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Modeling and control of the rodwheel

Luc Jaulin

TL;DR

This work derives a complete state-space model for a rodwheel—a wheel with an axially actuated rod—by applying a Lagrangian framework to a frictionless plane, incorporating non-holonomic constraints via D'Alembert's principle and symbolic computation to obtain tractable expressions. The authors implement a Runge–Kutta simulator for the resulting dynamics and demonstrate that, with no control, the system exhibits unstable, potentially chaotic behavior; they then propose simple, single-input controllers to stabilize the rod in the upward position and drive forward motion, even accounting for stand-angle perturbations. Key contributions include the symbolic generation of the mass/inertia and constraint structures ($\mathbf{M}(\mathbf{q})$, $\mathbf{b}(\mathbf{q},\dot{\mathbf{q}})$) and a practical control demonstration that stabilizes speed and rod orientation under certain conditions. The work also identifies a heading-controllability limitation near $\theta=0$ and recommends adding an inertial element to achieve complete maneuverability, highlighting the rodwheel as a minimal yet capable mobile robot with potential for fast, lightweight operation in urban environments.

Abstract

The rodwheel is a wheel equipped with a rod motorized on the axle. This paper proposes a Lagrangian approach to find the state equations of the rodwheel rolling on a plane without friction. The approach takes advantage of a symbolic computation. A controller is proposed to stabilize the rodwheel with the rod upward and going straight at a desired speed.

Modeling and control of the rodwheel

TL;DR

This work derives a complete state-space model for a rodwheel—a wheel with an axially actuated rod—by applying a Lagrangian framework to a frictionless plane, incorporating non-holonomic constraints via D'Alembert's principle and symbolic computation to obtain tractable expressions. The authors implement a Runge–Kutta simulator for the resulting dynamics and demonstrate that, with no control, the system exhibits unstable, potentially chaotic behavior; they then propose simple, single-input controllers to stabilize the rod in the upward position and drive forward motion, even accounting for stand-angle perturbations. Key contributions include the symbolic generation of the mass/inertia and constraint structures (, ) and a practical control demonstration that stabilizes speed and rod orientation under certain conditions. The work also identifies a heading-controllability limitation near and recommends adding an inertial element to achieve complete maneuverability, highlighting the rodwheel as a minimal yet capable mobile robot with potential for fast, lightweight operation in urban environments.

Abstract

The rodwheel is a wheel equipped with a rod motorized on the axle. This paper proposes a Lagrangian approach to find the state equations of the rodwheel rolling on a plane without friction. The approach takes advantage of a symbolic computation. A controller is proposed to stabilize the rodwheel with the rod upward and going straight at a desired speed.
Paper Structure (13 sections, 37 equations, 6 figures)

This paper contains 13 sections, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Modeling
  3. State vector
  4. Orientation
  5. Lagrangian
  6. Euler-Lagrange equations
  7. Non holonomic constraints
  8. D'Alembert's principle
  9. State equations
  10. Control
  11. Simulation
  12. Controller
  13. Conclusion

Figures (6)

  • Figure 1: Disk rolling on a plane with a rod$$
  • Figure 2: Wheel rolling on a plane (the rod is not represented)
  • Figure 3: Evolution of rodwheel with no control
  • Figure 4: Rodwheel with $\theta(t)=0$ which accelerates to reach the speed of $\dot{\varphi}=2$
  • Figure 5: A tiny perturbation with respect to $\theta$ generates a fall of the robot
  • ...and 1 more figures

Theorems & Definitions (1)

  • Remark 1