Adaptive Nonlinear Control of Spacecraft Near
Sun-Earth L2 Lagrange Point
Hong Wong and Vikram Kapila
Department of Mechanical, Aerospace, and Manufacturing Eriginecring
Polytechnic University, Brooklyn, NY 11201
Abstract
In this paper, an adaptive control algorithm is presented to enable a spacecraft to track desired trajectories near the Lz Lagrange point in the Sun-Earth system. An adaptive full-state feedback control law, designed using a Lyapunov-type analysis, exhibits globally asymptotic position tracking in the presence of the
unknown spacecraft mass. The control law is simulated
for a Lyapunov orbit in the vicinity of the LZ Lagrange
point as the periodic reference trajectory.
1. I n t r o d u c t i o n
Space-based missions near the Lagrange points of the
Sun-Earth system have recently received significant attention for their strategic benefits in long-term astronomical observations. The Lagrange points are defined
as equilibrium positions in the restricted three body
problem (RTBP), see Figure 1 for details. The first
three Lagrange points in the RTBP (labeled as LI, Lz,
and L3) are points that are collinear with the two primary masses. The last two Lagrange points in the
RTBP (labeled as L4 and L5) are equilibrium points
such that each of these points combined with the two
primary masses yields an equilateral triangle. Each of
the five equilibrium positions can host a spacecraft for
an indefinite amount of time. However, in the SunEarth system, the collinear Lagrange points are unstable equilibrium points (i.e., the linearization of the
spacecraft dynamics on any of the callinear Lagrange
points result in a Jacobian matrix that has unstable
eigenvalues) thus necessitating station-keeping control.
The Lagrange points L4 and Ls are locally stable (i.e.,
the linearization of the spacecraft dynamics on either
of these Lagrange points result in a Jacobian matrix
that has two pairs of distinct, complex conjugate, purely
imaginary eigenvalues and two real, negative eigenvalnes). Furthermore, periodic and quasi-periodic orbits
neighboring the collinear Lagrange points can host an
arbitrary number of spacecraft for long time durations
using station-keeping control. For precise details on the
stability of Lagrange points, see [17].
One of the first space mission to utilize the LI Lagrange point was the International Sun-Earth Explorer3 [l],which studied the collection of data on solar wind,
Research supported in part by the National Aeronautics and
Space Administration-Goddard Space Flight Center under Grant
NAG5-11365 and the NASA/New York Space Grant Consortium under Grant 39555-6519.
0-7803-7896-2/03/$17.00
02003 IEEE
cosmic-rays, plasma waves, etc., with negligible disturbances from the Earth. The mission duration lasted less
than four-years and opened the path for future space
exploration to the Sun-Earth Lagrange points. Currently, the Genesis mission resides about the Li Lagrange point. Genesis mission objectives include obtaining precise solar isotopic measurements and obtaining highly sensitive measurements of solar wind, as described in [SI.
Future space missions, such as [Z], that intend to utilize the L2 Lagrange point as the location for deepspace observations and/or interstellar communication
have the advantage that solar influences on the spacecraft are minimal and space observations can be conducted on a frequent basis. In contrast, spacecraft that
are to perform the same types of missions in either Snnsynchronous or Low Earth Orbits about the Earth are
not suitable because these orbits expose the spacecraft
to harsh physical conditions (e.g., gravitational and/or
atmospheric disturbances, space debris, etc.).
Current spacecraft control designs that facilitate the
trajectory tracking of periodic or quasi-periodic orbits
about the collinear Lagrange points are classified as impulsive and continuous methods. The dominant literature in impulsive spacecraft control near the collinear
Lagrange points consists of the Floquet mode approach,
wherein tailoring control inputs cancel instability effects
for a spacecraft orbiting the Lagrange points [5,14,15],
and the target mode approach, wherein impulsive control inputs are generated using the error feedback and
by minimizing a cost function 17-91, However, the Floquet mode approach suffers from large transient errors
that arise from tailoring control input, which “tries” to
cancel instability. Additionally, the resulting control design for the target mode approach does not guarantee
the spacecraft tracking error to be globally stable for
the periodic or quasi-periodic orbit.
