EP Model -- Delphi version, revised -- again!

CSGnet Archive CSG2014
1

[From Bruce Abbott (2014.02.07.1535 EST)]

Previously I noted that the equations provided by Lan and Zhu (2007), who got them from St-Onge, Qi, and Feldman (1993), resulted in contraction of the extensor muscle when a downward load was applied to the forearm, whereas it should have resulted in contraction of the flexor muscle. I found that I could correct this problem by reversing the signs of certain computations. However, I suspected that the problem was somehow related to a difference in geometrical representation between the original model and my representation of it.

This intuition turns out to be correct. According to Pilon and Feldman (2006), in the EP model, joint angle increases as the joint opens, reaching 180 degrees when the arm is straight. My demo uses the way Delphi represents angles: clockwise rotations increase the angle, whereas counterclockwise rotations decrease it. Therefore the opening of the elbow joint angle is expressed as a decreasing angle. The zero-angle position of in Delphi occurs when the forearm is horizontal (with the pivot point to the left). The angle increases as the arm rises toward 90 degrees, which in the original EP model would be zero degrees. As the forearm falls below horizontal the angle becomes increasingly negative to a limit of -90 degrees, which in the original EP model corresponds to +180 degrees.

With this new understanding, I’ve revised the Delphi EP model once again. (I did say that the original version was preliminary!) The forearm now begins raised to the vertical position, which is now represented as zero degrees. As the forearm moves downward, joint angle increases to a maximum of 180 degrees (forearm lowered to the vertical position).

I’ve rescaled the graph to reflect the new joint-angle representation. As the forearm falls (opening the joint), the line on the graph representing joint angle rises. This can seem counterintuitive, because the representation of forearm movement and the graph of its movements move in opposite directions. I’ve done the graph this way so that its data can be compared directly to results shown on Lan and Zhu’s (2006) Figure 6. I’ve posted this revision on my website (https://sites.google.com/site/perceptualcontroldemos/home/other-demos ).

With this revision, all equations in the demo match those presented by Lan and Zhu (2006) and St-Onge, Qi, and Feldman (1993), so there can be no question that the demo represents this version of the EP model correctly.

Bruce

2

[From Rick Marken (2014.02.07.1540)]

Bruce Abbott (2014.02.07.1535 EST)--

...
With this new understanding, I've revised the Delphi EP model once again. (I
did say that the original version was preliminary!) The forearm now begins
raised to the vertical position, which is now represented as zero degrees.
As the forearm moves downward, joint angle increases to a maximum of 180
degrees (forearm lowered to the vertical position)...

With this revision, all equations in the demo match those presented by Lan
and Zhu (2006) and St-Onge, Qi, and Feldman (1993), so there can be no
question that the demo represents this version of the EP model correctly.

RM: So now it doesn't control.

I'm getting dizzy. I hope we stick with one of these. But I think
whatever the model turns out to be, we can say for sure that it either
controls, like the last version, so it's a PCT model, or it doesn't,
so it's irrelevant as a model of the behavior of living systems. Right
now it looks like it's the latter, which is what I thought all along
was true of EP models.

Best

Rick

···

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                   -- Bertrand Russell

3

[From Bruce Abbott (2014.02.07.1915 EST)]

Rick Marken (2014.02.07.1540) --

Bruce Abbott (2014.02.07.1535 EST)

...
With this new understanding, I've revised the Delphi EP model once
again. (I did say that the original version was preliminary!) The
forearm now begins raised to the vertical position, which is now

represented as zero degrees.

As the forearm moves downward, joint angle increases to a maximum of
180 degrees (forearm lowered to the vertical position)...

With this revision, all equations in the demo match those presented by
Lan and Zhu (2006) and St-Onge, Qi, and Feldman (1993), so there can
be no question that the demo represents this version of the EP model

correctly.

RM: So now it doesn't control.

BA: I hope you are basing that conclusion on the fact that the torques don't
appear to balance out at equilibrium. It's a bug in the program; see below.

BA: I'm getting dizzy. I hope we stick with one of these. But I think
whatever the model turns out to be, we can say for sure that it either
controls, like the last version, so it's a PCT model, or it doesn't, so it's
irrelevant as a model of the behavior of living systems. Right now it looks
like it's the latter, which is what I thought all along was true of EP
models.

BA: While changing the formula for computing net torque, I forgot to do the
same for the counter that displays net torque. I've corrected that and
(once again, sigh!) uploaded a corrected version. I hope this one finally
gets everything right . . .

Bruce

4

[From Rick Marken (2014.02.07.1655)]

Bruce Abbott (2014.02.07.1915 EST)--

RM: So now it doesn't control.

BA: I hope you are basing that conclusion on the fact that the torques don't
appear to balance out at equilibrium. It's a bug in the program; see below.

RM: No ,I was basing it on the fact that increasing the weight brings
the arm down and decreasing the weight allows it to go back up. The
new version behaves the same way: no control.

BA: While changing the formula for computing net torque, I forgot to do the
same for the counter that displays net torque. I've corrected that and
(once again, sigh!) uploaded a corrected version. I hope this one finally
gets everything right . . .

RM: All of them have their charms;-) I like the muscle length control
one (which I think of as version 2); that one worked quite well as a
control system. And the first versions as well as this new version
works well as equilibrium point systems; like all such systems, they
don't control.

Again, I think you've got great material for a paper no matter which
version of the model turns out to be the one intended by Feldman and
his cohorts. If it controls, then you can identify the controlled
variable(s) and show that what they call an EP model is a control of
perception (PCT) model; if it doesn't control you can show that such
models, while they appear to control if you just vary the equilibrium
point,(by varying R), really don't control, as can be demonstrated by
their failure to protect the controlled variable(s) from disturbance.

Best regards

Rick

···

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                   -- Bertrand Russell

5

[From Rick Marken (2014.02.07.2015 EST)]

Rick Marken (2014.02.07.1655) --

Bruce Abbott (2014.02.07.1915 EST)

RM: So now it doesn't control.

BA: I hope you are basing that conclusion on the fact that the torques
don't appear to balance out at equilibrium. It's a bug in the program; see

below.

RM: No, I was basing it on the fact that increasing the weight brings the
arm down and decreasing the weight allows it to go back up. The new version
behaves the same way: no control.

BA: Other than the change of scale, it works exactly like the previous
version (before reverting to the original equations of Lan and Zhu (2007).
The arm drops when the weight is added, but the drop becomes less and less
as you increase C, analogous to the gain of the system. You said that the
previous version DOES control.

