[From Bill Powers (950606.1115 MDT)]

Hans Blom (950606)--

I have always taken it to be the task of the basic PCT controller

to bring about a best match between perceptions AT THE LOWEST

LEVEL, i.e. the intensities that directly impinge upon our sensors,

and internal reference levels. Now that might not necessarily be

true, but somehow I have always thought so.

This does explain some disconnects between us.

[Here follows an essay that is germane to the subject, and which has

also been requested by Dag Forssell as further explanation of the Little

Man v. 2 model. Dag can clip this essay and put it where he wants it.

This is called killing two birds with one stone.]

The hierarchical concept of control in HPCT (hierarchical PCT) is built

on the idea that perceptions of various levels of derivation from the

world are controlled by means of varying reference signals for sets of

lower-level control systems. Only at the lowest level do the control

system outputs produce forces that act on the physical environment.

I exploited this concept in the Little Man model. The lowest level of

control in this model is force control.

> ref force

>

--->[Comparator] ---

> >

Tendon Muscle

force tension (model of muscle)

> >

<--[local effects]<--

This control system contains a transport lag of about 10 milliseconds,

and the muscle model puts in a time constant of about 50 milliseconds

(the muscle model works through shortening a contractile element and

thus stretching a passive series spring). The closed-loop control has

the effect of making the tendon force very nearly a function of the

reference force alone, with the system time constant greatly reduced

below that of the muscle alone. This thus becomes an _angular

acceleration_ control system which makes angular acceleration very

nearly inversely proportional to the moment of inertia about the joint,

and equal to the reference-force setting.

The next level of control acts by varying the reference force, and

senses the angular velocity about the joint.

[start new page]

> reference angular velocity

>

----------->[Comparator] ---> second-order error signal

> >

> amplifier

sensed angular vel |

> > ref force

[rate of change] |

sensor --->[Comparator] ---

> > >

> Sensed tendon Muscle

> force tension

> > >

> <--[local effects]<-

> >

<---[velocity = f(torque)]<-

The sensed angular velocity is derived from the first derivative of the

muscle-spindle length signal. Since angular velocity of the arm about

the joint is essentially the integral of torque divided by moment of

inertia, we have a direct angular velocity control system at the second

level, which acts through a force control system that is very fast and

makes applied torque nearly proportional to the force reference signal.

Disturbances of tendon force due to friction, mechanical forces, and

inertial effects ("jerk") are removed by the first-level system, so the

second-level system sees a much more ideal environment that it would

without the first-level system.

The third level of control is an angular position control system, this

time using the proportional signal from muscle spindles and acting by

varying the second-level velocity reference signal:

[Start new page]

position reference signal

>

--------->[comparator] -->

>

amplifier

>

angular | reference angular velocity

position sensor V

----------->[Comparator] ---> second-order error signal

> >

> amplifier

sensed angular vel |

> > ref force

[rate of change] V

sensor --->[Comparator] ---

> > >

> Sensed tendon Muscle

> force tension

^ | | |

> <--[local effects]<-

> >

<---[velocity = f(torque)]<-

>

>

<-pos = f(velocity)<---

The third-level system now sees an environment that responds almost

ideally to a velocity reference signal. It senses angular position as

represented by muscle length, compares sensed muscle length with a

reference muscle length, and corrects the error by varying the second-

level velocity reference signal. Because the lower-level systems remove

system dynamics almost completely, the highest-level system sees a world

in which position follows the reference position with only a single

first-order lag, which requires no added frequency compensation.

Each degree of freedom of the 3-df arm is controlled by a three-level

system of this kind. The feedback functions shown as vel = f(torque) and

pos = f(velocity) are actually computed in a physical model of the arm

which receives torque inputs and returns angles, angular velocities, and

angular positions, recomputed for each degree of freedom on each

iteration of the model.

There is no special compensation for coriolis forces and other

interactions. This way of organizing control counteracts such

interactions as if they were external disturbances, and does so at the

appropriate level for the type of interaction. Direct force interactions

are countered at the force-control level; dynamic interactions like

coriolis forces due to conservation of angular momentum are countered at

the velocity control level. As a result, the position control level

experiences almost no dynamic disturbances as angular position follows

changes in the independent position reference signals for each degree of

freedom.

Actually this unconventional way of representing a three-level control

system exists anatomically collapsed into a very clever two-level system

with rate feedback providing damping. The entire model could be

represented as a single control system with suitably complex functions,

but by breaking out the various levels of control this way, we get a far

simpler picture of how the whole system works. Basically, every variable

is controlled at the lowest level possible, where control involves the

simplest possible loop. No individual control system becomes very

complex, even though the overall system is quite complex, involving

nonlinear second-order differential equations.

I should mention that the second level, velocity control, is not one of

the levels in the semi-official hierarchy. We have mulled over adding

this level, but never reached any final decision. Since this level can

be viewed as just adding damping to a position control system, perhaps

we should not assign a general level of this sort to the overall model.

