of the halo orbit is constant

due to the time-independent system of the CR3BP. Define

US :¼ �ð1� �SÞ=rS as the potential energy for the Sun’s

gravity, and define UE-M :¼ �ð1� �SÞ=krk as the potential

energy for the Earth’s and lunar gravities. Furthermore, de-

fine VK :¼ ð _xx2 þ _yy2 þ _zz2Þ=2 as the relative kinetic energy,

and define UC :¼ �ðx2 þ y2Þ=2 as the Coriolis potential en-

ergy. Then the Hamiltonian function is the sum of all the

variables defined above:

H0 ¼ US þ UE-M þ UC þ VK: ð21Þ

The energy allocation of the four different variables for the

halo orbit with H0 ¼ �1:5004 (with the magnitude of

Ax ¼ 260;000 km) is shown in Fig. 3.

However,US,UE-M, VK andUC all change over time, with

the exception of H0 only, and the time averages of them are

�3� 10�4, �1:0092, 4� 10�5 and �0:4909, respectively,

where the time average is calculated as

h f i ¼1

THalo

Z THalo

0

f ðtÞdt; f ¼ US;UE-M;VK;UC: ð22Þ

3.2. Stability analysis on halo orbits

Poincare mapping ðPÞ is defined as

ðPÞ ¼ ’THaloðPÞ; 8P 2 � ðAx; #Þ:

Then according to the Hamiltonian dynamical system, the

derivative function of ðPÞ: � ¼ DPðPÞ is a symplectic

matrix with four plural eigenvalues j�ji ¼ 1, i ¼ 1; 2; 3; 4

and two real eigenvalues �5 ¼ ��16 > 1 (the eigenvalues

of � are called characteristic exponents of the flow). �5and �6 indicate the stability of halo orbits.

The maximal real eigenvalue �5 is called the Floquet mul-

tiplier, and the periodic orbits will diverge at the exponential

rate of �5 � 1. Figure 4 shows the relationship between the

magnitude Ax of halo orbits and the function ln of the Flo-

quet multiplier.

From the view of stability, the Floquet multiplier should

be selected to be near 1 in order to avoid divergence. How-

ever, the closer to 1 the Floquet multiplier is, the larger the

magnitude of the halo orbit is. For example, a halo orbit with

a Floquet multiplier of 20 has a magnitude of 480,000 km,

so that the Halo orbit is too large for astronautical engi-

neering. Sometimes a halo orbit with small magnitude is

preferred even at the cost of more fuel consumption for

stationkeeping.

In this paper, the magnitude Ax ¼ 260;000 km of Genesis

is adopted with a Floquet multiplier of 1,219.3 and period of

174.684 days.

So the actual trajectory will diverge because of the two

divergence sources: one is the perturbation between the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2HU

E-MUSV

KU

C

Ene

rgy

allo

catio

n

Time [period of halo orbit]

Fig. 3. Energy allocation of the halo orbit.

0.2 0.25 0.3 0.35 0.4 0.45 0.52.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Ln o

f the

Flo

quet

mul

tiplie

r

Magnitude of the halo orbit [106 km]

Fig. 4. Relationship between the magnitude Ax (106 km) and the ln of the

Floquet multiplier.

0.990.992

0.9940.996

0.9981

1.002

-0.01

0

0.01

-5

0

5

10

15

x 10-3

x [RS-E/M]y [RS-E/M]

z [R

S-E

/M]

EL1

Earth

Fig. 2. Halo orbit family at EL1.


nominal and natural models, and the other is the instability

of the halo orbit caused by its large Floquet multiplier. For

the initial insertion errors in position and velocity listed in

section 6, it is no more than half an orbital period (approx-

imately three months) that the spacecraft can keep flying

near the nominal halo orbit, as illustrated in Fig. 5.

4. Problem Formulation for Stationkeeping

4.1. Nominal reference model and natural simulation

model

For the control problem, the nominal trajectory is created

by the known model which also acts as the nominal refer-

ence model to design the controller. The controller will then

be implemented in the natural simulation model to evaluate

its performance. Actually, the natural model should remain

unknown when constructing the controller. Therefore, the

CR3BP acts as the nominal model opening to the controller

in this paper, while the SBCM acts as the natural model.

