View Full Version : Re: symbolic expression for angular velocitie

03-18-2004, 10:00 AM
Dear Ton and all,

> M*Wi + M*Wj + M*Wk = mx.wx + my.wy + mz.wz, [4]
>Would there be a 3rd way to decompose power into "JCS" related
> P = Mi*W + Mj*W + Mk*W [4b]

This is an interesting point. In theory, [4b] has no problem. What will
matter, though, is the physical meaning of the equation. [4b] basically
shows how the angular velocity is associated with the decomposed moment
sub-vectors, while [4] shows how the moment is associated with the
angular velocity sub-vectors. The question is: What would be the
physical meanings of the non-orthogonal moment sub-vectors decomposed
along the Cardan axes?

I'd like to remind you at this point of something more fundamental. The
moment (torque) is the cause of angular motion and it will affect both
segments forming a joint. As I stated in my last posting, the JCS is not
a real perspective. It is just a hybrid system defined by two
intermediate systems involved in three successive rotations. It is in
reality neither the perspective that the pelvis is using, nor that used
by the thigh. If your focus is on how the moment affects the segment
motions, the moment must be described either in the pelvis frame or in
the thigh frame, not in something intermediate such as the JCS. This is
my line of thinking. In conclusion, I would not try to decompose the
moment vector in the JCS system.

The reason why we decompose the angular velocity into three sub-vectors
along the Cardan axes is that these sub-vectors represent three
independent rotations. The funny thing about the angular velocity is
that the components do not really represent independent rotations. For
example, the angular velocity has three orthogonal components:

W = |W|.N = Wx + Wy + Wz [5]

where N = unit vector of the angular velocity. Vector N shows the
orientation of the axis of rotation while |W| shows how fast the
rotation is. Now, what is the actual meanings of the components?
Anything more than the orientation of the axis of rotation? Wx, Wy, and
Wz DO NOT really represent three independent rotations. The
decomposition of the angular velocity vector along the Cardan axes
allows us to break down the angular motion into three independent
rotations. Which means we will have to rethink the merit of breaking the
joint power into three orthogonal terms (right side of [4]).

Another problem I see in [4b] is the difficulty associated with the
decomposition of the moment vector into three sub-vectors. Since the
system is not orthogonal, you cannot compute the sub-vectors by
dot-producting the moment vector and each of the axis unit vector. If
you do this, it will create a completely different monster. Rather it
must be computed in a system-of-linear-equation approach:

M = Mi.I + Mj.J + Mk.K [6a]

Mx = Mi.Ix + Mj.Jx + Mk.Kx [6b]

My = Mi.Iy + Mj.Jy + Mk.Ky [6c]

Mz = Mi.Iz + Mj.Jz + Mk.Kz [6d]

[ Mx ] [ Ix Jx Kx ][ Mi ]
[ My ] = [ Iy Jy Ky ][ Mj ] [6e]
[ Mz ] [ Iz Jz Kz ][ Mk ]

So the computation requires the inverse matrix of a 3x3 matrix formed by
the x, y, z components of the Cardan axis unit vectors.

>First, let's take the example of a hypothetical hip joint movement
where there is an extensor and adductor moment, and the motion is a
combination of flexion and adduction, such that total power M*W is zero.
Now, if we know that there is no single muscle that can simultaneously
produce the extensor and adductor moment, we know that there is positive
work in one muscle and equal and opposite negative work in another
muscle. We could try to estimate these amounts by decomposing the
(zero) total power into positive and negative components. Is equation
[4] best, or equation [4b]?

In this case, there will be no difference between [4] and [4b] because
the first (flexion/extension) and second (abduction/adduction) rotation
axes are always perpendicular to each other in the XYZ-type rotations.
What causes the troubles is the third rotation axis.

>The other question is one of interpretation. Can we interpret the JCS
as a cardanic mechanism where each axis is driven my a moment (Mi, Mj,
and Mk), with axes rotating at speeds Wi, Wj, and Wk? If so, the
amounts of mechanical power delivered by the motors are Mi*Wi, Mj,Wj,
and Mk*Wk. We now know that these do not add up to the total joint
power (M*W) if the i and k axes are not orthogonal. So is my
interpretation wrong, or is the calculation wrong?

As I stated earlier, I don't think JCS is a true physical perspective.
Looking at how the moment vector is associated with the three
independent rotations (left side of [4]) is meaningful enough to me but
I am not quite sure about any attempts beyond that. That depends on the
physical meanings of Mi, Mj, and Mk. We all know that the angular
momentum and angular velocity are not necessarily in the same direction.
That is because the inertia tensor is involved in the computation of the
angular momentum. Torque is the first time-differential of the angular
momentum and similar realtionship exists between torque (moment) and
angular acceleration (torque = inertia tensor * angular acc). The effort
to compute Mi, Mj, Mk and to relate these to Wi, Wj, Wk, respectively,
may not be as meaningful as it appears to be because of the magic work
of the inertia tensor.

- Young-Hoo Kwon, Ph.D.
- Biomechanics Lab, Texas Woman's University
- kwon3d@kwon3d.com
- http://kwon3d.com

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