August
1999, Issue 109
Where
in the World (Part 1):
GPS Introduction
ERROR
AND PRECISION
Although
GPS is extremely precise when it comes to navigation
equipment, it has some limitations. But first, let me
explain the quality measurements that describe the accuracy
of location fixes statistically.
The
first term, the circle error of probability (CEP), is
defined as the size of the circle that encompasses 50%
of all the location fixes. So, if you collect 100 position
fixes and select the 50 closest ones, the circle that
includes these is how certain you can be of your location.
A CEP of 100 m means that 50% of all position fixes
are within 100 m.
Similar
to the CEP, the distance root mean square (DRMS) describes
63% of all the fixes, 2 DRMS describes 97% of the fixes,
and 3 DRMS describes 100% of the fixes. Figure 1 illustrates
this pattern.
For
military use, the figures in Table 1 are much better.
For security reasons, the military has turned on selective
availability (SA) to purposely introduce uncertainty
into the position signal. There’s much discussion
in
GPS
mailing lists and usenet newsgroups about whether the
SA error gets better or worse during times of conflict,
but all we need to know is that SA exists and has some
peculiarities to be aware of.
In
the RMC message I showed you, you saw that there was
a speed and heading in the stationary fix. This happens
because of SA. The SA dither changes over time, so the
position fix seems like it’s slowly drifting. The
speed that the GPS receiver measures is actually the
rate at which the time is changing.
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Figure 1—Here’s
a graphical representation of CEP and the
various degrees of RMS distances. The CEP
(50%) has half of the fixes, DRMS (63%), 2
DRMS (90%), and 3 DRMS (100%).
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If
you average a stationary position over a long time (two
or more days), you can achieve high accuracy because
the random-number generator used to dither the signal
for SA has a zero mean over several days. Figure 2 shows
a plot of SA for 1 h.
Besides
the long-term mean of the SA dither, there’s another
effect we can use. The SA dither is the same for locations
within a locality (i.e., the error of the position is
the same for positions that are apart). If you use a
base station with a known accurate location, you can
measure the current error of another measurement of
which you do not know the location.
Figure 2—As
this plot shows, SA causes the fixes on a
stationary GPS receiver to wander over time.
This track was taken for ~1 h. Each longitude
tick mark of (0.02¢) is about 28.8 m, while
each latitude tick mark (0.01¢) is 18.5 m.
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For example, if you
know the current location (A[lat,lon]), and the
location that a GPS receiver reports to you (B[lat,lon]),
you can compute the current error with:
err(lat,lon) = B(lat,lon)
– A(lat,lon)
If you take a GPS reading
somewhere in the field at C(lat,lon), you can
compute an accurate position, by adding in the offset:
real(lat,lon) = C(lat,lon)
– err(lat,lon)
Systems use this technique
and broadcast the current error information via radio
signals. The US Coast Guard uses such a system near
coastal waterways via longwave radio stations, and their
system is free to use. In many cities, differential
GPS may be available via FM broadcast station subcarrier.
These services usually require a subscription and are
not standardized.
In future articles, I’ll show you
how to use a stationary GPS receiver and a roving GPS
receiver and do differential GPS using postprocessing.
Basically, I’ll show you how to map out donut shops
with fairly high accuracy, in case you ever need to
put some LEDs on a map.
Ingo Cyliax has written
for Circuit Cellar on topics such as embedded systems,
FPGA design, and robotics. He is a research engineer
at Derivation Systems Inc., a San Diego–based formal
synthesis company, where he works on formal-method design
tools for high-assurance systems and develops embedded-system
products. You may reach him at cyliax@derivation.com.
