# Another Behavioral Illusion (was Re: the speed-curvature power law in humans and flies)

···

[From Rick Marken (2016.07.16.1415)]

There is a considerable body of research aimed at understanding the power law relationship between curvature and angular velocity that is observed when people (or other organisms) move (or move things , like pens) along curved trajectories. The power law can be expressed as follows:

V = K*Rb (1)

where V is angular velocity, defined as

V = (X2*Y2 )1/2 (2)

and R is curvature, defied as

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (3)

What equation (1) says is that when you are drawing a figure, such as an ellipse, for example, the speed of your movement at each instant (V) is proportional to the degree of curvature at that point ®; that is, you move faster through steep than shallow curves. According to Gribble and Ostry (Journal of Neurophysiology, v. 76, 1996) the value of the power coefficient, b, estimated under a “variety of experimental conditions” has been consistently found to be about 1/3.

Researchers have tried to explain the observed power relationship between V and R. These explanations have taken the form of open-loop, output generation models, such as the one described in Gribble and Ostry (1996, Fig. 1). The Gribble/Ostry model assumes that hand drawn figures, like ellipses, are the result of motor programs that send the appropriate signals to the muscles to generate the forces that move the hand in a elliptical pattern in such a way that there is a power relationship between the speed with which the hand moves and the curvature of the figure being drawn (an ellipse in this case). The Gribble/Ostry model successfully generates ellipses like those drawn by people and these ellipses show the power law relationship between V and R with a b coefficient of about 1/3. And, like people, the coefficient of the power law relationship remains about 1/3 whether the ellipse is drawn rapidly or slowly.

I developed a closed loop control model of ellipse drawing. It was an extremely simple model in the sense that there were no complex output calculations. The model simply produced outputs that kept a perception of the state of an input variable (qi.x, qi.y) matching elliptically varying references (r.x, r.y). When I computed the relationship between V and R for the ellipses drawn by the model I found that there was a perfect power relationship between V and R with a b coefficient value of exactly 1/3 (.33). This was a very puzzling result; here I was getting a very nice power relationship between V and R, just like what had been observed empirically and what had resulted from the output generation model of Gribble and Ostry, and I was getting it from what seemed like a trivially simple model. T**hen it struck me that this must mean that the observed power relationship between V and R could not possibly be a result of the way the ellipse was generated; it had to be a property of the ellipse itself. Which led me to take another look at the equations for V and R (equations 2 and 3 above). And I realized that it would be possible to combine the two equations and solve for V as a function of R. The result was:

V = K*R1/3 (4)

where K = |dXd2Y-d2XdY| . This is the power law relationship that has been observed to exist between V and R and the coefficient is 1/3, the same as the value of b which, according to Gribble and Ostry, is estimated under a “variety of experimental conditions”. So the relationship between V and R that is observed in movement studies simply reflects a mathematical relationship between these variables; it has nothing to do with how the movements are produced!!

The mathematical power relationship between angular velocity and curvature exists even if these variables are measured as angular velocity = A=V/R and curvature = C=1/R. The observed relationship between A and C is again a power relationship of the form:

A = K*Cb (5)

with b typically found to be approximately 2/3 rather than 1/3 as it is in the relationship between V and R. When you do the algebra that converts equation (4) into a form involving A and C you get the following:

A = K*C2/3 (6)

where K = D1/3 . Again, there is a power relationship between A and C as there is between V and R and the coefficient of this mathematical relationship is close to the empirically observed coefficient, 2/3.

This is a startling result. It means that students of the power law of movement control have been mistaking a mathematical fact for empirical evidence regarding how movement is produced. It is like doing research on how people draw circles and taking the observation that the circumference of the circles they draw are always close to being equal to p times the diameter as a fact that says something about how the circles are drawn.

So how could researchers in this area have missed the fact that there is a mathematical power relationship between V and R (with a power of 1/3) and between A and C (with a power of 2/3)? It can’t be because these researchers were not as good at math as I am; the papers I’ve been reading are bristling with high powered math and I had to get help from my math teacher son to derive equations 4 and 6. No, I think the reason I discovered this alarming fact and others didn’t is because the latter were looking at the behavior under study (drawing figures) through causal theory glasses while I was looking at the behavior through control theory glasses.

Through causal theory glasses, drawing an ellipse (for example) looks like a caused output – the result of precisely calibrated neural signals sent to the muscles; these signals are thought to create just the right forces over time so that the resulting hand movement is an ellipse. The observed power relationship between angular velocity and curvature is then taken as evidence of how the nervous system generates the neural signals that produce the ellipse; it generates signals so as to maintain a power relationship between speed of movement and the size of the curve through which the movement is being made.

But through control theory glasses, drawing an ellipse looks like a controlled (rather than a caused) result of muscle forces – a controlled variable. This means that the same result – an ellipse – cannot be consistently produced by the same neural signals (and resulting muscle forces), even if those signals are precisely calibrated. That is because the same neural signals will have different effects on the controlled result under difference circumstances. The different circumstances are the changing fatigue levels of the muscles that produce the result (ellipse), the changing conditions of the surface on which the ellipse is drawn, and so forth. These changing circumstances can be lumped together in my model of ellipse production as varying disturbances to the state of the controlled variable – the elliptical movement of the pen in the X and Y dimension, qi.x, qi.y.

The results of a simulation run with moderate level disturbances are should in the graphs below. The top graph shows the ellipse drawn by the two control systems, one controlling pen position in the X dimension (qi.x) and the other controlling pen position in the Y dimension (qi.y). The ellipse traced out is almost exactly the same as the ellipse specified by the time varying values of the reference signals, r.x and r.y, that are sent to the systems controlling qi.x and qi.y, respectively. This looked like a pretty trivial result when the ellipse was produced in a disturbance free environment. But in this case there were moderately intense disturbances present. So the outputs that produced this result were not elliptical, as can be seen in the next graph.

The graph below shows the output variations that produced the elliptical result above. The outputs are clearly not elliptical; they had to vary as necessary in order to compensate for disturbances that would have prevented production of the ellipse.

The chart below shows a plot of log V versus log R for the elliptical figure above (plot of pen movement; qi.x, qi.y). If there is a power law relationship between V and R it will show up as a good linear fit on a log-log plot. And indeed, there is an excellent linear fit (R2=.92) with a coefficient, corresponding to b of .32, almost exactly .33. This power law relationship was produced by outputs that result in a power law but one that is quite different than that for the ellipse.

The graph below shows the power law relationship between V and R for the output trace (o.x, o.y) that produced the ellipse (qi.x, qi.y). The fit to a power law is not as good for the outputs (o.x,o.y) as for the result of the controlled variable(qi.x,qi,y) but it is still a pretty good fit (R2 = .88). But the coefficient of the power relationship is not nearly as close to 1/3 as it was in the case of the ellipse. I believe this is because odd shaped curves, like those of the o.x,o.y) can result in variations in K that obscure the 1/3 power component of the relationship between V and R. The variations in K result from variations in dX, dY,d2X and d2Y since K = |dXd2Y-d2XdY|.

I think the most important lesson here is that you have to understand that behavior is a control process before you can start doing research to determine how this behavior is carried out. The problem with the velocity-curvature power law research is not that the researchers are using the wrong theory (although they are); the problem is that they are looking at the wrong phenomenon. They think they are studying caused output when, in fact, they are studying controlled input. You have to get the phenomenon right before you can get the theory right. In the case of the power law, the bias to see behavior as caused output seems to have blinded the researchers to the fact that they are seeing a mathematical law of curved lines rather than the psychophysiological law of movement that they think they were seeing. This is another version of what Powers called “The Behavioral Illusion” – the illusion of seeing causality where, in fact, there is control.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Hi Rick, I don’t understand this at all. How can there be a mathematical relationship between curvature and speed. It doesn’t make sense to me. Surely an ellipse canbe mathematically represented at any speed at any point. Maybe there is a relationship determined by physics, but maths is just a symbolic way of describing forces and processes in nature. Or am I missing something?

Warren

···

[From Rick Marken (2016.07.16.1415)]

There is a considerable body of research aimed at understanding the power law relationship between curvature and angular velocity that is observed w
hen people (or other organisms) move (or move things , like pens) along curved trajectories. The power law can be expressed as follows:

V = K*Rb (1)

where V is angular velocity, defined as

V = (X2*Y2 )1/2 (2)

and R is curvature, defied as

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (3)

What equatio
n (1) says is that when you are drawing a figure, such as an ellipse, for example, the speed of your movement at each instant (V) is proportional to the degree of curvature at that point ®; that is, you move faster through steep than shallow curves. According to Gribble and Ostry (Journal of Neurophysiology, v. 76, 1996) the value of the power coefficient, b, estimated under a “variety of experimental conditions” has been consistently found to be about 1/3.

Researchers have tried to explain the observed power relationship between V and R. These explanations have taken the form of open-loop, output generation models, such as the one described in Gribble and Ostry (1996, Fig. 1). The Gribble/Ostry model assumes that hand drawn figures, like ellipses, are the result of motor programs that send the appropriate signals to the muscles to generate the forces that move the hand in a elliptical pattern in such a way that there is a power relationship between the speed with which the hand moves and the curvature of the figure being drawn (an ellipse in this case). The Gribble/Ostry model successfully generates ellipses like those drawn by people and these ellipses show the power law relationship between V and R with a b coefficient of about 1/3. And, like people, the coefficient of the power law relationship remains about 1/3 whether the ellipse is drawn rapidly or slowly.

I developed a closed loop control model of ellipse drawing. It was an extremely simple model in the sense that there were no complex output calculations. The model simply produced outputs that kept a perception of the state of an input variable (qi.x, qi.y) matching elliptically varying references (r.x, r.y). When I computed the relationship between V and R for the ellipses drawn by the model I found that there was a perfect power relationship between V and R with a b coefficient value of exactly 1/3 (.33). This was a very puzzling result; here I was getting a very nice power relationship between V and R, just like what had been observed empirically and what had resulted from the output generation model of Gribble and Ostry, a
nd I was getting it from what seemed like a trivially simple model. T**hen it struck me that this must mean that the observed power relationship between V and R could not possibly be a result of the way the ellipse was generated; it had to be a property of the ellipse itself. Which led me to take another look at the equations for V and R (equations 2 and 3 above). And I realized that it would be possible to combine the two equations and solve for V as a function of R. The result was:

V = K*R1/3 (4)

<
/p>

where K = |dXd2Y-d2XdY| . This is the power law relationship that has been observed to exist between V and R and the coefficient is 1/3, the same as the value of b which, according to Gribble and Ostry, is estimated under a “variety of experimental conditions”. So the relationship between V and R that is observed in movement studies simply reflects a mathematical relationship between these variables; it has nothing to do with how the movements are produced!!

The mathematical power relationship between angular velocity and curvatur
e exists even if these variables are measured as angular velocity = A=V/R and curvature = C=1/R. The observed relationship between A and C is again a power relationship of the form:

A = K*Cb (5)

with b typically found to be approximately 2/3 rather than 1/3 as it is in the relationship between V and R. When you do the algebra that converts equation (4) into a form involving A and C you get the following:

A = K*C2/3 (6)

where K = D1/3 . Again, there is a power relationship between A and C as there is between V and R and the coefficient of this mathematical relationship is close to the empirically observed coefficient, 2/3.

This is a startling result. It means that students of the power law of movement control have been mistaking a mathematical fact for empirical evidence regarding how movement is produced. It is like doing research on how people draw circles and taking the observation that the circumference of the circles they draw are always close to being equal to p times the diameter as a fact that says something about how the circles are drawn.

