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CHAPTER 3
SATELLITES IN FORMATION FLYING
3.1 INTRODUCTION
The concept of formation flight of satellites is different from that of
a satellite constellation. As defined by the NASA Goddard Space Flight
Center, a constellation is composed of “two or more spacecraft in similar
orbits with no active control to maintain a relative position”. Station keeping
and orbit maintenance are performed based on geocentric states, so groups of
global positioning system (GPS) satellites or communication satellites are
considered constellations. In contrast, “formation flight involves the use of an
active control scheme to maintain the relative positions of the spacecraft”.
The difference lies in the active control of the relative states of the formation
flying spacecraft. A distinction must also be made between formation keeping
(referred to here as formation flying) and formation changes. Formation
keeping is the act of maintaining a relative position between spacecraft in the
presence of disturbances, while formation changing modifies the formation
type, changing the relative satellite dynamics. Using a number of small
satellites to fly in a formation than to use a single large satellite is the recent
trend in space technology. During experiment and analysis with
magnetosphere or aurora, high performance satellite is needed which may
require large aperture area in its antenna to provide a required coverage area.
In spite of using such a high performance satellite, many satellites having
small aperture area can be used to fly in circular formation to produce such a
large aperture area and this technology is called synthetic aperture radar
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(SAR). By using small satellites in formation flying better results can be
produced than single high performance satellite.
As satellites used in formation flying have more or less similar
configuration, they can be produced in bulk, thus reducing the manufacturing
cost of the satellites. Also there is no need for a greater industrial setup for
making these small satellites as these can be easily produced with university
collaboration.
As many satellites are used instead of a single large satellite, the
satellites can be configured according to the need i.e. circular formation or
hexagonal formation, etc., and there is greater chance for mission success and
flexibility.
In a single satellite when any instrument fails then total mission will
become a failure. But in satellite formation flying the probability of failure
occurring in all the satellites is very low. Even when there is failure in one
satellite, other satellites can be configured in such a formation to satisfy the
requirement.
A brief introduction and the types of formation flying and station
keeping are described in this chapter. Solution to the Euler Hill’s equation and
initial conditions for various relative orbits are discussed. The conversion
from relative position to orbital elements and the method for finding the
relative distance and velocity between chief and deputy satellites are shown.
The various perturbation forces affecting the relative motion are analyzed and
their simulations by different approaches are discussed in this chapter.
The absolute distance keeping between the satellites achieved by a
fuzzy logic based and GA based orbit controller is described. GA based orbit
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controller uses orbital element feedback method in which the gains of the
method are optimized using genetic algorithm.
3.2 TYPES OF SATELLITE FORMATION FLIGHT
There are different types of formation flying which are classified
according to the following category:
a) Based on orbits in which they are flown.
b) Presence of control.
Based on the presence of control they are classified as given below:
3.2.1 Independent Satellite Formations
In this type of formations, during the time of launch, the satellites
are placed at their positions for formation and after that there will be no
control present to maintain their positions. So whenever the orbits of the
constellation are affected by perturbation, the satellites will not maintain their
relative positions.
3.2.2 Master Slave Formation
In this type of formation, there will be one master and other
satellites will not have any control in them. In this, the master satellite will
control the relative position of all other satellite. There will be a need for
communication system in satellites to receive the control action from the
master satellite and also to send the signal to the slave satellites. Master
satellite will receive knowledge about slave satellite by inter-satellite
communication.
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3.2.3 Leader Follower Formation
In this type of formation, some control will be given to the slave
satellites and the satellite that is in the front will make the satellites to follow
the correct orbit. The satellites having some control to maintain their relative
position with respect to their leader satellite are called follower satellites.
3.2.4 Peer-Peer formation
In this type of formation, control will be distributed equally between
all satellites. Whenever there is a failure in any one satellite only that satellite
will fail, whereas the other satellites will remain in formation. Satellites will
have control for maintaining their position and also their relative position,
based on the orbital position in which they are placed
3.2.5 In-Plane Formation
In this type of formation, the orbit of the satellites will be same but
the satellites will differ only in their position i.e. both will be in same plane.
3.2.6 Out of Plane Formation
In this type of formation, the satellites will be in different orbits and
their right ascension and inclination may be different.
3.2.7 Circular Formation
In this type of formation, the satellites are placed in positions such
that they form a circle.
3.2.8 Projected Circular Formation
In this type of formation, the satellites are placed in positions such
that they form a projected circle like ellipse.
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3.3 STATION KEEPING
In the formation flying of satellites, it is very important to maintain
the relative position of satellites and also it is required to make them to follow
a correct orbit. This task of making the satellites to maintain their position
with respect to other satellites is called station keeping. There are two types of
station keeping namely:
Absolute station keeping
Relative station keeping
3.3.1 Absolute Station Keeping
In this method, the absolute positions of the satellites in formation
flying are maintained thus relative positions also will be maintained. In this
method, the frequency of thruster used is high such that the fuel consumption
will be high. Also in this method, the position of the satellites will be known
in advance and also can be easily monitored from the ground.
3.3.2 Relative Station Keeping
In this method, the thruster will be fired when the satellites move
out of their relative positions. As some satellites will experience same amount
of disturbance they will deviate from their orbit although their relative
position is maintained. At those times even the fuel consumption is reduced
and satellites will deviate from their orbit in course of time.
The satellites, which are in formation flying, have to know their
position in space and also their relative position with respect to other
satellites. This is done to control their position, as there is a chance for
collision. Usually the satellites will be controlled only from ground station by
using antennas. The position of satellites at any particular time is found by
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knowing azimuth and elevation angle of the antenna. This information is then
propagated to know the position of satellite after one week. Using this data,
control signals for firing the thrusters for appropriate number of times, at
appropriate positions are sent to the satellites and satellites will fire the
thrusters accordingly. In a satellite, which is not in formation with any other
satellites, orbital correction will not be done frequently and the satellite’s
earth viewing angle will be corrected to get the required ground track and to
point at the particular area. In case of geostationary satellites orbital
correction will be done only when there is huge error in its orbital motion. But
in the formation flying, the satellites have to control their position at correct
time, so as to maintain the formation; otherwise the satellites will deviate
from the formation. So there is a need for an onboard orbit controller, which
will take care of orbital corrections needed for the satellite to maintain it in
formation. This onboard orbit controller will replace human in the loop and
thus human workload will be reduced. This increases precision of the
position.