The nominal trajectory is designed on the reference

model that is an approximation of the natural model. To im-

prove the performance of the controller, the most exact

model developed is preferred to act as the nominal model

since the higher degree the approximation has, the less

the cost of control will be for stationkeeping on the nominal

trajectory and vice versa. In this paper, we will evaluate our

robust adaptive controller in the case of the biggest pertur-

bation in the actual solar system. For the halo orbit located

in EL1, the average value of the lunar gravity is at the order

of 10�6 m/s2, which is somewhat bigger than that of Jupiter

(maximum of gravitational perturbation less than 3� 10�7

m/s2) or any other celestial bodies. Therefore, we employed

the CR3BP and SBCM to act as the nominal and natural

models, respectively. The difference between the two mod-

els is the lunar gravitational perturbation, which is the

biggest effect in the actual solar system.

Although some ephemeris models have been developed

which are exact enough (such as JPL DE403 or DE405),

we still use the CR3BP and SBCM to evaluate the perfor-

mance of robustness and adaptive estimation, because they

contain the maximum perturbation in the actual solar sys-

tem. So the controller achieved in this paper could improve

further with the more exact ephemeris models that have

much less slight perturbation if it is able to keep the space-

craft flying near the nominal trajectory under the biggest

perturbation imposed by the CR3BP and SBCM.

4.2. Control formula

The dynamics of the spacecraft with stationkeeping con-

trol can be classified as

_�� ¼ �

_�� ¼ f ð�;�Þ þ d þ u=m

�ð23Þ

where � 2 R3 is the position vector; � 2 R3 is the velocity

vector; d is the disturbance acceleration; f is the right-hand

side of Eq. (4) without any information on disturbance; u

is the control acceleration; and m is the true mass of the

spacecraft.

The error variables are defined as

e1 :¼ �� d; e2 :¼ �� _��d ð24Þ

where �d 2 R3 is the excepted position vector of halo orbit,

and _��d 2 R3 is the velocity vector.

Then, we transform the dynamics of the spacecraft

referred to in Eq. (23) into the error dynamics as

_ee1 ¼ e2

_ee2 ¼ f ð�d þ e1; _��d þ e2Þ � €��d þ d þ u=m

�ð25Þ

where

~ff ðe1; e2; tÞ , f ð�d þ e1; _��d þ e2Þ � €��d ð26Þ

so a controller is required to achieve

limt!1

e1; e2 ¼ 0:

Actually, u=m cannot be treated as one control parameter

instead of two independent parameters for the following two

reasons: the thrust engine only provides the thrust forces, not

the accelerations, thus we cannot obtain the exact control

forces even though the required control acceleration u=m

has been obtained in the case of no information on the un-

known mass. Another approach to compute the control

forces from the acceleration u=m and the estimating mass

mm is also undesirable as mm converges to its true value after

two or three drive signals from the control output, as shown

in Fig. 9. So we treat the mass and the forces (including

control forces and disturbance forces) as two independent

parameters in the design of the robust adaptive controller,

especially in the case of unknown mass and disturbance,

which have been validated by the numerical simulation in

Figs. 9 and 11.

Here, continuous thrust is required to drive the estimation

on the unknown mass and boundary of perturbation forces.

Hence we need the variable thrust engine to act as the sta-

tionkeeping controller. Actually, some missions, such as

GP-B (NASA, launched on 20 April, 2004), GOCE (Gravity

field and steady-state Ocean Circulation Explorer, ESA,

launched on 17 March, 2009), SLATS (Super Low Altitude

1.475 1.48 1.485 1.49 1.495

x 108

-5

0

5

x 105

x [km]

y [k

m]

-10 -5 0 5

x 105

-5

0

5

x 105

z [km]y

[km

]

1.48 1.485 1.49

x 108

-4

-2

0

2

4x 10

5

x [km]

z [k

m]

1.481.4821.4841.4861.488x 10

8-50

5

x 105

-4-2024

x 105

z [km]y [km]

z [k

m]

Nominal trajectory

Uncontrolled trajectory

Fig. 5. The actual trajectory without control.


Test Satellite, JAXA [planned]) and LISA (NASA [plan-

ned]) can provide the variable thrust by adjusting the electric

currents of the gaseous conductor inside the engine. The

constant thrust engine also has more application in lunar

or interplanetary transfer trajectories, such as SMART-1

(ESA, 27 September, 2003), Hayabusa (JAXA, 9 May,

2003) and Akatsuki (21 May, 2010). So the variable thrust

engine has more advances in the thrust control system than

the constant engine.

