Proc. of 15th Int. Joint Conf. on Artificial Intelligence pp. 1194-1200, Nagoya, Japan, August 1997

Proc. of 15th Int. Joint Conf. on Artificial Intelligence pp. 1194-1200, Nagoya, Japan, August 1997

Vision-Motion Planning of a Mobile Robot

considering Vision Uncertainty and Planning Cost

Jun Miura and Yoshiaki Shirai

Dept. of Computer-ControlledMechanicalSystems, OsakaUniversity

Suita, Osaka565, Japan

Email: jun@mech.eng.osaka-u.ac.jp

URL: http://www-cv.mech.eng.osaka-u.ac.jp/~jun/

Abstract

This pap er prop oses a planning metho d for

a vision-guided mobile rob ot under vision un-

certaintyandlimitedcomputationalresources.

Themetho dconsidersthefollowingtwo trade-

os: (1) granularityin approximatinga prob-

abilistic distributionvs. plan quality, and (2)

searchdepthvs. plan quality. Therst trade-

o is managed by predicting the plan quality

for a granularity using a learned relationship

b etweenthem,and byadaptivelyselectingthe

b estgranularity. Thesecond trade-o is man-

agedbyformulatingthe planning pro cessas a

searchinthespaceoffeasibleplans,andbyap-

propriatelylimitingthesearchconsidering the

meritofeachstepofthesearch. Simulationre-

sultsand exp erimentsusing a real rob otshow

thefeasibilityofthemetho d.

1 Intro duction

There has been an increasing interest in autonomous

mobile rob ot which recognizes anenvironment with vi-

sion and moves without guidance of human op erators.

A key to realize such a rob ot is the ability to gener-

ate a plan of vision and motion operations so that a

robot may ecientlyreach the destination. To design

a planning algorithm for sucha rob ot, we haveto con-

siderthefollowingtwoissues: limitedcomputationalre-

sourcesanduncertaintyinvisualdata. Thesetwoissues

are closely related; planning based on uncertain data

usually requiresmore computationthan planningwith-

out uncertainty because multiple p ossible outcomes of

actionsshould beconsidered, and therefore,thelimita-

tionofcomputationalresourcestendstob ecritical.

Oneoftheusefulto olsforplanningunderuncertainty

is statistical decision theories

[Berger, 1985].

Several

worksappliedstatisticaldecisiontheorytovisionand/or

motionplanningtasks(e.g.,

[Hutchinson

andKak, 1989]

[Cameron

andDurrant-Whyte,

1990][Dean

et al., 1990]

[

Miura and Shirai, 1993 ]

). Onedrawback of these ap-

proachesisthatalargebranchingfactorofasearchtree,

whichisdeterminednotonlybythenumb erofpossible

totheuncertaintyofsensorydata,makesaplanningpro-

cesscomputationallyexp ensive.

Regarding thelimitation of computational resources,

manyworkshaverecentlyb eenfo cusingontheconcept

oflimitedrationality [

RussellandWefald,1991 ]

,inwhich

thecostofplanningisexplicitlyconsideredandthetime

for object-levelplanning is allo catedso that theoverall

utility including b oth plan eciency and planning cost

is maximized. Someof examples are: exible computa-

tion

[Horvitz,1990],

decision-theoreticmeta-levelcontrol

of (object-level) reasoning [

Russell and Wefald, 1991 ]

,

and exp ectation-driveniterative renement(EDIR)us-

inganytimealgorithms

[Bo ddy

andDean, 1989].

This pap er is concerned with a vision-motion plan-

ningofamobilerobotconsideringvisionuncertaintyand

planning cost. Fig. 1 showsan exampleproblem. Our

mobile rob ot isgoing to the destination while avoiding

obstacles. Thereisaroutewhichpassesthenarrowspace

(wecallitthegate)b etweentheb oardandthepartition;

howeverthepassability ofthegate isinitially unknown

duetotheuncertaintyofvisualdata. Thedetourpassing

throughthehallwayisknowntob epassable,althoughit

islonger. Therob otestimatesthegatewidthwithstereo

vision to determine thepassability. Theplanner deter-

mines a set ofobservationp ointswhichecientlynavi-

gates therob ottothedestination. Forthisproblem,we

proposeaplanningmethodcombinesadecision-theoretic

approachwithconsiderationofplanning cost.

table table

desks desks

hallway initial

position gate

detour destination

robot gate

detour

Figure1: Anexampleplanningproblem.

width of the robot (W ) robot distance between the objects

W robot

probability

distance passable (a)

probability

distance impassable (b)

W robot

probability

distance (c)

W robot

unknown

Figure2: Threepossiblestateofthegate.

