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Rigidity Theory in SE(2) for Unscaled Relative Position Estimation using only Bearing Measurements

Daniel Zelazo 1 Antonio Franchi 2 Paolo Robuffo Giordano 3 
2 LAAS-RIS - Équipe Robotique et InteractionS
LAAS - Laboratoire d'analyse et d'architecture des systèmes
3 Lagadic - Visual servoing in robotics, computer vision, and augmented reality
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
Abstract : This work considers the problem of estimating the unscaled relative positions of a multi-robot team in a common reference frame from bearing-only measurements. Each robot has access to a relative bearing measurement taken from the local body frame of the robot, and the robots have no knowledge of a common reference frame. An extension of rigidity theory is made for frameworks embedded in the special Euclidean group SE(2) = R2 × S1. We introduce definitions describing rigidity for SE(2) frameworks and provide necessary and sufficient conditions for when such a framework is infinitesimally rigid in SE(2). We then introduce the directed bearing rigidity matrix and show that an SE(2) framework is infinitesimally rigid if and only if the rank of this matrix is equal to 2|V| − 4, where |V| is the number of agents in the ensemble. The directed bearing rigidity matrix and its properties are then used in the implementation and convergence proof of a distributed estimator to determine the unscaled relative positions in a common frame. Simulation results are given to support the analysis.
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Submitted on : Thursday, May 22, 2014 - 2:39:28 PM
Last modification on : Wednesday, June 1, 2022 - 3:59:20 AM
Long-term archiving on: : Friday, August 22, 2014 - 12:31:34 PM


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  • HAL Id : hal-00994989, version 1


Daniel Zelazo, Antonio Franchi, Paolo Robuffo Giordano. Rigidity Theory in SE(2) for Unscaled Relative Position Estimation using only Bearing Measurements. European Control Conference, ECC'14, Jun 2014, Strasbourg, France. ⟨hal-00994989⟩



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