A Brief Introduction to the Global
Positioning System (GPS)
CMPE-118 Lecture
OF 62
11 OF
28
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
OF 62
22 OF
28
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
OF 62
33 OF
28
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.
OF 62
44 OF
28
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 “+”
OF 62
55 OF
28
Stability of Clocks
• Clock stability is
directly related to
Navigation
because Earth
rotates ~15°/hour.
Figure from Hewlett-Packard
Application Note 1289: The Science
of Timekeeping by D. W. Allan, Neil
Ashby and Cliff Hodge.
• Difference between
local “celestial”
time and reference
yields Longitude.
• Atomic clocks are
too big and too
expensive for
general use.
OF 62
66 OF
28
Position Fixing Methods
a)
Bearing and range
(r-q) position fixing
(DME-VOR)
b)
Dual bearing (q-q)
position fixing (VORVOR)
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.
OF 62
77 OF
28
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
r 32  x - x3 )2   y - y3 )2
OF 62
88 OF
28
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:
rˆ  r  b  ct2 - t1 )  cdt  ct2 - t1 )  b
OF 62
99 OF
28
GPS Pseudoranges
As a user located at point X, the
true range measurements to the
three GPS satellites are:
SV #1
SV #2
r2
r1
r1True  r1  cbu
r 2True  r 2  cbu
r 3True  r 3  cbu
Your GPS receiver, however,
measures r1, r2 and r3. These
range measurement are called
pseudoranges.
cbu
cbu
cbu
r3
SV #3
10 OF
OF 62
10
28
Psueodranges and Satellite Geometry
Pseudorange
Measurement
Error
Resulting
Position
Uncertainty
Areas
Geometry plays a role in
the accuracy of the final
solution.
11 OF
OF 62
11
28
GPS Position Fixing
12 OF
OF 62
12
28
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
r 32  x - x3 )2   y - y3 )2
13 OF
OF 62
13
28
Linearization by Expansion
r12  x - x1 )2   y - y1 )2
rˆ12  xˆ - x1 )2   yˆ - y1 )2
r 22  x - x2 )2   y - y2 )2
rˆ 22  xˆ - x2 )2   yˆ - y2 )2
r 32  x - x3 )2   y - y3 )2
rˆ 32  xˆ - x3 )2   yˆ - y3 )2
Exact Equations you
would solve in an ideal
world
Equations the you can or will solve
rˆ i  ri  dri
xˆ  x  dx
yˆ  y  dy
14 OF
OF 62
14
28
Linearization by Expansion (2)
For the range measurements,
rˆ i2  ri  dri )2  ri2  2 ridr  dr 2  ri2  2 ridr (dropped higher order terms)
For the position coordinate x,
xˆ - xi )2  x  dx - xi )2  x - xi )2  2x - xi )dx
For the position coordinate y,
 yˆ - yi )2   y  dy - yi )2   y - yi )2  2 y - yi )dy
Where,
i  1,2, n
15 OF
OF 62
15
28
Linearization by Expansion (3)
Taking the difference between the true and estimated values,
ri2 - rˆ i2  ri2 - ri2  2 ridr )  -2 ridr  -2x - xi )dx - 2 y - yi )dy
Normally you have more equations than unknowns. Thus, you can do a
least squares solution. That is,
 x - x1 )

dr1   r1
dr   x - x2 )
 2   r
2
   
   
drn  x - xn )

 r n

dr
 y - y1 ) 
r1 
 y - y2 ) dx 
r2   
dy
  
 y - yn ) 

r n 
G

dx
16 OF
OF 62
16
28
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 )
 r
1

dr1 
 x - x2 )

dr 
 2    r2
   
   
drn  
  x - xn )
 r
n


dr
 y - y1 )
r1
 y - y2 )
r2
1


 y - yn )
1
rn
G
1



 dx 
 
 dy 
  dt 
 




dx
17 OF
OF 62
17
28
Least Squares Solution
For the moment, without proof, we state that the least
squares solution is given by,

