1. 6/21/2013
GENERATION OF
NAVIGATION
PARAMETERS IN RINEX
FORMAT FOR IRNSS
SATELLITES
By
Nagarji Javed,
Shripad Thakur,
Aditya Singh,
Ayush Gupta
ISRO Summer Internship 2013 Project
Report

2. 1
Declaration
We declare that this written submission represents our ideas in our own words and
where others ideas or words have been included, we have adequately cited and
referenced the original sources. We also declare that we have adhered to all
principles of academic honesty and integrity and have not misrepresented or
fabricated or falsified data/idea/fact/source in our submission. We understand that
any violation of the above will be cause of disciplinary action by the institute and
can also evoke penal action from the sources which have thus not been properly
cited or from whom proper permission has not been taken when needed.
By
Nagarji Javed,
Shripad Thakur,
Aditya Singh,
Ayush Gupta

3. 2
Contents
1. Introduction to Orbital mechanics 3
2. Geostationary Satellite 4
3. The Two Body and addition of perturbations 4
 The Non-homogeneity and Oblateness 0f the Earth 5
 Sun and Moon perturbations … (Third body perturbations ) 13
 Solar Radiation Pressure 15
 Atmospheric Drag 18
 Earth Tides 20
 Ocean Tides 21
 Relativistic Effects 23
 Plots and Graphs of simulated results 24-29
4. Orbit determination-Obtaining the measured state 30
 Obtaining the state vector through range and angles measurement 31
5. Batch Processing and Sequential Processing 33
 Kalman Filtering 33
 Kalman Filter Algorithm 34
 Plots and Graphs of simulated Results 35
6. Application of RINEX format 39
 Calculation of satellite position 40
 Signals, Services and Data Format 43
 A sample RINEX file 46
7. Acknowledgments 48
8. Reference 49

4. 3
Orbital mechanics, as applied to artificial spacecraft, is based on celestial mechanics. In
studying the motion of satellites, quite elementary principles are necessary. Kepler provided
three basic empirical laws that describe motion in unperturbed planetary orbits. Newton
formulated the more general physical laws governing the motion of a planet, laws that were
consistent with Kepler's observations. In this report, ideal, unperturbed Keplerian orbits will be
seen briefly and the dynamical equations of motion for realistic, perturbed orbits - will be
analyzed. Kepler's laws of motion describe ideal orbits that do not exist in nature. Perturbing
forces and physical anomalies cause spacecraft orbits to have strange properties; in most cases
these cause difficulties for the space control engineer, but in other cases these properties may be
of enormous help.
Kepler provided three empirical laws for planetary motion, based on Brahe's
Planetary observations.
1. The orbit of each planet is an ellipse with the sun located at one focus.
2. The radius vector drawn from the sun to any planet sweeps out equal areas in equal time
intervals (the law of areas).
3. Planetary periods of revolution are proportional to the [mean distance to sun]3/2
Newton stated in his law of gravitation that any two particles attract each other with a force
given by the expression 𝐹 = 𝐺
𝑚1 𝑚2
𝑟3
𝒓
where r is a vector of magnitude r along the line connecting the two particles with masses m1 and
m2 and G = 6.669 X 10-11 m3/kg-s2
is the universal constant of gravitation. This is the famous
inverse square law of force.
The two-body problem is an idealized situation in which only two bodies exist that are in relative
motion in a force field described by the inverse square law. In order to obtain simple analytical
results for the motion of celestial bodies or spacecraft, it is assumed that additional bodies are
situated far enough from the two-body system, thus no appreciable force is exerted on them from
a third body.
For a mass of unity we find that
𝑑ℎ
𝑑𝑡
= 0 and h = r x v = const. The vector h is called the specific
angular momentum. It is perpendicular to both r and v, and is also constant in space. This means
that the motion of the particle takes place in a plane.
There are several parameters that represent the orbital motion in a plane. The physical
parameters sufficient to define an orbit are its total energy E and its momentum h. The geometric
parameters sufficient to define the orbit are the semi-major axis a and the eccentricity e. The
mean anomaly M enables finding the location of the moving body in the orbit with time. There
are three more parameters are necessary to define an orbit in space.
The angle between the plane of revolution of the particle around earth and the X-Y plane - which
is also the equatorial plane of the earth ( since all measurements are in the eci coordinate system)
is an angle i, the inclination of the orbit. The orbital plane and the equatorial plane intersect at the
node line. The angle in the equator plane that separates the node line from the X axis is called the
right ascension, Ω. In the orbit plane, r is the radius vector to the moving body; rp is the radius
vector to the perigee of the orbit from the focus of the orbit. The angle between rp and the node
line is 𝜔, the argument of perigee. These three parameters, together with the three parameters (a,

5. 4
e, and M) of the orbit in the plane, complete a system of six parameters that suffices to define the
location in space of a body moving in any Keplerian orbit. These parameters are known as the
classical orbit parameters, which are redefined as follows:
1. Semi-major axis, a
2. Eccentricity, e
3. Inclination angle, i
4. Argument of perigee, 𝜔
5. Right ascension of the ascending node, Ω
6. Mean anomaly, M = n( t-t0)
GEOSTATIONARY SATELLITE
A geostationary satellite moves under the gravitational force of the earth. Ideally it is a two body
system where the satellite revolves around the earth in an elliptical orbit . Since the eccentricity
of the orbit is very small (around 0.00045) it can be assumed as a near circular orbit. Also the
inclination of an ideal geostationary satellite is zero degrees but in real cases it is a very small
non zero value.
Ideal Keplerian orbits were treated under the basic assumptions that the motion of a body in
these orbits is a result of just the gravitational attraction between two bodies. This ideal situation
does not exist in practice. The two-body problem of motion is an idealization, and additional
forces acting on any moving body must be taken into account. The perturbing forces are both
conservative and non-conservative in nature. The perturbing forces caused by the additional
bodies are conservative field forces. In the case of geostationary satellites which are high-orbit
satellites, the effects of the conservative perturbing forces of the sun and the moon on the motion
of the satellite cannot be ignored.
In general the perturbing forces are due to these following reasons:
1. Non-homogeneity and Oblateness 0f the Earth
2. A third body gravitational force
3. Solar radiation pressure of the sun
4. Atmospheric drag
5. Infrared radiations from earth
6. Earth’s albedo
7. Relativistic effect due to earth
8. Solid tides
Geostationary satellites are high-altitude satellites so they do not experience atmospheric drag as
this drag force is only encountered by low-orbit satellites. Forces 1-3 are major contributors to
the satellite orbit position and velocity so they will be dealt with first, while 5-8 are minor forces
which have very little effect (order of magnitude 4-5 less than major ones) so they will be dealt
with later.

6. 5
The Non-homogeneity and Oblateness 0f the Earth
(reference no. 1,3 and 7)
One of the most important perturbing forces on earth-orbiting satellites arises from the non-
homogeneity of the earth. The earth globe is not a perfect sphere, and neither is its mass
distribution homogeneous. These physical facts produce perturbing accelerations on the moving
body. The consequences of these accelerations are variations of the orbital parameters of earth-
orbiting satellites.
Because the force exerted by the earth on a body outside its sphere is a conservative force, it can
be derived from a gradient of a scalar potential function U(r) = -pJr. This would be completely
true if the earth were modeled as a mass concentrated in a single point, or as a homogeneous
sphere. Unfortunately, this is not the case: the earth is an oblate body, and its mass distribution is
not homogeneous. So we need to write the potential at the satellite due to the non-spherical earth
and then take its gradient to find the perturbing force due to the non-homogeneity of the earth.
𝑟̈ = ∇𝑈
This expression for the potential may easily be generalized to an arbitrary mass distribution by
summing up the contributions in the potential by individual mass elements dm = p(s)ds3
, (here
p(s) is density) .
𝑈 = 𝐺 ∫
𝜌( 𝑠) 𝑑3
𝑠
| 𝑟 − 𝑠|
Here 𝜌( 𝑠) means the density at some pint s inside the earth, and | 𝑟 − 𝑠| is the satellite’s
distance from this place.

7. 6
The above expression for the potential due to earth on the satellite is very general and cannot be
used in this form. So it is expanded into appropriate spherical harmonics
1
|𝑟 − 𝑠|
= ∑(
𝑠
𝑟
) 𝑛
𝑃𝑛(cos 𝛾)
∞
𝑛=0
𝑤𝑖𝑡ℎ cos(𝛾) =
𝐫. 𝐬
𝑟𝑠
Here ,
𝑃𝑛 ( 𝑢) =
1
2 𝑛 𝑛!
𝑑 𝑛
𝑑𝑢 𝑛
( 𝑢2
− 1) 𝑛
Is the Legendre Polynomial of degree n and 𝛾 is the angle between r and s vectors.
BY the use of addition theorem of Legendre polynomials we get
𝑃𝑛(cos 𝛾) = ∑( 2 − 𝛿0𝑚)
( 𝑛 − 𝑚)!
( 𝑛 + 𝑚)!
𝑃𝑛𝑚(sin ∅) 𝑃𝑛𝑚(sin ∅′) cos(𝑚(𝜆 − 𝜆′))
𝑛
𝑚=0
Here Pnm is the associated Legendre polynomial of degree n and order m and is defined as
Illustration taken from
Montenbruck page no. 56

8. 7
𝑃𝑛𝑚 ( 𝑢) = ( 1 − 𝑢2
) 𝑚/2
𝑑 𝑚
𝑑𝑢 𝑚
𝑃𝑛(𝑢)
one can write earth’s gravitational potential in the form, (Montenbruck chapter 3)
𝑈 =
𝐺𝑀
𝑟
∑ ∑
𝑅 𝑛
𝑟 𝑛
𝑃𝑛𝑚(sin ∅)(𝐶 𝑛𝑚 cos( 𝑚𝜆) + 𝑆 𝑛𝑚 sin( 𝑚𝜆))
𝑛
𝑚=0
.
∞
𝑛=0
with coefficients
𝐶 𝑛𝑚 =
2 − 𝛿0𝑚
𝑀
(𝑛 − 𝑚)!
( 𝑛 + 𝑚)!
∫
𝑠 𝑛
𝑅 𝑛
𝑃𝑛𝑚(sin∅′) cos(𝑚𝜆′) 𝜌(𝑠)𝑑3
𝑠
𝑆 𝑛𝑚 =
2 − 𝛿0𝑚
𝑀
(𝑛 − 𝑚)!
( 𝑛 + 𝑚)!
∫
𝑠 𝑛
𝑅 𝑛
𝑃𝑛𝑚(sin∅′) sin(𝑚𝜆′) 𝜌(𝑠)𝑑3
𝑠
These geopotential coefficients describe the dependence of the potential on earth’s internal mass
distribution.
The harmonics can be divided into 3 groups:
 Zonal harmonics
 Sectorial harmonics
 Tesseral harmonics
Zonal harmonics are the harmonics that are symmetric about the earth rotation axis. They have
only latitude dependence and not on the longitude. The coefficients with m=0 are called zonal
Table copied from
Montenbruck page no. 57

