# Behavior Synthesis in

Behavior Synthesis in

Self
-
Organizing Robotic
Systems

Eric Klavins

Assistant Professor

Electrical Engineering

University of Washington

55 min

In collaboration with
Rob Ghrist

(UIUC Mathematics) and
Josh Bishop
,
Sam Burden
,
JM McNew

and
Nils Napp

(UW Students).

Robotics: Science and Systems, MIT, 2005

Workshop on Modular and Reconfigurable Robots

The Idea

How do local interactions give
rise to global phenomena?

How do we program using
only local interactions?

e.g. Proteins
assemble into a
nuclear pore
structure (Beck et
al, Science vol 206,
p. 1387, 2004.

Tile Self
-
Assembly Seems Like a
Good Place to Start

Can compute:
Wang (1975)

Mesoscale (PDMS, Si, ...):
Whitesides, (1999 etc.) and others

Can make from DNA:
Winfree (1998)

passive

tiles

foam tiles, magnets, air hockey table.

Electrical connections

Parviz et al.

active

tiles

?

e.g. conformational
switching

A Model

Particles have
comlimentary
conformations

A Model

Particles have
comlimentary
conformations

A Model

Binding causes a
conformational
change

A Model

The new
conformation
makes the particle
complimentary to
different particles

A Model

And the process
continues ...

A Model

And the process
continues ...

Throw Away Geometry To Get A
Tractable Model

= a

= b
1

= b
2

= c
1

= c
2

a) Use symbols to denote
conformations

a

b
2

c
1

a

b
2

c
2

=

=

b) Denote assemblies
by
labeled graphs

c) Model the dynamics non
-
deterministically
(for now)
.

a

c

a

b

a

c

d

b

c

a

b

a

b

a

a

A system is modeled by a
simple labeled graph:

parts

connections

states

Graph Grammars

Klavins, Ghrist and Lipsky, IEEE TAC (To Appear)

b

d

a

Graph Grammars

a

c

a

b

a

c

d

b

c

a

b

a

b

a

a

e

f

g

An
rule

and a
monomorphism

Klavins, Ghrist and Lipsky, IEEE TAC (To Appear)

a

c

a

e

a

c

f

b

c

a

b

a

b

g

a

b

d

a

e

f

g

Small rules model local interactions.

Commutative rules model parallelism via interleaving.

An
rule

and a
monomorphism

Graph Grammars

Klavins, Ghrist and Lipsky, IEEE TAC (To Appear)

Example: Strands and Cycles

a a

b b

a b

b c

b b

c c

a

a

a

a

a

a

a

a

a

a

a

a

b

b

a

a

a

a

a

a

a

a

a

a

c

b

b

a

a

a

a

a

a

a

a

a

c

b

c

b

a

a

a

a

a

a

a

a

c

c

c

c

a

a

a

a

a

a

a

a

c

c

c

c

b

a

b

b

b

b

a

b

c

c

c

c

c

a

b

b

b

b

a

c

c

c

c

c

c

a

b

c

b

c

a

c

stable

unstable

=

For now
: Rules are applied nondeterministically, leading to a set of
possible trajectories T(G
0
,

).

Warnings:

a

b

a

a

a

a

a

a

a

b

b

b

A rule applicable in G
0

Embedding of lhs(r
1
)

into
G
0

G
0

with h
1

lhs

(r
1
) replaced by h
1

rhs(r
1
).

Graphs Describe
Topology Only!

Grammars say what
may

happen, but not what
will

happen.

A Physical Embedding

Damped nonlinear
spring (e.g. cappilary
force)

(with some physics)

Example II: Self
-
Replication

Klavins, ICMENS 2004

Break strands

Move Left

Attach raw materials

Move right

Close loops

And So Can Robots!

motor

motor mount

rotating

magnet

assembly

fixed magnet

IR transceiver

custom

electronics

Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005

Internal State = Conformation?

Version 1.0

1cm

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).

The Setup

Ming Wang

(idealistic

Automated
mixing system

Leaf blowers!

Camera

Fabrication
station

Robots

A Test of the Latching Mechanism

Four parts interacting on
an air table

(from 30fps vision tracking
system)

So how do you program them?

Example III: Hexagons

dimers

the “right” 4mers

“break rules”

4mers

hexagons

Bishop, Burden, Klavins, Kreisberg, Malone, Napp and Nguyen, IROS2005

Example IV: Hexagons

2x real speed.

