Calculating GPS Coordinates

[jgiffen]

Did you ever wonder how GPS coordinates are calculated? Here I have worked out the math for trilateration and provided a spreadsheet to do all of the calculations. GPS Math

[+fizzymagic]

Too bad that's not how the GPS unit calculates position, then, huh?

[+Walts Hunting]

This gives the correct answer

 

https://www.courses.psu.edu/aersp/aersp055_r81/satellites/gps_details.html

[seldom_sn]

To elaborate for more visually inclined readers:

 

The locus of points shared by two spheres is a circle.

The locus of points shared by three spheres is the intersection of two circles (two points, the circles being created by each sphere intersecting one other).

The locus of points shared by four spheres is one of the two points defined by the two intersecting circles.

So you need 4 satellites at minimum.

[+phlatlander]

Well, there goes my theory...

[John E Cache]

To elaborate for more visually inclined readers:

 

The locus of points shared by two spheres is a circle.

The locus of points shared by three spheres is the intersection of two circles (two points, the circles being created by each sphere intersecting one other).

The locus of points shared by four spheres is one of the two points defined by the two intersecting circles.

So you need 4 satellites at minimum.

...plus an accurate time of day for longitude(just like sailors since the the invention of accurate portable clocks).

[+fizzymagic]

To elaborate for more visually inclined readers:

 

The locus of points shared by two spheres is a circle.

The locus of points shared by three spheres is the intersection of two circles (two points, the circles being created by each sphere intersecting one other).

The locus of points shared by four spheres is one of the two points defined by the two intersecting circles.

So you need 4 satellites at minimum.

...plus an accurate time of day for longitude(just like sailors since the the invention of accurate portable clocks).

 

Nope. That is incorrect. The GPS satellites need accurate time, but the GPS receiver does not need any time information apart from that provided by the satellites.

[John E Cache]

To elaborate for more visually inclined readers:

 

The locus of points shared by two spheres is a circle.

The locus of points shared by three spheres is the intersection of two circles (two points, the circles being created by each sphere intersecting one other).

The locus of points shared by four spheres is one of the two points defined by the two intersecting circles.

So you need 4 satellites at minimum.

...plus an accurate time of day for longitude(just like sailors since the the invention of accurate portable clocks).

 

Nope. That is incorrect. The GPS satellites need accurate time, but the GPS receiver does not need any time information apart from that provided by the satellites.

The geometry gives you a point on a spinning globe. How do you get the longitude if you don't know angle the globe is on its axis(commonly referred to as time of day)? Can you compute the time of day with a GPS clock?

[jholly]

To elaborate for more visually inclined readers:

 

The locus of points shared by two spheres is a circle.

The locus of points shared by three spheres is the intersection of two circles (two points, the circles being created by each sphere intersecting one other).

The locus of points shared by four spheres is one of the two points defined by the two intersecting circles.

So you need 4 satellites at minimum.

...plus an accurate time of day for longitude(just like sailors since the the invention of accurate portable clocks).

 

Nope. That is incorrect. The GPS satellites need accurate time, but the GPS receiver does not need any time information apart from that provided by the satellites.

The geometry gives you a point on a spinning globe. How do you get the longitude if you don't know angle the globe is on its axis(commonly referred to as time of day)? Can you compute the time of day with a GPS clock?

GPS really does not care about time of day. It only cares about how long it took the signal to get from the satellite to the receiver.

[+Walts Hunting]

Picture the satellites as three/four objects of a known location and with a rod of known distance from each to you. They only intersect in one place.

 

Wikipedia and Garmin both have pages that go into detail about how this is done. Time of day is only relevant if measuring the angle of the sun for longititude.

[+odx]

I have worked out

 

Really? Have you? Did you write the Wikipedia page, then?

Edited May 24, 2012 by seangs

[seldom_sn]

They only intersect in one place.

True, but John E. Cache's point is well taken. To get longitude (since the sphere is spinning) the satellites (or the GPSr) have to be able to associate the fixed point created by the 4 satellites with a longitude on the spinning surface. My guess this is by some atomic clock corrected lookup table (offset local time from Zulu), but I welcome correction.

[John E Cache]

True, but John E. Cache's point is well taken. To get longitude (since the sphere is spinning) the satellites (or the GPSr) have to be able to associate the fixed point created by the 4 satellites with a longitude on the spinning surface. My guess this is by some atomic clock corrected lookup table (offset local time from Zulu), but I welcome correction.

It is more a random thought than a point. fuzzylogic says the thought is wrong. :-) Another thought. Both time and longitude are commonly expressed as hexagessimal numbers. I wonder if the ancient sailors knew they were using base sixty math to compute longitude? Number systems seem like a modern computer thing, to me.

[seldom_sn]

It is more a random thought than a point. fuzzylogic says the thought is wrong. :-) Another thought. Both time and longitude are commonly expressed as hexagessimal numbers. I wonder if the ancient sailors knew they were using base sixty math to compute longitude? Number systems seem like a modern computer thing, to me.

Huh? Ancient sailors thought they were sailing on a flat surface and were worried about what happened when they came to the edge. Except for those who noticed that the ships masts appeared before hulls, and they wouldn't have known how to calculate the circumference unless they read Eratosthenes. (Too bad modern Greek politicians didn't learn how to do the math.)

