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Behavior Synthesis in Self-Organizing Robotic Systems

Explore the use of local interactions to program self-organizing robotic systems. Study behavior synthesis and self-assembly using modular robots.

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Behavior Synthesis in Self-Organizing Robotic Systems

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  1. Robotics: Science and Systems, MIT, 2005 Workshop on Modular and Reconfigurable Robots Behavior Synthesis inSelf-Organizing Robotic Systems Eric Klavins Assistant Professor Electrical Engineering University of Washington In collaboration with Rob Ghrist (UIUC Mathematics) and Josh Bishop, Sam Burden, JM McNew and Nils Napp (UW Students). 55 min

  2. e.g. Proteins assemble into a nuclear pore structure (Beck et al, Science vol 206, p. 1387, 2004. The Idea • How do local interactions give rise to global phenomena? • How do we program using only local interactions?

  3. ? active tiles e.g. conformational switching Tile Self-Assembly Seems Like a Good Place to Start foam tiles, magnets, air hockey table. passive tiles • Can compute: Wang (1975) • Mesoscale (PDMS, Si, ...): Whitesides, (1999 etc.) and others • Can make from DNA: Winfree (1998) Electrical connections Parviz et al.

  4. A Model Particles have comlimentary conformations

  5. A Model Particles have comlimentary conformations

  6. A Model Binding causes a conformational change

  7. A Model The new conformation makes the particle complimentary to different particles

  8. A Model And the process continues ...

  9. A Model And the process continues ...

  10. = a b2 a = c1 = b1 = b2 = c1 = c2 b2 a = a) Use symbols to denote conformations c2 b) Denote assemblies by labeled graphs Throw Away Geometry To Get A Tractable Model c) Model the dynamics non-deterministically (for now). See also: K. Saitou, 1999: Conformational Switching

  11. Graph Grammars Klavins, Ghrist and Lipsky, IEEE TAC (To Appear) See Also, Courcelle: Handbook of TCS, Vol B A system is modeled by a simple labeled graph: c a c b parts connections a a a b a c a d b b a states

  12. b e a g d f Graph Grammars Klavins, Ghrist and Lipsky, IEEE TAC (To Appear) See Also, Courcelle: Handbook of TCS, Vol B An rule and a monomorphism lead to a new graph. c a c b a a a b a c a  d b b a

  13. b e a g d f Graph Grammars Klavins, Ghrist and Lipsky, IEEE TAC (To Appear) See Also, Courcelle: Handbook of TCS, Vol B An rule and a monomorphism lead to a new graph. c a c b a a a e a c g  f b b a • Small rules model local interactions. • Commutative rules model parallelism via interleaving.

  14. stable a a a a a a a a c c c b a b c b c c b c c c a c a a a a a a a a a c a c c b a c c b c c c c a a a a a a b b b a a a b a b a a b c c a a a b a a a a c a b a a b a a b a a a c c a a a a c b a b unstable Example: Strands and Cycles a a  b b a b  b c b b  c c  = For now: Rules are applied nondeterministically, leading to a set of possible trajectories T(G0,).

  15. Embedding of lhs(r1) into G0 A rule applicable in G0 b a a a G0 with h1 lhs(r1) replaced by h1  rhs(r1). b a a b a a a b Warnings: Graphs Describe Topology Only! Grammars say what may happen, but not what will happen.

  16. A Physical Embedding (with some physics) Damped nonlinear spring (e.g. cappilary force)

  17. Example II: Self-Replication Klavins, ICMENS 2004 Attach raw materials Move right Close loops Move Left Break strands

  18. rotating magnet assembly motor motor mount custom electronics fixed magnet IR transceiver And So Can Robots! Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005 Internal State = Conformation?

  19. 1cm Version 1.0 Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005 Initial test of latching mechanism Compare: Modular and Reconfigurable robots (e.g. Yim, Chirickjian, Murata, Rus, Lipson and many others).

  20. The Setup Overhead Camera Ming Wang (idealistic undergrad) Fabrication station Robots Automated mixing system Leaf blowers!

  21. A Test of the Latching Mechanism Four parts interacting on an air table (from 30fps vision tracking system) So how do you program them?