In the formation flying of satellites, in order to have the satellites
sense their position in the space, Global Positioning System (GPS) can be
used. Relative distance between the satellites can be known from the inter
satellite communication. GPS will give the position in latitude, longitude and
altitude from which satellite position in earth inertial frame can be obtained.
This information will be communicated between satellites and deviations
from the actual position and velocity can be used to compute the orbital
elements and firing of the thrusters can correct the deviations at appropriate
time. The satellite will hence be positioned using the control algorithm.
In the present work, the initial positions and velocities of the
satellites are determined using the Euler-Hill’s equations for three types of
formation flying configurations viz. leader follower, projected circular orbit
44
configuration and general circular orbit configuration. A method for
converting orbital elements to relative distance of satellites and the relative
distance to orbital elements of the satellites is also given. The variation of
relative distance for the satellites with initial conditions determined by using
Euler-Hill’s equations, due to eccentricity of the master orbit is simulated.
The variations in the orbital elements of the satellites due to various
perturbing forces are found out by using Gaussian planetary equation, general
perturbation effect equation and by using the Newton’s gravitational law. The
various perturbing forces considered are atmospheric drag, earth’s oblateness,
earth’s triaxiality and solar flux. The amount of variation in the orbital
elements are found and thus used for determining the amount of thrust needed
to correct the orbital elements. Genetic algorithm is used to determine the
optimized gains for the proposed orbital element feedback controller to reduce
the deviation in orbital elements and thus to maintain the relative distance. A
fuzzy controller using 2 orbital elements for controlling the orbit of the
satellite and hence distance between satellites is also designed.
3.4 SOLUTION OF EULER-HILL’S EQUATION
The solution of the Euler Hill’s equation given in equation (2.19)
can be given in time as,
sin
nxcos
ny2x3
nyx22)t(x 00
00
0
sin
ny2x32cos
nx2
nyx23
nx2y)t(y 0
000
00
0
sinnz
cosz)t(z 00
Taking Laplace transform of equation (2.19) and sorting gives
equation for x(s), y(s) and z(s). Inverse Laplace of x(s), y(s) and z(s) yields
(3.1)
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equation (3.1). Here is a product of mean motion and time. The Euler-Hill’s
equations admit bounded periodic solutions, which are suitable for formation
flying missions. The solutions to Euler-Hill’s solutions are as follows:
x = x0 cos + y0 sin / 2
y = y0 cos - 2x0 sin
z = z0 cos + 0z sin / n
2/cosnysinnxx 00
y = -ny0 sin - 2nx0 cos
coszsinnzz 00
The above equations on reduction give following solutions,
)ntsin(2cx 0
1
y = c1 cos (nt + 0) + c3
z = c2 sin (nt + 0)
)nt(cosn2cx 0
1
)ntsin(ncy 01
)nt(cosncz 02
where 21.2
0201 4
yx2c
(3.4)
21.
202
02 nzzc
(3.5)
(3.2)
(3.3)
46
2/y
xTan0
00 (3.6)
0
00 z
nzTan
(3.7)
The above constants are determined by initial conditions of the
satellite position i.e. based on the relative distance between the satellites. The
solutions are obtained by satisfying the following constraint on initial
conditions.
0nx2y 00 (3.8)
The initial conditions, which satisfy the above equation, are referred
to as Hill-Clohessy-Wiltshire initial conditions. The initial conditions (Vadali
et al 1999) are obtained by substituting t=0.
01
0 sin2cx
y0 = c1 cos 0 + c3
z0 = c2 sin 0
01
0 cosn2cx
010 sinncy
020 cosncz
3.5 INITIAL CONDITIONS FOR VARIOUS RELATIVE ORBITS
Bounded relative orbits of various shapes and sizes can be obtained
by choosing arbitrary values for c1, c2, c3, α0, β0.
(3.9)
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3.5.1 Projected Circular Orbit
The relative orbit that is obtained by choosing c1=c2=, c3=0 and
α0= β0 is known as Projected Circular Orbit (PCO). The formation is called so
because when the motion of slave satellite around the chief satellite is
projected on the local horizontal plane (y-z plane), the relative orbit is
circular.
)nt(sin2
x 0
)nt(cosy 0
)nt(sinz 0
From the above equations,
222 zy (3.11)
This represents a circle in the y-z plane with radius . α0
characterizes the position of the deputies along the circumference of the circle
and is a measure of the size of the formation.
3.5.2 General Circular Orbit
There exists another circular orbit of interest for the choice of c1 =,
c2= (√3/2) , c3 = 0 and α0= β0. This results in a circle in three-dimensional
space.
)nt(sin2
x 0
)nt(cosy 0
)nt(sin23z 0
(3.10)
(3.12)
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Equation (3.12) results in
2222 zyx (3.13)
This is an equation of the circle in three-dimensional space. This
relative orbit is known as the General Circular Orbit (GCO). The GCO is used
to simulate a large satellite with a circular geometry.
3.5.3 Leader Follower Formation
Choosing c1= c2=0 and c3=d yields x=0, y=d and z=0, results in
constant along-track separation. This is known as the leader follower
configuration because, the deputies either lead or follow the chief by a
constant distance in the along track direction.
The initial conditions corresponding to the projected circular orbit,
general circular orbit and the leader follower configuration will be referred to
as PCO initial conditions, GCO initial conditions and the LFC initial
conditions, respectively. It should be noted that PCO initial conditions, GCO
initial conditions and the LFC initial conditions are three particular cases of
Hill-Clohessy-Wiltshire solutions.
Of the three formations flying configurations having both the chief
and deputy satellites in same inclination and right ascension can form the
leader follower configuration. But the projected circular and general circular
orbits can be formed only by having the chief and deputy satellites in different
right ascension and (or) inclination.