9. 8
coefficients because they are not dependant on longitude. Legendre polynomial with m=0 only
have sine terms.
Sectorial harmonics are a function of longitude only. The coefficients with m=n are called
sectorial coefficients because of their latitude independence. Legendre polynomial with m=n
only have cosine terms.
Tesseral harmonics are dependent on both longitude and latitude. The coefficients with m>0
and m<n are tesseral coefficients. Legendre polynomial with m=0 only have both sine and
cosine terms.
The net effects of these perturbations are periodic but short term changes in semi major axis,
eccentricity and inclination angle but these changes do not show in the long run and their average
value remains constant. . Since the earth is neither a perfect sphere nor a perfect ellipse, every
time the satellite crosses the perigee its velocity is greater than expected at that point which
means it is more closer to earth than expected and every time it reaches its apogee its speed is
less than expected at that point which means it is more farther to earth than expected. This
happens in every revolution which is responsible for the periodic changes in e and a even though
their mean remains same for a long time interval. The zonal harmonics are responsible for a very
small shift in inclination angle also. The earth can be modeled as a sphere with an extra bulge at
the equator with additional radius of 20 km. A geostationary satellite ideally should be exactly at
the equator all the time but in reality its latitude may vary about 0 degree by 0.1 degrees. So
whenever our geostationary satellite is slightly above 0 degrees a component of the force from
the extra bulge will attract it downwards causing the angular momentum of the satellite to shift
because of the torque produced by the force due to this extra bulge is in east direction. Now
when the satellite is below 0 degrees then the bulge will have an upward force and this will cause
a torque in west direction. This happens in every revolution. The important thing here is that this
torque is because of the tangential component of the extra force due to bulge. So now we have a
rough idea that the inclination changes are due to the tangential forces exerted by the bulge while
the variation in apogee is due to radial components.
The tesseral and sectorial terms cause the forces in direction tangential to the longitudes which
causes longitude acceleration. This acceleration depends on the longitude we are on. Writing the
potential due to these terms and taking its gradient with respect to the longitude gives and
expression for the longitudinal acceleration. This gradient shows that the force is zero at 4
longitudes: two stable and twp unstable equilibrium points. Our geostationary satellite is at a
stable longitude at about 74 degrees east over Bhopal.
But on the other hand the effects on RAAN, argument of perigee and mean anomaly are secular
i.e. they are significant in the long term analysis. This is because the earth’s geopotential changes
the RAAN which in turn changes the other two parameters
(Re is taken as the mean radius of the earth at equator.)

10. 9
The geo-potential coefficients and the Legendre polynomial when normalized are denoted by ~
above them.(Montenbruck chapter 3)
𝑃𝑛𝑚
̃ = √
( 2 − 𝛿0𝑚)(2𝑛 + 1)(𝑛 − 𝑚)!
(𝑛 + 𝑚)!
𝑃𝑛𝑚
{
𝐶 𝑛𝑚
𝑆 𝑛𝑚
} = √
( 2 − 𝛿0𝑚)(2𝑛 + 1)(𝑛 − 𝑚)!
(𝑛 + 𝑚)!
{
𝐶 𝑛𝑚
̃
𝑆 𝑛𝑚
̃
}
Geopotential coefficients up to m=n=3..
Here we define 2 new functions Vnm and Wnm as,
𝑉𝑛𝑚 = (
𝑅𝑒
𝑟
)
𝑛+1
. 𝑃𝑛𝑚(sin ∅). cos(𝑚𝜆)
𝑊𝑛𝑚 = (
𝑅𝑒
𝑟
)
𝑛+1
. 𝑃𝑛𝑚(sin ∅). sin(𝑚𝜆)
So now our potential function takes the form
𝑈 =
𝐺𝑀
𝑅𝑒
∑ ∑(𝐶 𝑛𝑚 𝑉𝑛𝑚 + 𝑆 𝑛𝑚 𝑊𝑛𝑚)
𝑛
𝑚=0
.
∞
𝑛=0
The above functions can be evaluated at different values of n and m by using the following
recurrence relations, ( Montenbruck chapter 3)
Table copied from
Montenbruck page no. 72

11. 10
𝑉𝑛𝑚 = (2𝑚 − 1) {
𝑥𝑅𝑒
𝑟2
𝑉 𝑚−1,𝑚−1 −
𝑦𝑅𝑒
𝑟2
𝑊 𝑚−1,𝑚−1}
𝑊𝑛𝑚 = (2𝑚 − 1) {
𝑥𝑅𝑒
𝑟2
𝑊 𝑚−1,𝑚−1 +
𝑦𝑅𝑒
𝑟2
𝑉 𝑚−1,𝑚−1}
and
𝑉𝑛𝑚 = (
2𝑛 − 1
𝑛 − 𝑚
) .
𝑧𝑅𝑒
𝑟2
. 𝑉𝑛−1,𝑚 − (
𝑛 + 𝑚 − 1
𝑛 − 𝑚
) .
𝑅𝑒2
𝑟2
. 𝑉𝑛−2,𝑚
𝑊𝑛𝑚 = (
2𝑛 − 1
𝑛 − 𝑚
) .
𝑧𝑅𝑒
𝑟2
. 𝑊𝑛−1,𝑚 − (
𝑛 + 𝑚 − 1
𝑛 − 𝑚
) .
𝑅𝑒2
𝑟2
. 𝑊𝑛−2,𝑚
𝑉00 =
𝑅𝑒
𝑟
𝑎𝑛𝑑 𝑊00 = 0
The acceleration can be directly calculated from the expression given in Montenbruck page no.
68.
If m = 0 then
𝑥 𝑛0̈ =
𝐺𝑀
𝑅𝑒2
. {−𝐶 𝑛0 𝑉𝑛+1,1}
𝑦 𝑛0̈ =
𝐺𝑀
𝑅𝑒2
. {−𝐶 𝑛0 𝑊𝑛+1,1}
Else when 0<m <n
𝑥 𝑛𝑚̈ =
𝐺𝑀
𝑅𝑒2
.
1
2
. {(−𝐶 𝑛𝑚 𝑉𝑛+1,𝑚+1 − 𝑆 𝑛𝑚 𝑊𝑛+1,𝑚+1)
+
(𝑛 − 𝑚 + 2)!
(𝑛 − 𝑚)!
. (𝐶 𝑛𝑚 𝑉𝑛+1,𝑚−1 + 𝑆 𝑛𝑚 𝑊𝑛+1,𝑚−1)}
𝑦𝑛𝑚̈ =
𝐺𝑀
𝑅𝑒2
.
1
2
. {(−𝐶 𝑛𝑚 𝑊𝑛+1,𝑚+1 + 𝑆 𝑛𝑚 𝑉𝑛+1,𝑚+1)
+
(𝑛 − 𝑚 + 2)!
(𝑛 − 𝑚)!
. (−𝐶 𝑛𝑚 𝑊𝑛+1,𝑚−1 + 𝑆 𝑛𝑚 𝑉𝑛+1,𝑚−1)}

12. 11
𝑧 𝑛𝑚̈ =
𝐺𝑀
𝑅𝑒2
{( 𝑛 − 𝑚 + 1). (−𝐶 𝑛𝑚 𝑉𝑛+1,𝑚 − 𝑆 𝑛𝑚 𝑊𝑛+1,𝑚)}
The above formulae are given in r = r(x,y,z) where x ,y and z are in e.c.e.f. (earth fixed )
coordinates so we need to convert these coordinated into e.c.i coordinates to get accelerations in
inertial frame.
All the values of Legendre polynomials were taken and verified from Vallado chapter on special
perturbation methods.
CODE
Converting ECI Coordinated To ECEF And Then Calculating The Latitude And
Longitude Of The Satellite
ecef= [ cos(hh) sin(hh) 0 ; -sin(hh) cos(hh) 0 ; 0 0 1]*[X2-X1 Y2-Y1 Z2-Z1]';
glong2=atan2(ecef(2),ecef(1));
% longitude of the satellite
lat2 = atan2((Z2-Z1),(norm(X2-X1,Y2-Y1)));
% latitude of the satellite
% coefficients Vnm and Wnm ( referred from MONTENBRUCK)
V33 = (Re/r)^4*15*(cos(lat2))^3*cos(3*glong2);
W33 = (Re/r)^4*15*(cos(lat2))^3*sin(3*glong2);
V32 = (Re/r)^4*15*sin(lat2)*(cos(lat2))^2*cos(2*glong2);
W32 = (Re/r)^4*15*sin(lat2)*(cos(lat2))^2*sin(2*glong2);
V31 = (Re/r)^4*(cos(lat2))*((15*(sin(lat2))^2 - 3)/2)*cos(glong2);
W31 = (Re/r)^4*(cos(lat2))*((15*(sin(lat2))^2 - 3)/2)*sin(glong2);
V30 = (5/6)*ecef(3)*(Re^4/r^5)*(3*(sin(lat2))^2 -1) - (2/3)*((Re/r)^4)*sin(lat2);
W30 = 0;
V42 = (105/2)*(ecef(3))*(Re^5/r^6)*sin(lat2)*(cos(lat2))^2*cos(2*glong2) - (15/2)*(Re/r)^5 *
(cos(lat2))^2 * cos(2*glong2);

