Multi-agent rigid formations: A study of robustness to the loss of multiple agents

In this paper we study the robustness of information architectures to control a formation of autonomous agents. If agents are expected to work in hazardous environments like battle-fields, the formations are prone to multiple agent/link loss. Due to the higher severity of agent loss than link loss, the main contribution of this paper is to propose information architectures for shape-controlled multi-agent formations, which are robust against the loss of multiple agents. A formation is said to be rigid if by actively maintaining a designated set of inter-agent distances, the formation preserves its shape. We will use the rigidity theory to formalize the robust architecture problem. In particular we study the properties of formation graphs which remain rigid after the loss of any set of up to k-1 vertices. Such a graph is called k-vertex rigid. We provide a set of distinct necessary and sufficient conditions for these graphs. We then show that 3-vertex rigidity is the highest possible robustness one can achieve by just adding a small number of edges to a minimally rigid graph, i.e. retention of rigidity given the loss of 3 or more agents of a formation requires many more inter-agent distances to be specified than when maintaining rigidity with no, one or two agent losses. Based on this result, we further focus on 3-vertex rigid graphs and characterize a class of information architectures (with minimum number of control links) which are robust against the loss of up to two agents.