[From Rick Marken (2016.09.18.1230)]
RM: I see everyone is off talking about Roger’s model (or non-model, as Martin would have it) so I’m probably just talking to myself but, what the heck, this is supposed to be a list about PCT so I’ll sprinkle in a little just for my own satisfaction.
Martin Taylor (2016.09.17.22.56) –
MT: True. Apart from misrepresenting what Roger wrote, they referred to
Roger’s words as if he had proposed a model,
RM: Well, it looks like a model to me so I’ll treat it as though it is one. In fact, this is a perfectly fine PCT model of a control system that controls velocity, Vp. A linear version of the system side of Roger’s model looks like this:
Vm = k*(Vr-Vp)
So the output of the control system, Vm, is proportional to the difference between the controlled perception, Vp, and the reference for that perception, Vr. What’s missing is a description of the perceptual function that produces Vp and the environmental feedback connection from system output, Vm, to controlled input. I will assume that Vp is the perception of the tangential velocity of the object that is moving. So I’ll define Vp is a function of the X, Y position of that object. So
Vp = k.i*Sqrt((dX/dt)^2 +(dY/dt)^2)
RM: That is, the perception of the velocity of the moved object is proportional (by the input constant k.i) to V as calculated in the power law studies. Now we have to figure out how Vm relates to Vp. In order to have an effect on Vp, the system must be able to affect dX/dt and dY/dt. That is, the system has to be able to change the position of the object in the X and Y dimensions independently. This means that the output of the control system can’t be a single variable, Vm. And a single control system can’t independently vary two variables. So it looks like we’re going to have to model control of velocity using two control systems, one controlling movement in the X dimension and the other controlling it in the Y dimension. But this would mean each system could control only a component of Vp, one controlling dX/dt and the other dY/dt. But that looks like the way it has to be. So velocity control (in two dimensions) requires two control systems, that could be defined as follows:
Px = dX/dt : Py = dY/dt
Ox = kx* (Rx-Px) : Oy = dY/dt
dX/dt = Ox= dx; dY/dt = Oy + dy
RM: So we have two control systems, one controlling movement in the X direction (Px) and the other controlling movement in the Y dimension (Py). The state of the variables controlled by each system is affected by the output of each system (Ox, Oy) and environmental disturbances (dx,dy).
RM: Neither of these two control systems alone controls tangential velocity, Vp. This has to be done by a higher level system that sends references to these systems controlling dX/dt and dY/dt). But a single higher level control system cannot vary the references for two lower level system independently and we know that people make movements where the movements in the X and Y dimensions are different. So how can we get the model to behave this way.
RM: This is where curvature might come in. Curvature, like tangential velocity,Vp, is a function of dX/dt and dY/dt. And it is a variable aspect of the movement that might also be controlled! So why not have curvature as a second, higher level controlled variable, Cp. So now we have two higher level control systems, one controlling Vp and the other Cp, where both Vp and Cp are functions of the perceptions controlled by the two lower level systems, Px and Py.
Vp =kv* sqrt(Px^2+Py^2)
Cp = kc* [|PxdY2/dt-dX2/dtPy)/sqrt(Px^2+Py^2)]
RM: Note that the perception of both velocity and curvature depend on Px and Py but curvature also depends on dX2/dt and dY2/dt. This suggests that I may need a couple more lower level systems to control these variables. Bad news. Clearly this requies more thought. But I think I will try to build this two level system and see what happens.
RM: But there are some important things that we have learned from this exercise. First, we need at least two control system in order to control the velocity of movement. In order to make the movement move at the desired velocity in the right direction we need another system to control curvature (or, now that I think of it, perhaps just a system that controls instantaneous direction rather than curvature). And, finally, curvature is a consequence of the same outputs (movement in the X and Y dimension) that affect velocity. So curvature is not a disturbance to the movement; it’s a consequence of output, as is velocity.
RM: I’ll try to make a diagram of this model and write a simulation asap. Again, just for my benefit; I now return you to your regularly schedule high integrity PCT program.
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
Let Vr be the intended (reference) velocity.
Let Vp be the perceived velocity.
Let Vm be the output velocity, where Vm is
some function of e.
If Vp=Vr, then e=0 and Vm will remain
If Vp>Vr, then Vm will be reduced until
RM: I guess my comments on Roger's model didn't make
much of an impression in this new, high integrity world of