13. 12
W42 = (105/2)*(ecef(3))*(Re^5/r^6)*sin(lat2)*(cos(lat2))^2*sin(2*glong2) - (15/2)*(Re/r)^5 *
(cos(lat2))^2 * sin(2*glong2);
V41 = (7/2)*(ecef(3))*(Re^5/r^6)*cos(lat2)*(5*(sin(lat2))^2-1)*cos(glong2) - (4)*(Re/r)^5 *
cos(lat2) * sin(lat2) * cos(glong2);
W41 = (7/2)*(ecef(3))*(Re^5/r^6)*cos(lat2)*(5*(sin(lat2))^2-1)*sin(glong2) - (4)*(Re/r)^5 *
cos(lat2) * sin(lat2) * sin(glong2);
V40 = (Re/r)^5*(3*(sin(lat2))^2 - 1)*((35/24)*((ecef(3))/r)^2 - 3/8) -
(7/6)*(ecef(3))*(Re^5/r^6)*sin(lat2);
W40 = 0;
V43 = 105*(ecef(3))*(Re^5/r^6)*(cos(lat2))^3*cos(3*glong2) ;
W43 = 105*(ecef(3))*(Re^5/r^6)*(cos(lat2))^3*sin(3*glong2) ;
V44 = 105*(Re^5/r^6)*((cos(lat2))^3)*((ecef(1))*cos(3*glong2) - (ecef(2))*sin(3*glong2));
W44 = 105*(Re^5/r^6)*((cos(lat2))^3)*((ecef(1))*sin(3*glong2) + (ecef(2))*cos(3*glong2));
Acceleration terms from various harmonics
%% all accelerations due to geopotential changes are taken from montenbruck
% acc due to zonal term n=2,m=0
AXz1 = ( G*m1/Re^2 )*( - C20*V31 ) ;
AYz1 = ( G*m1/Re^2 )*( - C20*W31 ) ;
AZz1 = ( G*m1/Re^2 )*(3*( -C20*V30 - S20*W30) ) ;
%for SECTORAL harmonic effect due to n=2, m=2
%
AXs1 = ( G*m1/Re^2 )*(1/2)*(( -C22*V33- S22*W33 )+2*( C22*V31+S22*W31 )) ;
AYs1 = ( G*m1/Re^2 )*(1/2)*(( -C22*W33+ S22*V33 )+2*( -C22*W31- S22*V31 )) ;
AZs1 = ( G*m1/Re^2 )*(1)*( -C22*V32- S22*W32 ) ;
% acc due to zonal term n=3, m=0
%
AXz2 = ( G*m1/Re^2 )*( - C30*V41 ) ;
AYz2 = ( G*m1/Re^2 )*( - C30*W41 ) ;

14. 13
AZz2 = ( G*m1/Re^2 )*(4*( -C30*V40 - S30*W40) ) ;
% acc due to tesseral terms n= 3, m=1
AXt1 =( G*m1/Re^2 )*(1/2)*(( -C31*V42 - S31*W42 ) + 12*( C31*V40 + S31*W40 )) ;
AYt1 =( G*m1/Re^2 )*(1/2)*(( C31*W42 + S31*V42 ) + 12*( -C31*W40 + S31*V40 )) ;
AZt1 =( G*m1/Re^2 )*(3*( -C31*V41- S31*W41) ) ;
% acc due to tesseral terms n=3, m=2
%
AXt2 = ( G*m1/Re^2 )*(1/2)*((-C32*V43 - S32*W43 ) + 6*( C32*V41 + S32*W41 )) ;
AYt2 = ( G*m1/Re^2 )*(1/2)*(( C32*W43 + S32*V43 ) + 6*(-C32*W41 + S32*V41 )) ;
AZt2 = ( G*m1/Re^2 )*(2*( -C32*V42 -S32*W42) ) ;
% acc due to sectoral terms n=3,m=3
AXs2 = ( G*m1/Re^2 )*(1/2)*(( -C33*V44 - S33*W44 ) + 2*( C33*V42 + S33*W42 )) ;
AYs2 = ( G*m1/Re^2 )*(1/2)*(( C33*W44 + S33*V44 ) + 2*( -C33*W42 + S33*V42 )) ;
AZs2 = ( G*m1/Re^2 )*( -C33*V43 - S33*W43) ;
These acceleration terms are in ecef coordinates so we need to convert them back into eci
coordinate system to add them to the 2-body acceleration.
Sun and Moon perturbations … (Third body perturbations )
As the distance from earth increases, perturbations due to non-spherical earth become
comparatively less important and the perturbations from the gravitational forces of the sun and
moon and other planets become important. Such third body interactions are the source of major
orbital perturbations for interplanetary flights and dominate the motion entirely in some
circumstances. In practice, most work involving such third body interactions is done by
numerical integration of the equations of motion.
The two-body problem treated in the beginning of this chapter can be generalized to the much
more difficult n-body problem in the following way.
In a system consisting of n bodies, the sum of the forces acting on the ith
body will be

15. 14
𝐹𝑖 = 𝐺 ∑
𝑚 𝑖 𝑚 𝑗
𝑟 𝑖𝑗
3
𝑛
𝑗=1 . (𝑟𝑗 − 𝑟𝑖) ; 𝑖 ≠ 𝑗
From Newton’s second law of motion acceleration of the body due to this force is
𝑟𝑖̈ = 𝐺 ∑
𝑚 𝑗
𝑟 𝑖𝑗
3
𝑛
𝑗=1 . (𝑟𝑗 − 𝑟𝑖) ; 𝑖 ≠ 𝑗
The accelerations of the earth and the moon extracted from these equations are ( m1 and m2 are
masses of earth and satellite respectively.)
𝑟1̈ = 𝐺
𝑚2
𝑟12
3
( 𝑟2 − 𝑟1) + 𝐺 ∑
𝑚 𝑗
𝑟1𝑗
3
𝑛
𝑗=3 . (𝑟𝑗 − 𝑟1)
𝑟2̈ = 𝐺
𝑚1
𝑟12
3
( 𝑟1 − 𝑟2) + 𝐺 ∑
𝑚 𝑗
𝑟2𝑗
3
𝑛
𝑗=3 . (𝑟𝑗 − 𝑟2)
These are the equations of motion with respect to an inertial coordinate axes in space. If we want
the acceleration and the position vector of the satellite with respect to earth we need to subtract
the acceleration of the earth from that of the satellite since earth is also moving in the inertial
frame.
Subtracting the 2 equations we get,
𝑟2̈ − 𝑟1̈ = −𝐺
(𝑚1 + 𝑚2)
𝑟12
3
( 𝑟2 − 𝑟1)
+ 𝐺 ∑ 𝑚𝑗 {
1
𝑟2𝑗
3
𝑛
𝑗=3
. (𝑟𝑗 − 𝑟2) −
1
𝑟1𝑗
3
(𝑟𝑗 − 𝑟1)}
This equation can be simplified to look identical to the one for 2-body problem but with an extra
term added to it (perturbation)
𝒓̈ + 𝐺( 𝑚1 + 𝑚2)
𝒓
𝑟3 = 𝛾𝑝
Here r2 – r1 is replaced by r and 𝛾𝑝 is the additional acceleration term for perturbation from the
additional bodies (here the sun and moon)
𝛾𝑝 = 𝐺 ∑ 𝑚𝑗 {
(𝑟 𝑗 − 𝑟2)
𝑟2𝑗
3
𝑛
𝑗=3 −
(𝑟 𝑗− 𝑟1)
𝑟1𝑗
3 }

16. 15
For the effect from sun and moon perturbations put j = 3 for sun and j = 4 for moon. The system
of 4 bodies can then be solved by numerical integration.
The sun and moon perturbations are majorly responsible for the inclination change of the satellite orbit.
The change in inclination due to the sun is approximately 0.3 degrees annually while that because of
moon is about 0.6 degrees annually. The effect of moon is about double that of sun because even though
the sun has a huge size and mass as compared to moon, the distance of sun from earth more than
compensates for it. Now let us see how it affects the inclination in the long term. The equator of the earth
is inclined to the elliptic plane by 23.5 degrees. During mid winter of one of the hemispheres the
geostationary satellite (assuming it is constantly exactly above equator) is above the elliptic plane for one
half of the day while below for the other half.(refer Mattias Soop ). The part where it is crosses the elliptic
plane is when the ‘y’ coordinate in the eci system turns zero. This means the sun is located at the south
side of the orbit. Similarly in mid-summers it will be in the north side. Exactly in the mid-autumn and
mid-spring the sun will be in the orbital plane.
At midwinter and midsummer, the sun is in the Y-Z plane of the inertial system, above or below
the geostationary orbit plane. The sun exerts a mean gravity force F on the satellite directed
toward its center of mass. At half the day, when the sun is to the north of the orbit plane, a
positive average attracting force F + ∆F is exerted on the satellite. During the second half of the
day, an average force F - ∆F is exerted on the satellite, since the satellite is now more distant
from the sun and so less attracted to it (refer to M. Soop). The net effect is that the remaining
average moment on the orbit about the X axis will lead to a movement of the orbital pole about
the Y axis, toward the positive X axis. This is equivalent to the increase of the inclination i. In
the other half of the year, when the sun is to the south of the orbit plane, the net moment applied
on the orbit has the same direction as in the first part of the year, and an additional increase of
the inclination follows. Exactly in midspring and midautumn, the sun is in the equatorial plane,
so the force it exerts on the satellite now is not perpendicular to the direction of motion but lies
in the orbital plane so that no moment is applied on the orbit and the change in the inclination is
null. Between the seasons, smaller changes of the inclination are induced, but the gravitational
force of the sun exerted on the satellite causes an overall net increase of the inclination.
The same theory can be applied to the moon just the period here will not be one year. Clearly it
can be seen that they also change the RAAN of the orbit.
Solar Radiation Pressure
We all know that electromagnetic radiations contain photons that carry momentum with them
and thus are force carriers. The effect of force exerted by photons on absorption or reflection
from a surface can be seen on both microscopic and macroscopic levels. Solar radiations
comprise of all the electro-magnetic waves radiated by the sun with wavelengths from x rays to
radio waves. When these photons in the solar radiation are absorbed into or get reflected back
from the satellite surface the satellite experiences a pressure and hence a small force. This force
is non-conservative in nature. The pressure is proportional to the momentum flux (momentum per
unit area per unit time) of the radiation. The mean solar energy flux of the solar radiation is
proportional to the inverse square of the distance from the sun. The mean integrated energy flux
at the earth's position is given by

