[OT] Solving systems of quadratic equations in 3 variables

The problem I'm interested in is solving for position when knowing 
distance from >= 3 points in 3D space (Specifically where am I knowing 
how far from set of satellites).

Canned programs available *BUT*:
1. I want to understand what I'm doing.
2. I want to pick which satellites I use.

Numeric methods would be fine for my purposes as I "know" approximate 
location to begin with.

Google searches on permutations of  "quadratic", "equation(s)" and 
"system(s)" were not fruitful. Primarily got hits on intersection of 
straight line with a parabola.

Suggested search term(s)?
Other suggestions?

Thanks

>The problem I'm interested in is solving for position when knowing 
>distance from >= 3 points in 3D space (Specifically where am I knowing 
>how far from set of satellites).
>
>Canned programs available *BUT*:
>1. I want to understand what I'm doing.
>2. I want to pick which satellites I use.
>
>Numeric methods would be fine for my purposes as I "know" approximate 
>location to begin with.
>
>Google searches on permutations of  "quadratic", "equation(s)" and 
>"system(s)" were not fruitful. Primarily got hits on intersection of 
>straight line with a parabola.
>
>Suggested search term(s)?
>Other suggestions?
>
>Thanks
>
***********************************
Hello,

first of all, is your system of equations well defined, i.e. is there an
equal number of unknowns and equations?

Manolis C. Tsakiris

Manolis C. Tsakiris wrote:
>>The problem I'm interested in is solving for position when knowing 
>>distance from >= 3 points in 3D space (Specifically where am I knowing 
>>how far from set of satellites).
>>
>>Canned programs available *BUT*:
>>1. I want to understand what I'm doing.
>>2. I want to pick which satellites I use.
>>
>>Numeric methods would be fine for my purposes as I "know" approximate 
>>location to begin with.
>>
>>Google searches on permutations of  "quadratic", "equation(s)" and 
>>"system(s)" were not fruitful. Primarily got hits on intersection of 
>>straight line with a parabola.
>>
>>Suggested search term(s)?
>>Other suggestions?
>>
>>Thanks
>>
> 
> ***********************************
> Hello,
> 
> first of all, is your system of equations well defined, i.e. is there an
> equal number of unknowns and equations?
> 
> Manolis C. Tsakiris

Yes. Some might object that saying it is over defined as more than 3 
satellites are visible at a time. BUT I am explicitly interested in 
solutions involving EXACTLY 3 satellites.

Richard Owlett wrote:
> The problem I'm interested in is solving for position when knowing 
> distance from >= 3 points in 3D space (Specifically where am I knowing 
> how far from set of satellites).
> 
> Canned programs available *BUT*:
> 1. I want to understand what I'm doing.
> 2. I want to pick which satellites I use.
> 
> Numeric methods would be fine for my purposes as I "know" approximate 
> location to begin with.
> 
> Google searches on permutations of  "quadratic", "equation(s)" and 
> "system(s)" were not fruitful. Primarily got hits on intersection of 
> straight line with a parabola.
> 
> Suggested search term(s)?
> Other suggestions?
> 
Have you tried just writing them out and solving them?

This sounds like the GPS solution, except with GPS you need a fourth 
satellite because absolute time is also an unknown.  Searching on 
combinations with "GPS" in the title may yield useful answers.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html

On Mar 3, 10:17 am, Richard Owlett <rowl...@atlascomm.net> wrote:
> The problem I'm interested in is solving for position when knowing
> distance from >= 3 points in 3D space (Specifically where am I knowing
> how far from set of satellites).
>
> Canned programs available *BUT*:
> 1. I want to understand what I'm doing.
> 2. I want to pick which satellites I use.
>
> Numeric methods would be fine for my purposes as I "know" approximate
> location to begin with.
>
> Google searches on permutations of  "quadratic", "equation(s)" and
> "system(s)" were not fruitful. Primarily got hits on intersection of
> straight line with a parabola.
>
> Suggested search term(s)?
> Other suggestions?
>
> Thanks

Add "range only" to tracking, trackers, measurement, etc.

I would expect DTIC / STINET to have some of it.

Dale B. Dalrymple
http://dbdimages.com

Tim Wescott wrote:
> Richard Owlett wrote:
> 
>> The problem I'm interested in is solving for position when knowing 
>> distance from >= 3 points in 3D space (Specifically where am I knowing 
>> how far from set of satellites).
>>
>> Canned programs available *BUT*:
>> 1. I want to understand what I'm doing.
>> 2. I want to pick which satellites I use.
>>
>> Numeric methods would be fine for my purposes as I "know" approximate 
>> location to begin with.
>>
>> Google searches on permutations of  "quadratic", "equation(s)" and 
>> "system(s)" were not fruitful. Primarily got hits on intersection of 
>> straight line with a parabola.
>>
>> Suggested search term(s)?
>> Other suggestions?
>>
> Have you tried just writing them out and solving them?

