Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization

TL;DR

This paper extends bearing rigidity theory to arbitrary dimensions, establishing conditions for unique formation determination and proposing control laws for bearing-only formation stabilization with almost global convergence.

Contribution

It generalizes bearing rigidity results to higher dimensions and develops nonlinear control laws for bearing-only formation stabilization.

Findings

Frameworks are uniquely determined by bearings if and only if infinitesimally bearing rigid.

Proposed control laws achieve almost global stabilization of formations.

Numerical simulations validate the theoretical results.

Abstract

A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings. While many previous works focused on the bearing rigidity of two-dimensional frameworks, a first contribution of this paper is to extend these results to arbitrary dimensions. It is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid. In this paper, the proposed bearing rigidity theory is further applied to the bearing-only formation stabilization problem where the target formation is defined by inter-neighbor bearings and the feedback control uses only bearing measurements. Nonlinear distributed bearing-only formation control laws are proposed for the cases with and without a global orientation. It is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations. Numerical simulations are provided to support the analysis.