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Re: Quaternions vs. Euler angles

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  • Re: Quaternions vs. Euler angles

    as Ton already pointed out, there is an extensive explanation of the use
    of quaternions (in their "euler parameter" representation) in "Computer
    Aided Kinematics and Dynamics of Mechanical Systems", by E.J. Haug. In
    this Book Haug introduces the theory the multi-body-simulation software
    DADS is based on.
    >From the theoretical point of view euler parameters
    e0 = cos(A/2)
    e1 = Ux*sin(A/2)
    e2 = Uy*sin(A/2)
    e3 = Uz*sin(A/2)
    are an excellent tool to describe orientations and numerically solve
    equations of motion of mechanical multi-body-systems, since there is no
    gimbal lock, they are well defined on the unit-sphere (any orientation
    of a rigid body can be uniquely/continuesly described by the set of (Ux,
    Uy, Uz, A).
    Whereas there is no way of integrating angular velocity w to obtain
    orientation (since it's not integrable) one may integrate the time-
    derivative of euler parameters to calculate e0(t), e1(t), e2(t), e3(t).

    On the other hand euler parameters cannot be used to describe multiple
    revolutions (uniqueness only on the unit-sphere) which is not too much
    of a problem in biomechanics, since no joint range in biological systems
    is greater. A problem with euler parameters is though, that it's almost
    impossible to set up 3D-torque elements (e.g. for spherical joints).
    Except for the trivial case where the joint torque always acts along
    (Ux,Uy,Uz) it is very hard to describe a certain experimental behavior
    with an euler parameter torque element. I tried this when I wanted to
    limit the range of motion for a spherical joint but finally gave up,
    since it was simply impossible (at least for me) to find the equations
    using euler parameters. So I ended up using angles again.

    The interpretaton of euler parameters is simple as long as one looks at
    orientation only. But as soon as torque elements or experimental data
    are to be described, interpretation seems to be impossible.

    Arnim Henze.
    ================================================== ====================
    Institut f"ur Astronomie und Astrophysik
    Arnim Henze Abt. Computational Physics - Biomechanik -

    Universit"at T"ubingen Tel.: ++49 7071 29 78654
    Auf der Morgenstelle 10 Fax : ++49 7071 29 5889
    D-72076 T"ubingen, Germany email:
    ================================================== ====================

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