So how could researchers in this area have missed the fact that there is a mathematical power relationship betw
een V and R (with a power of 1/3) and between A and C (with a power of 2/3)? It can’t be because these researchers were not as good at math as I am; the papers I’ve been reading are bristling with high powered math and I had to get help from my math teacher son to derive equations 4 and 6. No, I think the reason I discovered this alarming fact and others didn’t is because the latter were looking at the behavior under study (drawing figures) through causal theory glasses while I was looking at the behavior through control theory glasses.

Through causal theory glasses, drawing an ellipse (for example) looks like a caused output – the result of precisely calibrated neural signals sent to the muscles; these signals are thought to create just the right forces over time so that the resulting hand movement is an ellipse. The observed power relationship between angular velocity and curvature is then taken as evi
dence of how the nervous system generates the neural signals that produce the ellipse; it generates signals so as to maintain a power relationship between speed of movement and the size of the curve through which the movement is being made.

But through control theory glasses, drawing an ellipse looks like a controlled (rather than a caused) result of muscle forces – a controlled variable. This means that the same result – an ellipse – cannot be consistently produced by the same neural signals (and resulting muscle forces), even if those signals are precisely calibrated. That is because the same neural signals will have different effects on the controlled result under difference circumstances. The different circumstances are the changing fatigue levels of the muscles that produce the result (ellipse), the changing conditions of the surface on which the ellipse is drawn, and so forth. These changing circ
umstances can be lumped together in my model of ellipse production as varying disturbances to the state of the controlled variable – the elliptical movement of the pen in the X and Y dimension, qi.x, qi.y.

The results of a simulation run with moderate level disturbances are should in the graphs below. The top graph shows the ellipse drawn by the two control systems, one controlling pen position in the X dimension (qi.x) and the other controlling pen position in the Y dimension (qi.y). The ellipse traced out is almost exactly the same as the ellipse specified by the time varying values of the reference signals, r.x and r.y, that are sent to the systems controlling qi.x and qi.y, respectively. This looked like a pretty trivial result when the ellipse was produced in a disturbance free environment. But in this case there were moderately intense disturbances present. So the outputs that produced this result were not ell
iptical, as can be seen in the next graph.

<image.png>

The graph below shows the output variations that produced the elliptical result above. The outputs are clearly not elliptical; they had to vary as necessary in order to compensate for disturbances that would have prevented production of the ellipse.

<image.png>

The chart below shows a plot of log V versus log R for the elliptical figure above (plot of pen movement; qi.x, qi.y). If there is a power law relationship between V and R it will show up as a good linear fit on a log-log plot. And indeed, there is an excellent linear fit (R2=.92) with a coefficient, corresponding to b of .32, almost exactly .33. This power law relationship was produced by outputs that result in a power law but one that is quite different than that for the ellipse.

<image.png>

The graph below shows the power law relationship between V and R for the output trace (o.x, o.y) that produced the ellipse (qi.x, qi.y). The fit to a power law is not as good for the outputs (o.x,o.y) as for the result of the controlled variable(qi.x,qi,y) but it is still a pretty good fit (R2 = .88). But the coefficient of the power relatio
nship is not nearly as close to 1/3 as it was in the case of the ellipse. I believe this is because odd shaped curves, like those of the o.x,o.y) can result in variations in K that obscure the 1/3 power component of the relationship between V and R. The variations in K result from variations in dX, dY,d2X and d2Y since K = |dXd2Y-d2XdY|.

<image.png>

I think the most important lesson here is that you have to understand that behavior is a control process before you can start doing research to determine how this behavior is carried out. The problem with the velocity-curvature power law research is not that the researchers are using the wrong theory (although they are); the problem is that they are looking at the wrong phenomenon. They think they are studying caused output when, in fact, they are studying controlled input. You have to get the phenomenon right before yo
u can get the theory right. In the case of the power law, the bias to see behavior as caused output seems to have blinded the researchers to the fact that they are seeing a mathematical law of curved lines rather than the psychophysiological law of movement that they think they were seeing. This is another version of what Powers called “The Behavioral Illusion” – the illusion of seeing causality where, in fact, there is control.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

I am sorry, Rick, but the general equation for curvature, when combined with the equation for speed, does not give rise to the power law. Please revise your math.

A second reason (for those not versed in math) is a physical one: curvature is a geometric quantity (thus, only in space), whereas speed is a kinematic one (in space and time). The shape of a scribble in space does not say anything in principle about how it should be drawn in time.

That is why, by the way, why the power law constraint is so interesting biologically, because physics alone would not enforce it. So, something in the biology (be it the nervous system, be it evolution embedded as morphological computation, be it other things) constraints such basic degrees of freedom.

I know you think that PCT is the ultimate microscope, but those who don’t have it are not so stupid to work for 40 years on something and have missed a simple kindergarden calculation just because they do not think of control as PCT does. I am sorry. The paradigm shift will need to wait here.

And, again, the PCT demos you present do not generate the law, they simply mimic it from certain very particular conditions that prescribe the references, or distort with disturbances.

So, the challenge for PCT saying something new and useful and true about non-PCT motor control power-law researchers is still up for whoever wishes to join me.

Cheers,

Alex

···

On Sat, Jul 16, 2016 at 11:38 PM, Warren Mansell wmansell@gmail.com wrote:

Hi Rick, I don’t understand this at all. How can there be a mathematical relationship between curvature and speed. It doesn’t make sense to me. Surely an ellipse canbe mathematically represented at any speed at any point. Maybe there is a relationship determined by physics, but maths is just a symbolic way of describing forces and processes in nature. Or am I missing something?

Warren

On 16 Jul 2016, at 22:13, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.16.1415)]

There is a considerable body of research aimed at understanding the power law relationship between curvature and angular velocity that is observed w
hen people (or other organisms) move (or move things , like pens) along curved trajectories. The power law can be expressed as follows:

V = K*Rb (1)

where V is angular velocity, defined as

V = (X2*Y2 )1/2 (2)

and R is curvature, defied as

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (3)

What equatio
n (1) says is that when you are drawing a figure, such as an ellipse, for example, the speed of your movement at each instant (V) is proportional to the degree of curvature at that point (R); that is, you move faster through steep than shallow curves. According to Gribble and Ostry (Journal of Neurophysiology, v. 76, 1996) the value of the power coefficient, b, estimated under a “variety of experimental conditions” has been consistently found to be about 1/3.

Researchers have tried to explain the observed power relationship between V and R. These explanations have taken the form of open-loop, output generation models, such as the one described in Gribble and Ostry (1996, Fig. 1). The Gribble/Ostry model assumes that hand drawn figures, like ellipses, are the result of motor programs that send the appropriate signals to the muscles to generate the forces that move the hand in a elliptical pattern in such a way that there is a power relationship between the speed with which the hand moves and the curvature of the figure being drawn (an ellipse in this case). The Gribble/Ostry model successfully generates ellipses like those drawn by people and these ellipses show the power law relationship between V and R with a b coefficient of about 1/3. And, like people, the coefficient of the power law relationship remains about 1/3 whether the ellipse is drawn rapidly or slowly.

I developed a closed loop control model of ellipse drawing. It was an extremely simple model in the sense that there were no complex output calculations. The model simply produced outputs that kept a perception of the state of an input variable (qi.x, qi.y) matching elliptically varying references (r.x, r.y). When I computed the relationship between V and R for the ellipses drawn by the model I found that there was a perfect power relationship between V and R with a b coefficient value of exactly 1/3 (.33). This was a very puzzling result; here I was getting a very nice power relationship between V and R, just like what had been observed empirically and what had resulted from the output generation model of Gribble and Ostry, a
nd I was getting it from what seemed like a trivially simple model. T**hen it struck me that this must mean that the observed power relationship between V and R could not possibly be a result of the way the ellipse was generated; it had to be a property of the ellipse itself. Which led me to take another look at the equations for V and R (equations 2 and 3 above). And I realized that it would be possible to combine the two equations and solve for V as a function of R. The result was:

V = K*R1/3 (4)

<
/p>

where K = |dXd2Y-d2XdY| . This is the power law relationship that has been observed to exist between V and R and the coefficient is 1/3, the same as the value of b which, according to Gribble and Ostry, is estimated under a “variety of experimental conditions”. So the relationship between V and R that is observed in movement studies simply reflects a mathematical relationship between these variables; it has nothing to do with how the movements are produced!!

The mathematical power relationship between angular velocity and curvatur
e exists even if these variables are measured as angular velocity = A=V/R and curvature = C=1/R. The observed relationship between A and C is again a power relationship of the form:

A = K*Cb (5)

with b typically found to be approximately 2/3 rather than 1/3 as it is in the relationship between V and R. When you do the algebra that converts equation (4) into a form involving A and C you get the following:

A = K*C2/3 (6)

where K = D1/3 . Again, there is a power relationship between A and C as there is between V and R and the coefficient of this mathematical relationship is close to the empirically observed coefficient, 2/3.

This is a startling result. It means that students of the power law of movement control have been mistaking a mathematical fact for empirical evidence regarding how movement is produced. It is like doing research on how people draw circles and taking the observation that the circumference of the circles they draw are always close to being equal to p times the diameter as a fact that says something about how the circles are drawn.

So how could researchers in this area have missed the fact that there is a mathematical power relationship betw
een V and R (with a power of 1/3) and between A and C (with a power of 2/3)? It can’t be because these researchers were not as good at math as I am; the papers I’ve been reading are bristling with high powered math and I had to get help from my math teacher son to derive equations 4 and 6. No, I think the reason I discovered this alarming fact and others didn’t is because the latter were looking at the behavior under study (drawing figures) through causal theory glasses while I was looking at the behavior through control theory glasses.

Through causal theory glasses, drawing an ellipse (for example) looks like a caused output – the result of precisely calibrated neural signals sent to the muscles; these signals are thought to create just the right forces over time so that the resulting hand movement is an ellipse. The observed power relationship between angular velocity and curvature is then taken as evi
dence of how the nervous system generates the neural signals that produce the ellipse; it generates signals so as to maintain a power relationship between speed of movement and the size of the curve through which the movement is being made.

But through control theory glasses, drawing an ellipse looks like a controlled (rather than a caused) result of muscle forces – a controlled variable. This means that the same result – an ellipse – cannot be consistently produced by the same neural signals (and resulting muscle forces), even if those signals are precisely calibrated. That is because the same neural signals will have different effects on the controlled result under difference circumstances. The different circumstances are the changing fatigue levels of the muscles that produce the result (ellipse), the changing conditions of the surface on which the ellipse is drawn, and so forth. These changing circ
umstances can be lumped together in my model of ellipse production as varying disturbances to the state of the controlled variable – the elliptical movement of the pen in the X and Y dimension, qi.x, qi.y.

The results of a simulation run with moderate level disturbances are should in the graphs below. The top graph shows the ellipse drawn by the two control systems, one controlling pen position in the X dimension (qi.x) and the other controlling pen position in the Y dimension (qi.y). The ellipse traced out is almost exactly the same as the ellipse specified by the time varying values of the reference signals, r.x and r.y, that are sent to the systems controlling qi.x and qi.y, respectively. This looked like a pretty trivial result when the ellipse was produced in a disturbance free environment. But in this case there were moderately intense disturbances present. So the outputs that produced this result were not ell
iptical, as can be seen in the next graph.

<image.png>

The graph below shows the output variations that produced the elliptical result above. The outputs are clearly not elliptical; they had to vary as necessary in order to compensate for disturbances that would have prevented production of the ellipse.

<image.png>

The chart below shows a plot of log V versus log R for the elliptical figure above (plot of pen movement; qi.x, qi.y). If there is a power law relationship between V and R it will show up as a good linear fit on a log-log plot. And indeed, there is an excellent linear fit (R2=.92) with a coefficient, corresponding to b of .32, almost exactly .33. This power law relationship was produced by outputs that result in a power law but one that is quite different than that for the ellipse.