2 Basic Planning Strategy

2.1 Plan Representation

Anactioniscomposedofamovementtothenextobser-

vationpoint andan observationat that p oint. A state

isrepresentedbythecurrentestimateofthegatewidth

andthe currentrob ot p osition. Due to theuncertainty

in observation results, the rob ot cannot determine the

gatewidthdeterministicallybutobtainsitsprobabilistic

distribution.

After anobservation, therob ot classies thestateof

the gate into one of the three states (see Fig. 2): if

the robot width is smallerthan the minimum valueof

theprobabilisticdistributionofthegatewidth,thegate

is passable; ifthe rob ot width is larger than the maxi-

mum valueof theprobabilisticdistribution, thegate is

impassable;otherwisethepassabilityisunknown.

Since the actual state after an observation dep ends

ontheobservationresultandcannotb edeterminedb e-

forehand,asubplanisgeneratedforeachpossiblestate,

whichispredictedusingtheuncertaintymo delofvision.

Fig. 3 shows an example plan for the problem shown

in Fig. 1. Sucha plan is represented by an AND/OR

tree;an ORno decorresp onds toselectionofan action;

anANDno decorresp ondstoap ossiblestate. Thequal-

ityofa planis measuredin termsofitsexecutioncost,

which isthe expectation ofthe totalexecution timefor

movementandobservation.

The leaves of an AND/OR tree are either terminal

nodeor open node. At aterminal no de, the passability

is decided without uncertainty and, thus, the nal ac-

tion(i.e., passing thegate ortaking thedetour) is also

decided. At an op en no de, since thepassability is un-

known,thenalactionhasnotb eendecidedyet. Aplan

candidateisrenedbyexpanding(makingsubplansfor)

one of its op en no des. In expansion of an open no de,

the p ossiblerange ofthe gate width is discretizedwith

some granularity, and a subplan is generated for each

passable unknown impassable

passable

impassable, unknown

Figure3: Anexampleplan. Dottedarrowsindicatep os-

siblemovementsafterobservation. Boldarrowsindicate

observationofthegate.

2.2 Computational Trade-os to b e

Considered

The planning metho d is designed to dealwith the fol-

lowingtwocomputationaltrade-os:

searchdepthvs. planquality: Thistrade-ohas

beeninvestigatedbyseveralresearchers(e.g., DTA 3

by

Russell and Wefald

[1991]).

We consider this trade-o

in generatingamulti-step plan.

granularity vs. plan quality: A nergranularity

fordiscretization improvesplanquality,butitincreases

theplanning cost. Thistrade-o haslittlebeenconsid-

ered, althoughitis imp ortantin planning underuncer-

tainty.

Thersttrade-oismanagedbyformulatingtheplan-

ning pro cess as an iterative renementpro cess

[Bo ddy

andDean,1989 ]

,andbyappropriatelylimitingtheiter-

ation. To cop e with the second trade-o, we represent

therelationshipb etweengranularityandtheexp ectation

of thereductionof aplan cost(wecall thisexp ectation

aplan improvement)asaperformanceprole [Dean

and

Boddy,1988 ][

Zilb erstein,1993 ]

,andthendeterminethe

best granularity by examining the p erformance prole

and thecost ofno de expansion. With consideration of

these trade-os, the plannertries tominimize thetotal

costofplan generationandplanexecution.

