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[DPRG] how is GPS covariance estimation done?

Subject: [DPRG] how is GPS covariance estimation done?
From: Chris Jang cjang at ix.netcom.com
Date: Sat May 19 18:15:39 CDT 2007

Hello,

I've been reading about the Kalman filter and wonder where the
covariance matrices come from? The theory is interesting. But
eventually to use it, the filter requires statistics for both
the system process and the measurements. The measurements can
be many things - but for most robot builders, they will be GPS
position readings.

Allow me to provide a concrete example so it makes more sense.

If a robot has odometry, then when it drives around, it has a
good idea where it is. This location belief will have drift in
it. UMBmark is a way of figuring out the systematic and random
errors. Ok, let's make this simpler and assume the robot has
been tuned so there is no systematic error anymore. So it can
drive around in a square (both clockwise and counterclockwise)
and on average comes back right to where it started. If these
final positions are recorded, we can estimate the covariance
experimentally.

This is a lot of work. But now there is a statistical
representation of uncertainty for the "system process" that is
based on odometry readings from the robot's driving. As it
drives around, an ever widening Gaussian distribution blob
represents the robot position. On average, the robot is really
located at the tallest point of the blob.

Now say the robot has a GPS receiver. With some assumptions,
the Kalman filter combines the odometry based belief as to
location with the GPS measurements for an optimal location.
To do this, the statistical covariances of the GPS measurement
is necessary, just as it was for the odometry.

How is this done? I've googled around and notice that as soon
as "GPS" and "covariance" are combined, there are more patents.
It looks like some methods measure jitter in satellite readings
from a fixed position and then try to estimate covariance. But
then we know that terrestial GPS is prone to large systematic
errors (buildings block the signal). Then there is differential
GPS, especially the WAAS corrections common in commercial
products. And to make it more complicated, I know the US armed
forces has a "GPS weather forecast". Depending on conditions,
GPS accuracy can be better or worse. Is this information
uploaded into ordnance like the JDAM (GPS guided bomb) on the
day it is used?

I'm wondering if anyone has experience with real GPS systems to
know how this actually works. I have no experience. I'm only
reading theory. And it may be that covariance estimation is the
really valuable part that no one will reveal. Everyone knows
the same theory. But usually in practice, reality decides how
things must work.

By the way, I'm not using GPS myself. But I do plan on using
Kalman filters for computer vision. I'm trying to get my mind
around how this will work and hope the GPS case provides some
insight.

Chris

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