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3.6 CONVERSION FROM RELATIVE POSITION TO ORBITAL
ELEMENTS
For finding the initial positions of the deputy satellites with respect
to the chief satellite, the position of the chief satellite in orbital frame in terms
of orbital elements are assumed. Then they are converted to the position in
orbital frame in terms of Cartesian coordinates. From the solution of Hill’s
equations and according to our requirements like size of the formation and
type of formation, the initial conditions (initial position and velocity) of the
deputy satellite with respect to chief satellite can be found out (in chief
satellite’s reference frame). From the conditions in chief satellite’s frame of
reference, they can be converted to the orbital elements corresponding to the
deputy’s position and velocity. Thus the deputy satellite’s position is obtained
in terms of orbital elements.
For getting the orbital elements of the deputy satellite from its
relative position with respect to chief satellite, first the deputy satellite’s
orbital frame has to be found out, mainly its inclination and its right ascension
in particular. In most cases this can be found out by assuming difference in
inclination between chief and deputy as zero i.e. deputy has the same
inclination as that of the chief satellite. The governing equation for finding the
right ascension is given by,
c
c
c
zyx
)icos()cos()sin()isin()cos()icos()sin()cos()isin()sin(
)isin(0)icos(
zyx
(3.14)
In the matrix equation (3.14), xc, yc, zc represents the deputy’s
position in the chief’s orbital frame. This equation is obtained from the basic
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equation for converting variables from one frame to another. From the above
matrix, the value of z is equal to zero, which leads to following equation,
0z)icos()cos(y)isin(x)isin()(cos ccc (3.15)
But since
0i (3.16)
Equation (3.15) becomes,
c
c
zytanarc (3.17)
From the above equation, the difference in right ascension between
the chief and deputy satellite can be found out.
It is known that deputy satellite’s position is expressed in chief
satellite’s coordinate frame of reference, which is rotating with the chief
satellite. So, at first the deputy satellite’s position is converted to position in
its own orbital frame in Cartesian coordinates. This can be done from the
anomaly angle of the chief satellite in orbital frame. The direction cosine
matrix can be found, which is used for finding the deputy’s position in chief’s
coordinate frame.
The direction cosine matrix is,
1000cossin0sincos
DCMcd (3.18)
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The position and velocity of the deputy in its orbital frame is given
by,
c
1cd
zyx
)DCM(zyx
c
1cd
zyx
)DCM(zyx
By knowing the right ascension, inclination and assuming the
argument of perigee as zero (as it is a circular orbit), the position and velocity
of deputy satellite in earth centered inertial frame of reference can be found
out in Cartesian coordinates.
1000cossin0sincos
)(A Z (3.20)
icosisin0isinicos0
001)i(A x (3.21)
1000cossin0sincos
)(A Z (3.22)
(3.19)
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zyx
)(A)i(A)(AZYX
1z
1x
1z
zyx
)(A)i(A)(AZYX
1z
1x
1z
From the position and velocity in earth-centered inertial frame in
cartesian coordinates, the orbital elements of the satellite can be found out.
3.7 CONVERSION FROM ORBITAL ELEMENTS TO THE
POSITION AND VELOCITY IN CHIEF SATELLITE’S
COORDINATE FRAME OF REFERENCE
The orbital elements of both chief and deputy satellite are converted
to their position and velocity in their orbital frame in Cartesian coordinates.
The deputy satellite’s position and velocity are converted to the chief’s orbital
frame by rotating by an angle of difference in right ascension and the
inclination. The deputy satellite’s position and velocity are further rotated to
coincide with the direction of revolution of chief satellite (i.e. to convert to
chief’s coordinate frame).
The orbital elements can be converted to their position and velocity
in their orbital frame in Cartesian coordinates by set of equations (3.24).
(3.23)
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aecosax
2e1sinay
z = 0
sin
yxa
ax
22
2
3
cose1
yxa
ay 2
22
2
3
0z
The position and velocity of the deputy in its orbital frame is
converted to chief’s orbital frame by the following conversions.
For cd (3.25)
If 0 (3.26)
Then
cos0sin010
sin0cos)(A y (3.27)
If 0 (3.28)
Then
cos0sin010
sin0cos)(A y (3.29)
For cd iii (3.30)
If 0i (3.31)
Then
icosisin0isinicos0
001)i(A x (3.32)
If 0i (3.33)
(3.24)
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Then
icosisin0isinicos0
001)i(A x (3.34)
d
yx
c zyx
)(A)i(Azyx
d
yx
c zyx
)(A)i(Azyx
Thus equation (3.35) gives the position and velocity of deputy
satellite in chief’s orbital frame. The deputy’s position and velocity in chief’s
coordinate frame can be found out by rotating the deputy’s position and
velocity in chief’s orbital frame in the direction of velocity vector of the chief
satellite. This is done by the following equations.
1000cossin0sincos
DCMcd (3.36)
d
cd
zyx
)DCM(zyx
d
cd
zyx
)DCM(zyx
Thus the orbital elements of the deputy satellite are converted to the
position and velocity of the deputy in chief’s coordinate frame. The relative
distance and velocity vectors can be obtained by subtracting deputy’s position
and velocity from chief’s position and velocity vector.
(3.35)
(3.37)
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3.8 RELATIVE MOTION SIMULATION
3.8.1 Introduction
In this, the initial conditions are determined for leader follower,
projected circular and general circular configuration of two satellites. The
distance of separation is considered to be 20 km. The initial conditions
(relative distance) in x, y and z-axis are given below.
3.8.2 Initial Conditions of Various Configurations
The initial conditions of various configurations are given below:
Leader Follower Configuration
c1 = c2 = 0, c3 = 20
n = 0.0594 deg/sec
x0 = 0 km
y0 = 20 km
z0 = 0 km
0x = 0 km/s
0y = 0 km/s
0z = 0 km/s
Projected Circular Orbit (PCO) Configuration
c1 = c2 = 20, c3 = 0, = = 90
n = 0.0594 deg/sec
x0 = 10 km
y0 = 0 km
z0 = 20 km
0x = 0 km/s
0y = -0.0207 km/s
0z = 0 km/s
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General Circular Orbit (GCO) Configuration
c1 = 20, c2 = 310 , c3 = 0, = = 90
n = 0.0594 deg/sec
x0 = 10 km
y0 = 0 km
z0 = 17.3205 km
0x = 0 km/s
0y = -0.0207 km/s
0z = 0 km/s
The above initial conditions for all the three configurations are used
in the solution of Euler-Hill’s equations to know the relative distance
behavior of two satellites at all times.
