Rigidity theory in SE(2) for unscaled relative position estimation using only 
bearing measurements

Zelazo, D; Franchi, A; Robuffo Giordano, P

doi:10.1109/ECC.2014.6862558

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Rigidity theory in SE(2) for unscaled relative position estimation using only bearing measurements

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https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6862558
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Citation

Zelazo, D., Franchi, A., & Robuffo Giordano, P. (2014). Rigidity theory in SE(2) for unscaled relative position estimation using only bearing measurements. In 13th European Control Conference (ECC 2014) (pp. 2703-2708). Piscataway, NJ, USA: IEEE.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0027-80A4-F

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 or inertial reference frame. A corresponding extension of rigidity theory is made for frameworks embedded in the ˘005Cemph\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 ˘005Cemph\{infinitesimally rigid\} in SE(2). Analogous to the rigidity matrix for point formations, we introduce the ˘005Cemph\{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|minus;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. Some simulation results are also given to support the analysis.