Juggling a Devil-Stick: Hybrid Orbit Stabilization Using the Impulse Controlled Poincaré Map

Nonprehensile Manipulation of a Stick Using Impulsive Forces

“…Section VII considers the special cases of planar juggling and steady precession. It is shown that previously derived expressions for planar symmetric juggling [9], [10] can be recovered from the current, more general, formulation. The steady precession of the stick is shown to be a limiting case of the hybrid dynamics.…”

“…2.1]. To stabilize the fixed point Ȳ * , we use the ICPM approach [8], which was developed for continuous orbits, and later extended to hybrid orbits associated with planar stick juggling [10]. The ICPM approach is explained with the help of Fig.…”

“…The choice ∆α * = π reduces the problem to that of planar symmetric juggling, which was discussed in [9], [10]. Setting ∆α * = π in (36), yields h *…”

“…An open-loop strategy for planar stick-juggling was presented by Schaal and Atkeson [24]; the dynamics of the system was however not completely modeled. The complete dynamic model and closed-loop control designs for planar stick-juggling were presented by Kant and Mukherjee in [9], [10]. Nonprehensile manipulation of a stick in three-dimensional space has not appeared in the literature and this work is the first to present a hybrid dynamic model that lends itself to closed-loop control design.…”

“…The dynamic model represents an underactuated system with five generalized coordinates and three control inputs; the three-dimensional juggling problem is more challenging than the planar case where the stick is described by three generalized coordinates and two control inputs. The Impulse Controlled Poincaré Map (ICPM) approach [8], [10], in which impulsive forces are intermittently applied on a Poincaré section, is used for stabilizing the hybrid orbit that describes the desired juggling motion.…”