1. USRG 2002 Elizabeth M. Tsai Jennifer E. Walter Nancy M. Amato Swarthmore College Vassar College Texas A&M University Concurrent Reconfiguration of Hexagonal Metamorphic Robots: Algorithms for Fast Execution and Obstacle Envelopment

2. USRG 2002 Metamorphic Robotic Systems What are metamorphic robots? robots with the capability to change shape i.e. Transformers What are Transformers? fighting robots that transform into everyday objects (e.g. cars, planes, appliances) Sunstreaker Soundwave

3. USRG 2002 Transformer Background Two types of transformers… 1)Autobots “good guys” lead by Optimus Prime 2) Decepticons “bad guys” lead by Megatron

4. USRG 2002 Metamorphic Robotic Systems We model robots like those developed by Chirikjian (ICRA94) Metamorphic modules are... 1)Uniform in structure and capability homogenous with regular symmetry modules fit together with minimal gaps 2)Individually mobile to allow system to change shape modules can connect, disconnect, and move over adjacent modules System composed of masses or clusters of robots (modules)

5. USRG 2002 1 2 3 Determine sequence of moves to reconfigure modules from an initial configuration I to a final configuration G Motion Planning Problem Statement time 1 2 3 I G | I | = |G| = n (number of modules in system) any module can fill any cell in G Step 1: move 3 CCW 1 2 3 Step 2: move 3 CCW 1 2 3 Step 3: move 2 CCW 1 2 3 Step 4: move 2 CCW 1 2 3 Step 5: move 2 CCW Additionally, we want as many modules as possible to move concurrently.

6. USRG 2002 2D hexagonal modules move by... Our Approach SSSSS A chain of unmoving modules that other modules move across during reconfiguration is called the substrate path. A combination of rotation and changing joint angles, disconnecting and connecting sides at appropriate times Modules “crawl” over unmoving neighbors ( S for substrate) Centralized motion plan for efficient concurrent reconfiguration that avoids deadlock and collision without message passing

7. USRG 2002 2)Select an admissible substrate path that approximately bisects the goal configuration. General Reconfiguration Strategy 3)Fill in the goal portion of the substrate path first, then fill in rest of goal cells above and below substrate path. I and G initially intersect in some goal cell in the westernmost column of G 1)Determine if G is admissible. If not, report failure.

8. Admissible Structures Pockets like this occur frequently in systems of hexagonal modules due to module shape. Our admissible structures are defined to eliminate configurations that contain such pockets

9. Admissible Structures Pockets like this occur frequently in systems of hexagonal modules due to module shape. Our admissible structures are defined to eliminate configurations that contain such pockets

10. Admissible Structures Pockets like this occur frequently in systems of hexagonal modules due to module shape. Our admissible structures are defined to eliminate configurations that contain such pockets

11. Admissible Structures Pockets like this occur frequently in systems of hexagonal modules due to module shape. Our admissible structures are defined to eliminate configurations that contain such pockets

12. Admissible Structures Pockets like this occur frequently in systems of hexagonal modules due to module shape. Our admissible structures are defined to eliminate configurations that contain such pockets

13. Admissible Structures Pockets like this occur frequently in systems of hexagonal modules due to module shape. Our admissible structures are defined to eliminate configurations that contain such pockets c 1 c 2 c 3 i Hierarchy of Admissible Structures viable cell – cell with clearance of three on each side cell i has SE-clearance

14. Admissible Structures Pockets like this occur frequently in systems of hexagonal modules due to module shape. Our admissible structures are defined to eliminate configurations that contain such pockets c 1 c 2 c 3 i Hierarchy of Admissible Structures viable cell – cell with clearance of three on each side cell i has SE-clearance admissible surface – surface composed of viable cells Goal cell Obstacle cell

15. Admissible Structures admissible substrate path – an east- monotone admissible surface allows traversal on both sides without collision or deadlock and spans G Substrate path cell Goal cell

16. Admissible Structures admissible substrate path – an east- monotone admissible surface allows traversal on both sides without collision or deadlock and spans G Admissible G Inadmissible G Substrate path cell Goal cell admissible goal – contains an admissible substrate path

17. Admissible Structures admissible substrate path – an east- monotone admissible surface allows traversal on both sides without collision or deadlock and spans G Admissible G Inadmissible G Substrate path cell Goal cell admissible goal – contains an admissible substrate path Our admissibility definitions are directly related to the degree of parallelism possible – i.e. how closely moving modules can be spaced without becoming deadlocked

18. USRG 2002 Selecting Substrate Paths Our method for finding the best admissible substrate path for reconfiguration is summarized as follows: 1) Convert G to an acyclic graph, H, and direct the edges 2)Use a graph traversal algorithm combined with a weighting heuristic to rank all candidate paths by straightness 3)Use a second heuristic to select a path that most evenly bisects the goal Example Substrate Path Selection: (1) Goal G converted to H (2a) Cost 1 path (2b) Cost 0 path (3) Selected path best bisects goal

19. USRG 2002 Simulation Results The effectiveness of our strategy to choose the “best” path was verified using a simulator to count the number of rounds needed to reconfigure different goal shapes. Path chosen Other paths 77 89 91 96 92 0 20 40 60 80 100 120 Number of Rounds (1) (2) (3) (1)

20. USRG 2002 Simulation Results The running time of the TraverseGraph algorithm was also verified by our simulator by counting the total number of vertex visits for a given graph.

21. USRG 2002 Reconfiguration with a Single Obstacle We consider the presence of a single obstacle in the environment that must… be enclosed completely inside the goal be admissible not involve purple swingy-weapons or water What is an admissible obstacle? an obstacle that contains an admissible surface How to check for obstacle admissibility For each obstacle cell on the perimeter of the obstacle… …for each side of the cell that is a goal cell… …check two and three cells over for another goal cell (i.e. pocket of size 1 or 2) Obstacle with pocket of size 1 Megatron is an inadmissible obstacle

22. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments

23. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

24. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

25. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

26. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

27. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

28. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

29. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

30. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

31. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

32. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

33. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

34. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

35. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

36. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

37. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

38. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

39. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

40. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

41. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

42. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

43. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

44. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

45. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

46. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

47. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

48. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

49. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

50. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

51. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell

52. USRG 2002 Determining Substrate Path with a Single Obstacle Original Idea 1)Direct the edges west of the goal to determine the “entrance” point for the path 2)Direct the edges inside the obstacle to determine the “exit” point for the path 3)Direct the edges east of the goal, going out of the exit point 4)Form the final substrate path by concatenating the above path segments But wait! We have a problem! Small pockets that modules can’t crawl through can form where the substrate path meets the obstacle These pockets are a result of the East-To-West filling-in strategy Goal cell Substrate goal cell Pocket formed by obstacle and filled goal cells

53. USRG 2002 Repairing the Obstacle To remedy the “pocket problem,” we “repair” the obstacle surface to make it traversable. Why? want to avoid modules getting trapped when filling in from east to west want modules to crawl over obstacle surface as a substrate path during reconfiguration How do we “repair” an obstacle? form a cone shape with the eastern-most column of the obstacle as its base fill in this cone from south to north, and from west to east Unrepaired obstacle Repaired obstacle Goal cell Obstacle cell Repaired cell

54. USRG 2002 Future Work 1.Develop algorithms for dealing with multiple obstacles in the environment. 2.Algorithmic work: asynchronous reconfiguration algorithms. procedures for deadlock and collision resolution. “complete” reconfiguration, from arbitrary initial to arbitrary goal. 3. Build fighting robots.