Generic Distributed Algorithms for Self-Reconfiguring Robots Keith Kotay and Daniela Rus MIT Computer Science and Artificial Intelligence Laboratory

RSS 2005MIT CSAIL Self-Reconfiguring Robot Multiple functionalities  Form follows function Advantages  Versatile  Robust  Extensible

RSS 2005MIT CSAIL Outline Generic distributed approach Previous work: locomotion Extension to non-locomotion

RSS 2005MIT CSAIL Challenges  Hardware implementation  Control algorithms Motivation

RSS 2005MIT CSAIL M-TRAN – Murata et al.Polypod – Yim et al. Challenges  Hardware implementation  Control algorithms Motivation Metamorphic – Chirikjian et al. Fractum – Murata et al. Crystal – Rus et al. 3-D Fractum – Murata et al. Molecule – Rus et al. CONRO – Shen et al. ATRON – Lund et al. Stochastic and self-replicating cubes – Lipson et al.

RSS 2005MIT CSAIL Challenges  Hardware implementation  Control algorithms Motivation Goals 1. Generic 2. Distributed 3. Correct

RSS 2005MIT CSAIL Generic distributed algorithms  Cellular automata paradigm Non-persistent modules  Proposed for self-reconfiguring robots by Hosokawa et al. (ICRA 1998) Synchronous update model Methodology

RSS 2005MIT CSAIL Methodology Approach 1. Use abstract module with simple motions 2. Create rule sets using only local information 3. Prove rule sets produce correct reconfigurations 4. Instantiate rule sets onto real systems

RSS 2005MIT CSAIL Methodology Approach 1. Use abstract module with simple motions 2. Create rule sets using only local information 3. Prove rule sets produce correct reconfigurations 4. Instantiate rule sets onto real systems

RSS 2005MIT CSAIL Methodology Approach 1. Use abstract module with simple motions 2. Create rule sets using only local information 3. Prove rule sets produce correct reconfigurations 4. Instantiate rule sets onto real systems Proof methods 1. Logical argument 2. Graph properties 3. Statistical argument Bounds size of error region with some confidence

RSS 2005MIT CSAIL Metamorphic Module – Chirikjian et al. Fracta Module – Murata et al.Crystal Module – Rus et al. Methodology Approach 1. Use abstract module with simple motions 2. Create rule sets using only local information 3. Prove rule sets produce correct reconfigurations 4. Instantiate rule sets onto real systems

RSS 2005MIT CSAIL Locomotion Rule Set (ICRA 2002)

RSS 2005MIT CSAIL Locomotion Example (ICRA 2002)

RSS 2005MIT CSAIL Simulation Details Evaluation models  Sequential evaluation  D k where k = relative cell actuation delay D 0 -- every module evaluated in each “round” in a fixed order D 1 -- every module evaluated in each “round” in random order D  -- no constraint on evaluation order synchronous asynchronous

RSS 2005MIT CSAIL Correctness Proof outline (ICRA 2002) 1. A rule can always be applied 2. Rule applications  east movement 3. The cell array remains connected Graph equivalence 1. No leaves 2. Cycles  eastward displacement 3. Nodes are connected cell arrays  Automated proofs can be produced for a given rule set and cell array

RSS 2005MIT CSAIL Methodology PAC proof  Statistical argument for correctness  Pr[n correct random simulations ] = (1 -  ) n  n = 1/  ln(1/  )  = size of the error region  = confidence PAC example   = 0.001,  =  n = 1000 ln(1000) ≈ 6908  99.9% confidence in error region < 0.1% activation sequences 1 -  

RSS 2005MIT CSAIL Methodology PAC caveats  Simulations must be unique Check strings of cell:rule pairs (activation sequences)  Simulations must be random Longest common subsequence Levenshtein distance

RSS 2005MIT CSAIL Self-Assembly Rule Set

RSS 2005MIT CSAIL Self-Assembly Example 1 Rule set  19 rules: 9 x 2 (east, west), 1 other  Internal state: direction, location  Rows act independently

RSS 2005MIT CSAIL 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

RSS 2005MIT CSAIL Self-Assembly Correctness SizeNodesEdgesTime (s) 2x21316< 1 3x266104< 1 3x x x x x x x Graph Proving Method Graph Properties 1. One leaf—the desired goal state 2. No cycles 3. No disconnection

RSS 2005MIT CSAIL Self-Assembly Correctness SizeIterationsAvg. Actuation Sequence Length Time (h) 3x3100, x4100, x5100, x6100, x7100, x8100, x97, x107, PAC Proving Method 100,000 runs = 99.99% confidence in error region < 0.01% of all actuation sequences 7,000 runs = 99.9% confidence in error region < 0.1% of all actuation sequences

RSS 2005MIT CSAIL Reconfiguration Algorithm Two-phase algorithm 1.Non-local phase Reconfigure so that each row has the correct number of modules Align rows with the goal shape 2.Local phase Locomotion to the goal shape location Self-assembly into the goal shape

RSS 2005MIT CSAIL Reconfiguration Algorithm Rule set for non-convex shapes  33 rules  2½-D start and goal shapes Layers must be connected components

RSS 2005MIT CSAIL Algorithm Correctness Non-convex shape rule set StartGoalModulesIterationsPAC Bounds SquarePyramid255,000, % % SquarePyramid81100, % % Random 92,000,000Not significant Random 161,000,000Not significant Random 255,000,000Not significant Random 49300,000Not significant

RSS 2005MIT CSAIL Reconfiguration Algorithm Ruleset developed by Kohji Tomita, AIST

RSS 2005MIT CSAIL Reconfiguration Algorithm Old A-2 RuleNew A-2 Rule New Stopping Rule

RSS 2005MIT CSAIL 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

RSS 2005MIT CSAIL 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

RSS 2005MIT CSAIL Algorithm Correctness New non-convex shape rule set StartGoalModulesIterationsPAC Bounds SquarePyramid251,000, % % SquarePyramid49200, % % SquarePyramid81100, % % SquareHollow Pyramid25100, % % Random 251,000,000Not significant Random 49200,000Not significant Random 8120,000Not significant

RSS 2005MIT CSAIL 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

RSS 2005MIT CSAIL Acknowledgements Boeing National Science Foundation  Awards IRI , EIA , IIS , IIS , and EIA Project Oxygen at MIT Intel Office of Naval Research  Award N Zack Butler and Kohji Tomita