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RE: Kinematics & Kinetics

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  • RE: Kinematics & Kinetics

    Dear Biomch-L readers,

    In reply to Fabio Catani's Biomch-L posting last Monday on how to assess 3-D
    ICR's, I have the following comments. Some of this is (to my knowledge) new
    information, and it seems interesting to announce this in a non-refereed,
    electronic bulletin rather than to wait for a comprehensive review process.

    (a) Utility:

    An instantaneous, 3-D centre of rotation for a moving rigid body (e.g., a
    distal segment w.r.t. a proximal segment in the case of 3-D joint motion)
    may be useful in a dynamics context for the following reasons:

    - If you are looking at net joint moments without regarding individual force
    and moment contributions in ligaments, muscles, and contact points, the
    numerical values and the graphical results will strongly depend on the
    chosen reference point at which the net moment M is assessed. Cappozzo in
    Rome/Italy has amply demonstrated this (Bristol ESB Meeting, 1987).
    - It is sometimes proposed that it is better to use as a reference point one
    which is stationary w.r.t. the anatomy. While this is possible within one
    segment, such a point will generally not be stationary w.r.t. the other
    segment. Thus, which of the two segments comprising a joint should be
    - The net power transmitted through a joint can be assessed from the rotatio-
    nal and translational contributions as M'omega + F'v, where F is the net
    force, and omega and v the rotation and translation velocity vectors of the
    joint (i.e., distal w.r.t. proximal or vice-versa). Of these entities, M
    and v are position-dependent, and v is minimal at the Instantaneous Helical
    Axis. The common practise to neglect the linear term F'v is appropriate
    if v == 0 (planar movement at the IHA) which includes the case of a fixed
    rotation axis or centre. While this may be the case for the healthy hip
    joint, it is certainly debatable for other joints. Note: while the net
    power is invariant with the position along the IHA, the net moment varies.
    - If interest is oriented to assessing the efficacy of individual muscles, the
    muscle's moment arm can be suitably assessed with respect to the IHA or ICR
    as defined below.

    (b) Definiton and Calculus/Estimation:

    A 3-D ICR can be defined by looking for the point with minimal, instantaneous
    displacement. Given the general rigid-body equation

    y(t) = R(t).x + p(t), R(t) orthonormal, x arbitrary but fixed

    one could try and find the point x for which the absolute velocity |dy/dt| and
    acceleration |d(dy/dt)/dt| are minimal in some weighted sense. For example,
    one might choose from all points with minimal velocity that particular one which
    has minimal acceleration.

    The class of points with minimal velocity is defined by the Instantaneous
    Helical Axis about and along which the body is instantaneously translating
    and rotating; it is defined by the projection s = p + omega * dp/dt / omega^2
    of p onto it, and by its direction vector omega, where * denotes the external
    vector product, and where omega = (omx,omy,omz)' follows from Poisson's equa-

    [ 0 -omz omy]
    A(omega) == [ omz 0 -omx] = dR/dt . R' (' denotes transposition).
    [-omy omx 0 ]

    The point q on the IHA with minimal acceleration then follows by minimizing
    |d(dy/dt)/dt|^2 as a function of the parameter r in the equation y =
    for an arbitrary point y(r) on the IHA. [N.B.: when working this out, be sure
    not to reverse the sequence of differentiating and substitution; like the Car-
    danic rotations of this spring's joint angle debate, these operations are not

    The measurement system and protocol must be sufficiently accurate to allow
    reliable assessment of both first and second derivatives. However, these data
    are required anyhow if one plans to assess inverse dynamics including inertial
    effects. Thus, skin motion artefacts should be minimized if external markers
    are utilized, and the marker distribution should be sufficiently non-collinear.

    Note also that the 3-D ICR is ill-defined unless the rotational velocity
    OMEGA == omega * d(omega)/dt / |omega|^2 with which the IHA changes its own
    direction is sufficiently large. Thus, the 3-D ICR is ill-defined if the
    movement is nearly planar. Presumably, though, the net moment will in that
    case have a negligible tendency to rotate the IHA into an other direction
    (this might be an interesting research question).

    It turns out that the ICR as defined above coincides with the so-called central
    point c about which the IHA itself executes an instantaneous, helical movement
    with rotation velocity vector OMEGA (cf. Suh & Radcliffe, Kinematics and Mecha-
    nisms Design, Wiley 1978, Ch. 10). Fischer (1907) and Chao & An (ESB Nijmegen/
    NL, 1982) have proposed that the central point may be a useful descriptor of
    spatial movement.

    The formal proof of the equivalence between ICR and c is quite tedious and
    error-prone; fortunately, the recent computer-algebra posting on this list has
    resulted in access to the Maple package (Maple Software, Waterloo, Canada) at
    the Computer Algebra Netherlands Expertise Centre in Amsterdam by means of
    which this relation could be verified [N.B.: SURFnet users in The Netherlands
    can obtain access to this system; for details, send a note to].

    I hope that this answers Dr Catani's question to some extent (a paper is in
    preparation, which hopefully will contain some experimental results).

    Sincerely -- Herman J. Woltring, Eindhoven/NL.