  22. 4mers dimers the “right” 4mers hexagons “break rules” Example III: Hexagons Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005

  23. Example IV: Hexagons 2x real speed.

  24. Local Minima • To avoid deadlock, add rules of the form (V,E,l)  (V, ,  x.a) that are executed randomly (Klavins, ICRA2002) • Note: Statistical mechanics takes care of this, but the above rules make for bigger energy differences (Compare: DNA Free Energy Landscapes)

  25. ... ... More Examples

  26. The Synthesis Problem • Assembly Synthesis Problem: Make rules that produce a desired graph as uniquely stable. • 2)Reachable Set Dynamics: Make rules that produce a desired transition system over a desired reachable set. • 3)Stochastic Process over Reachable Set: Given rates for rule application, define rules and G0 so that specified pathways are most likely. e g  c — h a — c  d e b — h  f — b d — f  g — a

  27. a b a a  b—c b c d a c  d—e a e a b d k e  m—n m n l a a g k l i g a f  l—k a a  g—f f i h g a f  i—h a g i a Rule Set for an Acyclic Graph

  28. d g  o — p b b d m m stable n n o l l p g i i b * stable i o m p n l l n p m o i b Stable Set Synthesis Klavins, Ghrist and Lipsky, TAC (Under review) Find  so that S(G0,)={Gdesired}. • ALG1[KGL]: For Gdesired acyclic, O(n) binary rules, O(h) concurrent steps. • ALG2[KGL]: For Gdesired arbitrary, O(cn) binary and ternary rules and O(cn) concurrent steps. # cycles Thm[KGL]: If  contains only acyclic rules, C(G0,) is closed under covers. Cor[KGL]: If, in addition, E(L)= for each LR   then the S(G0,) is closed under covers. See Also: I. Litovsky, Y. Metevier, and W. Zielonka. The power and limitations of local computations on graphs and networks. LNCS 657, pp 333-345, 1992.

  29. The Covering Problem(Proof Sketch)

  30. RFF= Synthesis for Boolean Functions Bishop and Klavins, In preparation Compare “Graph Recognition” problems in Graph Grammar Lit. Find  so that p(G0) ,9 k such that Gn is connected for all n¸ k, otherwise have Gn completely disconnected. ALG3[BK]: Requires O(2n) binary rules. = :B Ç A subscript records interaction history

  31. Specify Behaviors From the Point of View of the Particles Idea: Consider the transition system induced by a graph grammar modulo neighborhood equivalence. Graphs are U0 and U1 equivalent, but not U2 equivalent.

  32. Example

  33. Relation to Metabolic Networks • Each molecule follows a characteristicsequence of states. • Transitions between states are shared with other species, as in PetriNets. The citric acid cycle Compare: R. Hofestädt. A Petri Net Application to Model Metabolic Processes. Systems Analysis Modeling Simulation, 16(2):113–122, October 1994.

  34. CAS Generation Problem: Given (G0,) find its characteristic automata set. • (Dumb) Algorithm • Produce trajectories 1, 2, ... of (G0,). • Build the neighborhood automaton for each particle for each trajectory. • Take equivalence classes modulo neighborhood isomorphism. • Compare: Metabolic network identification.

  35. The CAS Can Tell You Things Implies that the stable set is not a singleton.

  36. The CAS as a Specification Problem: Given the desired structure of the CAS, find the system (G0,) that produces it. ... has a solution = ... has no solution. Solution: Search!?!

  37. Rates Issue 1: Our systems are usually stochastic. Issue 2: The number of states is (usually) exponential in the number of particles. Rate from G to G’ Rate out of G Rate out of G

  38. trimers first one at a time dimers first Rates for Programmable Parts The parts undergo random walks with fast diffusion. It is hard to determine the assembly rate for two assemblies. Compare: Ab initio studies of reaction rates.

  39. An Approach toProbabilistic Solutions Klavins, In Preparation Problem: Optimize the probability that the system behaves like the specification F. (G0,,) Find the best initial graph (e.g. change the base pair sequence in a DNA strand). Change the rules (e.g. reprogram the robots). Tweak the rates (e.g. optimize binding efficiencies). p(F|) is continuous in . p(F|G0) and p(F|) are discrete.

  40. F: (k) Optimizing  Example: Find k so that (k) behaves like the specification F half of the time. • Approach: • Sample behaviors of (k) to estimate |p(F|(k))-0.5| for a given k. • Use these samples to estimate the gradient and do gradient descent. J. Spall, Ch. 14: Simulation-Based Optimization, Introduction to Stochastic Search and Optimization: Estimate, Simulation and Control, Wiley & Sons, 2003.

  41. noise due to small sample size Results of Optimization • Only “finitely generated” properties can be checked. • Need a good model and an efficient simulation. • Can put  = (k1) [(k2) to combine/compare grammars.

  42. CAREER: Programmed Robotic Self-Assembly Acknowledgements COLLABORATORS Karl Böhringer Robert Ghrist STUDENTS Josh Bishop Sam Burden Richard Kreisberg William Malone Nils Napp Tho Nguyen Fayette Shaw Ming Wang

  43. Ongoing Work Modeling Synthesis Complexity Statistical Dynamics Robots Swarming MEMs DNA

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