3.8.3 Simulation of Various Configurations for Given Initial
Conditions
The simulations of various configurations for the given initial
conditions are given below:
The relative distance in leader follower configuration is shown in
Figure 3.1. Figure 3.2 shows the motion of the deputy satellite around master
in orbit in PCO configuration. Orbital motion is projected in y-z plane.
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Figure 3.1 Relative distance in leader follower configuration
Figure 3.2 Motion of the deputy satellite around master in orbit
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The relative distance between satellites in projected circular orbit
configuration and general circular orbit configuration are given in Figures 3.3
and 3.5 respectively. Figure 3.4 shows the motion of the deputy satellite
around master during orbit in GCO configuration.
Figure 3.3 Relative distances between satellites
Figure 3.4 Motion of the deputy satellite around master during orbit
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Figure 3.5 Relative distances between satellites in general circular orbit
configuration
From the above plots, it can be seen that the relative distance is
maintained for all three configurations, given the initial conditions of the
deputy satellite in all three configurations.
3.8.4 Simulation for Elliptic Chief Orbit
The solutions for Euler-Hill’s equations and the initial conditions of
the deputy satellite are found with the assumption that the chief satellite is in
the circular orbit. The above found solutions will not be satisfied, when the
orbit of the chief satellite has an eccentricity of 0.002 (Vadali 2002).
For the orbit with eccentricity, the position of the deputy satellite
with respect to chief, in chief satellite’s coordinate frame (Inalhan et al 2002)
is given by equations (3.38):
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j32
j22j2j1j d
)cose1(ed
cos)](Hed2ed[sin)(x
j3
j3j2
j4j1j d
)cose1(d
sin)(eHd2cose1
dd)(y
)](Hed2ed[cos 2j2j1
)cose1(
dcos
cose1d
sin)(z j6j5j
where 25
23 )e1(d
)cose1(cos)(H
0
E
2 dEcosEsin2eEsin)e1(
2eE3 (3.39)
cose1
coseEcos (3.40)
where E is the eccentric anomaly and θ is true anomaly.
The values of dij are given by following equation,
j4441
3332
2322
11
j4
3
2
1
)0(y)0(y)0(x)0(x
p00p0pp00pp0000p
dddd
(3.41)
(3.38)
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The values of matrix elements are,
P11 = 1/e,
P22 = (2+e) (1+e) 2/e2,
P23 = (1+e3)/e2
P32 = -[2(1+e)/e]
P33 = -[(1+e)/e]
P41 = -[(1+e) 2/e]
P44 = (1+e)
d5j = (1 + e) z (0)
d6j = z(0)
Substituting the above found values for an eccentricity of 0.002, the
position of the deputy satellite has a variation with respect to that of the chief
satellite, as given in Figure 3.6.
Figure 3.6 Motion of the deputy satellite around master in orbit
The relative distance between the satellites has variation in
Projected Circular Orbit (PCO) configuration as shown in Figure 3.7.
(3.42)
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Figure 3.7 Relative distances between satellites in elliptic chief orbit
3.9 PERTURBATION FORCES AFFECTING THE MOTION OF
SATELLITE
3.9.1 Atmospheric Drag
For the satellites, which are above thousand kilometers from the
earth surface, the density of atmosphere is very low. The effect of atmosphere
on the satellite i.e. the drag created by the atmosphere will be insignificant
and can be ignored. The important effect of drag on the satellite is to reduce
its altitude. The atmospheric drag is given by (Daniel et al 1996),
Fdrag = (1/2)ρcdv2A/m (3.43)
ρ - Atmospheric Density (kg/m3)
cd - Coefficient Of drag (2.2)
v - Velocity of satellite (√GM/a km/s)
m - Mass of the satellite (kg)
a - Position of satellite from earth’s centre
A - Projected area perpendicular to velocity vector (m2)
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3.9.2 Earth Asphericity
The earth is not spherical as it is assumed and its poles are flattened
and bulged in the equator. Due to this the gravitational potential exerted by
the earth is not uniform throughout its surface and is varying according to its
mass distribution. The earth’s gravitational potential is given by (Bong and
Carlos 2001),
U(r,λ,Φ)=µ/r[1+n∑(2-∞)m∑(0-n)(R/r)nPnmsinΦ(Cnmcosmλ+Snmsinmλ)]
(3.44)
where R - Radius of earth
r - Altitude of the satellite from the centre of earth
λ - Longitude
Φ - Latitude
The asphericity of the earth is classified into two types namely
oblateness and triaxiality of the earth. The oblateness of the earth is due to the
fact that the equator of the earth is not circular and it is elongated in one side.
The triaxiality of the earth is due to the effect of earth’s gravitational
potential. The gravitational potential can be found by earth’s radius, the
latitude and longitude.
3.9.3 Solar Flux
The force will be created due to solar flux by the impingement of
photons emitted by the sun on the surface of satellite.
Fsolar = KPAs/m (3.45)
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where K - Dimensionless constant between 1 and 2 (K=1: surface
perfectly absorbent, K=2: surface reflects all light)
P - Momentum flux from the sun (Nm-2s-1)
As - Surface area of satellite normal to solar flux (m2)
m - Mass of the satellite (kg)
3.9.4 Solar and Lunar Gravity
In addition to the gravity exerted by the earth, there will be
perturbation due to the gravitational force exerted by sun and moon. These
effects are called third body perturbations (Daniel et al 1996). The
acceleration due to third body is given by,
ad = (µd /rds3 ) (rs + f (q) rd) (3.46)
where µd - Gravitational constant due to third body
rds - Distance between satellite and sun (Km)
rs - Distance between earth and satellite (Km)
rd - Distance between earth and third body (Km)
q = rs( rs - 2 rd)/ rd2
(3.47)
f (q) = q((3+3q+q2)/(1+(1+q)3/2)) (3.48)
3.10 DETERMINATION OF EFFECTS DUE TO
PERTURBATION FORCES
In this work, the perturbations due to atmospheric drag, earth’s
asphericity and the solar flux are considered and the effect due to each force
on the orbital elements of the satellite are calculated using the following
methods.
