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Planar Juggling of a Devil-Stick using Discrete VHCs

Aakash Khandelwal, Ranjan Mukherjee

TL;DR

This work tackles planar devil-stick juggling under impulsive actuation by formulating discrete virtual holonomic constraints (DVHCs) that tie the center-of-mass trajectory to the stick orientation at impulse instants. The resulting discrete zero dynamics (DZD) provide stability conditions, enabling the design of an impulse-based controller that enforces the DVHC and stabilizes a desired 2-periodic juggling orbit within the inertial frame. A Poincaré-map-based orbital stabilization method (ICPM) with discrete LQR gains ensures convergence to the target orbit, demonstrated through simulations that enforce the DVHC and stabilize both generic and symmetric juggling orbits. The approach yields a rich set of stable juggling motions without requiring symmetric configurations, with potential for extension to robot underactuated manipulation under unilateral constraints and impact laws. The results establish DVHCs as a robust design tool for trajectory shaping and stability analysis in impulsive, underactuated hybrid systems.

Abstract

Planar juggling of a devil-stick using impulsive inputs is addressed using the concept of discrete virtual holonomic constraints (DVHC). The location of the center-of-mass of the devil-stick is specified in terms of its orientation at the discrete instants when impulsive control inputs are applied. The discrete zero dynamics (DZD) resulting from the choice of DVHC provides conditions for stable juggling. A control design that enforces the DVHC and an orbit stabilizing controller are presented. The approach is validated in simulation.

Planar Juggling of a Devil-Stick using Discrete VHCs

TL;DR

This work tackles planar devil-stick juggling under impulsive actuation by formulating discrete virtual holonomic constraints (DVHCs) that tie the center-of-mass trajectory to the stick orientation at impulse instants. The resulting discrete zero dynamics (DZD) provide stability conditions, enabling the design of an impulse-based controller that enforces the DVHC and stabilizes a desired 2-periodic juggling orbit within the inertial frame. A Poincaré-map-based orbital stabilization method (ICPM) with discrete LQR gains ensures convergence to the target orbit, demonstrated through simulations that enforce the DVHC and stabilize both generic and symmetric juggling orbits. The approach yields a rich set of stable juggling motions without requiring symmetric configurations, with potential for extension to robot underactuated manipulation under unilateral constraints and impact laws. The results establish DVHCs as a robust design tool for trajectory shaping and stability analysis in impulsive, underactuated hybrid systems.

Abstract

Planar juggling of a devil-stick using impulsive inputs is addressed using the concept of discrete virtual holonomic constraints (DVHC). The location of the center-of-mass of the devil-stick is specified in terms of its orientation at the discrete instants when impulsive control inputs are applied. The discrete zero dynamics (DZD) resulting from the choice of DVHC provides conditions for stable juggling. A control design that enforces the DVHC and an orbit stabilizing controller are presented. The approach is validated in simulation.

Paper Structure

This paper contains 20 sections, 2 theorems, 73 equations, 7 figures.

Table of Contents

  1. Introduction
  2. System Dynamics
  3. System Description
  4. Hybrid Dynamic Model
  5. Discrete Virtual Holonomic Constraints
  6. Constraints on Positions
  7. Constraints on Velocities
  8. Control Design Enforcing the DVHC
  9. Discrete Zero Dynamics
  10. Derivation
  11. Stable Juggling
  12. Stabilization of Juggling Motion
  13. Orbit Selection
  14. Orbital Stabilization
  15. Simulation
  16. ...and 5 more sections

Key Result

Theorem 1

The dynamical system in eq:zerodyn is periodic with period-2, i.e., $\theta_{k+2} = \theta_k$ and $\omega_{k+2} = \omega_k$, if and only if where $\sigma$ is a constant, and $\beta \in \mathbb{R}$ is an arbitrary constant.

Figures (7)

  • Figure 1: A devil-stick in the vertical plane with configuration variables $(h_x, h_y, \theta)$, and control variables $(I, r)$.
  • Figure 2: Stabilization of 2-periodic juggling of a devil-stick from arbitrary initial conditions: (a)-(b) show the components of $\rho_k$, (c)-(d) show the components of $D\rho_k$, (e) shows the pre-impact angular velocity $\omega_k$, (f) shows the time-of-flight $\delta_k$ (g) shows the applied impulse $I_k$, and (h) shows the point of application $r_k$ of the impulsive force.
  • Figure 3: Trajectory of the center-of-mass of the devil-stick corresponding to the results in (a) Fig. \ref{['fig:sim-vhc']} and (b) Fig. \ref{['fig:sim-orbit']}.
  • Figure 4: Orbital stabilization of 2-periodic juggling of a devil-stick from arbitrary initial conditions; $\delta_k = 0.5442$ s $\forall k$ corresponding to stable juggling: (a)-(b) show the components of $\rho_k$, (c)-(d) show the components of $D\rho_k$, (e) shows the pre-impact angular velocity $\omega_k$, (f) shows the time-of-flight $\delta_k$ (g) shows the applied impulse $I_k$, and (h) shows the point of application $r_k$ of the impulsive force.
  • Figure 5: Stabilization of 2-periodic juggling of a devil-stick between orientations symmetric about the vertical from arbitrary initial conditions: (a)-(b) show the components of $\rho_k$, (c)-(d) show the components of $D\rho_k$, (e) shows the pre-impact angular velocity $\omega_k$, (f) shows the time-of-flight $\delta_k$ (g) shows the applied impulse $I_k$, and (h) shows the point of application $r_k$ of the impulsive force.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 5
  • Remark 6
  • ...and 1 more