Re: GPS : Basic pseudo-distance computation

From: Iwo Mergler (Iwo_dot_Mergler_at_soton.sc.philips.com)
Date: 03/09/05 

Date: Wed, 09 Mar 2005 10:44:24 +0000

Randolph J. Herber wrote:
>
>
> Your description seems adequate and accurate.
>

Largely correct, but I'll nitpick anyway... :^)

> The navigation problem is determing your location in 4-space (3D+T).
>
> When the receiver knows nothing about its 4-space location, it tries for
> all possible satellites until it avails an almanac from one of them.

The search goes over all satellites (SVs) *and* over all possible
doppler shifts and local oscillator errors. The Almanac is not strictly
required and doesn't help with the initial search unless you know where
you are.

The 'blind' search continues until 4 SVs are found. Some tricks can be
used to shrink the search space and reduce the initially required number
of SVs for an initial inaccurate position.

> Then the receiver knows the correct time within a few seconds and that
> the satellite from which the almanac was availed is above the horizon.

After receiving and demodulating the first SV signal for about 6 seconds,
the GPS time is known within better than 0.07 seconds.

> The satellite almanac is a rough approximation of the satellite
> constellation orbits such that the receiver can spend its efforts on
> satellites which are near the satellite from which the almanac was
> availed as these satellites also are likely to be above the horizon.
> If the receiver has some idea of its location, such as assuming that
> it is near (~1Mm) where it was turned off, then it can the current
> time (such as from an internal quartz clock), assumed location and
> the almanac (whether a saved almanac or one just availed from a
> satellite) to do a more effective search.
>
Yes, these are the most common tricks to speed up the initial search.
It relies on creative definitions of "cold start". :^)

> As each satellite is detected, the receiver starts searching for that
> satellite's ephemeris data which is an extremely accurate description
> of that satellite's orbit. Normally, that accuracy is in millimeters.

Each satellite transmits its own Ephemerides data every ~20seconds and
the almanac of the full constellation every ~12 minutes. Ephemeris is a
very accurate "best fit", but only in the short term (4 hours). Almanac
information is inaccurate, but degrades much more slowly.

> It is kept to that accuracy by ground station monitoring of the
> satellites by radar and lidar from known locations on the ground and
> updating the satellites ephemeris several times a day.

As far as I know, the ground stations use (military) GPS to measure the
orbits. I have a feeling that the required Radar baseline may be larger
than the ground stations. And Lidar is normally used for aerial ground
mapping, I'm not sure it can handle atmospheric scatter at sufficient
accuracy.

Do you have any references for this?

>
> Once four non-coplanar satellites are located, or the equivalent from
> other known data, such as an accurately known time (such as a cesium
> clock in the receiver) or altitude (e.g., known to be a specified
> distance from sea level (consider a GPS receiver on a vessel at sea)),
> it is possible to solve the four, non-linear equations in four unknowns
> to determine the receiver's 4-space location. If more than 4 data
> can be availed, then the various combinations generally different
> solutions. These solutions could combined statistically to produce
> a hopefully more accurate solution. If the receiver stays at a fixed
> location in 3-space, then the time-separated solutions can be combined
> statistically, to produce a hopefully more accurate solution also.
>
The most common practice is to combine all the available measurements
into a so called over determined solution. This gives you a
"best position", usually according to least squares criteria. The
time solution is integral to this, no special treatment needed.

There are more advanced ways of doing this, usually with an enormous
increase in required computing power.

> The solution surfaces for two satellite signals are hyperboloids
> of two sheets in 4-space, for 3 satellites it is a closed curve
> in 4-space and for 4 satellites, a point in 4-space (some equivalent
> of latitude, longitude, altitude and time, all at the equivalent of
> a few nanoseconds accuracy).

Two points actually, but one is outside the satellite orbits.

>
> Finally, the pseudo-range from a satellite is conversion of the
> (possibly, assumed) time difference between the when a satellite
> sent its signal and when the receiver received the signal converted
> to distance.

Thats why it's called 'pseudo' range. it is subjected to the common
mode error of the receiver clock.

> Several corrections are commonly applied: an assumed
> base delay from passing through the atmosphere (particularly, the
> ionosphere) and if a correction signal can be availed, corrections
> for known errors in the satellites' signals. The correction signals
> may come from Differential GPS from either a governmental source
> (the US Coast Guard provides such a service), from a commercial
> service or your own base station at a known location; WAAS, EGNOS,
> LAAS or other equivalent, etc.

The almanac contains global average ionospheric corrections which are
always applied. WAAS/EGNOS DGPS normally corrects more accurately for
a large region (USA/Europe), DGPS beacons normally for an area of a
few 100Km. The smaller the region, the more accurate the correction.

Survey equipment normally consists of a reference receiver at a known
position and a mobile unit. This is a very small scale DGPS system
with added tricks (carrier phase measurement) which can achieve sub
cm accuracies.

Military GPS can measure the atmospheric delay for the exact position,
because the two frequencies have different delays trough the atmosphere.
Future GPS and the upcoming Gallileo system will provide dual-frequency
operation for civilian receivers.

>
> You might find it interesting or useful to study the LORAN system.
> (http://en.wikipedia.org/wiki/LORAN)
>
> Receivers that combine GPS, Galileo, GLONASS, LORAN, INS or a miniature
> cesium clock (http://www.nist.gov/public_affairs/releases/miniclock.htm)
> are reasonable devices to consider building as the navigation technologies
> complement each other. In particular, GPS and Galileo are designed
> so that building a combined receiver would not be much more expensive
> than a receiver for GPS or Galileo alone.

Kind regards,

Iwo

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