Fundamental Principles of Global Positioning Systems (GPS) · PDF fileLouie P. Balicanta...

Louie P. Balicanta

Fundamental Principles of Global Positioning Systems

(GPS)

1

TRAINING CENTER FOR APPLIED GEODESY AND PHOTOGRAMMETRY

UNIVERSITY OF THE PHILIPPINES DILIMAN

A Short Course on Basic and Advanced GPS Data Analysis and

Modeling for Scientific and Advanced Applications

Fundamental Principles of Global Positioning System (GPS)

• GPS Framework

• Motion of Satellites

• Positioning

• Error Sources

OUTLINE:

2Fundamental Principles of Global Positioning System (GPS)

Control Segment1 Master Station5 Monitoring Stations

Space SegmentNAVSTAR GPS30 Satellites20200 Km

User SegmentReceive Satellite Signal

GPS Framework


Space Segment• satellite system (SV or

satellite vehicles)

• GPS satellite like an orbiting radio station running on solar power collected by two solar collectors with an area of 7.2 sq m each

• uses 4 atomic clocks (2 cesium & 2 rubidium) to generate signals transmitted to a GPS receiver


GPS Framework

Space Segment


GPS Framework

• ground stations around the worldmonitoring the health of each satelliteand uploading orbital parameters to thesatellites

• Master Control Station (MCS): FalconAir Force Base, Colorado Springs,Colorado, USA

• five (5) Monitor Stations are locatedworldwide (Colorado Springs, Hawaii,Kwajelein, Ascension Island and DiegoGarcia)

• three (3) Ground ControlStations/Antennas located at Kwajalein,Ascension Island and Diego Garcia, areused for uploading data to the satellites

Control Segment


GPS Framework

The Monitor Stations compute signals from all GPS

satellites in-view every 1.5 seconds as well as collect

meteorological data and transmit them to the MCS.

The MCS then collects and processes these tracking data to compute satellite ephemeris and clock

parameters. It also carries out satellite control such as orbit maneuvers.

The Ground Control Stations then uploads these

corrections to the satellites once or twice per day.

Because of the global distribution of the Ground Control Stations, a

minimum of three (3) contacts per day can be established between the control segment and the satellites.


GPS Framework

• worldwide community of civilian & military users equipped with GPS receivers

• Typical information sought: 2D or 3D position, navigation parameters (position, velocity, heading), time

User Segment


GPS Framework

Types of receivers:

1. Navigation/Mapping Grade Receivers – (C/A-code)

2. Single Frequency/Survey Grade Receivers – (C/A-code and L1 carrier

3. Dual Frequency/Geodetic Receivers – (C/A-code, P-code and L1 & L2 carriers

User Segment


GPS Framework

Orbit is a function of:

• Earth’s Gravitational

Attraction

• Sun and moon’s

gravitational attraction

• Solar radiation pressure

• Atmospheric drag (low

orbits only)


Motion of Satellites

10

Normal Orbits

• Earth is a point mass (radial gravity)

• Satellite mass is negligible

• Satellite travels through a vacuum (no atmospheric drag or solar radiation)

• No sun, moon or other external gravitational fields

• Satellite path is elliptical

• The earth lies at one foci of the ellipse



11

a

b

Apogee Perigeeae

Focus

Orbital Path

Normal Orbits



12

where:

W-right ascension of the ascending

node

w- argument of perihelion

f – true anomaly (a function of time)

i – orbital inclination

a – semi-major axis of orbit ellipse

(m)

e – eccentricity of orbit ellipse

(unitless)

satellite

= f

Perigee



13

Deviations from Keplerian (Normal) orbits:

1. A non-central gravity field causes:

a. Rotation of the orbital plane in a direction opposite to the satellites

east-west motion.

(W) = dW/dt

b. Rotation of the major axis in the orbital plane.

(i )= di/dt



14

2. Other perturbations caused by the gravitational attraction of the

sun and moon, solar pressure and variations in the earth’s gravity

field are modelled by:

a. C u c , C u s corrections to argument of latitude

b. C r c , C r s corrections to orbital radius

c. C i c , C i s corrections to orbital inclination



15

Re: Six Kepler elements:

a, e define the size of the orbital ellipse

W, w, i position the orbital ellipse in celestial space (RA system)

f locates the satellite in its orbit relative to the perigee



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l k = W 0 + (W –we)t k – w e t 0e

where:

w e is the angular velocity of the earth

t 0e is the ephemeris reference time

• Perturbations to the Keplerian orbit cause the following

parameters to vary with time:

u k argument of latitude

r k orbital radius

i k orbital inclination



17

• Contents of the broadcast

ephemeris:

t 0e – ephemeris reference time

M 0 – Mean anomaly at time t 0e

W 0 - right ascension at t 0e

i 0 – orbital inclination at t 0e

Dn – mean motion difference

(a)1/2 – square root of semi-major axis

e – orbit eccentricity

w - argument of perihelion

(W-dot) – rate of ascension

(i-dot) – rate of inclination

C u c, C u s – corrections to argument of latitude

C r c, C r s – corrections to orbital radius

C i c, C i s – corrections to orbital inclination

f

C r c, C r s

(Dn)

