|
|
This is a joint project with Matthew T. Mason and Michael A. Erdmann at Carnegie
Mellon University. We model a multi-body collision in the impulse space as a
state transition diagram, where each state represents a phase during which
impacts are ``active'' at only a subset of the contact points. In the course of
the collision, an impact may become active and inactive multiple times,
depending on whether the two involved bodies are instantaneously penetrating
into each other or not. A state transition happens whenever an active impact
finishes restitution, or an inactive impact gets reactivated. The elastic
energy due to an impact is not only affected by the impulse at the corresponding
contact point, but also by other impulses exerted on the two involved bodies in
the duration of this impact. Poisson's impulse-based law of restitution could
result in negative energy. Our new law of restitution governs the loss of
elastic energy during the phase. The outcome of a collision depends on the
ratios of the contact stiffnesses rather than on their individual values.
In the representative case of a ball falling onto another on the table (shown to
the left below), two impacts, one between the two balls and the other between
the lower ball and the table. The curve describing the growth of the two
impulses, plotted red in the figure below to the right for a collision instance,
is bounded within a (cyan) ellipse due to that the elastic energy during the
collision stays non-negative. Also drawn in the figure are the two compression
lines, one for each impact. The impulse curve is monotonic in the sense that
neither of the two impulses will decrease in the course of the collision. The
sequence of impulse values output by the states thus either is finite or converges.
|
|
|
Billiard Robot
The collision model is applied to billiard shooting in which the cue stick
impacts the cue ball, which in turn impacts the pool table. The long term goal
is to design a robot able to play billiards with human-level skills based on
understanding of the mechanics. We will integrate work from other areas
including vision, control, mechanics, and planning. Existing billiard robot
systems focus either on the vision task of recognizing and localizing balls and
positioning the cue stick, or on performing simple shots under the guidance of a
learning algorithm, a fuzzy expert system, or gray decision theory. To our
knowledge, none of the developed systems perform shots based on the mechanics of
billiards, or have exhibited preliminary shooting skills.
A billiard with some initial velocity and angular velocity generally slides along a parabolic arc on the pool table and before it starts rolling along a straight trajectory. At the initial stage of this research, we are studying the motion of the cue ball imparted by the cue stick. This motion is a result of simultaneous collisions between the cue stick and the ball, and the ball and the pool table. Up til now, our effort has focused on developing an impact model for simultaneous frictional impacts. The topic has been a subject of controversy in order to be consistent with Coulomb's law of friction, Poisson's hypothesis of restitution, and the law of energy conservation.
Applying the collision model to billiard shooting, the system is driven by the normal impulses at the two contacts with the tangential impulses determined via contact mode analysis. The figure below on the left shows the trajectory generated by a modeled massé shot. The second figure illustrates the impact between the cue tip and the ball, and that between the ball and the table. The two figures on the right are of a shooting mechanism, developed with the help from Amir Degani and Ben Brown, both at Carnegie Mellon University. It includes a steel cue stick constrainted to linear motions by ball bearings inside an aluminum box. The cue stick can be elevated by adjusting the slope of the attached incline.
|
|
|
|
Last updated on Jul 26, 2008.