• navigation math

    From Hul Tytus@21:1/5 to All on Fri Nov 19 20:28:37 2021
    Anyone know the method for calculating a reciever's position from the time difference between three rf pulse transmiters of known positions? This has apparantly been in use since the second world war but a description of the mathematics involved is hiding. Maybe a text on navagation methods?

    Hul

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  • From Dennis@21:1/5 to Hul Tytus on Fri Nov 19 16:12:33 2021
    On 11/19/21 2:28 PM, Hul Tytus wrote:
    Anyone know the method for calculating a reciever's position from the time
    difference between three rf pulse transmiters of known positions? This has apparantly been in use since the second world war but a description of the mathematics involved is hiding. Maybe a text on navagation methods?

    Hul

    The search term is LORAN. I think the last ones were decommissioned
    years ago - replaced by GPS.

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  • From Grant Edwards@21:1/5 to Grant Edwards on Fri Nov 19 23:03:21 2021
    On 2021-11-19, Grant Edwards <invalid@invalid.invalid> wrote:
    On 2021-11-19, Hul Tytus <ht@panix.com> wrote:

    Anyone know the method for calculating a reciever's position from the time >> difference between three rf pulse transmiters of known positions?

    Yes. If you know the delta between the distances to two known
    locations, that places you on a hyperbola whose focii are those two
    known points. Plot the hyperbola on your map.

    Actually coastal navigational maps had/have those LORAN hyperbolas
    printed on them. So all you really had to do was read the delta value
    off the receiver, and then interpolate between two lines with
    dividers... rinse, repeat.

    https://www.penobscotmarinemuseum.org/pbho-1/collection/loran-lines-penobscot-bay-chart

    Modern LORAN receivers know where the transmitters are, do all the
    math internally, and just show you longitude/lattitude or display a
    map just like a GPS receiver.

    --
    Grant

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  • From Grant Edwards@21:1/5 to Hul Tytus on Fri Nov 19 22:54:25 2021
    On 2021-11-19, Hul Tytus <ht@panix.com> wrote:

    Anyone know the method for calculating a reciever's position from the time difference between three rf pulse transmiters of known positions?

    Yes. If you know the delta between the distances to two known
    locations, that places you on a hyperbola whose focii are those two
    known points. Plot the hyperbola on your map.

    Repat for the other two pairs of points. Hopefully there's one point where all three
    intersect.


    This has apparantly been in use since the second world war but a
    description of the mathematics involved is hiding. Maybe a text on
    navagation methods?

    https://www.sparknotes.com/math/precalc/conicsections/section4/

    You can do it analytically instead of graphically if you want:

    https://www.analyzemath.com/HyperbolaProblems/hyperbola_intersection.html https://math.stackexchange.com/questions/1920147/intersection-of-two-hyperbolas

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  • From Don Y@21:1/5 to Grant Edwards on Fri Nov 19 16:28:48 2021
    On 11/19/2021 3:54 PM, Grant Edwards wrote:
    On 2021-11-19, Hul Tytus <ht@panix.com> wrote:

    Anyone know the method for calculating a reciever's position from the time >> difference between three rf pulse transmiters of known positions?

    Yes. If you know the delta between the distances to two known
    locations, that places you on a hyperbola whose focii are those two
    known points. Plot the hyperbola on your map.

    Repat for the other two pairs of points. Hopefully there's one point where all three
    intersect.

    There may be *two* places where the hyperbolae intersect.
    If you understand the geometry of the pairs of foci, you
    may be able to rule out one case as "not possible" (in
    your application).



    This has apparantly been in use since the second world war but a
    description of the mathematics involved is hiding. Maybe a text on
    navagation methods?

    https://www.sparknotes.com/math/precalc/conicsections/section4/

    You can do it analytically instead of graphically if you want:

    https://www.analyzemath.com/HyperbolaProblems/hyperbola_intersection.html https://math.stackexchange.com/questions/1920147/intersection-of-two-hyperbolas




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  • From Don Y@21:1/5 to Hul Tytus on Fri Nov 19 16:19:56 2021
    On 11/19/2021 1:28 PM, Hul Tytus wrote:
    Anyone know the method for calculating a reciever's position from the time
    difference between three rf pulse transmiters of known positions? This has apparantly been in use since the second world war but a description of the mathematics involved is hiding. Maybe a text on navagation methods?

    You will be dealing with families of "concentric" hyperbolae
    (as the equation for a hyperbola involves maintaining a constant
    difference between lengths of vectors to foci).

    LORAN was renowned for using this -- on a global scale. It has
    since been decommissioned in the wild but there's an abundance
    of information regarding its use and deployment.

    Note, however, that there are many subtleties buried in the
    LORAN implementation that make it differ from a theoretical
    approach. E.g., there are intentional delays introduced
    to make the numbers cleaner.