17. 16
𝑒𝑛𝑒𝑟𝑔𝑦 𝑓𝑙𝑢𝑥 =
1,358𝑊
1.004 + 0.0334 cos(𝐷)
𝑊/𝑚2
( Spacecraft Dynamics and Control, Sidi)
Here D is the "phase" of the year, which is calculated as starting on July 4, the day of earth
aphelion . This is equivalent to a mean momentum flux of
𝑃 =
𝑒𝑛𝑒𝑟𝑔𝑦 𝑓𝑙𝑢𝑥
𝑐
= 4.5 × 10−6
𝑘𝑔 𝑚−1
𝑠𝑒𝑐−2
Here c is the velocity of light.
The solar radiation pressure F is proportional to P, to the cross-sectional area A of the satellite
perpendicular to the sun line, and to a coefficient Cp that is dependent on the absorption
characteristic of the spacecraft.
F = P.A.Cp
The value of Cp is ( 1 + 𝜀 and lies between 0 and 2; Cp = 1 for a black body, a perfectly
absorbing material, whereas Cp = 2 for a body reflecting all light back toward the sun, ∈ is the
reflectivity of the surface of the satellite. For typical materials used in construction of satellites
the value of 𝜀 varies between 0.2 and 0.9.
For a more general case, for a surface that reflects the fraction 𝜀of the incident radiation and
absorbs the fraction (1 + ∈ ) of it the force due to solar radiation is
𝐹 = −𝑃. cos(𝜃) . 𝐴[(1 − 𝜀) 𝐞 + 2𝜀 cos(𝜃) 𝒏] .
Here Acos(𝜃) is the cross sectional area of the satellite perpendicular to the sun line, e is the
vector from the satellite to the sun and n is the normal area vector of the surface of the satellite.
(figure on next page)
Due to the eccentricity of the earth’s orbit around the sun the distance between the earth orbiting
satellite and the sun varies between 147x106
km and 152 x 106
km during the course of the year.
This results in an annual variation of the solar radiation pressure by about ∓ 3.3%, since the
solar flux decreases with the square of the distance from the sun. accounting for this dependence,
Illustration taken
from Montenbruck
page no. 78

18. 17
one finally obtains the following expression for the acceleration of the satellite due to the solar
radiation pressure.(Montenbruck eq. 3.73)
𝑟̈ = −𝑃
𝐴𝑈2
𝑟2
𝐴
𝑚
cos(𝜃) [(1 − 𝜀) 𝐞 + 2𝜀 cos(𝜃)𝐧]
Here m means the mass of the satellite and cos( 𝜃) = 𝒏 𝑻
𝒆
AU is the Astronomical Unit = mean distance between the sun and earth (approx. 1.5 x 108
km)
For many applications it can suffice to assume that the normal vector n points in the direction of
the Sun. This means that the above equation can be simplified to
𝑟̈ = −𝑃 (1 + 𝜀)
𝒓
𝑟3
𝐴
𝑚
𝐴𝑈2
In computing the effect of solar radiation on satellite another function called the shadow function
is also used. Because the sun may not be completely visible from the satellite i.e. the satellite
may not be in complete sunlight the shadow function 𝝂 is multiplied to the equation above. It
considers the fact that the shadow of the sun on the satellite due to the earth changes the amount
of solar radiation reaching the satellite surface. In this project we have considered the shadow
function to be 1 i.e. the satellite is in complete sunlight.
The SRP perturbation affects the eccentricity in a major way. The sun applies radiation pressure
on the satellite that results in an acceleration of the satellite in the sun-satellite direction. This
solar pressure on the geostationary satellite produces a long-period cyclic perturbation in the
orbit eccentricity, with no change in the semi-major axis. Suppose that the orbit is initially
circular. The effect of the solar pressure on the satellite, integrated over one part of the orbit, may
be approximated by a differential increment ∆V at point 1 added to the initial circular velocity
Vc1. The solar pressure exerted on the satellite during the opposite part of the orbit produces the
same increment in velocity and in the same direction as before , but now it is located at point 2,
so that ∆V is now opposite to Vc2. The increment of velocity at point 1 tends to increase the
altitude of the orbit at point 2, which is similar to creating an apogee at point 2. On the other
hand the decrease in velocity at point 2 tends to reduce the altitude of the orbit at point 1. This
decrease of altitude has the effect of creating a perigee. The net result is that a small eccentricity
vector has been created with its direction perpendicular to the solar radiation direction. Assuming
that the area/mass ratio and reflectance of the satellite are constant over a long annual period that
the sun is in the equatorial plane and that the distance of the sun to the satellite is constant over
the year. With these assumptions the eccentricity vector will increase by constant increments
perpendicular to the sun pressure direction over constant time periods. Because the sun has an
apparent circular motion around the earth the tip of the eccentricity vector will describe a circle
called the eccentricity circle. The radius of this circle depends on the solar pressure P and also
on the physical properties of the reflectance Cr = (1 + 𝜀).

19. 18
Atmospheric Drag
An earth-orbiting satellite when in the atmosphere experiences a drag force from the air. For our
discussion we need to focus only on geostationary satellites which are high-altitude satellites and
are above the atmospheric layers so they do not encounter any major drag force and we can
safely neglect its effect.
The perturbations mentioned above were all major perturbations that one needs to look into for a
geostationary satellite. But if we want more accurate and precise determination of the orbit (e.g.:
for geodetic satellites) we need to consider minor perturbations also.
High Precision Modeling
High precision modeling of the satellite orbit takes into account very small perturbations which
can otherwise be neglected, if not needed. These perturbations change the radial position of the
satellite by around 10 cm or less.
Earth Radiation Pressure
(reference no. 3 and 6)
In addition to the direct solar radiation pressure, the radiation emitted by the earth also leads to a
small pressure on the satellite. Two components are distinguished: the shortwave optical
radiation and the long-wave infrared radiation in both cases the magnitude of the acceleration of
the satellite decreases with increasing altitude because of the inverse square law of the emitted
radiation pressure. As the altitude increases, the illuminated surface of the earth from where the
radiation is coming also increases which means there is more radiation coming from the
increased illuminating surface section giving an increase in acceleration but it only partially
compensates for the decrease in the acceleration due to inverse square dependence. The amount
of radiation falling on a satellite in vicinity from earth depends on altitude, the solar angle and
the position over earth including both latitude and longitude of the satellite. Also on earth
different parts have different reflectivity. Maximum reflection is caused over and around the
poles because the ice is a good reflector while the tropical regions tend to have low reflectivity
The shortwave optical radiation is the radiation from the sun that gets reflected and scattered
from the earth’s surface back to the satellite. It is called earth’s albedo and so is the effect
named the earth’s albedo effect. This reflection is described by the albedo factor a that is the
fraction of the shortwave radiation that is reflected back from earth into space to the total
shortwave solar radiation falling on earth . Many satellites are influenced by earth’ albedo and
yet very few model schemes exist, in order to predict this phenomenon. Earth albedo is
sometimes treated as noise or ignored completely. For instance, while applying solar cells in the

20. 19
attitude hardware, earth albedo can cause the attitude estimation to deviate with as much as 20
degrees. The mean reflectivity calculate for the earth is around 30.4 %. We will use the value of
a = .34 which is equivalent to a radiation of 459 W/m2
of the earth’s surface elements. It is
emitted only by the daylight side of the earth and may vary due to cloud coverage too.
Some part of the radiation that falls on the earth surface from sun is reflected back but some
fraction of it is absorbed by the surface. This energy is used to maintain the temperature of the
crust and the atmosphere and along with the energy from the core, it is also used for the chemical
and physical changes in and around earth. So earth has a temperature and thus keeps on radiating
infrared radiations just to maintain balance in the energy cycle. This radiation, in contrast to
the albedo effect, is a near isotropic remission of the direct solar radiation. So it can be safely
asserted that the amount of radiation that the earth absorbs is all emitted as infrared radiation into
space uniformly. The average emissivity 𝜖 of earth’s surface is approximately 0.68. Since the
irradiated earth cross section i.e. the projection area of the earth that receives the solar radiation
is 𝝅𝑹𝒆 𝟐
and the area that radiates the infrared radiation is 4 𝝅𝑹𝒆 𝟐
the contribution of this
radiation to the flux is reduced by a factor of 4. Hence the effective radiation of earth surface
elements to infrared emission is approx. 230W/m2
.
So now the acceleration of the spacecraft due to all the Earth radiation can be summed up from j
( j = 1,2,3………,N) individual terms, corresponding to different earth area elements dAj
(Montenbruck chapter 3)
𝑟̈ = ∑(1 + 𝜀)
𝑁
𝑗=1
(𝑎𝑗 cos(𝜃𝑗
𝐸
) +
1
4
𝜖𝑗 ) 𝑃
𝐴
𝑚
cos(𝜃𝑗
𝑆
)
𝑑𝐴𝑗
𝜋𝑟𝑗
2
𝒆𝒋
𝜃𝑗
𝐸
and 𝜃𝑗
𝑆
are the angles of the earth’s surface and the satellite surface normal vectors to the
incident radiations on them respectively. The unit vector 𝒆𝒋 points in the diraection from the
earth surface element to the satellite, while the distance is 𝑟𝑗 . 𝜺 is the reflectivity if the
satellite and 𝝐𝒋 is the emissivity of the surface element on earth.
In this project instead of summing up for entire earth surface we took a mean value of earth
surface emissivity as 0.68. The term due to albedo becomes
𝑟̈ = (1 + 𝜀) 𝑃𝑎
𝐴
𝑚
𝑅𝑒2
𝑟3
𝒓
And the term due to infrared radiation from earth is

