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)
.
See also: K. Saitou, 1999: Conformational Switching
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)
See Also, Courcelle: Handbook of TCS, Vol B
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
lead to a new graph.
Klavins, Ghrist and Lipsky, IEEE TAC (To Appear)
See Also, Courcelle: Handbook of TCS, Vol B
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
lead to a new graph.
Graph Grammars
Klavins, Ghrist and Lipsky, IEEE TAC (To Appear)
See Also, Courcelle: Handbook of TCS, Vol B
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
undergrad)
Automated
mixing system
Leaf blowers!
Overhead
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
•
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)
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]:
If, in addition, E(L)=
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(FG
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
gradient and do gradient descent.
(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
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