[FOSS-GPS] Hello, i am new to the list

Glenn Thorpe glenn at thorpies.com
Fri Sep 27 05:28:11 PDT 2013

Hello
I am Glenn, in Australia.  I have just joined the list.
I have been looking at hyperbolic intersections and it seems to me that
in there ought to be sufficient information contained in the fifth
satellite to fix errors better than they are fixed.
Using 2D diagrams with four satellites my reasoning is as follows:
1 -- from four satellites there are six hyperbolae (detailed explanation
excluded).  If distance information is perfect the six hyperbolae will
intersect at one point.
2 - if distance information is imperfect there are four points where
three of the hyperbolae intersect.  Each of these points is the fix
point from a triplet of satellites.
3 -- adjusting the distance from one satellite to the receiver results
in:  a) - the distance difference between the adjusted satellite and the
three other satellites varying, which b) - results in three of the
hyperbolae moving which c) - results in the movement of three of the
four triplet fix points.  The three hyperbolae that are NOT calculated
from the distance to the moved satellite remain static, as does the fix
point that is the intersection of these three hyperbolae.
4 -- the three fix points that move converge on the fix point that does
not move.  The change in distance to the one satellite that is being
varied can be set so that all four fix points become co-pointal.
5 -- refer to the statement in point 1 -- if the distance information is
perfect the six hyperbolae will intersect at one point.  With the
variance in the distance of one satellite the fix points can become
co-pointal and the six hyperbolae intersect at one point.  So the
distance information is therefore perfect at this point.  Isn't it?
6 -- steps 3 -- 5 can be repeated for the other three triplet fix
points.  The outcome is four distance variances (one for each
satellite), each of which results in a point location that has perfect
measurements.  The initial errors in the satellite distances are still
unknown, however how much this error has to be adjusted to give a
perfect location IS known, as are the co-ordinates of this perfect location!
7 -- four equations can be expressed, each of which has one of the known
distance variances and its related perfect co-pointal co-ordinates, as
well as the four unknown distance errors.  All other inputs are known
values.  These four equations can be simultaneously solved to provide
the four error values.
This method seems substantially different from the least squares
approach, which I have trouble comprehending with references to CORDIC
angles, QR, iterative QRD & the like.  It is my understanding, and
please correct me if I am wrong, that least squares produces a position
in the middle of all the triplet fix locations.
With the approach I have outlined the errors are being used to provide
the solution.  The solution provided will be accurate.
Would anybody know if this raw approach of using hyperbolae directly has
been examined recently?  Also, any comments on this approach, whether my
logic is faulty and whether you consider it worthwhile pursuing would be
appreciated.
Glenn
PS A spreadsheet & chart that demonstrates the above method is
attached.  The chart displays the hyperbolae and triplet fix points.
Inputs of satellite angles and range errors are reflected in the chart.
The chart is also useful in demonstrating and explaining why relatively
small range errors can produce large errors in fixes.

--
Glenn Thorpe
8 Dixon Drive, Holder, ACT, 2611, Australia
ph +61 2 6288 5748
glenn at thorpies.com

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