3 Formulation as Iterative Renement

3.1 Easily-Obtainable Feasible Plan

In an iterative renement framework, the planner

searches the space of feasible plans (executable plans)

for the nal plan. This formulation entails an easily-

obtainablefeasibleplanforanyop enno de. Therearetwo

such feasible plans. One of them is to take the detour

from the current p osition; this feasible plan is usually

costly. Thusweusetheotherfeasibleplan:

The robot moves from the current position to

the position just before the gate 1

. If the gate

1

At this p osition, the rob ot is assumed to b e able to

measure the gate width without uncertainty; this p osition

3

action

state

state state

passable impassable

action

state state

passable impassable

state state 1 ... i ... state n

action i

state

state state

passable unkown impassable ...

...

unkown

(a) before expansion (b) after expansion

Figure4:Expansionofanop enno deofaplancandidate.

Ellipsesdrawnwithboldlinesindicate op enno des.

ispassable,the robotpassesit;ifnot,the robot

takes the detourfromthatposition.

Each plan candidate has the temporary cost, C temp

,

whichisobtainedbytemporarilyassigningtheab ovefea-

siblesubplantoallofitsop enno des.

3.2 Selection of Open No de to Expand

Intheplanningpro cess,themostpromisingop enno deis

selectedand expanded. Beforeexpansion, anunknown

stateistreatedasoneop ennodeandhasafeasibleplan

withit(seeFig. 4(a)). Theexpansionoftheop enno de

consists of searching for the b est action for each dis-

cretizedstateandassigningthefeasiblesubplantonewly

generatedop enno des(seeFig. 4(b)).

The op en no de to b e expanded is determined as fol-

lows. Supp ose we can predict the maximum plan im-

provement(i.e.,costreduction)1Cofaplancandidate,

whichwillb eobtainedbyexpandingitsb estop enno de 2

.

Then, themerit ofexpandingtheno deis givenbysub-

tractingtheexpansioncostC exp

fromthepredictedplan

improvement1C.

LetC temp

i

,1C

i ,andC

exp

i

denotethetemporarycost,

thepredictedplanimprovement,andtheexpansioncost

of a plan candidate PC

i

, resp ectively. Then, wedene

thenewcostC new

i as:

C new

i

=C temp

i

0(1C

i 0C

exp

i

): (1)

LetC

FP 3

b ethecostoftheincumbent(theb estfeasi-

bleplanamongthosewhichhaveb eenobtainedso far).

Duringtheplanning,theplancandidatesarekeptwhose

new costs are less than C

FP 3.

Among thecandidates,

the plan candidatePC

i

3 which is to b e expanded next

isdeterminedas:

PC

i 3;

i 3

=argmin

i C

new

i

: (2)

The planneriteratively selects PC

i

3 and expands its

bestop en no de. Thisplanning pro cess hastheanytime

[Dean

and Bo ddy, 1988]

prop erty. The iteration stops

whenC new

i 3

is largerthanC

FP 3.

2

Themetho dtopredicttheplanimprovementwillb ede-

Theab ovealgorithmwillexpandanodeaslongasC new

i 3

is less than C

FP 3

even iftheir dierence is very small.

Expansioninsuchacase,however,mayb euselessifthe

meta-planningcostishigh.

Thus, we slightly mo dify the termination condition.

That is, if thedierenceislessthan themeta-planning

cost, the iteration pro cess stops and the b est feasible

plan is returned as the nal plan. The cost of meta-

planning is,atpresent,consideredtob econstant.

3.4 Deciding Only the NextAction

When the objective of the planner is not to generate

a whole plan but to select the b est next action (in a

dynamicenvironment,forexample),thealgorithmisal-

tered asfollows.

Let C

FP 3

A

b e thecost ofthebest feasibleplan which

starts with action A. If the smallest value of C new

i of

plancandidateswhichstartwithactionsotherthanAis

larger thanC

FP 3

A

,the planning pro cessterminates and

returnsAastheb estnextaction. Thisstrategyissimilar

to thatofDTA 3

[Russell

andWefald, 1991].

4 Determining the Best Granularity

Wehavementionedthat thegranularityin discretizing

the ranges of random variables (the gate width in our

case)directlyaectsboththeplanimprovementandthe

expansion cost. The selection of the best granularity

is, therefore,crucial to managing thetrade-o b etween

planning costandplaneciency.