Reference position (t0e)C u c, C u s

C i c, C i s

Perigee

(W) (i)



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• Ephemeris algorithm:

The order of computation is as follows:

1. t k – time since reference epoch

2. M k – Mean anomaly

3. E k – Eccentric anomaly (by iteration)

4. f k – True anomaly

5. u k – argument of latitude

6. r k – radius of orbit

7. i k – inclination of orbit

8. q1, q2 – position of satellite in orbit

9. l k - longitude of ascending node

10. x k ,y k , z k - ECEF coordinates

f

C r c, C r s

(Dn)


C i c, C i s

Perigee

(W) (i)



19

• Ephemeris algorithm:

Equations for the computation are:

1. t k = t – t 0e

2. M k = M 0 + n t k where n = n 0 + Dn

3. M k = E k – e sin E k (by iteration)

4. f k = tan -1 ((1-e2)1/2 sin E k)

( cos E k – e)

5. u k = (f k + w) + C uc cos 2(f k + w) +

C us sin 2(f k + w)

6. r k = a(1-e cos E k) + C rc cos 2(f k + w) +

C rs sin 2(f k + w)

7. i k = i 0 + (i-dot) t k + C ic cos 2(f k + w) +

C is sin 2(f k + w)

8. q1 = r k cos u k

q2 = r k sin u k

9. l k = W 0 + ((W -dot) – we)t k – we t 0e

10. x k = q1 cos l k – q2 cos i k sin l k

y k = q1 sin l k + q2 cos i k cos l k

z k = q2 sin i k

f

C r c, C r s

(Dn)


C i c, C i s

Perigee

(W) (i)



20

Navigation Message

• broadcast message transmitted at 50 bits per second

• contains the following information:

– correction parameters for clock errors and ionospheric

models

– broadcast ephemeris, a set of 16 parameters allowing

precise SV location as a function of time

– the almanac which shows the location of the other

satellites with relative to a given satellite

– satellite “health”.



21

The types of codes are the following• C/A-code – Coarse/Acquisition Code (1.023 MHz)

• P(Y)-code – encrypted military codes

At present L1 carrier is modulated with C/A and P(Y) codes

L2 carrier is modulated with P-code only

Modern GPS transmit a second civil code on L2 and a 3rd civil code

on a new L5 (1176.45 MHz) carrier


GPS Positioning

Carrier Frequencies

λ f

L1 19 cm 1575.42 MHz

L2 24 cm 1227.60 MHz

• Carrier Frequencies – uses the L1 and L2 of the L-band of microwave frequencies

• the other signals are modulated or “carried” into the L1 & L2 frequencies

• based on the equation, λ = c/f, where c = vacuum speed of light

• These are modulated with two types of codes and navigation message


GPS Positioning

Doppler System

• Doppler Effect: Imagine the continuously changing pitch of a car as it approaches and passes you.

• This change in frequency is a function of range or distance and is the underlying principle of the Doppler positioning systems.


GPS Positioning

• In the Doppler system, a precisely controlled radio frequency

is continuously transmitted from a satellite as it orbits past an

observer’s station.

• As the satellite draws nearer the receiver, the received

frequency increases. Then as the satellite passes the receiver,

the frequency decreases below the transmitted level.

• With the transmitting frequency, satellite orbit, and precise

timing of observations known, you can then compute the

position of the receiving station


GPS Positioning

Doppler System

Two Types of Observables

1. Pseudo-range – equals the distance between the satellite and the

receiver plus small corrective terms due to receiver and satellite

clock errors, impact of ionosphere and troposphere on signal

propagation and multipath

2. Carrier Phase – the difference between the received phase and

the phase of the receiver oscillator at the epoch of measurement


GPS Positioning

• Only possible using the C/A-code (or the P-code for authorized users)

• Solution is available in real-time

• Requires a knowledge of satellite position and satellite clock

• Satellite position is available from the broadcast ephemeris

• Satellite clock (signal transmit time) is carried on the PRN code

A

Pseudo-range

GPS Positioning

Fundamental Principles of Global Positioning System (GPS) 27

• C/A-code – clear access/coarse acquisition code– wavelength: 300 meters

– carried on L1 frequency

– it’s generated at a rate of 1.023 million bits per second but is very short and lasts only for one millisecond

• P-code – Precise/Private Code– wavelength: 30meters– carried on both L1 & L2 and generated at a rate of 10.23

million bits per seconds– a very long code that repeats itself every 267 days (since its

long, the code is divided into portions of week & each satellite generates its one week portion; ex: SV14 generates the 14th week of the code)