Local Minima

(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)

More Examples

The Synthesis Problem

1)
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 G
0

so that specified
pathways are most likely.

e g

c

h

a

c

d e

b

h

f

b

d

f

g

a

Rule Set for an Acyclic Graph

a

a

a

a

a

a

a

g

i

k

l

a f

l

k

g

i

l

d

b

m

n

k e

m

n

g

h

i

a f

i

h

d

e

b

a c

d

e

f

g

a a

g

f

b

c

a a

b

c

Stable Set Synthesis

Find

so that S(G
0
,

)={G
desired
}.

ALG
1
[KGL]:
For G
desired

acyclic,
O(n) binary rules, O(h)
concurrent
steps.

ALG
2
[KGL]:
For G
desired

arbitrary,
O(cn) binary
and ternary

rules
and O(cn)
concurrent steps.

Klavins, Ghrist and Lipsky, TAC (Under review)

Thm[KGL]:

If

contains only acyclic rules, C(G
0
,

) is closed
under covers.

Cor[KGL]:

for each L

R

then the
S(G
0
,

) is closed under covers.

g

i

l

d

b

m

n

d g

o

p

o

i

l

p

b

m

n

stable

b

o

i

l

p

m

n

o

i

l

p

b

m

n

*

stable

# cycles

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.

The Covering Problem

(Proof Sketch)

Synthesis for Boolean Functions

Bishop and Klavins, In preparation

Compare “Graph Recognition” problems in Graph Grammar Lit.

B

A

Q

F

F

T

F

T

T

T

F

F

T

T

T

R
FF

R
TF

)

R
TF

R
TF

R
TF

R
FT

)

R
TT

R
TT

A

R
FF

)

A

R
FT

subscript records
interaction history

R
FF
=

Find

so that p(G
0
)
,

9

k such that G
n

is connected for all n
¸

k, otherwise
have G
n

completely disconnected.

ALG
3
[BK]:

Requires O(2
n
) binary rules.

=
:
B
Ç

A

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 U
0

and U
1

equivalent,
but not U
2

equivalent.

Example

Relation to Metabolic Networks

Each molecule follows a
characteristic

sequence

of
states.

Transitions between states
are shared with other species,
as in
Petri

Nets
.

The citric
acid cycle

Compare: R. Hofest
ä
dt. A Petri Net Application to Model
Metabolic Processes.
Systems Analysis Modeling
Simulation
, 16(2):113

122, October 1994.

CAS Generation

Problem
: Given (G
0
,

) find its characteristic automata set.

(Dumb) Algorithm

1)
Produce trajectories

1
,

2
, ... of (G
0
,

).

2)
Build the neighborhood automaton for each particle for
each trajectory.

3)
Take equivalence classes modulo neighborhood
isomorphism.

Compare: Metabolic network identification.

The CAS Can Tell You Things

Implies that the
stable set is not
a singleton.

The CAS as a Specification

Problem
: Given the desired structure of the CAS, find the
system (G
0
,

) that produces it.

... has a solution =

... has no solution.

Solution: Search!?!

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

Rates for Programmable Parts

The parts undergo random
walks with fast diffusion.

It is hard to determine the assembly
rate for two assemblies.

dimers

first

one at

a time

trimers

first

Compare: Ab initio studies of reaction rates.

An Approach to

Probabilistic Solutions

Klavins,
In Preparation

(G
0
,

,

)

Problem
: Optimize the probability that the system behaves
like the specification
F
.

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|G
0
)

and p(F|

) are discrete.

Optimizing

Example
: Find
k

so that

(k) behaves like the specification
F

half of the time.

Approach
:

1)
Sample behaviors of

(k) to estimate
|p(F|

(k))
-
0.5| for a given k.

2)
Use these samples to estimate the

(k)

J. Spall, Ch. 14: Simulation
-
Based Optimization,
Introduction to Stochastic Search and
Optimization: Estimate, Simulation and Control
, Wiley & Sons, 2003.

F:

Results of Optimization

noise due to small
sample size

Only “finitely generated” properties can be checked.

Need a good model and an efficient simulation.

Can put

=

(k
1
)
[

(k
2
) to combine/compare grammars.

Acknowledgements

STUDENTS

Josh Bishop

Sam Burden

Richard Kreisberg

William Malone

Nils Napp

Tho Nguyen

Fayette Shaw

Ming Wang

CAREER: Programmed
Robotic Self
-
Assembly

COLLABORATORS

Karl B
ö
hringer

Robert Ghrist

Ongoing Work

Modeling

Synthesis

Complexity

Statistical Dynamics

Robots

Swarming

MEMs

DNA