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Chris Coakley
Math G
Ian Walton, Instructor
The GARMIN 296
When I am p=
lanning
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 airpla=
ne
in fine mechanical condition is always
important too; food and water are carried just in case o=
f an
emergency. Current sectionals
(maps,) and my new Garmin 296 global positioning system,=
or
yoke and the antenna mounted up on the windscreen. The m=
odern
day
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
The Garma=
n 296
is one of my favorite items for any flight because it has so much
information stored into a small package. I tell people t=
hat:
“my 296 is the best thing since sliced
bread.”Purchasing the 296 about two years ago, I have ye=
t to
use all the data and or functions
offered in the device because it has so much. At this po=
int
of ownership, I use about twenty-five
percent of its capacity, which for now, does most operat=
ions
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
9 small function buttons are on the right side-face of t=
he
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 m=
odes, for
example, when in the aviation mode, gives you
a map page which displays a little airplane and your cur=
rent
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 cli=
mb,
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 equ=
al
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 t=
ime
to your destination at the current speed
the airplane is flying at.
Favorite and Details
My favorit=
e 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 whe=
re 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
because of the heavy commercial operations of airliners.=
Airspace
classifications are invisible
layers of airspace depicted only by a sectional, or with=
a
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 t=
he
way out it would show the earth and
where I am on its face. This page also has a current rea=
ding
of my altitude, distance to
destination, and the track and airport listings.
<= o:p>
“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 push=
ed,
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 nig=
ht
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 especi=
ally
in an emergency situation and
you’re not sure of an airports location.
<= o:p>
What is a
Now that I ha=
ve
shared some of the details and uses for the
to describe what the
information from twenty-four orbiting satellites 11,000
miles in space, and five ground stations
th=
at
make sure the satellites are working properly. The
the earth.
few meters. In fact,
with advanced forms of
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 th=
e
speed of light. Even=
at
this speed, the signal takes a measurable amount of time to reach the
receiver. The differ=
ence
between the time the signal is sent and the time it is received, multiplied=
by the speed of ligh=
t,
enables the receiver to calculate the distance to the satellite.
Her=
e's
how
1. The basis of
2. To "triangulate,&q=
uot; a
3. To measure travel time,=
4. Along with distance, yo=
u need
to know exactly where the satellites are in space.
5. Finally you must correc=
t for
any delays the signal experiences as it travels through the atmosphere. You
must also correct for the clock differences between the
Note:
Please see page numbers 7- 9 for illustrations of the triangulation of =
<= o:p>
ORBITS
The satelli=
tes
used for the
the earth’s gravitational pull and the other counteracti=
ng
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
coverage. It was interesting to find out that no satelli=
tes
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 sta=
ted
important information on the circular orbit and the equa=
tion
associated with that orbit. I found
that it would be important to quote the information verb=
atim
because of its importance in the
understanding of
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, ex=
cept
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 const=
ant
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.<= o:p>
Using
F=
=3D g (
M M
) /R
Where M is the Mass of the earth or 5.983E24 Kg,=
<=
span
style=3D'mso-spacerun:yes'> M<=
span
style=3D'mso-spacerun:yes'> is the mass of the satellite, g is the universal gravitati=
on
constant, and R=
=
is the
distance from the satellite to the center of the earth.
The Mathematics of
The
receiver to work properly, the receiv=
er
needs to be in contact with at least three active orbiting
satellite sat one time. In order for =
a
satellites and the receiver are neces=
sary
to pinpoint the coordinates of your choice. More
specifically, the time it takes for a
satellite signal to reach the
with the coordinates reading the rece=
iver
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 communicatio=
n 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 inter=
est
(in this paper we will use N=3D3).In a perfect scenario, the
lines of sight from the three satelli=
tes
will all intersect at the point of interest, providing the
object’s position, but in real life t=
he
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 restric=
ted
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.
<=
span
style=3D'mso-spacerun:yes'>
very economical. And that makes the technolo=
gy
accessible to virtually everyone. These days
machinery, even laptop computers. Soon
triangulation step is something like this. Y=
ou
have at hand your trusty
Satellite 1
miles away, then you can only localize yours=
elf on
a circle on the earth that is the locus of all
points 11,000 miles from
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, obtain=
ing
the distance from a third satellite (
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
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.e=
du/~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 observat=
ions
from four satellites can give the exact position.
<= o:p>
*In each event, your position is on the locu=
s of
result points.
Now most
We now know that if we have the locations of=
the
satellites and distances from us, we can
accurately determine our position. But we ne=
ed to
know how the distance is computed. In fact
there are a couple of ways. Here are some me=
thods
to compute distance.
·<=
span
style=3D'font:7.0pt "Times New Roman"'> =
&nb=
sp;
Direct measurement with a "ruler."
·<=
span
style=3D'font:7.0pt "Times New Roman"'> =
&nb=
sp;
Inferred distances by measuring angles in triangles.
·<=
span
style=3D'font:7.0pt "Times New Roman"'> =
&nb=
sp;
Distance measurement using the speed of light (light propagati=
on
time).
As mentioned above
directly. How to measure distance by light? =
There
are a couple of methods. First, distance can be
measured directly by sending a pulse and mea=
suring
how it takes to travel between two points.
This most common method is to reflect the si=
gnal
and the time between when the pulse was
transmitted and when the reflected signal re=
turns.
Such systems, called bi-directional, are used in
radar and satellite laser ranging that requi=
re
single millimeter accuracy. They require a clock capable of timing accuracy=
of
Such equipment is expensive, costing for
satellites about $1m.
M=
ore Equations
and Conclusion
Assuming=
we have
the distances from four satellites and we know
exact po=
sitions
of the satellites, we must then solve the following system equations where =
the
.
where
the diff=
erence
between
is that =
there is
just one clock offset time. This means the all the satellites must have
perfectly
synchron=
ized
clocks, and this is just one of the tasks of the control sites. The unknowns
above are
perfectly
synchronized, the time offset would be zero.
www.math.tamu.edu/~dallen/physics/gps/gps.htm
Conclusion
As
you can see, the
=
Works Cited
I=
nternet
Sources
The Mathematics of GPS through Linear Algebra
File Format: PDF/Adobe Acr=
obat
- Quick
View
by WO Pence -=
2001 - Related articles
online.redwoods.cc.ca.us/instruct/darnold/laproj/.../Linear.pdf –
File Format: PDF/Adobe Acr=
obat
to take the picture unless switched over to manual operation. Photographers=
no
longer must think carefully about the physical pro-. The. Mathematics of
gauss.gge.unb.ca/gpsworld/.../Innov.1991.07-08.pdf
The
www.math.tamu.edu/~dallen/physics/gps/gps.htm
P=
roquest
Garmin Aera
Global positioning system: The mathematics of GPS receiv=
ers
Richard B Thompson. Mathematics Magazine.
"AutoGraph: How
B=
ooks
GARMIN. GPSMAP 296. <=
/span>
=
Works Cited Continued
Helfrick, Albert D. =
Principles
of Avionics. Leesburg, VA, USA: Avionics Communications, 2007. 120-22.
Print.
Coakley 6