So I'm confused as to your criteria for deciding whether a system does or
does not control. A low-gain negative feedback control system would behave
essentially the way the present model does when C is low.

Bruce

6

[From Rick Marken (2014.02.07.1840)]

Rick Marken (2014.02.07.2015 EST)--

I think we have a boundry issue here, Brice;-). I'm Rick Marken, not
you. So I think you meant to type Bruce Abbott (2014.02.07.2015 EST)--

RM: So now it doesn't control.

BA: I hope you are basing that conclusion on the fact that the torques
don't appear to balance out at equilibrium. It's a bug in the program; see

below.

RM: No, I was basing it on the fact that increasing the weight brings the
arm down and decreasing the weight allows it to go back up. The new version
behaves the same way: no control.

BA: Other than the change of scale, it works exactly like the previous
version (before reverting to the original equations of Lan and Zhu (2007).
The arm drops when the weight is added, but the drop becomes less and less
as you increase C, analogous to the gain of the system. You said that the
previous version DOES control.

RM: You're right. It does seem to control, though much more poorly
than the previous version. It seems to control perceptions of muscle
length, as before, so somewhere in the code the lambda*'s (controlled
perceptions) must be subtracted from the lambda's (references) (or
vice versa depending on the polarity of the connection of the output
torques to muscle lengths). So it looks like the story is that the EP
model is a control model that controls poorly. We'd have to figure out
why that is true -- why it controls so poorly.

BA: So I'm confused as to your criteria for deciding whether a system does or
does not control. A low-gain negative feedback control system would behave
essentially the way the present model does when C is low.

RM: Even when C is at it's maximum control is very poor. That's what
fooled me; the weight is still pretty effective at the highest level
of C; so I thought there was no control.. Maybe you could let C
increase to a level that would allow the system to control well. But
it does seem to control, even though the angle varies with weight;
when the weight is added, the flexor torque increases but, for some
reason, not quickly or strongly enough to compensate for the
disturbance the way a person would if someone applied a varying torque
force to the person 's hand while the person was controlling for
keeping the forearm at a certain fixed angle relative to the body.

So if this is the right version of the EP model, I think you can say
that it is a control model, that perceptions of muscle length are the
controlled variables and that it controls much more poorly that the
real system; and we have to find out why.

Best

Rick

···

Bruce

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                   -- Bertrand Russell

7

[From Bruce Abbott? (2014.02.08.0910 EST)]

Rick Marken (2014.02.07.1840) --

Rick Marken (2014.02.07.2015 EST)--

RM: I think we have a boundry issue here, Brice;-). I'm Rick Marken, not
you.

BA (or is it RM?): Hmm -- are you sure?

RM?: So I think you meant to type Bruce Abbott (2014.02.07.2015 EST)

BA?: Possibly -- or one of us is getting delusional. Had to happen
sometime, I suppose.

RM: So now it doesn't control.

BA: I hope you are basing that conclusion on the fact that the
torques don't appear to balance out at equilibrium. It's a bug in the
program; see below.

RM: No, I was basing it on the fact that increasing the weight brings
the arm down and decreasing the weight allows it to go back up. The
new version behaves the same way: no control.

BA: Other than the change of scale, it works exactly like the previous
version (before reverting to the original equations of Lan and Zhu (2007).
The arm drops when the weight is added, but the drop becomes less and
less as you increase C, analogous to the gain of the system. You said
that the previous version DOES control.

RM: You're right. It does seem to control, though much more poorly than the
previous version. It seems to control perceptions of muscle length, as
before, so somewhere in the code the lambda*'s (controlled
perceptions) must be subtracted from the lambda's (references) (or vice
versa depending on the polarity of the connection of the output torques to
muscle lengths). So it looks like the story is that the EP model is a
control model that controls poorly. We'd have to figure out why that is true
-- why it controls so poorly.

BA: So I'm confused as to your criteria for deciding whether a system
does or does not control. A low-gain negative feedback control system
would behave essentially the way the present model does when C is low.

RM: Even when C is at it's maximum control is very poor. That's what fooled
me; the weight is still pretty effective at the highest level of C; so I
thought there was no control.. Maybe you could let C increase to a level
that would allow the system to control well. But it does seem to control,
even though the angle varies with weight; when the weight is added, the
flexor torque increases but, for some reason, not quickly or strongly enough
to compensate for the disturbance the way a person would if someone applied
a varying torque force to the person 's hand while the person was
controlling for keeping the forearm at a certain fixed angle relative to the
body.

RM: So if this is the right version of the EP model, I think you can say
that it is a control model, that perceptions of muscle length are the
controlled variables and that it controls much more poorly that the real
system; and we have to find out why.

BA?: I think we've finally got it nailed. It seems to be reproducing the
data in Lan and Zhu (2007), Figure 6 and handles the effect of the added
weight correctly. The only important change from the previous iteration is
that the angles are now represented in a way that is consistent with the way
joint angles are reported in the literature. With the correct
representation, I was able to revert to the original equations as there is
no longer a need to adjust the equations for the difference in angle
representation.

BA?: Mark Latash sent me a paper, Gribble et al. (1998), that presents a
more fleshed-out version of the EP lambda model, based on physiological data
on such things as dynamic muscle stiffness and "the graded development of
muscle force due to calcium kinetics." The model presented includes both the
shoulder and elbow joints, constrained to motion in the horizontal plane,
and represents each of the six muscles involved in that motion. The paper is
entitled "Are complex control signals necessary for human arm movement?";
it's available for free on the web at
https://homes.cs.washington.edu/~todorov/courses/amath533/EPcomplex.pdf . If
I can figure out how to translate certain equations to their numerical
equivalents I may try to program this one as a next step, although perhaps
including only the elbow joint.

A thesis of the authors of this paper is that a complex pattern of control
signal changes, proposed as necessary by some researchers to account for
reaching data, can be replaced by a simple pattern once the true muscle
dynamics are simulated.

Bruce?

