Kalman filters
Ejo Schrama
>Hello,
>
>I wonder if someone could please explain to me pretty generally what a Kalman
>filter is for and how it works. A good reference book would also be helpful.
>I appreciate your input.
>
>Reply to crob...@dfd.dlr.de
>
>Craig Roberts
>
You may want to try: Gelb, Applied Optimal Estimation, MIT press, 1974
(reprint 1992)
Craig Roberts
Ejo Schrama
You may want to try: Gelb, Applied Optimal Estimation, MIT press, 1974
(reprint 1992)
Ejo Schrama
l2...@unb.ca
>From: Ejo Schrama <sch...@geo.tudelft.nl>
>Subject: Re: Kalman filters
>Date: 22 Feb 1996 11:53:59 GMT
How about 'Introduction to Random Signals and Applied Kalman Filtering' by RG
Brown and PYC Hwang. This includes a chapter on the application of Kalman
Filtering to GPS.
=================================================================
Peter Stewart
Dept. of Geodesy and Geomatics Engineering
University of New Brunswick
Fredericton
CANADA
===========================L2...@unb.ca===========================
Dr. Lawrence W. James
> Hello,
>
> I wonder if someone could please explain to me pretty generally what a Kalman
> filter is for and how it works.
Kalman filters in modern GPS receivers make them more accurate than they
otherwise would be. Early GPS receivers, such as the Sony IPS-360 Pyxis
simply display the position every 2 seconds from the data they have
received from the satellites in the last 2 seconds. Each reading is
independent of the last, and contains the results from the satellite
signals plus received noise. The noise causes the position to bounce
around somewhat, even without SA. The Kalman filter reduces the effect of
this noise by using information from the previous readings.
To see how this might work, consider this greatly simplified
one-dimensional example:
A GPS receiver is moving on a line. It gets its position from the
pseudo-range of 4 satellites, and its velocity from the Doppler shift. At
time 1 second it measures a position of 370 +/- 50 meters (marks on the
line) and a velocity of 50 +/- 10 meters per second. At time 3 seconds (2
seconds later) it measures a position of 530 +/- 50 meters and a velocity
of 50 +/- 10 meters per second. Thus according to the second position
measurement, the location is between 480 and 580 meters. But, if we use
the first position measurement and the two identical velocity
measurements, the second position is between (370-50+(50-10)*2 = 400
meters) and (370+50+(50+10)*2 = 540 meters). Thus using all the
information available, we can say that the second postion is between 480
and 540 meters, or 510 +/- 30 meters. We have cut our position
uncertainty almost in half by using all of the information available.
A real Kalman filter does the math correctly using normal distribution
functions for the noise statistics, and using either a 7 state (time
offset, x, y and z positions, and x, y and velocities) or a 10 state (the
previous, plus x, y and z accelerations) filter. The result is the same,
each reading after the first is more accurate than would be possible for
that reading alone by not ignoring the previous readings in calculating
the current reading.
A big win for the Kalman filter is in the time difference between the
receiver's internal clock and the clocks on the satellites (one of the 4
parameters solved for in the pseudo-range equations). In 2 seconds, the
receiver's internal crystal-controlled clock cannot drift that much. The
receiver can store internally its drift rate and its drift rate
temperature coefficient. If the receiver has a built-in temperature
sensor, then the short-term error of the corrected internal clock can be
extremely small.
Kalman filter design is about half science, and about half art. The
quality of the filter depends on the skill of the engineer. When choosing
a GPS receiver, consider that the quality of its Kalman filter is likely
to be related to the quality of more easily-observed parts of its internal
software.
What is the practical significance of all this? If you want to mark your
car's position and find it when you get back, leave your GPS receiver on
for a few minutes before marking the position rather than marking
immediately as soon as you get a position. Not only will you have a better
position from having more satellites used in the solution, you will have a
better position because the Kalman filter will have time to work.
Because two identical GPS receivers have at least a different noise
history if not a different position and velocity history, and the current
reading is based in part on earlier readings through the Kalman filter,
one should not expect the two receivers to read the same thing at the same
place, or to be usable as a "poor man's DGPS" even if the same satellites
are used. The Kalman filter means that on the average, the position read
by each receiver is more accurate than it would otherwise be, but the
difference between the 2 receivers is larger than it otherwise would be.
Larry James
James Associates -- your favorite GARMIN dealer
Bob Martin N6MZV
<sch...@geo.tudelft.nl> wrote:
> Craig Roberts <croberts> wrote:
> >Hello,
> >
> >I wonder if someone could please explain to me pretty generally what a Kalman
> >filter is for and how it works. A good reference book would also be helpful.
> >I appreciate your input.
> >
> >Reply to crob...@dfd.dlr.de
> >
> >Craig Roberts
> >
>
> You may want to try: Gelb, Applied Optimal Estimation, MIT press, 1974
> (reprint 1992)
Kalman filters are adaptive, tracking a signal through noise.
Try Kalman Filtering - Theory and Practice, Grewal & Andrews, Prentice Hall
1993 (includes a disk with code on it- makes learning this stuff a lot easier).
cheers-
--
Bob Martin N6MZV * r...@netgate.net
Jack W. Borninski
>Craig Roberts <croberts> wrote:
>>
>>I wonder if someone could please explain to me pretty generally what a Kalman
>>filter is for and how it works. A good reference book would also be helpful.
>You may want to try: Gelb, Applied Optimal Estimation, MIT press, 1974
>(reprint 1992)
KF is a stochastic (random process) estimator algorithm. It estimates the
state of a dynamic system driven by random disturbances, based on the
noise-corrupted measurements of some combination of the system's states.
The Gelb book above is probably as good an intro to KF as you can find.
*********************************************************
Jack W. Borninski, jb...@ti.com, expressing his own views
*********************************************************