1. ICRA 2002 Jennifer E. Walter Elizabeth M. Tsai Nancy M. Amato Vassar College Swarthmore College Texas A&M University Choosing Good Paths for Fast Distributed Reconfiguration of Hexagonal Metamorphic Robots

2. ICRA 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)

3. ICRA 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.

4. ICRA 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

5. ICRA 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.

6. Admissible Goal Configurations Informally, an admissible substrate path is a chain of modules that oallows traversal on both sides without collision or deadlock and ospans G G is an admissible goal configuration if it contains an admissible substrate path. Admissible G Inadmissible G Pockets like this occur frequently in systems of hexagonal modules due to module shape.

7. Fast Parallel Reconfiguration Fastest reconfiguration occurs when the substrate path…...is a straight chain or a chain with a single obtuse angle bend why? Because modules can move with minimal separation, rotating in alternate directions.... bisects G why? Because after the substrate path is filled in, modules can fill in both top and bottom of G simultaneously, without possibility of collision.

8. ICRA 2002 Converting G to directed graph H Direct some of the edges of G so that all “goal-spanning paths” can be traversed, from west to east. Edges that are unsuitable for inclusion in an admissible substrate path are not directed H Goal cells blue, non-goal cells gray

9. ICRA 2002 Weighting Vertices v = 10 v = 0 v = 1 v with an incoming vertical edge v with incoming edge in different direction than parent’s incoming edge v with incoming edge in same direction as parent’s incoming edge  Our graph traversal algorithm finds all goal-spanning paths in H. During execution, the vertices are weighted as shown below:

10. Example Graph Traversal with Path Weighting cost of A – B – C – E – G = 1 A A B B C C D D E E F F A B C D E F G G A B C D E F G G A B C D E F G A B C D E F G cost of A – B – D – F – G = 2 0 0 0 0 1 1 3 0 1 2 0 1 0 1 2 cost of A – B – D – E – G = 3 0 0 0 1 0 0 Root marked Backtracking to vertex B. E and G are unmarked. C remains marked since D hasn’t been visited Backtracking to vertex D. G is unmarked. E remains marked since F hasn’t been visited

11. ICRA 2002 Selecting The Best Substrate Path For each goal-spanning path of H, a path cost is evaluated. Candidate paths in H are considered in the following order: 1) Paths with cost 0 ( straight ) or cost 1 ( single bend ) that equally (or almost equally) bisect H 2) Higher cost paths that equally (or almost equally) bisect H Cost 0 path that equally bisects H Cost 1 path that does not equally bisect H Cost 2 path that partially bisects H Cost 5 path that doesn’t bisect H Selected Not selected Selected Not selected

12. ICRA 2002 Summary 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 the graph traversal algorithm to traverse all paths and determine path costs in H, and choose from a set of lowest cost paths a substrate path that most equally divides H Examples of good substrate paths found:

13. ICRA 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. 0 10 20 30 40 50 squarebridgebuttress chosen path other paths # of rounds Rounds of Reconfiguration by Goal Shape

14. ICRA 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 have an admissible surface What is an admissible surface? an admissible substrate path – i.e. all pockets must have clearance of at least three cells 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

15. ICRA 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 goal 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 Pocket formed by obstacle and filled-in goal cells Filled goal cell Filled substrate path cell

16. ICRA 2002 Repairing the Obstacle To remedy the “pocket problem,” we “repair” the obstacle surface. 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

17. ICRA 2002 Future Work 1.Change existing simulation to include internal and external obstacles. 2.Algorithmic work: asynchronous reconfiguration algorithms. procedures for deadlock and collision resolution. “complete” reconfiguration, from arbitrary initial to arbitrary goal.