02.Coordinate and Time Systems

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MIT OpenCourseWarehttp://ocw.mit.edu

12.540Principles of Global Positioning SystemsSpring 2008

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12.540 Principles of theGlobal Positioning System

Lecture 02Prof. Thomas Herring


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Coordinate Systems

Today we cover: Definition of coordinates

Conventional realization of coordinates

Modern realizations using spaced basedgeodetic systems (such as GPS).


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Coordinate system definition

To define a coordinate system you needto define:

Its origin (3 component)

Its orientation (3 components, usually thedirection cosines of one axis and one

component of another axes, and definitionof handed-ness)

Its scale (units)


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Coordinate system definition

In all 7 quantities are needed touniquely specify the frame.

In practice these quantities are

determined as the relationship betweentwo different frames

How do we measure coordinates How do we define the frames


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Measuring coordinates

Direct measurement (OK for graph paper) Triangulation: Snell 1600s: Measure angles

of triangles and one-distance in base triangle

Distance measured with calibrated chain orsteel band (about 100 meters long)

Baseline was about 1 km long Triangles can build from small to larges ones.

Technique used until 1950s.


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Small errors in the initial lengthmeasurement, would scale the whole network

Because of the Earth is nearly flat,

measuring angles in horizontal plane onlyallows horizontal coordinates to bedetermined.

Another technique is needed for heights.


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In 1950s, electronic distancemeasurement (EDM) became available(out growth of radar)

Used light travel times to measuredistance (strictly, travel times of

modulation on either radio, light or near-infrared signals)


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Advent of EDM allowed directmeasurements of sides of triangles

Since all distances measured lessprone to scale errors.

However, still only good for horizontal

coordinates


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Accuracies

Angles can be measured to about 1 arcsecond (5x10-6 radians)

EDM measures distances to 1x10-6 (1

part-per-million, ppm) Atmospheric refraction 300 ppm

Atmospheric bending can be 60 (moreeffect on vertical angles)


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Height coordinates

Two major techniques: Measurement of vertical angles

(atmospheric refraction)

Leveling measurement of heightdifferences over short distances (


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Other methods

Maps were made with plotting tables(small telescope and angular distancemeasurements-angle subtended by a

known distance

Aerial photogrammetry coordinates

inferred from positions in photographs.Method used for most maps


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Other methods

What is latitude and longitude Based on spherical model what

quantities might be measured

How does the rotation of the Earthappear when you look at the stars?

Concept of astronomical coordinates


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Geodetic coordinates: LatitudeNorth

Equator

Geoid

gravity direction

Normal to ellipsoid

ga

Local equipotenital surface

Earth's surface

P


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Longitude

x

Rotation of Earth

Longitude measured by time difference of astronomical events


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Astronomical coordinates

Return to later but on the global scalethese provide another method ofdetermining coordinates

They also involve the Earths gravityfield

Enters intrinsically in triangulation andtrilateration through the planes anglesare measured in


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Web sites about geodeticmeasurements

http://sco.wisc.edu/surveying/networks.phpGeodetic control for Wisconsin

http://www.ngs.noaa.gov/is web page of

National Geodetic Survey which coordinatesnational coordinate systems
http://sco.wisc.edu/surveying/networks.phphttp://www.ngs.noaa.gov/http://www.ngs.noaa.gov/http://sco.wisc.edu/surveying/networks.php


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Earths Gravity field

All gravity fields satisfy Laplacesequation in free space or material of

density . If V is the gravitational

potential then

2

V = 0

2V = 4G


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Solution to gravity potential

The homogeneous form of this equationis a classic partial differential equation.

In spherical coordinates solved byseparation of variables, r=radius,

=longitude and =co-latitude

V(r,,)=R(r)g()h(


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Summary

Examined conventional methods ofmeasuring coordinates

Triangulation, trilateration and leveling

Astronomical positioning uses externalbodies and the direction of gravity field

Continue with the use of the gravityfield.

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02.Coordinate and Time Systems