8

Hi both, I may be barking up the wrong tree here, but when there is a weight in hand, why should co-activating both muscles be necessary - why doesn't just the flexor muscle get the force it needs correct to keep the weight still? I can see why two muscles are necessary so that forces in either direction is acted upon and the angle of the arm can be restored once it has moved in one direction. But why do they both need to be co activated to maintain the position of an arm when the force is in one direction. Maybe this is what they actually do though in real life?
Warren

···

Sent from my iPhone

On 8 Feb 2014, at 14:12, Bruce Abbott <bbabbott@FRONTIER.COM> wrote:

[From Bruce Abbott? (2014.02.08.0910 EST)]

Rick Marken (2014.02.07.1840) --

Rick Marken (2014.02.07.2015 EST)--

RM: I think we have a boundry issue here, Brice;-). I'm Rick Marken, not
you.

BA (or is it RM?): Hmm -- are you sure?

RM?: So I think you meant to type Bruce Abbott (2014.02.07.2015 EST)

BA?: Possibly -- or one of us is getting delusional. Had to happen
sometime, I suppose.

RM: So now it doesn't control.

BA: I hope you are basing that conclusion on the fact that the
torques don't appear to balance out at equilibrium. It's a bug in the
program; see below.

RM: No, I was basing it on the fact that increasing the weight brings
the arm down and decreasing the weight allows it to go back up. The
new version behaves the same way: no control.

BA: Other than the change of scale, it works exactly like the previous
version (before reverting to the original equations of Lan and Zhu (2007).
The arm drops when the weight is added, but the drop becomes less and
less as you increase C, analogous to the gain of the system. You said
that the previous version DOES control.

RM: You're right. It does seem to control, though much more poorly than the
previous version. It seems to control perceptions of muscle length, as
before, so somewhere in the code the lambda*'s (controlled
perceptions) must be subtracted from the lambda's (references) (or vice
versa depending on the polarity of the connection of the output torques to
muscle lengths). So it looks like the story is that the EP model is a
control model that controls poorly. We'd have to figure out why that is true
-- why it controls so poorly.

BA: So I'm confused as to your criteria for deciding whether a system
does or does not control. A low-gain negative feedback control system
would behave essentially the way the present model does when C is low.

RM: Even when C is at it's maximum control is very poor. That's what fooled
me; the weight is still pretty effective at the highest level of C; so I
thought there was no control.. Maybe you could let C increase to a level
that would allow the system to control well. But it does seem to control,
even though the angle varies with weight; when the weight is added, the
flexor torque increases but, for some reason, not quickly or strongly enough
to compensate for the disturbance the way a person would if someone applied
a varying torque force to the person 's hand while the person was
controlling for keeping the forearm at a certain fixed angle relative to the
body.

RM: So if this is the right version of the EP model, I think you can say
that it is a control model, that perceptions of muscle length are the
controlled variables and that it controls much more poorly that the real
system; and we have to find out why.

BA?: I think we've finally got it nailed. It seems to be reproducing the
data in Lan and Zhu (2007), Figure 6 and handles the effect of the added
weight correctly. The only important change from the previous iteration is
that the angles are now represented in a way that is consistent with the way
joint angles are reported in the literature. With the correct
representation, I was able to revert to the original equations as there is
no longer a need to adjust the equations for the difference in angle
representation.

BA?: Mark Latash sent me a paper, Gribble et al. (1998), that presents a
more fleshed-out version of the EP lambda model, based on physiological data
on such things as dynamic muscle stiffness and "the graded development of
muscle force due to calcium kinetics." The model presented includes both the
shoulder and elbow joints, constrained to motion in the horizontal plane,
and represents each of the six muscles involved in that motion. The paper is
entitled "Are complex control signals necessary for human arm movement?";
it's available for free on the web at
https://homes.cs.washington.edu/~todorov/courses/amath533/EPcomplex.pdf . If
I can figure out how to translate certain equations to their numerical
equivalents I may try to program this one as a next step, although perhaps
including only the elbow joint.

A thesis of the authors of this paper is that a complex pattern of control
signal changes, proposed as necessary by some researchers to account for
reaching data, can be replaced by a simple pattern once the true muscle
dynamics are simulated.

Bruce?

9

[From Rick Marken (2014.02.08.0920)]

WM: Hi both, I may be barking up the wrong tree here, but when there is a weight in hand, why should co-activating both muscles be necessary - why doesn't just the flexor muscle get the force it needs correct to keep the weight still?

RM: That's one question. Another (possibly related) is "why is the
joint angle not moving back toward the reference angle after the
disturbance is applied"? You will notice that even when the weight is
just 1 kg, this weight disturbance causes the arm to move to a new
angle and remain there; there is not even the slightest sign of
resistance to the disturbance; no sign that the angle is being
returned to a reference state.

While the size of the deviation produced by a 1 kg weight disturbance
is smaller as "gain" (the value of C) increases, there is no evidence
that there is a system working to counter this small disturbance. The
decreased size of the deviation produced by a weight of 1 kg is
exactly what is expected if the same torque were applied to an
increasingly "stiff" joint. Compare this to what happens when limb
position is controlled; the 1 kg weight disturbance is almost
instantly countered, even when the gain of the control system is
relatively low.

What you see with a low gain (poor control) system after a 1 kg weight
is applied is the limb returning toward but not quite reaching the
original angle. There is no sign of this at all in the EP model at any
setting of "gain" (C). When you apply the weight the limb moves to a
new position and stays there; with higher "gain" the deviation
produced by the weight is smaller but it is still a fixed deviation.
This is exactly what is expected if one were applying a constant 1 kg
force to a stick anchored in the ground. If the stick is thin the
deviation produced by the constant force will be large but constant;
if the stick is thick, the deviation will be small but still constant.
This is what I think is going on with the EP model; increasing C is
equivalent to increasing the thickness of a stick in the ground. No
control at all.

So I retract my judgement that this "final" version of the EP model
controls. There is no behavioral evidence that the system controls;
the decreasing effect of a 1 kg weight disturbance to the limb is
exactly what is _expected_ if the weight disturbance is being applied
to a joint that is increasingly stiff.

I had said to Bruce:

RM: So if this is the right version of the EP model, I think you can say
that it is a control model, that perceptions of muscle length are the
controlled variables and that it controls much more poorly that the real
system; and we have to find out why.

And Bruce replied:

BA?: I think we've finally got it nailed.

I guess not. I believe I was wrong in concluding that the model is a
control model. This "final" version of the EP model is not controlling
at all. I was fooled by looking at the inner workings of the model,
which suggests that the flexor is increasing its upward torque in
response to the downward torque produced by the weight. It may be
doing that but this doesn't seem to be compensating for the
disturbance at all.

