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1
Biomimicry and Fuzzy Modeling: A Match Made in Heaven
Michael MargaliotSchool of Electrical EngineeringTel Aviv University, Israel
SCIS&ISIS’08, Nagoya University, Japan, Sep. 2008.
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OutlineBiomimicryFuzzy modeling: from words to
equationsFuzzy modeling of animal behavior: two
examplesAdvantages of fuzzy modeling:
A synergy between words, a fuzzy rule-base, and the mathematical model
Interpretability Verifying the verbal description
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Biomimicry
Definition: Biomimicry is the
development of artificial products or
machines that mimic (or are inspired
by) biological phenomena.
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Motivation for BiomimcryLiving systems developed efficient
solutions to various problems they
encounter in their natural habitat.
For example, foraging animals learned
how to address the challenge of
efficiently navigating and searching in
an unknown terrain.
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Motivation for BiomimicryScientists are interested in many
problems that living systems address.
For example: navigation in an unknown
terrain is a major challenge in the design
of autonomous robots. A natural idea is to
follow the solutions already developed by
living systems.
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Examples of Biomimicry Biological Agent
foraging animals
insects
evolution
trees
immune system
social insects
Artificial Design
autonomous robots
walking robots
genetic algorithm
artificial structures
computer security
clustering algorithms
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Biomimcry & Fuzzy Modeling Biomimcry requires “reverse engineering.”
In many cases, biologists have already provided a verbal description and explanation of the relevant biological behavior. This reduces biomimicry to the following problem.
Problem 1 Transform a given verbal description into a mathematical model or algorithm.
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Problem 1 & Fuzzy ModelingExtensive research suggests that fuzzy
modeling is the most suitable tool for
addressing Problem 1.
verbal description
mathematical model
fuzzy rule-base
simulation/analysis
Fuzzy modeling process:
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Fuzzy Modeling of Animal Behavior
Input: Verbal description of the behavior.
1. Identify the state variables
2. Restate the verbal data as If-Then rules
3. Define the fuzzy terms
4. Inference the fuzzy rule base to obtain a well-defined mathematical model
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Fuzzy Modeling of Animal Behavior
1. Territorial behavior of fish (Tron & Margaliot, 2004).
2. Flocking behavior (Lebar Bajec, Zimic, & Mraz, 2004).
3. Orientation to light in a planarian (Tron & Margaliot, 2005).
4. Foraging behavior of ants (Rozin & Margaliot, 2007).
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Fuzzy Modeling of Animal Behavior
5. Population dynamics in flies (Rashkovsky & Margaliot, 2007).
6. The Lambda switch (Laschov & Margaliot, 2008).
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Two Detailed Examples 1. Territorial behavior in the stickleback
(Lorenz)
2. Orientation to light in the Dendrocoleum lacteum (flat worm) (Ullyott, Fraenkel & Gunn)
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"a real stickleback fight can be seen only when two males are kept together in a large tank where they are both building their nests. The fighting inclinations of a stickleback, at any given moment, are in direct proportion to his proximity to his nest… The vanquished fish invariably flees homeward and the victor chases the other furiously, far into its domain. The farther the victor goes from home, the more his courage ebbs, while that of the
vanquished rises in proportion.
Arrived in the precincts of his nest, the fugitive gains new strength, turns right about and dashes with gathering fury at his pursuer.” (King Solomon’s Ring, p. 44)
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Fuzzy Modelling
• • • •
c1 x1 x2 c2
1 1w 1 1w
1 2 1x x x 1 1 1x c x
If If
If If
1 1( , )near x c Then 1 1( , )far x c Then
Then Then
1 1( , )near x c and and 1 1( , )far x c
1( )high w1( )low w
State variables:
Fuzzy rule-base:
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( ) (1 tanh( )) / 2
( ) 1 ( )
i
i
i i
whigh wa
low w high w
2 2( , ) exp( / )
( , ) 1 ( , )i i
i i
near x y x y k
far x y near x y
2 22 exp( ( ) / )( )( )
i i
i i i i
i i i j i
i i
w p x c k px c x high w x c
Inferencing yields the mathematical model:
Fuzzy Modelling
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Simulations
“The pursuit is repeated a few times in alternating directions, swinging back and forth like a pendulum which at last reaches a state of equilibrium at a certain point.” [Lorenz]
territory 1
territory 2
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Simulations (3D)
oscillatory behaviour convergence to equilibrium (proof via
linearization and eigenvalue analysis)
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Orientation to Light in the Dendrocoleum lacteum
dim light bright light
After a couple of hours:
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Rate of Change of Direction (r.c.d)
. . defelections (in a given time unit)r c d
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r.c.d. and Light Intensity
adaptation
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Klino-Kinesis(1)“An increase in stimulating intensity produces
an increase in r.c.d.
