# GPS A

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```					A Brief Introduction to the Global
Positioning System (GPS)

CMPE-118 Lecture

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Global Positioning System (GPS)
system
– Multilateration based
on one-way ranging
signals from 24+
satellites in orbit
– Operated by the
United States Air Force
– Nominal Accuracy
• 10 m (Stand Alone)
• 1-5 m (Code
Differential)
• 0.01 m (Carrier
Todd Walter
and Per Enge

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– Answer the to the question ―Where am I?‖
– Implies the use of some agreed upon coordinate system
• Related Terminology
– Control: Orienting yourself/vehicle to follow out the guidance
decision.
• Area of Study: GNC

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Latitude (Parallels) are        Longitude (Meridians) are
formed by the intersection of   formed by the intersection of
the surface of the earth with   the surface of the earth with
a plane parallel to the         a plane containing the earths
equatorial plane                axis.
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Latitude, Longitude and Altitude

• One of many coordinate
systems used to described a
location on the surface of the
earth
• Latitude — parallels
measured from the Equator.
– North is ―+‖
• Longitude — meridians
measured from Greenwich
Observatory.
– East is ―+‖
• Altitude — measured above
reference datum: MSL
– Normally Up is ―+‖

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Stability of Clocks

• Clock stability is
directly related to
because Earth
rotates ~15°/hour.
Figure from Hewlett-Packard
Application Note 1289: The Science
• Difference between
of Timekeeping by D. W. Allan, Neil                           local ―celestial‖
Ashby and Cliff Hodge.
time and reference
yields Longitude.
• Atomic clocks are
too big and too
expensive for
general use.

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Position Fixing Methods

a)   Bearing and range
(r-q) position fixing
(DME-VOR)
b)   Dual bearing (q-q)
position fixing (VOR-
VOR)
c)   Range (r-r) position
fixing (DME-DME,
GPS)
d)   Hyperbolic position
fixing (LORAN,
Omega)

From Kyton and Fried, Avionics Navigation
Systems, 2nd Ed., pp. 113.
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r-r Position Fixing (2-D)

Assuming you can make the
range measurements ri , where
i = 1,2,3, then the following
three equations can be
formed:

r12  x - x1 )2   y - y1 )2
r 22  x - x2 )2   y - y2 )2
r32  x - x3 )2   y - y3 )2

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Fundamentals of Position Fixing
•   The figure on the previous page raises to important questions:
– How do you estimate or measure the ranges?
– How do you solve the equations for the unknown x and y?

•   The range based on measuring the time-of-flight of a RF signal that
leaves the transmitter at t = t1 and arrives at the user at t = t2 is given
by:
r  ct2 - t1 )
•   In the presence of a clock error, dt (= b/c), the range estimate (or
measurement) becomes:

r  r  b  ct2 - t1 )  cdt  ct2 - t1 )  b
ˆ

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GPS Pseudoranges

SV #1                          SV #2
As a user located at point X, the
true range measurements to the
three GPS satellites are:
r1                       r2

r1True  r1  cbu
r 2  r 2  cbu
True
cbu              cbu

r 3  r 3  cbu
True                            cbu
r3
SV #3
measures r1, r2 and r3. These
range measurement are called
pseudoranges.
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Psueodranges and Satellite Geometry

Pseudorange
Measurement      Resulting     Geometry plays a role in
Error           Position      the accuracy of the final
Uncertainty   solution.
Areas
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GPS Position Fixing

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• Solve the r-r equations
– Easy and give you insight into the linearization process
• The r-r position fixing system of equations where three
independent range measurements are available was given
as:

r12  x - x1 )2   y - y1 )2
r 22  x - x2 )2   y - y2 )2
r32  x - x3 )2   y - y3 )2
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Linearization by Expansion

r12  x - x1 )2   y - y1 )2                    r12  x - x1 )2   y - y1 )2
ˆ      ˆ             ˆ
r 22  x - x2 )2   y - y2 )2                   r 22  x - x2 )2   y - y2 )2
ˆ       ˆ             ˆ
r32  x - x3 )2   y - y3 )2                    r32  x - x3 )2   y - y3 )2
ˆ      ˆ             ˆ