Alternatively, a nonlinear control algorithm has been
presented in 141 that utilizes the nonlinear dynamics of
a spacecraft relative to the L2 Lagrange point to track
a quasi-halo orbit. Ref. [4] accounts for the eccentricity of the Sun-Earth system by treating the elliptical
Sun-Earth motion as perturbations from an ideal circular Sun-Earth motion and by canceling these perturbations as a disturbance force. However, uncertainty in
the spacecraft mass has not been accounted for in this
approach.
In this paper, an adaptive control algorithm is presented to enable a spacecraft to track desired trajectories near the L2 Lagrange point in the Sun-Earth sys1116
Proceedings of the American Control Conference
Denver, Colorado June 4-6, 2003
tem. First, in Section 2, the mathematical model for a
spacecraft relative to the Lz Lagrange point in the SunEarth system is described. Next, a trajectory tracking
control problem is formulated in Section 3. In Section
4, using a Lyapunov-based approach, a full state feedback control law and a parameter update algorithm are
designed, which facilitate the tracking of given reference trajectories in the presence of unknown spacecraft
mass. Illustrative simulations are included in Section 5
to demonstrate the efficacy of the proposed controller.
Finally, some concluding remarks are given in Section
6.
2.
System M o d e l
In this section, we present the nonlinear model characterizing the position dynamics of a spacecraft relative
to the Lagrange point Lz in the Sun-Earth system. We
assume that the Earth and the Sun rotate in a circular
orbit about the Sun-Earth system barycenter (center of
mass). In addition, we consider that a n inertial coordinate system { X ,Y,Z ) is attached t o the Sun-Earth
system barycenter. Finally, we assume that a rotating
right-handed coordinate frame {zb, Yb, z b } is attached
to the Sun-Earth system barycenter with the zb-axis
pointing along the direction from the Sun to the Earth,
the zb-axis pointing along the orbital angular moment u m of the Sun-Earth system, and the yb-axis being mutually perpendicular to the x b and Zb axes, and pointing
in the direction that completes the right-handed coordinate frame.
T h e equation of motion of a spacecraft is derived by a
direct application of Newton's second law to the spacecraft, such that the only external forces influencing the
spacecraft motion are the gravitational forces from the
Sun and the Earth, thus yielding
where m is the mass of the spacecraft, a l ( i ) E R3 is
the inertial acceleration vector of the spacccrak, G is
the universal gravitational constant, Ms is the mass of
the Sun, RsLS(t)E R3 is the position vector from the
Sun to the spacecraft, M E is the mass of the Earth, and
RE,.(^) E R3 is the position vector from the Earth to
the spacecraft.
Next, the inertial acceleration of the spacecraft is expressed in the rotating coordinate frame {xb, &> Zb} as
follows (see [19]for details)
al
ab
+ R x (0 x Tb) i- 2R x Wb,
be expressed in the coordinate frame
spectively as
rb = fB+
ab
ub = 5B
yj+ z i 3
R
= d i + y i + ik,
{Zb,yb,zb},
re-
+ y j + ik,
= wk,
(3)
where { Z ( t ) , y(t),z(t)} E W3 are the components of the
position vector of the spacecraft expressed in the coordinate frame {zb, yb, zb} and w is the constant angular
speed of the Sun-Earth system. Note that i , j , and
idenote the unit vectors in the Z b , Y b , and zb directions, respectively. Furthermore, Rs-. and RE-^ can
be expressed in the coordinate frame {zb,yb,Zb}, respectively as
+
+ +
+ + zk,
Rs-. = (5 Rs) B yj
RE+^ = (f - RE)i yj
(4)
(5)
where Rs is the distance from the origin of the inertial
coordinate frame to the Sun and RE is the distance from
the origin of the inertial coordinate frame to the Earth.