Writing them out is no problem. Solving is something else.
I know ho to attack the *LINEAR* case.
But this has all 3 variables as linear AND squared terms.

> 
> This sounds like the GPS solution,

It is. Sort of ;)

> except with GPS you need a fourth 
> satellite because absolute time is also an unknown.

Except I sortof "cheat" ;o)

My input data are pseudoranges in RINEX format from CORS stations so 
"time" is already accounted for.

>  Searching on 
> combinations with "GPS" in the title may yield useful answers.
> 

That tends to get the the canned programs I referred to or to 
differential GPS - which isn't quite what I'm looking for.

dbd wrote:

> On Mar 3, 10:17 am, Richard Owlett <rowl...@atlascomm.net> wrote:
> 
>>The problem I'm interested in is solving for position when knowing
>>distance from >= 3 points in 3D space (Specifically where am I knowing
>>how far from set of satellites).
>>
>>Canned programs available *BUT*:
>>1. I want to understand what I'm doing.
>>2. I want to pick which satellites I use.
>>
>>Numeric methods would be fine for my purposes as I "know" approximate
>>location to begin with.
>>
>>Google searches on permutations of  "quadratic", "equation(s)" and
>>"system(s)" were not fruitful. Primarily got hits on intersection of
>>straight line with a parabola.
>>
>>Suggested search term(s)?
>>Other suggestions?
>>
>>Thanks
> 
> 
> Add "range only" to tracking, trackers, measurement, etc.

That jogs my thought pattern in choosing keywords.

> 
> I would expect DTIC / STINET to have some of it.

OBOY can I see interesting rabbit trails in my future. It's an obvious 
place -- once pointed out ;)


> 
> Dale B. Dalrymple
> http://dbdimages.com

Tim Wescott wrote:
> Richard Owlett wrote:
> 
>> The problem I'm interested in is solving for position when knowing 
>> distance from >= 3 points in 3D space (Specifically where am I knowing 
>> how far from set of satellites).
>>
> 
> This sounds like the GPS solution, except with GPS you need a fourth 
> satellite because absolute time is also an unknown.  Searching on 
> combinations with "GPS" in the title may yield useful answers.

GPS-like system of equations is a nasty problem. It takes a lot of 
computation and the solution has many special cases. The development of 
a good algorithm for that type of problem could be a good assignment for 
the graduate student level work.



Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

Vladimir Vassilevsky wrote:

> 
> 
> Tim Wescott wrote:
> 
>> Richard Owlett wrote:
>>
>>> The problem I'm interested in is solving for position when knowing 
>>> distance from >= 3 points in 3D space (Specifically where am I 
>>> knowing how far from set of satellites).
>>>
>>
>> This sounds like the GPS solution, except with GPS you need a fourth 
>> satellite because absolute time is also an unknown.  Searching on 
>> combinations with "GPS" in the title may yield useful answers.
> 
> GPS-like system of equations is a nasty problem. It takes a lot of 
> computation and the solution has many special cases. The development of 
> a good algorithm for that type of problem could be a good assignment for 
> the graduate student level work.
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Design Consultant
> http://www.abvolt.com

We agree. That's a large part of why I specified the 3 satellite case. 
With more satellites would I not be guaranteed a solution exists due to 
measurement errors, but I would be overdetermined as well.

An eventual goal is looking at "error" introduced by water vapor. I'll 
probably experiment first with ionospheric effects first as they are 
larger. I'll retire soon, need something to keep me out of mischief ;)

Tim Wescott wrote:

> Richard Owlett wrote:
> 
>> The problem I'm interested in is solving for position when knowing 
>> distance from >= 3 points in 3D space (Specifically where am I knowing 
>> how far from set of satellites).
>>
>> Canned programs available *BUT*:
>> 1. I want to understand what I'm doing.
>> 2. I want to pick which satellites I use.
>>
>> Numeric methods would be fine for my purposes as I "know" approximate 
>> location to begin with.
>>
>> Google searches on permutations of  "quadratic", "equation(s)" and 
>> "system(s)" were not fruitful. Primarily got hits on intersection of 
>> straight line with a parabola.
>>
>> Suggested search term(s)?
>> Other suggestions?
>>
> Have you tried just writing them out and solving them?


OK already. Just how big was GRIN when you wrote that?
If I had actually *LOOKED* at the equations ;/

But the hint to add GPS to search got the hit that pounded answer in.

"The pseudo-range equations are a system of quadratic equations. ... 
Each quadratic equation has the same quadratic terms, so differences of 
the quadratic equations give linear equations."

"Initial Solution of Pseudo-Range Equations"
  Mike Elgersma, Honeywell Technology Center
  www.ima.umn.edu/gps/abstract/elgersma1.html






> This sounds like the GPS solution, except with GPS you need a fourth 
> satellite because absolute time is also an unknown.  Searching on 
> combinations with "GPS" in the title may yield useful answers.
>