Generic Distributed Algorithms for Self-Reconfiguring Robots

1
Generic Distributed Algorithms for
Self-Reconfiguring Robots

• Keith Kotay and Daniela Rus
• MIT Computer Science and Artificial Intelligence
  Laboratory

2
Self-Reconfiguring Robot

• Multiple functionalities
• Form follows function

• Advantages
• Versatile
• Robust
• Extensible

3
Outline

• Generic distributed approach

• Previous work locomotion

• Extension to non-locomotion

4
Motivation

• Challenges
• Hardware implementation
• Control algorithms

5
Motivation

• Challenges
• Hardware implementation
• Control algorithms

Crystal Rus et al.
Molecule Rus et al.
ATRON Lund et al.
6
Motivation

• Challenges
• Hardware implementation
• Control algorithms

• Goals
• Generic
• Distributed
• Correct

7
Methodology

• Generic distributed algorithms
• Cellular automata paradigm
• Non-persistent modules
• Proposed for self-reconfiguring robots by
  Hosokawa et al. (ICRA 1998)
• Synchronous update model

8
Methodology

• Approach
• Use abstract module with simple motions
• Create rule sets using only local information
• Prove rule sets produce correct reconfigurations
• Instantiate rule sets onto real systems

9
Methodology

• Approach
• Use abstract module with simple motions
• Create rule sets using only local information
• Prove rule sets produce correct reconfigurations
• Instantiate rule sets onto real systems

10
Methodology

• Approach
• Use abstract module with simple motions
• Create rule sets using only local information
• Prove rule sets produce correct reconfigurations
• Instantiate rule sets onto real systems

• Proof methods
• Logical argument
• Graph properties
• Statistical argument
• Bounds size of error region with some confidence

11
Methodology

• Approach
• Use abstract module with simple motions
• Create rule sets using only local information
• Prove rule sets produce correct reconfigurations
• Instantiate rule sets onto real systems

Metamorphic Module Chirikjian et al.
Fracta Module Murata et al.
Crystal Module Rus et al.
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Locomotion Rule Set (ICRA 2002)
13
Locomotion Example (ICRA 2002)
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Simulation Details

• Evaluation models
• Sequential evaluation
• Dk where k relative cell actuation delay
• D0 -- every module evaluated in each round in a
  fixed order
• D1 -- every module evaluated in each round in
  random order
• D -- no constraint on evaluation order

synchronous
asynchronous
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Correctness

• Proof outline (ICRA 2002)
• A rule can always be applied
• Rule applications Þ east movement
• The cell array remains connected

• Graph equivalence
• No leaves
• Cycles Þ eastward displacement
• Nodes are connected cell arrays
• Automated proofs can be produced
  for a given rule set and cell array

16
Methodology

• PAC proof
• Statistical argument for correctness
• Prn correct random simulations (1 - ?)n
• n 1/? ln(1/?)
• ? size of the error region
• ? confidence

1 - ? ?
activation sequences

• PAC example
• ? 0.001, ? 0.001
• n 1000 ln(1000) 6908
• 99.9 confidence in error region lt 0.1

17
Methodology

• PAC caveats
• Simulations must be unique
• Check strings of cellrule pairs (activation
  sequences)
• Simulations must be random
• Longest common subsequence
• Levenshtein distance

18
Self-Assembly Rule Set
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Self-Assembly Example 1

• Rule set
• 19 rules 9 x 2 (east, west), 1 other
• Internal state direction, location
• Rows act independently

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Self-Assembly Example 2

• Rule set
• 19 rules 9 x 2 (east, west), 1 other
• Internal state direction, location, goal shape
• Rows act independently
• Works for convex 2½-D shapes

21
Self-Assembly Correctness
Graph Proving Method
Size Nodes Edges Time (s)
2x2 13 16 lt 1
3x2 66 104 lt 1
3x3 609 1372 1
3x4 3460 9215 33
4x3 3756 10159 37
4x4 31920 103938 1031
3x6 89830 317012 7063
6x3 119920 432940 12993
5x4 279464 1081364 110520

• Graph Properties
• One leafthe desired goal state
• No cycles
• No disconnection

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Self-Assembly Correctness
PAC Proving Method
Size Iterations Avg. Actuation Sequence Length Time (h)
3x3 100,000 25.3 0.93
4x4 100,000 76.1 1.06
5x5 100,000 181.8 1.66
6x6 100,000 372.5 6.93
7x7 100,000 682.4 12.03
8x8 100,000 1153.3 38.35
9x9 7,000 1831.3 8.19
10x10 7,000 2773.0 19.22
100,000 runs 99.99 confidence in error
region lt 0.01 of all actuation sequences 7,000
runs 99.9 confidence in error region lt 0.1 of
all actuation sequences
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Reconfiguration Algorithm

• Two-phase algorithm
• Non-local phase
• Reconfigure so that each row has the correct
  number of modules
• Align rows with the goal shape
• Local phase
• Locomotion to the goal shape location
• Self-assembly into the goal shape

24
Reconfiguration Algorithm

• Rule set for non-convex shapes
• 33 rules
• 2½-D start and goal shapes
• Layers must be connected components

25
Algorithm Correctness
Non-convex shape rule set
Start Goal Modules Iterations PAC Bounds
Square Pyramid 25 5,000,000 99.9997 -- 0.0003
Square Pyramid 81 100,000 99.99 -- 0.01
Random Random 9 2,000,000 Not significant
Random Random 16 1,000,000 Not significant
Random Random 25 5,000,000 Not significant
Random Random 49 300,000 Not significant
26
Reconfiguration Algorithm
Ruleset developed by Kohji Tomita, AIST
27
Reconfiguration Algorithm
Old A-2 Rule
New A-2 Rule
New Stopping Rule
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Reconfiguration Algorithm

• New non-convex shape rule set
• 66 rules
• 2½-D start and goal shapes
• Layers must be connected components
• Reduction in structure voids

29
Reconfiguration Algorithm

• New non-convex shape rule set
• 66 rules
• 2½-D start and limited 3-D goal shapes
• Layers must be connected components
• Reduction in structure voids

30
Algorithm Correctness
New non-convex shape rule set
Start Goal Modules Iterations PAC Bounds
Square Pyramid 25 1,000,000 99.999 -- 0.001
Square Pyramid 49 200,000 99.995 -- 0.005
Square Pyramid 81 100,000 99.99 -- 0.01
Square Hollow Pyramid 25 100,000 99.99 -- 0.01
Random Random 25 1,000,000 Not significant
Random Random 49 200,000 Not significant
Random Random 81 20,000 Not significant
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Conclusion

• Generic, distributed approach
• Abstract module
• Local rules
• Algorithm correctness
• Instantiation to real hardware

• Algorithms
• Self-assembly of convex 2½-D shapes
• Self-assembly of non-convex 2½-D shapes
• Extension to limited 3-D goal shapes

32
Acknowledgements

• National Science Foundation
• Awards IRI-9714332, EIA-9901589, IIS-9818299,
  IIS-9912193, and EIA-0202789

• Office of Naval Research
• Award N00014-01-1-0675

• Project Oxygen at MIT

• Intel

• Boeing

• Zack Butler and Kohji Tomita