Substituting (2) into (l),using (3)-(5), and dividing
by the spacecraft mass m, produces the mathematical
model describing the position of the spacecraft relative
to the Sun-Earth system barycenter expressed in the
rotating coordinate frame {Xb, Yb, Zb} as follows [17]
2 - z W -~ wZz -
esir+Ra)
I1
11
ra(z-R$,
11 RE-~II
-*-e>
ii+2wi-wZy
2
(6)
=
= -1ISfi-llf
IIRs-~II
IIRE-~II
'
where f i s EGMs and PE eGME.
R e m a r k 2.1. The collinear Lagrange points relative
t o the Sun-Earth system barycenter are obtained by setting the y and z position components and all velocity
and acceleration components in (6) to aero. This yields
a set of quintic polynomial equations for f which needs
t o be solved independently t o obtain each collinear Lagrange point (see (171 for further details). The collinear
Lagrange points, as shown in Figure (l),are ordered as
L1, Lz, and LB,where LI is located between the Earth
and the Sun, Lz is located on the far side of the Earth
in the positive Zb direction, and L3 is located on the far
side of the Sun in the negative Z b direction. In addition,
we denote the distance between the Sun-Earth system
barycenter and the Lagrange point Lz as R L ~ .
Next, we perform a translational coordinate transformation of the form f = x R L to
~ transform the
+
k.+RL2
(2)
where we are assuming that the Sun-Earth system is rotating at a constant rate about its barycenter, ab(t) E
R3 is the acceleration vector of the spacecraft relative
to the coordinate frame {zb,yb,zb), fi is the angular
velocity vector of the Sun-Earth system, ~ b ( t )t R3
is the position vector of the spacecraft relative to the
coordinate frame {Zb,yb, Zb}, and ub E R3 is the velocity vector of the spacecraft relative t o the coordinate
frame {Zb, Yb, Zb}. In addition, Tb, ob, ab, and R , can
coordinate frame {Xb, yb, z b }
H
{ X L ~ y, ~ Z L~ ~ },.
Now, (6) can be expressed in the new coordinate frame
{ Z L ~ , Y L ~ , Z Las
~}
2
- ZWY - w Z (Z+ RL>)=- W SI(lZR+sR+ L~+l 1R3s )
y+2wx-w2y
uEiZ+RL2-RE)
'
-*-*
= i1Rs-J
= - PF'
IIRE-~II
IIRE-~II'
'
(7)
vcii-*,
where Rs-. and RE-^ can be expressed in the coordin a t e f r a m e { 2 ~ ~ , Y ~ ~ . z ~ ~=
} ,(x+&
a s R ~ - ~ Rs)z^+
+
1117
Proceedings of the American Control Conference
Denver, Colorado June 45.2003
+
and RE-^ = (z
- R ~ ) i + y j + z R ,respectively.
After pre-multiplying the nonlinear dynamics of (7)
with the spacecraft mass m and inserting the thrusting
control forces for the spacecraft, the nonlinear position
dynamics of the spacecraft relative to the L2 Lagrange
point can be arranged in the following compact form
yj+&
q and q ) of the spacecraft relative to the LZ Lagrange
point are available for feedback.
To facilitate the control development, we assume that
the desired trajectory qd and its first two time derivatives are bounded functions of time. Next, we define
the spacecraft mass estimation error %(t)E R as
fh-m,
in
mq+ Cq f N ( q ) = U ,
where q ( t ) E R3 is defined as q
5
(8)
[z y
ro
=IT,
-1
C is a
01
Coriolis-like matrix defined as C 52mw
N is a nonlinear term consisting of gravitational effects
and inertial forces
where h ( t ) E R is the spacecraft mass estimate.
4.
A d a p t i v e Position Tracking Controller
In this section, we design an adaptive feedback control law that asymptotically tracks a pre-specified spacecraft position trajectory, in the presence of the unknown
constant spacecraft mass m. In order to state the main
result of this section, we define the following notation.