    If you are truly looking to navigate on a *large* scale
    (hundreds of miles), then you will have to consider things
    like changes in propagation delays over different types
    of terrain and the "shape" of that terrain (e.g., the Earth
    is an oblate sphere). Again, LORAN has these covered but
    you'll have to dig for details.

    Similar problems exist "in the small" for position
    resolution within a structure! (I use similar technology
    to determine where, in an "arena" -- home or office, in
    my case -- the user is sited)

    [If you look at a preprinted maritime map augmented with
    LORAN "lines of constant time difference", you'd see
    that they differ from what you would otherwise expect
    from a more naive mathematical/geometric treatment]

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  • From Paul Rubin@21:1/5 to Hul Tytus on Fri Nov 19 18:17:32 2021
    Hul Tytus <ht@panix.com> writes:
    Anyone know the method for calculating a reciever's position from
    the time difference between three rf pulse transmiters of known
    positions?

    See if this helps:

    https://en.wikipedia.org/wiki/True-range_multilateration

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  • From Robert Roland@21:1/5 to All on Sat Nov 20 14:27:42 2021
    On Fri, 19 Nov 2021 20:28:37 -0000 (UTC), Hul Tytus <ht@panix.com>
    wrote:

    Anyone know the method for calculating a reciever's position from the time >difference between three rf pulse transmiters of known positions?

    The word to look for is multilateration. Wikipedia has an article on
    it:

    https://en.wikipedia.org/wiki/Multilateration

    Calculating the time differences based on a known position is
    relatively simple. Once you try to solve the equations to go the other
    way, the math gets ugly.

    Years ago, when I was playing with this, I resorted to a numerical
    solution.
    --
    RoRo

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  • From Hul Tytus@21:1/5 to Grant Edwards on Sat Nov 20 20:50:57 2021
    Thanks Grant - that's whats needed.

    Hul

    Grant Edwards <invalid@invalid.invalid> wrote:
    On 2021-11-19, Hul Tytus <ht@panix.com> wrote:

    Anyone know the method for calculating a reciever's position from the time difference between three rf pulse transmiters of known positions?

    Yes. If you know the delta between the distances to two known
    locations, that places you on a hyperbola whose focii are those two
    known points. Plot the hyperbola on your map.

    Repat for the other two pairs of points. Hopefully there's one point where all three
    intersect.


    This has apparantly been in use since the second world war but a description of the mathematics involved is hiding. Maybe a text on navagation methods?

    https://www.sparknotes.com/math/precalc/conicsections/section4/

    You can do it analytically instead of graphically if you want:

    https://www.analyzemath.com/HyperbolaProblems/hyperbola_intersection.html https://math.stackexchange.com/questions/1920147/intersection-of-two-hyperbolas

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  • From Hul Tytus@21:1/5 to Don Y on Sat Nov 20 20:55:49 2021
    Don - At this point just the basics are of need, but the various subleties
    are worth looking at. Thanks.

    Hul

    Don Y <blockedofcourse@foo.invalid> wrote:
    On 11/19/2021 1:28 PM, Hul Tytus wrote:
    Anyone know the method for calculating a reciever's position from the time
    difference between three rf pulse transmiters of known positions? This has apparantly been in use since the second world war but a description of the mathematics involved is hiding. Maybe a text on navagation methods?

    You will be dealing with families of "concentric" hyperbolae
    (as the equation for a hyperbola involves maintaining a constant
    difference between lengths of vectors to foci).

    LORAN was renowned for using this -- on a global scale. It has
    since been decommissioned in the wild but there's an abundance
    of information regarding its use and deployment.

    Note, however, that there are many subtleties buried in the
    LORAN implementation that make it differ from a theoretical
    approach. E.g., there are intentional delays introduced
    to make the numbers cleaner.

    If you are truly looking to navigate on a *large* scale
    (hundreds of miles), then you will have to consider things
    like changes in propagation delays over different types
    of terrain and the "shape" of that terrain (e.g., the Earth
    is an oblate sphere). Again, LORAN has these covered but
    you'll have to dig for details.

    Similar problems exist "in the small" for position
    resolution within a structure! (I use similar technology
    to determine where, in an "arena" -- home or office, in
    my case -- the user is sited)

    [If you look at a preprinted maritime map augmented with
    LORAN "lines of constant time difference", you'd see
    that they differ from what you would otherwise expect
    from a more naive mathematical/geometric treatment]

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  • From Hul Tytus@21:1/5 to Dennis on Sat Nov 20 20:45:05 2021
    Thanks Dennis. That points me in the right direction.

