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   Re: How to calculate position with "too many" sate...
| GSV Three Minds... | 04-05-2008 |
 Re: How to calculate position with "too many" sate...
| GSV Three Minds... | 04-05-2008 |
 Re: How to calculate position with "too many" sate...
| David L. Wilson | 04-05-2008 |
  Re: How to calculate position with "too many" sate...
| GSV Three Minds... | 04-06-2008 |
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One of rules of math for solving systems of equations is that you want
EXACTLY the same number of equations as unknowns. Too few and no
possible solution. Too many and you are overdetermined. So in theory we
want only four satellites as we have 4 unknowns (x, y, z, time). But all
our data has measurement errors so we use more satellites to compensate.
How? What's the keyword I'm looking for to search for the appropriate
math? TIA
I guess the keywords would be the overdetermined least squares problem
and Moore-Penrose.
See e.g. http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
In short the solution you get is not an exact one, but the one that
fits best the overdetermined system in terms of the least squares.
Bitstring
>> One of rules of math for solving systems of equations is that you want
>> EXACTLY the same number of equations as unknowns. Too few and no
>> possible solution. Too many and you are overdetermined. So in theory we
>> want only four satellites as we have 4 unknowns (x, y, z, time). But all
>> our data has measurement errors so we use more satellites to compensate.
>> How? What's the keyword I'm looking for to search for the appropriate
>> math? TIA
>> EXACTLY the same number of equations as unknowns. Too few and no
>> possible solution. Too many and you are overdetermined. So in theory we
>> want only four satellites as we have 4 unknowns (x, y, z, time). But all
>> our data has measurement errors so we use more satellites to compensate.
>> How? What's the keyword I'm looking for to search for the appropriate
>> math? TIA
>I guess the keywords would be the overdetermined least squares problem
>and Moore-Penrose.
>See e.g. http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
>In short the solution you get is not an exact one, but the one that
>fits best the overdetermined system in terms of the least squares.
>and Moore-Penrose.
>See e.g. http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse
>In short the solution you get is not an exact one, but the one that
>fits best the overdetermined system in terms of the least squares.
Or to put it another way, you can use any 4 of the 'N' satellites you
can see, and get (N choose 4) calculated positions. You then pick an
answer which minimises the total distance from all the calculated
answers (which iirc, is the centre of mass of the group of dots?)
You could also tart things up a bit by throwing away any obvious
outliers (preferably on a satellite by satellite basis) , and making
allowances for the satellite geometries (i.e. each calculated position
has an associated error circle/ellipse (or 3D equivalent thereof), and
they are not all the same size).
I don't think any of the current crop of consumer handsets is smart
enough to cope with any of that in real time. 8<,
--
GSV Three Minds in a Can
11,020 Km walked. 2,118 Km PROWs surveyed. 38.3% complete.
On Sat, 5 Apr 2008 16:56:54 +0100, GSV Three Minds in a Can
>You could also tart things up a bit by throwing away any obvious
>outliers (preferably on a satellite by satellite basis) , and making
>allowances for the satellite geometries (i.e. each calculated position
>has an associated error circle/ellipse (or 3D equivalent thereof), and
>they are not all the same size).
>I don't think any of the current crop of consumer handsets is smart
>enough to cope with any of that in real time. 8<,
>outliers (preferably on a satellite by satellite basis) , and making
>allowances for the satellite geometries (i.e. each calculated position
>has an associated error circle/ellipse (or 3D equivalent thereof), and
>they are not all the same size).
>I don't think any of the current crop of consumer handsets is smart
>enough to cope with any of that in real time. 8<,
Would it not be simpler to consider the satellites' positions when
determining the best geometry?
>On Sat, 5 Apr 2008 16:56:54 +0100, GSV Three Minds in a Can
>>You could also tart things up a bit by throwing away any obvious
>>outliers (preferably on a satellite by satellite basis) , and making
>>allowances for the satellite geometries (i.e. each calculated position
>>has an associated error circle/ellipse (or 3D equivalent thereof), and
>>they are not all the same size).
>>I don't think any of the current crop of consumer handsets is smart
>>enough to cope with any of that in real time. 8<,
>>outliers (preferably on a satellite by satellite basis) , and making
>>allowances for the satellite geometries (i.e. each calculated position
>>has an associated error circle/ellipse (or 3D equivalent thereof), and
>>they are not all the same size).
>>I don't think any of the current crop of consumer handsets is smart
>>enough to cope with any of that in real time. 8<,
>Would it not be simpler to consider the satellites' positions when
>determining the best geometry?
>determining the best geometry?
That's what most handsets do, to just pick the 4 best satellites to use.
However it ignores all the other information which was available, and
the fact that the satellites in 'the best geometrical position' may
actual be the worst as far as clock and orbital parameters accuracy
goes.
--
GSV Three Minds in a Can
11,020 Km walked. 2,118 Km PROWs surveyed. 38.3% complete.
> EXACTLY the same number of equations as unknowns. Too few and no
> possible solution. Too many and you are overdetermined. So in theory we
> want only four satellites as we have 4 unknowns (x, y, z, time). But all
> our data has measurement errors so we use more satellites to compensate.
> How? What's the keyword I'm looking for to search for the appropriate
> math? TIA