I think this may have to do with your point, Warren.There's something
funny going on with the flexor/extensor torques. First, both are
always positive. What's that about? And the increase in C ("gain")
leads to an increase in _both_ torques, suggesting that what C is
doing is not increasing the gain of a control system but increasing
the stiffness of the joint, in terms of the strength of the offsetting
torques. In other words, increasing C looks like it does something
equivalent to tensing your arm muscles.

But just based on the behavior of this model I think it's clear that
it doesn't control at all, in the sense the the observed effect of
disturbances are exactly what are _expected_ on physical grounds; a
disturbance is completely effective; there is no resistance to the
disturbance at all.

So I think we can conclude that the EP model can move the limb to
reference positions but it doesn't do this in a controlled manner. So
the EP model can't be considered a serious model of real limb
movement.

Best

Rick

It seems to be reproducing the

···

On Sat, Feb 8, 2014 at 8:05 AM, Warren Mansell <wmansell@gmail.com> wrote:

data in Lan and Zhu (2007), Figure 6 and handles the effect of the added
weight correctly.

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                   -- Bertrand Russell

10

Hi Rick, that's what I was thinking. I guess the plan would be, after programming the next threshold control model, to reproduce their anatomy exactly but use a PCT model instead and compare?
Warren

···

Sent from my iPhone

On 8 Feb 2014, at 17:17, Richard Marken <rsmarken@GMAIL.COM> wrote:

[From Rick Marken (2014.02.08.0920)]

On Sat, Feb 8, 2014 at 8:05 AM, Warren Mansell <wmansell@gmail.com> wrote:
WM: Hi both, I may be barking up the wrong tree here, but when there is a weight in hand, why should co-activating both muscles be necessary - why doesn't just the flexor muscle get the force it needs correct to keep the weight still?

RM: That's one question. Another (possibly related) is "why is the
joint angle not moving back toward the reference angle after the
disturbance is applied"? You will notice that even when the weight is
just 1 kg, this weight disturbance causes the arm to move to a new
angle and remain there; there is not even the slightest sign of
resistance to the disturbance; no sign that the angle is being
returned to a reference state.

While the size of the deviation produced by a 1 kg weight disturbance
is smaller as "gain" (the value of C) increases, there is no evidence
that there is a system working to counter this small disturbance. The
decreased size of the deviation produced by a weight of 1 kg is
exactly what is expected if the same torque were applied to an
increasingly "stiff" joint. Compare this to what happens when limb
position is controlled; the 1 kg weight disturbance is almost
instantly countered, even when the gain of the control system is
relatively low.

What you see with a low gain (poor control) system after a 1 kg weight
is applied is the limb returning toward but not quite reaching the
original angle. There is no sign of this at all in the EP model at any
setting of "gain" (C). When you apply the weight the limb moves to a
new position and stays there; with higher "gain" the deviation
produced by the weight is smaller but it is still a fixed deviation.
This is exactly what is expected if one were applying a constant 1 kg
force to a stick anchored in the ground. If the stick is thin the
deviation produced by the constant force will be large but constant;
if the stick is thick, the deviation will be small but still constant.
This is what I think is going on with the EP model; increasing C is
equivalent to increasing the thickness of a stick in the ground. No
control at all.

So I retract my judgement that this "final" version of the EP model
controls. There is no behavioral evidence that the system controls;
the decreasing effect of a 1 kg weight disturbance to the limb is
exactly what is _expected_ if the weight disturbance is being applied
to a joint that is increasingly stiff.

I had said to Bruce:

RM: So if this is the right version of the EP model, I think you can say
that it is a control model, that perceptions of muscle length are the
controlled variables and that it controls much more poorly that the real
system; and we have to find out why.

And Bruce replied:

BA?: I think we've finally got it nailed.

I guess not. I believe I was wrong in concluding that the model is a
control model. This "final" version of the EP model is not controlling
at all. I was fooled by looking at the inner workings of the model,
which suggests that the flexor is increasing its upward torque in
response to the downward torque produced by the weight. It may be
doing that but this doesn't seem to be compensating for the
disturbance at all.

I think this may have to do with your point, Warren.There's something
funny going on with the flexor/extensor torques. First, both are
always positive. What's that about? And the increase in C ("gain")
leads to an increase in _both_ torques, suggesting that what C is
doing is not increasing the gain of a control system but increasing
the stiffness of the joint, in terms of the strength of the offsetting
torques. In other words, increasing C looks like it does something
equivalent to tensing your arm muscles.

But just based on the behavior of this model I think it's clear that
it doesn't control at all, in the sense the the observed effect of
disturbances are exactly what are _expected_ on physical grounds; a
disturbance is completely effective; there is no resistance to the
disturbance at all.

So I think we can conclude that the EP model can move the limb to
reference positions but it doesn't do this in a controlled manner. So
the EP model can't be considered a serious model of real limb
movement.

Best

Rick

It seems to be reproducing the

data in Lan and Zhu (2007), Figure 6 and handles the effect of the added
weight correctly.

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                  -- Bertrand Russell

11

[From Rick Marken (2014.02.08.1050)]

WM: Hi Rick, that's what I was thinking. I guess the plan would be, after programming the next threshold control model, to reproduce their anatomy exactly but use a PCT model instead and compare?

RM: Yes, I agree. But I think the PCT model was already written by
Bruce. It was the second (or third, I've lost track) version of
Bruce's attempts to reproduce the EP model; it's the version of the
model that clearly controlled perceptions of muscle length (the
lambda*'s) relative to references for these lengths (the lambda's). If
Bruce can reproduce that model we can look and see why it controls and
the "correct" EP model doesn't.

Another interesting result of this modeling exercise is that it shows
that the muscle length control version of the EP model (the PCT
version) and the non-control (current) version of the model behave
exactly the same and can't be distinguished without doing the test for
the controlled variable (ie. applying varying weight -- torque--
disturbances to limb). Without disturbances, both models change their
angles smoothly in response to smooth changes in R. But in the control
version of the model, R determines the values of the lambda references
that "command" _input_ (desired muscle length); in the non-control
version these lambda "references" command outputs (flexor/extensor
torque).

