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.

翻译：非均匀运动约束在机器人应用中普遍存在。地理围栏控制便是这样一种范式，要求机器人的运动必须限制在预定义的边界内。本文研究了在将单轮机器人稳定在期望圆形轨道的同时，将其运动限制在非同心外部圆形边界内的问题。我们的解决方案依赖于所谓的莫比乌斯变换概念，该变换在某些实际条件下可将两个非同心圆映射为一对同心圆，从而产生均匀的空间运动约束。此类莫比乌斯变换的选择由后设计分析中的一个二次方程的根决定，该方程决定了两个圆所包围的区域如何映射到两个平面上。我们证明，根据这些根的不同，该问题在变换后的平面上可表述为轨迹约束问题或避障问题。利用障碍李雅普诺夫函数的思想，我们提出了一种独特的控制律，该控制律可在变换后的平面上同时解决这两个对比鲜明的问题，并在实际平面上为原始问题提供解决方案。通过关联莫比乌斯变换及其逆映射下两个平面的参数，我们进一步建立了两个平面上控制律之间的联系，并确定了在实际平面上应用的控制律。文中提供了仿真和实验结果以阐明关键的理论进展。