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					A Brief Introduction to the Global
    Positioning System (GPS)


       CMPE-118 Lecture




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         Global Positioning System (GPS)
• Satellite Navigation
  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
         Differential)                      ©2000 by
                                           Todd Walter
                                           and Per Enge



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               Navigation Terminology
• Navigation
   – Answer the to the question ―Where am I?‖
   – Implies the use of some agreed upon coordinate system
• Related Terminology
   – Guidance: Deciding what to do with your navigation information
   – Control: Orienting yourself/vehicle to follow out the guidance
     decision.
• Area of Study: GNC
   – Guidance, Navigation, Control




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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
                                                              Navigation
                                                              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
Your GPS receiver, however,
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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           Solving Navigation Equations

• Solve the r-r equations
   – Easy and give you insight into the linearization process
   – GPS navigation equations.
• 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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       GPS Signal De-Spreading

• 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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Graphical Depiction of De-Spreading




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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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   Things to remember about GPS

• 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
  and your time bias.
• 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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        Instruments of Navigation




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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            Fundamentals of Radionavigation




•   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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            Classification of Radio Frequencies
           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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              INS and Radionavigation Systems

                                                                                 Application
             Navigation System
                                                                             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.
    ―Terrestrial Radionavigation, pp. 81)
     – 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
•   Based on Radar Principle
     – Airborne unit sends a pair of pulses
     – Ground Station receives 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
    complete one revolution about its
    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
         broadcasts them as the navigation message
           – 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
         navigation, etc.
     –   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
         navigation, guidance and control applications
     –   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
     navigation applications (especially
     inertial navigation)
• 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
•   Additional 9 parameters
     – 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
         • Orbit radius
         • 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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posted:9/14/2010
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