<image.png>

The graph below shows the power law relationship between V and R for the output trace (o.x, o.y) that produced the ellipse (qi.x, qi.y). The fit to a power law is not as good for the outputs (o.x,o.y) as for the result of the controlled variable(qi.x,qi,y) but it is still a pretty good fit (R2 = .88). But the coefficient of the power relatio
nship is not nearly as close to 1/3 as it was in the case of the ellipse. I believe this is because odd shaped curves, like those of the o.x,o.y) can result in variations in K that obscure the 1/3 power component of the relationship between V and R. The variations in K result from variations in dX, dY,d2X and d2Y since K = |dXd2Y-d2XdY|.

<image.png>

I think the most important lesson here is that you have to understand that behavior is a control process before you can start doing research to determine how this behavior is carried out. The problem with the velocity-curvature power law research is not that the researchers are using the wrong theory (although they are); the problem is that they are looking at the wrong phenomenon. They think they are studying caused output when, in fact, they are studying controlled input. You have to get the phenomenon right before yo
u can get the theory right. In the case of the power law, the bias to see behavior as caused output seems to have blinded the researchers to the fact that they are seeing a mathematical law of curved lines rather than the psychophysiological law of movement that they think they were seeing. This is another version of what Powers called “The Behavioral Illusion” – the illusion of seeing causality where, in fact, there is control.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2016.07.16.17.54]

``````Linear velocity has the dimensionality Lt^-1 (distance per unit
``````

time. Angular velocity has the dimension t^-1 , since angle has a
dimensionality distance/distance. Your first definition of V has
dimensionality the dimensionality of K times L^b, while your second
definition of V has the dimensionality L^2, which is not like either
version of velocity. Nor do the two definitions have the same
dimensionality unless K has dimensionality tL^(2-b). So to the
extent that anything that follows depends on V, it is unreliable.

``````It wasn't a puzzling result, as Alex pointed out (my time stamp for
``````

his message is 2016/07/13 4:06 AM, since Alex didn’t provide one).
Here’s a quote from the paper he posted: “One interesting
observation by Lacquaniti et al. (1983) was that the one-third
condition is satisfied if curved motion is produced by coupling two
orthogonal (i.e., x, y) sinusoids, independent of phase”, in other
words producing an ellipse. “Your reference trajectory and your
output trajectory were ellipses (in your case a Lissajous 2:1
figure, according to your diagram, but he also noted " it was later
shown that the law trivially holds for combinations of sinusoidal
patterns.”). The answer given to you was (and I may quote your “new”
discovery, since it is just what Alex had said):

``````True, as Alex had pointed out.

Nope. Not startling. From Alex's messages, the opposite is true.
``````

Unless I have totally misread him, he has been asking how, given the
mathematical relationship, it might be possible to move in ways that
are inconsistent with the geometrical law.

``````That might be difficult, if it is true that " it was later shown
``````

that the law trivially holds for combinations of sinusoidal
patterns." The reason it might be difficult is that ANY waveform in
x or in y can be linearly decomposed into a sum of sinusoidal
patterns, so the geometrical relationship holds for any reference
trajectory. The really interesting questions (interesting to me at
least) is why the power moves from 1/3 to 1/4 if the motion is in a
viscous medium, and is this change related to the slight bow in the
plotted data?

``````Or is the bow just due to the approximations involved when one
``````

computes a polynomial approximation to a transcendental function?
The power law is derived from such an approximation. As I suggested
in [Martin Taylor 2016.07.09.23.47] as a possible explanation, the
polynomial approximation (for perception) may have a lower degree in
the viscous case than in the free-running case. The same could be
true around sharp (small radius) curves as compared to long slow
curved parts of a trajectory.

``````Martin
``````
···

[From Rick Marken (2016.07.16.1415)]

``````      There is a considerable body of
``````

research aimed at understanding the power law relationship
between curvature and angular velocity that is observed when
people (or other organisms) move (or move things , like pens)
along curved trajectories. The power law can be expressed as
follows:

V = K*R b

``````  </sup>(1)
``````

where V is angular velocity, defined as

V = (X2*Y2 ) 1/2
(2)

I developed a closed
loop control model of ellipse drawing. It was an extremely
simple model in the sense that there were no complex
output calculations. The model simply produced outputs
that kept a perception of the state of an input variable
(qi.x, qi.y) matching elliptically varying references
(r.x, r.y). When I computed the relationship between V and
R for the ellipses drawn by the model I found that there
was a perfect power relationship between V and R with a b coefficient value of exactly
1/3 (.33). This was a very puzzling result;

* that the observed power
relationship between V and R could not possibly be a
result of the way the ellipse was generated*; * it
had to be a property of the ellipse itself.
*

``````        So the relationship between
``````

V and R that is observed in movement studies simply reflects
a mathematical relationship between these variables; it has
nothing to do with how the movements are produced!!

• ``````           This is a startling result. It means
``````

that students of the power law of movement control have
been mistaking a mathematical fact for empirical
evidence regarding how movement is produced.*

[Martin Taylor 2016.07.16.18.34]

[Martin Taylor 2016.07.16.17.54]

... The really interesting questions (interesting to me at least) is why the power moves from 1/3 to 1/4 if the motion is in a viscous medium, and is this change related to the slight bow in the plotted data?

Or is the bow just due to the approximations involved when one computes a polynomial approximation to a transcendental function? The power law is derived from such an approximation. As I suggested in [Martin Taylor 2016.07.09.23.47] as a possible explanation, the polynomial approximation (for perception) may have a lower degree in the viscous case than in the free-running case. The same could be true around sharp (small radius) curves as compared to long slow curved parts of a trajectory.

Martin

I ought perhaps to have reiterated that the implication is that the controller is perceiving the target as being a little further around the curve than the current position of the pen, with the assumption that there is at least one lower-level controller that produces linear motion in a reference direction supplied by the one controlling for following the track. I don't have a functioning model or a dynamic analysis so I can't say whether this would work, but at the moment something like that seems plausible to me. When the curve is sharp or the medium viscous (allowing for more abrupt changes of direction at any speed), the look-ahead would be shorter, and the polynomial approximation to the transcendental geometry would be closer. One might guess that reorganization would reduce the drawing error at any speed as one practiced the task. Anecdotally, if you watch a good artist drawing precisely designed swirls and arcs, it is hard to imagine it being done by a novice at that speed.

Martin

[From Rick Marken (2016.07.16.1550)]

···

On Sat, Jul 16, 2016 at 2:38 PM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Rick, I don’t understand this at all. How can there be a mathematical relationship between curvature and speed.

RM: Here’s my derivation (V = speed, R = curvature):

V = (X2*Y2 )1/2