)
-1 T 

T
dx  G G G dr
•
Algorithm for solving the navigation equation:
–
–
–
–
–
1)
2)
3)
4)
5)
Pick an initial guess for x and y
Compute r̂ i for as many measurements as you have
Form dri for all measurements and then form G

Solve for dx
Update your initial guesses for x and y as follows:
x (  )  x ( - )  dx
y (  )  y ( - )  dy
– 6) Repeat until convergence
18 OF
OF 62
18
28
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
19 OF
OF 62
19
28
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
20 OF
OF 62
20
28
GPS L1 Signal Generation
21 OF
OF 62
21
28
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( -  )
22 OF
OF 62
22
28
Graphical Depiction of De-Spreading
23 OF
OF 62
23
28
PRN Auto- and Cross-Correlation
24 OF
OF 62
24
28
PRN Correlation Example
25 OF
OF 62
25
28
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
26 OF
OF 62
26
28
Typical Search Results
27 OF
OF 62
27
28
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.
28 OF
OF 62
28
28
Questions?
29 OF
OF 62
29
28
30 OF
OF 62
30
28
31 OF
OF 62
31
28
Latitude Determination Using Polaris
Actual location of Polaris is
89o 05’
The Sky Above Palo Alto on Jan 6, 2002
32 OF
OF 62
32
28
Instruments of Navigation
An Astrolabe
A Sextant
33 OF
OF 62
33
28
View Through a Sextant
Easier to “align” Sun’s (or other
celestial body’s) limb with the
horizon.
34 OF
OF 62
34
28
Latitude Determination Using the Sun
  900 - Sun' s Altitude  Sun' s Declinatio n
35 OF
OF 62
35
28
The Longitude Problem
•
Celestial map changes because of Earth’s 15 o/hr (approximately) rotation
rate.
36 OF
OF 62
36
28
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
37 OF
OF 62
37
28
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
38 OF
OF 62
38
28
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
39 OF
OF 62
39
28
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.
40 OF
OF 62
40
28
INS and Radionavigation Systems
Navigation System
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
* INS is not a radionavigation system but is normally used in conjunction with such systems
X
X
41 OF
OF 62
41
28
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.
42 OF
OF 62
42
28
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
43 OF
OF 62
43
28
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
2
r  c(DT - 50ms)
r
44 OF
OF 62
44
28
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)
45 OF
OF 62
45
28
Time Scales
•
•
•
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)
An
Apparent
Solar Day
Earth
Sun
Earth’s
Orbit
46 OF
OF 62
46
28
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
47 OF
OF 62
47
28
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
48 OF
OF 62
48
28
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.
49 OF
OF 62
49
28
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
–
–
–
Earth Centered Earth Fixed (ECEF)
East-North-Up (ENU)
Geodetic Coordinates
–
North-East-Down (NED) – used widely in aircraft
navigation, guidance and control applications
Wander-Azimuth
Coordinate systems you will mostly encounter in
GPS are
Other coordinate systems used in navigation
–
50 OF
OF 62
50
28
Coordinate Frame Relationships
• Geodetic coordinates (f, l, h) to ECEF
a  6378137 m
e  0.08181919
N
a
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
51 OF
OF 62
51
28
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
52 OF
OF 62
52
28
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
53 OF
OF 62
53
28
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’
where f  f’
tan( f ' )  (1 - f ) 2 tan( f )
f
f = Geodetic Latitude
f’ = Geocentric Latitude
54 OF
OF 62
54
28
Geoidal Heights
55 OF
OF 62
55
28
56 OF
OF 62
56
28
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
r
r  G ( M E  mS ) r  r  GM E r  r  m r  0
r3
r3
r3
  
r  rS - rE
57 OF
OF 62
57
28
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
58 OF
OF 62
58
28
GPS Orbital Parameters
•
Perturbed Orbits - quasi-Keplerian 15
element set
Figure from Bate, Mueller and White,
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
59 OF
OF 62
59
28
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.
60 OF
OF 62
60
28
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
61 OF
OF 62
61
28
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.
62 OF
OF 62
62
28