Exact Equations you                            Equations the you can or will solve
would solve in an ideal
world
ri  ri  dri
ˆ
x  x  dx
ˆ
y  y  dy
ˆ
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Linearization by Expansion (2)

For the range measurements,

ri2  ri  dri )2  ri2  2ridr  dr 2  ri2  2ridr (dropped higher order terms)
ˆ
For the position coordinate x,

x - xi )2  x  dx - xi )2  x - xi )2  2x - xi )dx
ˆ
For the position coordinate y,

 y - yi )2   y  dy - yi )2   y - yi )2  2 y - yi )dy
ˆ
Where,
i  1,2, n
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Linearization by Expansion (3)

Taking the difference between the true and estimated values,
r i2 - r i2  r i2 - r i2  2 r idr )  -2 r idr  -2x - xi )dx - 2 y - yi )dy
ˆ

Normally you have more equations than unknowns. Thus, you can do a
least squares solution. That is,
  x - x1 )  y - y1 ) 
                r1 
dr1   r1                     
dr    x - x2 )  y - y2 )  dx
 2   r                         
                     r2   
dy
  
2

           
dr n  x - xn )  y - yn )
                       
 rn
                rn    
                                   
dr                  G               dx                        16 OF 28
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Linearization by Expansion (4)

Because we don’t have true ranges, but pseudo-ranges, we augment the G
matrix with a column of ones for the time bias. We need at least 3
measurements for the 2-D solution.

 x - x1 )     y - y1 )       
 r                         1   
r1
                               
  x - x2 )
1
dr1                   y - y2 )        dx 
dr                                1
 2    r2               r2            
                                     dy 
                                    dt 
drn                                   
  x - xn )    y - yn )       
 r               rn
1   
      n                        
                                        
dr                     G                  dx           17 OF 28
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Least Squares Solution
For the moment, without proof, we state that the least
squares solution is given by,
dx  GT G ) GT dr
              -1       

•    Algorithm for solving the navigation equation:
–   1)   Pick an initial guess for x and y
–   2)   Compute r i for as many measurements as you have
ˆ
–   3)   Form dr i for all measurements and then form G

–   4)   Solve for dx
–   5)   Update your initial guesses for x and y as follows:
x (  )  x ( -)  dx
y (  )  y ( -)  dy
– 6) Repeat until convergence

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Iterated Solution Numerical Example

• Solution is
done in
MATLAB
• Assumes an
initial
position of
[0,0,0]
• Walks
solution in to
the final
position
• Redraws the
range circles
at each
iteration
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GPS Signal Structure

• GPS broadcasts a modulated carrier on L1
(1575.42 MHz)
• Pseudo-Random Noise (PRN) sequence of
1023 ―chips‖ used to spread the signal
• PRN is carefully chosen to have unique
auto— and cross—correlation properties
• All signal components generated from the
same 10.23 MHz satellite clock

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GPS L1 Signal Generation

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• In order to use the PRN code correlation
properties to de-spread the GPS signal, need to
recover code down to baseband (no carrier)
• Use trigonometric identities to mix down and
remove the carrier

            
cos(   )  cos( ) cos( ) - sin( ) sin(  )
            
cos( -  )  cos( ) cos( )  sin( ) sin(  )

                            
2 cos( ) cos( )  cos(   )  cos( -  )
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PRN Auto- and Cross-Correlation

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PRN Correlation Example

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Initial Acquisition Search

• Assume 1 channel & 1 ms
dwell period
• Exhaustive search (if real
time) requires:
– (32) x (2046) x (20) x 1ms
= 1309 seconds
• 12 channel assumption
requires:
– (1309) / 12 = 109 seconds

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Typical Search Results

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• Navigation is a hard problem, and only
recently has GPS made this easy!
• GPS is a r-r system that has precise
clocks on board that give you position
• PRN signal has correlation properties that
allow you to find the signal in the noise
even without any knowledge of position.

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Questions?

