Numerical study of a billiard in a gravitational field

Stabilization through weak and occasional interactions : a billiard benchmark

“…The uncontrolled wedge billiard model we consider in this paper leads to stabilization problems of various complexity depending on the value of the angle θ. The variety of dynamical phenomena the wedge billiard displays as a function of the angle θ has been first studied by Lehtihet et al (1986) (see also subsequent studies by Szeredi andGoodings (1993), Milner et al (2001)). The value of the wedge angle determines integrable, KAM (Kolmogorov-Arnold-Mauser, see Ott (1993) for an introduction) and chaotic regions in the phase space.…”

“…Following Lehtihet et al (1986), we use the state variables V r = vr cos θ , V n = vn sin θ and E, the discrete state vector being We first review the derivation of the uncontrolled billiard map (Lehtihet et al (1986)). The flight map is then entirely determined by the wedge geometry, that is by the parameter α = tan θ.…”

Open-Loop stabilization of 2d impact juggling

“…The ball is assumed to be an unit mass point, let v = v r e r + v n e n denotes its velocity. Following (Lehtihet and Miller, 1986), the state variables are V r = vr cos θ , V n = vn sin θ and E, the total energy of the ball, the discrete state vector being:…”

“…It is still amenable to mathematical analysis but presents new features compared to the 1D bouncing ball. The rich dynamical properties of the elastic wedge billiard were first studied in (Lehtihet and Miller, 1986). A model including actuation of the edges has recently been studied by (Sepulchre and Gerard, 2003) for feedback stabilization of (unstable) periodic orbits.…”

Robotics and neuroscience: A rhythmic interaction

“…In an unactuated (fixed) and elastic (e = 1) Wiper, the periodic orbits previously described do exist but are not stable (Lehtihet & Miller, 1986). Real Wiper is obviously not elastic, and requires therefore an actuation to maintain a steady-state pattern.…”