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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 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

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

• 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.

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Position Fixing Methods

From Kyton and Fried, Avionics Navigation Systems, 2nd Ed., pp. 113.

a) Bearing and range ( position fixing (DME-VOR)

b) Dual bearing () position fixing (VOR-VOR)

c) Range ( position fixing (DME-DME, GPS)

d) Hyperbolic position fixing (LORAN, Omega)

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Position Fixing (2-D)

23

23

23

22

22

22

21

21

21

yyxx

yyxx

yyxx

Assuming you can make the range measurements i , 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, t (= b/c), the range estimate (or measurement) becomes:

12 ttc

bttctcttcb 1212ˆ

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SV #1 SV #2

SV #3

3

21

cbucbu

cbu

As a user located at point X, the true range measurements to the three GPS satellites are:

uTrue

uTrue

uTrue

cb

cb

cb

33

22

11

Your GPS receiver, however, measures 1, 2 and 3. These range measurement are called pseudoranges.

GPS Pseudoranges

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Psueodranges and Satellite Geometry

PseudorangeMeasurement Error

ResultingPositionUncertaintyAreas

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 equations

– Easy and give you insight into the linearization process– GPS navigation equations.

• The position fixing system of equations where three independent range measurements are available was given as:

23

23

23

22

22

22

21

21

21

yyxx

yyxx

yyxx

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Linearization by Expansion

23

23

23

22

22

22

21

21

21

yyxx

yyxx

yyxx

23

23

23

22

22

22

21

21

21

ˆˆˆ

ˆˆˆ

ˆˆˆ

yyxx

yyxx

yyxx

Exact Equations you would solve in an ideal

world

Equations the you can or will solve

yyy

xxxiii

ˆ

ˆ

ˆ

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Linearization by Expansion (2)

iiiiiii 22ˆ 22222 (dropped higher order terms)

For the range measurements,

For the position coordinate x,

xxxxxxxxxx iiii 2ˆ 222

For the position coordinate y,

yyyyyyyyyy iiii 2ˆ 222

ni ,2,1Where,

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Linearization by Expansion (3)

yyyxxx iiiiiiii 2222ˆ 2222

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,

y

x

yyxx

yyxx

yyxx

n

n

n

nn

2

2

2

2

1

1

1

1

2

1

G x

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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.

t

y

x

yyxx

yyxx

yyxx

n

n

n

n

n

1

1

1

2

2

2

2

1

1

1

1

2

1

G x

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Least Squares Solution

TT GGGx

1

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

ii G

x

yyy

xxx

)()(

)()(

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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()cos(2

)sin()sin()cos()cos()cos(

)sin()sin()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 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

The Sky Above Palo Alto on Jan 6, 2002

Actual location of Polaris is 89o 05’

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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

nDeclinatio sSun' Altitude sSun' 900

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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 FrequenciesName 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

Land Sea Air

X X

X X

X

X

X

X

X X X

NDB – Non Directional Beacon

LORAN – Long RAnge Navigation

VOR – VHF Omni-directional Range

DME – Distance Measuring Equipment

ILS – Instrument Landing System

MLS – Microwave Landing System

INS – Inertial Navigation System*

Navigation SystemApplication

* 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 () 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• between the phases of

signal 1st and 2nd signal is

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Distance Measuring Equipment (DME)

• Measures Slant Range ()• 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 s) ground station resends the pulses back– Airborne unit receives the signal and calculates range by using the following

equation:

)50(2

1sTc

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• 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)

Instrument Landing System (ILS)

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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)

Sun

Earth

Earth’s Orbit

An Apparent Solar Day

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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

• 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 tr 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 sec per day– Compensated for by setting the satellite fundamental frequency of

10.23 MHz 0.00455 Hz lower.

rocfcff tttattaat 22010 )()(

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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 (h) to ECEF

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

)sin(1

)sin()cos(

)cos()cos(

)sin(1

081819190

m 6378137

2

2

heNz

hNy

hNx

e

aN

.e

a

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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 =

– REW =

2)(sin31 2 fa

)(sin1 2 fa

where’ )tan()1()tan( 2' f

= Geodetic Latitude’ = Geocentric Latitude

’

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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 ,2 r

r

r

mGMF SE

ES rrr

0)(

333

r

rrr

r

GMrr

r

mMGr ESE

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• 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 ()• Characterize ellipse’s

orientation in orbital plane– Argument of Perigee ()

• Position of the satellite in the orbit

– True anomaly ()

Six Keplerian Elements

• Sometimes it is convenient to sum and 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

(-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

Figure from Bate, Mueller and White, Fundamentals of Astrodynamics (1971), pp. 156

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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, , 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, k

– Compute the argument of latitude k

– Use k 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 (0)• Rate of right ascension (-dot)• Argument of perigee ()• 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.