Constrained Stabilization on the n-Sphere

TL;DR

This paper addresses the stabilization problem on the n-sphere with conic constraints by transforming it into a navigation problem in Euclidean space, enabling the use of existing obstacle avoidance algorithms.

Contribution

It introduces a novel approach using stereographic projection to solve stabilization on the n-sphere with constraints, leveraging Euclidean navigation algorithms.

Findings

Effective stabilization demonstrated on the 2-sphere.

The method handles conic constraints successfully.

Navigation algorithms are adaptable to spherical stabilization.

Abstract

We solve the stabilization problem on the $n-$sphere in the presence of conic constraints. We use the stereographic projection to map this problem to the classical navigation problem on $\mathbb{R}^n$ in the presence of spherical obstacles. As a consequence, any obstacle avoidance algorithm for navigation in the Euclidean space can be used to solve the given problem on the $n-$sphere. We illustrate the effectiveness of the approach using the kinematics of the reduced attitude model on the $2-$sphere.