Best

Rick

···

On Sat, Feb 8, 2014 at 10:29 AM, Warren Mansell <wmansell@gmail.com> wrote:

Warren

Sent from my iPhone

On 8 Feb 2014, at 17:17, Richard Marken <rsmarken@GMAIL.COM> wrote:

[From Rick Marken (2014.02.08.0920)]

On Sat, Feb 8, 2014 at 8:05 AM, Warren Mansell <wmansell@gmail.com> wrote:
WM: Hi both, I may be barking up the wrong tree here, but when there is a weight in hand, why should co-activating both muscles be necessary - why doesn't just the flexor muscle get the force it needs correct to keep the weight still?

RM: That's one question. Another (possibly related) is "why is the
joint angle not moving back toward the reference angle after the
disturbance is applied"? You will notice that even when the weight is
just 1 kg, this weight disturbance causes the arm to move to a new
angle and remain there; there is not even the slightest sign of
resistance to the disturbance; no sign that the angle is being
returned to a reference state.

While the size of the deviation produced by a 1 kg weight disturbance
is smaller as "gain" (the value of C) increases, there is no evidence
that there is a system working to counter this small disturbance. The
decreased size of the deviation produced by a weight of 1 kg is
exactly what is expected if the same torque were applied to an
increasingly "stiff" joint. Compare this to what happens when limb
position is controlled; the 1 kg weight disturbance is almost
instantly countered, even when the gain of the control system is
relatively low.

What you see with a low gain (poor control) system after a 1 kg weight
is applied is the limb returning toward but not quite reaching the
original angle. There is no sign of this at all in the EP model at any
setting of "gain" (C). When you apply the weight the limb moves to a
new position and stays there; with higher "gain" the deviation
produced by the weight is smaller but it is still a fixed deviation.
This is exactly what is expected if one were applying a constant 1 kg
force to a stick anchored in the ground. If the stick is thin the
deviation produced by the constant force will be large but constant;
if the stick is thick, the deviation will be small but still constant.
This is what I think is going on with the EP model; increasing C is
equivalent to increasing the thickness of a stick in the ground. No
control at all.

So I retract my judgement that this "final" version of the EP model
controls. There is no behavioral evidence that the system controls;
the decreasing effect of a 1 kg weight disturbance to the limb is
exactly what is _expected_ if the weight disturbance is being applied
to a joint that is increasingly stiff.

I had said to Bruce:

RM: So if this is the right version of the EP model, I think you can say
that it is a control model, that perceptions of muscle length are the
controlled variables and that it controls much more poorly that the real
system; and we have to find out why.

And Bruce replied:

BA?: I think we've finally got it nailed.

I guess not. I believe I was wrong in concluding that the model is a
control model. This "final" version of the EP model is not controlling
at all. I was fooled by looking at the inner workings of the model,
which suggests that the flexor is increasing its upward torque in
response to the downward torque produced by the weight. It may be
doing that but this doesn't seem to be compensating for the
disturbance at all.

I think this may have to do with your point, Warren.There's something
funny going on with the flexor/extensor torques. First, both are
always positive. What's that about? And the increase in C ("gain")
leads to an increase in _both_ torques, suggesting that what C is
doing is not increasing the gain of a control system but increasing
the stiffness of the joint, in terms of the strength of the offsetting
torques. In other words, increasing C looks like it does something
equivalent to tensing your arm muscles.

But just based on the behavior of this model I think it's clear that
it doesn't control at all, in the sense the the observed effect of
disturbances are exactly what are _expected_ on physical grounds; a
disturbance is completely effective; there is no resistance to the
disturbance at all.

So I think we can conclude that the EP model can move the limb to
reference positions but it doesn't do this in a controlled manner. So
the EP model can't be considered a serious model of real limb
movement.

Best

Rick

It seems to be reproducing the

data in Lan and Zhu (2007), Figure 6 and handles the effect of the added
weight correctly.

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                  -- Bertrand Russell

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                   -- Bertrand Russell

12

[Martin Taylor 2014.02.08.14.10]

[From Rick Marken (2014.02.08.0920)]

WM: Hi both, I may be barking up the wrong tree here, but when there is a weight in hand, why should co-activating both muscles be necessary - why doesn't just the flexor muscle get the force it needs correct to keep the weight still?

RM: ...what happens when limb
position is controlled; the 1 kg weight disturbance is almost
instantly countered, even when the gain of the control system is
relatively low.
What you see with a low gain (poor control) system after a 1 kg weight
is applied is the limb returning toward but not quite reaching the
original angle. There is no sign of this at all in the EP model at any
setting of "gain" (C).

You see the return after the initial move only if the output stage of the controller is an integrating stage. Not all control systems have an integrating output stage. A PID controller has position, integration, and derivative. The "standard" PCT controller has only "I". A P controller will exhibit different behaviour, as in: "You will notice that even when the weight is just 1 kg, this weight disturbance causes the arm to move to a new angle and remain there;". Is that new position what would be expected if both muscle tensions were left unchanged? If not, the system is probably acting like a "P" controller with a gain calculable from the G/(1+G) factor in the equations (which for a "P" system are the same as for the asymptotic values of the standard "I" controller). Is that G value the same for all disturbance weights?

I haven't looked at the program, but in the paper I didn't see any sign of integration. Did I miss it? Is there something in the actual physiology but not in the paper that integrates the error value?

The real question is whether it does what real arms do, not whether the model is a standard PCT controller.

Martin

···

On Sat, Feb 8, 2014 at 8:05 AM, Warren Mansell <wmansell@gmail.com> wrote:

13

[From Rick Marken (2014.02.08.1205)]

Martin Taylor (2014.02.08.14.10) --

MT: You see the return after the initial move only if the output stage of the
controller is an integrating stage.

RM: Not according to my simulations. After the onset of a step
disturbance, the variable controlled by a proportional (P) control
system immediately starts moving back to the reference state; how far
back towards the reference state it moves depends on gain but even a
low gain P controller will start moving back toward the reference
state immediately after the onset of the disturbance. If the EP model
were a P controller of arm angle we would see the trace for arm angle
start to move back toward the reference state immediately after
applying the weight disturbance. We don't see that at all. Indeed, the
response of the EP model to a step disturbance of weight is exactly
what is expected of a P controller with 0 gain, ie. the behavior of a
non-control system.

Best

Rick

···

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                   -- Bertrand Russell

14

[Martin Taylor 2014.02.08.16.46]

[From Rick Marken (2014.02.08.1205)]

Martin Taylor (2014.02.08.14.10) --
MT: You see the return after the initial move only if the output stage of the
controller is an integrating stage.

RM: Not according to my simulations.

