Research on formation control and cooperative localization for multirobot 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 not only 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, and 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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June 2019
Research-Article
Physical Interpretation of Rigidity for Bearing Formations: Application to Mobility and Singularity Analyses
Sébastien Briot,
Sébastien Briot
1
Laboratoire des Sciences du Numérique de Nantes (LS2N),
UMR CNRS 6004, Nantes 44321,
e-mail: sebastien.briot@ls2n.fr
Centre National de la Recherche Scientifique (CNRS)
,UMR CNRS 6004, Nantes 44321,
France
e-mail: sebastien.briot@ls2n.fr
1Corresponding author.
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Paolo Robuffo Giordano
Paolo Robuffo Giordano
Centre National de la Recherche Scientifique (CNRS),
Campus de Beaulieu, Rennes 35000,
e-mail: prg@irisa.fr
University of Rennes 2
, Inria, IRISA,Campus de Beaulieu, Rennes 35000,
France
e-mail: prg@irisa.fr
Search for other works by this author on:
Sébastien Briot
Laboratoire des Sciences du Numérique de Nantes (LS2N),
UMR CNRS 6004, Nantes 44321,
e-mail: sebastien.briot@ls2n.fr
Centre National de la Recherche Scientifique (CNRS)
,UMR CNRS 6004, Nantes 44321,
France
e-mail: sebastien.briot@ls2n.fr
Paolo Robuffo Giordano
Centre National de la Recherche Scientifique (CNRS),
Campus de Beaulieu, Rennes 35000,
e-mail: prg@irisa.fr
University of Rennes 2
, Inria, IRISA,Campus de Beaulieu, Rennes 35000,
France
e-mail: 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 April 9, 2019. Assoc. Editor: Raffaele Di Gregorio.
J. Mechanisms Robotics. Jun 2019, 11(3): 031006 (10 pages)
Published Online: April 9, 2019
Article history
Received:
October 8, 2018
Revision Received:
February 18, 2019
Accepted:
February 20, 2019
Citation
Briot, S., and Giordano, P. R. (April 9, 2019). "Physical Interpretation of Rigidity for Bearing Formations: Application to Mobility and Singularity Analyses." ASME. J. Mechanisms Robotics. June 2019; 11(3): 031006. https://doi.org/10.1115/1.4043050
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