21. 20
𝑟̈ = (1 + 𝜀)
1
4
𝜖 𝑚𝑒𝑎𝑛 𝑃
𝐴
𝑚
𝑅𝑒2
𝑟3
𝒓
Earth Tides
The gravitation of the sun and moon exert a direct force on earth satellites, as discussed earlier.
In addition, those forces are also acting on the body of the earth and thus lead to a time-varying
deformation of the earth. The small periodic deformations of the solid body of the earth are
called solid earth tides. The oceans respond n a different way to the luni-solar tidal
perturbations, known as ocean tides. As a consequence, the earth’s gravity field is no longer
static in nature, but exhibits small periodic variations, which also affect the motion of satellites.
The earth’s tidal potential leads to an elastic deformation of the earth. This may mathematically
by a linear relation of the tidal potential U2 and the resulting perturbed gravity potential UT, the
ratio of both potentials being the love number k. a completely stiff body would therefore have a
vanishing love no.
The earth is, however, only elastic to first order. Deviations from an elastic tidal response are due
to the rate-dependent behavior of the terrestrial fluids, like earth’s inner core and the oceans, as
well as friction. These cause phase lags of the tidal bulge with respect to the position of the sun
and the moon.
The changes induced by the solid Earth tides in the free space potential are most conveniently
modeled as variations in the standard geopotential coefficients Cnm and Snm . The contributions
Δ𝐶 𝑛𝑚 and Δ𝑆 𝑛𝑚 from the tides are expressible in terms of the k Love number. The effects of
ellipticity and rotation of the Earth on tidal deformations necessitates the use, in general, of three
k parameters, 𝑘 𝑛𝑚
(0)
𝑎𝑛𝑑 𝑘 𝑛𝑚
(±)
to characterize the changes produced in the free space
potential by tides of spherical harmonic degree and order (nm). Anelasticity of the earth’s mantle
causes 𝑘 𝑛𝑚
(0)
𝑎𝑛𝑑 𝑘 𝑛𝑚
(±)
to acquire small imaginary parts (reflecting a phase lag in the
deformational response of the Earth to tidal forces), and also gives rise to a further variation with
frequency which is particularly pronounced within the long period band.
The degree 2 tides produce time dependent changes in C2m and S2m, through 𝑘2𝑚
(0)
which can
exceed 10−8
in magnitude. They also produce changes exceeding a cutoff of 3x10-12
in C4m and
S4m through 𝑘2𝑚
(+)
but in this project it was safely assumed to be negligible.
The computation of the tidal contributions to the geo-potential coefficients is most efficiently
done by a two-step procedure. In Step 1, the (2m) part of the tidal potential is evaluated in the
time domain for each m using lunar and solar ephemerides, and the corresponding changes Δ𝐶2𝑚
and Δ𝑆2𝑚 are computed using frequency independent nominal values k2m for the respective
𝑘2𝑚
(0)
.
Step 2 corrects for the deviations of the 𝑘21
(0)
of several of the constituent tides of the diurnal
band from the constant nominal value k21 assumed for this band in the first step. Similar
corrections need to be applied to a few of the constituents of the other two bands also. These are

22. 21
frequency dependent corrections and are of the order of 10-12
and below range and this report
neglects them too but for further accuracy they can be considered too (refer Doodson McCarthy).
With frequency-independent values knm (Step 1), changes induced by the (nm) part of the tide
generating potential in the normalized geo-potential coefficients having the same (nm) are given
in the time domain by ( chapter 6, McCarthy Doodson, IERS Technical Note 21)
∆𝐶̃ 𝑛𝑚 + 𝑖∆𝑆̃ 𝑛𝑚 =
𝑘 𝑛𝑚
2𝑛 + 1
∑
𝐺𝑀𝑗
𝐺𝑀
(
𝑅𝑒
𝑟𝑗
)
𝑛+1
𝑃̃𝑛𝑚(𝑠𝑖𝑛 ∅ 𝑗)𝑒−𝑖𝑚𝜆 𝑗
3
𝑗=2
(with Sn0 = 0), where
knm = nominal degree Love number for degree n and order m,
Re = equatorial radius of the Earth,
GM = gravitational parameter for the Earth,
GMj = gravitational parameter for the Moon (j = 2) and Sun (j = 3),
rj = distance from geocenter to Moon or Sun,
∅ 𝑗 = body fixed geocentric latitude of Moon or Sun,
𝜆𝑗 = body fixed east longitude (from Greenwich) of Moon or Sun,
and 𝑃̃𝑛𝑚is the normalized associated Legendre function related to the classical (unnormalized)
one by 𝑃̃𝑛𝑚 = 𝑁 𝑛𝑚 𝑃𝑛𝑚 i.e.
𝑃𝑛𝑚
̃ = √
( 2 − 𝛿0𝑚)(2𝑛 + 1)(𝑛 − 𝑚)!
(𝑛 + 𝑚)!
𝑃𝑛𝑚
{
𝐶 𝑛𝑚
𝑆 𝑛𝑚
} = √
( 2 − 𝛿0𝑚)(2𝑛 + 1)(𝑛 − 𝑚)!
(𝑛 + 𝑚)!
{
𝐶 𝑛𝑚
̃
𝑆 𝑛𝑚
̃
}
These values of the changes in geo-potential coefficients Is converted to unnormalized form and
then added to the coefficients used to evaluate the geo-potential perturbations (earth’s
oblateness). The values of k20 and k21 and k22 used are 0.29525, 0.29470, 0.29801
respectively.(McCarthy Doodson chapter 6)
Ocean Tides have a more dynamic and periodic effect with amplitudes one order of magnitude
less than the solid tides, so we did not consider these as perturbations but their effects can be
counted as noise.
CODE
% degree 2 love numbers for solid tides
k20 = 0.29525;
k21 = 0:29470;
k22 = 0.29801;

23. 22
% multiplication factor of normalized and unnormalizd geopotential coeff
N20 = 2.236054;
N21 = 0.64363756;
N22 = 0.6443;
%solid tides
slat2 = atan2((Z3-Z1),(norm(X3-X1,Y3-Y1)));
% latitude of the sun
mlat2 = atan2((Z4-Z1),(norm(X4-X1,Y4-Y1)));
% latitude of the moon
ecefs= [ cos(hh) sin(hh) 0 ; -sin(hh) cos(hh) 0 ; 0 0 1]*[X3-X1 Y3-Y1 Z3-Z1]';
slong2=atan2(ecefs(2),ecefs(1)); % longitude of sun
ecefm= [ cos(hh) sin(hh) 0 ; -sin(hh) cos(hh) 0 ; 0 0 1]*[X4-X1 Y4-Y1 Z4-Z1]';
mlong2=atan2(ecefm(2),ecefm(1)); %longitude of moon
LEGENDRE POLYNOMIALS AS FUNCTION OF SUN AND MOON LATITUDE
Ps20 = (3*(sin(slat2))^2 - 1)/2;
Ps21 = 3*sin(slat2)*cos(slat2);
Ps22 = 3*(cos(slat2))^2;
Pm20 = (3*(sin(mlat2))^2 - 1)/2;
Pm21 = 3*sin(mlat2)*cos(mlat2);
Pm22 = 3*(cos(mlat2))^2;
%l = 2 , m=0
q20 = (G*m3/R13)*N20*Ps20 + (G*m4/R14)*N20*Pm20;
u20 = 0;
% normalized geopotential coefficients
delta_nC20 = (k20/5)*(Re^3/(G*m1))*q20;
delta_nS20 = (k20/5)*(Re^3/(G*m1))*u20;
% l=2, m=1;
q21 = (G*m3/R13^3)*Ps21*cos(slong2) + (G*m4/R14^3)*Pm21*cos(mlong2);
u21 = (G*m3/R13^3)*Ps21*sin(slong2) + (G*m4/R14^3)*Pm21*sin(mlong2);
%
delta_C21 = (k21/5)*(Re^3/G*m1)*q21;
delta_S21 = (k21/5)*(Re^3/G*m1)*u21;
% l= 2, m=2;

24. 23
q22 = (G*m3/R13)*N22*Ps22*cos(2*slong2) + (G*m4/R14)*N22*Pm22*cos(2*mlong2);
u22 = (G*m3/R13)*N22*Ps22*sin(2*slong2) + (G*m4/R14)*N22*Pm22*sin(2*mlong2);
%
delta_nC22 = (k22/5)*(Re^3/G*m1)*q22;
delta_nS22 = (k22/5)*(Re^3/G*m1)*u22;
C20 = N20*(-484.165368e-6 + delta_nC20);
S20 = N20*(0 + delta_nS20);
C22 = N22*(2.439e-6 + delta_nC22);
S22 = N22*(-1.400266e-6 + delta_nS22);
Relativistic Effects
A rigorous treatment of the satellite’s motion should be formulated in accordance with the theory
of general relativity. While the special theory of relativity considers a flat four-dimensional
space-time, this is no longer true in the vicinity of the earth. Instead, the earth’s mass M with a
potential of U = GM / r and angular momentum lead to a curvature of the four-dimensional
space-time. Since the moon is much smaller than earth in size and mass and the sun is too far
away to have a considerable influence their relativistic effects on the satellite are neglected. The
calculation procedure is out of the scope of this project and thus we will only use a simplified
relation for the relativistic correction term in the Newtonian equation of motion under gravity.
The additional post-Newtonian space-time correction term of the acceleration is (ref. Montenbruck
chapter 3)
𝑟̈ = −
𝐺𝑀
𝑟2
((
4𝐺𝑚
𝑐2 𝑟
−
𝑣2
𝑐2
) 𝒆 𝒓 + 4
𝑣2
𝑐2
( 𝒆 𝒓 . 𝒆 𝒗 ) 𝒆 𝒗 )
here 𝒆 𝒓 and 𝒆 𝒗 denote the unit position and unit velocity vector of the satellite in the e.c.i. frame.
For geostationary satellite the orbit is a near circle so 𝒆 𝒓 and 𝒆 𝒗 are perpendicular to each other
and the velocity square is 𝑣2
=
𝐺𝑀
𝑟
. so the terms simplifies to
𝑟̈ = −
𝐺𝑀
𝑟2
(3
𝑣2
𝑐2
) 𝒆 𝒓
This means the additional relativistic term is the Newtonian acceleration term multiplied by a
factor of 3 𝑣2
/𝑐2
which is roughly 3x10-8
for a typical satellite. Typically the effect of this
term on the radial position of the satellite is in centimeters.