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Latitude Determination Using Polaris

Actual location of Polaris is
89o 05’

The Sky Above Palo Alto on Jan 6, 2002                           32 OF 28
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An Astrolabe               A Sextant
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View Through a Sextant

Easier to “align” Sun’s (or other
celestial body’s) limb with the
horizon.

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Latitude Determination Using the Sun

  900 - Sun's Altitude  Sun's Declination
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The Longitude Problem

•   Celestial map changes because of Earth‘s 15 o/hr (approximately) rotation
rate.
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Longitude Determination

• Longitude Determination Methods
– Methods based on time
• Compare the time between a clocks at the current location
and some other reference point.
• Requires Stable Clocks
– Celestial Methods
• Eclipses of Jupiter‘s Moons
• Lunar Distance Method

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•   Radio Frequency (RF) signals emanating from a source or sources.
•   The generators of the RF signal are at known locations
•   RF signals are used to determine range or bearing to the known location

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Name of Band                 Frequency Range           Wavelength

Very Low Frequency (VLF)               < 30 kHz                > 10 km

Low Frequency (LF)                     30 – 300 kHz            1 - 10 km

Medium Frequency (MF)                  300 kHz – 3 MHz         100 m – 1 km

High Frequency (HF)                    3 – 30 MHz              10 – 100 m

Very High Frequency (VHF)              30 – 300 MHz            1 – 10 m

Ultra High Frequency (UHF)             300 MHz – 3 GHz         10 cm – 1 m

Super High Frequency (SHF)             3 – 30 GHz              1 – 10 cm

Propagation characteristic of RF signals is a function of their frequency
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Line of Sight Transmission

•   VHF (VOR, ILS Localizer) and UHF (ILS Glide Slope, TACAN/DME) are line of sight systems.
– Limited Coverage area
•   LORAN and OMEGA are over the horizon systems
– Large coverage area
– In the case of Omega, coverage was global
•   Frequency band in which GPS operates makes it a line of sight system. However, because
of the location of the satellites, it is able to cover a large geographic area .

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Application
Land        Sea      Air
NDB – Non Directional Beacon                                                                  X    X
LORAN – Long RAnge Navigation                                                                 X    X
VOR – VHF Omni-directional Range                                                                   X
DME – Distance Measuring Equipment                                                                 X
ILS – Instrument Landing System                                                                    X
MLS – Microwave Landing System                                                                     X
INS – Inertial Navigation System*                                               X             X    X
* INS is not a radionavigation system but is normally used in conjunction with such systems
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Phases of Flight

•   The required navigation accuracy and reliability (i.e., integrity,
continuity and availability) depend on the phase of flight
•   Currently, as well as in the past, this meant that an aircraft had to
be equipped with various navigation systems.
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VHF Omni-directional Range (VOR)

•   Provides Bearing (Y) Information
•   Operates 112 – 118 MHz
•   Accuracy 1o to 2 o.
•   Principles of Operation (Enge et. al.
– Transmits 2 Signals
• 1st Signal has azimuth
dependent phase
• 2nd Signal is a reference
• D between the phases of
signal 1st and 2nd signal is Y

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Distance Measuring Equipment (DME)
•   Measures Slant Range (r)
•   Operates between 962 and 1213 MHz
– Airborne unit sends a pair of pulses
– After short delay (50 ms) ground station resends the pulses back
– Airborne unit receives the signal and calculates range by using the following
equation:

1
r  c(DT - 50ms)
2
r

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Instrument Landing System (ILS)

•   Used extensively during approach and landing to provides vertical and lateral guidance
•   Principle of Operation
– Lateral guidance provided by a signal called the Localizer (108-112 MHz)
– Vertical guidance provided by another signal called the Glide Slope (329-335 MHz)
•   Distance along the approach path provided by marker beacons (75 MHz)

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Time Scales
•   Sidereal Time – Based on the                    An        Earth
time required by Earth to                    Apparent
axis relative to distant stars.              Solar Day
•   Apparent Solar Day - Time
required for Earth to complete
one revolution with respect to
the sun
•   Mean Solar Time - Same as
apparent solar day except it is                     Sun
based on
– Hypothetical earth
– Rotating in a circular orbit around
the sun.
– Axis of rotation perpendicular to
the orbital plane
– Same as Greenwich Mean Time                      Earth’s
(GMT)
Orbit

