Optimal Shape Changes for Robot Teams

R 3 for a fixed shape orientation. Using a self-dual IPM solver, we show that optimal solutions for large-scale formations (e.g. thousands of robots) can be computed in less than 1 second using a standard desktop PC. We also investigate the computational complexity of our approach by solving the SOCPs using a logarithmic-barrier (central-path following) method. The most computationally expensive step of this method is solving the associated KarushKuhn-Tucker (KKT) linear system of equations. Through augmenting and restructuring the KKT matrix, we show that the system can be solved in O(m) time, where m denotes the number of robots in the formation. Employing this result, we demonstrate through extensive empirical simulations (10,000 trials for systems of up to 1,000 robots) that optimal solutions for a formation can be computed in O(m log m) time in practice. It is anticipated that these results will prove useful for extending the mission lives of both large and small-scale mobile robot formations and Mobile Ad-hoc Networks (MANETs).