``````                       </sup>(1)
``````

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (2)

V2 = (X2*Y2 ) (3)

Substituting (3)
into the numerator of (2) and setting |dXd2Y-d2XdY| = D we get

R = [(V2)3/2]/D (4)

R = V3//D (5)

Rearranging terms we
get:

V = DR1/3 (6)

WM: It doesn’t make sense to me. Surely an ellipse can be mathematically represented at any speed at any point. Maybe there is a relationship determined by physics, but maths is just a symbolic way of describing forces and processes in nature. Or am I missing something?

RM: Does the above derivation help? And here is the log R vs log V plot for a perfect ellipse:

RM: Looks to me like equation 6 is a pretty good description of the relationship between R and V for an ellipse. The results of the log-log regression are exactly the same regardless of the frequency with which the ellipse is drawn, by the way.

RM: And aren’t you up kind of late;-)

Best

Rick

Warren

On 16 Jul 2016, at 22:13, Richard Marken rsmarken@gmail.com wrote:

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.07.16.1415)]

There is a considerable body of research aimed at understanding the power law relationship between curvature and angular velocity that is observed when people (or other organisms) move (or move things , like pens) along curved trajectories. The power law can be expressed as follows:

V = K*Rb (1)

where V is angular velocity, defined as

V = (X2*Y2 )1/2 (2)

and R is curvature, defied as

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (3)

What equation (1) says is that when you are drawing a figure, such as an ellipse, for example, the speed of your movement at each instant (V) is proportional to the degree of curvature at that point (R); that is, you move faster through steep than shallow curves. According to Gribble and Ostry (Journal of Neurophysiology, v. 76, 1996) the value of the power coefficient, b, estimated under a “variety of experimental conditions” has been consistently found to be about 1/3.

Researchers have tried to explain the observed power relationship between V and R. These explanations have taken the form of open-loop, output generation models, such as the one described in Gribble and Ostry (1996, Fig. 1). The Gribble/Ostry model assumes that hand drawn figures, like ellipses, are the result of motor programs that send the appropriate signals to the muscles to generate the forces that move the hand in a elliptical pattern in such a way that there is a power relationship between the speed with which the hand moves and the curvature of the figure being drawn (an ellipse in this case). The Gribble/Ostry model successfully generates ellipses like those drawn by people and these ellipses show the power law relationship between V and R with a b coefficient of about 1/3. And, like people, the coefficient of the power law relationship remains about 1/3 whether the ellipse is drawn rapidly or slowly.

I developed a closed loop control model of ellipse drawing. It was an extremely simple model in the sense that there were no complex output calculations. The model simply produced outputs that kept a perception of the state of an input variable (qi.x, qi.y) matching elliptically varying references (r.x, r.y). When I computed the relationship between V and R for the ellipses drawn by the model I found that there was a perfect power relationship between V and R with a b coefficient value of exactly 1/3 (.33). This was a very puzzling result; here I was getting a very nice power relationship between V and R, just like what had been observed empirically and what had resulted from the output generation model of Gribble and Ostry, and I was getting it from what seemed like a trivially simple model. T**hen it struck me that this must mean that the observed power relationship between V and R could not possibly be a result of the way the ellipse was generated; it had to be a property of the ellipse itself. Which led me to take another look at the equations for V and R (equations 2 and 3 above). And I realized that it would be possible to combine the two equations and solve for V as a function of R. The result was:

V = K*R1/3 (4)

where K = |dXd2Y-d2XdY| . This is the power law relationship that has been observed to exist between V and R and the coefficient is 1/3, the same as the value of b which, according to Gribble and Ostry, is estimated under a “variety of experimental conditions”. So the relationship between V and R that is observed in movement studies simply reflects a mathematical relationship between these variables; it has nothing to do with how the movements are produced!!

The mathematical power relationship between angular velocity and curvature exists even if these variables are measured as angular velocity = A=V/R and curvature = C=1/R. The observed relationship between A and C is again a power relationship of the form:

A = K*Cb (5)

with b typically found to be approximately 2/3 rather than 1/3 as it is in the relationship between V and R. When you do the algebra that converts equation (4) into a form involving A and C you get the following:

A = K*C2/3 (6)

where K = D1/3 . Again, there is a power relationship between A and C as there is between V and R and the coefficient of this mathematical relationship is close to the empirically observed coefficient, 2/3.

This is a startling result. It means that students of the power law of movement control have been mistaking a mathematical fact for empirical evidence regarding how movement is produced. It is like doing research on how people draw circles and taking the observation that the circumference of the circles they draw are always close to being equal to p times the diameter as a fact that says something about how the circles are drawn.

So how could researchers in this area have missed the fact that there is a mathematical power relationship between V and R (with a power of 1/3) and between A and C (with a power of 2/3)? It can’t be because these researchers were not as good at math as I am; the papers I’ve been reading are bristling with high powered math and I had to get help from my math teacher son to derive equations 4 and 6. No, I think the reason I discovered this alarming fact and others didn’t is because the latter were looking at the behavior under study (drawing figures) through causal theory glasses while I was looking at the behavior through control theory glasses.

Through causal theory glasses, drawing an ellipse (for example) looks like a caused output – the result of precisely calibrated neural signals sent to the muscles; these signals are thought to create just the right forces over time so that the resulting hand movement is an ellipse. The observed power relationship between angular velocity and curvature is then taken as evidence of how the nervous system generates the neural signals that produce the ellipse; it generates signals so as to maintain a power relationship between speed of movement and the size of the curve through which the movement is being made.

But through control theory glasses, drawing an ellipse looks like a controlled (rather than a caused) result of muscle forces – a controlled variable. This means that the same result – an ellipse – cannot be consistently produced by the same neural signals (and resulting muscle forces), even if those signals are precisely calibrated. That is because the same neural signals will have different effects on the controlled result under difference circumstances. The different circumstances are the changing fatigue levels of the muscles that produce the result (ellipse), the changing conditions of the surface on which the ellipse is drawn, and so forth. These changing circumstances can be lumped together in my model of ellipse production as varying disturbances to the state of the controlled variable – the elliptical movement of the pen in the X and Y dimension, qi.x, qi.y.

The results of a simulation run with moderate level disturbances are should in the graphs below. The top graph shows the ellipse drawn by the two control systems, one controlling pen position in the X dimension (qi.x) and the other controlling pen position in the Y dimension (qi.y). The ellipse traced out is almost exactly the same as the ellipse specified by the time varying values of the reference signals, r.x and r.y, that are sent to the systems controlling qi.x and qi.y, respectively. This looked like a pretty trivial result when the ellipse was produced in a disturbance free environment. But in this case there were moderately intense disturbances present. So the outputs that produced this result were not elliptical, as can be seen in the next graph.

<image.png>

The graph below shows the output variations that produced the elliptical result above. The outputs are clearly not elliptical; they had to vary as necessary in order to compensate for disturbances that would have prevented production of the ellipse.

<image.png>

The chart below shows a plot of log V versus log R for the elliptical figure above (plot of pen movement; qi.x, qi.y). If there is a power law relationship between V and R it will show up as a good linear fit on a log-log plot. And indeed, there is an excellent linear fit (R2=.92) with a coefficient, corresponding to b of .32, almost exactly .33. This power law relationship was produced by outputs that result in a power law but one that is quite different than that for the ellipse.

<image.png>

The graph below shows the power law relationship between V and R for the output trace (o.x, o.y) that produced the ellipse (qi.x, qi.y). The fit to a power law is not as good for the outputs (o.x,o.y) as for the result of the controlled variable(qi.x,qi,y) but it is still a pretty good fit (R2 = .88). But the coefficient of the power relationship is not nearly as close to 1/3 as it was in the case of the ellipse. I believe this is because odd shaped curves, like those of the o.x,o.y) can result in variations in K that obscure the 1/3 power component of the relationship between V and R. The variations in K result from variations in dX, dY,d2X and d2Y since K = |dXd2Y-d2XdY|.

<image.png>

I think the most important lesson here is that you have to understand that behavior is a control process before you can start doing research to determine how this behavior is carried out. The problem with the velocity-curvature power law research is not that the researchers are using the wrong theory (although they are); the problem is that they are looking at the wrong phenomenon. They think they are studying caused output when, in fact, they are studying controlled input. You have to get the phenomenon right before you can get the theory right. In the case of the power law, the bias to see behavior as caused output seems to have blinded the researchers to the fact that they are seeing a mathematical law of curved lines rather than the psychophysiological law of movement that they think they were seeing. This is another version of what Powers called “The Behavioral Illusion” – the illusion of seeing causality where, in fact, there is control.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.07.16.16o0)]

···

On Sat, Jul 16, 2016 at 3:24 PM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

AGM: I am sorry, Rick, but the general equation for curvature, when combined with the equation for speed, does not give rise to the power law. Please revise your math.

RM: knew this would cause you heartburn. I am sosorry. Anyway, I post my math just now so you can check it.

RM: And it’s not PCT that is the issue here; I made a point not to even mention PCT in my post. I didn’t need to. As I said, the problem with the power law research is that it is based on an incorrect understanding of the phenomenon under study. The curved lines (or curved paths) that are traced out by organisms are controlled results of action; that is an objectively observable fact. Therefore, they cannot possibly be the result of computations of output, as in the Gribble and Ostry (1996, Fig. 1) model. Such a model can’t draw ellipses in a disturbance prone world; people can.

RM: Anyway, check out my math. Maybe I’m all wrong. For the sake of the careers of all those people who have been working on this for 40 years I hope I am.

Best

Rick

A second reason (for those not versed in math) is a physical one: curvature is a geometric quantity (thus, only in space), whereas speed is a kinematic one (in space and time). The shape of a scribble in space does not say anything in principle about how it should be drawn in time.

That is why, by the way, why the power law constraint is so interesting biologically, because physics alone would not enforce it. So, something in the biology (be it the nervous system, be it evolution embedded as morphological computation, be it other things) constraints such basic degrees of freedom.

I know you think that PCT is the ultimate microscope, but those who don’t have it are not so stupid to work for 40 years on something and have missed a simple kindergarden calculation just because they do not think of control as PCT does. I am sorry. The paradigm shift will need to wait here.

And, again, the PCT demos you present do not generate the law, they simply mimic it from certain very particular conditions that prescribe the references, or distort with disturbances.

So, the challenge for PCT saying something new and useful and true about non-PCT motor control power-law researchers is still up for whoever wishes to join me.

Cheers,

Alex

On Sat, Jul 16, 2016 at 11:38 PM, Warren Mansell wmansell@gmail.com wrote:

Hi Rick, I don’t understand this at all. How can there be a mathematical relationship between curvature and speed. It doesn’t make sense to me. Surely an ellipse canbe mathematically represented at any speed at any point. Maybe there is a relationship determined by physics, but maths is just a symbolic way of describing forces and processes in nature. Or am I missing something?

Warren

On 16 Jul 2016, at 22:13, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.16.1415)]

There is a considerable body of research aimed at understanding the power law relationship between curvature and angular velocity that is observed w
hen people (or other organisms) move (or move things , like pens) along curved trajectories. The power law can be expressed as follows:

V = K*Rb (1)

where V is angular velocity, defined as

V = (X2*Y2 )1/2 (2)

and R is curvature, defied as

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (3)

What equatio
n (1) says is that when you are drawing a figure, such as an ellipse, for example, the speed of your movement at each instant (V) is proportional to the degree of curvature at that point (R); that is, you move faster through steep than shallow curves. According to Gribble and Ostry (Journal of Neurophysiology, v. 76, 1996) the value of the power coefficient, b, estimated under a “variety of experimental conditions” has been consistently found to be about 1/3.

Researchers have tried to explain the observed power relationship between V and R. These explanations have taken the form of open-loop, output generation models, such as the one described in Gribble and Ostry (1996, Fig. 1). The Gribble/Ostry model assumes that hand drawn figures, like ellipses, are the result of motor programs that send the appropriate signals to the muscles to generate the forces that move the hand in a elliptical pattern in such a way that there is a power relationship between the speed with which the hand moves and the curvature of the figure being drawn (an ellipse in this case). The Gribble/Ostry model successfully generates ellipses like those drawn by people and these ellipses show the power law relationship between V and R with a b coefficient of about 1/3. And, like people, the coefficient of the power law relationship remains about 1/3 whether the ellipse is drawn rapidly or slowly.

I developed a closed loop control model of ellipse drawing. It was an extremely simple model in the sense that there were no complex output calculations. The model simply produced outputs that kept a perception of the state of an input variable (qi.x, qi.y) matching elliptically varying references (r.x, r.y). When I computed the relationship between V and R for the ellipses drawn by the model I found that there was a perfect power relationship between V and R with a b coefficient value of exactly 1/3 (.33). This was a very puzzling result; here I was getting a very nice power relationship between V and R, just like what had been observed empirically and what had resulted from the output generation model of Gribble and Ostry, a
nd I was getting it from what seemed like a trivially simple model. T**hen it struck me that this must mean that the observed power relationship between V and R could not possibly be a result of the way the ellipse was generated; it had to be a property of the ellipse itself. Which led me to take another look at the equations for V and R (equations 2 and 3 above). And I realized that it would be possible to combine the two equations and solve for V as a function of R. The result was:

V = K*R1/3 (4)

<
/p>

where K = |dXd2Y-d2XdY| . This is the power law relationship that has been observed to exist between V and R and the coefficient is 1/3, the same as the value of b which, according to Gribble and Ostry, is estimated under a “variety of experimental conditions”. So the relationship between V and R that is observed in movement studies simply reflects a mathematical relationship between these variables; it has nothing to do with how the movements are produced!!

The mathematical power relationship between angular velocity and curvatur
e exists even if these variables are measured as angular velocity = A=V/R and curvature = C=1/R. The observed relationship between A and C is again a power relationship of the form:

A = K*Cb (5)

with b typically found to be approximately 2/3 rather than 1/3 as it is in the relationship between V and R. When you do the algebra that converts equation (4) into a form involving A and C you get the following:

A = K*C2/3 (6)

where K = D1/3 . Again, there is a power relationship between A and C as there is between V and R and the coefficient of this mathematical relationship is close to the empirically observed coefficient, 2/3.

This is a startling result. It means that students of the power law of movement control have been mistaking a mathematical fact for empirical evidence regarding how movement is produced. It is like doing research on how people draw circles and taking the observation that the circumference of the circles they draw are always close to being equal to p times the diameter as a fact that says something about how the circles are drawn.