25. 24
PLOTS
For one day

26. 25

27. 26
For one year

28. 27

29. 28

30. 29

31. 30
Orbit Determination section
Obtaining the measured state
Absence of direct telemetry data from ground stations in our code led us to obtain the measured
state data from the computed force model (with all the perturbations) data by the following
suitable relations
𝜌 = √𝑟2 + 𝑅2 − 2𝑟𝑅𝑐𝑜𝑠(𝑙 − 𝜆)𝑐𝑜𝑠𝜙 1
𝑠𝑖𝑛𝜀 =
𝑟𝑐𝑜𝑠(𝑙−𝜆)𝑐𝑜𝑠𝑅
𝜌
2
tan α = tan(l − λ)/sinϕ 3
tan (φ’)=tan(φ)*(1-f2
) 4
Here ϱ = the measured range r = distance of satellite from center of the Earth
R= radius of earth l= the longitude of the station
λ = longitude of the satellite φ’= geocentric latitude of the station
α = Azimuthal angle of the satellite from the local topocentric frame of the ground station
ε = Elevation angle of the satellite from the local topocentric frame of the ground station
φ = geodetic latitude of the station f = oblateness of earth (= 1/298.257)
We need to first find the geodetic latitude from the geocentric by using the (4) formula. ‘r’ is
obtained from the force model data and rest all the other terms are known. Taking norm of
position of satellite with respect to earth at every instant of time step we get arrays of range,
elevation, azimuthal, measurements. Their respective rates can be obtained numerically.
𝜌̇ = (𝛒(𝐭) − 𝛒(𝐭 − 𝟏)) /𝐝𝐭 .̇
This gives us the measured state vector

32. 31
Calculation of the state vector from measurements of range,
angular position and their rates
The fundamental vector triangle is formed by the topocentric position vector ϱ of
a satellite relative to a tracking station, the position vector R of the station relative
to the center of attraction C and the geocentric position vector r . The relationship among
these three vectors is given below. All the vectors are with respect to inertial coordinates
𝑟 = 𝑅 + 𝜌𝜌̂
𝑣 = 𝑟̇ = 𝑅̇ + 𝜌𝜌̂̇ + 𝜌𝜌̂̇
= unit vector along range vector
= angular velocity vector of earth
R=position vector of station
𝜌̇ = range rate 𝜌̂̇= unit range rate vector
V=velocity
Calculating Formulas and terms
Using the altitude H, latitude φ of the site and local sidereal time θ of the site, Re radius of
earth calculate its geocentric position vector R of the site.
𝑅 = [
𝑅𝑒
√(1 − (2𝑓 − 𝑓2)𝑠𝑖𝑛2 𝜙)
+ 𝐻] 𝑐𝑜𝑠𝜙(𝑐𝑜𝑠𝜃𝐼̂ + 𝑠𝑖𝑛𝜃𝐽)̂
+ [
𝑅𝑒(1 − 𝑓)2
√(1 − (2𝑓 − 𝑓2)𝑠𝑖𝑛2 𝜙)
+ 𝐻]𝑠𝑖𝑛𝜙𝐾̂

33. 32
Since topocentric right ascension ( )/ declination ( ) are related to the azimuth (A) /elevation
(a) are related. They are calculated as follows
𝑠𝑖𝑛𝛿 = 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝑎 + 𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝑎
For simplifying the equations for ( ) calculations we introduce the hour angle h,
ℎ = 𝜃 − 𝛼
h is the angular distance between the satellite and the local meridian (site longitude) . If h is
positive, the object is west of the meridian; if h is negative, the object is east of the meridian.
ℎ = {2𝜋 − 𝑐𝑜𝑠−1
((𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝑎 − 𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝐴cos a)/cos𝛿) 0 < 𝐴 < 𝜋
𝑐𝑜𝑠−1
(
𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝑎−𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝐴 𝑐𝑜𝑠 𝑎
𝑐𝑜𝑠𝛿
) 𝜋 < 𝐴 < 2𝜋
Therefore 𝛼 = 𝜃 − ℎ
The direction cosine unit vector is calculated as follows
𝜌̂ = 𝑐𝑜𝑠𝛿(𝑐𝑜𝑠𝛼𝐼̂ + 𝑠𝑖𝑛𝛼𝐽̂) + 𝑠𝑖𝑛𝛿𝐾̂
Geocentric position vector r is finally calculted by 𝑟 = 𝑅 + 𝜌𝜌̂
And the inertial velocity ˙R of the site 𝑅̇ = ΩxR
Declination and right ascension angle rates and rate of change of unit range vector
𝛿̇ =
1
𝑐𝑜𝑠𝛿
[−𝐴̇ 𝑐 𝑜𝑠𝜙𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝑎 + 𝑎̇(𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝑎 − 𝑐𝑜𝑠𝜙𝑐𝑜𝑠𝐴𝑠𝑖𝑛𝑎)]
𝛼̇ = 𝜔 𝐸 +
𝐴̇ 𝑐 𝑜𝑠𝐴𝑐𝑜𝑠𝑎−𝑎̇ 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝑎+𝛿̇ 𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝑎 𝑡𝑎𝑛𝛿
𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝑎−𝑠𝑖𝑛𝜙𝑐𝑜𝑠𝐴𝑐𝑜𝑠𝑎
𝜌̂̇ = (−𝛼̇ 𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛿 − 𝛿̇ 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛿)𝐼̂ + (𝛼̇ 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛿 − 𝛿̇ 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛿)𝐽̂ + 𝛿̇ 𝑐𝑜𝑠𝛿𝐾̂
Finally the velocity of satellite in the ECI frame is 𝑉 = 𝑅̇ + 𝜌̇ 𝜌̂ + 𝜌𝜌̂̇

34. 33
Batch Processing and Sequential Processing
Batch processing: We use it to find the initial position of the satellite which we have taken to be
known throughout the high fidelity force model .This is a method in which we have more
equations than the no of variables in measurements taken at different instants of time over a
period of time. We use it to find the initial position using least squares method at a particular
epoch at one of the measured instants.
Sequential Processing: The best known method of this type is Kalman filter –Classical and
Extended. The Extended is used in filtering in case of Non-linear equations. Here we use
extended since orbit determination model is non-linear .We first linearize the state space form
using the Taylor’s expansion. We first assume that we know the error covariance matrix initially
is known. The noise in the measurements is assumed to be of the white noise (Gaussian) form
that is with a mean of zero and a standard deviation of 1. The Kalman estimated value depends
whether there is more noise in the process or the measurements as it takes more weight of the
one with less covariance i.e. the one with less noise and it therefore it follows either process or
measurements. Unlike batch processing it gives estimated value at each measurement.
The Kalman Filtering
Theory: Linearizing the Non-Linear model of satellite Orbit for Kalman filtering
𝑥̇ = 𝑓(𝑥, 𝑢 𝑑, 𝑡) + 𝑢(𝑡)
Z=h(x,t) +v(t)
f and h are known functions relating state to state derivatives
ud is a deterministic forcing function and u and v are white noises (whose initial estimates we assumed
to be known)
x(t)=x*
(t) +∆𝑥(𝑡) x*
(t)=approximate state as obtained from by force model 3
∆𝑥=addition to x*
to get true (corrected) state x (it’s assumed
small )
𝑥∗̇ + ∆𝑥̇ = 𝑓(𝑥∗
+ ∆𝑥, 𝑢 𝑑, 𝑡) + 𝑢(𝑡)
Z=ℎ(𝑥∗
+ ∆𝑥, 𝑡) + 𝑣(𝑡)
Applying Taylor’s expansion to first order
𝑥̇∗
+ ∆𝑥̇ ≈ 𝑓(𝑥∗,̇ 𝑢 𝑑, 𝑡) + [
𝜕𝑓
𝜕𝑥
] 𝑥=𝑥∗. ∆𝑥+u(t)

35. 34
𝑧 ≈ ℎ(𝑥 ∗, 𝑡) + [
𝜕ℎ
𝜕𝑥
] 𝑥=𝑥∗. ∆𝑥+v(t)
Where F=
𝜕𝑓
𝜕𝑥
= ⌈
𝜕𝑓1/𝜕𝑥1 𝜕𝑓1/𝜕𝑥2 … … . .
𝜕𝑓2/𝜕𝑥1 𝜕𝑓2/𝜕𝑥2 … . .
… … . … . .
⌉ H=
𝜕ℎ
𝜕𝑥
= ⌈
𝜕ℎ1/𝜕𝑥1 𝜕ℎ1/𝜕𝑥2 … … . .
𝜕ℎ2/𝜕𝑥1 𝜕ℎ2/𝜕𝑥2 … . .
… … . … . .
⌉
Since, 𝑥̇ = 𝑓(𝑥∗,̇ 𝑢 𝑑, 𝑡) the linearized model of state (8) and measurement (9) as
follows
∆𝑥̇ ≈̇ + [
𝜕𝑓
𝜕𝑥
] 𝑥=𝑥∗. ∆𝑥+u(t) 8
[𝑧 − ℎ(𝑥 ∗, 𝑡)] = [
𝜕ℎ
𝜕𝑥
] 𝑥=𝑥∗. ∆𝑥+ v(t) 9
The Extended Kalman Filtering Algorithm
Convention: ̂ 𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 −
𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑝𝑟𝑒𝑣𝑖𝑜𝑠 𝑠𝑡𝑎𝑡𝑒
Theory : 𝑧 − ℎ(𝑥 ∗) = 𝐻∆𝑥 + 𝑣 𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑠𝑒𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
H=measurement matrix at time ‘k’
𝑧 𝑘̂−
= 𝐻 𝑘 × 𝑥 𝑘̂−
= predicted measurement value at Kth time instant
𝑥 𝑘̂−
=predicted state value at Kth time instant
𝑥 𝑘̂ = 𝑥 𝑘̂−
+ 𝐾𝑘(𝑧 𝑘 − 𝑧 𝑘̂−
) 𝐾𝑘 = 𝐾𝑎𝑙𝑚𝑎𝑛 𝐺𝑎𝑖𝑛
𝐾𝑘 = 𝑃𝑘
−
𝐻(𝐻𝑃𝑘
−
𝐻 + 𝐸𝑧)−1
𝑃𝑘 = (𝐼 − 𝐾𝑘 𝐻 𝑘)𝑃𝑘
−
𝑃𝑘 = 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 𝐸𝑟𝑟𝑜𝑟 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥
𝑃𝑘
−
= 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝐸𝑟𝑟𝑜𝑟 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑚𝑎𝑡𝑟𝑖𝑥
Ez or 𝑄 𝑘= Expected noise in measurements