This pap erprop oses torepresenttherelationshipb e-

tweenthe granularityand the predictedplan improve-

mentasap erformanceprole,andtocalculatetheb est

granularityandthemeritofexpansionsimultaneously.

Currently we equally divide the range of a variable;

thus,thegranularityissp eciedwiththenumberofdivi-

sions. LetPI(n)denotethepredictedplanimprovement

forgranularityn.

Theb estgranularityn 3

isgivenby

n 3

=argmax

n

fPI(n)0C exp

(n)g; (3)

where C exp

(n)is thecostof expansionwithgranularity

n,whichisdened asfollows:

C exp

(n)=N

cand 3C

exam

3n; (4)

where N

cand

isthenumberof actioncandidates;C exam

isthecostrequiredforexaminingoneactioncandidate.

Fig. 5 illustrates the determination of the b est granu-

larity. Once the b est granularityn 3

is determined, the

meritofexpansion iscalculatedasPI(n 3

)0C exp

(n 3

).

IfthereisatleastoneORno deb etweenthero otno de

and anop enno de, theprobabilityofreachingtheop en

nodeis lessthan one. Inthis case, wemultiply PI(n),

whichisoriginallygeneratedforthecasethatthereach-

ingprobabilityisone,bythecurrentreachingprobabil-

PI(n)−C (n) plan improvement

best granularity

exp

n*

granularity n

− C (n) exp

PI(n)

Figure 5: Determining theb estgranularity. This gure

isbasedon [

Horvitz, 1990 ]

.

Every timean openno de is generated (byexpansion

of its parent), the b est granularity for the no de is de-

termined. Weinclude thecost of determining the b est

granularityinthecostofexpandingtheparentno de.

5 Deriving Performance Prole

Derivationofap erformanceprole(PP)isoneoftheim-

portant issues in anytime algorithm-based approaches.

In some cases, PPs may b e obtained from the struc-

tural analysis of the problem; in other cases, PPs may

beobtainedfromexperimentaldata. Itis,however,usu-

ally dicult to obtain PPs for complex problems only

with one of these metho ds. We, thus, derive a PP

throughstructuralandexp erimentalanalysisoftheplan-

ningproblemin thefollowingsteps:

1. Analyzethestructure oftheplanningproblem and

extract imp ortant problem parameters which can

reasonablycharacterizethePP.

2. Construct a generalized PP using the parameters

obtainedab ove;ageneralizedPPhasco ecientsto

b eestimated.

3. CalculateactualPPsforanenoughnumberofprob-

lemparametersets,andadjusttheco ecientsofthe

generalizedPPsothatthePPtswelltotheactual

dataset.

5.1 Problem Analysis

The plan improvement is the dierence of temp orary

costsofaplancandidateb eforeandafterexpandingone

ofitsop ennodes. Letuscalculatetheplanimprovement

usinganexamplesituationshownin Fig. 6.

Supposethat therobotisinitiallyatx

0

andthenext

observationp ointisx

1

. Theop enno deunderconsider-

ationisthestatethatthegate's passabilityisunknown

aftertheobservationatx

1

. Thefeasibleplanb eforeex-

pansionistogofromx

1

directlytothezero-uncertainty

point x 3

, where therob ot can measure the gate width

withoutuncertainty. Expansion ofthis op en no de, i.e.,

selectionof a second observationp oint x i

2

for each dis-

D e a i

D x

x i

x 0

D d b i

D D b

D e i

detour to destination

(current position)

x *

gate

1 2

x d

Figure6: Anexamplesituation.

Usingthecostofmovementb etweenp ointsdenedin

Fig. 6(forexample,D i

a

isthecostofmovementfromx

1

to x i

2

), the plan improvement 1C is describ ed as (see

AppendixA):

1C= n

X

i=1 0

@ P

x i

2

;ng (D

b +D

d 0D

i

a 0D

i

e )

0P

x i

2

;ud C

v

0(P

x i

2

;ok +P

x i

2

;ud )(D

i

a +D

i

b 0D

b )

1

A

; (5)

where C

v

isthecostofoneobservation;P

x i

2

;ok (P

x i

2

;ng ,

P

x i

2

;ud

)istheprobabilitythatthegate'sstateispassable

(impassable,unknown)afterobservationatx i

2

. Wehere

supposethatx i

2

indicates thebestsecond viewp ointfor

theithdividedstate.

