Re: GPS : Basic pseudo-distance computation

From: Randolph J. Herber (herber_at_ncdf107.fnal.gov)
Date: 03/09/05 

Date: 9 Mar 2005 16:10:23 GMT

In article <dsAXd.126$826.8619@ns2.gip.net>,
Iwo Mergler <Iwo_dot_Mergler@soton.sc.philips.com> wrote:
>Randolph J. Herber wrote:

>> Your description seems adequate and accurate.

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

        Now, my turn.

>> 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.

        1) I had simplified the description of the search by not
           describing all the issues of the methodology of the search.
        2) ``until it avails an almanac'' At that stage the receiver
           does not have an almanac, therefore the almanac can not
           be used. I was describing a ``search the sky'' cold start
           with a unset internal clock.

>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.

        Only if one assumes that one is on or near the surface of
        earth, otherwise, one must only assume some location within
        the GPS service volume (roughly within the Lunar orbit).
        That gives the several seconds accuracy --- which is
        sufficient to make effective use of the almanac.

>> 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". :^)

        ``assuming ....''

>> 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.

        Exactly, what I meant and exactly what I thought I had said!

>> 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?

        I would have to look them up again. I found them the first
        time at the USAF web site that is publically accessable and
        is managed by the same group that operate the GPS ground
        stations. That site mentions that military GPS receivers
        are used to measure the signal errors and radar is used to
        cross-check the distances to the satellites. I had seen a
        mention that lidar has been considered and tested for this
        purpose as well. I do not remember a mention that lidar is
        in production use.

>> 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.

        ``least squares'' is a statistical method for combining
        overdetermined data. I agree that ``least squares'' is
        probably the most common method. Again, I was simplifying
        the description for the intended benefit of the original
        questioner.

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

        Not necessarily. And, DSPs can supply enormous computing power
        and many GPS receivers have (d)igital (s)ignal (p)rocessors.

>> 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.

        Touche (I considered my LORAN example ... Your point
        is at least possible if not probable.)

>> 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.

        Correct. I just did not use the language you used for
        the delay value included in the GPS signal itself (I
        said assumed base delay). You added more detailed
        descriptions of the corrections.

>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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