So how could researchers in this area have missed the fact that there is a mathematical power relationship betw
een V and R (with a power of 1/3) and between A and C (with a power of 2/3)? It can’t be because these researchers were not as good at math as I am; the papers I’ve been reading are bristling with high powered math and I had to get help from my math teacher son to derive equations 4 and 6. No, I think the reason I discovered this alarming fact and others didn’t is because the latter were looking at the behavior under study (drawing figures) through causal theory glasses while I was looking at the behavior through control theory glasses.

Through causal theory glasses, drawing an ellipse (for example) looks like a caused output – the result of precisely calibrated neural signals sent to the muscles; these signals are thought to create just the right forces over time so that the resulting hand movement is an ellipse. The observed power relationship between angular velocity and curvature is then taken as evi
dence of how the nervous system generates the neural signals that produce the ellipse; it generates signals so as to maintain a power relationship between speed of movement and the size of the curve through which the movement is being made.

But through control theory glasses, drawing an ellipse looks like a controlled (rather than a caused) result of muscle forces – a controlled variable. This means that the same result – an ellipse – cannot be consistently produced by the same neural signals (and resulting muscle forces), even if those signals are precisely calibrated. That is because the same neural signals will have different effects on the controlled result under difference circumstances. The different circumstances are the changing fatigue levels of the muscles that produce the result (ellipse), the changing conditions of the surface on which the ellipse is drawn, and so forth. These changing circ
umstances can be lumped together in my model of ellipse production as varying disturbances to the state of the controlled variable – the elliptical movement of the pen in the X and Y dimension, qi.x, qi.y.

The results of a simulation run with moderate level disturbances are should in the graphs below. The top graph shows the ellipse drawn by the two control systems, one controlling pen position in the X dimension (qi.x) and the other controlling pen position in the Y dimension (qi.y). The ellipse traced out is almost exactly the same as the ellipse specified by the time varying values of the reference signals, r.x and r.y, that are sent to the systems controlling qi.x and qi.y, respectively. This looked like a pretty trivial result when the ellipse was produced in a disturbance free environment. But in this case there were moderately intense disturbances present. So the outputs that produced this result were not ell
iptical, as can be seen in the next graph.

<image.png>

The graph below shows the output variations that produced the elliptical result above. The outputs are clearly not elliptical; they had to vary as necessary in order to compensate for disturbances that would have prevented production of the ellipse.

<image.png>

The chart below shows a plot of log V versus log R for the elliptical figure above (plot of pen movement; qi.x, qi.y). If there is a power law relationship between V and R it will show up as a good linear fit on a log-log plot. And indeed, there is an excellent linear fit (R2=.92) with a coefficient, corresponding to b of .32, almost exactly .33. This power law relationship was produced by outputs that result in a power law but one that is quite different than that for the ellipse.

<image.png>

The graph below shows the power law relationship between V and R for the output trace (o.x, o.y) that produced the ellipse (qi.x, qi.y). The fit to a power law is not as good for the outputs (o.x,o.y) as for the result of the controlled variable(qi.x,qi,y) but it is still a pretty good fit (R2 = .88). But the coefficient of the power relatio
nship is not nearly as close to 1/3 as it was in the case of the ellipse. I believe this is because odd shaped curves, like those of the o.x,o.y) can result in variations in K that obscure the 1/3 power component of the relationship between V and R. The variations in K result from variations in dX, dY,d2X and d2Y since K = |dXd2Y-d2XdY|.

<image.png>

I think the most important lesson here is that you have to understand that behavior is a control process before you can start doing research to determine how this behavior is carried out. The problem with the velocity-curvature power law research is not that the researchers are using the wrong theory (although they are); the problem is that they are looking at the wrong phenomenon. They think they are studying caused output when, in fact, they are studying controlled input. You have to get the phenomenon right before yo
u can get the theory right. In the case of the power law, the bias to see behavior as caused output seems to have blinded the researchers to the fact that they are seeing a mathematical law of curved lines rather than the psychophysiological law of movement that they think they were seeing. This is another version of what Powers called “The Behavioral Illusion” – the illusion of seeing causality where, in fact, there is control.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

again: your math is wrong because D in your last derivation is not a mathematical constant parameter K. it depends on speed and acceleration at every point in time and along the curve. so it is NOT a trivial mathematical relation.

and the fact that simple ellipses give you power law has been known for 30 years and i have sent you some of those papers.

and that ellipse falls into power law does not explain all the other (infinite!) possible ways in which one could draw a non-ellipse, and do so at different speeds, and the Fact that they still follow the powe law.

and no pct generative model can explain that yet.

my heartburn is at your blind insistence to prove others wrong when you are shown you are wrong at the basis, before the speech for control versus linear causality begins…!

and by the way, we don’t need to prove “them” wrong, we just need to find one way to get that law from pct loops. but we can’t so far because we spend more time with wrong interpretation of flawed and incomplete findings than actually digging in old and new.

···

On Sat, Jul 16, 2016 at 3:24 PM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

AGM: I am sorry, Rick, but the general equation for curvature, when combined with the equation for speed, does not give rise to the power law. Please revise your math.

RM: knew this would cause you heartburn. I am sosorry. Anyway, I post my math just now so you can check it.

RM: And it’s not PCT that is the issue here; I made a point not to even mention PCT in my post. I didn’t need to. As I said, the problem with the power law research is that it is based on an incorrect understanding of the phenomenon under study. The curved lines (or curved paths) that are traced out by organisms are controlled results of action; that is an objectively observable fact. Therefore, they cannot possibly be the result of computations of output, as in the Gribble and Ostry (1996, Fig. 1) model. Such a model can’t draw ellipses in a disturbance prone world; people can.

RM: Anyway, check out my math. Maybe I’m all wrong. For the sake of the careers of all those people who have been working on this for 40 years I hope I am.

Best

Rick

A second reason (for those not versed in math) is a physical one: curvature is a geometric quantity (thus, only in space), whereas speed is a kinematic one (in space and time). The shape of a scribble in space does not say anything in principle about how it should be drawn in time.

That is why, by the way, why the power law constraint is so interesting biologically, because physics alone would not enforce it. So, something in the biology (be it the nervous system, be it evolution embedded as morphological computation, be it other things) constraints such basic degrees of freedom.

I know you think that PCT is the ultimate microscope, but those who don’t have it are not so stupid to work for 40 years on something and have missed a simple kindergarden calculation just because they do not think of control as PCT does. I am sorry. The paradigm shift will need to wait here.

And, again, the PCT demos you present do not generate the law, they simply mimic it from certain very particular conditions that prescribe the references, or distort with disturbances.

So, the challenge for PCT saying something new and useful and true about non-PCT motor control power-law researchers is still up for whoever wishes to join me.

Cheers,

Alex

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Sat, Jul 16, 2016 at 11:38 PM, Warren Mansell wmansell@gmail.com wrote:

Hi Rick, I don’t understand this at all. How can there be a mathematical relationship between curvature and speed. It doesn’t make sense to me. Surely an ellipse canbe mathematically represented at any speed at any point. Maybe there is a relationship determined by physics, but maths is just a symbolic way of describing forces and processes in nature. Or am I missing something?

Warren

On 16 Jul 2016, at 22:13, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.16.1415)]

There is a considerable body of research aimed at understanding the power law relationship between curvature and angular velocity that is observed w
hen people (or other organisms) move (or move things , like pens) along curved trajectories. The power law can be expressed as follows:

V = K*Rb (1)

where V is angular velocity, defined as

V = (X2*Y2 )1/2 (2)

and R is curvature, defied as

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (3)

What equatio
n (1) says is that when you are drawing a figure, such as an ellipse, for example, the speed of your movement at each instant (V) is proportional to the degree of curvature at that point (R); that is, you move faster through steep than shallow curves. According to Gribble and Ostry (Journal of Neurophysiology, v. 76, 1996) the value of the power coefficient, b, estimated under a “variety of experimental conditions” has been consistently found to be about 1/3.

Researchers have tried to explain the observed power relationship between V and R. These explanations have taken the form of open-loop, output generation models, such as the one described in Gribble and Ostry (1996, Fig. 1). The Gribble/Ostry model assumes that hand drawn figures, like ellipses, are the result of motor programs that send the appropriate signals to the muscles to generate the forces that move the hand in a elliptical pattern in such a way that there is a power relationship between the speed with which the hand moves and the curvature of the figure being drawn (an ellipse in this case). The Gribble/Ostry model successfully generates ellipses like those drawn by people and these ellipses show the power law relationship between V and R with a b coefficient of about 1/3. And, like people, the coefficient of the power law relationship remains about 1/3 whether the ellipse is drawn rapidly or slowly.

I developed a closed loop control model of ellipse drawing. It was an extremely simple model in the sense that there were no complex output calculations. The model simply produced outputs that kept a perception of the state of an input variable (qi.x, qi.y) matching elliptically varying references (r.x, r.y). When I computed the relationship between V and R for the ellipses drawn by the model I found that there was a perfect power relationship between V and R with a b coefficient value of exactly 1/3 (.33). This was a very puzzling result; here I was getting a very nice power relationship between V and R, just like what had been observed empirically and what had resulted from the output generation model of Gribble and Ostry, a
nd I was getting it from what seemed like a trivially simple model. T**hen it struck me that this must mean that the observed power relationship between V and R could not possibly be a result of the way the ellipse was generated; it had to be a property of the ellipse itself. Which led me to take another look at the equations for V and R (equations 2 and 3 above). And I realized that it would be possible to combine the two equations and solve for V as a function of R. The result was:

V = K*R1/3 (4)

<
/p>

where K = |dXd2Y-d2XdY| . This is the power law relationship that has been observed to exist between V and R and the coefficient is 1/3, the same as the value of b which, according to Gribble and Ostry, is estimated under a “variety of experimental conditions”. So the relationship between V and R that is observed in movement studies simply reflects a mathematical relationship between these variables; it has nothing to do with how the movements are produced!!

The mathematical power relationship between angular velocity and curvatur
e exists even if these variables are measured as angular velocity = A=V/R and curvature = C=1/R. The observed relationship between A and C is again a power relationship of the form:

A = K*Cb (5)

with b typically found to be approximately 2/3 rather than 1/3 as it is in the relationship between V and R. When you do the algebra that converts equation (4) into a form involving A and C you get the following:

A = K*C2/3 (6)

where K = D1/3 . Again, there is a power relationship between A and C as there is between V and R and the coefficient of this mathematical relationship is close to the empirically observed coefficient, 2/3.

This is a startling result. It means that students of the power law of movement control have been mistaking a mathematical fact for empirical evidence regarding how movement is produced. It is like doing research on how people draw circles and taking the observation that the circumference of the circles they draw are always close to being equal to p times the diameter as a fact that says something about how the circles are drawn.

So how could researchers in this area have missed the fact that there is a mathematical power relationship betw
een V and R (with a power of 1/3) and between A and C (with a power of 2/3)? It can’t be because these researchers were not as good at math as I am; the papers I’ve been reading are bristling with high powered math and I had to get help from my math teacher son to derive equations 4 and 6. No, I think the reason I discovered this alarming fact and others didn’t is because the latter were looking at the behavior under study (drawing figures) through causal theory glasses while I was looking at the behavior through control theory glasses.

Through causal theory glasses, drawing an ellipse (for example) looks like a caused output – the result of precisely calibrated neural signals sent to the muscles; these signals are thought to create just the right forces over time so that the resulting hand movement is an ellipse. The observed power relationship between angular velocity and curvature is then taken as evi
dence of how the nervous system generates the neural signals that produce the ellipse; it generates signals so as to maintain a power relationship between speed of movement and the size of the curve through which the movement is being made.

But through control theory glasses, drawing an ellipse looks like a controlled (rather than a caused) result of muscle forces – a controlled variable. This means that the same result – an ellipse – cannot be consistently produced by the same neural signals (and resulting muscle forces), even if those signals are precisely calibrated. That is because the same neural signals will have different effects on the controlled result under difference circumstances. The different circumstances are the changing fatigue levels of the muscles that produce the result (ellipse), the changing conditions of the surface on which the ellipse is drawn, and so forth. These changing circ
umstances can be lumped together in my model of ellipse production as varying disturbances to the state of the controlled variable – the elliptical movement of the pen in the X and Y dimension, qi.x, qi.y.

The results of a simulation run with moderate level disturbances are should in the graphs below. The top graph shows the ellipse drawn by the two control systems, one controlling pen position in the X dimension (qi.x) and the other controlling pen position in the Y dimension (qi.y). The ellipse traced out is almost exactly the same as the ellipse specified by the time varying values of the reference signals, r.x and r.y, that are sent to the systems controlling qi.x and qi.y, respectively. This looked like a pretty trivial result when the ellipse was produced in a disturbance free environment. But in this case there were moderately intense disturbances present. So the outputs that produced this result were not ell
iptical, as can be seen in the next graph.

<image.png>

The graph below shows the output variations that produced the elliptical result above. The outputs are clearly not elliptical; they had to vary as necessary in order to compensate for disturbances that would have prevented production of the ellipse.

<image.png>

The chart below shows a plot of log V versus log R for the elliptical figure above (plot of pen movement; qi.x, qi.y). If there is a power law relationship between V and R it will show up as a good linear fit on a log-log plot. And indeed, there is an excellent linear fit (R2=.92) with a coefficient, corresponding to b of .32, almost exactly .33. This power law relationship was produced by outputs that result in a power law but one that is quite different than that for the ellipse.

<image.png>

The graph below shows the power law relationship between V and R for the output trace (o.x, o.y) that produced the ellipse (qi.x, qi.y). The fit to a power law is not as good for the outputs (o.x,o.y) as for the result of the controlled variable(qi.x,qi,y) but it is still a pretty good fit (R2 = .88). But the coefficient of the power relatio
nship is not nearly as close to 1/3 as it was in the case of the ellipse. I believe this is because odd shaped curves, like those of the o.x,o.y) can result in variations in K that obscure the 1/3 power component of the relationship between V and R. The variations in K result from variations in dX, dY,d2X and d2Y since K = |dXd2Y-d2XdY|.

<image.png>