Thisequationrepresentsthetrade-ob etweentheef-

fectofvisualinformationtob eobtained byobservation

atx i

2

andthecostofobservationincludingthatofmove-

menttotheobservationp oint.Therstterminsidethe

parenthesesis the eect of visualinformation, which is

gained by decidingto takethedetour atx i

2

, insteadof

decisionat x 3

, incasethat thegate isimpassable. The

second term is the cost of the extra observation at x 3

in casethat thepassabilityis stillunknownat x i

2 . The

third termistheextramovementcostto visitx i

2 .

5.2 Designing Generalized Performance

Prole

Weexamineequation(5)inordertodesignageneralized

PP whichcanreasonably approximatetheequation.

The actual values whichdep end onx i

2

(e.g., P

x i

2

;ng )

cannotb eobtainedb eforep erformingthesearchforx i

2

;

it may also b e hard to reliably predict these values.

Thus, we would like to predict the plan improvement

usingonlyparameters whosevaluesareavailable.

We use an upp er bound of plan improvement for

constructing a generalized PP. To calculate the upp er

bound, we rst employ the concept of the assumption

of perfect sensor information [Miura

and Shirai, 1993],

which is basedon the fact that a plan generated using

certain information is alwaysmore ecient than plans

generatedusing uncertaininformation.

Assuming that the observation result at x i

2

includes

no uncertainty (i.e., P

x i

2

;ud

! 0), a mo died plan im-

0

1C 0

= n

i=1 P

x i

2

;ng (D

b +D

d 0D

i

a 0D

i

e )

0 X

n

i=1 P

x i

2

;ok (D

i

a +D

i

b 0D

b ):

(6)

Since1C 0

stillincludesvaluesdep endingonx i

2

,wethen

nd its maximum value. 1C 0

is maximized by letting

x i

2

!x

1

(i.e., D i

b

!D

b , D

i

e

!D

e

,and D i

a

!0); nally,

weobtainthefollowingupp erb ound:

1C

ub

=P

x1!x 3

;ng (D

b +D

d 0D

e

); (7)

whereP

x

1

!x 3

;ng

istheprobabilitythatthegate isim-

passable after observations at x

1 and x

3

, and is equal

to P

n

i=1 P

x i

2

;ng

under theassumption of p erfectsensor

information. Wesupp osethat theactualplan improve-

ment1C increasesas 1C

ub

increases.

Wealsoreasonablyassumethattheplanimprovement

isamonotonicallyincreasingfunctionofthegranularity

nandasymptoticallyapproachessome limitvalue.

Based on the above examination, we eventually de-

cided to use the following generalized PP (function

PI(n)) asanapproximationof equation(5):

PI(n) = K(10e 0k

1 n

); (8)

K = k

2 1 1C

ub k3

; (9)

wherek

1 ,k

2 ,andk

3

areco ecientstob eestimatedfrom

theexp erimentaldata.

5.3 Determining PP through Exp eriments

We found that the co ecients in PI(n) varies as the

initial p osition x

0

changes. Thus, we divided thework

space of the rob ot, which is the triangle comp osed of

x

0 , x

3

, and x

d

(see Fig. 6), into several regions and

generateda PPforeachdividedregion. Thesizeofone

regionisdeterminedempirically.

Ineachregion, itscenteris selectedas x

0

. Then, we

calculated actual PPs ab out a hundred problems with

various x

1

's and gate widths. We limited n to one of

f1;3;5;7;9;11g.

Inestimating theco ecients,werstttedequation

(8) to each PP and estimated K and k

1

. Fig. 7 shows

an example of actual PP and the tted curve. Then,

weexamined the relationship b etween K and 1C

ub in

equation (9). Fig. 8 shows a log-scaled plot of these

values.Byttingalinetotheplotteddata,weobtained

k

2

=0:033 and k

3

=1:319 in this case. As for k

1 , we

adopted themeanof allk

1

's b ecausewe couldnot nd

any strong correlation b etween k

1

and other problem

parameters.

