From: [Permission pending]
Newsgroups: sci.math
Subject: Need position of point from multiple known distances.....
Date: 14 Apr 1995 06:38:09 GMT

I'm new to the group and have only attained the rank of cookbook   
mathematician.  I have a problem which seems easy but the more I dig into   
it the more it digs into me.  I'm sure it has been solved a million   
times, but none of my college text books (which were always short on real   
world solutions) seem to even get close.  I don't even know how to   
classify the problem (linear equations, polynomial roots, minimization,   
etc).  Here goes..... 
 
The simple version: 
I have 3 known points in 3-space, and the 3 distances from each known   
point to an unknown point.  How do I find the position of the unknown   
point? 
 
When I setup 3 simultaneous equations for a sphere, and expand the   
equations, I end up with x^2 and x terms (and y and y^2 terms, etc.)   
which I can't eliminate or subtract out.  If I carefully choose the known   
point positions I can eliminate most of the single terms, and thereby   
solve for the squared terms.  I'd like to set this up as a matrix   
equation, but the solution for this simple case seems too complex to put   
into a matrix equation (at least for me).  I'd also like to get it into a   
matrix equation so that I can get a better estimate of the unknown point   
position through additional distance measures from other known points. 
 
The complicated version: 
I have 3 known points in 3-space, and the 9 distances from each known   
point to three unknown points.  The unknown points are in a known   
geometric relationship (co-planar, isoceles triangle, known lengths   
of sides, unknown orientation).  How do I find the unknown position   
(centroid?) and orientation (euler angles) of the set of unknown points? 
 
This seems like an overspecified problem with 9 equations and only 6   
(3-position, 3-orientation) unknowns.  The simple way would be to use the   
simple solution to solve the three unknown points independently, and then   
work out the geometry for orientation, but I was hoping for a less noise   
sensitive solution. 
 
Any help would be appreciated. 
[sig deleted]

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From: [Permission pending]
Newsgroups: sci.math
Subject: Re: Need position of point from multiple known distances.....
Date: 14 Apr 1995 17:56:16 GMT

Just in case anyone is interested, the problem I'm trying to solve   
involves an acoustic headtracker, with three transmitters in a stationary   
triangle, and three receivers in a triangle that is attached to the   
object being tracked. 
 
This problem should also be similar to that of radar and acoustic imaging   
(i.e. from a time series of known return times at known transmitter   
locations, what is the shape of the thing being imaged), as well the one   
about the motion platform.  I thought this might be a rather common   
problem, but I guess I'm wrong. 
 
I'm still trying to integrate all the help everyone has provided (and I'm   
giving the math dictionary a real work out).  Thanks. 
[sig deleted]

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From: [Permission pending]
Newsgroups: sci.math
Subject: Re: Need position of point from multiple known distances.....
Date: 17 Apr 1995 13:24:43 GMT

     The Stewart reference is:
D. Stewart, "A platform with six degrees of freedom", Proceedings of
the Institution of Mechanical Engineers, v. 180, Part 1, No. 15,
pp. 371-368 (1965/66).

If I had a copy I'd send it along to you, but I've never actually seen the
paper - just many references to it.  You can find many relevant references
in the journal Mechanism and Machine Theory, as well as in the robotics
literature.   
[sig deleted]

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From: [Permission pending]
Subject: Re: Need position of point from multiple known distances.....
To: rusin@math.niu.edu (Dave Rusin)
Date: Mon, 17 Apr 1995 10:44:30 -0400 (EDT)

> 
> Thank you for posting this!
> 

    You're welcome.  I have a dozen or two relevant references.
Unfortunately, I haven't compiled a bibliography of these papers.  But
I could provide pointers to a few selected recent references if you'd
like.

    Stewart platforms have been used in flight simulators and for
controlling optical elements of telescopes (both optical and radio).
Here we're working on one which will control the 8-meter diameter
subreflector (which will weigh 3000 or 4000 pounds) of a big new radio
telescope under construction in West Virginia.  I'm working with
H. Zhuang of Florida Atlantic University on algorithms for the
kinematic calibration of our Stewart platform mechanism.

    By the way, I'm originally from Sterling, 60 miles west of Dekalb.

    For the problem of estimating the position of a point, by
least-squares, from three or more (possibly noise contaminated)
distance measurements, I wrote the Fortran code included below.  Not
sure if that was your interest or that of one of the other posters (I
haven't sent anything to the other(s)).

[sig deleted]

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[Code included as a separate file in this directory. -- djr]
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