If you did the simulation in a spreadsheet, could you send it to me? I'm not clear how to do it (in Libre Office) without introducing some kind of integration to prevent an exponentially growing oscillation with a cycle time of 2*loop-transport-lag.

Martin

15

Hi Martin

I’m visiting my granddaughter (and her mom and dad) in Seattle but I’ll try to get a copy of the spreadsheet to you ASAP. Best Rick

···

On Saturday, February 8, 2014, Martin Taylor mmt-csg@mmtaylor.net wrote:

[Martin Taylor 2014.02.08.16.46]

[From Rick Marken (2014.02.08.1205)]

Martin Taylor (2014.02.08.14.10) –

MT: You see the return after the initial move only if the output stage of the

controller is an integrating stage.
RM: Not according to my simulations.

If you did the simulation in a spreadsheet, could you send it to me? I’m not clear how to do it (in Libre Office) without introducing some kind of integration to prevent an exponentially growing oscillation with a cycle time of 2*loop-transport-lag.

Martin

–
Richard S. Marken PhD
www.mindreadings.com
The only thing that will redeem mankind is cooperation.

                                               -- Bertrand Russell
16

Hi Rick, I disagree. I think the PCT model should be built from first principles of PCT, not inferred from the mistaken workings of a completely different theory....
Warren

···

Sent from my iPhone

On 8 Feb 2014, at 18:48, Richard Marken <rsmarken@GMAIL.COM> wrote:

[From Rick Marken (2014.02.08.1050)]

On Sat, Feb 8, 2014 at 10:29 AM, Warren Mansell <wmansell@gmail.com> wrote:
WM: Hi Rick, that's what I was thinking. I guess the plan would be, after programming the next threshold control model, to reproduce their anatomy exactly but use a PCT model instead and compare?

RM: Yes, I agree. But I think the PCT model was already written by
Bruce. It was the second (or third, I've lost track) version of
Bruce's attempts to reproduce the EP model; it's the version of the
model that clearly controlled perceptions of muscle length (the
lambda*'s) relative to references for these lengths (the lambda's). If
Bruce can reproduce that model we can look and see why it controls and
the "correct" EP model doesn't.

Another interesting result of this modeling exercise is that it shows
that the muscle length control version of the EP model (the PCT
version) and the non-control (current) version of the model behave
exactly the same and can't be distinguished without doing the test for
the controlled variable (ie. applying varying weight -- torque--
disturbances to limb). Without disturbances, both models change their
angles smoothly in response to smooth changes in R. But in the control
version of the model, R determines the values of the lambda references
that "command" _input_ (desired muscle length); in the non-control
version these lambda "references" command outputs (flexor/extensor
torque).

Best

Rick

Warren

Sent from my iPhone

On 8 Feb 2014, at 17:17, Richard Marken <rsmarken@GMAIL.COM> wrote:

[From Rick Marken (2014.02.08.0920)]

On Sat, Feb 8, 2014 at 8:05 AM, Warren Mansell <wmansell@gmail.com> wrote:
WM: Hi both, I may be barking up the wrong tree here, but when there is a weight in hand, why should co-activating both muscles be necessary - why doesn't just the flexor muscle get the force it needs correct to keep the weight still?

RM: That's one question. Another (possibly related) is "why is the
joint angle not moving back toward the reference angle after the
disturbance is applied"? You will notice that even when the weight is
just 1 kg, this weight disturbance causes the arm to move to a new
angle and remain there; there is not even the slightest sign of
resistance to the disturbance; no sign that the angle is being
returned to a reference state.

While the size of the deviation produced by a 1 kg weight disturbance
is smaller as "gain" (the value of C) increases, there is no evidence
that there is a system working to counter this small disturbance. The
decreased size of the deviation produced by a weight of 1 kg is
exactly what is expected if the same torque were applied to an
increasingly "stiff" joint. Compare this to what happens when limb
position is controlled; the 1 kg weight disturbance is almost
instantly countered, even when the gain of the control system is
relatively low.

What you see with a low gain (poor control) system after a 1 kg weight
is applied is the limb returning toward but not quite reaching the
original angle. There is no sign of this at all in the EP model at any
setting of "gain" (C). When you apply the weight the limb moves to a
new position and stays there; with higher "gain" the deviation
produced by the weight is smaller but it is still a fixed deviation.
This is exactly what is expected if one were applying a constant 1 kg
force to a stick anchored in the ground. If the stick is thin the
deviation produced by the constant force will be large but constant;
if the stick is thick, the deviation will be small but still constant.
This is what I think is going on with the EP model; increasing C is
equivalent to increasing the thickness of a stick in the ground. No
control at all.

So I retract my judgement that this "final" version of the EP model
controls. There is no behavioral evidence that the system controls;
the decreasing effect of a 1 kg weight disturbance to the limb is
exactly what is _expected_ if the weight disturbance is being applied
to a joint that is increasingly stiff.

I had said to Bruce:

RM: So if this is the right version of the EP model, I think you can say
that it is a control model, that perceptions of muscle length are the
controlled variables and that it controls much more poorly that the real
system; and we have to find out why.

And Bruce replied:

BA?: I think we've finally got it nailed.

I guess not. I believe I was wrong in concluding that the model is a
control model. This "final" version of the EP model is not controlling
at all. I was fooled by looking at the inner workings of the model,
which suggests that the flexor is increasing its upward torque in
response to the downward torque produced by the weight. It may be
doing that but this doesn't seem to be compensating for the
disturbance at all.

I think this may have to do with your point, Warren.There's something
funny going on with the flexor/extensor torques. First, both are
always positive. What's that about? And the increase in C ("gain")
leads to an increase in _both_ torques, suggesting that what C is
doing is not increasing the gain of a control system but increasing
the stiffness of the joint, in terms of the strength of the offsetting
torques. In other words, increasing C looks like it does something
equivalent to tensing your arm muscles.

But just based on the behavior of this model I think it's clear that
it doesn't control at all, in the sense the the observed effect of
disturbances are exactly what are _expected_ on physical grounds; a
disturbance is completely effective; there is no resistance to the
disturbance at all.

So I think we can conclude that the EP model can move the limb to
reference positions but it doesn't do this in a controlled manner. So
the EP model can't be considered a serious model of real limb
movement.

Best

Rick

It seems to be reproducing the

data in Lan and Zhu (2007), Figure 6 and handles the effect of the added
weight correctly.

