Redundantly Rigid Topologies in Decentralized Multi-Agent Networks

In this paper, we consider the problem of decentralized evaluation of redundant rigidity of a multi-agent network topology. Redundancy is an important restriction of a graph's rigidity property as it is a necessary condition for the global rigidity of a graph, a combinatorial notion that is vital for graph realizability with consequences for example in localization. Furthermore, redundant rigidity enforces additional structure which allows for link loss or failure while maintaining an underlying rigid topology. First, we map the combinatorial conditions for determining redundant rigidity to the context of a centralized pebble game. By exploiting intuition from a bipartite matching characterization, we demonstrate decentralized conditions based in graph search and agent agreement. Then we provide a complete description of a decentralized algorithm for evaluating redundant rigidity, exploiting auctions and boolean consensus. To illustrate our methods, we provide a brief simulation study of redundant rigidity evaluation and nonholonomic control that closes the paper.