|For tasks such as collective transport, a multi-robot formation must preserve its global shape in order to prevent damage to the carried object. We focus on persistent leader- follower formations, which maintain local distance constraints in order to preserve the global shape of the formation. In this theoretical model of persistence theory, robots are modeled as vertices and dependencies between robots are represented as directed edges. To allow persistent formations to maintain persistence after mechanical failure, we incorporate redundancy into the existing theoretical model. We define persistence circuits to be persistent formations which become minimally persistent after the loss of any edge, present one method for creating these formations, and prove that persistent leader-follower formations cannot be redundantly persistent. For persistent leader- follower formations, we focus on a more restricted notion of redundancy and present 3 methods for constructing these formations. Finally, we present simulation results for 3 multi-robot formations simulated using Webots to evaluate behavior between formations with and without redundancy.