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                 -- Bertrand Russell

--
Richard S. Marken PhD
www.mindreadings.com

The only thing that will redeem mankind is cooperation.
                                                  -- Bertrand Russell

17

[From Bruce Abbott (2014.02.09.10:45 EST)]

···

-----Original Message-----
From: Control Systems Group Network (CSGnet)
[mailto:CSGNET@LISTSERV.ILLINOIS.EDU] On Behalf Of Warren Mansell
Sent: Saturday, February 08, 2014 11:05 AM
To: CSGNET@LISTSERV.ILLINOIS.EDU
Subject: Re: EP Model -- Delphi version, revised -- again!

WM: Hi both, I may be barking up the wrong tree here, but when there is a
weight in hand, why should co-activating both muscles be necessary - why
doesn't just the flexor muscle get the force it needs correct to keep the
weight still? I can see why two muscles are necessary so that forces in
either direction is acted upon and the angle of the arm can be restored once
it has moved in one direction. But why do they both need to be co activated
to maintain the position of an arm when the force is in one direction. Maybe
this is what they actually do though in real life?
Warren

BA: The muscles are a bit like rubber bands that become stiffer as they are
contracted. The increased stiffness is equivalent to raising the spring
constant of a spring, which relates the degree of the spring's extension
beyond its resting length to the restoring force the spring generates. By
co-contracting the flexor and extensor muscles, you increase stiffness
(resistance to being stretched).

In the EP model, the equilibrium point is the joint angle (really, the pair
of opposing muscle lengths) at which, under no-load conditions, neither
muscle is being contracted via motor neuron activity, and thus no torque is
being exerted by them on the joint. With the C (co-contraction) command at
zero, the R (reciprocal) command sets the threshold lengths at which each
muscle begins to recruit motor neuron impulses as it is stretched. In other
words, it sets the muscle length at which lengthening a muscle will begin to
activate the muscle stretch reflex.

Research on the elbow joint usually involves having the elbow rotating the
forearm in a horizontal plane while the forearm is resting on a support that
is pivoted at the elbow. The participant grabs and holds onto a vertical
handle located on the support. A torque is exerted by the support mechanism
that would tend to rotate the forearm in a given direction (clockwise or
counterclockwise) and the participant is instructed to hold the forearm at a
specified angle against the torque. (The actual angle of the joint is given
by a readout from the support.)

When this initial condition is realized, the torque is suddenly reduced.
The participant has been instructed not to attempt to hold the initial joint
angle, but to allow the joint angle to change as it will. The angle at which
the forearm stops is a measure of the equilibrium point.

If the torque working against the joint is reduced to zero, then the
equilibrium point will be that position at which both the extensor and
flexor muscles are relaxed. If when torque is being exerted externally on
the joint, it is the point at which the torque generated by the muscles
exactly opposes the external torque. By varying the external torque, one
can sweep out a function relating EP to external torque. Whether the flexor
or extensor muscle is involved in resisting the torque depends on whether
the torque is tending to rotate the joint counterclockwise or clockwise.

Increasing the magnitude of co-contraction has been shown to stiffen the
joint without changing the final angle -- R and C act independently. But as
C increases (and thus muscle stiffness), The joint becomes increasingly
resistant to being rotated by an external force. This is because the
recruitment of muscle fibers during stretching follows the "size principle":
motor neurons that have motor end-plates connected to relatively few muscle
fibers are activated under low levels of stretch, but as stretching
continues, motor neurons having progressively larger end-plate "fields" are
activated, producing larger and larger increments in muscle force. This is
represented in the Lan and Zhu (2007) model by an exponential function in
which the exponent is a function of the difference between the "dynamic
lambda" (a function of the current equilibrium point for that muscle,
specified in degrees, and the actual joint angle.) With greater
co-contraction, the force of muscle contraction ramps up faster, so the
muscle stretches less under a given external torque. Thus the stretch reflex
is activated more strongly, the muscle generates more force at a smaller
difference between commanded and actual angle, and the forearm droops less
when a weight is added that generates a torque against the elbow joint.

By the way, to allow the demo to conform to research findings based on
motion on the horizontal plane, I did not include a gravitational pull on
the forearm due to the weight of the arm.

A common situation in which co-contraction is quite noticeable is when one
walks on a slippery surface. By stiffening the hip and knee joints, one
reduces the change in leg angle that occurs when a foot begins to slip. If
you are not aware that the surface is about to get slippery (e.g., you
encounter a small patch of ice and don't notice it, the lower level of
co-contraction being maintained (because you do not fear slipping) may allow
the leg to slip out from under you, leading to a fall.

Bruce

18

Thank you Bruce, this is highly detailed and I am beginning to understand it better. I guess I am intrigued as to how much of the is so unequivocally within the physiology of the muscles that it is factual and how much is theory? In turn, I am intrigued as to whether a PCT model is has the same physiological model, an insufficient physiological model, or a different physiological model. To what degree are there different labels for the same thing (e.g. Lambas and reference values), and at which point might the contribution of various perceptual control loops as modelled in PCT improve control... Or not? I guess that might be for a bit later on?
Warren

···

Sent from my iPhone

On 9 Feb 2014, at 15:46, Bruce Abbott <bbabbott@FRONTIER.COM> wrote:

[From Bruce Abbott (2014.02.09.10:45 EST)]

-----Original Message-----
From: Control Systems Group Network (CSGnet)
[mailto:CSGNET@LISTSERV.ILLINOIS.EDU] On Behalf Of Warren Mansell
Sent: Saturday, February 08, 2014 11:05 AM
To: CSGNET@LISTSERV.ILLINOIS.EDU
Subject: Re: EP Model -- Delphi version, revised -- again!

WM: Hi both, I may be barking up the wrong tree here, but when there is a
weight in hand, why should co-activating both muscles be necessary - why
doesn't just the flexor muscle get the force it needs correct to keep the
weight still? I can see why two muscles are necessary so that forces in
either direction is acted upon and the angle of the arm can be restored once
it has moved in one direction. But why do they both need to be co activated
to maintain the position of an arm when the force is in one direction. Maybe
this is what they actually do though in real life?
Warren

BA: The muscles are a bit like rubber bands that become stiffer as they are
contracted. The increased stiffness is equivalent to raising the spring
constant of a spring, which relates the degree of the spring's extension
beyond its resting length to the restoring force the spring generates. By
co-contracting the flexor and extensor muscles, you increase stiffness
(resistance to being stretched).