5. Robust Adaptive Controller Designing for Station-

keeping

Here the proposal for stationkeeping control is that: the

control law u is designed to let limð�;�Þ ! ð�d; _��dÞ basedon the backstepping idea associated with the unknown

spacecraft mass m and disturbance d 6¼ 0.

5.1. Backstepping control

Backstepping control, usually associated with Lyapunov

stability theory, is used to make the whole closed-loop sys-

tem to meet the expected static and dynamic performance.

The method, which was first proposed by Kokotovic in

199120) has attracted the interest of many researchers in

the past few years.

The work procedure for backstepping control is to first

divide a complex nonlinear system into subsystems with

their order no more than the order of the entire system.

The Lyapunov functions and virtual controls are then de-

signed for each subsystem. Subsequently, the design for

the subsystems retreats to the entire system, and they are in-

tegrated together to complete the design of the control law.

According to the standard backstepping control algo-

rithm,20) virtual feedbacks can be used to define the error

variables. The definitions of error variables s1 and s2 are

s1 ¼ e1

s2 ¼ e1 þ e2

�ð27Þ

then the error dynamics have the new form:

_ss1 ¼ �s1 þ s2

_ss2 ¼ ~ff þ d þ u=m� s1 þ s2:

�ð28Þ

According to the standard algorithm, the control u can be

designed as

u ¼ �m ~ff � md � ms2 � cðs2Þ ð29Þ

where the function cð�Þ can be classified into the sgn func-

tion as

8v ¼ ½v1 v2 v3�T 2 R3

sgn v ¼ diag ½sgn ðv1Þ sgn ðv2Þ sgn ðv3Þ� ð30Þ

sgn vi ¼þ1; vi � 0

�1; vi < 0, i=1, 2, 3.

�ð31Þ

Hence, the Lyapunov function can be included as

V0 ¼1

2msT1 s1 þ

1

2msT2 s2: ð32Þ

5.2. Robust adaptive controller

The disturbance acceleration caused by the difference be-

tween the nominal reference and natural simulation models

is bounded, that is,

d �dd ð33Þ

where �dd 2 R3 is the non-negative constant vector unknown.

The adaptive estimation errors is defined as

~mm :¼ mm� m ð34Þ

E :¼ m �dd ð35Þ

~EE :¼ EE�E: ð36Þ

Due to the high specific impulse provided by the low-thrust

engine, the mass of the spacecraft will vary slowly during

flight. Thus, we can assume _mm ¼ 0 (viz. _~mm~mm ¼ _mmmm), which will

be checked and verified in section 6. Since �dd is defined as the

non-negative constant vector, we attain the fact that _EE ¼ 0

and _~EE~EE ¼ _EEEE.

The control law presented in Eq. (29) will fail in the sta-

tionkeeping for including the unknown variables of d andm.

So some improvements in the controller are necessary to

keep its robustness.

Proposition 2. For the control problem described by

Eqs. (27) and (28), the robust adaptive control law, consist-

ing of

u ¼ �mm ~ff � mms2 � sgn s2 � EE� cðs2Þ ð37Þ

_~mm~mm ¼ �ðsT2 � ~ff þ sT2 s2Þ ð38Þ

_~EE~EE ¼ � � sgn s2 � s2 ð39Þ

ensures asymptotic convergence of the position and velocity

tracking errors as delineated by

limt!1

ðe1; e2Þ ¼ 0:

Proof. We define a positive definite, candidate Lyapu-

nov function as

V ¼ V0 þ1

2�~mm2 þ

1

2~EET��1 ~EE ð40Þ

and then differentiate the function V with respect to time to

yield

_VV ¼ msT1 _ss1 þ msT2 _ss2 þ1

�~mm _~mm~mmþ ~EE

T��1 _~EE~EE: ð41Þ

Substitute the dynamics of Eqs. (27) and (28) and the con-

trol law of Eqs. (37)–(39) into the above differentiating

function to obtain

_VV ¼ msT1 ð�s1 þ s2Þ þ msT2

�~ff þ d þ

u

m� s1 þ s2

þ1


T��1 _~EE~EE

¼ �msT1 s1 þ sT2 ðm ~ff þ md þ uþ ms2Þ

þ1


T��1 _~EE~EE


and then substitute the control law of Eq. (37) to yield

_VV ¼ �msT1 s1 þ sT2 ð� ~mm ~ff � ~mms2 þ md � sgn s2EEÞ � sT2#ðs2Þ

þ1


T��1 _~EE~EE

¼ �msT1 s1 � sT2#ðs2Þ þ sT2 ðmd � sgn s2 �EÞ

þ ~mm

�� sT2

~ff � sT2 s2 þ1

�_~mm~mm

þ ~EE

Tð��1 _~EE~EE� sgn s2 � s2Þ:

Furthermore, we substitute the parameter update law of

Eqs. (38) and (39) to obtain

_VV ¼ �msT1 s1 � zT2#ðs2Þ þ sT2 ðmd � sgn s2 �EÞ

sT2 ðmd � sgn s2 �EÞ:Obviously, the inequality sT2 ðmd � sgn s2 �EÞ 0 is estab-

lished constantly. The necessary and sufficient condition

for _VV ¼ 0 is s1 ¼ 0, s2 ¼ 0. According to the Lyapunov the-

orem, we can conclude that

limt!1

ðe1; e2Þ ¼ 0:

5.3. Improvements in adaptive estimation

The stationkeeping strategy proposed in the above section

is weak in terms of adaptive estimation (the numerical sim-

ulation for the primary controller is implemented in section

6, and the estimation for the unknown mass is shown in

Fig. 10).

In fact, the weakness in estimating parameters is inherited

from the initial theory of adaptive control. To estimate the

unknown parameters on a line requires the drive signals

from the control output. However, the control output will

decline quickly upon the decreasing errors in position and

velocity, and it will weaken the capability of estimation

on the line especially after the errors of states have been

converged to zero.

In order to maintain the vitality of the capability of esti-

mation, continuous drive signals from the control output

are required. So we attempt to introduce the on/off schedule

for the thrust. The thrust will be controlled by the robust

adaptive law during the on time intervals, and has no output

during the off time intervals. The trigger for the controller is

illustrated in Fig. 6.

For the first trigger of the controller, the longest interval t0is required to reduce the initial errors of position and veloc-

ity. The thrust then has null output during the ‘‘off’’ interval

toff . Subsequently, the thrust needs to work for a longer ‘‘on’’

interval ton to keep the actual trajectory within a certain dis-

tance from the nominal one.

Due to the perturbations, the ballistic flight during toff will

deviate from its nominal trajectory, and thus the controller

should work well under a suitable value of toff . If toff is

too long, the controller will lose its stability for stationkeep-

ing. If toff is too short, however, the controller will lose its

adaptive estimations. Here, some preliminary discussions

on the choice of toff are demonstrated under two constraints,

which are the fixed first trigger duration t0 and the ‘‘on’’

interval ton depending only on the variable of toff , i.e., ton ¼1� toff (year). Due to the contribution of the first trigger

with the longest ‘‘on’’ interval t0, the initial deviation inher-

ited from the insertion of halo orbit is stabilized by the work-

ing thrust, and the maximum position errors may appear

during this interval because of the unknown perturbations,

whose boundaries will be estimated by the controller. In

practical engineering, the stationkeeping controller faces

the main challenge of stabilizing the initial insertion into

halo orbit, i.e., the maximum error is only required during

the first trigger of the switch thrust in practical engineering.

So a value of toff is considered unstable if the residual con-

trol error achieved is larger than the critical error.

The ergodic representation of the controller’s perfor-

mance is demonstrated in Fig. 7 over the values of toff , in-

cluding the stability, fuel cost and estimation. The controller

arrives at the critical point with the toff of 0.254 years, and

will lose its stability after this point. toff has a slight influ-

ence on fuel cost from 0 to the critical point since the cost

at the critical point is only 20m/s (or 0.6 kg) more than that

at the value of 0. However, the estimation has been greatly

improved from 445 to 499.5 kg over the values from 0 to

the critical point. After the critical point, this capability is

reduced. Thus, the optimal toff should be chosen from the

values less than the critical value.

During the ‘‘off’’ intervals toff , the uncontrolled spacecraft

cannot stay on the nominal orbit due to the perturbations,

and will leave for the Earth or other regions along the unsta-

ble manifolds (inherited from the fixed libration point, there

exist stable and unstable manifolds associated with the halo

orbit due to the hyperbolic type of topology). Therefore, the

stationkeeping controller is required to keep the controlled

trajectory near the nominal orbit as shown in Fig. 8.

