From Stochastic Motion Generation and

Recognition to Geometric Symbol Development

and Manipulation

Tetsunari Inamura�� Hiroaki Tanie�� and Yoshihiko Nakamura���

� University of Tokyo� ����� Bunkyo�ku Hongo� Tokyo� Japan� Japan Science and Technology Corporation� CREST� Saitama� Japan

Abstract� Mimesis theory is one of the primitive skill of imitative learn�ing� which is regarded as an origin of human intelligence because imita�tion is fundamental function for communication and symbol manipula�tion� When the mimesis is adopted as learning method for humanoids�loads for designing full body behavior would be decrease because bottom�up learning approaches from robot side and top�down teaching approachesfrom user side involved each other� Therefore� we propose a behavior ac�quisition and understanding system for humanoids based on the mimesistheory� This system is able to abstract observed others� behaviors intosymbols� to recognize others� behavior using the symbols� and to generatemotion patterns using the symbols� In this paper� we extend the mime�sis model to geometric symbol space which contains relative distanceinformation among symbols� We also discuss how to generate complexbehavior by geometric symbol manipulation in the symbol space� andhow to recognize novel behavior using combination of symbols by knownsymbols�

� Introduction

Recently� the human behavioral science and the human intelligence have becomeconspicuous as a real research issue of robotics� Although the motivation of thearti�cial intelligence originated there� the physical limitations have forced orjusti�ed the researchers to carry on their research in a limited scope and scaleof complexity� It ought to be the major challenge of contemporary robotics tostudy robotic behaviors and intelligence in the full scale of complexity mutuallysharing research outcomes and hypotheses with the human behavioral scienceand human intelligence�

The discovery of mirror neurons��� have been a notable topic of brain sciencewhich have been found in primates� brain and humans� brain� �re when thesubject observes a speci�c behavior and also �re when the subject start to actthe same behavior� Furthermore� it is located on Broka�s area which has closerelationship between language management� The fact suggests that the behaviorrecognition process and behavior generation process are combined as the sameinformation processing scheme� and the scheme is nothing but a core engine of

symbol manipulation ability� Indeed� in Donald�s �Mimesis Theory������ it issaid that symbol manipulation and communication ability are founded on thebehavior imitation� that is integration of behavior recognition and generation�We believe that a paradigm can be proposed taking advantage of the mirrorneurons� with considerations of Deacon�s contention��� that the language andbrain had evolved each other�

So far� we have proposed a mathematical model that abstracts the wholebody motions as symbols� generates motion patterns from the symbols� anddistinguishes motion patterns based on the symbols� In other words� it is afunctional realization of the mirror neurons and the mimesis theory� For theintegration of abstract� recognition and generation� the hidden Markov model�HMM is used� One as observer would view a motion pattern of the other as theperformer� the observer acquires a symbol of the motion pattern� He recognizessimilar motion patterns and even generates it by himself� One HMM is assignedfor a kind of behavior� We call the HMM as symbol representation�

Symbols are required to represent similarity or distance between each sym�bol� An example application is symbol manipulation based on the similarity ordistance information� However� our conventional method have no way to repre�sent similarity of relationship between each behavior� Therefore� in this paper�we extend the mimesis model to geometric symbol space which contains relativedistance information among symbols� We also discuss how to generate complexbehavior by geometric symbol manipulation in the symbol space� and how torecognize novel behavior using combination of symbols by known symbols�

� Hierarchical Mimesis Model

��� Usual mimesis models and its defects

Figure � shows the con�guration of usual mimesis models������� The model con�sists of three processing� perception part� generation part and development part�In the perception part� observed motion patterns are analyzed into basic motionelements� Motion elements are low level physical parameter for short period oftime� like joint angle� angular velocity or torque� Others� motion are representedby the sequence of the element� then the dynamics in the motion is abstracted assymbol representations� namely HMMs� We have call such symbol representationas �proto�symbols��

Here� one defect arises that the relationship between two similar behaviorhave been lost when the behavior transfered into proto�symbols� An aspect ofthe symbol is that semantic relation exists among symbols� contrary to the iconsand labels which have no relation representation between each elements� Tosolve this defect� a hierarchy structure is needed in which the relation in thelower motion pattern layer will be kept in the higher symbol layer� We call suchstructure as �Hierarchical Mimesis Model�� In following sections� we introducean mathematical framework for the Hierarchical Mimesis Model�

Observation

Proto-symbols

CommunicationConcept Formation

Angular data level

Behavior of

Humanoids and Human

Embodiement

Evaluation

G e

n e

r a

t i o

n P e r c e p t i o n

( , )

Database

Motion Elements

HMM ( , )

D e v e l o p m

e n t

Motion sequence level

Ps

BD

BC

D

Fig� �� Outline of Usual Mimesis Models

��� Hierarchical Mimesis Models

To construct such a hierarchical model� the mathematical framework ought tobe had an ability to treat distance relationship between each behavior and sym�bol� and to structure the hierarchic� An conceptual image of such heirarchicalmimesis model is shown in Fig�� In the model� each proto�symbol is representedby continuous vector� that is� continuous proto�symbol space exists above thelow level behavior pattern� Using the proto�symbol space� human�s long termbehavior are transfered into trajectory in the space� We can also abstract thetrajectory as a symbol�like representation using the same HMM� In other words�hierarchical abstract framework can be established easily based on the HMM�Continous HMM is the adequate way to realize such a hierarchical structure�

WalkKick

Run

ShotAbstract with HMMs

Generation with HMMs

Proto-Symbol Space

Meta Proto-Symbol

Observe Behavior

Generate Behavior

Fig� �� Outline of Hierarchical Mimesis Model

� Construction of hierarchical proto�symbol space

��� Continuous HMMs and humanoids� motion

We focused on Hidden Markov Models �HMMs as mathematical backbone forsuch an integration� HMMs are one of stochastic processes which takes time seriesdata as an input� then outputs a probability that the data is generated by themodel� HMMs is most famous tool as a recognition method for time series data�especially in speech recognition �eld� HMMs consist of a �nite set of states Q �fq�� � � � � qNg� a �nite set of output vectors S � fo�� � � � � oMg� a state transitionprobability matrix A � faijg �the probability of state transition from qi to qj �an output probability matrix B � fbijg� and an initial distribution vector � �f�ig� that is a set of parameter � � fQ�S�A�B��g� In this framework� statetransition and output processes are performed probabilistically� then sequence

q1 q2 qN

q3 qN-1

o

a11 aN-1N-1a22 a33

a12 a23 aN-1 N

o o o oO

time

1uk 2

uk 3uk 4

uk Tuk

1k 2k 3k k Tk 4

b1k1b2k2 b3k4 bN-1 kT

Fig� �� Discrete Hidden Markov Models and Humanoids� Motion

of vectors are output during the transition as shown in Fig�� The vector maybe a discrete label� or be a vector� In the case of label� the HMMs are called asdiscrete HMMs �DHMMs � In another case of vector� the HMMs are called ascontinuous HMMs �CHMMs as shown in Fig���

��� De�nition of distance between HMMs

Through distance is needed for construction of space� distance between twoHMMs is not able to be de�ned easily because it is stochastic model� For suchstochastic modes� there is a method in order to express the distance informa�tion� In this paper� we adopt Kullback�Leibler information as the representationof distance between HMMs� To say strictly� the Kullback�Leibler information isnot distance because it does not satisfy the property of the distance� triangleinequality and symmetry� therefore we call the Kullback�Leibler information asdegree of similarity of the HMMs� The Kullback�Leibler information against twostochastic models p� and p� is de�ned as follows�

D�p�� p� �

Z�

��

�p��x log

p��x

p��x

�dx ��

To apply the Eq��� to HMMs� following equation is usually used��� �

D ���� �� �Xn

�

Tn

�logP �yT� j�� � logP �yT� j��

��

q1 q2

qN

q3

qN-1

a11 aN-1N-1a22 a33

a12 a23 aN-1 N

a13 aN-2 N

time

u(1) u(2) u(3) u(4) u(t)

b ( )1 b ( )3b ( )2

b ( )N-1

Fig� �� Continuous Hidden Markov Models and Humanoids� Motion

As the Eq�� does not satisfy the distance axiom� we use following improvedinformation�

Ds���� �� ��

�D ���� �� �D ���� �� �

��� Construction of space based on similarity

In order to construct proto�symbol space from the distance information� mul�tidimensional scaling �MDS is used� MDS is a method that accepts distanceinformation among elements and outputs position of each element in the gener�ated space� Let the similarity between i�th element and j�th element as fij � thedistance between i�th and j�th element as dij � MDS makes the following errorto be minimum for the space construction�

S� �Xi�j

�fij � dij � ��

In the case of HMMs� Kullback�Leibler information Ds is used for the simi�larity fij � Let the position of each HMMs as x � fx�� x�� � � � � xng where n is thenumber of dimension of the space� Using least�squares method� each position xis calculated�

We con�rmed the performance of the proposed space construction methodagainst six kinds of motion shown in Fig��� At �rst� we gave �� dimensional vectorfor each fx�� x�� � � � � xng� Figure � shows the constructed space and locationof each proto�symbol �HMM � As the diagram indicates� from �rst to fourthdimensions are e�ectively used for the space construction� however� the rest of

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0

10

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30

cossack

crawl kicksquat

swing

walk

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0

10

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30

cossackcrawl

kicksquat swingwalk

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0

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cossackcrawlkicksquatswingwalk

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0

10

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cossackcrawlkicksquatswingwalk

-30 -20 -10 0 10 20-10

0

10

20

30

cossackcrawlkicksquatswingwalk

3rd dimension1st dimension

7th dimension5th dimension

9th dimension

2nd

dim

ensi

on

4th

dim

ensi

on

6th

dim

ensi

on

8th

dim

ensi

on

10th

dim

ensi

on

Fig� �� Result of proto�symbol space

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0

20

-10

-5

0

5

10

walk stretchkick squat throw stoop

first dimension second dim

ension

third

dim

ensi

on

Fig� �� Six motions in �D proto�symbol space

�a�

�b�

�c�

�d�

�e�

�f�

Fig� �� Six motions performed by human

the dimension are not well used� Therefore� we adopted three dimensional proto�symbol space as shown in Fig���

� Behavior Manipulation by Proto�symbol Manipulation

��� proto�symbol manipulation

In this paper� symbol manipulation is de�ned as following�

� Generation of novel behavior using known basic motions� Recognition of novel behavior using known basic motions� Abstract of novel behavior using know basic motions

From the view point on the proto�symbol space� above de�nitions can be inter�preted as followings�

�a�

�b�

�c�

�d�

�e�

�f�

Fig� �� Six motions imitated by humanoid

� Generation� creating a novel state point and motions using existing statepoint of proto�symbols�

� Recognition� recognize a novel sequence of state point� using existing statepoint of proto�symbols�

� Abstract� abstract of a novel sequence of state point into meta�proto�symbol�

A mutual conversion method between motion patterns and symbol represen�tation in proto�symbol space have been established up to the previous section�In this section� proto�symbol manipulation method where generation and recog�nition of novel motion are de�ned by simple structure�

��� Generation of novel proto�symbol

Creating a novel proto�symbol is equal to create a novel state point on the proto�symbol space� To create a novel state point� following composition regulation isused�

bi�o �

MXm��

�cimAN��imA

���imA

�MXm��

��� � cimBN��imB

���imB ��

aij � �aijA � ��� � aijB ��

Eq��� and Eq��� is applied when a novel state point is located on a straightline connecting the two points ��A and �B � When a novel state point is not onany straight lines connecting known proto�symbols� the parameter is composedaccording to the distance ratio among each known proto�symbol as follows�

bi�y �

MXm��

�

dlP

l�dl

cN��lim��lim�

aij �MXm��

�

dlP

l�dl

alij ��

where dl is the distance between a novel state point and known proto�symbol�l� Finally� law�level motion pattern is generated from the state point� using themethod proposed in the paper����

Novel motion generation from state sequence in proto�symbol space

Furthermore� motion generation can be performed against the state transitionsequence in the proto�symbol space� In this paragraph� a motion generationmethod in which a state transition sequence have been given�

Let be the state transition as x����x��� � � � �x�n�� As a generation methodin which a �xed state point is given in the proto�symbol space is introduced

