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Chris Coakley

Math G

Ian Walton, Instructor

May 10,
2010

The GARMIN 296 GPS, Space and Mathematics

When I am planning
a long cross-country trip to…let’s say… the Mojave Desert in my

Cessna 172 airplane, I plan at least a day in advance so I’m
prepared for the tasks at hand.

Keeping track of the weather patterns is probably the top
priority as with making sure the

airplane is filled with fuel and oil. Keeping the airplane
in fine mechanical condition is always

important too; food and water are carried just in case of an
emergency. Current sectionals

(maps,) and my new Garmin 296 global positioning system, or GPS, is
safely installed on the

yoke and the antenna mounted up on the windscreen. The modern
day GPS is a
remarkable tool

that receives information from orbiting satellites many
thousands of miles away, and the

GARMIN 296, and the mathematics behind this space based
system has changed the way most

pilots fly today.

The GARMIN 296 GPS

The Garman 296
is one of my favorite items for any flight because it has so much

information stored into a small package. I tell people that:
“my 296 is the best thing since sliced

bread.”Purchasing the 296 about two years ago, I have yet to
use all the data and or functions

offered in the device because it has so much. At this point
of ownership, I use about twenty-five

percent of its capacity, which for now, does most operations
for my type flying.  The 296

measures 5 ½” long x 3 ¼” wide, 1 ½” thick, it’s black in
color, has a  4” 265 TFT color display,

9 small function buttons are on the right side-face of the
unit with one large curser button. It

weighs one pound, and with all its accessories (antenna,
yoke clip and wiring,) it weighs a total

of two and half pounds.

MODES

The 296 has a variety of different modes, for
example, when in the aviation mode, gives you

a map page which displays a little airplane and your current
location. Pressing the “page” button

allows one to advance to the next page and so on. The
terrain page gives you elevations to

mountains and obstacles; a panel page displays the
instruments just like in the cockpit, giving a

very accurate airspeed indicator, altimeter, rate of climb,
heading indicator and a turn

coordinator. These small instruments on the 4” screen are
very accurate too. I have made many

flights with the GARMIN 296 and they are shown to be equal
to the airplanes factory-built in

gauges. As one clicks through the page via the page mode,
you can find the current time and

date, the time of sunrise and sunset, battery life and time
to your destination at the current speed

the airplane is flying at.

Favorite and Details

My favorite page,
and the one I spend the most time with as I’m flying, is the map page. I

like this page because it displays a detailed map of where I
am at the current time and give me

information of the types of airspace that I should avoid or
may be concerned with; certain

airspaces such as military operating areas (MOA’s.)  Class BRAVO (B) airspace, for

example, is  the area around
San Francisco International Airport class
B should be avoided

because of the heavy commercial operations of airliners. Airspace
classifications are invisible

layers of airspace depicted only by a sectional, or with a GPS. The
information displayed on the

GPS is
great because the unit tells you with a flag if you are getting to close. Using
the scan “in

or out” mode is on this map page as well, this page lets you
scan “in” for better detail, and scan

“out” for a more objective reference. If I scanned all the
way out it would show the earth and

where I am on its face. This page also has a current reading
of my altitude, distance to

destination, and the track and airport listings.

“NRST FIND”

One of the features that can be quickly
accessible from this page, or any page for that matter,

is the button labeled “NRST FIND” This feature when pushed,
tells you the nearest airport from

your current location, the frequencies of that airport, time,
and miles to that airport. This feature

alone is very helpful especially when I am flying at night
or when visibilities are so poor that I

can’t see an airport or its rotating beacon. When I push
that button a little arrow pops-up and

points to where it is, miles, and en-route time at your
current speed. This button is a truly

amazing option if you should ever need an airport especially
in an emergency situation and

you’re not sure of an airports location.

What is a GPS?

Now that I have
shared some of the details and uses for the GPS, I
think it is only appropriate

to describe what the GPS is and
how it works. In simple terms, the GPS
receives data and

information from twenty-four orbiting satellites 11,000
miles in space, and five ground stations

that
make sure the satellites are working properly. The GPS satellites each take 12
hours to orbit

the earth. GPS uses these satellites as reference points to
calculate positions accurate to just a

few meters. In fact,
with advanced forms of GPS you can make measurements to better than a

centimeter! Each
satellite is equipped with an accurate clock to let it broadcast signals
coupled

with a precise time
message. The ground unit receives the satellite signal, which travels at the

speed of light. Even at
this speed, the signal takes a measurable amount of time to reach the

receiver. The difference
between the time the signal is sent and the time it is received, multiplied

by the speed of light,
enables the receiver to calculate the distance to the satellite.

How does it work?

