Stabilization of periodic orbits in a wedge billiard

Abstract

Abstract-This paper introduces a stabilization problem for an elementary impact control system in the plane. The rich dynamical properties of the wedge billiard, combined to the relevance of the associated stabilization problem for feedback control issues in legged robotics make it a valuable benchmark for energy-based stabilization of impact control systems.

Open-Loop stabilization of 2d impact juggling

“…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. (Schaal and Atkeson, 1993) mentioned the wedge billiard as an example of juggling device which can stabilize period-1 orbits (see Section 6) using open-loop control.…”

“…The present paper uses the model of (Sepulchre and Gerard, 2003) to show the existence of stable period-2 orbits in the (non-)elastic vibrating wedge, i.e. when a sinusoidal motion of the two edges is used to restore the energy dissipated in the impacts.…”

Robotics and neuroscience: A rhythmic interaction

“…In Section 3, we detailed the theoretical and experimental results obtained in robotics control of Wiper's periodic orbits (Gerard & Sepulchre, 2005;Ronsse et al, 2006Ronsse et al, , 2007Sepulchre & Gerard, 2003). Sensorless open-loop control is efficient for the simplest juggling pattern, revealing a 2D analogy with the stable bouncing ball under sinusoidal actuation (Bapat et al, 1986;Guckenheimer & Holmes, 1986;Holmes, 1982).…”

“…The impact model is only a crude approximate of real impact dynamics, since for example it does not capture spin effects of the puck at impact (Spong, 2001). The complete dynamics of Wiper under these simplifying assumptions has been derived in previous papers (Gerard & Sepulchre, 2005;Ronsse, Lefevre, & Sepulchre, 2006Sepulchre & Gerard, 2003).…”

“…Assuming an orthogonal wedge angle and parallel actuations of the edges (that is, the two edges are assumed to remain aligned with the two orthogonal axes of Fig. 4), the 2D motion of the juggled objects projects on each axis to a 1D motion that is unaffected by the bounces on the other axis (Ronsse et al, 2007;Sepulchre & Gerard, 2003), see Fig. 4.…”