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Universal & Atomic Time

• Universal Time (UT) – Time based on astronomical observations
– UT0 – Mean Solar Time measured at the prime meridian
– UT1 – UT0 Corrected for Earth‘s irregular spin rate and polar
motion
• International Atomic Time (TAI)
– Based on Ce-133 Atom
• Coordinated Universal Time (UTC)
– Set to agree with UT1 on January 1, 1958.
– Leap seconds introduced to keep it within 0.9 seconds of UT1

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GPS Time
• GPS Time (GPST) – A continuous time scale (no leap seconds)
– Based on Cesium and Rubidium standards
– ‗Steered‘ to be within fractions of a microsecond modulo one
second from UTC
• Thus GPST-UTC = whole number of seconds + a fraction of a
microsecond.
• GPS time information transmitted by the satellites include
– GPS second of the week - 604,800 seconds per week
– GPS week number – 1024 weeks per epoch

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GPS Time (2)
• GPS satellites carry atomic clocks
– Rubidium and/or Cesium frequency standards
– Satellite clocks monitored by MCS
• Clock bias is modeled as a quadratic
dt  a f 0  a f 1 (t - t0c )  a f 2 (t - toc )2  Dtr
• Parameters of the Quadratic are uploaded to Satellites which in turn
– Sub-frame 1 of the navigation message
• Clock correction term Dtr takes into account relativistic effects
– Account for speed and location in the gravitation potential of the clocks
– Net effect results in satellite clocks gaining ~38.4 msec per day
– Compensated for by setting the satellite fundamental frequency of
10.23 MHz 0.00455 Hz lower.

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GPS Coordinate Frames
•   Inertial Frame of Reference – Defined to
be a non-accelerating or rotating coordinate
frame of reference
–   e.g., Earth Centered Inertial (ECI)
–   Required for analysis of satellite motion, inertial
–   Not convenient for terrestrial navigation
•   Coordinate systems you will mostly encounter in
GPS are
–   Earth Centered Earth Fixed (ECEF)
–   East-North-Up (ENU)
–   Geodetic Coordinates
•   Other coordinate systems used in navigation
–   North-East-Down (NED) – used widely in aircraft
–   Wander-Azimuth

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Coordinate Frame Relationships
• Geodetic coordinates (f, l, h) to ECEF
a  6378137 m
e  0.08181919
a
N
1 - e sin(f ) )
2

x   N  h ) cos(f ) cos(l )
y   N  h ) cos(f ) sin(l )
        ) )
z  N 1 - e 2  h sin(f )

• ECEF to Geodetic coordinates
– Iterative algorithm
– See Wgsxyz2lla.m in toolbox

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Geometry of Earth (1)
• Crude Approximation
– A sphere
– R0 = 6378.137 km
– A spherical model is only good for
―back of the envelope‖ type of
calculations
– Need a more precise model for
• A more accurate model is an
ellipsoid
– Parameters of the mathematical
ellipsoid are defined in WGS-84

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Geometry of Earth (2)

• Topographic Surface
– Shape assumed by Earth‘s
crust.
– Very complicated shape not
amenable to mathematical
modeling
• Geoid
– An equipotential surface of
Earth's gravity field which
best fits, in a least squares
sense, global Mean Sea
Level (MSL).
• Reference Ellipsoid
– Mathematical fit to the geoid
that happens to be an
ellipsoid of revolution and
minimizes the mean-square
deviation of local gravity and
the normal to the ellipsoid

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WGS-84 Reference Ellipsoid
•   Some geometric facts about the WGS-84
Reference Ellipsoid
–   Semi-major axis ( a ) = 6378137 m
–   Semi-minor axis ( b ) = 6356752 m
–   Flattening ( f ) = 1-(b/a) = 1/(298.25722)
–   Eccentricity ( e ) = [f(2-f)]1/2 = 0.081819191
•   Given the WGS-84 Ellipsoid parameters, the
following are derived quantities:

–   RNS =            
a 1  f 3 sin 2 (f ) - 2   ))
–   REW =

a 1  f sin 2 (f )   )
f
f’
f = Geodetic Latitude
f’ = Geocentric Latitude
where f  f’              tan(f ' )  (1 - f ) 2 tan(f )
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Geoidal Heights

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Orbital Mechanics
• Kepler‘s Law
– Based on observations made by Tycho Brahe (1546-1601)
• First Law: Each planet revolves around the Sun in an elliptical path, with
the Sun occupying one of the foci of the ellipse.
• Second Law: The straight line joining the Sun and a planet sweeps out
equal areas in equal intervals of time.
• Third Law: The squares of the planets' orbital periods are proportional to
the cubes of the semi-major axes of their orbits.
• Explanation came later – Isaac Newton (1642-1727)
– Universal Law of Gravitation,                         where
combined with his second law leads to            
GM E mS r                   
F -     2
,               r  rS - rE
r     r

  G( M E  mS ) r  r  GM E r  r  m r  0

r
                    
r3                 r3           r3
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Six Keplerian Elements
•   Recast the two-body equation
of motion.
•   Characterize orbital ellipse
– Semi-major Axis (A)
– Eccentricity (e)
•   Characterize orbit‘s orientation
in space
– Inclination (i)
– Right Ascension of the
Ascending Node (W)
•   Characterize ellipse‘s
orientation in orbital plane
– Argument of Perigee (w)
•   Position of the satellite in the
orbit
– True anomaly (n)                •   Sometimes it is convenient to sum n
and w to form a new variable called
argument of latitude

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GPS Orbital Parameters
•   Perturbed Orbits - quasi-Keplerian 15          Figure from Bate, Mueller and White,
element set                                    Fundamentals of Astrodynamics (1971), pp. 156
– Non-central gravitational force
– gravitational fields of the sun and
moon
– solar pressure
– Three to account for the rate of
changes:
• Right Ascension of the Ascending Node
(W-dot)
• Inclination (i-dot)
• Mean motion (n-dot)
– Three pairs (6 parameter total) to
correct
• Argument of latitude
• Inclination angle

Pertubative Torque caused by
Earth’s Equatorial Bulge

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GPS Constellation and Orbits
•   Nominal Constellation – 24 Satellites.
– At present more than 24 satellites on
orbit.
•   Semi-major axis – 26,560 km
•   Eccentricity – less than 0.01
•   Period – approximately 11 h 58 min
•   Six orbital planes
– Planes designated A through F
– Inclination of 550 relative to the
equatorial plane
– RAAN, W, for the six orbital planes
separated by 600.
– Four Satellites per orbital plane.
Satellites in a given orbital plane are
distributed unevenly to minimize the
impact of a single satellite failure.

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GPS Ephemeris Calculation
•   Compute the satellites position in the orbital coordinate frame
– Solve Kepler's equation ( E = M + e sin E ) for eccentric anomaly at epoch k, Ek.
• Solution requires iteration if orbit is non-circular
– Compute the true anomaly, nk
– Compute the argument of latitude Fk
– Use Fk to compute the corrections for argument of latitude, radius and inclination
then apply the computed corrections.
– Compute the x and y coordinates (xk’ and yk’) of the satellite in it‘s orbit.
•   Covert the computed xk’ and yk’ position into ECEF coordinates
– Compute the correction for the longitude of the ascending node.
– Apply the correction to the longitude of the ascending node.
– Compute the ECEF coordinates

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GPS Almanac
• A subset of clock and ephemeris parameters.
– Limited to seven parameters and the associated reference time (toe)
•   Square root of semi-major axis ((A)1/2)
•   Eccentricity (e)
•   Inclination (i)
•   Longitude of ascending node (W0)
•   Rate of right ascension (W-dot)
•   Argument of perigee (w)
•   Mean anomaly (M)
– Reduced precision
– Allows determining approximate position of satellites
• All satellites broadcast almanac data for all other satellites in the
constellation
– Sub-frames 4 and 5 of the navigation message
– Updated less frequently than the ephemeris parameters in sub-frames 2
and 3.

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