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3.10.1 Gaussian Planetary Equation
The Gaussian planetary equation is a first order differential
equation, which gives the variation of all the orbital elements of the satellite
due to the forces are acting on the satellite in the coordinates of orbital frame.
The Gaussian planetary equation is given in the matrix equation (3.49) (Sidi
1997).
h
r
22
u
u
u
0ahe
sin)rp(bahe
)re2cosp(bisinh
icos)sin(rhe
sin)rp(hecosp
isinh)sin(r00
h)sin(r00
0h
recos)rp(h
sinp
0hr
pa2hsinea2
M
i
e
a
(3.49)
where a - Semi major axis b - Semi minor axis
e - Eccentricity of the orbit
i - Inclination of the orbit Ω - Right ascension of the orbit
ω - Argument of perigee
M - Mean anomaly h2 = a(1+e) µ (3.50)
µ = GM=398600.4405 (3.51)
p = h2/ µ (3.52)
r = P/(1+ecosθ) (3.53)
ur - Force acting on the r axis of orbital frame uθ - Force acting on the θ axis of orbital frame
uh - Force acting on the h axis of orbital frame
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Forces due to atmospheric drag are:
ur = (1/2) )ρcdVx2A/M (3.54)
Vx = Ve sinθ (3.55)
uθ = (1/2) )ρcdVθ2A/M (3.56)
Vθ = V (1+ecosθ) (3.57)
uh = 0 (3.58)
Forces due to oblateness of earth are:
ur = - (3µJ2R2/2r4)(1- 3sin2i sin2(ω+θ)) (3.59)
uθ = - (3µJ2R2/2r4)( sin2i sin2(ω+θ)) (3.60)
uh = - (3µJ2R2/2r4)( sin2i sin(ω+θ)) (3.61)
where J2 - 1082.63 10-6
Forces due to triaxiality of earth (Bong and Carlos 2001) are:
ur = - (9µR2/r4)( C22cos2λ + S22 sin2 λ)cos2 Φ (3.62)
uθ = - (6µR2/r4)( C22sin2λ + S22 cos2 λ ) cos Φ (3.63)
uh = - (6µR2/r4)( C22cos2λ + S22 sin2 λ) cosΦ sinΦ (3.64)
where C22 - 1.57432 10-6
S22 - -0.903593 10-6
λ - Longitude at the particular time
Φ - Latitude at the particular time.
The latitudes and longitudes are found out by first converting the
orbital elements into Cartesian coordinates in ECI reference frame. Then they
are converted from the Cartesian coordinates into spherical coordinates,
which gives the latitude and longitude at a particular time.
67
Solar flux is given by,
Fx = Fsin θ (3.65)
Fy = F cos θ (3.66)
Fz = 0 (3.67)
Force, F = KPAs (3.68)
where K - dimensionless constant between 1 and 2 (K=l: surface
perfectly absorbent, K=2: surface reflects all light)
(Taken as 1.5)
P - Momentum flux from the sun (4.4 10-6 kg/m/s2)
As - Surface area of satellite normal to solar flux
(48 10-8 km2)
3.10.2 General Equations of Orbital Elements Variation
The following set of equations (3.69) (Sidi 1997) gives the variation
of orbital elements due to all the perturbations.
2
D icosne1aAca
0e
0i
222
)e1(2icosnJ3
22
22
)e1(4)1icos5(nJ3
3
222
)e1(2))sinisin31(J31(nM
(3.69)
68
3.10.3 Gravitational Force Equation
The variation of the orbital elements due to various perturbing
forces can be found out from the basic force equation of Newton’s universal
law of gravitation and algebraically adding it with the perturbing forces,
which affect the satellite motion. This is also called as Cowell’s method.
m
V|V|c21r
|r|GMra D
3
(3.70)
In the present work, the effect of atmospheric drag perturbation is
only calculated by using the equation (3.70). By numerically integrating the
above equation, the velocity vector can be got, which on further integration
yields the position vector at all time. The position and velocity vector
obtained in the earth-centered inertial frame will be converted to the orbital
elements, from which the deviation in the orbital elements can be found out.
3.11 CONTROLLER DESIGN
In the formation flying of satellites, control is needed as the
satellites orbital parameters are varied by various disturbances, which alters
the position and velocity of the satellites. The effect of various disturbances
on the position and velocity of the satellite varies with the corresponding
variation in the shape and size of the orbit, in which satellite is orbiting. For
making the satellites in the formation to maintain their relative distance,
control is applied to fire thrusters in appropriate direction and correction is
done on orbital elements.
The controllers can be made to control the position and the velocity
of the satellites directly or can be used to control the orbital elements. There
are different types of controllers as continuous controller or impulsive type
69
controller. In the continuous type controller, thruster will be used
continuously to correct the small changes in the position. In this type of
control, the accuracy obtained will be more with reduced lifetime of mission
for the same quantity of fuel as in impulsive type control. In impulsive type
controller the thruster is used for orbital corrections only at particular times.
Thus the accuracy of the relative distance maintenance will be low, but the
lifetime of the mission will be more for impulsive type control. In the present
work, the control of orbital elements of the satellites is by Gaussian planetary
equation. The Gaussian planetary equation, which gives the dynamics of
spacecraft in orbit, is given in equation (3.49).
The equation (3.49) is used to find out the variation in orbital
elements due to effects of various forces acting on the satellite. For
controlling the relative distance between the satellites, it is needed to control
the orbital elements of the satellites. This in turn requires the amount of thrust
to be applied at various directions in orbital frame to maintain the orbital
elements of the satellite. The amount of force, which is to be applied at
various directions in the orbital frame, can be found out by equation (3.71)
(Naasz 2002).