Here's
how GPS works in five steps:

1.  The basis of GPS is "triangulation"
from satellites.

2.  To "triangulate," a
GPS receiver measures distance
using the travel time of radio signals.

3.  To measure travel time, GPS needs very accurate timing.

4.  Along with distance, you need
to know exactly where the satellites are in space.

5.  Finally you must correct for
any delays the signal experiences as it travels through the atmosphere. You
must also correct for the clock differences between the GPS receiver and the satellites.

Note:
Please see page numbers 7- 9 for illustrations of the triangulation of GPS

ORBITS

The satellites
used for the GPS system
are held in place by two balancing forces. One being

the earth’s gravitational pull and the other counteracting
force is the centripetal force which

“flings” the satellite into space. The orbits are in
circular patterns and the move relative to the

earth’s surface. The GPS orbits
are inclined fifty-five degrees to the equator to provide global

coverage. It was interesting to find out that no satellites
pass directly over the North or South

poles because it would require an inclination of ninety
degrees. With the curvature of the earth,

aircraft or other receivers can pick up the signals from the
current orbiting satellites at the fifty-

five degree inclinations.

Equation of an Orbiting Satellite Explained

One of the
resources used in this report was from a college aeronautics text which stated

important information on the circular orbit and the equation
associated with that orbit. I found

that it would be important to quote the information verbatim
because of its importance in the

understanding of GPS
systems. From the text Principles of Avionics, describes in
a circular

orbit, gravity is in the direction of a radial from the
center of the earth to the satellite, and

centripetal force is along the same radial direction, except
directed away from earth. The forces

act along the same radial line except in opposite
directions. Also, traversing a circular path

involves a constant magnitude of acceleration, not constant
acceleration, because the direction of

the acceleration vector changes. Therefore, we can write an
equation using only the magnitude of

the forces, realizing that forces are actually vectors.

Using Newton’s law,
the force on a satellite due to gravity is:

F = g (
M  M
) /R

Where  M  is the Mass of the earth or 5.983E24 Kg,  M  is the mass of the satellite, g is the universal gravitation
constant, and R is the
distance from the satellite to the center of the earth.

The Mathematics of GPS through Linear Algebra

The GPS should use an active
device to accurately pinpoint the position of choice. For a GPS

receiver to work properly, the receiver
needs to be in contact with at least three active orbiting

satellite sat one time. In order for a GPS receiver to be active,
effective communication between

satellites and the receiver are necessary
to pinpoint the coordinates of your choice. More

specifically, the time it takes for a
satellite signal to reach the GPS receiver will reduce
the error

with the coordinates reading the receiver
calculated (this is signal propagation delay).

A passive receiver, as opposed to an
active receiver, will receive information from satellites

concerning their position in relation to
the receiver’s position just as an active device would, but

there would be no active communication between
the receiver and satellites. This is referred to as

“spoofing.” The receiver would still
relay coordinates, but how accurate would those coordinates

be? This question has been labeled as the
N-Site problem. The first step with
developing the N-

Site solution is to analyze the geometry of
the situation. At this time we know the direction from

N passive sensors to a point of interest
(in this paper we will use N=3).In a perfect scenario, the

lines of sight from the three satellites
will all intersect at the point of interest, providing the

object’s position, but in real life the
lines will miss each other, and overlap in such a way

that a “gray” area exists. The
calculation for the point of interest within the gray area can be

found using triangulation. Our
mathematical model is a sample of using triangulation (our point

of interest exists within a triangle, and
it’s position can be found using information from the lines

forming the triangle). Although this
problem exists in three space, we will use a two-dimensional

model with vectors to demonstrate our
N-Site solution. The reason why a 2D model can be an

example for a realistic 3Dmodel is
because vectors will be used. Otherwise, if coordinates and

angles were used, we would be restricted
to a dimensional model only.

Here's a graphic to
illustrate the complete functionality.

This illustration above is a good
representation of the triangulation from ground based radar, satellites, and
ground monitor stations, also a good representation of the direct link to
aircraft.

GPS Triangulation Explained and Equation

GPS receivers have
been miniaturized to just a few integrated circuits and so are becoming

very economical. And that makes the technology
accessible to virtually everyone. These days

GPS is finding its way into cars, boats, planes, construction
equipment, movie making gear, farm

machinery, even laptop computers. Soon GPS will become almost
as basic as the telephone. The

triangulation step is something like this. You
have at hand your trusty GPS receiver. What the

GPS can do is precisely measure your distance from
any one of a number of satellites. From

Satellite 1

your device (GPS unit) measures your distance from the satellite to be
11,000

miles away, then you can only localize yourself on
a circle on the earth that is the locus of all

points 11,000 miles from

Now from Satellite 2 (

your GPS unit measures your distance

from the satellite to be 12,000 miles away. Now
you are localized to be on the intersection of the

spheres centered at the satellites and of the
respective radii. The intersection of the spheres is a circle. Next, obtaining
the distance from a third satellite (