6. Simulation Results

The halo orbit generated by the differential correction

method in section 3.1.2 has the parameters

Ax ¼ 260;000 km and �0 ¼ 0. The real mass of the space-

craft is 500 kg. The initial deviations inherited from the in-

Fig. 6. Trigger for the on/off schedule of the controller.

The critical error is approximately 2:83� 104 km for the initial devia-

tion in section 6.


sertion into the halo orbit are 1,464.6 km in position and

1.35m/s in velocity (with confidence level 99%).21)

Some parameters and functions in the controller are as

follows:

1. cðs2Þ ¼ sgn s2;

2. The SIGN function sgn ðvÞmay cause the discontinuity

of the control u, and can be replaced by

ð2=�Þ arctan ðG � vÞ for the purpose of smoothing,

where G ¼ 1� 104;

3. The parameters in the update law are taken as

� ¼ 1� 107; ¼5 0 0

0 5 0

0 0 5

264

375;

4. The initial value for the mass estimation of the space-

craft is mm0 ¼ 400 kg and there is no information on the

disturbance d, that is, EE0 ¼ ½0 0 0�T;5. The parameters t0, toff and ton for the on/off schedule

are chosen as 2 years, 0.25 years and 0.75 years,

respectively.

We implement the numerical simulations to achieve the

improved results. The 3D-controlled trajectory is shown in

Fig. 9, and compared with the nominal one. The controlled

errors in position along the axes of x, y, z are shown in

Fig. 10, with the maximum errors less than 3� 104 km.

The controlled trajectory under the improved strategy works

very well, and has great applications to missions such as

solar observation and early warning systems for solar wind.

The control forces of the three axes are shown in Fig. 11.

The maximum control forces (less than 3� 10�3 N) take

place during the first interval t0 due to the initial errors

caused by launching or inserting errors. Furthermore, the to-

tal velocity increment �V over five years is only 126m/s.

For the ion or nuclear engine with low thrust but high spe-

cific impulse Isp, the fuel cost will be low due to the high

specific impulse (for example, only fuel of 3.8 kg is required

for the Hall thrust with Isp ¼ 1;643:4 s carried by ESA’s

SMART-1 mission), so the mass m of the spacecraft will

vary so slowly that the assumption of _mm ¼ 0 in section 5.2

is accepted.

To check the adaptive estimation capacity from the im-

provement on the trigger for the on/off schedule, the numer-

ical simulations for initial and improved strategies are im-

1.478 1.48 1.482 1.484 1.486 1.488 1.49 1.492 1.494 1.496 1.498

x 108

-8

-6

-4

-2

0

2

4

6

8x 10

5

x [km]

y [k

m]

EL1 htraE

Unstable manifold

Nominal orbit

Controlled orbit

Uncontrolled orbit

Nominal orbit

Uncontrolled orbitUnstable manifold

Controlled orbit

Fig. 8. Unstable flight without thrust.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

4

Time [years]

Err

or in

pos

ition

[km

]

Error in x axis

Error in y axis

Error in z axis

Fig. 10. Control errors in position.

0 0.05 0.1 0.15 0.2 0.25 0.3

33.5

44.5

5

x 104

Max

imum

err

or [k

m]

0 0.05 0.1 0.15 0.2 0.25 0.3100

110

120

130

140

∆V [m

/s]

0 0.05 0.1 0.15 0.2 0.25 0.3

450

475

500

525

550

toff [years]

Est

imat

ion

for

mas

s [k

g]

Critical point toff = 0.254

Fig. 7. Ergodic representation of the controller’s performance over toff .

1.48

1.485

x 108

-5

0

5

x 105

-4

-2

0

2

4

x 105

x [km]

y [km]

z [k

m]

Nominal trajectory

Controlled trajectory

Fig. 9. Nominal and controlled trajectories for the halo orbit at EL1.


plemented for comparison in Figs. 12 and 13, respectively

(for the case of initial strategy, all the control parameters re-

main the same as the other case, with the exception of only

the parameter of t0 set as five years). For the initial strategy,

the spacecraft mass is converged to 445 kg, deviating more

than 10% from its true mass. However, for the improved

strategy, it is after approximately three iterations that the

mass converges to the true values, deviating only 1 kg from

its true mass. The similar case occurs in estimating the

boundary of perturbation forces, as shown in Fig. 14.

