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Mobius Transformation-Based Circular Motion Control for Unicycle Robots in Nonconcentric Circular Geofences

Shubham Singh, Anoop Jain

TL;DR

This paper tackles stabilizing a unicycle robot around a circular orbit while keeping its trajectory within a nonconcentric circular geofence. It leverages a Möbius transformation to map the nonuniform boundary problem to a pair of concentric circles in a transformed plane, where a barrier Lyapunov function-based controller is designed to achieve both trajectory-constraining and obstacle-avoidance objectives depending on the transformation root. The authors derive precise relations between the actual and transformed planes, relate the corresponding control laws, and prove uniform boundedness of post-design signals. Simulations and experiments on a Khepera IV validate the approach, demonstrating convergence to the desired orbit and confinement within the outer boundary. The work offers a principled method to handle nonuniform spatial constraints in geofencing applications by transforming them into uniform constraints in an auxiliary plane and then mapping the solution back to the real world.

Abstract

Nonuniform motion constraints are ubiquitous in robotic applications. Geofencing control is one such paradigm where the motion of a robot must be constrained within a predefined boundary. This paper addresses the problem of stabilizing a unicycle robot around a desired circular orbit while confining its motion within a nonconcentric external circular boundary. Our solution approach relies on the concept of the so-called Mobius transformation that, under certain practical conditions, maps two nonconcentric circles to a pair of concentric circles, and hence, results in uniform spatial motion constraints. The choice of such a Mobius transformation is governed by the roots of a quadratic equation in the post-design analysis that decides how the regions enclosed by the two circles are mapped onto the two planes. We show that the problem can be formulated either as a trajectory-constraining problem or an obstacle-avoidance problem in the transformed plane, depending on these roots. Exploiting the idea of the barrier Lyapunov function, we propose a unique control law that solves both these contrasting problems in the transformed plane and renders a solution to the original problem in the actual plane. By relating parameters of two planes under Mobius transformation and its inverse map, we further establish a connection between the control laws in two planes and determine the control law to be applied in the actual plane. Simulation and experimental results are provided to illustrate the key theoretical developments.

Mobius Transformation-Based Circular Motion Control for Unicycle Robots in Nonconcentric Circular Geofences

TL;DR

This paper tackles stabilizing a unicycle robot around a circular orbit while keeping its trajectory within a nonconcentric circular geofence. It leverages a Möbius transformation to map the nonuniform boundary problem to a pair of concentric circles in a transformed plane, where a barrier Lyapunov function-based controller is designed to achieve both trajectory-constraining and obstacle-avoidance objectives depending on the transformation root. The authors derive precise relations between the actual and transformed planes, relate the corresponding control laws, and prove uniform boundedness of post-design signals. Simulations and experiments on a Khepera IV validate the approach, demonstrating convergence to the desired orbit and confinement within the outer boundary. The work offers a principled method to handle nonuniform spatial constraints in geofencing applications by transforming them into uniform constraints in an auxiliary plane and then mapping the solution back to the real world.

Abstract

Nonuniform motion constraints are ubiquitous in robotic applications. Geofencing control is one such paradigm where the motion of a robot must be constrained within a predefined boundary. This paper addresses the problem of stabilizing a unicycle robot around a desired circular orbit while confining its motion within a nonconcentric external circular boundary. Our solution approach relies on the concept of the so-called Mobius transformation that, under certain practical conditions, maps two nonconcentric circles to a pair of concentric circles, and hence, results in uniform spatial motion constraints. The choice of such a Mobius transformation is governed by the roots of a quadratic equation in the post-design analysis that decides how the regions enclosed by the two circles are mapped onto the two planes. We show that the problem can be formulated either as a trajectory-constraining problem or an obstacle-avoidance problem in the transformed plane, depending on these roots. Exploiting the idea of the barrier Lyapunov function, we propose a unique control law that solves both these contrasting problems in the transformed plane and renders a solution to the original problem in the actual plane. By relating parameters of two planes under Mobius transformation and its inverse map, we further establish a connection between the control laws in two planes and determine the control law to be applied in the actual plane. Simulation and experimental results are provided to illustrate the key theoretical developments.
Paper Structure (21 sections, 18 theorems, 86 equations, 9 figures)

This paper contains 21 sections, 18 theorems, 86 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A Review of Möbius Transformation
  3. The Möbius Transformation
  4. Mapping of Two Nonconcentric Circles to Concentric Circles
  5. Illustrative Examples
  6. System Model and Problem Description
  7. System Model
  8. Problem Description
  9. Problem Formulation and Control Design in the transformed plane
  10. System Model and Problem Formulation in Transformed Plane
  11. Control Design in Transformed Plane
  12. Relating Actual and Transformed Planes Parameters and Designing Actual Control Law
  13. Transformed Plane in Terms of Actual Plane Parameters
  14. Actual Plane in Terms of Transformed Plane Parameters
  15. Stringent Bounds on Post-Design Signals
  16. ...and 6 more sections

Key Result

Lemma 1

For any positive constant $\varpi$, let $\Lambda \coloneqq \{\zeta \in \mathbb{R} \mid |\zeta| < \varpi\} \subset \mathbb{R}$ and $\mathcal{N} \coloneqq \mathbb{R}^\ell \times \Lambda \subset \mathbb{R}^{\ell+1}$ be open sets. Consider the system $\dot{{\mathcal{X}}} = {h}(t, \mathcal{X})$, where,

Figures (9)

  • Figure 1: Case I: Circle $\mathcal{C}'$ encircles circle $\mathcal{C}$ and $\mu > 1 + |\lambda|$.
  • Figure 2: Case II: Circle $\mathcal{C}$ encircles circle $\mathcal{C}'$ and $\mu < 1 - |\lambda|$.
  • Figure 3: Mapping of the circles $\mathcal{C}$, $\mathcal{C}'$ and their enclosed regions under Möbius transformation \ref{['mobius_transformation']} with roots $\alpha_s$ and $\alpha_{\ell}$, defined by \ref{['alpha_s']} and \ref{['alpha_l']}, respectively.
  • Figure 4: Unicycle robot tracking desired circular orbit $\mathcal{C}$ with its motion bounded within nonconcentric circular boundary $\mathcal{C}'$.
  • Figure 5: Problem in the transformed plane with choices of roots $\alpha_s$ and $\alpha_{\ell}$.
  • ...and 4 more figures

Theorems & Definitions (41)

  • Definition 1: Barrier Lyapunov Function tee2009barrier
  • Lemma 1: Convergence with BLFtee2009barrier
  • Theorem 1: Mapping of nonconcentric circles to concentric circles
  • Lemma 2: Non-collinear points turyn2014advanced
  • Corollary 1: Roots of \ref{['mobius_roots']}
  • proof
  • proof : Proof of Theorem \ref{['thm_main_theorem']}
  • Theorem 2: Mapping of enclosed regions
  • proof
  • Example 1: $\mathcal{C}'$ encircles $\mathcal{C}$
  • ...and 31 more