Dear subscribers,

recently, I have been discussing with some colleagues about the reason
why we call "orientation/attitude matrix of reference frame B
relative to A" the rotation matrix R used to transform any vector V from B
to A, using the following formula:

V(in A) = R * V(in B)

We were neither concerned about the obvious order of the nine unit
vector components (or direction cosines) contained in the rotation matrix
performing the above operation, nor about the many different symbols used in
the literature to indicate the reference frames and the rotation matrix. The
problem was just about terminology. Here's the complete question:

Why do most authors call the above matrix "the orientation/attitude
matrix of
reference frame B relative to A", rather than "the orientation/attitude
matrix of reference frame A relative to B"?

Some of you might think that the answer is easy, and indeed it is. Yet,
here's the names of those who didn't know the answer, initially: Paolo de
Leva, Ton van den Bogert, Aurelio Cappozzo, Jesus Dapena. Is that enough for
guessing that many subscribers will be interested in reading the end of this
e-mail message?

The answer was quickly found by Jesus Dapena. It's so simple and elegant
that I can summarize it in a few words:

1) a rotation of the reference system has the same effect, on the components
of a vector V, as a rotation of that vector IN THE OPPOSITE DIRECTION!!!!

2) If the orientation of frame B relative to A is +30°, to transform a
vector from A to B you need to rotate it by -30°, which is the orientation
of A relative to B.

With kind regards,

Paolo de LEVA

University Institute of Motor Sciences
Sport Biomechanics
P. Lauro De Bosis, 6
00194 ROME - ITALY

Telephone: (39) 06.367.33.522
FAX/AM: (39) 06.367.33.517
FAX: (39)


Tel./FAX/AM: (39) 06.336.10.218

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