6 Simulation Results

Fig. 9 showsexamplesof generatedplanfora planning

problem. Fromtheinitial observationresultofthegate

width, a complete plan, which is an AND/OR tree of

observation p oints, was planned o-line. Observation

points are limited only to grid points set on the work

spaceoftherob ot.

Inorder toshow that theplanner canadaptively de-

termine theplanning time (search depth and granular-

0.00 granularity

1 3 5 7 9 11

data fitted curve

1

10.00

8.00

6.00

4.00

2.00

plan improvement (sec.)

K = 9.945 k = 0.774

Figure7: Anexamplep erformanceproleandthetted

curve.

0.1 1 10 100

10 100

K

DCub

data points fitted curve

Figure8: Log-scaled plotofdataforequation(9).

simulations for the same problem with dierent com-

puting p owers(C exam

's, see equation(4)). Noticethat

the higherthe computing p ower is (the smaller C exam

is), the more precise and the less costly plan is gener-

ated. Thedierenceofthecostsis,however,rathersmall

comparedwiththatofthecomputingpowers.Thisisb e-

causethattheinitialfeasibleplanisalreadyago o dplan

in thiscase.

Wethen compared, in termsof the totalofplanning

cost andexecution cost,the prop osedmetho d withthe

xedmetho d, whichusesa xedgranularityanda xed

search depth. Fig. 10 showsa comparisonresult. The

proposedmethodgivestheb estp erformance.

Weconducted thiscomparison foraboutfortydier-

ent problems; in about two-thirds cases, the prop osed

method outp erformed others. In other cases, the dif-

ference b etween the b est result and the result by the

proposed metho dwas lessthanone p ercentofthe b est

result. Note that in order to obtain the b est result

withthexedmetho d,wehadto adjustthegranularity

and searchdepth for eachproblem, while theprop osed

methodalwaysp erformedwellwithoutanysuchadjust-

ments.

7 Exp eriments using Real Rob ot

Thissectiondescrib espreliminaryresultsforarealplan-

ning problem shownin Fig. 1. The gate width isesti-

mated by the two vertical segments on b oth sides; its

probabilistic distribution is calculated using an uncer-

[Miura 1993].

(a)C =1.1e-4,Cost=186.9,n =7 (b)C =1.1e-3,Cost=188.6,n =3 (c)C =1.1e-2,Cost=189.0,n =0

Figure9: Simulationresults: circlesindicateobservationp oints;solidarrowsindicatethepathtothenextobservation

point; dottedarrows indicate p ossiblepathsafter observations. n 3

is thegranularityusedfordiscretizing theop en

nodeatthenextobservationp oint. Incaseof(c), theinitialfeasibleplan isused.

230.0 260.0

proposed

0 1 2 3 4

search depth planning cost + execution cost

(sec.)

fixed (n=1) fixed (n=3) fixed (n=5) fixed (n=7) fixed (n=9)

fixed (n=11)

Figure 10: Comparison of the prop osed metho d

with xed-granularity and xed-search depth metho ds.

Changeofthe totalcost accordingto thechangeof the

searchdepth isindicatedforeachgranularity.

Intheexperiment,theplanner decidesonlythe next

action (see Section 3.4). Then the rob ot movesto the

planned observation p oint and observes the gate. The

planningandactionop erationsareiterativelyp erformed

untilthepassabilityofthegateisdetermined.

Fig. 11showstheresultofatrial. Fromtheinitialim-

age(atlower-rightinthegure),therob otestimatedthe

probabilisticdistributionofthegatewidth;theprobabil-

ityofthegateb eingpassablewas0.53. Thentheplanner

determined thenext observation pointas shown in the

map. The rob ot moved to the next observation p oint

(lower-left) and obtained the new image (upper-right).

After this observation, the gate was determined to b e

passable; thustherob ot movedforwardand passedthe

gate(upp er-left).