A filter tracking error variable r(t) E R3 is defined as
T
and u(t) E R3 is the thrust control input to the spacecraft. The following remarks further facilitate the subsequent control design and stability analysis.
R e m a r k 2.2. The Coriolis matrix C satisfies the
skew symmetric property of
vz E R3.
(10)
R e m a r k 2.3. The left-hand side of (8) produces an
affine parameterization
m h + CCZ+ N ( q ) = Y(<I,<z,d m ,
(11)
where <j(t) E R3,for j = { l , Z ) , are dummy variables
with components (j,, <jv,and
, m is the unknown,
constant mass of the spacecraft, and Y E R3 is a regression matrix, composed of known functions, defined
as
<,,
Y f [YZ yy
YzlT
1
(12)
f i+cye,
(15)
where cy E R3x3is a constant, diagonal, positivedefinite, control gain matrix, an augmented error variable 9 ( t ) E R6 is defined as
9
A
=
T
eTIT,
(16)
and a positive constant X is defined as
= 0,
ZTCZ
(14)
x f min {Amin
{ K }, Amin { ~ , a ) } ,
(17)
where A,,,;"{.} denotes the minimum eigenvalue of a matrix and K , K p E
are constant, diagonal, positivedefinite matrices. Additionally, we solve for i: in (15) to
produce
e = T - ae.
(18)
Finally, we define a new regression matrix Yd(.) E R3
as
yd(') 5 Y(qd-'&qd-ae,s),
(19)
where (11) has been used with <I = qd -cue and
=
qd - a e , in the definition of (12).
T h e o r e m 4.1. Let K , K , E R3x3be constant, diagonal, positive-definite matrices and E R be a positive
constant. Then, the adaptive control law
r
U
3. Problem F o r m u l a t i o n
In this section, we formulate a control design problem
such that the spacecraft position q tracks a desired position trajectory qd E R3, i.e., lim q ( t ) = qd(t). The eft-m
fectivenessof this control objective is quantified through
the definition of a position tracking error e(t) E R3 as
e
D
q-qd.
= Ydm-Kpe-KT,
&
=l
(13)
The goal is to construct a control algorithm that obtains the aforementioned tracking result in the presence
of the unknown constant spacecraft mass m. We assume that the position and velocity measurements (i.e.,
-rYzT,
(20)
(21)
ensures global asymptotic convergence of the position
and velocity tracking errors as delineated by
lim e(t), i(t) = 0.
t-m
(22)
Proof. We begin by rewriting the spacecraft position
dynamics (8) in terms of the filtered tracking error variable (15). To this end, differentiating (15) with respect
to time, multiplying both sides of the resulting equation
by rn, using e = q-qd from (13), and rearranging terms
yields
m? = -m(qd
cyd) + mq.
(23)
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Proceedings 01 the ~rnencanControl Conference
Denver, Colorado June 4-6, 2003
Since V ( t )is bounded, t 2 0, we conclude that q(t) E
Lz, t 2 0. Finally, using Barbalat's Lemma [3,
161, we conclude that
Substituting for mq from (8) in (23), we obtain
C,
m i = -m(qd - cue) - Cq
N ( q )+ U .
~
(24)
Next, we expand (15) by noting that e = q - q d . Then
solving for e, substituting the result into (24), and rearranging terms yields
lim q ( t )= 0.
t-m
+ u - Cr,
(25)
where the definition of (19) has been used. Eq. (25)
characterizes the open-loop dynamics of T . Now, substituting (20) into (25) results in the following closed-loop
dynamics for r
- Cr,
mi. = Ydm - Kpe - Kr
(26)
5.
where the definition of (14) has been used. Finally,
note that differentiating (14) with respect to time and
using (21), produces the closed-loop dynamics for the
spacecraft mass estimation error
A = -rY,r.
(27)
Now, we utilize the error systems of (26) and (27)
along with the positive-definite, candidate Lyapunov
function defined by
1
V ( t )$-mr'r
2
1
1 -2,
+ -eTKpe
+ -m
2
2r
(28)
t o prove the above stability result for the position and
velocity tracking errors. Specifically, differentiating (28)
witb respect to time yields
V ( t )2rTrn?
+ eTKpd+ -mm.
r
1
-
L
(29)
Now, the substitution of the closed-loop dynamics of
(18) and (26) into (29) results in
2
+%
V ( t ) - r T K r - eTKpae
($1.
+ :&)
,
(30)
where the property of (10) has been used. Finally, substituting (27) into the parenthetical term of (30), we
obtain
V ( t ) = -rTKr - eTKpcue5 - X 1 1 ~ 1 ) ~IO,
(32)
Using the definition of T and 7 in (15) and (16), respectively, the limit statement of (32), and Lemma 1.6 of 131,
yield the result of ( 2 2 ) .
0
Remark 4.1. Similar t o the traditional adaptive
control designs in 1161, the stability result of (22) neither requires nor guarantees the convergence of m(t) to
m. However, Eqs. (21) and (32) with the signal chasing
arguments used in the proof of Theorem 4.1 yield the
convergence of m ( t ) t o some constant value.
m7: = -rn(qd - cue) - C(qd - ae) - N ( q ) - Cr + U ,
= -Ydm
n
(31)
where the definitions of (16) and (17) have been used.
Since V is a non-negative function and V is a negative
semi-definite function, 1' is a non-increasing function.
))
Thus V ( t ) E C, as described by V ( r ( t ) , e ( t ) , % ( t5
V(r(O),e(O),%(O)), t 2 0. Using standard signal chasing arguments, all signals in the closed-loop system can
now be shown to be bounded. Using (18) and (26) along
with the boundedness of all signals in the closed-loop
E Cm. Solving the
system, we now conclude that
differential inequality of (31) results in V ( 0 )- V(c0)2
As;
lls(t)l12dt.
Simulation R e s u l t s
In this section, the efficacy of the adaptive control
methodology is demonstrated for the desired trajectories of the spacecraft that are generated using the
ideal, thrust-free, periodic trajectories around the Lz
Lagrange point in the form of a Lyapunov orbit. A
succinct overview of a numerical algorithm t o generate
the thrust-free periodic trajectories around the L2 Lagrange point is given below. Additional details on the
generation of these periodic trajectories can be found
in [10-13,18l.
One numerical method, 1181, of generating thrustfree periodic orbits around the Lz Lagrange point in
the Sun-Earth system involves finding a proper set of
position and velocity initial conditions to propagate
the spacecraft dynamics of (7). First, the PoincarBLindstedt method is used t o find a high order analytic
approximation to a periodic trajectory in the neighborhood of the Lz Lagrange point. Next the initial conditions, based on the PoincarB-Lindstedt method, are
used as an initial seed in a numerical algorithm to find
a better set of initial conditions leading t o a periodic
trajectory. This numerical algorithm applies a Taylor
series expansion to the spacecraft states with respect to
the initial conditions and time and truncates higher order terms, such that the result is a set oflinear equations
(Lyapunov orbits: 2 equations) for a set of unknown
variables (Lyapunov orbits: 3 variables). Families of orbits can be characterized by fixing one of the unknown
variables so that the result gives an equal number of
equations to unknowns (Lyapunov orbits: 2 equations,
2 unknowns). Solving the aforementioned linear matrix
equation and using the result to update the previous
set of initial conditions provides a new initial condition
guess.
The spacecraft dynamics are then propagated using
the new updated set of initial conditions to verify trajectory periodicity. If the t,rajectory is sufficiently close to
being periodic, then the initial conditions can be used
for further simulation, else the above numerical algorithm is used to solve for a new set of initial conditions.
