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unknown user
06-11-1991, 01:00 AM
Dear fellow biomechanicians,

Some of you will (I hope) answer Herman Woltring's call, and give
comments on practical use of net forces/moments/powers in functional
movement analysis. I want to reply to some technical statements made
by Herman. Most differences of opinion seem to originate from different
areas of application; my interest is in estimation of 'internal forces',
which need not be part of 'functional movement analysis'.

(1) Are joint powers physical quantities? In case of only monoarticular
muscles, yes. In that case one joint power equals the sum of several
muscle powers. When there are polyarticular muscles, the only equality
is: sum of all joint powers = sum of all muscle powers. A joint power
is then just another mathematical transformation of measured variables,
without physical meaning. It may have practical value though, but
others should comment on that.

An example. If you say that ankle power is very large just before take-
off in a vertical jump, you are restating what was observed: a large
distance between GRF and joint center, simultaneous with a large
extension velocity. The information was there already. But again, the
information might be more useful (for pattern recognition?) in the
transformed form; you get one variable instead of the original three:
GRF vector and point of application, and joint angular velocity.
Mechanical interpretation of joint power is dangerous: if you assume
that all ankle power is generated by ankle muscles, you are wrong
because the gastrocnemius muscle (even when it does not contract) can
transmit power from the knee extensors to the ankle joint.

Herman is very right that interpretation of physical joint forces is
also not straightforward. Very little is known on how cartilage and
bone react to specific loading patterns. But, you can not use
differentiation of force to get the 'jerk' if you do not estimate the
force in the first place.

(2) I agree with Herman, that the ICR (or 3D IHA) should ideally be part
of a biomechanical analysis. But using the ICR as moment reference
point requires high-quality kinematics and processing. It would be a
pity if a cheap and simple analysis would be considered below standard.
Especially if there is no good reason to prefer this difficult
transformation of measured variables over another.

(3) I apologize for using confusing terminology. It is indeed logical
to reserve the term 'net joint force' for the force obtained from 'net
kinetic analysis', i.e. models with one force and one moment transmitted
by each joint. In other types of analysis, which include estimation of
muscle forces, the use of just 'joint force' seems more appropriate.
When a joint is a complex (powerless) kinematic connection, this 'joint
force' can be the resultant vector of several forces, e.g. contact and
ligament forces. In that case, one might be tempted to add 'net'. To
avoid confusion, the term 'constraint force' or 'constraint reaction
force' from theoretical mechanics could be an alternative.

(4) My discussion on muscle moment arms was, admittedly, not properly
generalized to 3D. I had only joints in mind with one degree of freedom
(DOF). In that case, the IHA (instantaneous helical axis) depends only
on the joint angle, and muscles cannot change it. Within this limi-
tation the moment arm is still d(length)/d(angle). When for example,
the knee joint is part of a model, one must decide on the number of DOF.
If it is simplified to one DOF, the above theory applies. If the laxity
is an essential part of the analysis, more DOF are required. The
removal of kinematic constraints means that the corresponding constraint
forces are also lost (see below). This produces incorrect muscle
forces, because only the muscles are assumed to be responsible for the
observed movement, unless the actual physical constraints (the joint
ligaments) are added to the model. This shows that it is best to
reduce the degrees of freedom as much as possible in a dynamic analysis.

In a truly general 3D-theory, the concept of generalized coordinates
is convenient. If a joint has N degrees of freedom, you need N
variables (generalized coordinates) to specify the position of body 2
relative to body 1. There are many ways to define such variables, as
was shown by the '3D joint angles' debate some time ago on Biomch-L, but
the theory always applies. Each of the N generalized coordinates is
associated with a generalized force (the 'moments', if the coordinates
are angles). The relationship between physical forces and generalized
forces is linear, and the coefficients (the 'moment arms') can be found
using the principle of virtual work:
SUM(F.dr) = Q.dq (for all dq).
Where dq is a small change in the N-vector q of generalized coordinates, dr
is the resulting (3D-vector) change in position of the point of
application of each force vector F. Q is the N-vector of generalized
forces. Actually, the principle of virtual work defines Q. From this,
we find for each component Qi of Q:
Qi = SUM(F.(dr/dqi)) (the d's mean partial derivative here)
For a muscle, the direction of the vector F is exactly opposite to the
direction of lengthening, so F.(dr/dqi) = -|F|dL/dqi. For a ground
reaction force, the full vector equation must be used.

The conditions for static equilibrium are now simply: Q=0. Dynamic
equations of motion can also be formulated in generalized coordinates:
Q = M(q).q"
I am not familiar with the method to find the inertia matrix M, which
may depend in a complex way on q. My dynamics software (DADS) does not
use generalized coordinates but 'cartesian' coordinates, which are more
suitable for general-purpose software.

Finally, there are 6-N (in 3D) constraint force variables (the 'joint
force'). Examples: A ball-and-socket joint (the hip) has N=3, and the
constraint force is a 3D force vector. A universal joint (in Dutch:
"kruiskoppeling"), as used in machines, has N=2 because it does not
allow internal/external rotation. The 4 constraint reaction
forces are one 3D force vector, plus one torque. Note that joints can
also have translational degrees of freedom ('slider' joints), where the
corresponding qi is best measured in meters, not in radians. The
generalized theory still applies.

Enough of theoretical mechanics now, let's get back to more practical
matters!

-- Ton van den Bogert
University of Utrecht, Netherlands.