Optimal partitioning for multi-vehicle systems using quadratic performance criteria

We consider the problem of characterizing a generalized Voronoi diagram that is relevant to a special class of area assignment problems for multi-vehicle systems. It is assumed that the motion of each vehicle is described by a second order mechanical system with time-varying linear or affine dynamics. The proposed generalized Voronoi diagram encodes information regarding the proximity relations between the vehicles and arbitrary target points in the plane. These proximity relations are induced by an anisotropic (generalized) distance function that incorporates the vehicle dynamics. In particular, the generalized distance is taken to be the minimum control effort required for the transition of a vehicle to an arbitrary target point with a small terminal speed at a fixed final time. The space we wish to partition corresponds to the union of all the terminal positions that can be attained by each vehicle using finite control effort. Consequently, the partition space has lower dimension than the state space of each vehicle. We show that, in the general case, the solution to the proposed partitioning problem can be associated with a power Voronoi diagram generated by a set of spheres in a five-dimensional Euclidean space for the computation of which efficient techniques exist in the relevant literature.