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Square Formation by Asynchronous Oblivious Robots CCCG 2016 Marcello Mamino, Giovanni Viglietta Vancouver August 3, 2016 Square Formation by Asynchronous Oblivious Robots Anonymous robots sensing and moving in the plane We consider a

  1. Why the general approach fails with 4 robots We can still identify a supporting square... Square Formation by Asynchronous Oblivious Robots

  2. Why the general approach fails with 4 robots ...But it is not unique! Square Formation by Asynchronous Oblivious Robots

  3. Why the general approach fails with 4 robots ...But it is not unique! Square Formation by Asynchronous Oblivious Robots

  4. Why the general approach fails with 4 robots The “central” supporting polygon may be chosen... Square Formation by Asynchronous Oblivious Robots

  5. Why the general approach fails with 4 robots ...But asynchronous robots may never manage to form a square Square Formation by Asynchronous Oblivious Robots

  6. Why the general approach fails with 4 robots ...But asynchronous robots may never manage to form a square Square Formation by Asynchronous Oblivious Robots

  7. Why the general approach fails with 4 robots ...But asynchronous robots may never manage to form a square Square Formation by Asynchronous Oblivious Robots

  8. Why the general approach fails with 4 robots ...But asynchronous robots may never manage to form a square Square Formation by Asynchronous Oblivious Robots

  9. How to solve the rectangle How do we solve the rectangular case? Square Formation by Asynchronous Oblivious Robots

  10. How to solve the rectangle Choose a supporting square that is tilted by 45 ◦ ... Square Formation by Asynchronous Oblivious Robots

  11. How to solve the rectangle ...And make the robots move to the midpoints of its edges Square Formation by Asynchronous Oblivious Robots

  12. How to solve the rectangle Again, the supporting square is preserved as the robots move Square Formation by Asynchronous Oblivious Robots

  13. How to solve the rectangle Again, the supporting square is preserved as the robots move Square Formation by Asynchronous Oblivious Robots

  14. How to solve the rectangle When they reach the midpoints, they form a square Square Formation by Asynchronous Oblivious Robots

  15. Identifying the supporting square (In general, we can also identify a supporting square...( Square Formation by Asynchronous Oblivious Robots

  16. Identifying the supporting square ...Having a robot on each (extended) edge Square Formation by Asynchronous Oblivious Robots

  17. Identifying the supporting square (But once again, the supporting square is not unique!( Square Formation by Asynchronous Oblivious Robots

  18. Identifying the supporting square (However, there is a geometric construction that identifies one( Square Formation by Asynchronous Oblivious Robots

  19. Identifying the supporting square (However, there is a geometric construction that identifies one( Square Formation by Asynchronous Oblivious Robots

  20. Identifying the supporting square (However, there is a geometric construction that identifies one( Square Formation by Asynchronous Oblivious Robots

  21. Identifying the supporting square (However, there is a geometric construction that identifies one( Square Formation by Asynchronous Oblivious Robots

  22. Identifying the supporting square (However, there is a geometric construction that identifies one( Square Formation by Asynchronous Oblivious Robots

  23. Identifying the supporting square (However, there is a geometric construction that identifies one( Square Formation by Asynchronous Oblivious Robots

  24. Identifying the supporting square (All robots automatically agree on the same supporting square!( Square Formation by Asynchronous Oblivious Robots

  25. Identifying the supporting square (All robots automatically agree on the same supporting square!( Square Formation by Asynchronous Oblivious Robots

  26. Identifying the supporting square (All robots automatically agree on the same supporting square!( Square Formation by Asynchronous Oblivious Robots

  27. Identifying the supporting square (No two robots have intersecting pathways!( Square Formation by Asynchronous Oblivious Robots

  28. Identifying the supporting square (No two robots have intersecting pathways!( Square Formation by Asynchronous Oblivious Robots

  29. Identifying the supporting square (No two robots have intersecting pathways!( Square Formation by Asynchronous Oblivious Robots

  30. Problem: orthogonal diagonals Suppose the two diagonals “accidentally” become orthogonal Square Formation by Asynchronous Oblivious Robots

  31. Problem: orthogonal diagonals Suppose the two diagonals “accidentally” become orthogonal Square Formation by Asynchronous Oblivious Robots

  32. Problem: orthogonal diagonals gThen our construction does not workg Square Formation by Asynchronous Oblivious Robots

  33. Problem: orthogonal diagonals The robots may not agree on a supporting square Square Formation by Asynchronous Oblivious Robots

  34. Special strategy for orthogonal diagonals If the diagonals are orthogonal, we use a different approach Square Formation by Asynchronous Oblivious Robots

  35. Special strategy for orthogonal diagonals If the diagonals are orthogonal, we use a different approach Square Formation by Asynchronous Oblivious Robots

  36. Special strategy for orthogonal diagonals The robots that are closest to the center move away from it Square Formation by Asynchronous Oblivious Robots

  37. Special strategy for orthogonal diagonals The robots that are closest to the center move away from it Square Formation by Asynchronous Oblivious Robots

  38. Special strategy for orthogonal diagonals The robots that are closest to the center move away from it Square Formation by Asynchronous Oblivious Robots

  39. Special strategy for non-convex configurations For non-convex configurations, our construction does not work... Square Formation by Asynchronous Oblivious Robots

  40. Special strategy for non-convex configurations ...Because the diagonals are not well defined Square Formation by Asynchronous Oblivious Robots

  41. Special strategy for non-convex configurations gIn this case, the internal robot moves...g Square Formation by Asynchronous Oblivious Robots

  42. Special strategy for non-convex configurations gIn this case, the internal robot moves...g Square Formation by Asynchronous Oblivious Robots

  43. Special strategy for non-convex configurations ...So to make the diagonals orthogonal... Square Formation by Asynchronous Oblivious Robots

  44. Special strategy for non-convex configurations ...And reduce the problem to the previous case Square Formation by Asynchronous Oblivious Robots

  45. Special strategy for collinear configurations If the robots are collinear, the previous approach does not work Square Formation by Asynchronous Oblivious Robots

  46. Special strategy for collinear configurations gIn this case, the internal robots move to either side of the lineg Square Formation by Asynchronous Oblivious Robots

  47. Special strategy for collinear configurations As they asynchronously move, their supporting square may change Square Formation by Asynchronous Oblivious Robots

  48. Special strategy for collinear configurations ◦ ◦ 50 50 So we must identify a “safe region”, e.g., a thin hexagon Square Formation by Asynchronous Oblivious Robots

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