# American Institute of Mathematical Sciences

September  2019, 9(3): 257-267. doi: 10.3934/naco.2019017

## 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

Received  April 2018 Revised  December 2018 Published  May 2019

Fund Project: The first author is supported by China Scholarship Council.

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 $SE(3)$.

Citation: Liangming Chen, Ming Cao, Chuanjiang Li. Bearing rigidity and formation stabilization for multiple rigid bodies in $SE(3)$. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 257-267. doi: 10.3934/naco.2019017
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##### References:
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 $SE(2)$
 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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