But we made decide otherwise, depending on how many other low-level

systems turn out to have a similar distinguishable level of control.

There is a fourth level of control in the Little Man which receives

position perceptual signals, and in effect changes the coordinate system

in which control takes place. Where the three 3-level systems control in

coordinates centered on the various arm joints, this fourth level senses

and controls angular position in x angle, y angle, and radius with the

origin at the shoulder. Each system at this level also involves an

integrating output function, which makes overall resistance to sustained

constant disturbances (like gravity) much stronger. The result is to

remove ambiguities from the standpoint of higher-level systems. The

fourth level of control receives three reference signals, one specifying

lateral angle, another specifying vertical angle, and the last

specifying radial distance of the end of the arm from the shoulder (all

measures at this level are relative to the shoulder). So these reference

signals are the points of action available to still higher-level

systems.

If you now imagine all these four levels of control as a set of three

output functions providing reliable effects in x, y, and radius, we can

add still another level, a visual control level. I won't try to draw it;

the block diagram is essentially the same as the block diagram that can

be displayed in Little Man version 1. Through ray-tracing we locate the

tip of the arm (the "fingertip") on the retinas of two eyes, and also

the position of a movable target on the two retinas. We assume that

there is a visual input function that can report two sets of x-y

position signals, one set from each eye. The eyes and head are

controlled in x and y by two control systems each of which centers the

image of the target on each retina by moving both eyes and head. The

distance of the fingertip relative to the target is then computed from

the discrepany in the lateral fingertip-to-target position in the two

retinal images. The right eye alone is used to provide x and y signals

representing the lateral and vertical angular distance between the

fingertip image and the image of the target. Needless to say, the model

does NOT describe the machinery by which these signals are produced from

the retinal images.

So we have three visual perceptual signals representing x, y, and radial

distances between target and fingertip relative to the eyes. We can

supply three reference signals for the desired visual distances, and

route the three error signals through appropriate amplifiers to vary the

three reference signal inputs to the fourth-level kinesthetic systems

already discussed. The center of the visual coordinate system is

displaced from the center of the kinesthetic coordinate system (the

shoulder), but over the whole range of possible positions within a

normal 180-degree field of view, there are no ambiguities between these

two coordinate systems. So even though there are some rather strong

nonlinearities in the connection between the visual control systems and

the kinesthetic ones, the visual systems can bring the fingertip to the

target and make it track the target, moving the arm as necessary in its

three degrees of freedom. With, I should add, gravity turned either on

or off at any time.

We can touch on one final level of control, which in the model is

represented by a pattern generator rather than a complete control

system. On command, the x and y visual reference signals can be made

into slow sine and cosine waves (normally they are zero, meaning that

the fingertip is to be held _at_ the target position). This pair of

varying reference signals represents a still-higher system that is

perceiving and controlling some variable relationship between fingertip

and target. With the sine and cosine waves going, the fingertip

describes a continuous circle around the target, still tracking the

target movements in three dimensions. Obviously, by choosing reference

signal patterns other than a sine and cosine wave, and varying all three

reference signals, we could make the fingertip describe any desired

trajectory in three-dimensional space relative to a stationary or moving

target. Note that such trajectories are not simply an expression of arm

dynamics. They are arbitrary and independent of arm dynamics.

This model was meant primarily as a test of the control-system model as

a way of explaining how arm position control works. The visual control

level was added to show how hierarchical control is built up level by

level, and also as an antidote to certain published models of how

visually-controlled pointing works. To this extent, the model is a

success.

However, the model is unfinished and needs several refinements. One

needed refinement is a way of compensating for the difference in origins

of the visual and kinesthetic perceptual control systems. For slow

movements of the target this difference has no important effects, but

when the target jumps suddenly to a new position, the initial movement

of the fingertip can be in a severely wrong direction. That would be all

right if the human system did the same thing, but it doesn't. I tried to

install an adaptive map which converts a visual error into the correct

direction of change of the three kinesthetic reference signals, but this

map works only approximately and takes days of continuous running to

converge to a reasonable form. Also, this map lies on the output side of

the visual control systems, whereas it would be nicer to have it on the

input side, making the perception of visual space map onto the

perception of kinesthetic space. The mathematical problems in achieving

a simple and smooth-working adaptive map have been beyond me, and I have

left the model in this state hoping that someone else with more skill

could carry it further.

Even with such improvements, the current model would be far from a

definitive explanation of human motor behavior. But it is a start on a

new approach, and for those specifically interested in movement control

might serve as a first approximation for further development.

## ···

---------------------------------

OK, Hans, back to you -- but not in this post, which is very long

already. I hope this has given you a better idea of how hierarchical

control is treated in PCT, even though it covers only a narrow range of

behaviors.

-----------------------------------------------------------------------

Best,

Bill P.