I think the most important lesson here is that you have to understand that behavior is a control process before you can start doing research to determine how this behavior is carried out. The problem with the velocity-curvature power law research is not that the researchers are using the wrong theory (although they are); the problem is that they are looking at the wrong phenomenon. They think they are studying caused output when, in fact, they are studying controlled input. You have to get the phenomenon right before yo
u can get the theory right. In the case of the power law, the bias to see behavior as caused output seems to have blinded the researchers to the fact that they are seeing a mathematical law of curved lines rather than the psychophysiological law of movement that they think they were seeing. This is another version of what Powers called “The Behavioral Illusion” – the illusion of seeing causality where, in fact, there is control.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From MK (2016.07.17.1250 CET)]

Rick Marken (2016.07.16.1600)--

The curved lines (or curved paths) that are traced
out by organisms are controlled results of action; that is an objectively
observable fact.

Do the virtual agents in the Crowd demonstration control the
appearance of their "trajectories" or "paths" as observed by the
person who runs the demonstration?

M

[From Bruce Abbott (2016.07.17.0800 EDT)]

MK (2016.07.17.1250 CET) --

Rick Marken (2016.07.16.1600)--

The curved lines (or curved paths) that are traced out by organisms
are controlled results of action; that is an objectively observable
fact.

MK: Do the virtual agents in the Crowd demonstration control the appearance
of their "trajectories" or "paths" as observed by the person who runs the
demonstration?

BA: Great question, Matti. No, they don't. The organisms control their
perceived proximity to the "goal" circle on the right side of the screen, to
the fixed-position obstacles between them and the goal, and to each other.
The observed trajectories emerge dynamically depending partly on chance and
partly on control actions that occur as needed to control perceptions of
these proximities.

BA: Although not illustrated by the crowd demo, one CAN control a
trajectory as seen by an external observer. A person drawing an ellipse (or
any other figure) would be controlling for seeing (and/or proprioceptively
feeling) a line being followed that matches a reference shape. The person
might also control the speed with which the figure is being traced,
especially if instructed to do so (e.g., "draw it as fast as you can"). One
control organization that could reproduce a desired shape would be a
hierarchy in which the upper level would act so as to perceive a figure
being drawn that matches a reference figure. The output of this system
would connect to the references of lower systems that control the angles of
the joints involved in the drawing motion. (The actual system would be more
complex than this, but I'm just sketching out the basic idea here.)

BA: It's an interesting and well-known fact that the figures a person
reproduces will have similar idiosyncratic characteristics regardless of
scale or of what muscles and joints are used to produce them. For example,
a person's signature looks basically the same whether the writing is done
using only finger and wrist movements or the whole arm.

BA: By the way, for those who may not know, the Crowd demo is part of the
suite of demos that accompany Bill Powers' book, Living Control Systems III:
The Fact of Control. The book explains in detail how the Crowd demo works.

Bruce

[From Rick Marken (2016.07.17.1030)]

···

On Sun, Jul 17, 2016 at 1:56 AM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

AGM: again: your math is wrong because D in your last derivation is not a mathematical constant parameter K. it depends on speed and acceleration at every point in time and along the curve. so it is NOT a trivial mathematical relation.

RM: You are absolutely right! D is a variable, not a constant, and it should also be raised to the 1/3 power. So the correct formula relating R to V is:

V = |dXd2Y-d2XdY|
1/3 *R1/3 (1)

RM: This implies that for any curved line (not straight lines since R is undefined and V is constant for straight lines) it should be true that:

log V = .33log(D) + .33log(R) (2)

where D = |dXd2Y-d2XdY|

RM: I tested this using multiple regression with log(D) and log (R) as the predictor variables and log (V) as the criterion variable. In every case – that is, for every curved trace I used as input – the coefficients for log (D) and log (R) were .33 (and the intercept was close to 0 and the R2 value was always 1.0).

AGM:and the fact that simple ellipses give you power law has been known for 30 years and i have sent you some of those papers.

AGM: and that ellipse falls into power law does not explain all the other (infinite!) possible ways in which one could draw a non-ellipse, and do so at different speeds, and the Fact that they still follow the power law.

RM: Actually, Equation 1 is a general expression for the relationship between V , R and D for all curved lines (it doesn’t hold for straight lines, as noted). I tested this using multiple linear regression on several curved lines using the log form of equation 1 (equation 2). Here’s one of the curved lines I used:

RM: This is an interesting one because if you just look at the relationship between log R and log V you get the curved plot that is often found in the power law studies (shown below). So if you just look at this relationship you might get the idea that V deviates in some interesting way from a power law relationship with R for this particular curve (above).

RM: But if you include log D along with log R as predictors of log V you get this:

RM: The regression accounts for all the variance in the curve (R2 = 1.0) and the coefficient of the power law relationship between R and V is .33, just as predicted by equation 2. The bending seen in the plot of V versus R above is clearly variance in V that is related to variance in D, which was not included in the analysis. Again, the regression comes out exactly the same – that is, just like equation 2 – for all curves that I’ve used. It only fails for straight lines.

AGM: my heartburn is at your blind insistence to prove others wrong when you are shown you are wrong at the basis, before the speech for control versus linear causality begins…!

RM: I am not trying to prove others wrong. I have simply made an observation about the relationship between V and R for curved lines that shows that this relationship will always be a power function with an exponent of .33 if variations in D are also taken into account. Therefore, observed relationships between V and R that are observed for curved lines (or paths) tell you nothing about how they are produced. If they are properly analyzed, the relationship between V and R will always be found to follow equation 2.

RM: I’m not presenting this result in order to show that some people were “wrong”. I’m presenting it in a spirit of helpfulness. I want to help people re-orient their research in a more fruitful direction – a direction that takes into account the fact that behavior, like drawing curved lines, is a control process. What I would love to discuss is how do do this; what should research on drawing curved lines look like based on an understanding that this behavior is a control process?

RM: I know that the chances of this happening – of people making fundamental changes in the way they go about doing their research – are small but from my point of view I would wonder why I should do the work if there is no hope. Bill Powers had such hope up to the end. But he knew that what he was describing was an approach to doing research that would be very disturbing to conventional behavioral scientists. He knew this, in particular, when he published his 1978 Psych Review paper (v.85, pp. 417-435) subtitled “Some Spadework at the Foundations of Scientific Psychology”. In that paper he showed that the conventional approach to studying the behavior of living systems could only produce misleading results and he says the following, which is relevant to our discussion here:

“The nightmare of any experimenter is to realize too late that his results were forced by his experimental design and do not actually pertain to behavior. This nightmare has a good chance of becoming a reality for a number of behavioral scientists”. – Powers, 1978

RM: Clearly, Powers was presenting the facts of control as a warning, not as a “gotcha”. Of course, very few behavioral scientists have heeded this warning. But Bill maintained his hope that eventually more would. I’m just carrying on the (possibly Sisyphean) task.

AGM: and by the way, we don’t need to prove “them” wrong, we just need to find one way to get that law from pct loops.

RM: I think I have shown how that is done. My finding that the power relationship between curvature and velocity is a mathematical property of the curve itself is based on the PCT-based understanding that a drawn curve is a controlled result of action.

RM: I hope we can continue this conversation towards a fruitful resolution. But whatever happens, I really think you for getting me in touch with this line of research.

Best regards

Rick

but we can’t so far because we spend more time with wrong interpretation of flawed and incomplete findings than actually digging in old and new.

On Sunday, 17 July 2016, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.16.16o0)]

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Sat, Jul 16, 2016 at 3:24 PM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

AGM: I am sorry, Rick, but the general equation for curvature, when combined with the equation for speed, does not give rise to the power law. Please revise your math.

RM: knew this would cause you heartburn. I am sosorry. Anyway, I post my math just now so you can check it.

RM: And it’s not PCT that is the issue here; I made a point not to even mention PCT in my post. I didn’t need to. As I said, the problem with the power law research is that it is based on an incorrect understanding of the phenomenon under study. The curved lines (or curved paths) that are traced out by organisms are controlled results of action; that is an objectively observable fact. Therefore, they cannot possibly be the result of computations of output, as in the Gribble and Ostry (1996, Fig. 1) model. Such a model can’t draw ellipses in a disturbance prone world; people can.

RM: Anyway, check out my math. Maybe I’m all wrong. For the sake of the careers of all those people who have been working on this for 40 years I hope I am.

Best

Rick

A second reason (for those not versed in math) is a physical one: curvature is a geometric quantity (thus, only in space), whereas speed is a kinematic one (in space and time). The shape of a scribble in space does not say anything in principle about how it should be drawn in time.

That is why, by the way, why the power law constraint is so interesting biologically, because physics alone would not enforce it. So, something in the biology (be it the nervous system, be it evolution embedded as morphological computation, be it other things) constraints such basic degrees of freedom.

I know you think that PCT is the ultimate microscope, but those who don’t have it are not so stupid to work for 40 years on something and have missed a simple kindergarden calculation just because they do not think of control as PCT does. I am sorry. The paradigm shift will need to wait here.

And, again, the PCT demos you present do not generate the law, they simply mimic it from certain very particular conditions that prescribe the references, or distort with disturbances.

So, the challenge for PCT saying something new and useful and true about non-PCT motor control power-law researchers is still up for whoever wishes to join me.

Cheers,

Alex

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Sat, Jul 16, 2016 at 11:38 PM, Warren Mansell wmansell@gmail.com wrote:

Hi Rick, I don’t understand this at all. How can there be a mathematical relationship between curvature and speed. It doesn’t make sense to me. Surely an ellipse canbe mathematically represented at any speed at any point. Maybe there is a relationship determined by physics, but maths is just a symbolic way of describing forces and processes in nature. Or am I missing something?

Warren

On 16 Jul 2016, at 22:13, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.16.1415)]

There is a considerable body of research aimed at understanding the power law relationship between curvature and angular velocity that is observed w
hen people (or other organisms) move (or move things , like pens) along curved trajectories. The power law can be expressed as follows:

V = K*Rb (1)

where V is angular velocity, defined as

V = (X2*Y2 )1/2 (2)

and R is curvature, defied as

R = [(X2Y2 )3/2]/|dXd2Y-d2X*dY| (3)

What equatio
n (1) says is that when you are drawing a figure, such as an ellipse, for example, the speed of your movement at each instant (V) is proportional to the degree of curvature at that point (R); that is, you move faster through steep than shallow curves. According to Gribble and Ostry (Journal of Neurophysiology, v. 76, 1996) the value of the power coefficient, b, estimated under a “variety of experimental conditions” has been consistently found to be about 1/3.

Researchers have tried to explain the observed power relationship between V and R. These explanations have taken the form of open-loop, output generation models, such as the one described in Gribble and Ostry (1996, Fig. 1). The Gribble/Ostry model assumes that hand drawn figures, like ellipses, are the result of motor programs that send the appropriate signals to the muscles to generate the forces that move the hand in a elliptical pattern in such a way that there is a power relationship between the speed with which the hand moves and the curvature of the figure being drawn (an ellipse in this case). The Gribble/Ostry model successfully generates ellipses like those drawn by people and these ellipses show the power law relationship between V and R with a b coefficient of about 1/3. And, like people, the coefficient of the power law relationship remains about 1/3 whether the ellipse is drawn rapidly or slowly.

I developed a closed loop control model of ellipse drawing. It was an extremely simple model in the sense that there were no complex output calculations. The model simply produced outputs that kept a perception of the state of an input variable (qi.x, qi.y) matching elliptically varying references (r.x, r.y). When I computed the relationship between V and R for the ellipses drawn by the model I found that there was a perfect power relationship between V and R with a b coefficient value of exactly 1/3 (.33). This was a very puzzling result; here I was getting a very nice power relationship between V and R, just like what had been observed empirically and what had resulted from the output generation model of Gribble and Ostry, a
nd I was getting it from what seemed like a trivially simple model. T**hen it struck me that this must mean that the observed power relationship between V and R could not possibly be a result of the way the ellipse was generated; it had to be a property of the ellipse itself. Which led me to take another look at the equations for V and R (equations 2 and 3 above). And I realized that it would be possible to combine the two equations and solve for V as a function of R. The result was:

V = K*R1/3 (4)

<
/p>

where K = |dXd2Y-d2XdY| . This is the power law relationship that has been observed to exist between V and R and the coefficient is 1/3, the same as the value of b which, according to Gribble and Ostry, is estimated under a “variety of experimental conditions”. So the relationship between V and R that is observed in movement studies simply reflects a mathematical relationship between these variables; it has nothing to do with how the movements are produced!!

The mathematical power relationship between angular velocity and curvatur
e exists even if these variables are measured as angular velocity = A=V/R and curvature = C=1/R. The observed relationship between A and C is again a power relationship of the form:

A = K*Cb (5)

with b typically found to be approximately 2/3 rather than 1/3 as it is in the relationship between V and R. When you do the algebra that converts equation (4) into a form involving A and C you get the following:

A = K*C2/3 (6)

where K = D1/3 . Again, there is a power relationship between A and C as there is between V and R and the coefficient of this mathematical relationship is close to the empirically observed coefficient, 2/3.

This is a startling result. It means that students of the power law of movement control have been mistaking a mathematical fact for empirical evidence regarding how movement is produced. It is like doing research on how people draw circles and taking the observation that the circumference of the circles they draw are always close to being equal to p times the diameter as a fact that says something about how the circles are drawn.

So how could researchers in this area have missed the fact that there is a mathematical power relationship betw
een V and R (with a power of 1/3) and between A and C (with a power of 2/3)? It can’t be because these researchers were not as good at math as I am; the papers I’ve been reading are bristling with high powered math and I had to get help from my math teacher son to derive equations 4 and 6. No, I think the reason I discovered this alarming fact and others didn’t is because the latter were looking at the behavior under study (drawing figures) through causal theory glasses while I was looking at the behavior through control theory glasses.

Through causal theory glasses, drawing an ellipse (for example) looks like a caused output – the result of precisely calibrated neural signals sent to the muscles; these signals are thought to create just the right forces over time so that the resulting hand movement is an ellipse. The observed power relationship between angular velocity and curvature is then taken as evi
dence of how the nervous system generates the neural signals that produce the ellipse; it generates signals so as to maintain a power relationship between speed of movement and the size of the curve through which the movement is being made.

But through control theory glasses, drawing an ellipse looks like a controlled (rather than a caused) result of muscle forces – a controlled variable. This means that the same result – an ellipse – cannot be consistently produced by the same neural signals (and resulting muscle forces), even if those signals are precisely calibrated. That is because the same neural signals will have different effects on the controlled result under difference circumstances. The different circumstances are the changing fatigue levels of the muscles that produce the result (ellipse), the changing conditions of the surface on which the ellipse is drawn, and so forth. These changing circ
