View Full Version : Re: 3D Joint Power

Richard Baker
01-26-2004, 11:17 AM
Ton, Jonas, subscribers

I'd like to express an opposing point of view to Ton's. I'm replying though
partly to try and sort out my own thoughts on what joint power actually is
in the light of the light of recent articles (principally those by Felix
Zajac, Rick Neptune and Steve Kautz, Gait and Posture 2002;16:215-232 and
2003;17:1-17) and I'd be most open to comment, criticism and correction of
anything I write.

Power is the increase or decrease in mechanical energy, that is kinetic,
(gravitational) potential and elastic energy. Most analyses ignore the
elastic energy so I'm happy to confine what I say here to kinetic and
potential energy. The first thing that interests us is where that energy
comes from? I think the unambiguous answer is that it comes from the
muscles. The second question of interest is where the energy goes to and I
think this is a source of some muddled thinking. The kinetic and potential
energy are those of the body segments. It is rarely stated but almost all
analyses assume that the joints are massless, and hence cannot possess
either kinetic or potential energy.

What Zajac et al. have really flagged up for us is that it is very
difficult to establish which segment gains the energy produced by any
specific muscle action. Every muscle action has the potential to contribute
energy (potential or kinetic) to any segment and a correspondence between
the "source" (muscle) and "destination" (segment) of the energy can only be
established if the system is considered as a whole. For example, under some
situations contraction of the soleus may accelerate the leg segment in
others it may raise the HAT segment. It is not possible to know which
without considering what is happening in all the other muscles (of both
lower limbs) and the position of the segments and their inertial
properties. Not only can any muscle contribute to the energy of any segment
but it can also do this by changing the height of its centre of mass or
changing the components of its velocity in any direction. 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.

So what can we know? If we know the velocity and position of the body
segments (and their inertial properties) then we can work out the total
mechanical energy of the system at any given time. If we know how the
length of every muscle is changing and the force it is generating then we
can work out the rate of change in the total mechanical energy contributed
to the system by each of the muscles (lets call this "muscle power").
Given measurement error and the effect of various assumptions in the
analysis (e.g. that we ignore elastic energy in the muscles) the change in
the total mechanical energy of the system will be equal to the work done by
all the muscles. We cannot, however,determine which muscle is contributing
to energy changes in any given segment without doing a far more complex
analysis which also requires a knowledge of the lines of action of all of
the muscles in the system, I think this can be seen as a modification of
Induced Acceleration Analysis (any comments?).

There is another approach to calculating the changes in the total
mechanical energy of the system. This is to work out the joint moments
using inverse dynamics and the joint angular velocities. The dot product of
these is the quantity we generally refer to as "joint power" (although I am
becoming increasingly convinced that this expression is highly misleading).
The sum of the joint powers for all of the joints also represents the total
rate of change of mechanical energy in the entire system and is thus equal
to this quantity as calculated by either of the two methods outlined in the
above paragraph.

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. Thus the stated aim of Jonas' analysis,
" I am ultimately using the joint power to get an estimate of the amount of
power that muscle-tendon units must generate or absorb at a joint" is not a
valid one.

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. (This is different to the relationship between moment and the
vector product of force and moment arm. As this is a relationship between
vectors, it can be broken up into an individual analysis of the components
in different directions.) I'd thus argue that it is not correct to separate
out the "components" of joint power as Ton recommends. I think this is
generally inappropriate but think that in this particular case where joint
powers and muscle powers are being compared it has the potential to be
extremely misleading.

I'd thus tip what Ton has said on its head. Rather than, "if you keep knee
and hip separate, you should also keep the degrees of freedom within each
joint separate", I suggest, particularly in looking at the relationships
between joint powers and muscle powers, that you cannot look at the hip in
isolation (because this will miss out the effects of bi-articular muscles
crossing both the hip and knee and knee and ankle) and that you cannot
consider the "components" of power separately (because it is a scalar and
does not have components).

The biggest question in my mind is why the association between "muscle
power" and "joint power" has persisted for so long in our collective
sub-conscious (including mine). Anyone like to defend it? (indeed has
anyone managed to read this far?!).


Richard Baker
Gait Analysis Service Manager, Royal Children's Hospital
Flemington Road, Parkville, Victoria 3052
Tel: +613 9345 5354, Fax +613 9345 5447

Adjunct Associate Professor, Physiotherapy, La Trobe University
Honorary Senior Fellow, Mechanical and Manufacturing Engineering, Melbourne

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