2 2
1
2r
3
4h
2a esin 2a p 0h hr
psin (p r)cos re 0 k ah hk er sin( )u 0 0 k ihu
r sin( ) k d0 0u h sin ipcos (p r)sin r sin( )cosi
he he h sin ib(pcos 2re) b(p r)sin 0
ahe ahe
5
6
kk M
(3.71)
70
In the above equation the various values of proportional gains viz.
k1, k2, k3, k4, k5 and k6, are found. Using these gains the required amount of
forces to be applied to the satellite to have desired position and velocity are
calculated. The forces are found and are applied in the direction vectors of
orbital frame of the satellite. The values of proportional gains can be found
out by using genetic algorithm, as the orbital dynamics of satellite is
nonlinear.
3.12 SIMULATION OF PERTURBATION EFFECTS ON
ORBITAL ELEMENTS
The simulation of the orbital elements variation of the satellites due to
various perturbations is done in Matlab. The amount of thrust required to
maintain the relative distance between the satellites is done by using orbital
element feedback control in which genetic algorithm is used to determine the
amount of proportional gains needed to calculate the required amount of
thrust. The genetic algorithm is implemented in Matlab.
3.12.1 Gaussian Planetary Equation
The perturbing forces that are taken into consideration for
determining their effects on orbital elements are atmospheric drag, earth’s
oblateness, triaxiality of earth and the solar flux from sun. The effect due to
each force is simulated separately with the forces converted to the force
vectors acting in the different directions in orbital frame of the satellite.
The effect on orbital elements due to atmospheric drag is given in
Figure 3.8 and the effect on orbital elements due to earth’s oblateness is given
in Figure 3.9.
71
Figure 3.8 Variations of orbital elements due to atmospheric drag
72
Figure 3.9 Variation of orbital elements due to earth oblateness
73
The effect on orbital elements due to earth’s triaxiality is given in
Figure 3.10.
Figure 3.10 Variation of orbital elements due to earth’s triaxiality
74
The effect on orbital elements due to solar flux is given in
Figure 3.11.
Figure 3.11 Variation of orbital elements due to solar flux
75
In the above simulation, for the LEO considered it is found that
there is secular effect on the satellite’s motion due to atmospheric drag and
oblateness of the earth. The other effects are either periodic or very low. The
effect of atmospheric drag on the semi-major axis is that the semi-major axis
is reduced. The effect of oblateness is on the right ascension of the orbit and is
causing the westward nodal regression of the satellite orbit at the rate of
0.1 deg/day. Due to atmospheric drag, the semi major axis is decreasing at the
rate of 3.8 m/day.
3.12.2 General Orbital Elements Variation Equation
The methods of general perturbations are well studied and are used
to calculate the effect of perturbative forces on the orbital parameters (Battin
1987, Danby 1962, Kaplan 1976 and Roy 1982). The effects of perturbation
forces on orbital elements are calculated using general orbital elements
variation equation and are given in Figure 3.12.
In the above simulation, it is found that there is variation in semi
major axis, right ascension and argument of perigee. The variation in semi
major axis is reduction of 3.8 m/day and that of right ascension is reducing by
nearly 0.15 deg/day, which is same as that found by using Gaussian variation
equation.
3.12.3 Newton’s Law of Gravitation
In this only atmospheric drag is taken into account. Using this
method the reduction in semi major axis is same as that found by using
Gaussian variation equation and general equation. It is found to be reducing
by 3.8 m/day.
The effect on orbital elements due to Cowell’s method is given in
Figure 3.13.
76
Figure 3.12 Variation of orbital elements using general variation
equation
77
Figure 3.13 Variation of orbital elements in Cowell’s method
78
3.13 GAINS OPTIMIZATION USING GENETIC ALGORITHM
Orbit controller can correct the deviations in the orbital elements.
The mostly used is the orbital element feedback method. The orbital element
feedback method has gains for the six orbital elements, five of which are
calculated by formulas. One gain has to be arbitrarily chosen. The gain values
need the length the firing time also apart from orbital parameters. By using
genetic algorithm, the optimized values of the gains are found for the orbital
element feedback controller. These gains will calculate the required amount of
thrust to be applied to the satellite, to keep them in the required position and
velocity by correcting their orbital elements.
3.13.1 General Procedure of Finding Optimized Gain Using GA
The procedure is in steps as explained below:
i) The fitness function design is very important to evaluate each
individual in one generation. For these problems,
minimization of orbital deviation is taken as the performance
criteria and tuning is done. So, E = Desired output - Actual
output is taken as error, E. Therefore, a smaller E represents a
higher fitness (GA maximizes performance). The E is
converted to a fitness value of a GA by using, Fitness = 1/ |E|
(Jinwoo et al 1994).
ii) In Genetic operations, the entire individuals are expressed as
binary strings, not the parameters themselves. In the coding
method, the scaling factors generated randomly, are first
coded into binary strings. While calculating the fitness values
for each individual, these binary strings are converted into
corresponding values in the parameter space by using a
79
decoding procedure. The mapping equation used to map the
binary string to the gain value is given below:
P = PMIN + b (PMAX-PMIN) / (2m -1) (3.72)
where P - Gain Value.
PMIN - Minimum Value of Gain.
PMAX - Maximum Value of Gain.
b - Integer Value Corresponding to the mth bit.
m - No: of Bits.
e.g.,
PMIN = 0.0,
PMAX = 1.0, Binary Value = 00010000
b = 16, m = 8 Bits
P = 0+16 (1-0) / (28-1)
P = 0.063.
iii) Once the fitness values of all individuals in the population are
evaluated, the fittest individuals are selected for survival and
reproduction. The selection process is based on proportional
selection method, i.e. an individual with a high fitness value
has a high probability of being selected, as described below,
a) pi = fi / Σ fi (3.73)
b) pi = pi suitable multiplying factor (3.74)
c) pi >1.0 is selected for the next generation.
80
The selected individuals are randomly mated to perform
genetic operations.
iv) Two mating parents exchange information through simple
crossover and are replaced with the new individuals. For a
simple crossover, the cut off position is randomly
determined. The crossover operation at the cut off position 6
i.e 6th bit is shown as follows:
X = [1 1 0 0 1 1 0 0] becomes [1 1 0 0 1 0 1 0]
Y = [0 1 1 1 0 0 1 0] becomes [0 1 1 1 0 1 0 0]
v) After completing the simple crossover, mutation
operation is performed. Mutation of 5th bit on X changes it to
[1 1 0 0 0 0 1 0].