umstances can be lumped together in my model of ellipse production as varying disturbances to the state of the controlled variable – the elliptical movement of the pen in the X and Y dimension, qi.x, qi.y.

The results of a simulation run with moderate level disturbances are should in the graphs below. The top graph shows the ellipse drawn by the two control systems, one controlling pen position in the X dimension (qi.x) and the other controlling pen position in the Y dimension (qi.y). The ellipse traced out is almost exactly the same as the ellipse specified by the time varying values of the reference signals, r.x and r.y, that are sent to the systems controlling qi.x and qi.y, respectively. This looked like a pretty trivial result when the ellipse was produced in a disturbance free environment. But in this case there were moderately intense disturbances present. So the outputs that produced this result were not ell
iptical, as can be seen in the next graph.

<image.png>

The graph below shows the output variations that produced the elliptical result above. The outputs are clearly not elliptical; they had to vary as necessary in order to compensate for disturbances that would have prevented production of the ellipse.

<image.png>

The chart below shows a plot of log V versus log R for the elliptical figure above (plot of pen movement; qi.x, qi.y). If there is a power law relationship between V and R it will show up as a good linear fit on a log-log plot. And indeed, there is an excellent linear fit (R2=.92) with a coefficient, corresponding to b of .32, almost exactly .33. This power law relationship was produced by outputs that result in a power law but one that is quite different than that for the ellipse.

<image.png>

The graph below shows the power law relationship between V and R for the output trace (o.x, o.y) that produced the ellipse (qi.x, qi.y). The fit to a power law is not as good for the outputs (o.x,o.y) as for the result of the controlled variable(qi.x,qi,y) but it is still a pretty good fit (R2 = .88). But the coefficient of the power relatio
nship is not nearly as close to 1/3 as it was in the case of the ellipse. I believe this is because odd shaped curves, like those of the o.x,o.y) can result in variations in K that obscure the 1/3 power component of the relationship between V and R. The variations in K result from variations in dX, dY,d2X and d2Y since K = |dXd2Y-d2XdY|.

<image.png>

I think the most important lesson here is that you have to understand that behavior is a control process before you can start doing research to determine how this behavior is carried out. The problem with the velocity-curvature power law research is not that the researchers are using the wrong theory (although they are); the problem is that they are looking at the wrong phenomenon. They think they are studying caused output when, in fact, they are studying controlled input. You have to get the phenomenon right before yo
u can get the theory right. In the case of the power law, the bias to see behavior as caused output seems to have blinded the researchers to the fact that they are seeing a mathematical law of curved lines rather than the psychophysiological law of movement that they think they were seeing. This is another version of what Powers called “The Behavioral Illusion” – the illusion of seeing causality where, in fact, there is control.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.07.17.1050)]

···

MK (2016.07.17.1250 CET)

Rick Marken (2016.07.16.1600)–

RM:>The curved lines (or curved paths) that are traced

out by organisms are controlled results of action; that is an objectively

observable fact.

MK: Do the virtual agents in the Crowd demonstration control the appearance of their “trajectories” or “paths” as observed by the person who runs the demonstration?

RM: No, in that case the paths are a side effect of the individuals controlling for collision avoidance, a destination position and proximity to a target individual. But if you analyzed any path (other than a straight line) in terms of the relationship between R and V for that path, you would find a power relationship between R and V with a coefficient of .33 (assuming you took variations in D into account). Actually, if we could get the X, Y coordinates of some of the paths taken by individuals in the CROWD demo, it would be interesting to show that this is true. It would certainly be a nice way of showing that the power law relationship between R and V tells you nothing about how the paths were generated.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

*RM: I have simply made an observation about the relationship between V and R for curved lines that shows that this relationship will always be a power function with an exponent of .33 if variations in D are also taken into account. *

Alex: you have simply numerically checked a mathematical equality by writing curvature as a function of speed. Every single scientist working on the power law has done that. But they don’t believe it is such a big achievement.

*RM: what should research on drawing curved lines look like based on an understanding that this behavior is a control process? *

Alex: that is what we need/could answer, but we haven’t yet. That is what I proposed from the very first email: that the challenge is to have a control system that generates output that follows the power law. This could be done by trying to keep smoothness (via jerk) to zero. But it is not trivial to do it. Manually plugging in reference signals is not the way.

*RM: My finding that the power relationship between curvature and velocity is a mathematical property of the curve itself is based on the PCT-based understanding that a drawn curve is a controlled result of action. *

Alex: “Your finding” is a two line re-writing of the equation for curvature; it has nothing to do with PCT-understanding. What is really new there, Rick?

*RM: I hope we can continue this conversation towards a fruitful resolution. But whatever happens, I really think you for getting me in touch with this line of research. *

Alex: As it sometimes happens, after a long spiralling, we are back to where we started in our first email: we have the fact of the power-law and we don’t have a physical or biological explanation.

···

On Sun, Jul 17, 2016 at 7:51 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.17.1050)]

MK (2016.07.17.1250 CET)

Rick Marken (2016.07.16.1600)–

RM:>The curved lines (or curved paths) that are traced

out by organisms are controlled results of action; that is an objectively

observable fact.

MK: Do the virtual agents in the Crowd demonstration control the appearance of their “trajectories” or “paths” as observed by the person who runs the demonstration?

RM: No, in that case the paths are a side effect of the individuals controlling for collision avoidance, a destination position and proximity to a target individual. But if you analyzed any path (other than a straight line) in terms of the relationship between R and V for that path, you would find a power relationship between R and V with a coefficient of .33 (assuming you took variations in D into account). Actually, if we could get the X, Y coordinates of some of the paths taken by individuals in the CROWD demo, it would be interesting to show that this is true. It would certainly be a nice way of showing that the power law relationship between R and V tells you nothing about how the paths were generated.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

Great, and I think this is what Alex is asking for…

Warren

···

MK (2016.07.17.1250 CET)

Rick Marken (2016.07.16.1600)–

RM:>The curved lines (or curved paths) that are traced

out by organisms are controlled results of action; that is an objectively

observable fact.

MK: Do the virtual agents in the Crowd demonstration control the appearance of their “trajectories” or “paths” as observed by the person who runs the demonstration?

RM: No, in that case the paths are a side effect of the individuals controlling for collision avoidance, a destination position and proximity to a target individual. But if you analyzed any path (other than a straight line) in terms of the relationship between R and V for that path, you would find a power relationship between R and V with a coefficient of .33 (assuming you took variations in D into account). Actually, if we could get the X, Y coordinates of some of the paths taken by individuals in the CROWD demo, it would be interesting to show that this is true. It would certainly be a nice way of showing that the power law relationship between R and V tells you nothing about how the paths were generated.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Bruce Abbott (2016.07.18.0945 EDT)]

Rick Marken (2016.07.17.1050) –

MK (2016.07.17.1250 CET)

Rick Marken (2016.07.16.1600)–

RM:>The curved lines (or curved paths) that are traced

out by organisms are controlled results of action; that is an objectively
observable fact.

MK: Do the virtual agents in the Crowd demonstration control the appearance of their “trajectories” or “paths” as observed by the person who runs the demonstration?

RM: No, in that case the paths are a side effect of the individuals controlling for collision avoidance, a destination position and proximity to a target individual. But if you analyzed any path (other than a straight line) in terms of the relationship between R and V for that path, you would find a power relationship between R and V with a coefficient of .33 (assuming you took variations in D into account). Actually, if we could get the X, Y coordinates of some of the paths taken by individuals in the CROWD demo, it would be interesting to show that this is true. It would certainly be a nice way of showing that the power law relationship between R and V tells you nothing about how the paths were generated.

BA:Â The Crowd demo does not simulate the physics:Â the individuals are essentially massless particles.Â If the power law is a product of the forces produced by objects with mass as they follow curved paths, it will not appear in this context.

BA:Â For what itâ€™s worth, many years ago when I worked as a technician in the Glass Science department at Owens-Illinois Glass Co., one of the researchers I worked with (a physical chemist) told me that just about anything can be fit to a straight line on when plotted on log-log paper.Â Thatâ€™s not literally true, of course, but the point was, that such a fit may not mean much – whole families of curves (when plotteed on linear scales) will follow an approximate straight line on log-log plots.

Bruce

[From Rick Marken (2016.07.18.1000)]

···

On Sun, Jul 17, 2016 at 1:47 PM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

*RM: I have simply made an observation about the relationship between V and R for curved lines that shows that this relationship will always be a power function with an exponent of .33 if variations in D are also taken into account. *

Alex: you have simply numerically checked a mathematical equality by writing curvature as a function of speed. Every single scientist working on the power law has done that. But they don’t believe it is such a big achievement.

RM: If they know this then I don’t understand why they still think that any observed relationship between R and V would tell them anything about how the curves are produced. There is simply no way for them to find any relationship between R and V other than

V = |dXd2Y-d2XdY| 1/3 *R1/3 (1)

*RM: what should research on drawing curved lines look like based on an understanding that this behavior is a control process? *

Alex: that is what we need/could answer, but we haven’t yet. That is what I proposed from the very first email: that the challenge is to have a control system that generates output that follows the power law.

RM: I believe that I already demonstrated a control system that generates output (o.x and o.y) that produces a result (curved line, qi.x,qi.x)) that follows the power law. Equation (1) shows that there was no way to find anything else. But perhaps I’m not understanding what you mean by “output”. In a control system, output is the variable driven by the error signal; this output (o.x and o.y) varies “as necessary” to produce the intended result (pattern of qi.x and qi.y) – the result specified by r.x and r.y. So it sounds to my like the challenge is to have a control system that generates o.x and o.y that follows a power law. And a simple control model does generate outputs that follow a power law simply because those outputs themselves trace out a curve (resulting in the power law relationship between V and R as per equation (1)). This is true even if disturbances are present do that the cure produced by the outputs, o.x and o.y, are quite different than the intended curve, qi.x and qi.y. And, of course, the intended curve also follows a power law (once again, per equation (1)).

RM: But perhaps this is not what you mean by “output”. I think I would understand the “challenge” you describe if you could show me an examples of a non-control system model that generates output that follows the power law.

*RM: My finding that the power relationship between curvature and velocity is a mathematical property of the curve itself is based on the PCT-based understanding that a drawn curve is a controlled result of action. *

Alex: “Your finding” is a two line re-writing of the equation for curvature; it has nothing to do with PCT-understanding. What is really new there, Rick?

RM: Clearly you’re not the only one who sees no “there” there. Maybe not. But I think I can better see the possibly trivial nature of what I found if you could show me an example of what you seem to want from PCT: a non-control system model that generates output that follows the power law.

*RM: I hope we can continue this conversation towards a fruitful resolution. But whatever happens, I really think you for getting me in touch with this line of research. *

Alex: As it sometimes happens, after a long spiralling, we are back to where we started in our first email: we have the fact of the power-law and we don’t have a physical or biological explanation.

RM: Isn’t the Gribble/Ostry model (in Fig 1 of their paper) an attempt at an explanation? Are there really no models (explanations) of the power law after all these years of research? If there really is no such model, maybe you could sketch out a diagram of what you think a model of the power law shouldlook like.

Thanks.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

On Sun, Jul 17, 2016 at 7:51 PM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.17.1050)]

MK (2016.07.17.1250 CET)

Rick Marken (2016.07.16.1600)–

RM:>The curved lines (or curved paths) that are traced

out by organisms are controlled results of action; that is an objectively

observable fact.

MK: Do the virtual agents in the Crowd demonstration control the appearance of their “trajectories” or “paths” as observed by the person who runs the demonstration?

RM: No, in that case the paths are a side effect of the individuals controlling for collision avoidance, a destination position and proximity to a target individual. But if you analyzed any path (other than a straight line) in terms of the relationship between R and V for that path, you would find a power relationship between R and V with a coefficient of .33 (assuming you took variations in D into account). Actually, if we could get the X, Y coordinates of some of the paths taken by individuals in the CROWD demo, it would be interesting to show that this is true. It would certainly be a nice way of showing that the power law relationship between R and V tells you nothing about how the paths were generated.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[From Rick Marken (2016.07.18.1020)]

···

Bruce Abbott (2016.07.18.0945 EDT)

MK: Do the virtual agents in the Crowd demonstration control theÂ appearance of their “trajectories” or “paths” as observed by theÂ person who runs the demonstration?

RM: No, in that case the paths are a side effect of the individuals controlling for collision avoidance, a destination position and proximity to a target individual. But if you analyzed any path (other than a straight line) in terms of the relationship between R and V for that path, you would find a power relationship between R and V with a coefficient of .33 (assuming you took Â variations in D into account). Actually, if we could get the X, Y coordinates of some of the paths taken by individuals in the CROWD demo, it would be interesting to show that this is true. It would certainly be a nice way of showing that the power law relationship between R and V tells you nothing about how the paths were generated.Â

Â

BA:Â The Crowd demo does not simulate the physics:Â the individuals are essentially massless particles.Â If the power law is a product of the forces produced by objects with mass as they follow curved paths, it will not appear in this context.

RM: Right!! And I’m am certain that the power law will appear in that context. So could you get me the X,Y coordinates of a few of the curved paths – like the one in Figure 9.3 of LCS III? About 1000 points for each path would be nice. I predict with great confidence that, unless the path is a perfect straight line, there will be a power relationship between V and R for every path that is precisely equal to:

log V = .33log(D) + .33log(R) Â Â Â Â Â Â

Â

Â BA:Â For what itâ€™s worth, many years ago when I worked as a technician in the Glass Science department at Owens-Illinois Glass Co., one of the researchers I worked with (a physical chemist) told me that just about anything can be fit to a straight line on when plotted on log-log paper.Â Thatâ€™s not literally true, of course, but the point was, that such a fit may not mean much – whoole families of curves (when plotted on linear scales) will follow an approximate straight line on log-log plots.

RM: It’s not the log-log regression aspect of the test that is important. It’s the fact that the coefficient of the power function – for every path – will be .33; and the R2 will be 1.0, showing that the observed power relationship between V and R is a mathematical property of curved paths and, thus, tells you nothing at all about how those paths were produced.

RM: If you could send those paths to me that would be great!

Thanks.Â

BestÂ

Rick

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

[Martin Taylor 2017.07.18.14.13]

[From Rick Marken (2016.07.18.1000)]

