Signed Angle Rigid Graphs for Network Localization and Formation Control

TL;DR

This paper introduces a comprehensive analysis of signed angle rigidity in graphs, establishing Laman graphs as key structures, and applies these findings to improve distributed network localization and formation control using signed angle measurements.

Contribution

It provides a complete combinatorial characterization of signed angle rigidity, linking it to Laman graphs, and proposes methods to construct minimal signed angle constraints for network applications.

Findings

Laman graphs characterize minimally signed angle rigid graphs.

A method to construct minimal signed angle constraints is proposed.

Applications to distributed network localization and formation control are demonstrated.

Abstract

Graph rigidity theory studies the capability of a graph embedded in the Euclidean space to constrain its global geometric shape via local constraints among nodes and edges, and has been widely exploited in network localization and formation control. In recent years, the traditional rigidity theory has been extended by considering new types of local constraints such as bearing, angle, ratio of distance, etc. Among them, the signed angle constraint has received extensive attention, since it is practically measurable and independent of the global coordinate frame. However, the relevant studies always consider special graph structures, which are sufficient but not necessary for signed angle rigidity. This paper presents a comprehensive combinatorial analysis in terms of graphs and angle index sets for signed angle rigidity. We show that Laman graphs equivalently characterize minimally signed angle rigid graphs. Moreover, we propose a method to construct the minimal set of signed angle constraints in a Laman graph to effectively ensure signed angle rigidity. These results are finally applied to distributed network localization and formation stabilization problems, respectively, where each agent only has access to signed angle measurements.