    Hul

    Dennis <dennis@none.none> wrote:
    On 11/19/21 2:28 PM, Hul Tytus wrote:
    Anyone know the method for calculating a reciever's position from the time
    difference between three rf pulse transmiters of known positions? This has apparantly been in use since the second world war but a description of the mathematics involved is hiding. Maybe a text on navagation methods?

    Hul

    The search term is LORAN. I think the last ones were decommissioned
    years ago - replaced by GPS.

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  • From Hul Tytus@21:1/5 to Paul Rubin on Sat Nov 20 20:59:31 2021
    Thanks Paul, I'll take a look.

    Hul

    Paul Rubin <no.email@nospam.invalid> wrote:
    Hul Tytus <ht@panix.com> writes:
    Anyone know the method for calculating a reciever's position from
    the time difference between three rf pulse transmiters of known
    positions?

    See if this helps:

    https://en.wikipedia.org/wiki/True-range_multilateration

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  • From Don Y@21:1/5 to Hul Tytus on Sat Nov 20 15:38:11 2021
    On 11/20/2021 1:55 PM, Hul Tytus wrote:
    Don - At this point just the basics are of need, but the various subleties are worth looking at. Thanks.

    As folks depended on LORAN for their livelihood (no way to easily
    and reliably locate a particular spot in the middle of the ocean),
    a lot of effort was expended to make it as useful and useable as
    possible -- for "average joes".

    And, the technology required in the receiver was very low (think
    1970's and earlier). We didn't start putting MPUs into receivers
    until the late 70's. And, converting between TD & lat-lon wasn't
    practical -- in real time -- until the same time frame! Folks
    would talk in terms of TDs, not lat-lons!

    A few important implementation issues to consider. Because LORAN
    was designed so the Master drove the timing of its chain, there was
    less need for all stations to share a common sense of time. When
    each Slave received the Master's transmission (which would occur at
    different ABSOLUTE times because the propagation delays from Master
    to each Slave would differ based on geographical distance, etc.),
    it would initiate its own "local" transmission timing sequence
    (Slaves didn't immediately emit their beacons but waited for a
    specific coding delay). So, the RATE of time progressing was
    important to each Slave -- but not the *actual* time (of day).

    In addition to having the Slaves' transmissions sync'd to the
    arrival of the Master's beacon, the time between Master
    transmissions was fixed -- Group Repetition Interval (GRI).
    So, a receiver could find a particular chain's transmissions
    by looking for this GRI (in the time domain).

    Also, a station could act as Master for one chain -- and (a) Slave
    in another. Running the chains at different GRIs allowed their
    transmissions (and time differences) to be sorted out remotely.

    For example, the 9960 (99600 microseconds between Master transmissions)
    chain had a station on Nantucket Island (SSE of Massachusetts). As
    this is a prime area for maritime traffic (servicing NYC, Boston,
    Maine, etc.), it was heavily used.

    Note the geometry of the "lines of constant time-difference"
    ("grid lines") near Nantucket (for that LORAN chain):
    <http://afterthemap.info/images/5-14.jpg>
    Notice how common it is to have *two* locations which resolve to
    the same pair of TDs? For example, the brown "80" (13880 microseconds)
    crosses the green 6060 (6060 microseconds time difference) almost within *sight* of each other (an exaggeration as horizon is about 4 miles).

    Note, also, how the spacing between grid lines varies? E.g., along the baseline (the dashed line connecting slave to master) the distance between lines is at its minimum.

    So, a given change in time difference (delta-TD) correlates to the smallest physical distance (greatest positional resolution). As one moves off of
    this baseline, the distance between grid lines increases (lower positional resolution). Remember, you're *measuring* time-differences so you want a unit of TD to represent the smallest physical distance.

    Likewise, note how the angle between grid lines from different secondaries (slaves) varies. In some cases, they are close to "normal"; in others, they cross at very shallow angles. (Geometric Dilution of Precision -- GDoP).

    (Amusingly, you can also see how the LORAN overlay isn't perfectly
    aligned with the mercator projection overlayed!)

    In addition to the "basics", there are lots of subtle details in LORAN
    that made it usable even in really poor conditions. E.g., the "pulses"
    sent by the Master and Slaves were actually pulse *trains* and could
    encode information. Additionally, as each individual pulse was a
    burst of carrier in a tightly controlled envelope, the receiver could
    easily identify and track a specific portion of that burst (IIRC, third positive zero-crossing). And, could be told to track a different
    portion -- with a corresponding fixed temporal offset in the TD displayed
    for that Secondary.

    <shrug> *Lots* of design detail that was impressive for its time!
    Cherry-pick the aspects of the design that are most appropriate
    for your application.

    The US Coast Guard published a great reference on LORAN (decades ago).
    I'm too lazy to hunt for it in my collection... :<

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