In the EP model, the equilibrium point is the joint angle (really, the pair
of opposing muscle lengths) at which, under no-load conditions, neither
muscle is being contracted via motor neuron activity, and thus no torque is
being exerted by them on the joint. With the C (co-contraction) command at
zero, the R (reciprocal) command sets the threshold lengths at which each
muscle begins to recruit motor neuron impulses as it is stretched. In other
words, it sets the muscle length at which lengthening a muscle will begin to
activate the muscle stretch reflex.

Research on the elbow joint usually involves having the elbow rotating the
forearm in a horizontal plane while the forearm is resting on a support that
is pivoted at the elbow. The participant grabs and holds onto a vertical
handle located on the support. A torque is exerted by the support mechanism
that would tend to rotate the forearm in a given direction (clockwise or
counterclockwise) and the participant is instructed to hold the forearm at a
specified angle against the torque. (The actual angle of the joint is given
by a readout from the support.)

When this initial condition is realized, the torque is suddenly reduced.
The participant has been instructed not to attempt to hold the initial joint
angle, but to allow the joint angle to change as it will. The angle at which
the forearm stops is a measure of the equilibrium point.

If the torque working against the joint is reduced to zero, then the
equilibrium point will be that position at which both the extensor and
flexor muscles are relaxed. If when torque is being exerted externally on
the joint, it is the point at which the torque generated by the muscles
exactly opposes the external torque. By varying the external torque, one
can sweep out a function relating EP to external torque. Whether the flexor
or extensor muscle is involved in resisting the torque depends on whether
the torque is tending to rotate the joint counterclockwise or clockwise.

Increasing the magnitude of co-contraction has been shown to stiffen the
joint without changing the final angle -- R and C act independently. But as
C increases (and thus muscle stiffness), The joint becomes increasingly
resistant to being rotated by an external force. This is because the
recruitment of muscle fibers during stretching follows the "size principle":
motor neurons that have motor end-plates connected to relatively few muscle
fibers are activated under low levels of stretch, but as stretching
continues, motor neurons having progressively larger end-plate "fields" are
activated, producing larger and larger increments in muscle force. This is
represented in the Lan and Zhu (2007) model by an exponential function in
which the exponent is a function of the difference between the "dynamic
lambda" (a function of the current equilibrium point for that muscle,
specified in degrees, and the actual joint angle.) With greater
co-contraction, the force of muscle contraction ramps up faster, so the
muscle stretches less under a given external torque. Thus the stretch reflex
is activated more strongly, the muscle generates more force at a smaller
difference between commanded and actual angle, and the forearm droops less
when a weight is added that generates a torque against the elbow joint.

By the way, to allow the demo to conform to research findings based on
motion on the horizontal plane, I did not include a gravitational pull on
the forearm due to the weight of the arm.

A common situation in which co-contraction is quite noticeable is when one
walks on a slippery surface. By stiffening the hip and knee joints, one
reduces the change in leg angle that occurs when a foot begins to slip. If
you are not aware that the surface is about to get slippery (e.g., you
encounter a small patch of ice and don't notice it, the lower level of
co-contraction being maintained (because you do not fear slipping) may allow
the leg to slip out from under you, leading to a fall.

Bruce

19

I'm interested in this too. Are there counterparts to R and C in some of PCT models?
Adam

···

On Sun, Feb 9, 2014 at 6:56 PM, Warren Mansell <<mailto:wmansell@gmail.com>wmansell@gmail.com> wrote:

To what degree are there different labels for the same thing (e.g. Lambas and reference values).

20

[From Bruce Abbott (2014.02.09.1615 EST)]

···

-----Original Message-----
From: Control Systems Group Network (CSGnet)
[mailto:CSGNET@LISTSERV.ILLINOIS.EDU] On Behalf Of Warren Mansell
Sent: Sunday, February 09, 2014 12:57 PM
To: CSGNET@LISTSERV.ILLINOIS.EDU
Subject: Re: EP Model -- Delphi version, revised -- again!

WM: Thank you Bruce, this is highly detailed and I am beginning to
understand it better. I guess I am intrigued as to how much of the is so
unequivocally within the physiology of the muscles that it is factual and
how much is theory? In turn, I am intrigued as to whether a PCT model is has
the same physiological model, an insufficient physiological model, or a
different physiological model. To what degree are there different labels for
the same thing (e.g. Lambas and reference values), and at which point might
the contribution of various perceptual control loops as modelled in PCT
improve control... Or not? I guess that might be for a bit later on?
Warren

BA: All I know about this at present is what I've gleaned from a few papers
on the equilibrium point hypothesis -- and apparently there are competitor
models out there that propose somewhat different mechanisms. The EP
hypothesis (which Anatol Feldman has called an empirically established fact
in his more recent papers) is based on measuring electromyographic (EMG)
signals in the relevant muscles under experimental conditions such as the
one I outlined in my previous post, where the forearm reaches a position
after unloading in which the EMG is quiet or other positions depending on
the load remaining. I think the attempt has been made to model such things
as how the torque produced by a muscle varies with the length of the muscle,
the speed of its contraction, differences between muscles in their
cross-sections, changes in joint angle (which alter the mechanical advantage
of the muscle at its attachment points to the bones), rate of diffusion of
calcium to the muscle fibers within the muscle, and so on. Equations
representing these factors are derived from measurements made under various
experimental conditions both in vivo and in vitro. These equations simulate
the behavior of the muscle, in the same way that we employ, say, an output
function in a PCT model. Presumably a more detailed model would produce
these functions from the electro-mechanical properties of muscles, tendons,
neurons, and so on.

BA: Having not yet looked at competing models, I can't say which of these
relationships incorporated into a given EP model is an empirically
established and accepted relationship and which are theoretical.

BA: Bill presented a preliminary alternative model in his 1999 paper, "A
model of kinesthetically and visually controlled arm movement." This model
includes two control loops, the outer one controlling the perceived force
being generated by the muscle and the inner one the perceived length of the
muscle. Before commenting on the differences between Bill's model and the EP
model we've been exploring, I need to carefully review Bill's model.

BA: Bill apparently created a simulation based on his model, but I haven't
located a copy of it as yet; if it doesn't turn up I'll use the information
provided in Bill's paper to recreate it. This will be relatively easy
because I already have the EP model simulation, and to recreate Bill's model
will only need to substitute the relevant controller code -- no need to
change the graphical displays.

Bruce