) allows us to locate in the

intersection of the three spheres and this
intersection is two points (A circle intersected by a sphere is two points, in
general.) One of the points is where you are

. The other point is rarely

on the planet's surface and so by a little
reckoning, you can eliminate this possibility and select

your location. This is broad strokes is how its
done. But there are fine details and they lie in

computing those distances.

http://www.math.tamu.edu/~dallen/physics/gps/gps.htm#gps

Another view of the same situation shows the
spheres centered at the satellites. In the picture below, the planet earth is
the small circle toward the middle of the intersection of the spheres.

Here's the easy way to see that observations
from four satellites can give the exact position.

*In each event, your position is on the locus of
result points.

Now most GPS units search for and determine measurements from as many
as six satellites. This can only improve the accuracy using least squares
methods.

How exactly is the distance computed?

We now know that if we have the locations of the
satellites and distances from us, we can

accurately determine our position. But we need to
know how the distance is computed. In fact

there are a couple of ways. Here are some methods
to compute distance.

·
Direct measurement with a "ruler."

·
Inferred distances by measuring angles in triangles.

·
Distance measurement using the speed of light (light propagation
time).

As mentioned above GPS
methods are related to measuring light propagation time but not

directly. How to measure distance by light? There
are a couple of methods. First, distance can be

measured directly by sending a pulse and measuring
how it takes to travel between two points.

This most common method is to reflect the signal
and the time between when the pulse was

transmitted and when the reflected signal returns.
Such systems, called bi-directional, are used in

radar and satellite laser ranging that require
single millimeter accuracy. They require a clock capable of timing accuracy of

seconds (3 picoseconds). The clock stability need is

. A clock with this longtime
stability would gain or lose 0.03 seconds in a year.

Such equipment is expensive, costing for
satellites about $1m.

More Equations
and Conclusion

Assuming we have
the distances from four satellites and we know

,

are the

exact positions
of the satellites, we must then solve the following system equations where the
.

where

is the speed of light and

is the receiver clock offset time. The receiver clock offset is

the difference
between GPS time and internal receiver time.
Obviously, a key portion of all this

is that there is
just one clock offset time. This means the all the satellites must have
perfectly

synchronized
clocks, and this is just one of the tasks of the control sites. The unknowns
above are

and

. Let us reiterate: if the clocks of the GPS receiver and the satellite were

perfectly
synchronized, the time offset would be zero.

www.math.tamu.edu/~dallen/physics/gps/gps.htm

Conclusion

As
you can see, the GPS receiver is not just a fine electronic mechanism that helps me
and most of the pilot community these days; it can do a whole lot of
mathematics as well. My 296 does not have a “calculator” to do simple addition
and subtraction problem, but an intricate device in the complex calculations of
geometry, linear algebra and calculus. The next time I take to the air and look
down at my GPS I will think about all twenty-four satellites and the ground
based receivers that make the accuracy I interpret on my little 4” color
display.

Works Cited

Internet
Sources

The Mathematics of GPS through Linear Algebra

File Format: PDF/Adobe Acrobat
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by WO Pence - 2001 - Related articles

The Mathematics of GPS through Linear Algebra. William O. Pence and Robin R. Rumple. December 11, 2001. Abstract. This article will be introducing the ...

online.redwoods.cc.ca.us/instruct/darnold/laproj/.../Linear.pdf –

The Mathematics of GPS

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to take the picture unless switched over to manual operation. Photographers no
longer must think carefully about the physical pro-. The. Mathematics of
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gauss.gge.unb.ca/gpsworld/.../Innov.1991.07-08.pdf

gps.htm

The GPS receiver is not just a fine electronic mechanism, it can do a
whole lot of mathematics as well. Wow and they put it in such a small
package! ...

www.math.tamu.edu/~dallen/physics/gps/gps.htm

Proquest

Garmin Aera

Bill Cox. Plane and Pilot. Los
Angeles: May
2010. Vol. 46, Iss. 4; p. 70 (3 pages)

Global positioning system: The mathematics of GPS receivers

Richard B Thompson. Mathematics Magazine. Washington: Oct 1998. Vol. 71, Iss. 4; p. 260 (10 pages)

"AutoGraph: How GPS
works. " Newsday  5  Feb. 2010, Combined editions: Newsday, ProQuest.
Web.  2 May. 2010.

Books

GARMIN. GPSMAP 296. Kansas: GARMIN, 2005. Print.

Works Cited Continued

Helfrick, Albert D. Principles
of Avionics. Leesburg, VA, USA: Avionics Communications, 2007. 120-22.
Print.