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Genghis the Engineer
16th Jun 2006, 16:51
I've been doing a bit of work on the use of GPS for calibrating airspeed measurement systems (yes, I know the theory and the maths !).

The one thing that I've been struggling with is finding any rigorous estimate of the accuracy of velocity measurement using GPS - all the available references spend a lot of time considering 2D positional accuracy, a bit less worrying about geopotential altitude accuracy, but then pretty much ignore velocity. I've come across the catalogue for a company who sell GPS based units for measuring race-car performance who quote 0.1kph, but I suspect that this is most likely a bit of inspired guesswork.

So, can anybody point me at a reference document or paper anywhere that does a rigorous job of estimating the errors in velocity (groundspeed) measurement with GPS?

G

Eric T Cartman
16th Jun 2006, 17:20
I can't help directly but if you don't already know about it, there is an excellent forum for advanced users available at
http://www.pocketgpsworld.com/
where you might wish to post your query.
Rgds ETC

GOLF_BRAVO_ZULU
16th Jun 2006, 17:35
I spoke to a man at Garmin, many years ago when Selective Availability was permanently switched on. He said that they use a 1 per second sample rate and cumulative averaging. For Velocity this applied to the track and rate of displacement. The example he gave was that if, in a boat, you tracked to a buoy at a particular position for several minutes you would hit it, even though the indicated error radius was 75 feet. The limiting factor was that the track had to be constant and, if the ETA was to be accurate, so did the rate of displacement. This, of course, hasn't helped you much with the maths but perhaps a call to Garmin UK:
(The Quadrangle
Abbey Park
Romsey
Hampshire
SO51 9AQ
United Kingdom

01794 519944)

might still get a knowledgeable and friendly response.

GBZ

LH2
16th Jun 2006, 18:10
Genghis,

velocity is not an observable in a GPS (neither are coordinates, for that matter, but let's not get picky), it is simply derived as the difference between two positions divided by time. From there, and having a precision figure for your time and position "measurements" (again, let's consider the position as a direct measurement) it's trivial to assert the precision of a speed readout by a simple error propagation computation.

Let A1 be your GPS position at time T1
Let A2 be your GPS position at time T2

Your velocity is V = (A2-A1) / (T2-T1)

Now:

Let EP1 be a measure of the uncertainty of A1
Let EP2 be a measure of the uncertainty of A2
Let ET1 be a measure of the uncertainty of T1
Let ET2 be a measure of the uncertainty of T2

An uncertainty value "EV" for V can be calculated as

EP = SQRT(EP1^2 + EP2^2)
ET = SQRT(ET1^2 + ET2^2)

EV = SQRT((EP/(A2-A1))^2 + (ET/(T2-T1))^2)

Notes:

* EP[12] is a suitable measure of positional uncertainty, such as SEP (Spherical Error Probable) or CEP (Circular Error Probable)

* ET[12] can probably be considered negligible, in which case the formula above is reduced to EV = SQRT(EP1^2 + EP2^2)/(A2-A1). Note that this is a relative error. Use SQRT(EP1^2 + EP2^2) for absolute error (assuming ET[12] ~= 0)

* I've done the error propagation above from memory. Needless to say, check the formulas! (is "formulae" too pedantic, btw?)

Hope this helped and got nothing too wrong :\

archae86
16th Jun 2006, 18:21
Here is a link to the pdf for a formal paper. The application was (literally) agricultural. I can't vouch for the quality of the paper, but it is reasonably recent. At least it will get you past the serious error of ignoring the fact that commercially available GPS receivers (down to inexpensive handhelds) in fact use Doppler as a component of their velocity estimation, not just position differencing.
GPS velocity error paper (http://gauss.gge.unb.ca/papers.pdf/ionntm2004.serrano.pdf)

Genghis the Engineer
16th Jun 2006, 18:23
LH2

Two obvious issues there:

(1) Are we certain that's how GPS determines velocity, and it's not using some quasi-doppler function? (Archae86 certainly believes so, and that was my general belief also).

(2) This assumes the same random variation in errors at each position report, and that the reported position at T2 isn't some function of the reported position at T1, as well as of the actual position?

- I do agree with the error propogation method however, at-least assuming that the total errors are normally distributed, which seems a fair assumption.

G

Genghis the Engineer
16th Jun 2006, 18:24
Here is a link to the pdf for a formal paper. The application was (literally) agricultural. I can't vouch for the quality of the paper, but it is reasonably recent. At least it will get you past the serious error of ignoring the fact that commercially available GPS receivers (down to inexpensive handhelds) in fact use Doppler as a component of their velocity estimation, not just position differencing.
GPS velocity error paper (http://gauss.gge.unb.ca/papers.pdf/ionntm2004.serrano.pdf)

Thank you, that might just be what I was after - but I think that I need the weekend to go and think about it!

G

Sirius Flying
16th Jun 2006, 18:48
Did a little work on the subject.
How the velocity is determined depends on the receiver model. On high-grade models you may find a velocity determination through doppler-effect, on the low-grade ones, frankly, I don't know, but probably not.
LH2's approach will not get you far. GPS can be horribly complicated, and most GPS marketing guys have no idea (sorry).
Manufacturer data MAY be correct, but as usual, check for yourself before planning to do anything important with the data.
On a good GPS model, accuracy of velocity will actually be a lot higher than position accuracy, but that's comparing apples and oranges...
If you have a GPS capable of relative positioning, you probably will get very good accuracy (more than enough for most aviation-type applications)
Precise numbers will actually depend on time of day etc... (satellite positions...)

LH2
16th Jun 2006, 19:13
Genghis,


(1) Are we certain that's how GPS determines velocity, and it's not using some quasi-doppler function? (Archae86 certainly believes so, and that was my general belief also).


That's how survey grade receivers do it. OTOH, those are not specifically designed for navigation. Now that you mention, it is possible that aviation Garmins & similar do use carrier-phase Doppler measurements. Your best bet is to ring the manufacturer of your instrument to find out (in fact, I will do so, out of curiousity)


(2) This assumes the same random variation in errors at each position report, and that the reported position at T2 isn't some function of the reported position at T1, as well as of the actual position?


Position errors can be assumed uncorrelated for this application, even though in fact they aren't. However, that could still be taken care of by using the formula for correlated error propagation if you feel so inclined (that will make your afternoon miserable though). For ET[12], as long as the elapsed period is not too big (i.e. long term clock drift not applicable), those are uncorrelated.


- I do agree with the error propogation method however, at-least assuming that the total errors are normally distributed, which seems a fair assumption.


As you say, it's a "fair" assumption, i.e., close enough to the truth for most practical purposes. In reality, the distribution of 3D autonomous GPS fixes looks a bit like a slightly flattened rugby ball.

Note that the PDF above deals with a different issue, as far as I can tell from the abstract. What they are doing is using raw GPS data (carrier wave measurements) for calculating speed. Whether your receiver works in a similar way or not will determine how applicable their research is to your question. Looks interest nonetheless, so that's my bedtime reading for tonight :E

Let me know how you get on. I needed to look into this a few years ago for a similar application, but unfortunately I don't think I keep a copy of my notes anymore.

ICT_SLB
17th Jun 2006, 03:57
Genghis,
Won't make your problem any easier but Rockwell-Collins GPS-4000A units (the common Transport unit) have a current problem that they give random Ground Speeds at rest of up to 14kt - the newer IRUs with automatic initialisation had to be be modified to cope. Apparently will not be fixed until after the WAAS-capable unit comes out in 2007. Wonder if this isn't deliberate wrt to non-navigational uses.