Stabilization through weak and occasional interactions : a billiard benchmark

Abstract

The paper addresses the stabilization of periodic orbits in a wedge billiard with actuated edges. It is shown how the rich dynamical properties of the open-loop dynamics, e.g. ergodicity properties and KAM curves, can be exploited to design robust stabilizing feedbacks with large basins of attraction.

Rhythmic Feedback Control of a Blind Planar Juggler

“…the steady-state angle (µ(t) θ), see Fig. 10 [11], [13], [24], [32], [34], [35]. Despite this simplifying assumption, the model predicted a parametric stability region of the period-one orbit, which is in excellent agreement with the experiments [24].…”

Sensorless stabilization of bounce juggling

“…Eliminating the radial velocities and from their update equations (16) (17) we find another relation between the normal velocities (18) Equations (15) and (18) yield (19) Thanks to (18) and (19), one finds and . These four variables being nonnegative, (18) must be equal to zero to be satisfied, such as (15) (20) The radial velocity and the impact position should then satisfy [see (16) and (3)] (21) (22) Geometrically, the period-two solutions are therefore characterized by two symmetrical parabolas.…”

“…They exploit the rhythmical feature of the task to design controllers with low-rate (discrete) feedback. In ongoing work, we use the planar device described in this paper for feedback control of the shower [18], [19], as well as for investigations on human bounce juggling.…”

Feedback Control of Impact Dynamics: the Bouncing Ball Revisited

“…Feedback control algorithms have also been proposed for the Bouncing Ball, including the mirror law algorithms developed in [2], [3], [4]. Thereafter, the Bouncing Ball has been used as a benchmark to study controllability and stabilization of impact dynamics [14], [15], [5], [16], [6] [17], [7], [18], [8], [19].…”