Bearing rigidity and formation stabilization for multiple rigid bodies in $ SE(3) $

Liangming Chen; Ming Cao; Chuanjiang Li

doi:10.3934/naco.2019017

Article Contents

2019, Volume 9, Issue 3: 257-267. Doi: 10.3934/naco.2019017

This issue Previous Article Preface Next Article A resilient convex combination for consensus-based distributed algorithms

Bearing rigidity and formation stabilization for multiple rigid bodies in $ SE(3) $

1.
Faculty of Science and Engineering, University of Groningen, 9747 Groningen, The Netherlands
2.
Department of Control Science and Engineering, Harbin Institute of Technology, 150001, China

^* Corresponding author: L. M. Chen
^* Corresponding author: L. M. Chen

Received: April 2018

Revised: December 2018

Published: May 2019

The first author is supported by China Scholarship Council

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Abstract

In this work, we first distinguish different notions related to bearing rigidity in graph theory and then further investigate the formation stabilization problem for multiple rigid bodies. Different from many previous works on formation control using bearing rigidity, we do not require the use of a shared global coordinate system, which is enabled by extending bearing rigidity theory to multi-agent frameworks embedded in the three dimensional $ special \; Euclidean \; group $ $ SE(3) $ and expressing the needed bearing information in each agent's local coordinate system. Here, each agent is modeled by a rigid body with 3 DOFs in translation and 3 DOFs in rotation. One key step in our approach is to define the bearing rigidity matrix in $ SE(3) $ and construct the necessary and sufficient conditions for infinitesimal bearing rigidity. In the end, a gradient-based bearing formation control algorithm is proposed to stabilize formations of multiple rigid bodies in $ SE(3) $.

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Mathematics Subject Classification: Primary: 93D15; Secondary: 93D05.

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Table 1. Comparison of different definitions for bearing and bearing rigidity

Definitions for bearing	Measurement variable	Rigidity
Angle in 2D space	$\theta_{ij}$	Parallel bearing rigidity
Unit vector in a global frame	$\frac{p_j-p_i}{\|\|p_j-p_i\|\|}$	Bearing rigidity in $\mathbb{R}^d$
Unit vector in $SE(2)$	$T(\theta_i)\frac{p_j-p_i}{\|\|p_j-p_i\|\|}$	Bearing rigidity in $SE(2)$

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