Re: Simple question about solving simultaneous equations

On Dec 11, 5:20 pm, Zerex71 <Zere...@xxxxxxx> wrote:
On Dec 10, 10:42 am, 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?

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

I actually need to correct what I am saying about these frames. C_bE
would actually take five angles, not two. The relationship is as
follows:

C_bE = C_nE * C_bn

where:

C_bn is the DCM relating the body frame to a nav frame (North-East-
Down, or NED, sometimes called LTP for local tangent plane)
C_nE is the DCM relating the nav frame to the ECEF frame
C_bE is the combined DCM relating body coordinates to ECEF coordinates

I know how to convert or obtain C_bn as a quaternion, and that works
correctly. The second part I'm having a lot of trouble figuring out.
In surveying a lot of literature, I can find no example where the
author(s) actually use (lat,lon) to turn C_nE into a quaternion. My
suspicion is that this *may* have something to do with the fact that
converting body to nav coordinate frames is "easy" because both of
those frames are oriented in the same way (forward-right-down),
however, the ECEF frame is not so (it's forward-left-up). I had a
similar problem earlier when I was trying to measure points on a
vehicle and realized that the way I measured the points from
engineering drawings was in a frame that was oriented differently than
a typical forward-right-down frame. I may need to come up with some
correction factor when going from nav to Earth. Just a thought...

I was in a hurry to write this and made a mistake. Also, I know this
has veered somewhat off-topic into domain knowledge, but am unaware of
an appropriate group to cross-post to.

Mike

More clarification - C_bn takes (roll, pitch, yaw) and C_nE takes
(lat, lon, wander) but we can always set the wander angle to zero for
our purposes. Hence, five angles.
.