View Full Version : Re: symbolic expression for angular velocitie

03-17-2004, 04:13 AM
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
- kwon3d@kwon3d.com
- http://kwon3d.com

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