View Full Version : -no subject-

08-10-2003, 11:20 AM
Dear Biomch-L Readers,

I thank everyone for spending the time to give quality response. Special thanks
to Ton van den Bogert for his very detailed response that answered almost all my
queries on the subject. I also agree that this topic is an extremely important
one for biomechanics, and can be the subject of confusion. So I do think that
subsequent questions should be addressed directly to Biomch-L and not myself so
that these matters are brought out into open discussion.

Thanks also to Karen Siegel and the others for their informative responses.

Note that in Zajac's article he referred to inverse solution as the Netwon-Euler
iterative method only. Other methods such as Lagrange's equations,
Newton-Lagrange multiplier method, etc are referred to as coupled dynamic
equations of motion. As I understand it, this is not a terminology that I
favour, but the principle behind differentiating the two approaches is

I look forward to further discussion on this topic.

Thank you.

Rene Ferdinands
Department of Physics &
Electronic Engineering
University of Waikato
Private Bag 3105
New Zealand

Rene Ferdinands formulated some interesting questions on power analysis
using inverse dynamics.

As moderator, I would like to suggest that this topic is of general
interest and quite suitable for a public discussion. So, please post your
responses directly to Biomch-L@nic.surfnet.nl.

Below follows my own response.

> (a) From the inverse dynamics solution of two or more coupled rigid bodies,
> would the value of just the net joint torque multiplied the corresponding
> segment angular velocity compute (i.e. just P = Tw)? Anything meaningful?

This quantity would represent the mechanical power delivered by a hypothetical
motor that drives the joint. If the joint is driven by muscles that span only
one joint, the quantity P = Tw therefore directly represents the total
mechanical power delivered by those muscles. If some of the driving muscles
are biarticular, muscle mechanical power is much harder to estimate. It will
require estimates of individual muscle forces and lengh changes.

The fact that any muscle, even a single-joint muscle, causes motion in
all other joints, is not relevant here. P = Tw is part of an inverse
analysis, and still accurately represents the power required at that one
joint. When doing an inverse analysis, the accelerations induced elsewhere will
also be measured and thus automatically be accounted for in the power
calculations for those other joints.

> (b) If the net joint torque (from inverse dynamics) multiplied by the
> in angular velocities of the adjacent segments was calculated would this
> satifactorily give the values of THE TOTAL SUM OF active muscle power flows in
> or out of the segment (i.e. P=T(w2-w1)). Is this always necessary or could
> sometimes P=Tw used?

If I understand your definition of these variables, the joint angular
velocity w is simply the relative rotation between the two segments (i.e. w =
w2-w1). So both equations would give the same value for joint power. I
do not have Winter's book here to confirm this definition.

Power flow (in and out of the segment) is a tricky concept. There is also
power flow due to the resultant joint force (P = F.v). Results will depend
on which reference frame is used to measure v. This was discussed on
Biomch-L some time ago in relation to analysis of treadmill locomotion. I
prefer to stay away from the concept of power flow. It can be useful but
only if interpreted carefully.

> (c) Can joint muscle power or absorption be calculated accurately using
> approach using the joint torques found from an inverse dynamics solution? Is
> this what Winter meant or did he mean as in (ii) above?

Apart from the issue of bi-articular muscles, yes, I think that net muscle
power generation and absorption is equal to the joint power measured from
inverse dynamics. Note that I use "net muscle power". If there is
of antagonistic muscles, there is simultaneous positive and negative muscle
power. Only the total net value is "seen" by the inverse dynamic analysis.

> Is the methodology for
> calculating power flows correct in Winter?

Again, without looking at Winter's book, I do not think that Winter proposed
calculating power flow. I think his analysis was limited to calculation of
joint power.

> (d) Can the power flow equations easily applied to the 3D case since power is
> scalar quantity?

If you mean joint power, yes. One way to do this is to model the joint as a
mechanism with three successive hinge joints. This is the classic Grood-Suntay
joint coordinate system (JCS) which gives cardanic angles from the kinematic
analysis. The time derivatives of these angles are the joint angular
If you use the same JCS to represent the joint moments, you will have a scalar
moment in in each of the three hinge joints and the power calculation
works exactly as in 2-D. You will get three joint powers, which can be added up
if you want the total for the entire cardanic joint complex. The three
individual joint powers can be interpreted as the power required for,
respectively, flexion, abduction, and rotation.