36. 35
𝑃𝑘+1
−
= 𝜑 𝑘 𝑃𝑘 𝜑 𝑘
𝑡
+ 𝑄 𝑘 𝑃𝑘+1
−
= 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑢𝑠𝑖𝑛𝑔
the previous instant’s corrected error covariance
And the state transition matrix
𝜑 𝑘= Is obtained by the MATLAB command expm on the Fdt i.e. expm(dt)
F = state matrix i.e
𝜕𝑓
𝜕𝑥
dt = step size
Figure : Kalman filter estimate of satellite orbit for less measurement noise

37. 36
Figure : Kalman filter estimate of satellite orbit for less process noise
Figure : Relative range of second satellite from reference satellite

38. 37
Figure : Estimated 7 satellite orbits
Figure : Relative range of 4 geosynchronous satellites from reference GEO satellite

39. 38
Figure : Latitude vs Longitude for 3 GEO satellites
Figure : Latitude vs Longitude for 4 geosynchronous satellites

40. 39
APPLICATION OF THE RINEX FORMAT
IRNSS
Introduction :
The Indian Regional Navigation Satellite System (IRNSS) is planned to be a constellation of 7 satellites – 3
in GEO and 4 in GSO orbit. This constellation is expected to provide position accuracies similar to GPS in
a region centered on India with a coverage extending up to 1500 km around India. It is a basically
position determining system. From the satellite constellation, the user position can be solved. However
the equations required for solving the user position turned out to be nonlinear simultaneous equations,
which are difficult to solve. In addition, some practical considerations (the inaccuracy of the user clock)
will be included in the equations. These equations are solved through a linearization and iteration
method. The solution is in a Cartesian coordinate system. The user position is then translated into the
earth-based coordinate system.
Basic equations for finding user position
𝝆 = √(𝒙𝟏 − 𝒙) 𝟐 + (𝒚𝟏 − 𝒚) 𝟐 + (𝒛𝟏 − 𝒛) 𝟐
with three such equations from three receivers and three variables, satellite coordinates can be found.
Measurement of pseudo range
𝝆 𝒓 = 𝒄 ∗ (𝒕 𝒖 − 𝒕 𝒔𝒊)
where, c is speed of light, 𝝆 𝒓 is pseudo range from user to satellite, 𝒕 𝒔𝒊 is referred as true transmission
time and 𝒕 𝒖 is referred as true receiver time.
Doppler frequency shift
Doppler frequency is caused by the satellite velocity component towards the user. It arrives due to
difference in solar and sidereal day. The time difference of 3min, 55.91 sec in one day causes the
satellite to move 914 km more which is around 2 to 2.6 degree angle . if the satellite is closer to the
horizon, this angle is around 2 degrees. Doppler velocity is expected to be maximum when satellite is
closer to the horizon.

41. 40
Satellite position Calculations
The position of satellites plays a major role in Relative Kinematic Positioning so it becomes very
important to determine satellite positions correctly. Satellite positions can be determined with the help
of IRNSS message that a satellite sends to the receiver. Following is a step-wise procedure to find the
satellite positions
1. Compute Mk (mean anomaly), given e, 𝑀𝑜, ∆n, 𝛼1/2
, toe(time of reference epoch):
𝒏 𝟎 = √ 𝝁/𝒂 𝟑
- For near circular orbits, there is little difference between ∆𝒏 and 𝒅𝝎/𝒅𝒕.over the time
interval of interest. Absorbing the effect of either in ∆𝒏 and rather than in 𝒅𝝎/𝒅𝒕
simplifies the users algorithm by reducing sine and cosine computations.
-
∆𝒏 = 𝒅𝝎/𝒅𝒕
𝒏 = 𝒏 𝟎 + ∆𝒏
𝒕 𝒌 = 𝒕 − 𝒕 𝒐𝒆
𝑴 𝒌 = 𝑴 𝟎 + 𝒏𝒕 𝒌
2. Compute Eccentric Anomaly (Ek) by Iterative solution
𝑬 𝒌 − 𝒆 𝐬𝐢𝐧 𝑬 𝒌 = 𝑴 𝒌
3. Compute true anomaly as follows:
𝐭𝐚𝐧(𝜽 𝒌/𝟐) = √
𝟏+𝒆
𝟏−𝒆
𝐭𝐚𝐧
𝑬 𝒌
𝟐
4. Compute periodic corrections:
𝝋 𝒌 = 𝜽 𝒌 + 𝝎 𝒌 Arg. of latitude
𝜹𝝁 𝒌 = 𝑪 𝝁𝒔 𝐬𝐢𝐧 𝟐𝝋 𝒌 + 𝑪 𝝁𝒄 𝐜𝐨𝐬 𝟐𝝋 𝒌
𝜹𝒓 𝒌 = 𝑪 𝒓𝒔 𝐬𝐢𝐧 𝟐𝝋 𝒌 + 𝑪 𝒓𝒄 𝐜𝐨𝐬 𝟐𝝋 𝒌
𝜹𝒊 𝒌 = 𝑪𝒊𝒔 𝐬𝐢𝐧 𝟐𝝋 𝒌 + 𝑪𝒊𝒄 𝐜𝐨𝐬 𝟐𝝋 𝒌

42. 41
5. Apply the Schuler and periodic corrections to 𝜇 𝑘 , 𝑟 and 𝑖.
𝝁 𝒌 = 𝝋 𝒌 + 𝜹𝝁 𝒌
𝒓 𝒌 = 𝒂(𝟏 − 𝒆 𝐜𝐨𝐬 𝑬 𝒌) + 𝜹𝒓 𝒌
𝒊 𝒌 = 𝒊 𝟎 + 𝑰𝑫𝑶𝑻 ∗ 𝒕 𝒌 + 𝜹𝒊 𝒌
6. Compute satellite position in orbit plane (𝑥 𝑘 , 𝑦 𝑘)
𝒙 𝒌 = 𝒓 𝒌 𝐜𝐨𝐬 𝝁 𝒌
𝒚 𝒌 = 𝒓 𝒌 𝐬𝐢𝐧 𝝁 𝒌
7. Rotate from satellite frame to ECEF frame:
[
𝑿 𝑬,𝒌
𝒀 𝑬,𝒌
𝒁 𝑬,𝒌
] = [
𝒓 𝐜𝐨𝐬 𝛀 𝒆𝒓 𝐜𝐨𝐬 𝝋 𝒌 − 𝒓 𝐬𝐢𝐧 𝛀 𝒆𝒓 𝐬𝐢𝐧 𝝋 𝒌 𝐜𝐨𝐬 𝒊 𝒌
𝒓 𝐬𝐢𝐧 𝛀 𝒆𝒓 𝐜𝐨𝐬 𝝋 𝒌 + 𝒓 𝐜𝐨𝐬 𝛀 𝒆𝒓 𝐬𝐢𝐧 𝝋 𝒌 𝐜𝐨𝐬 𝒊 𝒌
𝒓 𝐬𝐢𝐧 𝝋 𝒌 𝐬𝐢𝐧 𝒊 𝒌
]
These satellite positions will be used to calculate the unit vectors from receivers to satellite. Then by
vector addition, the baseline vector can be determined.
Correction in IRNSS time
The IRNSS System Time start epoch shall be 00:00 UT on Sunday August 22nd 1999 (midnight between
August 21st and 22nd). At the start epoch, IRNSS System Time shall be ahead of UTC by 13 leap seconds.
In IRNSS time, leap seconds are not added. Also 𝑡 𝑜𝑒 is always changed for each Saturday-Sunday
midnight to 00.00
𝒕 𝒌 = 𝒕 − 𝒕 𝒐𝒆
𝒕 𝒌 = 𝒕 𝒌 + 𝟔𝟎𝟒𝟖𝟎𝟎 If 𝒕 𝒌 < −302400
𝒕 𝒌 = 𝒕 𝒌 − 𝟔𝟎𝟒𝟖𝟎𝟎 If 𝒕 𝒌 > 302400

43. 42
Ephemeris data
𝑴 𝟎 : mean anomaly at reference time.
∆𝒏 : mean motion difference from computed value.
√ 𝒂 : Square root of the semi major axis of the satellite orbit.
𝒆 : Eccentricity of the satellite orbit.
𝑻 𝑮𝑫 , 𝒕 𝟎𝒆 , 𝒂 𝒇𝟎 , 𝒂 𝒇𝟏 , 𝒂 𝒇𝟐 : clock correction parameters.
𝒕 𝟎𝒆 : Reference time ephemeris.
𝑪 𝝁𝒔 , 𝑪 𝝁𝒄 : Amplitude of the sine and cosine harmonic correction term to the argument of latitude,
respectively.
𝑪 𝒓𝒔 , 𝑪 𝒓𝒄 : Amplitude of the sine and cosine harmonic correction term to the orbit radius,
respectively.
𝑪𝒊𝒔 , 𝑪𝒊𝒄 : Amplitude of the sine and cosine harmonic correction term to the angle of inclination,
respectively.
𝛀 𝒆 : Longitude of the ascending node of orbit plane at weekly epoch.
Ω̇ : Rate of right ascension.
𝒊 : Inclination angle at reference time.
𝝎 : Argument of perigee.
IDOT: rate of inclination angle.

44. 43
SIGNAL, SERVICES AND DATA FORMAT
SIGNALS AND SREVICES
- The IRNSS will provide basically two types of services
1. Standard positioning system (SPS)
2. restricted services (RS)
-IRNSS signals contains two frequency components. L5 band and S band.
L5 = 1176.4 MHz
S = 2492.4 MHz
-total code period contains 1023 chips
-C/A code is 1ms long and repeats every millisecond.
DATA FORMAT
- the first row shows a C/A code with 1023 chips; the total length is 1 ms.
- The second row shows a navigation data bit that has a data rate of 50 Hz; thus a data bit is 20
ms long and contains 20 C/A code.
- Thirty data bits make a word that is 600 ms long in the third row.
- Ten words make a sub frame that is 6 seconds long
- The fifth row shows a page that is 30 seconds long and contains 5 sub frames.
- Twenty- five pages make a complete data set that is 12.5 min long
- The 25 pages of data can be referred to as a superframe
SUBFRAMES
- The first two words of each frame are TLM & HOW.
- TLM begins with 8 preamble bits 16 reserved and 6 parity bits.
- HOW – first 17 bits are the truncated Time OF Week (TOW) count
Next two bits are flag bits
Following 3 bits are sub frame ID to identify one of the sub frames
Last 8 bits used for parity.

