A Brief Introduction to the Global Positioning System (GPS)

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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)

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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 formed by the
intersection of the surface of the earth with a
plane parallel to the equatorial plane
Longitude (Meridians) are formed by the
intersection of the surface of the earth with a
plane containing the earths 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.
• Difference between local celestial time and
  reference yields Longitude.
• Atomic clocks are too big and too expensive for
  general use.

Figure from Hewlett-Packard Application Note
1289 The Science of Timekeeping by D. W. Allan,
Neil Ashby and Cliff Hodge.
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Position Fixing Methods

• Bearing and range (r-q) position fixing (DME-VOR)
• Dual bearing (q-q) position fixing (VOR-VOR)
• Range (r-r) position fixing (DME-DME, GPS)
• 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
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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
• In the presence of a clock error, dt ( b/c), the
  range estimate (or measurement) becomes

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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
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
Geometry plays a role in the accuracy of the
final solution.
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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

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Linearization by Expansion
Exact Equations you would solve in an ideal world
Equations the you can or will solve
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Linearization by Expansion (2)
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Linearization by Expansion (3)
Taking the difference between the true and
estimated values,
Normally you have more equations than unknowns.
Thus, you can do a least squares solution. That
is,
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Linearization by Expansion (4)
Because we dont 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.
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Least Squares Solution
For the moment, without proof, we state that the
least squares solution is given by,

• Algorithm for solving the navigation equation
• 1) Pick an initial guess for x and y
• 2) Compute for as many measurements as you
  have
• 3) Form for all measurements and then form
• 4) Solve for
• 5) Update your initial guesses for x and y as
  follows
• 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
  crosscorrelation 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

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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
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Instruments of Navigation
An Astrolabe
A Sextant
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View Through a Sextant
Easier to align Suns (or other celestial
bodys) limb with the horizon.
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Latitude Determination Using the Sun
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The Longitude Problem

• Celestial map changes because of Earths 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 Jupiters 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
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
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

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
Earth
An Apparent Solar Day

• Sidereal Time Based on the time required by
  Earth to complete one revolution about its axis
  relative to distant stars.
• 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 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 (GMT)

Sun
Earths 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 Earths 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
• 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
• 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 Earths 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
• REW

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Geoidal Heights
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Orbital Mechanics

• Keplers 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

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Six Keplerian Elements

• Recast the two-body equation of motion.
• Characterize orbital ellipse
• Semi-major Axis (A)
• Eccentricity (e)
• Characterize orbits orientation in space
• Inclination (i)
• Right Ascension of the Ascending Node (W)
• Characterize ellipses 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 element set
• 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

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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 its 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.