The specifications for GA are summarized below:
Population size : 10
Each string represents the set of solution to the optimization
problem.
Selection method: Proportionate selection method
Probability of crossover: 1 (All the string will undergo
crossover)
Probability of mutation: 0.1 (One out of 10 string will undergo
a change in bit)
During the process of iteration, the genetic algorithm maintains a
constant population of individuals. Each individual will undergo evaluation
and selection. The surviving individuals will undergo crossover and mutation
81
operations. The iteration process is repeated until the termination conditions
are satisfied.
3.13.2 Simulation Result
The result of the simulation gives the values of the gains optimized
by genetic algorithm. The gains are used to find out the amount of thrust
required for correcting the orbital deviations. The thrust is applied to the
satellite dynamics to correct for the orbital perturbations. The amount of
corrections obtained is also found out.
The gains optimized using genetic algorithm are:
K1 = 3.6233e-11
K2 = 1
K3 = 1
K4 = -55.8
K5 = -2.5833e-8
K6 = -4.5933e-8
The thrusts to be applied are:
Ur = 1.6038 µN
Uθ = 1.774 mN
Uz = 0.018721 N
82
The corrections in orbital elements made by applying the calculated
amount of thrust in corresponding directions are:
Correction in semi major axis = 3.741 m
Correction in Eccentricity = -1.2977e-7
Correction in inclination = -0.00051116 rad
Correction in right ascension = 0.0024609 rad
Correction in mean anomaly = -0.0002291 rad
The actual amounts of secular variation in the orbital elements are:
For semi major axis = 3.8 m
For eccentricity ≈ 0
For Inclination ≈ 0 rad
For right ascension = 0.002 rad
For argument of perigee ≈ 0 rad
For mean anomaly ≈ 0 rad.
The response of the orbital elements to the controller is given in
Figure 3.14. It gives a picture of how the orbital elements deviations are
corrected by the proposed controller. The initial values of the orbital
elements, their variation due to atmospheric drag and their corrected values by
the proposed controller can be visualized.
83
Figure 3.14 Deviation and correction of orbital elements
84
The control law used is a hybrid continuous feedback control law
for a local Cartesian relative orbit frame and is a function of differential
orbital elements. So, these control laws are valid for both circular and
elliptical orbits. Hence the orbital element feedback controller works well for
both circular and elliptic orbits. The GA used is just an optimization tool, so
the proposed work is applicable for different orbits. The deviations in the
orbital elements are reduced to approximately zero, just like that of a circular
orbit. This has been proved through results of simulation and is given in
Table 3.1.
Table 3.1 Performance for various eccentricities
Semi -major Axis (km)
Eccentricity Inclina-
tion (rad)
Mean Anomaly (rad)
Right Ascension
(rad)
Argument of Perigee
(rad)
Satellite
with e=0.001
Actual 7185 0.001 1.5551 0 0.7854 0.14 Perturbation 0.0038 2.880910-7
0 -4.485510-4 0 4.485510-4
Correction 0.0038 Approx 0 0 0 0 0
Satellite
with e=0.01
Actual 7185 0.01 1.5551 0 0.7854 0.14 Perturbation 0.0039 2.934310-7
0 -4.527910-5 0 4.527910-5
Correction 0.0039 Approx 0 0 0 0 0
Satellite
with e=0.05
Actual 7185 0.05 1.5551 0 0.7854 0.14 Perturbation 0.0040 3.178210-7
0 -9.445210-6 0 9.444310-6
Correction 0.0042 Approx 0 0 0 0 0
Satellite
with e=0.1
Actual 7185 0.1 1.5551 0 0.7854 0.14
Perturbation 0.0042 3.508410-7 0 -4.970210-6 0 4.966410-6
Correction 0.0046 Approx 0 0 0 0 0
The amount of thrust required for the orbital correction is of the
order of micro-Newton. Hence, advanced chemical and low power electric
propulsion offers attractive options for small satellite propulsion. Applications
include orbit raising, orbit maintenance, attitude control, repositioning, and
85
de-orbit of both earth-space and planetary spacecraft. Potential propulsion
technologies for these functions include high pressure Ir/Re bipropellant
engines, very low power arcjets, Hall thrusters, and pulsed plasma thrusters,
all of which are shown to operate in manners consistent with currently
planned small satellites (Roger and Steven 1994 and Yashko and Hastings
1996). Any suitable propulsion system can be designed to achieve the
distance keeping function.
3.14 FUZZY CONTROLLER
In certain low Earth orbit (LEO) satellite missions, it is required that
two or more satellites must operate in a certain special configuration relative
to each other. This section introduces a simple concept of utilizing
aerodynamic drag to achieve this type of constellation control. The key
feature is utilizing aerodynamic drag, a natural phenomenon that is normally
considered as an unwanted disturbance, especially for low Earth orbit
missions. The perturbation due to aerodynamic drag is designed using
Gaussian Planetary equation as explained in section 3.10.1. The satellites
considered for simulation are micro-satellites with a semi-major axis of
7185 Km and eccentricity of 0.001 and are 1000 Km apart.
The simulations of the effect of atmospheric drag on the low-Earth
orbiting satellite shows deviations in their orbital elements, for a day is as
given in Table 3.2.
Table 3.2 Deviation in orbital elements
a e i M
3.8m 5.3366e-007 rad 0 0 0 0
86
The deviations in the orbital elements are corrected by producing
the required amount of thrusts. Fuzzy controller determines the amount of
thrust required for the satellite. For formation flying and constellation station
keeping in near-circular orbits, the deployment and maintenance of the
formation or constellation can be done by closely controlling two mean
orbital elements (Bainum and Duan 2004). This concept has been
implemented in this paper by the use of fuzzy control.
The inputs for fuzzy controller are semi-major axis a, and
eccentricity e. The outputs of the controller are the thrusts in the three
directions. The fuzzy controller is represented as a block diagram in
Figure 3.15.
uh
a ur
e u
Figure 3.15 Block diagram of fuzzy controller
The change in the semi-major axis and eccentricity after a period of
time (e.g. one day) is calculated using the Gaussian planetary equation. These
two elements are given as inputs to the fuzzy controller. The input variables
are then mapped into fuzzy sets. The fuzzy set values are obtained from the
triangular membership function. The membership functions are shown in
Figures 3.16 and 3.17.
Fuzzy
Controller
87
Figure.3.16 Membership function, a
Figure.3.17 Membership function, e
The amount of overlap between the different fuzzy sets is optimized
through simulation. The saturation point of each input variable is set using an
engineering knowledge of the system and optimized using simulation trails.
Sample of output membership function for the thrusts is given in Figures 3.18.
The rules of the controller are given in Table 3.3.
Figure 3.18 Membership function for output, thrust
88
Table 3.3 Rules
Da de ur uh u R1 Z Z Z Z Z
R2 Z P Z Z Z
R3 Z MP P P P
R4 P Z Z Z Z
R5 P P P P P
R6 P MP P P P
R7 MP Z P P P
R8 MP P P P P
R9 MP MP MP MP MP
Rule evaluation is performed using correlation-product encoding,
i.e. the conjunctive (AND) combination of the antecedent fuzzy sets. When
the result of all the rules is known, the final value is obtained by disjunctively
(OR) combining the rule values:
N Ni i i
i 1 i 1y (y ) sgn y min 1, y
(3.75)
The disjunction method can be described as a kind of signed
Lukasiewicz OR logic. It is chosen to maximally negatively correlate the rule
outputs. For example, opposing rule outputs (different in sign) cancel each
other to deliver a small rule base output. Defuzzification is done by centroid
method. The crisp output value x is the abscissa under the centre of gravity of
the fuzzy set,
)i
i
iii
x(
x)x(u
(3.76)
89
Here xi is a running point in a discrete universe, and µ(xi) is its
membership value in the membership function. The expression can be
interpreted as the weighted average of the elements in the support set. The
controller corrects the deviation in the orbital elements due to atmospheric
drag. The value of orbital elements at the initial position, their deviation after
a day and the corrected values of orbital elements are given in Table 3.4.
Table 3.4 Simulation results
Semi-major Axis
Eccentricity Inclination Right
Ascension Argument of perigee
Mean Anomaly
During Launch Orbit of Sat 1
7185 0.001 1.5551 0.7854 0.14 0
Perturbed orbit after a day, Sat 1
7184.9962 0.0009994 1.5551 0.7854 0.14 0
Orbit Corrected by Fuzzy controller,Sat1
7185 0.001 1.5551 0.7854 0.134 0.006
During Launch Orbit of Sat 2
7185 0.001 1.5551 0.7854 0.25 0.1117
Perturbed orbit after a day, Sat 2
7184.9962 0.0009994 1.5551 0.7854 0.2499 0.11176
Orbit Corrected by Fuzzy controller,Sat2
7185 0.001 1.5551 0.7854 0.2439 0.1117
The result of the simulation for the proposed fuzzy controller is given
in Figure 3.19. The initial orbital elements their perturbation due to
atmospheric drag and the correction are indicated in the result.
90
Figure 3.19 Orbital elements deviation and correction
3.15 CONCLUSION
In this work, a method of finding the initial conditions of the
satellites in formation flying in various orbital configurations like leader-
91
follower, projected circular orbit and general circular orbit configuration were
presented using the solutions of Euler-Hill’s equations of relative motion.
Simulation of relative distance between the satellites with the given initial
conditions was also shown for various configurations. The movement of the
satellites away from the formation was simulated for the chief satellite in
elliptic orbit. The variations in orbital elements of the satellites due to various
perturbation forces were calculated and simulated using various methods of
orbital propagation and they were found to agree with each other.
A fuzzy and Genetic Algorithm approach has been made to control
the orbit and hence to achieve distance keeping for formation flying. The
solution proposed in this thesis can be used for formation flying and
constellation station keeping and maintaining the relative distance between
spacecrafts. Moreover, the deployment or maintenance of the formation or
constellation can be done by closely controlling two mean orbital elements,
semi-major axis and eccentricity. The method works under the influence of
the atmospheric drag. Therefore, it can be effectively used on the low and
mid-altitude orbits where this represents the main perturbation effect.
A new method to find out the gains of the orbital element feedback
controller, to develop a required amount of thrust for absolute station keeping
of satellites was done using genetic algorithm. The gains of the classical
orbital element controller are generally found by using the orbital elements,
length of thruster firing time and other orbital parameters. There is no formula
for finding the mean motion gain, and this is chosen depending on how
aggressively we want to correct the argument of latitude error. So instead of
using the formulas for finding five gains and assuming the sixth gain, the
proposed GA based gain optimization technique finds the optimal value for
all the gains. The deviations were reduced to almost zero by the proposed
control algorithm. The fuzzy controller and the genetic controller are effective
92
in correcting the orbit and hence to maintain distance keeping. The
performance of both the controllers, GA optimized orbital element feedback
and fuzzy for the same orbital elements are given in Table 3.5.
Table 3.5 Comparison of proposed controllers
Semi-major Axis Eccentricity Inclination Right
Ascension Argument of perigee
Mean Anomaly
Initial Orbital Elements 7185 0.001 1.5551 0.7854 0.14 0
Perturbed orbit after a day
7184.9962 0.0009994 1.5551 0.7854 0.14 0
Orbit Corrected by Fuzzy controller
7185 0.001 1.5551 0.7854 0.134 0.006
Orbit Corrected by GA
7185 0.001 1.5551 0.7854 0.14 0
On comparison the performance of the GA optimized orbital
feedback controller is found to be better than fuzzy controller. For the same
orbital elements the GA optimized orbital element feedback controller
corrects almost all the orbital deviations to zero whereas there is some amount
of error in case of fuzzy controller for argument of perigee and mean
anomaly. Hence, it can be concluded that the genetic algorithm is time
consuming in optimizing the gains for orbital element feedback controller.
The fuzzy logic has an advantage over GA in terms of time. But the
performance of GA optimized orbital element feedback controller is found to
be marginally better than the fuzzy controller. Optimization being an offline
process the time consumption is not a problem and the performance obtained
suggests that the GA optimized orbital element feedback controller is chosen
as a better candidate over the fuzzy controller.