Physical Interpretation of Rigidity for Bearing Formations: Application to Mobility and Singularity Analyses

[+] Author and Article Information
Sebastien Briot

Ecole Centrale de Nantes 1 rue de la Noë Nantes, 44321 France Sebastien.Briot@ls2n.fr

Paolo Robuffo Giordano

campus de Beaulieu Rennes, 35000 France prg@irisa.fr

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received October 8, 2018; final manuscript received February 18, 2019; published online xx xx, xxxx. Assoc. Editor: Raffaele Di Gregorio.

ASME doi:10.1115/1.4043050 History: Received October 08, 2018; Accepted February 20, 2019


Research on formation control and cooperative localization for multi-robot systems has been an active field over the last years. A powerful theoretical framework for addressing formation control and localization, especially when exploiting onboard sensing, is that of formation rigidity (mainly studied for the cases of distance and bearing measurements). Rigidity of a formation depends on the topology of the sensing/communication graph but also on the spatial arrangement of the robots, since special configurations (‘singularities’ of the rigidity matrix), which are hard to detect in general, can cause a rigidity loss and prevent convergence of formation control/localization algorithms based on formation rigidity.

The aim of this paper is to gain additional insights into the internal structure of bearing rigid formations by considering an alternative characterization of formation rigidity using tools borrowed from the mechanical engineering community. In particular, we show that bearing rigid graphs can be given a physical interpretation related to virtual mechanisms, whose mobility and singularities can be analyzed and detected in an analytical way by using tools from the mechanical engineering community (Screw theory, Grassmann geometry, Grassmann-Cayley algebra). These tools offer a powerful alternative to the evaluation of the mobility and singularities typically obtained by numerically determining the spectral properties of the bearing rigidity matrix (which typically prevents drawing general conclusions). We apply the proposed machinery to several case formations with different degrees of actuation, and discuss known (and unknown) singularity cases for representative formations. The impact on the localization problem is also discussed.

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