Distributed optimization on proximity network rigidity via robotic movements

This paper considers a rigidity optimization problem for mobile robotic teams modeled in a proximity network with state-dependent network topology. The aim is to move all robots´ positions to reach a configuration such that the worst-case rigidity metric can be maximized. Key properties of a Gramian matrix involving a weighted rigidity matrix are discussed for solving this optimization problem. We design a decentralized algorithm to update all robots´ positions to maximize the eigenvalue function, which requires local information from each robot itself and its neighbors. Furthermore, a distributed eigenvector estimation scheme based on inverse shifted power iteration method and averaging consensus algorithm is devised to allow each robot to estimate the global eigenvector information. Simulation results are also provided to demonstrate the effectiveness of the estimation and optimization scheme.