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unknown user
02-25-1990, 06:09 AM
Excerpt From "A Dynamic Parable"

"Mr. Cartesian was very unhappy. ... He had an invincible attachment to the
x,y,z, which he regarded as the "ne plus ultra" of dynamics. `Why will you
burden the science,' he sighs, `with all these additional names? Can you not
express what you want without talking about ... twists, and wrenches, ...
instantaneous screws, and all the rest of it?' `No,' said Mr. One-to-One,
`there can be no simpler way of stating the results than the natural method
we have followed. ... "We are dealing with questions of perfect generality,
and it would involve a sacrifice of generality were we to speak of the
movement of a body except as a twist, or a system of forces except as a
wrench.'"

Ball, R.S., 1900, "A Treatise on the Theory of Screws"
Cambridge University Press

I suppose in a debate one good parable deserves another. Hoffman's parable
on arrows is quite appropriate and its point (pun intended) supports my
position. Rigid body rotations, like arrows, are not vectors. Rotations,
but not arrows, are, as Ball states, twists (ie. screws or helices). My
objection to your proposal, Herman, is that you want to decompose a screw as
if it were a vector. The resulting components might be independent in the
vector sense, but they are not independent rotations. You can not obtain the
total rotation by performing the three component rotations obtained this way.

Is it possible I'm a stronger supporter of the screw axis than you? I
believe the component rotations must also be screws and follow the rules of
screw addition to produce the total screw. Here, again, I make an analogy
to the simpler case of particle displacements. The total displacement is a
vector, the components are vectors and follow the rules of vector addition
to produce the total displacement.

You referred to my analogies as beguiling, which Webster defines as "to lead
by deception". Am I deceptive when I teach my students they must specify the
independent coordinates (ie. establish the coordinate system) they plan to
use prior to solving a problem in particle kinematics? I think not. Nor is
it deceptive for me to argue the same practice for the description of rigid
body motions.

For both particle displacements and rigid body rotations the independent
degrees of freedom are specified using lines (or axes). In both cases one
degree-of-freedom is obtained by the cross-product of unit vectors along the
other two degrees-of-freedom. For particle displacements the three axes so
formed are mutually orthogonal and are either fixed to a point on the path
(path coordinates) or to a stationary system the motion is referred to. For
rigid body rotations one axis is fixed to each of the bodies whose relative
rotation are to be described and the third axis is there mutual
perpendicular.

It still surprises me that so little attention is paid to the physical axes
that correspond to the rotational degrees-of-freedom while so much attention
is paid to the axis used for particle displacements.

Your program PRP is a nice way to examine the differences between the various
Euler/Cardanic systems. I recommend the following question be asked whenever
it is used, "What physical axes correspond to the independent degrees-of-
freedom for each of the systems?" The limitations of electronic mail prevent
me showing figures. However, the figure in Goldstein's book "Classical
Mechanics" which is used to show the angles is excellent for this purposes.
All you need to do is also focus on the axes the rotations are performed
about. These axes are clearly shown in the figure and are perpendicular to
the plane of the rotation

I will try to describe the figure in words for those who do not have a copy
of the figure readily available. I will also mail a copy to any who send an
Email request to "Grood@UCBEH.SAN.UC.EDU". This figure shows two circular
planes which intersect along a common diameter. The normals to each plane
and the line formed by the intersection of the planes (line of nodes) are the
three axes which define the three independent degrees-of-freedom. In this
figure, one plane and its normal are in the fixed system while the other
plane and normal are in the moving system. The magnitude of the rotations
preformed about the body fixed axes are the angle between the line of nodes
and reference lines located in the plane of rotation for each body. The
reference lines are attached to the body and move with the rotation. The
rotation about the line of nodes is given by the angle between the two body
fixed axes.

The different Euler/Cardanic systems correspond to different orientations of
the planes in the fixed and moving bodies (ie. to different sets of three
independent degrees-of freedom). All of the different systems have the path
descriptive properties that you discussed in you last response.

Analogy time again. The different orientations of the planes produced by the
various Euler\Cardanic systems is equivalent to specifying different
Cartesian coordinate systems to characterize particle displacements. The
particle displacement produced by an x,y,z triple is invariant to changes in
sequence only when the same set of axes are always employed. It is not
invariant when both the sequence of the component displacements and the
orientation of the Cartesian system change.

The fact that each Euler/Cardanic system produce a different set of
independent coordinates is reflected by the fact that each set is acceptable
for use as generalized coordinates in writing the Lagrangian (or Hamiltonian)
for 3-D dynamical problems. Such generalized coordinates must have the
property of being independent. The solution of a single problem with
different conventions will produce different, but equivalent, descriptions
of the same motion. This is exactly what happens when a problem in particle
kinematics is solved using Cartesian systems with different orientations.

