[Avery Andrews (9208060]
(Bill Powers 920805)
The linear position -> reference relationships works, but doesn't make
efficient use of the thrusters. Intuitively, the reason is this: when
you are far from the goal, the reference velocity is high. So you if
you are moving at it, it is also changing fast (since the reference
velocity is a linear function of position, and you are moving fast).
But when you are near the goal, you are moving slowly, so the reference
velocity is changing slowly. So although the target velocity is a
linear function of position, it is a nonlinear function of time, with
a steep slope far from the goal and a shallow one nearby. So the
acceleration, which is what the thrusters produce, will be way beyond
their maximum capacity far away, and way below it nearby (and the
results look pretty dumb on the screen). I in fact have a calculus
argument the acceleration will be k1^2*x, where k1 is the gain factor
in the second diagram, but there's a step I'm unsure of in the
Motivating the square-root taking: suppose you want to get from point A
to point B in minimum time (assuming newtonian physics), and energy
expenditure & avoiding obstacles is not an issue. Then you want to
go at maximum acceleration on the first half of your trip, and maximum
deceleration on the second (the time-position plot of the second.
half will look like that of a falling coffee-cup, with time reversed).
So, for x = position, v = velocity, and t = time, we have:
x = k*t^2 (i)
x = v*t (ii)
where k is determined by the acceleration, and we have cleverly chosen
our coordinate system so that x and t are both 0 at the moment of
So, solving for v as a function of x, we get:
t = (x/k)^(1/2)
v = t/x
v = k1*x^(1/2)
Of course, it would be interesting to get the exponent to be selected by
a reorganization process, but since it follows from fixed laws of
nature, it would be sensible for it to be hardwired (but maybe I will
experiment with getting it set by reorganization as well). As for where
it goes, I'm currently putting it into the function that turns a
position error signal into a velocity reference level, but maybe it
could be put into the relative position perception system as well.
I agree that when reorganization is multidimensional (one intrinsic
error controlling change in several control system parameters),
something like the method in Bill's simultaneous equational-solver is
needed - I just want to get a feel for the 1-dimensional case first.
The point is that what you want is random choice of direction, rather
than of the total magnitude of delta.
I'm not at all happy with the way the Alife discussion is going.
Diehard PCT-ers believe that living systems have to be made out
of control systems, but nobody really *knows* this yet. The only
way to actually find out is for models based on this idea to be
compared with others that aren't, and this is exactly the opportunity
that Alife provides. But PCT-ers will have to show up at the party
bearing goodies, rather than stand outside sneering at it.
What seems to me to be the most compelling criticism of Alife is that
the success of their models is judged by their producing behavior that
`merely' looks lifelike, rather than fitting any principled
specifications of what lifelikeness is. But our judgements of the
behavior of lifelike agents are not just an exiguous construction
based on our current culture, but the end result of 600 million years
or so of competitive R&D by multicellular organisms trying to fake
each other out, and so are not to be taken lightly, in my opinion.
A line I am thinking along goes like this: what the nervous system of
a living creature is is a continuous vector transducer whose outputs
influence its inputs. The expected behavior of such a system is a
non-viable combination of chaos and catatonia (outputs being forced to
extreme values by positive feedback effects), and the (putatively) only
way to avoid this consequence is to load the systems to the gills with
negative feedback loops. So the PCT perspective ought to improve the
design of Alife systems, and the analysis of Rlife ones (organisms).
I think this ought to be an adequate rationale for the pursuing the PCT
approach within Alife circles, but it's got to be followed up by flash demos.
Gatherings (nee Crowd) looks like a fine start in this direction, and
one of my motivations for astro was to see if one could construct a
system that could learn to handle itself with thrusters in a
frictionless environment, which is a rather challenging problem (as I
discovered while playing with one of my kids' toy hovercraft).