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Summary re: Rationale for defining rotation matrixes

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  • Summary re: Rationale for defining rotation matrixes

    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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