Here’s how mathematicians might
define the sport of wrestling. A system composed of two mechanical agents
coupled via mechanical actions such as contact and collision. The aim of the
contest is for one agent to floor the other while maintaining its own balance.
The rest is just show business. That’s more or less exactly how researchers at
Utsunomiya University in Japan describe the sport in developing a mathematical
model of wrestling which they go on to test in a numerical simulation. The end
result is a pair of autonomous mechanical wrestlers that compete to topple each
other. Their model is simple in principle. Each wrestler is an inverted
pendulum on a cart that can move backwards and forwards, a bit like balancing a
pencil on your finger. These robot ‘wrestlers’ are joined at the tips by a
spring that can stretch and compress. That means one wrestler can pull or push
the other over. However, the opposing wrestler can take evasive action by
moving in a way that stabilises itself and unbalances its opponent. The contest
is over when one wrestler or the other falls to the ground. The question that researchers
tackle is how best to design an intelligent controller that outperforms its
opponent. The only action this controller can take is to move its cart
backwards or forwards. Although simple in principle, this problem turns out to
be hugely complex. In creating a mathematical model of the contest, researchers
identify 17 different parameters that influence the behaviour of the wrestlers.
These include the mass and length of the pendulum, the mass of the cart,
acceleration due to gravity, the various properties of the spring, friction and
so on.
Each wrestler can end the bout in
one of three configurations: standing up, having been pushed over or having
been pulled over. So it’s not hard to see that there are nine possible outcomes
in this contest. Of these, five permutations correspond to a draw, with both
wrestlers having been pushed or pulled to the ground or with both remaining
upright. The other four permutations correspond to a win for one side or the
other. Each controller knows its position and the position of its opponent. It
also knows how its own movement will produce a turning force that tends to
unbalance the inverted pendulum. The question that the controller must solve is
how to move in a way that maintains the upright position of its own pendulum
while exerting a turning force that unbalances the opponent. To simplify
matters, researchers assumed that the spring is rigid. But to avoid the trivial
situation in which one controller simply drags the other over using brute
force, they limit the impulse that each can produce. This turns the context
into something of a chess game. One problem is that the solution space becomes
so complex that the controllers cannot simulate it successfully and the
contests end with the winner more or less chosen at random. But it turns out
that when one controller has a short delay built in to its calculations; it
becomes about twice as successful as an opponent that does not have this delay.
That’s because undelayed controllers drive the system into complex states that
they can no longer control but the delayed controllers never reach these levels
of complexity and so turn out to be more successful. So far, all of this work
has been pure numerical simulation but they have ambitious plans.
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