Since the collinear Lagrange points are inherently unstable, long-term propagation of spacecraft dynamics
1119
Proceedings of the American Control Conference
Denver, Colorado June 4 6 , 2 0 0 3
References
using the initial conditions obtained in the above manner is futile. However, we can artificially obtain close
to a periodic orbit by computing trajectory information
during half of a period and reusing this trajectory data
throughout other simulations.
Lyapunov orbits are classified as periodic trajectories
set in the orbital plane { z ~ ~ , y(i.e.,
~ . ~ } = 0). An
initial seed for the numerical algorithm of [IS] consists
of a spacecraft starting on the X L % axis with a nonzero
initial y~~ velocity (i.e., q d ( 0 ) = [zd(O) 0 0IT and
& ( O ) = [0 yd(0) OlT). Updates to the initial y~~ velocity contributes to finding a closed periodic trajectory
in the orbital plane. In addition, the initial I L ~position
determines the size of the Lyapunov orbit.
Applying the nunierical algorithm presented in [IS]
results in a set of initial conditions for a desired
Lyapunov orbit given as qd(0) = [-1.501026450400631
n 01 x 103km, qd(o) = [o 2.692047849889458 01 x
loay
p, I n addition, we initialize the spacecraft
with the set of initial conditions given as q(0) =
[-1.9513343855208~0 n.~5113n79351zoi893 01 x 103km ,
q(0) = 1-0.8076143549668373 3.499662204856295 01 x
[11 http://nssdc.gsfc.nasa.gov/space/isee.html.
(21 http://www.stsci.edu/ngst/.
131 D. M. Dawson, J. Hu, and T. C. Burg, Nonlinear
Control of Electvic Machinery. New York, NY: Marcel
Dekker, 1998.
(41 P. Di Giarnberardino and S. Monaco, "Nonlinear Regulation in Halo Orbits Control Design," Proc. COG, pp.
536-541, 1992.
151 G. Gomez, K. C. Howell, J. Masdemont, and C. Simo,
"Station-Keeping Strategies for Translunar Libration
Point Orbits,'' Adv. in the Astr. Sciences, AAS 98-168:
pp. 949-967, 1998.
B.T. Barden, R. S . Wilson, M. W. Lo,
"Trajectory Design Using a Dynamical Systems Approach with Application to Genesis," A A S / A I A A Astr.
Spec. Conf., AAS Paper 97-709, 1997.
K. C. Howell,
K. C. Howell and S . C. Gordon, "Orbit Determination
Error Analysis and a Station Keeping Strategy for Sun
Earth LI Libration Point Orbits," J . Astr. Sciences,
42(2): pp. 207-228, April-June, 1994.
K . C. Howell and H. J. Pernicka, "Station-Keeping
Method For Libration Point Trajectories" J . Guid. and
Contr., 16(1): pp. 151-159, 1993.
l o ay
p.
The adaptive control law of (20) and (21) has been
simulated for the nonlinear dynamics of (8) such that
t h e spacecraft outputs are q(t) and q(t),with the SunE a r t h system circular orhit parameters [lS, 191: G =
6.671 x in-11&,
M S = 1.9891 x
ME =
T. M. Keeter, Station-Keeping Strategies For Libration Point Orbits: Target Point and Floguet Mode Approaches, Master's thesis, School of Aeronautics and
Astronautics, Purdue University, West Lafayette, Indiana, 1994.
M. Kim, Periodic Spacecraft Orbits for Future Spacebased Deep Space Observations, diplomarbeit, Technische Universitat Wien, Austria, 2001.
5.974 x 1024kg,
1 AU = 1.496 x 108km,
RLZ =
1.010033599267463 AU, where 1AU stands for 1 astronomical unit denoting the distance between the Sun
and the Earth and we consider t h a t t h e spacecraft has
a mass of m = 1000kg. In addition, the distances R s
and RE can be computed as Rs =
x 1AU and
RE = & x l A U .
