Allocation of Formation Control Tasks Using Delaunay Triangulation and Tetrahedralization

This paper studies the task allocation for the formation control of multi-agent systems with directed graph topologies, where agents are required to form and preserve a specific geometric shape. A directed graph is said to be persistent if it is constraint-consistent and the underlying undirected graph is rigid. A formation of agents is constraint-consistent if every agent can satisfy all its distance constraints. In this context, based on Delaunay triangulation in 2-D and Delaunay tetrahedralization in 3-D space, we propose a novel task allocation algorithm applicable to distance-based formation topologies that guarantees constraint consistency of a desired formation of agents. Given the desired shape - a framework in space - the algorithm uses the Delaunay operations to build a network of rigid triangles/tetrahedrons representing the desired formation topology. The algorithm uses the shortest Euclidean distance-based task assignment to establish a desired topology in the form of directed triangulated or directed trilateral Laman graphs. The source agent of each edge is responsible for preserving the desired distance to the sink agent. The output of the proposed algorithm is a minimally persistent graph in 2-D or a minimally structurally persistent graph in 3-D. Examples are provided to demonstrate and verify the effectiveness of the proposed method.