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Hello<br>
I am Glenn, in Australia. I have just joined the list.<br>
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Sorry if I annoy you but I am seeking your advice.<br>
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. <br>
Using 2D diagrams with four satellites my reasoning is as follows:<br>
1 – from four satellites there are six hyperbolae (detailed
explanation excluded). If distance information is perfect the six
hyperbolae will intersect at one point. <br>
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.<br>
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. <br>
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.<br>
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?<br>
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!<br>
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. <br>
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. <br>
With the approach I have outlined the errors are being used to
provide the solution. The solution provided will be accurate. <br>
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.<br>
Thank you for your time<br>
Glenn<br>
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. <br>
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<pre class="moz-signature" cols="72">--
Glenn Thorpe
8 Dixon Drive, Holder, ACT, 2611, Australia
ph +61 2 6288 5748
<a class="moz-txt-link-abbreviated" href="mailto:glenn@thorpies.com">glenn@thorpies.com</a>
</pre>
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