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Re: Quaternions vs. Euler angles

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  • Re: Quaternions vs. Euler angles


    In animation, the problems with Euler angles mainly arise when doing
    motion editing. As long as you leave them alone they are fine.
    Motion editing could be: resampling (interpolation), amplification,
    motion blending etc. If you perform these operations on Euler angles
    directly, you can get strange results, especially near gimbal lock.
    The quaternion representation seems to behave better.

    It seems that quaternions are the same as the "euler parameters" which
    are often used in computational kinematics:

    I have also heard the term "angle-axis representation", i.e. the rotation
    is represented as a rotation of magnitude A about an axis (Ux,Uy,Uz).
    Euler parameters are defined as follows

    e0 = cos(A/2)
    e1 = Ux*sin(A/2)
    e2 = Uy*sin(A/2)
    e3 = Uz*sin(A/2)

    The sum of squares of these parameters is exactly one.
    See also "Computer Aided Kinematics and Dynamics of Mechanical Systems",
    by E.J. Haug.

    Note that this representation is closely related to the three "helical
    angles" proposed by Herman Woltring. The helical angle representation is:

    h1 = A*Ux
    h2 = A*Uy
    h3 = A*Uz

    Why quaternions are not used more in biomechanics? This probably has
    something to do with interpretation. Euler angles can be associated
    with the rotations in a mechanical linkage or 3-D goniometer (Grood
    and Suntay, J Biomech Eng, 1983). The other representations work well
    for computation but are not so easily interpreted.

    On the other hand, Woltring makes some good points on error propagation
    in his 1994 paper (J Biomech 27:1399-1414). Near gimbal lock, Euler angles
    become increasingly sensitive to measuring errors.

    For the newcomers on Biomch-L, I also recommend reading the debate between
    Grood and Woltring, about 10 years ago on Biomch-L:

    Ton van den Bogert


    A.J. (Ton) van den Bogert, PhD
    Department of Biomedical Engineering
    Cleveland Clinic Foundation
    9500 Euclid Avenue (ND-20)
    Cleveland, OH 44195, USA
    Phone/Fax: (216) 444-5566/9198

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