GPS Orbits
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Transcript of GPS Orbits
CSED, BAHIR DAR UNIVERSITY
GPS ORBITS
CSED, BAHIR DAR UNIVERSITY
•What is the requirement of orbital information in GPS positioning.
•The nature of satellite orbits, design And forces acting on them.
•Orbit broad cast ephemeris..
•Tracking methodology
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Orbital information is required for four major tasks involved in the GPS positioning process:
1. receiver position determination,
2. planning,
3. receiver aiding, and
4. satellite selection.
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Orbital information can be accurate and approximate
• For receiver positioning accurate data is required
positioning can be of two types :-
1. Point positioning
2. Relative positioning
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Point positioning
The positioning With respect to defined coordinate system usually by three coordinate values.
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In the case of point position computations,
there strict requirement for precise orbital information
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Relative Positioning
The positioning with respect to another point, taking
one point as the origin of a local coordinate system.
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Shown above is Relative positioning Using local
astronomical system.
If R1 of point Pj is known in one coordinate system
& interstation vector ΔR12 (ΔX, ΔY, ΔZ) is determined in
the same coordinate system. Then the vector equation
gives us the position of P2 in the same coordinate
system
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For relative positioning, the effects of any uncertainties
in the ephemeris data on the final position are less
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The vector error db in the baseline vector b, is given by
Here db is determined as a function of dri ,in
position of i satellite used to determine b
ρi = range to ith satellite
ei = unit vector to ith satellite
db depends not only on the magnitudes of p, b,
and dr but also on their directions
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Approximate information of orbits are required
in some other tasks:-
Planning the use of GPS. Knowing when to use GPS
Signal acquisition. first GPS receiver must acquire the
satellite signals to be used.
If nothing is known cold search is done
This involves selecting each of the 32 possible C/A-codes
(requires lot of time)
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Almanac' data:- information provided by the visible
satellite regarding the tentative location of other sats.
Satellite alerts :- computing approximate satellite
positions for planning or receiver aiding.
*For planning , various graphical representation are there•linear plot -elevations and azimuths are plotted
against time for each satellite, with respect to a
specified location
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• polar plot- elevations and azimuths , are predicted
and plotted, as a function of time for each
satellite, with respect to a specified location
•Track plot, satellite subpoint is plotted on a map.
•Visibility plot, plot against time of day,of the period
during which each satellite is visible from a specified
location
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COORDINATE SYSTEMS
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Dynamics of satellites is governed by
Newton's law of motion stating
Acceleration α C.O.M of free body
The position, velocity, acceleration, and force
components refer to a special coordinate system called
Inertial
#Inertial coordinate system, which is either stationary
or in uniform motion in space
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For this purpose we adopt Right ascension (RA) system,
(which is an appropriate inertial coordinate system).
The orientation of the RA-system relative to the fixed
stars changes by about 1" per year, due to the forced
periodic motion of the earth's rotation axis.
this motion is called precession and nutation,
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But for terrestrial positioning purposes, RA-system is not
a convenient reference frame.
#Here we define the coordinate system which is 'fixed to
the solid earth' i.e (stationary point on earth will have
Fixed coordinates ever).
Conventional terrestrial system (CT- also known as the
average terrestrial system)
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•The 1st axis of the CT-system passes through the
intersection of the Greenwich meridian and the equatorial
Plane.
•The 3rd axis of the CT-system is defined by the
Conventional International Origin (CIO).
•The 2nd axis is orthogonal to the first and third axes .
[Until July 1985, the realization of the CT-system used to describe the GPS
broadcast ephemeris was the World Geodetic System 1972 (WGS-72)]
**(Since January 1987 the WGS-84 is adopted ww)
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Forces acting on satellites
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#Forces that contributes to satellite motion comprise:
•Gravitational attraction of the earth,
•Gravitational attractions of the sun, moon, and planets (called the third body effects),
• Atmospheric drag force,
•Solar radiation pressure, both direct and albedo effects,
•Magnetic forces, and
• The variable part of the earth's gravitational field
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Above all forces the most prominent is gravitational
attraction of the earth.
Earth’s gravitational attraction can be further divided into:-
•Central part, the radial gravitational attraction.
•The non-central part.
#Perturbations The non-central part and remaining forces
introduce some small variations into the motion of satellite
these variations are called as perturbations.
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KEPLER'S FIRST LAW
It states that “THE ORBIT IS AN ELLIPSE WITH ONE
OF THE FOCI LOCATED AT THE EARTH‘S CENTRE OF
MASS”
Which depicts that “It is not possible to launch
satellite directly into an orbit with an angle of
inclination less than the latitude of the launch site”
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Kepler's first law is very usefull today in determining
the orbits which are achievable by satellites launched
from various launch sites.
*To obtain inclinations lower than the launch site latitude,
the satellite must first be launched into a 'parking orbit,'
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Basically Keplerian motion is the idealized satellite motion
caused only by the earth's central gravitational field.
The main aspects of Keplerian motion are:
1. The motion relative to the RA-system occurs in a
stationary plane which contains the centre of mass of the
Earth.
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2. The closest and farthest points of the orbit to the
earth's centre of mass, called perigee and apogee.
3. The size and shape of the elliptical orbit are constant.
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KEPLER'S SECOND LAW
It states that “GEOCENTRIC POSITION VECTOR OF A
SATELLITE SWEEPS EQUAL AREA IN EQUAL TIME”
Basically it(2nd law) follows energy conservation law
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As satellite is in motion possesses two kinds of energy —1. potential 2. kinetic.
