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  • Joint centers (was: Wobbling mass model)

    Dear BIOMCH-L readers,

    In his discussion of the 'wobbling mass' model, Herman Woltring has
    raised the question whether the assumption of fixed joint centers would
    cause significant errors in the estimation of joint contact forces. I
    think it will, because an essential step before joint forces are
    calculated, is the calculation of muscle forces. This involves two
    basic problems: the distribution criterion and the moment arms of the
    muscles. The moment arms have usually been calculated with respect to a
    fixed joint center, and we all know that such a thing does not exist in
    the knee joint. Estimated forces will contain relative errors as large
    as the error in the joint center location, divided by the moment arm.

    An elegant way to get rid of fixed joint centers has been demonstrated
    in the December (1990) issue of the Journal of Biomechanics by Cees Spoor
    and co-workers from the University of Leiden, Netherlands. The
    instantaneous moment arm of a muscle (conventionally defined as the
    perpendicular distance from the line of action to the instantaneous
    joint center) can also be defined as the partial derivative of the
    muscle length with respect to the angle of the joint. So, you only need
    to know how the length of a muscle depends on joint angles (from cadaver
    measurements), and *not* the location of the actual joint center at each
    instant in time. The method also solves another problem: that of a
    muscle not following the shortest path from origin to insertion when it
    curves around bone or other muscles. Conventional models have great
    difficulty describing this.

    If this principle (of virtual work) is also used to determine the
    moment arm of an external (ground reaction) force, assumptions on the
    kinematical properties of joints will no longer be a source of error in
    inverse dynamics analysis. Inertial forces require some special
    attention though.

    Sometimes, e.g. in a 'forward' simulation, we *do* want to describe
    exactly how the segments are kinematically coupled, but modern multibody
    software packages (e.g. DADS) contain a rather extensive library of
    kinematic connections.

    -- Ton van den Bogert
    Faculty of Veterinary Medicine
    University of Utrecht

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