However, inverse dynamics software usually does not give joint moments
decomposed along the JCS axes. Instead, you may get Mx,My,Mz expressed
in an orthogonal segment-fixed XYZ reference frame, or in a global reference
frame. To use this for calculation of joint power, you need an angular
velocity vector expressed in the same reference frame. 3-D kinematic
analysis gives, at each time, a rotation matrix R that describes the relative
orientation between the two segments. The angular velocity vector w =
(wx,wy,wz) can be estimated from R and its time-derivative Rdot:

( 0 -wz wy )
( wz 0 -wx ) = inv(R)*Rdot
( -wy wx 0 )

Now joint power will be the dot product of M and w: P = Mx.wx + My.wy + Mz.wz.
You can interpret these three terms of the dot product as flexion, abduction,
and rotation power, as long as you are aware that these components do *not*
correspond to standard JCS definitions.

I worry a bit that some people may be mixing these two approaches and
multiply JCS angular velocities by the "corresponding" segment-fixed XYZ
components of the joint moment vector. This is only correct when the
JCS axes are aligned with the XYZ axes, i.e. when two of the three
joint rotations are zero.

The ISB has proposed standards for the reporting of joint motion (Wu
et al., J Biomech 35:543-548, 2002; http://www.isbweb.org/standards ) but has
not directly made recommendations on the reporting of joint moments. This
issue needs to be resolved. My suggestion would be to report joint moments
in the same JCS as joint motion, so that power calculations can be done as in my
first suggestion above. I would welcome some comments on this, especially from
the ISB standardization committee. Please change the "Subject" line when
responding to this standardization issue rather than to Rene Ferdinands'

> (e) How should power flows into or out of a segment be described taking into
> account statement (iv) by Zajac et al. Does this mean that P=T(w2-w1) give the
> net joint power which represents the SUMMED power by the net joint moment
> to/from ALL the segments?

I agree with Zajac's statement and I think this is one more reason why the
concept of "power flow into a segment" is not useful. Joint power itself is
still perfectly valid, I think, as long as it is interpreted with an awareness
of biarticular muscles and antagonistic co-contraction.


Ton van den Bogert


A.J. (Ton) van den Bogert, PhD
Department of Biomedical Engineering
Cleveland Clinic Foundation
9500 Euclid Avenue (ND-20)
Cleveland, OH 44195, USA
Phone/Fax: (216) 444-5566/9198



I think it might be easier to answer all your questions in a general way
rather than on a point-by-point basis because of the overlap across your

I agree with Zajac et al that coupled dynamics is the only way to estimate
the power delivered by a muscle to a segment. The P=Tw term reflects only a
portion of the energy transferred in or out of a segment by a given joint
moment. Coupled equations of motion must be used to calculate the
intersegmental forces and power associated with that joint moment, and this
term must be included when determining a joint moment's contribution to
segmental power. So if you wish to use Winter's segmental power equations,
the forces in the F*v terms must be obtained using coupled equations of
motion, not inverse dynamics.

If you want to study power but are only using inverse dynamics, you can
still calculate joint power P=T(w2-21). But as you noted, joint power
represents the net effect of the joint moment on the mechanical energy of
the whole body, not any one particular body segment. You can't use either
P=T(w2-21) or P=Tw to make any inferences about the effect of the joint
moment on the power of an individual segment. Our study using coupled
dynamics showed that a given joint moment can be associated with segmental
power that is several times larger or even opposite in sign than the joint
power (Gait & Posture, in press).

You are welcome to contact me if you would like to discuss this in more

Karen Lohmann Siegel, PT, MA
CDR, US Public Health Service

National Institutes of Health
Bldg 10, Rm 6s235
10 Center Drive, MSC 1604
Bethesda, MD 20892-1604

phone: 301-496-9890
fax: 301-480-9896
e-mail: karen_siegel@nih.gov
Dear Rene:

Determining mechanical power and work in human movement is discussed in
detail in a book entitled Kinetics of Human Motion by V. Zatsiorsky (Human
Kinetics, 2002). I hope that at least some of the questions that you have
raised are answered there.


Vladimir Zatsiorsky


First, I am an orthopedic surgeon, not a physicist, so you will not believe a
word I write.

You are obviously at point where you are questioning the accepted paradigms, as
there are conflicts.

Take a look at:

Zajac is right. It is impossible to contract only one muscle. (See some recent
work by Peter Huijling who shows that all muscles are connected). Biologic
structures are trusses (Thompson 1917, Gordon 1988). As in all trusses, there
are only tension and compression elements and zero joint moments. See my
website for a paper or two on the subject . Even
if you don't believe me, I would appreciate some constructive arguments.


Stephen M. Levin, MD
Potomac Back Center


To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl
For information and archives: http://isb.ri.ccf.org/biomch-l