Potential energy is governed only by the gravitational field
and is lowest when the satellite is closest to the attracting
body (i.e., at the perigee), and highest at apogee.
Total energy must remain constant
Therefore the kinetic energy must be the largest at perigee
and lowest at apogee.
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Practical implications
1. spy satellites
2. communication satellites
3. lifetime of a satellite
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KEPLER'S THIRD LAW
It states that the ratio between the square of the orbital
period and the cube of the semi-major axis, a,
is the same for all satellites
i.e,
Where , μ=GM
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Generally if, Often the period is expressed in terms of itsreciprocal, the mean motion
n = 2π/T.
Then we have,
n2/a3=μ
Generally a & μ are known to us so,
n=(μ/a3)1/2
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Practically it means that “Two satellites with the
same semi-major axis length have equal orbital
periods regardless of eccentricity”
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KEPLERIAN ORBITAL ELEMENTS
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A particular set of parameters called the
Keplerian elements is commonly used in
satellite positioning, Out of them only one is a
function of time. They are:-
•(Ω). Right ascension of the ascending node.
•(i) Inclination
Ω and i, define the orientation of the orbital plane inspace
•
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•(ω) Argument of perigee. It defines the location of
the perigee
•(a) Semi-major axis of the elliptical orbit
•(e) Eccentricity of the orbit
a and e define the size and shape of the orbit.
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THIRD-BODY EFFECTS AND TIDES
Direct effectT.B.E
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It is nothing but the gravitational force per unit mass
of a satellite due to the attraction of a third body i.e
moon, sun and stars etc..
*(IMP# T.B.E of the moon and the sun need be
modelled for short to medium orbital arcs)
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T.B.E (Ftb)is given by:-
Where, M* =Mass of TB r* = geocentric position vector of TB G = ?
∵ M* / |r*|3 for the sun is 0.46 times of moon, the solar TBE of sun=1/2 moon
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For the planets, the masses M* are far smaller
a.c.t sun, and distances |r*| much greater than for
the moon; the Planetary T.B.E on GPS satellites are
negligible.
Point to be noted is:
* TBE is greatest due to moon.
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Tides :The earth and the oceans undergo tidal
deformations at primarily the semi-diurnal and
diurnal frequencies.The tidal deformations cause a
periodic re-arrangement of the earth's mass which in
turn alters the earth's gravitational attraction.
These effects are insignificant for GPS satellites over
orbital periods.
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SOLAR RADIATION PRESSURE AND
ATMOSPHERIC DRAG
D.EI.D.E
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Direct Effect:-The photons emitted by the sun create a
repulsive pressure upon their impact on the satellite.
Indirect Effect:- The photons striking satellite after being
reflected from the earth. This I.E isalso called as
Albedo effect.
The direct radiation pressure is zero when the
satellite is in the earth's shadow.
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Force of solar radiation/unit mass on a satellite varies
In accordance with
Fsr = v p A r / m
Where,
v = eclipse factor (1 in sunlight, 0 in shadow) p = solar radiation pressure A = C.S.A of the sat on which the radiation falls r = reflectivity of the satellite, and m = satellite mass.
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Atmospheric drag:- It is a non-conservative force
arising due to friction betn the satellite & surrounding
atmosphere. This results in loss of satellite
energy which in turn results in secular changes which
are non-recoverable in a and M.
Where,
m = satellite mass.
A = C.S.A of the sat on which the radiation falls
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The Fad per unit mass on a satellite varies in
Accordance with
Fad =ρ v2 A / mWhere,
ρ = the atmospheric density at satellite v = the sat's velocity relative the atmosphere A & m we know
The ρ at GPS satellite altitudes, ~20 000 km, is so
small that the Fad are negligible
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EPHEMERIS PREDICTION
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GPS MCS computes and controlles the broadcast
ephemeris information transmitted by the satellites & is
Pridicted in two ways:-
1. off-line least-squares to produce a reference
ephemeris.
2. on-line 1st-order correction by a Kalman filter &
additional measurements to predict current states of
satellite which, in turn, predicts future states
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The reference ephemeris is an initial estimate of the
satellite trajectory,
The on-line ephemeris prediction is driven by pseudo-
range and integrated Doppler measurements from
each satellite in view at each monitor station
##Post computed ephemeris
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BROADCAST EPHEMERIS MESSAGE PARAMETERS
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Broadcast ephemerides for the GPS satellites are readily
available in the navigation message modulated on the
GPS carrier signals.
The 6 parameters (√a, e, iQ, ω, Ω0, Mo) describe a
smooth, elliptical orbit with the satellite position being
a function of time since t.
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The parameters (Δn, Ω-dot, i-dot, and the six
sine and cosine coefficients) describe the deviations of
the actual satellite motion from this smooth ellipse.
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GPS ORBIT DESCRIPTION
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There are 4 points of importance along the orbital
ellipse. These are:
• The ascending node. This is the intersection of
the orbital plane, and the
equatorial plane,.
• Perigee. This is point at which the satellite most
closely approaches the earth..
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The reference position. This is position of the
satellite at the reference time toe .
• The satellite position. This is what we want to
determine. It is separated from perigee by the true
anomaly f, and from the ascending node by the
Argument of latitude u = ω + f.
##Satellite coordinate computation