GPS Math

[Jon G.]

Here is the math that calculates position from 3 satellites. There are 2
points of intersection of 3 spheres of known radii, which are determined by
time-stamped transmissions from the satellites.

http://www.geocities.com/jongiff2000/GPS_math.xls

Jon Giffen
***@peoplepc.com

[junoexpress]

Here is the math that calculates position from 3 satellites.  There are 2
points of intersection of 3 spheres of known radii, which are determined by
time-stamped transmissions from the satellites.
http://www.geocities.com/jongiff2000/GPS_math.xls
Jon Giffen

Pick up a book (see Spilker, Tsui, or Enge & Misra) and do some
reading instead of relying on others or the Internet solely for your
info. As you are hopefully learning, this is not a very reliable
strategy.

But if you find this all too taxing for your mind, then at last read
http://en.wikipedia.org/wiki/Pseudorange
(*read*, don't *skim*, got it)

Cheers Quaker,

M

[Jon G.]

I saw it on the History Channel... three giagantic spheres looming over the
planet Earth, and the narrator said, "GPS is based on the intersection of
three spheres from three satellite transmissions, at two points."

All I did was calculate those two points. ALRIGHT? Include yourself among
the others in "instead on relying on others."

I made a life out of Bouguer anomalies and corrections. WHO NEEDS THEM?
--
Jon G.
***@peoplepc.com
http://mypeoplepc.com/members/jon8338/math/index.html

Here is the math that calculates position from 3 satellites. There are 2
points of intersection of 3 spheres of known radii, which are determined by
time-stamped transmissions from the satellites.
http://www.geocities.com/jongiff2000/GPS_math.xls
Jon Giffen

Pick up a book (see Spilker, Tsui, or Enge & Misra) and do some
reading instead of relying on others or the Internet solely for your
info. As you are hopefully learning, this is not a very reliable
strategy.

But if you find this all too taxing for your mind, then at last read
http://en.wikipedia.org/wiki/Pseudorange
(*read*, don't *skim*, got it)

Cheers Quaker,

M

[junoexpress]

Post by Jon G.
I saw it on the History Channel... three giagantic spheres looming over the
planet Earth, and the narrator said, "GPS is based on the intersection of
three spheres from three satellite transmissions, at two points."

Wow! The History Channel! What a solid scientific source. Dude, I am
like so impressed. Forgive me for ever doubting you for even a time-
stamped nano-second.

BTW, did you ever figure out what the pseudorange was?

M

[christian.bau]

Post by Jon G.
I saw it on the History Channel... three giagantic spheres looming over the
planet Earth, and the narrator said, "GPS is based on the intersection of
three spheres from three satellite transmissions, at two points."

History Channel probably has to simplify things a little bit for its
audience.

Each satellite transmits a time stamp. And it transmits enough
information so that your GPS receiver can figure out exactly where the
satellite was when it sent that time stamp. If your GPS receiver also
knew exactly what time it was when that information was received, then
it would know how long ago that signal was sent, and if it was sent
lets say x seconds ago, then the GPS receiver would know that it is on
a sphere around the satellite with a radius xc (where c = speed of
light).

Unfortunately, the GPS doesn't know the exact time. That makes things
more difficult. With three satellites, the GPS would know that those
satellites sent their time stamps x1 + d, x2 + d, x3 + d seconds ago,
but d is unknown. The radius of the free spheres isn't x1*c, x2*c,
x3*c but x1*c + cd, x2*c + cd and x3*c + cd, where d is unknown. The
GPS receiver could make a guess that it is on the surface of the
earth, and calculate its position from that additional information
(probably good enough for a car, but not for an airplane), but for the
correct position without such guesses you need a fourth satellite.

[Tim Smith]

In article

Post by christian.bau
Each satellite transmits a time stamp. And it transmits enough
information so that your GPS receiver can figure out exactly where the
satellite was when it sent that time stamp. If your GPS receiver also
knew exactly what time it was when that information was received, then
it would know how long ago that signal was sent, and if it was sent
lets say x seconds ago, then the GPS receiver would know that it is on
a sphere around the satellite with a radius xc (where c = speed of
light).
Unfortunately, the GPS doesn't know the exact time. That makes things
more difficult. With three satellites, the GPS would know that those
satellites sent their time stamps x1 + d, x2 + d, x3 + d seconds ago,
but d is unknown. The radius of the free spheres isn't x1*c, x2*c,
x3*c but x1*c + cd, x2*c + cd and x3*c + cd, where d is unknown. The
GPS receiver could make a guess that it is on the surface of the
earth, and calculate its position from that additional information
(probably good enough for a car, but not for an airplane), but for the
correct position without such guesses you need a fourth satellite.

Another way to do it would be to look at time differences. Say you
receive time stamp T from satellite S1 first. Note the time, say t1.
Then you receive that time stamp from S2, at t2.

From this, you know that you are (t2-t1)*c farther from S2 than you are
from S1. That puts you somewhere on the surface of revolution generated
by a hyperbola with its foci at the positions of S1 and S2 at time T.

Note that finding this hyperbola only depends on the difference between
t1 and t2, rather than on t1 and t2 themselves, so your notion of t1
does not have to be even remotely accurate, just as long as t2-t1 is
accurate.

Bring in a third satellite, S3, whose T arrives at t3, and now you've
got three hyperbolic surfaces of revolution (one for each pair of the 3
satellites), and you are on the intersection of all three. Bring in a
fourth, and you've got 6 surfaces.

Note that a receiver using this method would not even need a clock. It
could use a counter based on or synced to the carrier frequency of the
satellite signal to measure time differences.