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kwon3d
01-27-2004, 03:14 AM
Dear all,

Richard Baker stated:

>Zajac et al. show very neatly that if all muscles crossing a joint are
uniarticular then the joint power at any joint must be equal to the
muscle power. However in the presence of bi-articular muscles this is
not the case and the joint power at any joint is not equal to the
combined muscle power of all muscles crossing that joint.

I don't quite understand this statement. Joint power is the dot product
of the net joint force and the linear velocity of the joint. This power,
in other words, is the power caused by the linear motion of the joint
(the rate of energy flow through a joint). Energy can only flow through
a joint from one segment to another. For a given joint, the sum of the
joint powers for its proximal segment and distal segment must be zero
beacuse energy only flows from one segment to another.

On the other hand, the muscle power is the dot product of the net joint
torque (muscle moment) and the angular velocity of the segment. It
represents the power caused by the angular motion of the segment at the
joint. The muscle power for a joint-segment combination has two
components: (1) the rate of energy flow into the segment through the
muscles from the other segment (PF) + (2) the rate of work done by the
muscles to the segment (PW).

We typically add the muscle powers for both the proximal and distal
segments at a given joint which is in fact equal to the net joint torque
times the joint angular velocity (difference in the angular velocity
between the proximal and distal segments). Visit the
page (http://kwon3d.com/theory/jtorque/jen.html) at http://kwon3d.com
for details. Or

Pj = PFd + PWd + PFp + PWp, [1a]

where Pj = sum of the muscle powers at a joint, j = joint, d = distal
segment for a given joint, and p = paired proximal segment. PFd + PFp is
always 0 because if one segment gains energy, the other will lose energy
by the same amount due to the energy flow through the muscles. Thus Eq.
1a reduces to

Pj = PWd + PWp. [1b]

However, the sum of the PW's is not necessarily zero. If it is zero
there is only an energy flow from one segment to another through the
muscles and no work is done by the muscles. A positive value means that
a positive work was done by the muscles, vice versa. I don't quite
understand how the joint power can be equal to the muscle power at a
given joint regardless of whether the muscles are uniarticular or not.

For any movements, if the mechanical energy of the whole body changes,
it means this change is coming from the work done by the muscles.
Muscles convert chemical energy into mechanical energy to do a work. The
sum of the joint powers is always 0 because if one segment at a joint
loses mechanical energy, the other segment at the same joint will gain
the same amount through the joint and there is no net change in the
mechanical energy due to the energy flow through the joints.

>Leaving this aside, the question asked was how to interpret the
"components" of power in the different planes. I use the parenthesis
because power is a scalar (it is the time derivative of energy, another
scalar) and, unlike a vector, does not have components. The identity
that the scalar product of muscle force (F) and contraction velocity (v)
is equal to that of joint moment (M) and angular velocity (w) does not
imply the equivalent relationship for the individual components i.e.
>F.v = M.w
>can be expanded to
>Fx.vx + Fy.vy + Fz.vz = Mx.wx + My.wy + Mz.wz (1)
>but this does not imply that Fx.vx = Mx.wx and, indeed, this is
generally not the case.

For the reasons presented above, Fv = Mw is incorrect.

The main question in this discussion is perhaps how we interprete the 3D
muscle power. To me,

M dot w = Mx.wx + My.wy + Mz.wz 

is all right as long as the two vectors are transformed to a meaningful
local reference frame.

>Exactly what changes occur will depend on the mechanics of the entire
system and there is no a priori reason why contraction of a muscle in
the sagittal plane will lead only to changes in movement of segments in
the sagittal plane.

What's in Eq. 2 is the torque (moment), not the force. The effect of the
muscles in question in Eq. 2 is the torque produced by the muscles,
whether it is sagittal musclel or not. Dividing M dot w into three terms
(I intentionally try to avoid using the term 'component' here since this
term dictates people to think it as a vector component.) will simply
tell how the energy flow through the muscle and the energy generation by
the muscle are associated with the joint motion. If you stick to Pj in
Eq. 1, it only tells us how much energy (to be precise, the energy
generation rate) is generated in association with the joint motion about
which axis.

Likewise,

P = M dot w = M dot (w1 + w2 + w3)
= M dot w1 + M dot w2 + M dot w3 

is also all right to me, where w1, w2, and w3 are the three Kardanian
angular velocities. Again what we have in Eq 3 is not three vector
components, but three scalar terms. It still tells us how the work rate
is associated with the joint motion. Although the three angular velocity
vectors are not necessarily orthogonal, it is OK because what we want to
come up with is the three power terms, not three vector components.

In conclusion, I think the confusion is mainly coming from the fact that
we are trying to interpret the three power terms as vector components.
To me, both Eqs. 2 and 3 are legitimate in an effort to associate the
muscle energetics and the joint motion. The only question is which will
make more sense: to look at the power terms in terms of the three
orthogonal axes or three Kardanian rotation axes? I hope it made any
sense. Comment or criticism is welcome.

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

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