45. 44
Ephemeris parameters in sub frame 1 :
Parameter No. of bits Scale factor Units
WN : week number 10 1 Week
Satellite accuracy 4 1
Satellite health 6 1
IODC : issue of data clock 10
𝑻 𝑮𝑫 : satellite group delay differential 8 𝟐−𝟑𝟏 Seconds
𝒕 𝒐𝒆 : satellite clock correction 16 𝟐 𝟒 Seconds
𝒂 𝒇𝟐 : satellite clock correction 8 𝟐−𝟓𝟓 Sec/sec^2
𝒂 𝒇𝟏 : satellite clock correction 16 𝟐−𝟒𝟑 Sec/sec
𝒂 𝒇𝟎 : satellite clock correction 22 𝟐−𝟑𝟏 seconds
Ephemeris parameters in sub frame 2 :
Parameter No. of bits Scale factor Units
IODE 8
𝑪 𝒓𝒔 16 𝟐−𝟓 Meters
∆𝒏 16 𝟐−𝟒𝟑 Semicircles/sec
𝑴 𝟎 32 𝟐−𝟑𝟏 Semicircle
𝑪 𝝁𝒄 16 𝟐−𝟐𝟗 Radians
𝒆 32 𝟐−𝟑𝟑 Dimensionless
𝑪 𝝁𝒔 16 𝟐−𝟐𝟗 Radians
√ 𝒂 32 𝟐−𝟏𝟗
√ 𝒎𝒆𝒕𝒆𝒓𝒔
𝒕 𝟎𝒆 16 𝟐 𝟒 seconds
Ephemeris parameters in sub frame 3 :

46. 45
Parameters No. of bits Scale factor Units
𝑪𝒊𝒄 16 𝟐−𝟐𝟗 Radians
𝛀 𝒆 32 𝟐−𝟑𝟏 Semicircles
𝑪𝒊𝒄 16 𝟐−𝟐𝟗 Radians
𝒊 32 𝟐−𝟑𝟏 Semicircles
𝑪𝒊𝒄 16 𝟐−𝟓 Meters
𝝎 32 𝟐 𝟑𝟏 Semicircles
Ω̇ 24 𝟐−𝟒𝟑 Semicircles/sec
IODE
IDOT 14 𝟐−𝟒𝟑 Semicircles/sec
Sub frame 4 :
- Pages 2, 3, 4, 5, 7, 8, 9 and 10 contains the almanac data for the satellite 25 through 32.these
pages may be designed for the other functions. The satellite ID of that page defines the format
and content.
- Page 17 contains special messages.
- Page 18 contains ionospheric and universal coordinated time (UTC).
- Page 25 contains antispoof flag, satellite configuration for 32 satellites, and satellite health for
satellites 25-32.
- Pages 1, 6, 11, 12, 16, 19, 20, 21, 22, 23, and 24 are reversed.
- Page 13, 14, and 15 are spares.
Sub frame 5 :
- Page 1-24 contain almanac data for the satellite 1 through 24.
- Page 25 contains satellite health for the satellite 1 through 24, the almanac reference time, and
the almanac week number.

47. 46
TYPICAL IRNSS NAVIGATION FILE:
IRNSS Satellite Geo Synchronous 7
2011 4 13 0 0 0 0.000000000000e+000 0.000000000000e+000 -3.084822098103e+000
0.000000000000e+000 7.344119967303e-004 0.000000000000e+000 2.053386303493e+002
0.000000000000e+000 0.000000000000e+000 2.316808399561e+000 0.000000000000e+000
5.070790423285e-001 0.000000000000e+000 -8.290254572999e-003 0.000000000000e+000
0.000000000000e+000 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 2 0 0 0.000000000000e+000 2.267820571094e-012 -2.597388312076e+000
0.000000000000e+000 7.272579022461e-004 0.000000000000e+000 2.053384584210e+002
0.000000000000e+000 0.000000000000e+000 2.316807824048e+000 0.000000000000e+000
5.070798628016e-001 0.000000000000e+000 2.925643005978e-002 -7.993241414440e-011
1.139545882656e-010 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 4 0 0 0.000000000000e+000 5.267823544808e-012 -2.121080370934e+000
0.000000000000e+000 7.108471689363e-004 0.000000000000e+000 2.053370717359e+002
0.000000000000e+000 0.000000000000e+000 2.316792640891e+000 0.000000000000e+000
5.070737223565e-001 0.000000000000e+000 7.793138535476e-002 -2.108771782564e-009
-8.528395973687e-010 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 6 0 0 0.000000000000e+000 6.846725810783e-012 -1.644201288144e+000
0.000000000000e+000 6.750078706320e-004 0.000000000000e+000 2.053348759362e+002
0.000000000000e+000 0.000000000000e+000 2.316757088335e+000 0.000000000000e+000
5.070703254065e-001 0.000000000000e+000 1.260505056242e-001 -4.937854972255e-009
-4.717986034569e-010 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 8 0 0 0.000000000000e+000 6.599814623207e-012 -1.135910608897e+000
0.000000000000e+000 6.164816511813e-004 0.000000000000e+000 2.053326392654e+002
0.000000000000e+000 0.000000000000e+000 2.316717181535e+000 0.000000000000e+000
5.070764306519e-001 0.000000000000e+000 1.427558899343e-001 -5.542611167315e-009
8.479507498321e-010 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 10 0 0 0.000000000000e+000 4.662968868179e-012 -5.864515583264e-001
0.000000000000e+000 5.708990811197e-004 0.000000000000e+000 2.053307259829e+002
0.000000000000e+000 0.000000000000e+000 2.316697648534e+000 0.000000000000e+000
5.070877709637e-001 0.000000000000e+000 1.183140307573e-001 -2.712916706282e-009
1.575043304623e-009 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 12 0 0 0.000000000000e+000 1.801689510410e-012 -5.220363351863e-002
0.000000000000e+000 5.577715371692e-004 0.000000000000e+000 2.053297378418e+002
0.000000000000e+000 0.000000000000e+000 2.316692753139e+000 0.000000000000e+000
5.070950319529e-001 0.000000000000e+000 1.091527920273e-001 -6.799160073228e-010
1.008470716410e-009 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 14 0 0 0.000000000000e+000 -1.215274374249e-012 4.739094103416e-001
0.000000000000e+000 5.615360049533e-004 0.000000000000e+000 2.053300776748e+002
0.000000000000e+000 0.000000000000e+000 2.316686431254e+000 0.000000000000e+000
5.070876292704e-001 0.000000000000e+000 1.081453841988e-001 -8.780396195303e-010
-1.028150346342e-0090.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 16 0 0 0.000000000000e+000 -3.931362241079e-012 9.886172606830e-001
0.000000000000e+000 5.901031828199e-004 0.000000000000e+000 2.053326108305e+002
0.000000000000e+000 0.000000000000e+000 2.316653944376e+000 0.000000000000e+000
5.070743415829e-001 0.000000000000e+000 1.185281531626e-001 -4.512066383547e-009
-1.845512154378e-0090.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 18 0 0 0.000000000000e+000 -5.938547280089e-012 1.481047559878e+000
0.000000000000e+000 6.392412262897e-004 0.000000000000e+000 2.053361268546e+002
0.000000000000e+000 0.000000000000e+000 2.316590778925e+000 0.000000000000e+000

48. 47
5.070685160175e-001 0.000000000000e+000 1.511416183362e-001 -8.772979328183e-009
-8.091063035460e-0100.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 20 0 0 0.000000000000e+000 -6.848366941609e-012 1.952860757550e+000
0.000000000000e+000 7.020780314763e-004 0.000000000000e+000 2.053387410142e+002
0.000000000000e+000 0.000000000000e+000 2.316523325165e+000 0.000000000000e+000
5.070793246071e-001 0.000000000000e+000 2.042970787536e-001 -9.368577695475e-009
1.501193002792e-009 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000
2011 4 13 22 0 0 0.000000000000e+000 -6.180057924628e-012 2.416946443450e+000
0.000000000000e+000 7.479459555759e-004 0.000000000000e+000 2.053397456339e+002
0.000000000000e+000 0.000000000000e+000 2.316486589959e+000 0.000000000000e+000
5.071002824222e-001 0.000000000000e+000 2.651478848384e-001 -5.102111974329e-009
2.910807656868e-009 0.000000000000e+000 0.000000000000e+000 0.000000000000e+000

49. 48
ACKNOWLEDGMENTS
We would like to express my gratitude to The Group Director of Controls division
Mr. P.M.Natarajan , A.K.Kulkarni & our supervisor, Mr. Vinod Kumar, Control
Dynamics and Simulation Group, ISRO Satellite Centre, whose expertise,
understanding and patience, added considerably to my experience. We appreciate his
vast knowledge and skill in many areas and even his ability to make a friendly
environment (e.g., vision, ethics, interaction with participants). His guidance and
supervision throughout the internship are the reason this project could give some
positive results. Finally, we would like to thank Prof. Hari B. Hablani, Aerospace
Engineering Department, IIT Mumbai for giving us an opportunity to work at ISRO
Satellite Centre and learn what could not have learnt otherwise, elsewhere.

50. 49
REFERENCES
1. Spacecraft Dynamics and Control –A Practical Engineering Approach, Marcel J. Sidi
2. Curtis H., Orbital Mechanics[c] For Engineering Students (2005)(1st ed.)(en)(704s)
3. Satellite Orbits - Models, Methods and Applications, Montenbruck. E. Gill
4. Introduction to geostationary orbits, E Mattias Soop, European Space Operations Centre
5. Applied Orbit Perturbation and Maintenance , Chin-Chun “George” Chao
6. Spacecraft Attitude Determination and Control, Ed. By James R. Wertz
7. Fundamentals of Astrodynamics and Application, David A. Vallado
8. IERS Technical Note 21, IERS Conventions 1996, Denis D. McCarthy
9. Prospects for an improved Lense-Thirring Test with SLR and the GRACE Gravity
mission[paper], Glenn E. Peterson
10. Modeling Earth Albedo For Satellites In Earth Orbit[paper], Dan Bhanderi, Thomas Bak
11. Fundamentals of GPS receivers, a software approach, James Bao, Yen Tsui
12. Introduction to random signals and applied Kalman filtering by Robert Grover Brown
Y.C Hwang
13. IRNSS navigation message file Neetha