Consequently, the improvement on the adaptive estimation

in the controller is successful.

7. Conclusion

This paper dealt with the stationkeeping control for halo

orbits at EL1, and proposed an effective adaptive robust con-

troller for spacecraft with unknown mass and boundary of

perturbation forces. The controller developed in this paper

had to deal with two divergence sources: one was the insta-

bility of the halo orbit, and the other was the perturbation

between nominal and natural models. The former was

checked by the Floquet multiplier from the Poincare map-

ping. However, the latter was checked by the difference

in Hamiltonian functions between the nominal reference

model CR3BP and the natural simulation model SBCM

from the view of the Hamiltonian dynamical system.

The controller was designed based on the nominal refer-

ence model, and was then implemented in the natural simu-

lation model to evaluate its performance. The algorithm of

backstepping control theory was employed to generate the

initial controller. Some improvements were then imple-

mented for the estimations of the unknown parameters as

the boundary of perturbations from the nominal and natural

models and the mass of spacecraft, which may cause the

failure of the initial unimproved controller in stationkeep-

ing. The controller proved to be effective in terms of adap-

tive robust estimation and asymptotic stability from the

view of Lyapunov’s stability theory. Further improvements

of the trigger for the on/off schedule were proposed based

on the weakness in the capability of estimation of exces-

sively long (infinite) time required to converge.

It is after approximately three iterations that the mass and

boundary of the perturbation forces converged in the numer-

ical simulation. The estimated mass deviated 1 kg from its

true mass, while the deviation of the unimproved controller

was 55 kg. Furthermore, the total velocity increment over

five years was only 126m/s, equal to the fuel consumption

of 3.8 kg for the Hall thrust engine carried by SMART-1.

For GOCE and SLATS, feedback control logic takes the

measured acceleration from the sensor of the accelerometer

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6x 10

-4

Est

imat

ion

of E

x [N]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4x 10

-4

Est

imat

ion

of E

y [N]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4x 10

-4

Time [years]

Est

imat

ion

of E

z [N]

Estimate value

Actual value

Estimate value

Actual value

Estimate value

Actual value

Fig. 14. Estimate of E in the x, y and z axes.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3x 10

-3

Time [years]

Con

trol

forc

e [N

]

On Off Off Off On On On

Control in x axis

Control in y axis

Control in z axis

Fig. 11. Control forces in the x, y and z axes.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5400

410

420

430

440

450

460

470

480

490

500

Est

imat

ion

for

mas

s [k

g]

Time [years]

Fig. 12. Estimate of the spacecraft mass by the improved strategy.

0 1 2 3 4 5400

405

410

415

420

425

430

435

440

445

Est

imat

ion

for

mas

s [k

g]

Time [years]

Fig. 13. Estimate of the spacecraft mass by the initial strategy.


and computes commands sent to the thrusters to generate

tiny continuous forces to compensate for any drag that

GOCE and SLATS experience along their trajectories. For

the flight at super low altitude of approximately 200 km, it

is difficult to evaluate the exact atmospheric drag (even its

boundary) because of the uncertainty of the atmosphere

(especially at super low altitude). Thus, the robust adaptive

control law developed in this paper may have potential ap-

plications in stationkeeping for super low altitude satellites

without the assistance of measurements of the unknown

atmospheric drag from accelerometers.

Acknowledgments

This project was supported by the National Natural Science

Foundation of China (11172020), the Aerospace Science and Tech-

nology Innovation Foundation of the China Aerospace Science

Corporation, and the Talent Foundation Supported by the Funda-

mental Research Funds for the Central Universities.

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Appendix

The values of ai, bi, ci and ! for the Richardson expansion

of the halo orbit with Ax ¼ 260;000 km are listed in

Table A1.

Table A1. Coefficients for the Richardson expansion.

a0 9:89131506445114� 10�1

a1 �1:73799264365736� 10�3

a2 1:70870834774553� 10�4

a3 2:73210089081076� 10�5

b1 5:53934086594681� 10�3

b2 2:06832695501756� 10�4

b3 4:14890349892431� 10�5

c0 5:60766326302696� 10�4

c1 3:05775892311808� 10�3

c2 1:86922108767565� 10�4

c3 3:09420953976059� 10�5

! 2:0348403360716� 100


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