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 computationall y expensive step of this method is solving the associated Karu shKuhn-Tucker (KKT) linear system of equations. Through augmenting and restructuring the KKT matrix, we show that the system can be solved inO(m) time, where m denotes the number of robots in the formation. Employing this result, we demonstrate through extensive empirical simulations (10,00 trials for systems of up to 1,000 robots) that optimal soluti ons 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 mobi le robot formations and Mobile Ad-hoc Networks (MANETs).

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