Geometry, optimization and control in robot coordination

expressionism “Ocean Park No. 27” and “Ocean Park No. 129” by Richard Diebenkorn (1922-1993), inspired by aerial landscapes Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 6 / 42 Territory partitioning ... centralized district design California Voting Districts: 2008 Obama/McCain votes Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 7 / 42 Territory partitioning is ... animal territory dynamics Tilapia mossambica, “Hexagonal Territories,” Barlow et al, ’74 Red harvester ants, “Optimization, Conflict, and Nonoverlapping Foraging Ranges,” Adler et al, ’03 Sage sparrows, “Territory dynamics in a sage sparrows population,” Petersen et al ’87 Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 8 / 42 Territory partitioning: behaviors and optimality ANALYSIS of cooperative distributed behaviors 1 how do animals share territory? how do they decide foraging ranges? how do they decide nest locations? 2 what if each robot goes to “center” of own dominance region? 3 what if each robot moves away from closest robot? DESIGN of performance metrics 4 how to cover a region with n minimum-radius overlapping disks? 5 how to design a minimum-distortion (fixed-rate) vector quantizer? 6 where to place mailboxes in a city / cache servers on the internet? Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 9 / 42 Multi-center functions Expected wait time H(p, v) = ∫ v1 ‖q − p1‖dq + · · · + ∫ vn ‖q − pn‖dq n robots at p = {p1, . . . , pn} environment is partitioned into v = {v1, . . . , vn} H(p, v) = n ∑ i=1 ∫ vi f (‖q − pi‖)φ(q)dq φ : R2 → R≥0 density f : R≥0 → R penalty function Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 10 / 42 Optimal partitioning The Voronoi partition {V1, . . . ,Vn} generated by points (p1, . . . , pn) Vi = {q ∈ Q | ‖q − pi‖ ≤ ‖q − pj‖ , ∀j 6= i} = Q ⋂ j (half plane between i and j , containing i) Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 11 / 42 Optimal centering (for region v with density φ) function of p minimizer = center p 7→ ∫ v ‖q − p‖φ(q)dq centroid (or center of mass) p 7→ ∫ v ‖q − p‖φ(q)dq Fermat–Weber point (or median) p 7→ area(v ∩ disk(p, r)) r-area center p 7→ radius of largest disk centered at p enclosed inside v incenter p 7→ radius of smallest disk centered at p enclosing v circumcenter From online Encyclopedia of Triangle Centers Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 12 / 42 From optimality conditions to algorithms H(p, v) = ∫ v1 f (‖q − p1‖)φ(q)dq + · · · + ∫ vn f (‖q − pn‖)φ(q)dq 1 at fixed positions, optimal partition is Voronoi 2 at fixed partition, optimal positions are “generalized centers” 3 S. P. Lloyd. Least squares quantization in PCM. IEEE Trans Information Theory, 28(2):129– 137, 1982. Presented at the 1957 Institute for Mathematical Statistics Meeting Q. Du, V. Faber, and M. Gunzburger. Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review, 41(4):637–676, 1999 Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 13 / 42 From optimality conditions to algorithms H(p, v) = ∫ v1 f (‖q − p1‖)φ(q)dq + · · · + ∫ vn f (‖q − pn‖)φ(q)dq 1 at fixed positions, optimal partition is Voronoi 2 at fixed partition, optimal positions are “generalized centers” 3 alternate v -p optimization =⇒ local opt = center Voronoi partition S. P. Lloyd. Least squares quantization in PCM. IEEE Trans Information Theory, 28(2):129– 137, 1982. Presented at the 1957 Institute for Mathematical Statistics Meeting Q. Du, V. Faber, and M. Gunzburger. Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review, 41(4):637–676, 1999 Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 13 / 42 Voronoi+centering algorithm for robots Voronoi+centering law At each comm round: 1: acquire neighbors’ positions 2: compute own dominance region 3: move towards center of own dominance region Area-center Incenter Circumcenter F. Bullo, J. Cortés, and S. Mart́ınez. Distributed Control of Robotic Networks. Applied Mathematics Series. Princeton Univ Press, 2009. Available at http://www.coordinationbook.info Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 14 / 42 Incomplete literature S. P. Lloyd. Least squares quantization in PCM. IEEE Trans Information Theory, 28(2):129–137, 1982. Presented at the 1957 Institute for Mathematical Statistics Meeting J. MacQueen. Some methods for the classification and analysis of multivariate observations. In L. M. Le Cam and J. Neyman, editors, Proceedings of the Fifth Berkeley Symposium on Mathematics, Statistics and Probability, volume I, pages 281–297. University of California Press, 1965-1966 A. Gersho. Asymptotically optimal block quantization. IEEE Trans Information Theory, 25(7):373–380, 1979 R. M. Gray and D. L. Neuhoff. Quantization. IEEE Trans Information Theory, 44(6):2325–2383, 1998. Commemorative Issue 1948-1998 Q. Du, V. Faber, and M. Gunzburger. Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review, 41(4):637–676, 1999 J. Cortés, S. Mart́ınez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Trans Robotics & Automation, 20(2):243–255, 2004 J. Cortés and F. Bullo. Coordination and geometric optimization via distributed dynamical systems. SIAM JCO, 44(5):1543–1574, 2005 F. Bullo, J. Cortés, and S. Mart́ınez. Distributed Control of Robotic Networks. Applied Mathematics Series. Princeton Univ Press, 2009. Available at http://www.coordinationbook.info Francesco Bullo (UCSB) Robotic Coordination SIAM CT 2011 15 / 42