Distributed Laplacian Eigenvalue and Eigenvector Estimation in Multi-robot Systems

Abstract

In many multi-robot systems applications, obtaining the spectrum and the eigenvectors of the Laplacian matrix provides very useful information. For example, the second smallest eigenvalue, and the corresponding eigenvector, can be used for connectivity maintenance (see for example Freeman et al., Stability and convergence properties of dynamic average consensus estimators, 2006, [5]). Moreover, as shown in Zareh et al. (Decentralized biconnectivity conditions in multi-robot systems, 2016, [22], Enforcing biconnectivity in multi-robot systems, 2016, [23]), the third smallest eigenvalue provides a metric for ensuring robust connectivity in the presence of single robot failures. In this paper, we introduce a novel decentralized gradient based protocol to estimate the eigenvalues and the corresponding eigenvectors of the Laplacian matrix. The most significant advantage of this method is that there is no limit on the multiplicity of the eigenvalues. Simulations show the effectiveness of the theoretical findings.