Rigidity and Persistence of Meta-Formations

This paper treats the problem of the merging of formations, where the underlying model of a formation is graphical. We first analyze the persistence of meta-formations, which are formations obtained by connecting several persistent formations. Persistence is a generalization to directed graphs of the undirected notion of rigidity. In the context of moving autonomous agent formations, persistence characterizes the efficacy of a directed structure of unilateral distance constraints seeking to preserve a formation shape. We derive then, for agents evolving in a two- or three-dimensional space, the conditions under which a set of persistent formations can be merged into a persistent meta-formation, and give the minimal number of interconnections needed for such a merging. We also give conditions for a meta-formation obtained by merging several persistent formations to be persistent

[1]  Brian D. O. Anderson,et al.  Use of meta-formations for cooperative control , 2006 .

[2]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[3]  Vincent D. Blondel,et al.  Three and higher dimensional autonomous formations: Rigidity, persistence and structural persistence , 2007, Autom..

[4]  C. Moukarzel.  An efficient algorithm for testing the generic rigidity of graphs in the plane , 1996 .

[5]  A. Stephen Morse,et al.  Operations on Rigid Formations of Autonomous Agents , 2003, Commun. Inf. Syst..

[6]  R. Murray,et al.  Graph rigidity and distributed formation stabilization of multi-vehicle systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[7]  Walter Whiteley,et al.  Some matroids from discrete applied geometry , 1996 .

[8]  Changbin Yu,et al.  Principles to control autonomous formation merging , 2006, 2006 American Control Conference.

[9]  B. Anderson,et al.  Directed graphs for the analysis of rigidity and persistence in autonomous agent systems , 2007 .

[10]  T. Samad,et al.  Formations of formations: hierarchy and stability , 2004, Proceedings of the 2004 American Control Conference.

[11]  Changbin Yu,et al.  Merging Multiple Formations: A Meta-Formation Prospective , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[12]  John Baillieul,et al.  Information patterns and Hedging Brockett's theorem in controlling vehicle formations , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[13]  B. Hendrickson,et al.  An Algorithm for Two-Dimensional Rigidity Percolation , 1997 .

[14]  Tiong-Seng Tay,et al.  Rigidity of multi-graphs. I. Linking rigid bodies in n-space , 1984, J. Comb. Theory, Ser. B.

[15]  B. Hendrickson,et al.  Regular ArticleAn Algorithm for Two-Dimensional Rigidity Percolation: The Pebble Game , 1997 .

[16]  Vincent D. Blondel,et al.  Primitive operations for the construction and reorganization of minimally persistent formations , 2006, ArXiv.

[17]  Camillo J. Taylor,et al.  A vision-based formation control framework , 2002, IEEE Trans. Robotics Autom..

[18]  W. Whiteley,et al.  Generating Isostatic Frameworks , 1985 .