A decentralized spatial partitioning algorithm based on the minimum control effort metric

We consider the problem of characterizing a spatial partition of the position space of a team of vehicles with double integrator kinematics. The proximity relations between the vehicles and an arbitrary target point in the partition space is the minimum control effort required for each vehicle to reach the latter point with zero miss distance and exactly zero velocity at a prescribed final time (both the finite and the infinite horizon are considered). We show that the solution to the generalized Voronoi partitioning problem can be associated with a class of affine diagrams whose combinatorial complexity is comparable to the standard Voronoi diagram. For the computation of the latter class of affine diagrams, we utilize a partitioning algorithm, which is decentralized in the sense that each vehicle can compute an approximation of its own cell independently from the other vehicles from the same team. Numerical simulations that illustrate the theoretical developments are also presented.

[1]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[2]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[3]  Daniel Reem,et al.  An Algorithm for Computing Voronoi Diagrams of General Generators in General Normed Spaces , 2009, 2009 Sixth International Symposium on Voronoi Diagrams.

[4]  Sonia Martínez,et al.  Coverage control in constant flow environments based on a mixed energy-time metric , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[5]  Kokichi Sugihara,et al.  Direct Diffusion Method for the Construction of Generalized Voronoi Diagrams , 2007, 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007).

[6]  Efstathios Bakolas,et al.  Optimal partitioning for spatiotemporal coverage in a drift field , 2013, Autom..

[7]  Efstathios Bakolas,et al.  Optimal partitioning for task assignment of spatially distributed vehicles based on quadratic performance criteria , 2013, 2013 American Control Conference.

[8]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[9]  J. Boissonnat,et al.  Algorithmic Geometry: Frontmatter , 1998 .

[10]  Efstathios Bakolas,et al.  Optimal partitioning for multi-vehicle systems using quadratic performance criteria , 2013, Autom..

[11]  Mariette Yvinec,et al.  Algorithmic geometry , 1998 .

[12]  Kokichi Sugihara,et al.  Why Are Voronoi Diagrams so Fruitful in Application? , 2011, 2011 Eighth International Symposium on Voronoi Diagrams in Science and Engineering.

[13]  Efstathios Bakolas,et al.  The Zermelo-Voronoi Diagram: a dynamic partition problem , 2010, Proceedings of the 2010 American Control Conference.

[14]  Kokichi Sugihara,et al.  Voronoi diagrams in a river , 1992, Int. J. Comput. Geom. Appl..