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

[Glenn Thorpe]

Hello
I am Glenn, in Australia.  I have just joined the list.
Sorry if I annoy you but I am seeking your advice.
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.
Thank you for your time
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

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.osgeo.org/pipermail/foss-gps/attachments/20130927/caf23366/attachment-0001.html>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: hyperbolae.xlsx
Type: application/vnd.openxmlformats-officedocument.spreadsheetml.sheet
Size: 22791 bytes
Desc: not available
URL: <http://lists.osgeo.org/pipermail/foss-gps/attachments/20130927/caf23366/attachment-0001.xlsx>