Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery of curves: ellipses, quartics, sextics, circles, and even a stationary point. Secondly, one notices this family conserves an intriguing ratio: Inradius-to-Circumradius. In turn this implies three conservation corollaries: (i) the sum of bounce angle cosines, (ii) the product of excentral cosines, and (iii) the ratio of excentral-to-orbit areas. Monge's Orthoptic Circle's close relation to 4-periodic Billiard trajectories is wellknown. Its geometry provided clues with which to generalize 3-periodic invariants to trajectories of an arbitrary number of edges. This was quite unexpected. Indeed, the Elliptic Billiard did surprise us!

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We consider the problem of parallel part manipulation, i.e., the simultaneous position and orientation control of one or more p arts in a bounded r egion of the plane. We propose a novel, minimalist device, based on a single horizontallyvibrating at plate. We show that a closed rigid motion of the plate, involving its 3 dofs, can be c omputed which produces desired average forces at a nite number of points, e.g., parts' locations. This implies that one or more p arts can follow independent trajectories simultaneously, as they interact with a single vibrating plate. This is in sharp contrast with more c omplex designs such as massively-parallel actuator arrays and or prehensile manipulation. Dynamic simulation is used to test the current method in two parallel part manipulation examples. A prototype of the device has been built with inexpensive parts; physical implementation of the proposed method is currently underway.

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Abstract-We analyze the dynamics of part motion for a novel type of planar parts feeder consisting of a longitudinally vibrating flat plate and a part placed on its surface. For each vibration cycle, the plate's velocity is held positive (forward motion) for a longer time than it is held negative (backward motion). This type of asymmetric vibration combined with the non-linear nature of Coulomb friction causes the part to accelerate along a straight line to a terminal velocity called the "feed rate". The average force exerted by the plate on the part is shown to be proportional to the latter's deviation from the feed rate. In other words, the part behaves as if it were immersed in a forward moving viscous fluid. Expressions for the feed rate and viscosity constants are derived with respect to various physical and control parameters. Rigid-body dynamic simulation results are shown to be in good agreement with the analysis.

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