0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

angl

e[ra

d]

time[s]

Walk

Kick

Fig� � Generated motion from mixed HMMs

before� the continuous generation by transitional state points is equivalent tothe average of motions which generated by those state points�

Figure �� shows the outline of the generation process�In step �� motion patters are generated from each state point in the proto�

symbol space using the proposed method���� In step � the time length of allmotion patterns are set to the same value Tc in order to composition� In step and �� partial motion patterns are picked up based on the phase information foreach state point� that is� charging period of time for each state point� Finally instep �� composite motion pattern are generated�

��� Recognition of novel motion based on proto�symbol

manipulation

The proto�symbol based recognition is equivalent to decide a state point in theproto�symbol space� After the decision� the relationship between proto�symbolsbring about the symbol representation of the novel motion�

Di�erence between observed motion O and each proto�symbol is representedas the likelihood P �Oj�i � To convert the likelihood into distance information�following equation

D��k � �i �X �

T

�logP �oTk j�k � logP �oTk j�i

���

proto-symbolspace

step2

step1

step4

step3

Fig� �� Outline of motion generation

are used� however� the P �Oj�O is not decided before learning of the HMM �O �Here� we introduce an approximation for the P �Oj�O �

Table �� likelihood

dance hmm crawl hmm kick hmm squat hmm swing hmm walk hmm

dance �� ����� ���� ��� ���� �����

crawl ��� �� ���� ����� �� �� �� �����

kick ���� ��� ����� ���� �� ����

squat ����� ���� ����� �� � �� ����

swing � � ����� ����� ����� � � � ������

walk � �� ����� ���� ���� ����� ���

This table shows the logarithm of the likelihood for each motion and proto�symbol� Diagonal values indicate about from � to �� however� the rest show avery little value� According to the result� we had an assumption that a novelbehavior also follows above empirical rule that�

logP �Oj�O � � ��

Novel motion recognition as state sequence in proto�symbol space

The outline of novel motion recognition as state sequence in the proto�symbolspace is shown if Fig���

In step �� focusing on the period of time Tspan in the observed motion pattern

O ��o� o� � � � oT

�� Let the cut o� motion pattern beO� �

ho��� o�� � � � o�Tspan�

i�

In step � state point is decided using mentioned method in previous paragraph�Next� shift the focus point� and let the k�th focus point beOk � fo���k����Tstep � � � � �

o��Tspan��k����Tstepg� with increase the index as k � �� � � � � �T���Tspan

Tstep��� Fi�

nally in step � sequence of state point in the proto�symbol space is acquired�

� Experiments

�� Recognition of novel motion

First� recognition in which motion patterns are transfered into a state point inproto�symbol space� was performed� Two novel motion �throwing with kicking�and �stretching with walking� are target motions� Figure � shows the resultof recognition� The squares and triangles are known basic proto�symbols� Thesmall dots indicates result state points� As the diagram shows� two dot marksare located on the line between each basic proto�symbol�

Next� recognition in which novel motion pattern are transfered into a se�quence of state points in the proto�symbol space� was performed� The targetmotion is �walking �rst� then shift to kicking�� The result is shown in Fig���As the diagram shows� recognized dot marks starts from the proto�symbol of�walk�� ends at the proto�symbol of �kick��

Walk

Proto-Symbol Space

KickRun

Observed Behavior

Walk

Proto-Symbol Space

KickRun

Observed Behavior

step3

step1

step2

Tspan

Fig� ��� Procedure of projecting motion in proto�symbol space

�� Generation of novel motion

In this experiment� we have investigate the motion output when a trajectory isgiven in the proto�symbol space� As a given trajectory� we prepared simple linetrajectory from the �walking� state point to the �kicking� state point� Figure�� is the result of motion output�

As the �gure shows� motion of the humanoid is adequately controlled as thesymbol manipulation in the proto�symbol space�

� Discussion

�� Comparison with conventional research

Motion recognition using the HMM is famous method� thus many research areproposed for gesture recognition or behavior understanding ������� �������� how�ever� no research has been existed in which motion is generated from HMMs�Masuko et al������� have been proposed a speech parameter generation methodusing HMMs� however the generation process is not opposite direction of thespeech recognition process� The most important characteristic of our method isthat the motion recognition and motion generation process are integrated byonly a HMM�

Acquisition and symbolization of motion pattern of robots have been dis�cussed in Doya�s research��������� In their MOSAIC model� time�series data are

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-15

-10

-5

0

5

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15

20

first dimension

seco

nd d

imen

sion

walk stretch kick squat throw stoopkick + throw walk + stretch

Fig� ��� A recognition result of novel motions

acquired as sequence of symbol representations� however� the symbol represen�tation do not contains dynamics information in the time�series data� In contrast�our method� the proto�symbol representation contains dynamics information inmotion patterns� Another di�erent issue from Doya�s research is that the singleHMM is acts as recognition and generation model� In Doya�s MOSAIC model�two modules� prediction module �for generation and control module �for recog�nition � becomes a companion as motion primitive� We think that single HMMis smarter for abstract as symbol representation�

To integrate recognition process and generation process for time�series data�some dynamical approach have been proposed��������� In these research� explicitdynamics like attractor is designed in order to abstract the time�series data�In conventional research� recurrent neural network is one of the e�ective wayfor the dynamics design ���������� ��� However� it is di�cult to design symbolrepresentation if these method was adopted as the dynamics representation� TheHMM is e�ective for both dynamics representation and symbol representation�

Symbol emergence had been tried in conventional research of arti�cial intel�ligence� The most di�cult issue of the symbol emergence is how to manipulatethe created symbol representation� contrary to the easiness of symbol creation�Deacon have proposed the symbol development model ��� as shown in Fig���� Inhis theory� symbolic representation is developed from indexical level and iconiclevel� In the indexical level� simple relationship between a motion pattern anda symbol representation is established� however� relationship between each sym�bol and motion pattern is not considered� In the transitional level� relationshipbetween each symbol is developed as token combinations� then the relationshipbetween each motion pattern starts to be constructed� In the �nal level� logicalrelationship between symbol combined with the physical relationship between

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walk stretchkick squat throw stoop traject

first dimension

third

dim

ensi

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secon

d dim

ension

Fig� ��� A result of motion recognition in the proto�symbol space

motion pattern� Our approach follows the development model� At the presentmoment� our method achieved the transitional level and is going to achieve the�nal symbolic level�

� Conclusion

To solve the defect of usual mimesis model� that is� impossibility of the symbolmanipulation� we introduce hierarchical mimesis model� The hierarchical modelis build on a continuous Hidden Markov Model and distance representation us�ing Multidimensional scaling method� Continuous Hidden Markov Model enablesthe model to generate humanoids� motion naturally as contrary direction of themotion recognition� Multidimensional scaling method enables the model to de�scribing the relationship between each proto�symbol� namely continuous HMM�Owing to the distance representation� symbol manipulation is achieved as geo�metric state manipulation� Through experiments� following ability is realized� ��

Fig� ��� Generated motion� from walking to kick

novel motion can be recognized as combination of known motion�s proto�symbols�� novel motion can be generated by combination of known proto�symbols�

Remaining problem is the defectiveness of proto�symbol space� Even if themedian point in the proto�symbol space between two proto�symbols generatesnatural motions� the characteristics of the motion is not always similar to thecharacteristics of real two motions� The most frequent cause is the proto�symbolspace is not Euclid space� Toward the issue� we think that information geometrywould achieve the desired e�ect�

Acknowledgement

This research was supported by the Core Research for Evolutional Science andTechnology �CREST program of the Japan Science and Technology Corporation�PI� Y� Nakamura �

S3S8 S4

S1

S5S2

S6

S7

O3O8 O4

O1

O5O2

O6

O7

S3S8 S4

S1

S5S2

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3.SYMBOLIC

2.TRANSTIONAL

1.INDEXICAL

logical relationshipsbetween tokens

physical/pragmaticrelationships

patterns of tokencombinations

relationshipsbetween objects

sign stimuli(tokens)

objects ofreference

Fig� ��� Development of symbol representation from indexical level by Deacon� S in�dicates symbol� o indicates raw�level pattern like a motion patterns

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