1. Potential Fields for Maintaining Connectivity of Dynamic Graphs MEAM 620 Final Project Michael M. Zavlanos

2. Problem Formulation n mobile agents in an obstacle free workspace, with single integrator dynamics: dx i /dt = u i. state – dependent graph G(x): Nodes correspond to the agents and we draw an Edge between two nodes if their pairwise distance is smaller than some threshold R. Adjacency matrix: A(x). Graph Laplacian: L(x) = D(x) – A(x). λ 1 (L(x)) = 0 with corresponding eigenvector 1. λ 2 (L(x)) > 0 => G(x) is connected.

3. Background Yoonsoo Kim and Mehran Mesbahi, “On Maximizing the Second Smallest Eigenvalue of a State – Dependent Graph Laplacian”, IEEE Transactions on Automatic Control (to appear). Let P be a nx(n-1) projection matrix to the space perpendicular to the vector 1. L(x) positive semi-def. => P T L(x)P positive semi-def. λ 2 (P T L(x)P) > 0  P T L(x)P > 0

4. Potential Field Approach Eigenvalues of L(x): 0 = λ 1 <= λ 2 <= … <= λ n Eigenvalues of P T L(x)P : 0 <= λ 2 <= … <= λ n λ 2 (P T L(x)P) > 0  det(P T L(x)P) > 0 Control Law: Connectivity modeled as an obstacle. u i = d/dx i ( 1/det(P T L(x)P) )

5. Gradient of det(P T L(x)P) Let M(x) = P T L(x)P. We can show that: And hence: