Re: our work (fwd)

STUART JOSHUA MICHAEL (stuartj@rintintin.Colorado.EDU)
Tue, 14 Apr 1998 13:35:48 -0600 (MDT)

On Tue, 14 Apr 1998, Darren Kelly wrote:

> Why not simply impose contraints on the possible steps ? The chaotic
> selection must be made from a set anyway. Just introduce a mapping
> from each position to all next possible positions. The chaos then selects
> a path through possibilities. This, by the way, is exactly how chaotic
> "motion" through quantum processes proceeds. There are infinitely
> many ways to move through a finite set of possibilities. The possible
> outcomes (destinies) are constrained, yet chaotically random.
> This also affords the possibilty of weighting the probability of moving
> from one position to another (again exactly what happens with particles
> in chaotic quantum systems). Chaos doesn't imply the impossible. Why
> assume dancers should make impossible transitions ? Chaotic electrons
> don't.
> Darren


Thanks for your comments! You've raised an interesting point
and I'll try to address it. I agree with you in the sense that
your approach is asthetically and theoretically pleasing. It's just
hard for me to think of a practical way to implement it, and for
that reason I haven't pursued it.

The approach you mentioned is a good idea, but it has some tricky
aspects to it. What you described is exactly analogous to the
random walk of a markov process, only instead of using randomness
to choose from a set of successor states, chaos is somehow used.

One difficulty is to really make the decision genuinely chaotic
and not disguised randomness. Even if we can represent all
possible successor states of a given body position, how should
we order them? If their order is random then we're already lost.
So we need an ordering that is consistent throughout the walk, which
implies we need a metric to compare body positions. What metric do we
use to score the transitions from the current state to a successor
state? We could use a kinematic-based metric that estimates the
amount of work required to go from one state to the another, or
a probabilistic metric that estimates how likely it is to go from
one state to another or a combination of the two. Now, to realize
the decision aspect of the walk we simply map points from a
chaotic trajectory into the interval [1,N] and let the chaos choose
between the N ordered next states.

There only remains the question of determining
the successors of some body position. Since we could
never represent the entire space of body positions we would need to
come up with the successor states on-the-fly, which means we need
a function that takes in the state of the body as a parameter and
returns a list of possible next states. The rules we use cannot
treat joints independantly of one another since they would ignore
aspects of coordination and since the number of possible states they
would come up with would still be too large to handle (for instance,
if we're modelling a body that has 20 joints and each joint
can stay in its current possition or move to only one other position,
then we'd have 2^20 = milliion possible successor states even for this
limited "move/dont-move" character). If you wanted to use more
realistic models, you'd probably want to generate successor states
based also on the *history* of movement leading up to the current
position, not just the current position itself.

To be honest, this approach still seems attractive and interesting and
I was considering taking this attack myself in the early stages of the
project but was overwhelmed with the difficulties at the time. I
still have no ideas about how to formulate the successor-generating
rules, but would love to hear more of your thoughts on the subject!

Take care,

Josh Stuart
Professional Research Assistant for Dr. Elizabeth Bradley
Dept. of Computer Science / College of Engineering & Applied Science
University of Colorado
Campus Box 430
Boulder, Colorado 80309-0430
(303) 492-8425