GPS Always Overestimates Distances (i-programmer.info)

mikejuk writes: Have you had a suspicion that your GPS app is overestimating the distance traveled? It is something that runners and walkers complain about a lot. If so, you are probably correct -- but the reason isn't an algorithmic glitch. The answer lies in the statistics, and it is a strange story. If you make a measurement and it is subject to a random unbiased error, then you generally are safe in assuming that the random component will make the quantity larger as often as it makes it smaller. Researchers at the University of Salzburg (UoS), Salzburg Forschungsgesellchaft (SFG), and the Delft University of Technology have done some fairly simple calculations that prove that this is not the case for GPS distance measurement. Consider the distance between two points — this is along a straight line, and hence it is the shortest distance. Now add some unbiased random noise, and guess what? This tends to increase the distance. So unbiased errors in position give rise to a biased overestimate of the distance. There is an exact formula for the bias and in some cases it can be more than 20%. Is there a solution? Perhaps using velocity measurements and time to work out distance is better — it isn't biased in the same way, but how accurate it could be remains to be seen. So when your fitness band tells you you have run a 4-minute mile — don't believe it.

Comments

Simple problem with a simple solution

[carlhaagen]

As a sailor who use GPS alongside traditional navigation: stop using devices that poll and insert a new leg on your route every 5 seconds. If you lower the polling frequency to f.e. every 1 or 2 minutes this problem goes away. I realize that this doesn't work for people who insist on taking their morning run zig-zagging through city blocks.

Re:Simple problem with a simple solution

[Solandri]

That doesn't completely fix it. The problem is related to both polling frequency and positional error.

The reason is pretty obvious if your GPS device has ever been slow getting a fix on multiple satellites and has a large error circle. One moment it says you're at position x, 5 seconds later it says you've warped half a mile away. The larger the positional error, the greater the distance error. Assuming you're traveling a straight line path, the positional errors average out to zero only when the errors fall along the line you're traveling. Any positional error perpendicular to your direction of motion results in an overestimate of distance traveled by tracing your route as a zig-zag instead of straight.

That would suggest a smoothing algorithm could mostly fix this. Basically draw the simplest curve you can that goes through all the error circles around the breadcrumbs the GPS has recorded.

Re:Simple problem with a simple solution

[Aighearach]

http://www.adafruit.com/datash...

That's the cheapo GPS chip that makers are using. ~$40 on a breadboad module, but more like $5 as an OEM part. Calculates and stores "up to 3 days" of satellite position, then uses a lookup table. That is part of why new units still take a long time to get a cold fix; they're calculating satellite positions.

This has been a standard feature since around 10-15 years ago when they started advertising improved urban accuracy. If you're not predicting locations, you'd have to do a cold fix every time you walked past a building! It would be basically unusable downtown. Like in the old days! Back then it was also largely unusable in the forest; you'd have to find a clearing, or else remain stationary for a couple minutes, to get a fix on a cheap unit. That is why back then for serious outdoor use people were dropping $500 on dedicated GPS units that had the predictive routines, and regular GPS was intended mostly for highway auto use where the sky is usually visible.

Almost everything predictive uses Kalman filters. That is not a sign of low quality. It is a system of using predictions and expected error rates to overcome noise. If the result quality is low, it is not because of using a Kalman filter it is because of having a low quality prediction routine, or an inaccurate expected noise level. There is nothing about orbital mechanics or radio propagation that makes Kalman filters a poor fit; actually, it is nearly the ideal case for them; the noise will have a Gaussian distribution.

Understatement of the day

[Sqr(twg)]

"it can be more than 20%" -Yeah, it can be an infinite number of %, if the actual distance travelled is zero, and the random error is not.

Fantastic math there, guys

[macklin01]

This gem caught my eye:

To test the theory some experiments were performed. A consumer quality GPS was walked around a 10m square with segment lengths of 1m and 5m. The average measured segments were 1.2m and 5.6m. That is, an overestimate of between 20% and 60%. Clearly a smaller segment length is a good idea.

That's some amazing math there. (1.2 m - 1.0 m) / 1 m = 0.2, indeed a relative error of 20%. But (5.6 m - 5.0 m) / 5 m = 0.12, a relative error of 12%, not 60%! So, let's fix this thing, with of course the opposite conclusion:

To test the theory some experiments were performed. A consumer quality GPS was walked around a 10m square with segment lengths of 1m and 5m. The average measured segments were 1.2m and 5.6m. That is, an overestimate of between 20% and 12%. Clearly a longer segment length is a good idea.

Re:Fantastic math there, guys

[_Shorty-dammit]

It's not just that they can't even determine that it was 12% in that case. They're simply doing it wrong from the beginning. "Let's take a technology that gives us a measurement that contains a possible error of more than the length we are trying to measure, and then complain about how much of an error we got in that length!" The best accuracy I've ever seen reported on my GPS is 2 m. Why would I try to measure 1 m with that?! Go do the experiment with a 10 km square and see what your error is then. *sigh*

Consider a Fractal Cow

[goombah99]

Imagine for a moment that instead of going from point A to point B, you instead Zigged at 45 degrees then half way you Zagged back on the slant path to point B. Clearly then your total trip is sqrt(2) times the straigtline distance. But you went needlessly off the path. Instead we shall imagine that we can detect when we have deviated by 1 foot from the straight path. So we now execute a zig zag (45 degree) path turning each time we deviate by 1 foot from the true path. How far did we go? Oh my, it's still sqrt(2) times the straight line. We work hard and improve the duidance system so we now can detect when we are 1 micron from the straightline path. How far? still sqrt(2) times the straight line.

Indeed no matter how accurate we can detect the deviation so that the true path is beyond the capacity of any instrument to measure the deviation, say 1/100th of angstrom the same logic holds.

ergo all paths between A and B including the straight path are 1.41 times the straighpath distance.

Fractal hamburger.