(2) This initial increase in r.c.d. falls off under
constant stimulation owing to adaptation.
(3) There is a basal r.c.d., which is an expression of
the fact that turning movements occur even in
absolute darkness or at complete adaptation.”
(P. Ullyott, J. Experimental Biology, 1936.)
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The “Average Animal*”
light Increases r.c.d increases AB short
adaptation r.c.d. decreases CD long
(* Fraenkel & Gunn. The Orientation of Animals, 1961)
dim light
bright light
A B
CD
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Fuzzy Modeling L(t) – light intensity
A(t) – level of adaptation to light
R(t) – r.c.d. B – basal r.c.d.
1( )A t c
1( )A t c
2( )R t c
3( )R t c
If (L(t)-A(t)) is positive then
If (L(t)-A(t)) is negative then
If (R(t)-B) is large then
If (L(t)-A(t)) is high then
Fuzzy rule-base:
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Fuzzy Modeling1 1
2 2 3 3
tanh( ( ))
( ) ( )cos( ), sin( )
k k
A c k L A
R c S R B c S L Ax v y v
/ 21
i
i it t
( )i
t
t R d q
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Simulation 1
R(t) as a function of time. Light is switched on at t=1.
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Simulation 2
Trajectory (x(t),y(t)). Light intensity is L(x,y)=x
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Advantages of Fuzzy Modeling
The knowledge is represented in three forms:
1. The initial verbal description
2. The fuzzy rule-base
3. The mathematical model
This provides a synergistic overview of the
system.
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Advantage 1: Interpretability
A fuzzy model is interpretable; each
parameter has a perceivable meaning.
Example 1: Consider the parameter in the
stickleback model. Recall: 2 2( , ) exp( / ).i in x y x y k
ik
As decreases, the Gaussian becomes
more centered, so Fish becomes “less
aggressive.”
ik
i
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Advantage 1: Interpretability
This links the parameter with the verbal
description.
The equilibrium points of the mathematical
model are: 1 2
1 2ln 2, ln 2.x c k x c k
1 2 ,k kIf the equilibrium position is no
longer symmetric; eventually fish 1 will have
a larger territory than fish 2.
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k1 =1, k2 =0.5
Advantage 1: Interpretability
first fish is “more aggressive”
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“.. the relative fighting potential of the individual is shown by the size of the territory he keeps clear of rivals.” (Lorenz)
Advantage 1: Interpretability
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Advantage 2: Verification
The mathematical model can be examined
using both simulations and rigorous
analysis.
This can be used, to some extent,
to verify the original verbal description.
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Advantage 2: Verification
Example: The planarian model includes the
rule:3( ) .R t c
3 0.c
If is high, then
Consider the case The r.c.d. will not
increase, and we may expect that the
model’s behavior will change substantially.
( ) ( )L t A t
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Advantage 2: VerificationFor the mathematical model yields:
22 ( ).kR c S R B
IfRecall that the right-hand turns take place at
times such that:
3 0,c
(0) ,R B then 0,R so ( ) .R t B
1 ( ) .i
i
t
t R d q
Hence, a periodic trajectory without
gradually moving to the shadier parts.
kt
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Fuzzy Modeling and Animal Behavior1. Animal (and human) actions are “fuzzy”:
“… a class of objects with a continuum of
grades of membership.” (Zadeh, 1965)
“… no sharp distinction is possible between
intention movements and more complete
responses; they form a continuum.”
(Heinroth, 1910)
Compare with:
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Fuzzy Modeling and Animal Behavior2. Verbal (and therefore vague) information:
“Nor shall I here discuss the various definitions which have been given of the term species. No one definition has as yet satisfied all naturalists; yet every naturalist knows vaguely what he means when he speaks of a species.” (Darwin, 1859)
“A high degree of contact causes low activity.” (Fraenkel & Gunn, 1961)
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Summary
Fuzzy modeling seems very suitable for transforming words to equations.
Numerous potential applications in the “soft sciences”: psychology, economy, animal behavior and more.
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Summary Fuzzy modeling seems particularly suitable for modeling animal behavior and for biomimcry:Start with a verbal description of abiological system (e.g., foraging ants); use fuzzy modeling to derive an analytical model which can then be implemented byartificial systems (e.g., autonomous robots).
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The Humpback Flippers*“Flippers with tubercles produced as much as 32% lower drag than the sleek flipper.”
*Miklosovic, Murray, Howlea & Fish, Physics of Fluids, May 2004.