``````The question isn't about geometry, it's about velocity. V =
``````

d(distance)/d(time). Your V is just a measure of local curvature.
There’s no time in it at all. Alex keeps telling you as much. His
original question was about why people slow down at sharp curves,
not about how you describe curves in a Cartesian space.

``````V(angular velocity) = (d(distance travelled)/d(time))/R for a
``````

portion of a circular arc of radius R.

``````Martin
``````
···

On Sun, Jul 17, 2016 at 1:47 PM, Alex
Gomez-Marin agomezmarin@gmail.com
wrote:

``````                  Rick,
``````

• ``````                   RM:
``````

relationship between V and R for curved lines
that shows that this relationship will always be
a power function with an exponent of .33 if
variations in D are also taken into account. *

``````                  Alex:
``````

you have simply numerically checked a mathematical
equality by writing curvature as a function of
speed. Every single scientist working on the power
law has done that. But they don’t believe it is
such a big achievement.

``````          RM: If they know this then I don't understand why they
``````

still think that any observed relationship between R and V
would tell them anything about how the curves are
produced. There is simply no way for them to find any
relationship between R and V other than

``````            V
``````

= |dXd2Y-d2XdY| 1/3 *R1/3
(1)

[From Rick Marken (2016.07.18.1515)]

···

Martin Taylor (2017.07.18.14.13)

``````MT: The question isn't about geometry, it's about velocity. V =
``````

d(distance)/d(time). Your V is just a measure of local curvature.
There’s no time in it at all. Alex keeps telling you as much. His
original question was about why people slow down at sharp curves,
not about how you describe curves in a Cartesian space.

``````V(angular velocity) = (d(distance travelled)/d(time))/R for a
``````

portion of a circular arc of radius R.

RM: In determining the power relationship between V and R, V (velocity) is measured as:

where X.dot and Y.dot are Newton’s notation for the time derivatives of X (dX/dt) and Y(dY/dt) respectively. This measure of V is the variable that is actually computed from the curved paths that are observed in the velocity/curvature power law studies. This measure of velocity is then related to the R measure of curvature, which is computed as follows:

RM: Equation 1 is found by solving the two equations above, for V and R, simultaneously. X and Y are, of course, the coordinates of the curve (or path) that was observed to have been produced by the organism under study.

RM: So whatever the merits of your equation for V might be (and one of its demerits is that it doesn’t specify how to compute the value of the radius, which you call R) it is not the V that they are talking about in the power law research.

Best

Rick

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

``````                  Alex:
``````

you have simply numerically checked a mathematical
equality by writing curvature as a function of
speed. Every single scientist working on the power
law has done that. But they don’t believe it is
such a big achievement.

``````          RM: If they know this then I don't understand why they
``````

still think that any observed relationship between R and V
would tell them anything about how the curves are
produced. There is simply no way for them to find any
relationship between R and V other than

``````            V
``````

= |dXd2Y-d2XdY| 1/3 *R1/3
(1)

I am obviously systematically failing at making myself understandable to Rick (I don’t know about the rest).Â

First, Rick, your demo hardly proves anything because you inject ad hoc temporal dynamics in the references whose lawful (or unlawful) properties will simply be reflected by the control system, which does hardly more than integrating them.Â

Second, you are stubbornly confused about the difference between a mathematical relation (that allows to re-express curvature as a function of speed, plus another non-constant term that you insist in ignoring and treating like a constant), between a physical realization (the fact that one can in principle draw the same curved line at infinitely different speeds), and between a biological fact (that out of all possible combinations of speed and curvature, living beings are, for yet some unknown reason but there are tens if not hunddreds of papers making proposals constrained following the power laaw, which means that geometry interacts with kinematics in a very precise way; plus the more puzzling fact that changing medium dynamics, changes the exponent).

But you will now reply for the n-th time saying that everybody that has ever worked on the power-law misses the point of control systems and that your toy demo proves they don’t get it. But, again, you gloss over serious flaws interpreting the difference between mathematical equations, physical conditions and biological constraints as facts, and you magnify the relevance of a toy demos that, I wish could shed new light, but so far don’t shed much new to the problem.

So I encourage you (and everyone still reading these email exchanges) to say something new and relevant, because I still believe that asking what is being perceived and what is being controlled is worth-while in figuring out why speed and curvature are constrained they way they are.

Alex

···

On Tue, Jul 19, 2016 at 12:17 AM, Richard Marken rsmarken@gmail.com wrote:

[From Rick Marken (2016.07.18.1515)]

Martin Taylor (2017.07.18.14.13)

``````MT: The question isn't about geometry, it's about velocity. V =
``````

d(distance)/d(time). Your V is just a measure of local curvature.
There’s no time in it at all. Alex keeps telling you as much. His
original question was about why people slow down at sharp curves,
not about how you describe curves in a Cartesian space.

``````V(angular velocity) = (d(distance travelled)/d(time))/R for a
``````

portion of a circular arc of radius R.

RM: In determining the power relationship between V and R, V (velocity) is measured as:

Â

where X.dot and Y.dot are Newton’s notation for the time derivatives of X (dX/dt) and Y(dY/dt) respectively.Â This measure of V is the variable that is actually computed from the curved paths that are observed in the velocity/curvature power law studies.Â This measure of velocity is then related to the R measure of Â curvature, which is computed as follows:

RM: Equation 1 is found by solving the two equations above, for V and R, simultaneously.Â X and Y are, of course, the coordinates of the curve (or path) that was observed to have been produced by the organism under study.

RM: So whatever the merits of your equation for V might be (and one of its demerits is that it doesn’t specify how to compute the value of the radius, which you call R) it is not the V that they are talking about in the power law research.Â

Best

Rick

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

``````                  Alex:
``````

you have simply numerically checked a mathematical
equality by writing curvature as a function of
speed. Every single scientist working on the power
law has done that. But they don’t believe it is
such a big achievement.

``````          RM: If they know this then I don't understand why they
``````

still think that any observed relationship between R and V
would tell them anything about how the curves are
produced. There is simply no way for them to find any
relationship between R and V other thanÂ

``````            V
``````

= Â |dXd2Y-d2XdY|Â 1/3Â *R1/3Â Â Â
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (1)