View Full Version : Re: Power flow calculations using Inverse Dynamics

Ton Van Den Bogert
08-05-2003, 04:37 AM
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, what
> 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 difference
> 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 Winter's
> 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 co-contraction
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 a
> 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 velocities.
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

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