Locomotion of Animals,
Design of Robotsand Mathematics

R. Kobayashi, T. Nakagaki and A. Ishiguro
Hiroshima University, JST CREST

Amazing Creature - True Slime Mold
Physarum can solve a maze !

Physarum Polycepahlum（和名：モジホコリ）

Large single cell organismwith multi nuclei
Tubular structure

Nakagaki et al., Nature (2000)

Physarum Solver
Shortest path finding model on the graphiDij(piLijI0
I00pj ) =d
Dij = f (|Qij |)dt

Dij

for j = 1,
for j = 2,otherwise.
Qij =I0
N1Dij(pi − pj )LijN2I0Seatle

Tero, Kobayashi, Nakagaki, J. Theor. Bio. (2007)

HoustonPhysarum designs networks !
Real Railroad Network 	in Tokyo area

Network produced 	by Physarum

by S. Takagi
Efficiency

CostFault tolerance

Tero et al., Science(2010)

Extended Physarum Solver
CostEfficiency, Fault tolerance

by A.Tero

Physarum is completely decentralized system !

Ig Nobel Prize2008

Cognitive Science Prize

2010

Transportation Planning PrizeFirst make people laugh,
and then make them think

Locomotion of Animals

Locomotion of Robots

Completely centralized control
Decentralization
Autonomous Decentralized ControlControl policy which attains useful functions by the interactions between
local elements having simple ability of sense, judge and motor output

ADC in Animals

Neural ganglion in each
body segment

Gait transition of DC cat
on treadmill

Central Pattern Generator
Neural circuits which generate rhythm
Lamprey, Tadpole
Details are still unknown

Maｍmals

ADC is OK, but ....
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How can we achieve theemergence of function
from such systems ?

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1. Dynamics of each component
2. Interaction between components
3. Local sensory feedback
ad hoc design for each case example

Still missing a systematic way of designing ADC !

Outline of Our Project
Goal
Understand the animalʼs locomotion from the view
point of mechanics and control
Produce robots which move in supple, agile androbust manner like animals
Mathematics
R.Kobayashi

Who ?
Team consists of Biologists,
Mathematicians & Roboticists

How ?

Robotics
Biology
T.Nakagaki A.Ishiguro

Learn from the animals
Design robots with large DOF controlled by ADC

Which animal at ﬁrst ?

Go Back to Physarum !
Completelydecentralized system
Driven by distributed
oscillator system
High ability
Solving a maze anddesigning networks

Anti-phase Oscillations
Peripheral Phase Inversion

BodyBottleneck Phase Inversion

Phase of oscillators

Physarum Model ver.1
Real thickness

Kobayashi and Nakagaki (2003)
Rediscovered by Ishiguro (2008)

Darcyʼs Laws
sns+

Pressurep
Active spring whose natural lengthis driven by the phase oscillator

Stiﬀness
Natural thickness

snOscillatorsn ( ) = s̄(1Local Sensory Feedback⇤t = ⇥ +· (D)Ia cos )p = (ssn (⇥))

ts =· (sM p)

I=2p2 : Discrepancy
Function

Basic Design Schemei : the i -th controller

RobotEnvironment

S1BodyS2(phase oscillator)

Actuator

S i : State variable ofthe i -th actuator

12Ii (S i ,Sii: Frustration accumulated
in the i -th unitjSj⇤t i = ⇥i +gij ( i ,ji)

Discrepancy Functionj)

iIi

Indirect interaction
through the bodyOur Robots

Slimy

HAUBOT

OSCILLEXPENTABOTSnake Robot：HAUBOT２
Joint mechanism

Stiffness [a.u.]

T. Sato (2011)

η [rad]Elastic elements permit
discrepancy between themotor angle (targetangle)and the actual jointangle
Stiﬀness is also controllable
by twistingPhasic & Tonic Control⇤t i = ⇥ + D( i 1⌅t ⇤i = (⇥IiHead¯t = ¯i +ii+1i

Target angle¯i = a sin ⇥ii

TailIiDiscrepancy

Ii = ⇥| i¯b = ¯iii¯i |Real angle

Mechanical SystemControl Systemi)–⇤i )

i 1




i




HAUBOT 1 & 2
Phasic Control

Tonic Control

phase adjustment

stiﬀness adjustment

Energetic eﬃciency

Powerful motion

NTF Award Finalist for Entertainment Robots and Systems (IROS2011)

OSCILLEX series
24



1
3

Control rule⌅t ⇥i = ⇤Control parameterfrom the center

脚先位置の制御
⇡/2
A1BB

A2

3⇡/2

Load at the toe

No direct interaction between
controllers, but indirectinteraction through the bodySwing phasei⇡

Ni cos ⇥i

0Stance phase

Try to keep the leg instance phase if loaded

0SILLEX 1sinsiniLF

LF

LH

LH

RF

RFRH

0time(sec)
20

RH

0

itime(sec)20Why we call it 0SILLEX ?
Oscillatory⌅t ⇥i = ⇤Excitable

Ni cos ⇥i
⇥Ni

⇥
Ni

Go back and force
between Oscillatory &Excitable mode

Quick transition to thestationary walking

OSCILLEX 3.1

Signiﬁcance of OSCILLEX⌅t ⇥i = ⇤Control
SystemCompletelyExplicit Control

Ni cos ⇥iBraincomputation

Slide Bar

Implicit
Control

Explicit
ControlManmals

Morphologicalcomputation？

Insects

Successful example of attaining
Adaptive Behavior by the balance ofExplicit Control and Implicit Control

Body
SystemCompletely
Implicit Control

Summary
We extracted the concept of discrepancy function fromthe model of Physarum.
The design scheme of ADC using discrepancy function
was implemented to our robots, and it worked very well.
Do not design the complicated explicit control rule.
Listen to the voice of the body !