Re: Simple question about solving simultaneous equations
- From: Zerex71 <Zerex71@xxxxxxx>
- Date: Mon, 10 Dec 2007 09:10:12 -0800 (PST)
On Dec 10, 11:35 am, A N Niel <ann...@xxxxxxxxxxxxxxxxxxxxx> wrote:
In article
<30e88420-dc7a-4da0-a14f-a5bf948b5...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Zerex71 <Zere...@xxxxxxx> wrote:
Greetings,
I was wondering if I could get a little help on a problem I'm working
on. In trying to solve a system of simultaneous equations (more on
that later), I ran into two expressions that when compared have the
following form:
AB = DE
AC = DF
Recalling from wayyyyy back when as a kid that you could add both
sides and proceed from there, I tried to reason that I could divide
both sides by common factors to reduce the expression:
B/C = E/F
My first question is, Is this even legal or permissible?
permissible if A,C,D,F are not zero
I got some interesting results because all of the factors are trig
functions and when done this way, I found some useful substitutions in
my CRC book that reduced the expression nicely. So, a cool discovery,
even it is may be wrong. :)
Now for the hard part: What am I doing and why am I doing it?
I am trying to develop a relationship between a body coordinate system
(frame) and the Earth coordinate system (frame, specifically the ECEF
frame). I realize there are formulas for this, and these formulas
involve the use of fairly straightforward DCMs (direction cosine
matrices) but I am trying to develop a quaternion relationship between
them instead as part of a math methodology that I am developing.
The problem comes when I can't find the appropriate formulas anywhere,
and I'll keep checking, but in developing this methodology, I noticed
the following: The formula for a DCM gives you a 3x3 matrix (9
elements), as does the rotation matrix derived from a given quaternion
(called R_q) (in effect, converting 4 elements to 9). Seeing this
pattern, I reasoned that by setting all elements of the DCM equal to
the corresponding elements of R_q, I devised a system of 9
simultaneous equations. I then proceeded to group, substitute, etc.
as I noticed patterns and relationships between the 9 expressions,
trying to solve the four elements of the quaternion (q_bE = [qw qx qy
qz]) in terms of the DCM input angles lambda and gamma (lambda =
geodetic latitude and gamma = longitude). I was unable to do it, and
my original question was something I was scratchpadding and wondered
about.
(For anyone interested -- and still even reading at this point -- the
DCM relating body to Earth frames, call it C_bE -- is a function of
only latitude and longitude; that is, a body-fixed frame will, when
placed at different points on the globe, have different values in the
Earth system, e.g. vector v_B = [5 10 -4] will have different values
at different locations.)
Please help if you have any insight into this, I am eager to hear from
you folks. Thanks.
Mike
They are nonzero, but since they are expressions themselves (functions
of variables), they could be zero. Anyway, in looking over my notes,
it appears that under certain conditions, I can get a divide-by-zero,
which is definitely not good. So I will probably abandon my approach.
Mike
.
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