Open-Loop stabilization of 2d impact juggling

Abstract

The paper studies the properties of a sinusoidally vibrating wedge billiard as a model for 2D impact juggling. It is shown that some periodic orbits that are unstable in the elastic fixed wedge become exponentially stable in the (non-)elastic vibrating wedge. These orbits are linked with some classical juggling patterns, providing an interesting benchmark for the study of the frequency-locking properties in human rhythmic tasks.

Timing Feedback Control of a Rhythmic System

“…The planar wedge-billiard we consider has been previously presented in [5], [6], [7]. This robotic device is viewed as an idealization of a human juggler: the ball in a constant gravitational field undergoes collisions with two edges, acting the juggler arms.…”

Oscillators as Systems and Synchrony as a Design Principle

“…Because of the coupling, this transparent description of the dynamics is lost when the edges are not orthogonal to each other and when the actuation of the wedge is an oscillatory motion around the fixed vertex instead of an axial vibration of each edge separately. The analysis in [20,21] nevertheless shows that the exponential stabilization of period two orbits persists over a broad range of parameters, even in this generalized situation.…”

“…This sensorless stabilization phenomenon is rather surprising because an exponentially unstable periodic orbit of the unactuated wedge becomes exponentially stable in the actuated wedge in spite of any feedback measurement. The analysis in [20,21] shows that the phenomenon persists over a broad range of angles θ and, when the collisions are nonelastic, over a broad range of coefficients of restitution. Recent experimental validation of this sensorless stabilization suggests that the phenomenon is also quite robust.…”

“…In particular, for θ > 45 deg, the system possesses an infinite family of periodic orbits, each of which is exponentially unstable. In recent work [20,21], we have shown that period one and period two orbits can be stabilized by proper oscillatory actuation of the wedge. This sensorless stabilization phenomenon is rather surprising because an exponentially unstable periodic orbit of the unactuated wedge becomes exponentially stable in the actuated wedge in spite of any feedback measurement.…”

“…We describe here a manifestation of synchrony as a design principle in a very contrived but illuminating example: the stabilization of periodic orbits in the bounce juggler model illustrated in Figure 1. This toy stabilization problem captures several important issues of impact control problems and is the subject of ongoing research [23,3,20,21]. Here we only describe the problem in its simplest configuration and underline the phase-locking properties of the stabilized system.…”