Pseudo-range

GPS Positioning

Fundamental Principles of Global Positioning System (GPS) 28

• The measured distance from the receiver to the

satellite-based on code observations – is called

pseudo-range

• The pseudo-range is not a true range because it is

corrupted (biased) by the receiver clock error

Pseudo-range


GPS Positioning

29

r = c (DT)

• Pseudoranges (Code Observation)

– Each satellite sends a unique signal

which repeats itself approx. 1 msec

– Receiver compares self generated signal

with received signal

– From the time difference (DT) a range

observation can be determined

– Receiver clock needs to be synchronized

with the satellite clockDT

Received Code

from Satellite

Generated

Code from

Receiver

Pseudo-range


GPS Positioning

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Adopting the following notation:

t s - satellite clock-time signal transmission (carried on the C/A- code

t R - receiver clock time at signal reception

d s - satellite clock offset from GPS time

d R- receiver clock offset from GPS time

D t= t R - t s = (t R (GPS) - d R ) –(t s(GPS)- d s)

D t= D t(GPS) + Dd (where Dd = d s - d R)

Assuming that the satellite clock error can be modeled, Dd = - d

R(negative of the receiver offset)

Pseudo-range


GPS Positioning

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• The time interval multiplied by the speed of light gives the pseudo-range (in meters)

r =c Dt = (c Dt(GPS) + cDd) = (R + cDd)

Where R is the true range

r =/Xs – XR/ + c Dd

r S R

Y

Z

X

Pseudo-range


GPS Positioning

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3 Spheres intersect at a point

3 Ranges to resolve for Latitude, Longitude and Height

R1

R2

R3

Outline Principle : Position

Pseudo-range


GPS Positioning

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r =/Xs – XR/ + c Dd

r RS=((Xs – XR)2 +(Ys – YR)2 +(Zs – ZR)2)1/2+ c Dd

So for each satellite s say s= 1, 2, 3, 4

r R1=((X1 – XR)2 +(Y1 – YR)2 +(Z1 – ZR)2)1/2+ c Dd

r R2=((X2 – XR)2 +(Y2 – YR)2 +(Z2 – ZR)2)1/2+ c Dd

r R3=((X3 – XR)2 +(Y3 – YR)2 +(Z3 – ZR)2)1/2+ c Dd

r R4=((X4 – XR)2 +(Y4 – YR)2 +(Z4 – ZR)2)1/2+ c Dd

Pseudo-range


GPS Positioning

34

• There are four unknowns in the basic pseudo-range observation

equation:

• (XR, YR, ZR)) and Dd

• Four satellites are required to solve for the four unknowns

(minimum)

• More than four satellites allow a least squares solution to be

computed

• The equations are non-linear with respect to the unknowns

• Linearization is required for solution (Taylor’s expansion)

Pseudo-range


GPS Positioning

35

The importance of satellite geometry

• DOP factors (Dilution of Precision) are used to quantify the effect of satellite geometry on the navigation solution

• DOP factors are simple functions of the diagonal elements of the covariance matrix of the adjusted parameters from the navigational solution

QX = s2X s YX s ZX s TX

s XY s2Y s ZY s TY

s XZ s YZ s2Z s TZ

s XT s YT s ZT s2T

Pseudo-range


GPS Positioning

36

• A description of purely geometrical contribution to the

uncertainty in a position fix

• It is an indicator as to the geometrical strength of the satellites

being tracked at the time of measurement

– GDOP (Geometrical)

• Includes Lat, Lon, Height & Time

– PDOP (Positional)

• Includes Lat, Lon & Height

– HDOP (Horizontal)

• Includes Lat & Lon

– VDOP (Vertical)

• Includes Height only

Dilution of Precision (DOP)

Good GDOP

Pseudo-range


GPS Positioning

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Dilution of Precision (DOP)

Poor DOP

• A description of purely geometrical contribution to the

uncertainty in a position fix

• It is an indicator as to the geometrical strength of the satellites

being tracked at the time of measurement

– GDOP (Geometrical)

• Includes Lat, Lon, Height & Time

– PDOP (Positional)

• Includes Lat, Lon & Height

– HDOP (Horizontal)

• Includes Lat & Lon

– VDOP (Vertical)

• Includes Height only

Pseudo-range


GPS Positioning

38


• Transformation from the geocentric into a local cartesian system (east, north, height) gives:

QX = s2e s ne s he s Te

s en s2n s hn s Tn

s eh s nh s2h s Th

s eT s nT s hT s2T

Pseudo-range


GPS Positioning

39


• The relevant DOP factors are then given by:

VDOP = sh

HDOP = (s2e + s2

n)1/2

PDOP = (s2e + s2

n+ s2h) 1/2

TDOP = s T

GDOP = (s2e + s2

n+ s2h+c2s2

T) 1/2

Pseudo-range


GPS Positioning

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r = c DT + lN

Range Determination from Phase Observations

• Phase Observations

– Wavelength of the signal is 19 cm on L1 and 24 cm on L2

– Receiver compares self-generated phase with received phase

– Number of wavelengths is not known at the time the receiver is switched on (carrier phase ambiguity)

– As long as you track the satellite, the change in distance can be observed (the carrier phase ambiguity remains constant)

DT

Received Satellite

Phase

Generated

Phase from

Receiver

Carrier Phase Observables


GPS Positioning

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If the receiver and satellite clocks were perfectly synchronized and the signal passed through a vacuum, the transmit and receive times would be related by

T = t + r/c

• However this is not the case. Both the receiver and

satellite clocks will be offset from GPS time:

(T+dT) = (t+dt)+ r/c

• The signal travel time (r/c) will be affected by the ionospheric and tropospheric delay:

(T+dT) = (t+dt)+(r – dion + dtrop)/c

(T—t) = dt—dT+(r – dion + dtrop)/c



GPS Positioning

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j total = j(T) - j(t) = f(T-t)

= f(dt—dT)+f(r – dion + dtrop)/c

In terms of what a GPS receiver

actually observes, the total phase

consists of:

Fr(j) - measured fractional part

Int(j; to ,t) - measured integer

count of complete cycles

since the initial epoch to

N(to) - an unknown number of integer cycles

between the satellite and receiver at

the initial epoch

j total = Fr(j) + Int(j; to ,t) + N(to)



GPS Positioning

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j total = j measured + N(to)

j measured + N(to) = f(dt—dT)+f(r – dion + dtrop)/c

j measured = f(dt—dT)+f(r – dion + dtrop)/c - N(to)

Using the relation l=c/f and multiplying by l, convert the measured carrier beat phase into length units

F = l j measured

= c(dt—dT)+ r – dion + dtrop - l N(to)



GPS Positioning

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Considering the receiver multipath and noise:

F = c(dt—dT)+ r – dion + dtrop - l N(to) + dmult + n

Usually multipath and noise are combined so that:

F = c(dt—dT)+ r – dion + dtrop - l N(to) + n



GPS Positioning

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• Many errors that affect GPS observations are correlated

in a spatial sense

• Thus some errors affecting observations at one

receiver will be similar at a nearby receivers

• Measurement differencing is a technique to eliminate

or reduce the impact of common errors and

thereby simplify the observation equation



GPS Positioning

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Differences can be formed as follows :

• Between receivers (∆)

• Between satellites (▼)

• Between epochs (∂)



GPS Positioning

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Single Differences:

• Between receivers (common satellite and epoch)

• Between satellites (common receiver and epoch)



GPS Positioning

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Single Differences:

• Between epochs

(common receiver and satellite)

• Between receivers (commonsatellite and epoch)

• Between satellites (commonreceiver and epoch)



GPS Positioning

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Double Differences:

• Receiver - Satellite

• Satellite - Receiver

• Both satellite and receiver clock offsets are eliminated in the satellite-receiver double difference

• Ionospheric and tropospheric errors are further reduced

•Observational noise is potentially increased



GPS Positioning

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Double Differences:

• Receiver -Time



GPS Positioning

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Double Differences:

• Satellite - Time



GPS Positioning

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Triple Differences:

An identical equation for the triple difference is obtained by differencingany of the following double difference combinations:

• Receiver – satellite double difference against time (t1 and t2)

• Receiver – time double difference against satellites (p and q)

• Satellite – time double difference against receivers (i and j)



GPS Positioning

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Triple Differences:

• The triple difference completely eliminates the integer ambiguity, as long as therehave been no cycle slips

• If cycle slips have occurred, there will appear large residuals in the solution of thetriple difference observation equations

• Ionospheric and tropospheric errors are also reduced



GPS Positioning

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GPS Positioning

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• Two categories of error

Systematic errors (biases) - can usually be modelled

mathematically or eliminated.

Random errors (noise) –unavoidable

• Three sources of error:

Satellite

Propagation medium (atmosphere),

Receiver


Error Sources

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AN APPROXIMATE ERROR BUDGET


Error Sources

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Error Sources

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• combined effect of orbit, propagation and clock errors and

receiver noise, projected onto a line between receiver and

satellite is called the User Equivalent Range Error (UERE)

• The product of a DOP value by the UERE gives the expected

accuracy of positioning:

Accuracy =DOP x UERE (m)


Error Sources

59

Thank you for your

Attention!!!


Fundamental Principles of Global Positioning Systems (GPS) · PDF fileLouie P. Balicanta...

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Transcript of Fundamental Principles of Global Positioning Systems (GPS) · PDF fileLouie P. Balicanta...