Discrete VHCs for Propeller Motion of a Devil-Stick using purely Impulsive Inputs

TL;DR

This paper introduces a novel approach using discrete virtual holonomic constraints to control and stabilize propeller motion of a devil-stick with impulsive forces, addressing an underexplored problem in robotic juggling.

Contribution

It pioneers the use of discrete virtual holonomic constraints for orbital stabilization in impulsively controlled underactuated systems like the devil-stick.

Findings

Successful simulation validation of trajectory design.

Effective stabilization of propeller motion using proposed controllers.

Reduction to continuous forcing case in the limit of small rotation angles.

Abstract

The control problem of realizing propeller motion of a devil-stick in the vertical plane using impulsive forces applied normal to the stick is considered. This problem is an example of underactuated robotic juggling and has not been considered in the literature before. Inspired by virtual holonomic constraints, the concept of discrete virtual holonomic constraints (DVHC) is introduced for the first time to solve this orbital stabilization problem. At the discrete instants when impulsive inputs are applied, the location of the center-of-mass of the devil-stick is specified in terms of its orientation angle. This yields the discrete zero dynamics (DZD), which provides conditions for stable propeller motion. In the limiting case, when the rotation angle between successive applications of impulsive inputs is chosen to be arbitrarily small, the problem reduces to that of propeller motion under continuous forcing. A controller that enforces the DVHC, and an orbit stabilizing controller based on the impulse controlled Poincar\'e map approach are presented. The efficacy of the approach to trajectory design and stabilization is validated through simulations.