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 [2]

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 [3]

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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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 [2]

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 [3]

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

------------------------------------------------------

-----------------------------------------------------------------

To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl

For information and archives: http://isb.ri.ccf.org/biomch-l

Please consider posting your message to the Biomch-L Web-based

Discussion Forum: http://movement-analysis.com/biomch_l

-----------------------------------------------------------------