8 Conclusions and Discussion

Wehaveprop osedaplanningmetho dforavision-motion

planningofa mobilerobotundervisionuncertaintyand

limitedcomputationalresources. Wemanagedgranular-

ity vs. plan quality and search depth vs. plan quality

trade-os by: (1) considering the relationship b etween

granularityand planimprovementwhich is represented

asap erformanceprole,and(2)formulatingthepro cess

ofgeneratingamulti-stepplanasaniterativerenement

process. The prop osed metho d is always comparable

with the b est of the metho ds which uses xed granu-

larity and xed search depth. We havealso describ ed

amethodtoderiveap erformanceprolethroughstruc-

turalandexp erimentalanalysisoftheplanningproblem.

In the p erformance prole (PP)-based planning, the

qualityofPPhasthelargestimp ortance. Iftheparame-

ters(e.g.,congurationofobstacles)ofthecurrentprob-

lem arelargelydierentfromthose ofproblemsusedin

derivationofPPs,thepredictedplanimprovementmay

not b e accurate enough. In this pap er, in order to in-

crease the accuracy,we divided into theproblem space

intoseveralregions andobtained a PP for eachregion.

Sucha strategymightb e practicalifa PP whichisap-

plicabletoa wideproblemspaceishardtond.

This pap er has treated a relatively simple vision-

motion planning problem (a single-gate problem). A

future work is to apply the prop osed metho d to large

planning problems which have many gates to observe.

To solvesuchalargeproblem,itisnecessaryto decom-

posetheproblemintoaset ofsingle-gateproblems. We

arenowdevelopinganecientdecomp osition metho d.

Acknowledgments

This workissupp ortedinpart byGrant-in-AidforSci-

entic Research from Ministry of Education, Science,

Sports, and Culture, Japanese Government and bythe

OkawaFoundationforInformationandTelecommunica-

tions,Tokyo,Japan.

A Derivation of Plan Improvement 1C

Only the subplan for the op en no de at x

1

changes by

expansion. ThecostC ofthesubplanb eforeexpansion

initial position planned observation point

movements of robot obtained right images

Figure 11: Anexp erimentalresult: [center]the planned observation p oint and the trajectory takenby the rob ot;

[lower-right] the right image obtained at the initial p osition; [upp er-right] the right image obtained at the next

observationp oint;[lower-left]themovementfrom theinitialp ositionto thenextobservationp oint;[upp er-left]the

movementfrom thenextobservationpointintothegate.

isgivenby

C

1

= P

x

1

;ud (D

b +C

v )+P

x

1

!x 3D

g oal

+ P

x

1

!x 3

;ng (D

d +D

detour );

(10)

where D

g oal

and D

detour

is the cost of movement

from x 3

to the destination and from x

d

to the des-

tination through the detour, respectively; P

x1!x 3

;ok

(P

x

1

!x 3

;ng

) is the probability that the gate is pass-

able (impassable) at x 3

in case that the passability is

unknownatx

1

. ThecostC

2

afterexpansionisgivenby

C

2

= n

X

i=1 min

x i

2 C

i

2

; (11)

C i

2

= 0

B

B

@ P

i

x

1

;ud (D

i

a +C

v )+P

x i

2

;ok (D

i

b +D

g oal )+

P

x i

2

;ng (D

i

e +D

detour )+P

x i

2

;ud (D

i

b +C

v )+

P

x i

2

!x 3

;ok D

g oal +

P

x i

2

!x 3

;ng (D

d +D

detour )

1

C

C

A

;(12)

where P i

x1;ud

is the probability of the ith divided

state; P

x i

2

!x 3

;ok (P

x i

2

!x 3

;ng

) is similarly dened as

P

x1!x 3

;ok (P

x1!x 3

;ng

). Theb estx i

2

isselectedforeach

stateS i

.

The plan improvement is obtained as C

1 0C

2 . The

followingrelationsholdamongprobabilities:

P

x1;ud

= P

n

i=1 P

i

x1;ud

;

P

x1!x 3

;ok

= P

n

i=1 P

x i

2

;ok +

P

n

i=1 P

x i

2

!x 3

;ok

;

P

x1!x 3

;ng

= P

n

i=1 P

x i

2

;ng +

P

n

i=1 P

x i

2

!x 3

;ng

;

P i

x

1

;ud

=P

x i

2

;ok +P

x i

2

;ng +P

x i

2

;ud :

(13)

Afteraseriesofcalculationusingtheab overelations,we

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