Having spent a lot of time on the important fundamentals of our disagreement,
I will only briefly address the issues of gimbal lock and Codman's paradox.
Gimbal lock is not a serious problem and only happens when the two body fixed
axes are both parallel and co-linear. This is a rare occurrence for
biological joints and can be easily avoided when analyzing experimental data.
If the presence of a singularity in an equation were reason to reject an
approach we would have to throw out a lot of good physics and mathematics.
Use of the total helical axis alone solves the problem by refusing to look
at the component rotations. Using the vector components of the helical axis
to describe the motion does not provide a set of rotations that yield the
total helical axis when the rotations are performed.

Codman's paradox, which you cite as an argument against my point a view I
cite as an argument for my point of view. The paradox exists independent of
how the motion is described. I like to express the paradox as follows. Why
does an abduction of the shoulder by 180 degrees followed by an extension of
180 degrees result in no net abduction or extension, but an external rotation
of 180 degrees instead? I propose the value of a system be judged by its
ability to explain the paradox.

If the system proposed by Fred Suntay and I for the knee is applied to the
shoulder the following explanation can be deduced. First I specify the
degrees-of-freedom for a right arm. The flexion axis is embedded in the
glenoid and points away from the body (unit vector e1). The internal
rotation axis is in the humerus and points proximally when the arm is by your
side (unit vector e3). The orientation of the abduction axis is obtained by
the cross-product of unit vectors along the flexion and internal rotation
axes, e2 = e1 x e3 / magnitude(e1 x e3) . The sense of the abduction axis
is obtained from the right hand rule and by taking the cross-product from the
flexion to the internal rotation axis. With the arm at your side it points
posteriorly. The sense of the abduction axis is also given by the sine of
the angle alpha between the flexion and internal rotation axes through the
standard formula for the cross-product of two vectors. At the starting
position alpha is 90 degrees and the sin(90)=+1.

The first part of Codman's path is an abduction of 180 degrees. The arm goes
from being at your side (palm toward the body) to straight overhead (palm
away from the body). As the shoulder is abducted the angle alpha increases
from its initial value of 90 degrees. As the arm passes through the 90
degree abduction position (arm straight out with palm down) alpha becomes
greater than 180 degrees, the sin(alpha) becomes negative, and there is a
reversal in the sense of the abduction axis which now points anteriorly.
Since the sense of the abduction axis is reversed, motion of the arm from the
90 degree abducted position to the straight overhead position is actually an
adduction of 90 degrees that takes the net adduction back to zero.

The sense reversal of the abduction axis also affects both flexion and
internal rotation. Since the abduction axis is a reference line for both
motions a change in its sense changes both motions by 180 degrees. (This can
be seen by changing the direction of either of a pair of lines that forms any
angle.) No net displacement of the arm occurs when this happens because the
flexion and internal rotation axes are parallel and the corresponding
physcial rotations are in opposite directions. Because the axis have
opposite sense, internal rotation points toward the body and flexion away,
the signs of the rotations are the same. Both decrease and the arm becomes
extended and externally rotated by 180 degrees in addition to the 90 degrees
abduction. As the arm is raised to the overhead position, abduction returns
to 0 leaving the arm in a position of 180 degrees extension and 180 degrees
external rotation. The 180 degrees of flexion which return the arm to the
side of the body reduce the net flexion to zero leaving only the 180 degrees
of external rotation. Thus the angles of the system I support predict the
net result of motion along Codman's path and therefore explain the paradox.

I will leave it as a mental exercise to show that at the end of the motion
the abduction axis points posteriorly as it did at the start of the motion.
Hint: The abduction axis is always perpendicular to both the flexion and
internal rotation axes because it is formed by cross-product of unit vectors
along these axes.

What always amazes me about the above explanation is the apparent
mathematical quirk which occurs when the abduction axis changes sense is
actually expressed by the physical motion of the arm.

Summarizing:

1. I support the use of the helical (screw) axis to describe the
rotational motions between body segments.

2. I believe the proper component rotations are also screws and can not be
expressed as vector like the arrows in Hoffman's tale. To quote Ball
"We are dealing with questions of perfect generality, and it would
involve a sacrifice of generality were we to speak of the movement of
a body except as a twist, or a system of forces except as a wrench."

3. I do not believe the existence of gimbal lock is a serious problem.

4. Codman's paradox can be easily explained by the use of the component
screw approach I support.

Herman, I also agree with you it is now time to hear from the community.

Edward S. Grood
Grood@UCBEH.SAN.UC.EDU
Noyes-Giannestras Biomechanics Laboratories
University of Cincinnati