usrg 2002 elizabeth m. tsai jennifer e. walter nancy m. amato swarthmore college vassar college...
TRANSCRIPT
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
USRG 2002
Metamorphic Robotic SystemsWhat 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
USRG 2002
Transformer Background
Two types of transformers…
1) Autobots
• “good guys”
• lead by Optimus Prime
2) Decepticons
• “bad guys”
• lead by Megatron
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)
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
12
3I
G
• | I | = |G| = n (number of modules in system)
• any module can fill any cell in G
Step 1: move 3 CCW
12
3
Step 2: move 3 CCW
12
3
Step 3: move 2 CCW
12
3
Step 4: move 2 CCW
1
23
Step 5: move 2 CCW
Additionally, we want as many modules as possible to move concurrently.
USRG 2002
2D hexagonal modules move by...
Our Approach
SS S S S
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
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 cellin the westernmost column of G
1) Determine if G is admissible. If not, report failure.
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
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
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
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
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
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
c1c2
c3
i
Hierarchy of Admissible Structures
• viable cell – cell with clearance of three on each side
cell i has SE-clearance
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
c1c2
c3
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 cellObstacle cell
Admissible Structures
• admissible substrate path – an east-monotone admissible surface
• allows traversal on both sides without collision or deadlock and
• spans G Substrate path cellGoal cell
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 cellGoal cell
• admissible goal – contains an admissible substrate path
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 cellGoal 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
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
USRG 2002
Simulation Results
The effectiveness of our strategy to choose the “best” path was verifiedusing a simulator to count the number of rounds needed to reconfiguredifferent goal shapes.
Path chosen
Other paths
77
89 9196
92
0
20
40
60
80
100
120
Nu
mb
er
of
Rou
nd
s
(1) (1) (1) (1) (1)66
6768
6970
7172
73
76
60
62
64
66
68
70
72
74
76
78
Num
ber
of R
ound
s(2) (2) (2) (2)(3) (3) (3)(1
)(1)
USRG 2002
Simulation Results
The running time of the TraverseGraph algorithm was also verified by oursimulator by counting the total number of vertex visits for a given graph.
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 admissibilityFor 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 foranother goal cell (i.e. pocket of size 1 or 2)
Obstacle with pocket of size 1
Megatron is an inadmissible obstacle
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
USRG 2002
Future Work1. 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.