navigation math

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]

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

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...  :<