1. Beth Tsai Jennifer E. Walter Nancy M. Amato Department of Computer Science Texas A&M University, College Station Distributed Reconfiguration of Metamorphic Robot Chains

2. Metamorphic Robotic Systems Proposed by Chirikjian (ICRA94) and Murata et al. (ICRA94) Metamorphic modules are... 1)Uniform in structure and capability usually homogenous with regular symmetry (e.g., hexagons, cubes) desirable for modules to 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 small robots (modules)

3. 1 2 3 Motion planning problem Determine sequence of moves reconfigure modules from an initial configuration I to a final configuration G 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 1 2 3 Step 5: move 2 CCW

4. 2D hexagonal modules move by... Module Movement Complexity Modules move in synchronous rounds: SSSSS A chain of unmoving modules that other modules move across during reconfiguration is called the substrate path Complexity measures of interest = # rounds and # moves A combination of rotation and changing joint angles, disconnecting and connecting sides at appropriate times “Crawling” over unmoving neighbors ( S for substrate)

5. General shape changing: construction, e.g., bridges, buttresses Potential Applications Object envelopment: surrounding objects for recovery or removal, e.g., satellite recovery, tumor excision

6. Our Summer Work We can achieve a higher degree of parallelism when filling the goal if I is aligned with the substrate path Fastest reconfiguration occurs when the substrate path… 1)Is a straight chain or a chain with a single bend 2)Equally bisects G 3)Is aligned parallel to longest axis of G 4)Intersects I at an obtuse angle Determining the best substrate path We represent G as an acyclic, directed graph, H. We weight the vertices of H to favor straight paths and use a graph traversal algorithm to find the lowest cost (straightest) substrate path. A straight substrate path that equally bisects G and is parallel to the longest axis

7. Converting G to H We convert G into an acyclic graph, H, by mapping each cell in G to a vertex in H. To direct the edges of H… If two cells in G are adjacent in the SE or NE direction, their vertices are connected in H with a west-east edge If two cells in G are adjacent in the N or S direction, their vertices are connected in H only if there is a “clearance” of three columns between it and any edge in the opposite direction or it and any goal cells

8. Weighting Vertices A vertex, v, is assigned a weight of… 10, if any of v’s incoming edges are vertical 1, if v’s incoming edge is oriented a different direction than its parent’s incoming edge 0, if v’s incoming edge is oriented the same direction as its parent’s incoming edge 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 Once all vertices in H have been weighted, a graph traversal algorithm is employed to find the shortest paths

9. A Graph Traversal Algorithm Traverses all paths from root to leaves in a rooted, directed, acyclic graph Algorithm procedure 1) Initially, all vertices in H are black (unvisited) 2) Let v := root of H and color v red (visited) 3) While v has a black (unvisited) child Pick a child, c, and let parent := v Let v := c 4) If v is a leaf... …If v’s parent has a black (unvisited) child, back up to v’s parent and continue traversing …Else if v’s parent has no black (unvisited) children, color v and v’s sibling black (i.e. unvisit them) and back up to v’s parent. Continue backtracking until reaching the root.

10. Example of Graph Traversal Algorithm Root colored red Path from root to leaf G found 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 Backtracking to vertex B. E and G are colored black. C remains red since D hasn’t been visited Traversing continues from B to leaf G Backtracking to vertex D. G is colored black. E remains red since F hasn’t been visited Traversing continues from D to leaf F

11. Selecting The Best Path The graph traversal algorithm makes one pass over H for each vertex in the first column (i.e. traverses H with each first column vertex as the root) Paths in H are considered in the following order: 1) Paths with cost 1 (single bend) or cost 0 (straight) that equally (or almost equally) bisect H 2) Higher cost paths that equally (or almost equally) bisect H A cost 0 path that equally bisects H - suitable for substrate path A cost 1 path that doesn’t equally bisect H A high cost path that partially bisects H - suitable for substrate path A high cost path that doesn’t bisect H

12. Summary Our method for finding the best substrate path for reconfiguration is summarized as follows: 1) Convert G to an acyclic graph, H, and direct the edges 2) Weight the vertices of H using our simple weighting scheme 3) Use the graph traversal algorithm to traverse all paths in H 4) Select the lowest cost path that equally (or partially) bisects H to be the substrate path Examples of good substrate paths found:

13. Obstacles Conditions for obstacle admissibility Obstacles allowed in the plane around the goal or adjacent to the goal Obstacles prohibited in cells “inside” the goal Obstacles adjacent to goal must be… Perpendicular to substrate path, or At an obtuse angle to the substrate path All other obstacles must have a “clearance” distance of at least two cells from the substrate path or at least one cell from the goal perimeter Admissible - obstacles perpendicular to substrate path Admissible - obstacles have correct clearance Inadmissible - obstacles inside goal Inadmissible - obstacles without clearance of 2 cells Obstacles, or “forbidden cells,” that modules cannot enter or touch may be introduced into the problem

14. Dealing With Obstacles Obstacles may block I from intersecting G at the best substrate path Options: Bend I around obstacle Enter G from opposite end of substrate path Enter G on another axis Use alternate substrate path

15. Next Step: General Reconfiguration Algorithms Goal: Any arbitrary shape to any other arbitrary shape Our approach: 2 phase process 12