The control and adaptation gains, in the control law
of (20) and (21), are obtained through trial and error
in order to obtain good performance for the tracking
error response. The following resulting gains were used
in this simulation K p = diag (1,1,1) x 7.49 x lo@, K =
diag(1, i , l ) x 5 . 4 7 x IO3, a = d i a g ( l , l , l ) x 8 . 4 9 x lo-',
and = 8.89 x lo6. In addition, the spacecraft mass
parameter estimate was initialized to IjL(0) = 900 kg.
The actual trajectory q is shown in Figure 2. Figures
3 and 4 show the tracking error e and velocity tracking
error e , respectively. The control input U is shown in
Figure 5. Finally, the spacecraft mass estimate 61is
shown in Figure 6.
D. L. Richardson, "Periodic Orbits About the LI and
LZ Collinear Points in the Circular-Restricted Three
Body Problem," Techntcal Report CSC/TR-78/6002,
&
Computer Sciences Corporation, 1970.
D. L. Richardson, "Analytic Construction of Periodic
Orbits About the Collinear Points," Celestial Mechanics, 22: pp. 241-253, 1980.
D. L. Richardson, "Halo Orbit Formulation For the
[SEE-3 Mission," J . Guid. and Contr., 3(6): pp. 543548, 1980.
C. Simo et al., "Station-Keeping of a Quasi-Periodic
Halo Orbit using Invariant Manifolds," Proc. of Second 1nt. Sym. on Spacecraft Flight Dynamics, E u r e
pean Space Agency, Darmstadt, Germany, 1986.
r
6.
C. Simo et al., "On the Optimal Station-Keeping Control of Halo Orbits," Acta Astronautica, 15(6): pp. 391397, 1987.
J L . E. Slotine and W. Li, Applied Nonlinear Control.
Englewood Cliffs, NJ: Prentice-Hall, 1991.
Conclusion
V. Szebehely, Theory of Orbits. New York, NY: Academic Press, 1967.
In this paper, we designed an adaptive control algorithm for the position dynamics of a spacecraft to enable it to perform trajectory tracking relative to the L,
Lagrange point in the Sun-Earth system. A Lyapunovtype design was used to construct a full state feedback
control law and parameter estimates which facilitate the
tracking of periodic reference trajectories with global asymptotic convergence. Simulation results were given t o
illustrate the efficacy of t h e control design.
R. Thurman and P. A . Worfolk, The GeornetnJ of Halo
Orbits i n the Circular Restricted Three-Body Problem,
Geometry Center Research Report GCG95, University
of Minnesota, 1996.
D.A. Vallado, Fundamentals of Astrodynamics and Applications. McGraw-Hill, 1997.
1120
Proceedings of the American Control Conference
Denver, Colorado June 4-6, 2003
I ,..
........................
. ..:
.~..
6,
4
............
Figure 1: Sun-Earth system schematic diagram
i
-10.
/?-?I
.-I
,..:U
1
. . .
...........
.
.
...
....
,.
..;
:..
.-
.-.-
E"
:
..,;
. . . .. . . ... . . . .. . . .. . . . . . . . . . . . . .
.
~ - .
,,.W'
....
: . . : . .
.
.
.
--
.
. . . .
.
.I_
.,-
.
.
.
.................
.
._ *
.m
,-
3-
j , ...........
--
o
I
, ... ..:
m
m
. . . . . . . . . . . .
a
a
_
*
.
.
*II
Figure 2: Trajectory of the spacecraft relative to the
1 ;
I*.
LZ Lagrange point
.......................
to
. . . . . . . . .
.I*
I
l
:
m
.I.
....
z
i....
Q
a
m
. . !I
. . .
....:
....
..:. . . . . .
m
j.
w
--I
Figure 5: Control input t o spacecraft
...
Figure 3: Position tracking error
..
..
. :
%
I
r
.
.
n
.
.
.
m
i.
.
m
M
~
%C**
Figure 6: Mass parameter estimate of spacecraft
1121
Proceedings 01 the American Control Conference
Denver, Colorado June 4-6.2003
~