[From Rick Marken (930311.0800)
Avery Andrews (931103.0928) --
Thank you for that very helpful explanation of control using the
jacobian transform matrix. I would just like to add some even
more naive comments about it just in case anyone is actually
interested -- because I think it a very important topic for PCTers.
First, I would like to emphsize that the Jacobian approach you
describe is CLOSED LOOP. So it can be thought of as one approach to
the control of perception. You mentioned this in your post when you
said:
Another thing we can do with the Jacobian is find out a bit about which
joint movements will be helpful in reducing the distance between the
perceived and reference locations of the tip. Suppose we have an
`error vector', pointing from the tip to the reference location.
If the Jacobian component for a joint is more or less in the same
direction as the error-vector, then flexion of the joint will reduce
the size of the error vector,
but it might have slipped by. Let me try to paraphrase what you are
saying; feel free to correct me if I have it wrong. The system you
describe is controlling the perceived x,y,z coordinates of the finger
tip. There is a reference position -- x',y',z' -- and the current
perceived position ---x,y,z. The difference between reference and
perception is the "error vector" that you mention above -- x'-x,
y'-y,z'-z. Now the problem is to turn the errors in this vector
into outputs (changes in limb positions) that will reduce the error.
That's what the Jacobian is for; it is a means of transforming error
into output based on the facts of arm kinematics (as defined by the
jacobian matrix). So there is an "output computation" based on the
jacobian -- but it produces just a slight change in the outputs that
affect the arm, resulting in a slight change in the perception of finger
tip position and a slight change in the error matrix which is used
to compute the next change in output via the jacobian.
This is a legitimate "control of perception" approach to building a
robot. The jacobian is actually acting as a complex "output
function" that transforms error DIRECTLY into environmental
effects that alter the perception that is being controlled. This
is NOT a pure output generation approach -- and it WILL PRODUCE
CONTROL -- ie. resist disturbances to the controlled variable; for
example, if you "push" on the finger tip or any of the arm components
that influence the perceived position of the fingertip this system will
compensate.
The problems that Avery is running into with this approach are a
bit esoteric for me -- but I think it has to do with the fact that
certain fingertip targets (reference settings) result in
outputs of the jacobian that specifiy physically incorrect or
mathematically impossible states of the arm components.
Avery points to one advantage of the PCT approach:
So we might regard the Little-Man style controller (and Bill Powers'
14df, if I understand it more or less correctly) as a kind of
clever transpose Jacobian scheme, the cleverness residing in a good
choice of coordinate systems (really, perceptual dimensions),
I think another way of saying this is that the success of the Little
Man stems, in part, from incorporating appropriate perceptions to
be controlled. But an important implication of this is that the PCT
model is HIERARCHICAL; the "error vector" in PCT specifies references
for LOWER LEVEL PERCEPTIONS -- not for environmetal outputs, as
in the jacobian approach. So it is not just selection of clever
perceptual dimensions that characterizes the PCT approach -- it is the
fact that errors ARE specifications for lower level perceptions (except
at the lowest level, of course, where errors "command" physical effects
like muscle fiber contractions or glandular secretions). In the
jacobian approach there is only one perception being controlled,
finger tip position in Cartesean coordinates. You could change the
jacobian approach by having it control the fingertip in polar
coordinates, for example. But that would not change the structure of
the model in any important way (though it MIGHT improve performance
in some situations); the jacobean would still transform error (now
indicating deviation from polar references) into outputs.
So I think another moral of the jacobean experiments is that
there is virtue in having higher level control systems control by
setting specifications for lower level perceptions. Bill Powers
has mentioned several principles from which this moral can also
be derived (convergent rather than divergent control functions,
experiential dependence of "higher order" perceptions on existence
of "lower order" perceptions [can't see configurations without
sensations], etc) -- I mention some of these in my "Hierarchical
behavior of perception" paper. But the jacobean experiments seem like
a very good practical demonstration of this moral.
Great work, Avery -- especially for a linguist!
Best
Rick