[John E Cache]

It is more a random thought than a point. fuzzylogic says the thought is wrong. :-) Another thought. Both time and longitude are commonly expressed as hexagessimal numbers. I wonder if the ancient sailors knew they were using base sixty math to compute longitude? Number systems seem like a modern computer thing, to me.

Huh? Ancient sailors thought they were sailing on a flat surface and were worried about what happened when they came to the edge. Except for those who noticed that the ships masts appeared before hulls, and they wouldn't have known how to calculate the circumference unless they read Eratosthenes. (Too bad modern Greek politicians didn't learn how to do the math.)

lol My thoughts here come from two places. The ' " key on my scientific calculator that Sharp calls the sexagesimal key and a book I bought off the sale table.

 

http://www.powells.com/biblio/1-9780802715296-8

 

I haven't filled in the blanks, yet, because the book is still on my coffee table unread.

[+phlatlander]

Excellent book by the way!

 

Depends on your definition of "ancient sailors". I would hazard the "ancient navigators" were pretty much aware the world was not flat.

[seldom_sn]

Excellent book by the way!

 

Depends on your definition of "ancient sailors". I would hazard the "ancient navigators" were pretty much aware the world was not flat.

Great Bookstore too. But I'm pretty sure that it depends how old "ancient" is. I don't think Odysseus knew he was navigating on an oblate spheroid.

[John E Cache]

Excellent book by the way!

 

Depends on your definition of "ancient sailors". I would hazard the "ancient navigators" were pretty much aware the world was not flat.

I'm sure kids think I am ancient,. All joking aside, I do really believe the GPS needs the time of day clock for longitude calculation.

 

PS I forgot a third source of info that I found searching for OCR software. fuzzylogic

Edited May 27, 2012 by John E Cache

[+fizzymagic]

All joking aside, I do really believe the GPS needs the time of day clock for longitude calculation.

 

Not in the way you are thinking, it does not. You're just wrong there. The satellites know their positions and the solution does not depend on the time of day. Of course, for the satellites to know their position the ephemeris includes the rotation of the Earth's surface, so since the satellites know their position from the time and their ephemeris, the time is required to get the GPS position.

 

But here is the key point: none of this depends on the GPS unit having an accurate clock!

[+Team Periwinkle]

True, but John E. Cache's point is well taken. To get longitude (since the sphere is spinning) the satellites (or the GPSr) have to be able to associate the fixed point created by the 4 satellites with a longitude on the spinning surface. My guess this is by some atomic clock corrected lookup table (offset local time from Zulu), but I welcome correction.

 

Not really, but kind of. What we're talking about here is the Sagnac effect, which is accounted for in GPS calculations. The Sagnac effect is the difference in time of something traveling on a spinning object depending on the direction it is sent (with rotation or without). For GPS, that Sagnac effect is the angular velocity of the earth - which could be expressed (or understood) as a change in longitude, since longitude is the direction of angular velocity for the earth.

 

Allow me to paraphrase and expand on a really good paragraph from a really good article on Relativity in the GPS:

 

It should be emphasized that the transmitted navigation messages from the GPS satellites provide the user only with a function from which the satellite position can be calculated in the Earth Centered Earth Fixed (NOT ROTATING) reference geoid as a function of the transmission time. Usually, the satellite transmission times are unequal, so the coordinate system in which the satellite positions are specified changes orientation from one measurement to the next. Since the first step of GPS position solving is knowing where the satellites are, the receiver must generally perform a different rotation for each measurement made, into some common inertial frame, so that the position of the satellites can be solved. After solving the propagation delay equations, a final rotation must usually be performed to determine the receiver’s position. This can become exceedingly complicated and confusing. The actual steps that a GPS receiver takes to calculate a position and account for the various issues associated with these and other factors are described in this paper.

_____________

 

The big problem in GPS related to the Sagnac effect is not really the fact that the receiver moved a little bit during the transmission of the signal. The big problem for GPS with the Sagnac effect is keeping the clocks synchronized. Since the atomic clocks on the earth are moving (with the rotation of the earth), their synchronization with each other and with the GPS satellites is affected by the rotation of the earth. But that's a horse of a different color.

Edited May 28, 2012 by Team Periwinkle

[seldom_sn]

Thanks, Team Periwinkle. I can't get your links to work.

[+Glenn]

Thanks, Team Periwinkle. I can't get your links to work.

 

Team Periwinkle's links were a little messed up. I've fixed the mistake and reposted them here.

 

Relativity in the GPS

 

this paper

[+Glenn]

Excellent book by the way!

 

Depends on your definition of "ancient sailors". I would hazard the "ancient navigators" were pretty much aware the world was not flat.

I'm sure kids think I am ancient,. All joking aside, I do really believe the GPS needs the time of day clock for longitude calculation.

 

PS I forgot a third source of info that I found searching for OCR software. fuzzylogic

 

From what that I've read in training guides and documentation about how the GPS system works. I've always read that 4 satellites where needed for a fix. 3 for position and 1 for a time reference.

[+Team Periwinkle]

Thanks, Team Periwinkle. I can't get your links to work.

 

Team Periwinkle's links were a little messed up. I've fixed the mistake and reposted them here.

 

Relativity in the GPS

 

this paper

 

Thanks, I fixed the links in my original post also.

 

Dave