[From Bill Powers (2002.01.06.0536 KST)]
I'm sending this to CSGnet just to go on record.
If I were going to do a scientific study of arm movements, I would follow
a particular course that would lead from ignorance to understanding in a
systematic way.
First, I would have to ask how an arm would move if there were no muscle or
other internal torques acting on it at all. Suppose the arm were supported
in a forward horizontal position, straight out, the hand palm upward, and
released in the normal gravitational field, with the body held stationary.
The arm would begin to swing downward, the elbow joint probably bending
somewhat, and it would swing downward through the vertical and upward again
behind the body. After that it would swing forward and backward like a
compound pendulum, both shoulder angle and elbow angle changing with time,
and it would continue doing this indefinitely.
Of course the real arm will not behave this way, but that is the point. If
we knew how an arm would move with neither muscle nor other
internally-generated torques acting, then any deviations from this kind of
movement must be due to muscle and other torques. Of course in cases where
the arm might hit the body, we would study only the part of the movement
prior to the collision.
So the first step must be to understand exactly how the free-swinging arm
would move under whatever external influences are acting, with the muscles
and joints themselves producing no torques of any kind other than inertial.
Certain empirical facts can be of use here: the moments of inertia of the
arm segments, their physical dimensions of the segments, and the manner of
their linkages. The hands and wrist joints could be added with whatever
degree of detail proves important to consider. Also, we would need to know
the physical constraints on joint movement; the elbow is a hinge joint that
cannot hyperextend, and so on.
The one thing that _cannot_ be determined empirically is how the arm would
move with no internally generated torques (other than limits of motion)
created by muscles and joints. We cannot remove the dynamic and contractile
properties of the muscles or make the joints frictionless in a real human
arm. Even a totally-relaxed muscle has dynamic properties, and joints also
have properties we can't alter for the sake of baseline measurements. There
is only one way we can determine the baseline behavior of the arm, and that
is by the use of a physical model.
We could, of course, construct an artificial arm with the proper mass
distribution and joints, and measure its behavior under all the
circumstances of interest. But there is an easier way that would permit
investigating not just one arm, but any arm with any dimensions and any mix
of segment lengths. That is to calculate its behavior from basic physical
principles. If this is done correctly, the calculated arm movements can be
as close to those of a tangible physical model as we please.
I'm speaking, of course, of a computer simulation of the arm. A simulation
is done by evaluating the differential equations representing the physical
system over and over, as near to continuously as necessary to produce an
accurate picture of the behavior. The results can be communicated as tables
of numbers, as graphs, or as pictures of a moving arm with all the
positions and movements accurately portrayed.
Suppose we wish to compare the behavior of a real arm and the model arm in
a specific experimental situation. We could start with the condition
described above: palm up, arm straight and horizontal straight ahead. We
support the real arm in this position and at a certain time we suddenly
remove the supports and measure the joint angles at closely-spaced time
intervals. This will provide tables of numbers that represent what each
joint angle in the real arm did in the experiment.
Then we start the model in the same configuration and release it, producing
tables of numbers showing what the model's joint angles did, and we compare
the two tables of numbers. The differences contain the information needed
to reconstruct all muscle torques and joint frictions that were present in
the real arm and absent in the model.
What the real arm does will depend greatly on the instructions, which must
be described as part of the experimental situation. Suppose the instruction
is, "When the support is removed, keep your arm in the same position it is
in now, as closely as you can." After the release, the arm will probably
sag briefly, then come back close to its original configuration.
The model arm will simply drop and start swinging. The difference between
the real and model behavior will begin to appear almost -- but not quite --
immediately. In the first 100 milliseconds or so, the real arm and the
model arm will free-fall under the influence of gravity in exactly the same
way. But then the real arm will begin to decelerate relative to the model
arm; its downward swing will stop and then reverse, and it will return to
the original position. This relative (angular) acceleration can only be
produced by internally-generated torques acting at the joints of the real
arm.
We want to know what those internally-generated torques are at each
instant. There are two ways to determine them. One is to calculate the
angular accelerations, and by taking the inverse of the system of
differential equations, obtain the torques directly. The other is to
generate simulated torques at the joints, and adjust them as the model is
run over and over until the model arm behaves as exactly as possible like
the real arm. Both methods end up finding the inverse of the system
equations. The second method, however, can work when there are
nonlinearities and other complications that make an analytical solution
impossible.
When this much of the investigation is finished, we will have tables of
numbers showing the active torque at each joint as a function of joint
angle and time. These will be, in effect, _measurements_, because these
numbers will be what the torques MUST HAVE BEEN, by basic physical
principles, to produce the behavior that was observed.
The next step is to look for static and time-dependent relationships
between the torques and the joint angles. This will provide the information
needed to deduce the properties of the muscles and joint friction as
functions of joint angle. A muscle model can be built up from this kind of
information.
To deduce the role of reflexes, if any, we must compare the apparent
properties of muscle as deduced above with properties measured directly in
muscle preparations. For example, a muscle will have a certain passive
stiffness under constant neural-signal input. If the muscle stiffness
deduced by the above procedure is the same as what is measured in a valid
muscle preparation, then no _change_ in neural input signals is necessary.
to account for variations in muscle tensions. And this, in turn, would
rule out reflexes as having any role in creating the _apparent_ muscle
stiffness: the apparent and actual stiffness are the same.
On the other hand, if the deduced stiffness were larger than the measured
stiffness, or varied in a different way with joint angle, the only logical
conclusion would be that the neural signals entering the muscle vary with
joint angle. We could then deduce the dynamical equations relating the
joint angles to effective neural signals, and that would be a first
approximation to characterizing a reflex arc.
Once the reflexes were understood, we could look at other neural effects
such as the effects of different instructions, changes in task, and
adaptations, each step building on what was found in the previous step. The
whole procedure would be principled and systematic, with no random
experimentation just to see what inexplicable effect we might find next.
As I say, if I were interested in arm movements, this is how I would go
about studying them. Others can do as they please.
Best,
Bill P.
ยทยทยท
To: Isaac Kurtzer