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

  • * Corresponding author: L. M. Chen

    * Corresponding author: L. M. Chen 

The first author is supported by China Scholarship Council

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  • 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) $.

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


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

    Definitions for bearingMeasurement variableRigidity
    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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