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

TL;DR

This paper introduces a Mobius transformation-based control method for unicycle robots to maintain a circular orbit within a nonconcentric circular boundary, ensuring stable motion and obstacle avoidance.

Contribution

It proposes a novel control approach using Mobius transformations to map nonconcentric circles to concentric ones, simplifying the control problem for unicycle robots.

Findings

Effective stabilization within nonconcentric circular boundaries

Successful obstacle avoidance via transformed plane control

Validated through simulations and experiments

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.