No announcement yet.

Re: symbolic expression for angular velocitie

This topic is closed.
  • Filter
  • Time
  • Show
Clear All
new posts

  • Re: symbolic expression for angular velocitie

    Dear Tomislav and Ton,

    > If your result is correct, this brings us back to the original
    > question of joint power.

    Let me assure you one more time. The angular velocity can be computed
    from the first time-derivatives of the Cardan angles or from the
    transformation matrix as I demonstrated before. They yield identical

    > The two ways to compute joint power are 1.
    > Sum of moment-angular velocity products in Cardanic (joint coordinate
    > system) axis system 2. Dot product of moment and angular velocity in a

    > cartesian reference frame.

    Be careful in using the first method. As I tried to emphasize all along,

    Pj mi.wi + mj.wj + mk.wk, [1]

    where i, j, and k are the three non-orthogonal Cardanic axes, m = moment
    components, and w = ang vel components. This relationship only holds
    when the system is orthogonal. Rather, joint power must be computed as

    Pj = M*W
    = M*(Wi + Wj + Wk)
    = M*Wi + M*Wj + M*Wk, [2]

    where * = dot product operator, and Wi, Wj, and Wk = three angular
    velocity vectors decomposed along the three Cardanic axes. Or

    Wi = wi.I = da/dt.I
    Wj = wj.J = db/dt.J
    Wk = wk.K = dc/dt.K, [3]

    where I, J, and K = unit vectors of the Cardan axes, and a, b, and c =
    three Cardan angles. All these vectors (M, Wi, Wj, and Wk) must be
    described in a unified orthogonal reference frame, such as the global
    inertial frame or a particular local segmental reference frame. Each of
    the three terms in the above equation shows how the moment vector is
    associated with the three angular velocity vectors. One can easily see

    M*Wi + M*Wj + M*Wk = mx.wx + my.wy + mz.wz, [4]

    where x, y, and z = Cartesian axes. It is just a matter of how to break
    down the power into meaningful terms and Eq. 4 shows two different ways.

    Tomislav brought up an interesting issue: JCS. To me, a coordinate
    system just means how to describe a vector into components: rectangular,
    polar, cylindrical, hyperbolic, etc. What's more important in describing
    motions of the body parts is the reference frame or the perspective of
    description. The only entities that can carry perspectives to me are the
    segments, not the joints. As the orientation of a segment changes, the
    reference frame fixed to it also moves with the segment. However, the
    relative relationship between the segment and the local frame does not
    change. On the other hand, joint motion is simply the relative motion of
    the distal segment to its linked proximal segment. The relative
    orientation of the Cardanic axes (JCS) changes as the joint motion
    progresses. The JCS may be used to decompose the angular velocity vector
    into meaningful sub vectors, but JCS shouldn't be treated as a reference
    frame (or perspective). The confusion shown in Eq. 1 originates from the
    concept of JCS. In Eq. 2, the vectors can be described in any reference
    frame (global or segmental). However, the perspective used in the
    computation does not affect the three power terms to be obtained from
    Eq. 2.

    - Young-Hoo Kwon, Ph.D.
    - Biomechanics Lab, Texas Woman's University

    To unsubscribe send SIGNOFF BIOMCH-L to
    For information and archives:
    Please consider posting your message to the Biomch-L Web-based
    Discussion Forum: