(1995) Fuzzy Logic and Autonomous Vehicles - Experiments in Ultrasonic Vision

8/13/2019 (1995) Fuzzy Logic and Autonomous Vehicles - Experiments in Ultrasonic Vision

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E L S E V I E R Fuzzy Sets and Systems 69 (1995) 15 27U Z Z Ys e ts n d s y s t e m s

F u z z y l o g i c a n d a u t o n o m o u s v e h i c l e sE x p e r i m e n t s i n u l t r a s o n i c v i s i o n

M . P o l o n i , G . U l i v i , M . endittelliDipartimento di ln/ormatica e Sistemistica. Uniuersitlt degli Studi di Roma 'La Sapienza , Italy

Received September 1993: revised March 1994

A b s t r a c t

The opportunities offered by fuzzy logic to build maps for robot navigation are investigated. Characteristics of pointsof the space (occupied, free, uncertain, etc.) are easily expressed through set theoretical operations. Real-world experi-ments validate the approach. The experimental set-up is based on modified Polaroid ultrasonic sensors; however, theapproach can be easily extended to incorporate other kinds of sensors.Keywords. Data analysis methods; Linguistic modeling; Robotics

1 I n t r o d u c t i o n

Use of mobile r obots is spreading in many fields of application. Typical examples are submarine andspatial exp loratio n [21, 11, 9], surveillance of industri al installations, rescue missions in dange rous environ-ments (radiations, chemical contamination). More common applications are, for example, internal maildelivery, partially autonomous wheel-chairs for disabled [-20], intelligent Automatic Guidance Vehicles(AGVs) for floor-shops [2].

Despite the differences a mong these applications, an au tonomous vehicle always needs a navigation system todetermine a suitable path to its target and a sensory system to acquire knowledge about the crossed environment.

Cons ide r the simplified functional structure of a naviga tion system depicted in Fig. 1. At the beginning themap can be totally or partially unknown. The lowest level is constituted by the measuring processes thatsupply rough information coming from each sensor, typically the distance of the nearest obstacle (Sensors).Inf orma tio n is suitably merged and used to build or upgrade the map of the traveled environ ment (Mapreconstruction). The map is used to implement the navigation strategy. It can also be interpreted by otherlevels of the structure, trying to identify the shape and possibly the nature of the objects on the scene (Mapinterpretation) to devise more complex strategies [18]. In this case the map could contain other details on theenvironment as, for example, the colors of the obstacles.

* Corresponding author. Correspondence address: ELITE Foundation, Promenade 9. 52076 Aachen, Germany.0165-0114/95/ 09.50 CO 1995 ElsevierScience B.V. All rights reservedSSDI 0165-0114(94)00237-1


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16 M. Poloni et al. / Fuzzy Sets and Syste ms 69 1995) 15 27

Navigat ion S tra teg iesMapnterpretation

Map Reconstruct ionSensors

F i g 1 S t r u c t u r e o f a n a v i g a t i o n s y s t e m

W h e n f ul l a u t o n o m y i s p u r s u e d i n a h i g hl y u n s t r u c t u r e d e n v i r o n m e n t , s o p h i s t i c a te d s e n s o r s a r e ty p i c a ll yu s e d , e .g . v i d e o c a m e r a s ; h o w e v e r , t h i s s o l u t i o n r e q u i r e s a n e x p e n s i v e h a r d w a r e a n d a c o m p l e x s o f t w a r e .

I n s i m p l e r a p p l i c a t i o n s t h e e n v i r o n m e n t c a n b e a s s u m e d p l a n a r a n d s o m e a p r i o r i k n o w l e d g e o n t h eo b s t a c l e n a t u r e i s a v a i l a b l e , ty p i c a l l y t h a t t h e y a r e p r i s m a t i c . I n t h e s e c a s es , u l t r a s o n i c U S ) s e n s o r s a r e o f te nu s e d a s t h e y a r e c h e a p , l i g h t a n d r e l i a b l e .

B e s i d e t h e s e p o s i t i v e c h a r a c t e r is t i c s , U S s e n s o r s s h o w a l s o t w o m a i n d r a w b a c k s . F i r s t, t h e w i d e a n g l e o fr a d i a t i o n c a u s e s l a r g e u n c e r t a i n t i e s o n t h e a n g u l a r l o c a t i o n o f t h e o b s t a c l e t h a t o r i g i n a t e s t h e e c h o . S e c o n d ,t h e u l t r a s o n i c b e a m i s e a s i ly r e fl e c t e d w h e n i t h i t s a s u r f a c e w i th a l a r g e a n g l e o f i n c id e n c e . A f t e r s o mer e f le c t i o n s it c a n f i n d a p a t h t o r e t u r n b a c k t o t h e s e n s o r o r i t c a n g o l o s t. T h e r e f o r e , t h e m e a s u r e d d i s t a n c ec a n b e q u i t e l o n g e r t h a n t h e o n e o f th e n e a r e s t o b j e c t .

A w e l l - k n o w n a p p l i c a t i o n o f u l t r a s o n i c s e n s o r s i n t h i s a r e a is d e s c r i b e d i n [ 5 ] w h e r e t h e u n c e r t a i n t i e s h a v eb e e n m a n a g e d i n a s t o c h a s t i c s e t t i n g .

S t o c h a s t i c m e t h o d o l o g i e s [ 6, 4 ] c a n p r o d u c e r e l i ab l e r es u l ts f r o m u n c e r t a i n r e a d i n g s w h e n a h i g h n u m b e ro f d a t a i s a v a i l a b l e a n d t h e y a r e w e l l d i s t r i b u t e d i n t h e e x p l o r e d a r e a . I n t h i s c a s e a v e r a g i n g c a n r e d u c e t h eo v e r a l l u n c e r t a i n t y s h o w n b y t h e f i n a l m a p . C l e a r l y , w i t h a l i m i t e d n u m b e r o f m e a s u r e s t h e r e s u l t s a r e n o t s og o o d ; i n d e e d , f u r t h e r s t u d i e s h a v e b e e n c a r r i e d o u t t o i m p r o v e t h e e f fi c ie n c y o f m a p b u i l d i n g . I n [ 7 ] t h e a n g l eo f i n c i d e n c e i s e s t i m a t e d t o d e r i v e t h e p r o b a b i l i t y o f a m u l t i p l e r e f le c t io n , in [ 1 2 ] t y p i c a l e n v i r o n m e n tc o n f i g u r a t i o n s e .g . c o r n e r s ) a r e c l as s i fi e d a n d u s e d t o i m p r o v e th e m a p a c c u r a c y .

A d i f fe r e n t a p p r o a c h t o u n c e r t a in ty h a n d l i n g i s p r o v id e d b y t h e t h e o r y o f f u z z y s e ts , w h ic h a l l o w s a g r e a tf lex ib i li ty in the t r ea tm en t of in fo rm a t io n see e.g. [10 , 23 , 1 ]) . Th is theo ry has a l r e ady be en used in the f ie ld ofn a v ig a t i o n o f mo b i l e r o b o t s [ 1 3, 1 8 ], e s p e c ia l l y i n t h e h ig h e s t l a y e r s o f t h e n a v ig a t i o n s t r u c tu r e s c e n e i n t e r p r e t a -t i o n p a r t i c u l a r l y fo r s ce n e s d e r i v e d b y c a m e r a s ) a n d p a th p l a n n in g h a v e b e e n e x t en s iv e ly s t u d i e d s ee [ 14 , 1 5 , 8 ] ).

T h i s p a p e r i s a n e x p l o r a t i o n o f t h e p o s s ib i li t ie s o f fe r ed b y t h i s t h e o r y f o r m a p r e c o n s t r u c t i o n f r o m s o n a rd a t a b u t o t h e r k i n d s o f r a n g e f i n d e r s c a n b e c o n s i d e r e d s i m p l y m o d i f y i n g t h e d e s c r i p t i o n o f t h e u n c e r t a i n t i e st h e y i n t r o d u c e , n a m e l y t h e s e n s o r u n c e r t a i n t y m o d e l .

A f t e r a b r i e f i n t r o d u c t i o n o f t h e u s e d n o t a t i o n s a n d o p e r a t o r s , t h e s e n s o r c h a r a c te r i s ti c s a r e o u t l i n e d a n d a nu n c e r t a in ty m o d e l d e r i v e d . A s in [ 5 ] , me a s u r e s a r e u s e d t o b u i l d tw o ma p s . T h e f i r s t c o n c e r n s t h e f r e e a r e a s , t h eo t h e r t h e o c c u p i e d o n e s . B o t h m a p s s p e ci f y o u r d e g r e e o f b e l ie f a b o u t t h e s t a t u s o f e a c h p o i n t o f t h e s p a ce .F u z z y l o g ic is u s e d t o g i v e a f o r m a l d e s c r i p t i o n o f t h e m a n i p u l a t i o n s o f t h e m a p s ; t h is a l l o w s t o c a r r y o u t s e v e ra lc h a r a c t e ri s t i c s o f t h e e n v i r o n m e n t . T h e o p e r a t i o n s a r e d e s c r i b e d i n n a t u r a l l a n g u a g e a n d e a s il y f o r m u l a t e du s in g f u z z y s e t o p e r a to r s . A d e s c r i p t i o n o f t h e e x p e r im e n ta l s e t - u p a n d o f t h e o b t a in e d r e s u l ts c o n c lu d e t h e p a p e r .

2 N o t a t i o n s a n d o p e r a t or sG i v e n a c r i s p u n iv e r s a l s e t U a n d d e n o t i n g b y u i t s g e n e r i c e l e me n t , a f u z z y s e t A _ ~ U w i l l b e i d e n t i f ie d b y

i t s m e m b e r s h i p f u n c t i o n :A u ) : u ~ [ 0 , 1 ] .


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M . P o l o n i e t a l. / F u z z ) , S e t s a n d S y s t e m s 6 9 1 9 9 5 ) 1 5 - 2 7 7

I n t h e p a p e r , r e c t a n g u l a r b i d i m e n s i o n a l m a p s a r e c o n s i d e r e d . T h e r e f o r e , t h e u n i v e r s a l se t a n d i ts e le m e n t sa r e , r e s p e c t i v e l y , s p e c i f i e d a s U = X x Y a n d u = ( x , y ) , b e i n g x a n d y t h e Ca r t e s i a n c o o r d i n a t e s o f a p o i n t i nt h e m a p a n d X a n d Y t h e i r r a n g e s .

E a c h p o i n t c a n b e e i th e r e m p t y o r o c c u p i ed , ' a c ri s p p r o p e r t y . H o w e v e r , b e c a u s e o f t h e m e a s u r i n gu n c e r t a i n t i e s , w e c a n e x p r e s s o n l y a d e g r e e o f b e l i e f a b o u t t h e s e p r o p e r t i e s . A s e x p l a i n e d l a t er , t w o f u z z y s e tsa r e u s e d t o e x p r e s s e m p t i n e s s a n d f u ll n e ss :

E x , y ) : X Y ~ [ 0 , 1 ] , O x , y ) : X x Y - - , [ 0 , 1 ] .S e v e r a l l o g ic a l o p e r a t i o n s a r e d e f i n e d f o r t h e s e f u z z y s e ts b y t r i v i a l e x t e n s i o n o f t h e u n i d i m e n s i o n a l o n e s .

T h e s e t c o m p l e m e n t a t i o n i s i m p l e m e n t e d , a s u s u a l , b yc A u ) ) = 1 - A u ) . 1 )

A s f o r t h e i n t e r s e c t i o n , b o t h t h e a l g e b r a i c p r o d u c ti A u ) , B u ) ) = A u ) n B u ) = A u ) * B u ) (2)

a n d t h e b o u n d e d p r o d u c ti A u ) , B u ) ) = A u ) c~ B u ) = m a x ( 0 , A u ) + B u ) - 1) (3)

a r e u s e d .B y D e M o r g a n l a w , se t u n i o n i s d e f i n e d , r e s p e c t iv e l y , a s t h e a l g e b r a i c s u m :

u A u ), B u ) ) = A u ) w B u ) = A u ) + B u ) - A u ) * B u ) (4)a n d t h e b o u n d e d s u m

u A u ) , S u ) ) = A u ) w S u ) = mi n ( 1 , A u ) + B u ) ) . (5)I n t h e f o l l o w i n g , w e s h a l l r e f e r t o t h e c o u p l e s ( 2 , 4 ) a n d ( 3 , 5 ) , r e s p e c t i v e l y , a s A P a n d BS c o u p l e s .I n th i s p a p e r a l s o p a r a m e t e r i z e d u n i o n o p e r a t o r s h a s b e e n c o n s i d e r e d , d u e , r e s p e c ti v e l y , t o D o m b i a n d

Ya ger [3 , 10, 22] :1E 1+ A ~ - I -t- B ~ - - I

fo r 2 E (0 , ~ ) an dm i n [ 1, ~ /A ( u ) p + B u ) P 1

wi th p > 0 .

(6)

(7)

3 S e n s o r m o d e l i n g an d m a p b u i l d in gG e n e r a l l y a n u l t r a s o n i c s e n s o r i s c o m p o s e d b y t w o f u n d a m e n t a l e l e m e n t s : t h e a c o u s t i c t r a n s d u c e r a n d

a r a n g i n g c i r c u i t b o a r d . I t w o r k s i n a s i m p l e w a y : a p a c k e t o f u l t r a s o n i c w a v e s i s g e n e r a t e d a n d t h e r e s u l ti n ge c h o i s d e t e c t e d . T h e t i m e b e t w e e n t h e t r a n s m i s s i o n a n d t h e r e c e p t i o n o f t h e p a c k e t is p r o p o r t i o n a l t o t h ed i s t a n c e o f th e o b s t a c l e . M o r e s o p h i s t i c a t e d t e c h n i q u e s c o u l d b e u s e d , i n c l u d i n g r e a l - t im e s i gn a l p r o c e s s in g ;h o w e v e r , t h i s w o u l d r e q u i r e e x p e n s i v e c o m p u t a t i o n d e v i c e s .


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18 M . P o l o n i e t a l. / F u z z y S e t s a t t d S y s t e m s 6 9 1 9 9 5 ) 1 5 2 7We assume a typical office-like environment, occupied by prismatic obstacles extending indefinitely in the

vertical direction. Therefore, the poin ts of an ideal map are either empty or occupied .The measur ing process, however, introduces several kinds of uncertainties, mainly related to the character-

istics of the used sensor. One source of uncertainty is given by the non-zer o width of the r adiation cone (about30 for the used ones), making it impossible to know exactly the angular position of the object whichoriginated the echo; for example the first three cases in Fig. 2 all give the same result. Moreover, the beammay go easily lost or it may be reflected more than once before coming back to the sensor as shown by thelast case in the same figure. As a consequence, each point of the real map can be assumed empty or occupiedwith a certain degree of belief and the two conditions are not mutually exclusive. Very low degrees of belieffor both att ributes in a point denote a lacking of information, while two very high values imply a contradic -tion due, e.g. to a poor accuracy of the measures.

In the process of map building several bidimensional fuzzy sets are determined by iteratively mappi ng thesensor model on the grid. Indeed, the final map itself results to be a fuzzy set. All these are defined overa bidimensional universal set corresponding, in our experiments, to a square with 10 m sides. F or c ompu ta-tional reasons, the square is subdivided in 100 x 100 square cells. This approach is well suited to describe bysimple linguistic statements the ope rations necessary to derive a navigatio n map and m oreov er its grid-basednature complies with the implementation of path planning algorithms.

The choice of a suitable sensor model (more exactly: of the whole sensor-environment interaction) is ofparamount importance for the overall performance. It should describe all the phenomena involved in themeasuring process. In the authors ' opinion, a precise numeric appro ach e.g. based on an accurate analysisof the shape of the radiation lobe - is useless because of the level of the overall uncertainty. Here onlyqualitative (linguistic) descriptions of the different phenomena are used and expressed in terms of simplefunctions.a. Neglecting beam reflections, a single measure provides the inf ormation that one or more obstacles are

located somewhere on the 30 arc of circumfe rence of radius r (the measur ed distance). Hence, there isa possibility for each of the cells near the arc to belong to the set (o of the occup ied ones. A simpleparabola models this phenomenon:

I / 'qo p) =< \ ~p I J Vp: ]p -r I< A p ,otherwise, 8 )where r is the distance meas ured by the US sensor, with a range between 0.3 and 8 m, p is the radialdistance of the cell from the sensor, and A p is an estimate of the overall accuracy. Among the factorscontri butin g to the fuzziness of the concept of near are the errors affecting the measures and the mapdiscretiz ation, i.e. the sensor and the obstacle(s) are generally loca ted off the cell centers.

V V i/4 : /a ) b ) c ) d )

Fig. 2. Typical uncertainties with ultrasonic sensors.


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M. Pohm i e t a l . / Fuzz ) , S e t s and Sys tems 69 (199 5) 15 27 19T w o r e m a r k s a r e i n o r d e r : ( a) t h e s u m o f t h e o c c u p a t i o n p o s s i b i l it i e s o f t h e c el ls n e a r t h e a r c is n o t e q u a l t oo n e b e c a u s e o n e o r m o r e c el ls m a y b e o c c u p i e d ; (b ) t h e r e p e t i ti o n o f t h e s a m e m e a s u r e c a n r e d u c e o n l ya l i t t l e t h e o v e r a l l u n c e r t a i n t y w h i c h m a i n l y d e p e n d s o n n o n - s t o c h a s t i c f a c t o r s .

b . C e l l s i n s i d e t h e c i r c u l a r s e c t o r o f r a d i u s r s h o w a n e v i d e n c e t o b e f re e f r o m o b s t a c l e s , s o th e y b e l o n g t o t h es e t ~ o f th e e m p t y c e ll s. T h i s is s i m p l y e x p r e s s e d b y

10 V p < r - - A pg b ( P ) = o t h e r w i s e . ( 9)c . R e f l e c t i o n s m o d i f y o u r b e l i e f a b o u t t h e s t a t u s o f t h e e x p l o r e d c e l ls . A s re f l e c te d b e a m s m a k e a l o n g e r f ly

t h a n d i r e ct o n e s , th e s i m p l es t w a y t o t a c k l e w i th t h i s p h e n o m e n o n is t o g r a d u a l l y r e d u c e o u r b e l ie f f r o ma m a x i m u m t o a m i n i m u m a c c o r d i n g t o th e d is t an c e p :

g c ( p ) = r a i n ( l , h x e -h 2 p + h3) . ( 10)P a r a m e t e r h 3 r e p r e s e n t s t h e m i n i m u m d e g r e e o f b e li ef , h i , h 2 a r e u s e d t o i n t e r p o l a t e b e t w e e n t h e t w ov a l u e s a n d t a k e i n t o a c c o u n t a v e r a g e d i s t an c e s i n t h e e n v i r o n m e n t .

d . T h e w a v e i n t e n s i t y d e c r e a s e s f a r f r o m t h e b e a m a x i s a n d r e a c h e s z e r o a t t h e b o r d e r s o f t h e lo b e . T h e r e f o r e ,t h e b e l i e f o f e a c h a s s e r t i o n is h i g h e r f o r th e c e ll s n e a r e r t o t h e b e a m a x is . T h e c e l ls o u t s i d e t h e l o b e d o n o tc h a n g e t h e i r a t t r i b u t e s .

g a(~ 9 ) = ~ 1 - - ( 1 2 , 9 ) 2 V ~ 9 : ~ l < ~ / 1 2 , ( 11)~ o t h e r w i s e .e . E v e n i n i d ea l s i t u a t i o n a s in g l e m e a s u r e i s n o t s u f fi c ie n t t o g a i n c e r t a i n t y o n t h e s t a t u s o f a c el l. I n d e e d e a c h

c el l is t o b e v i e w e d s e v e ra l t i m e s a n d i ts o c c u p a n c y a n d e m p t i n e s s r e s u lt f r o m t h e u n i o n o f m a n ym e a s u r e s . T h e w e i g h t o f t h e e l e m e n t a r y a c q u i s i t i o n i s g i v e n b y t w o c o e f fi c ie n t s : k s a n d k ~ ,, r e s p e c ti v e l y , f o re m p t i n e s s a n d o c c u p a n c y .W i t h t h e u s e d s e n s o r s , t h e v a l u e s o f t h e p a r a m e t e r s a r e , r e s p e c ti v e l y : A p = 0 .15, k,o = 0 .5, ks = 0 .1,

hi = 1.2, h = 1, h 3 = 0.1 .G i v e n a c el l l o c a t e d a t a d i s t a n c e p f r o m t h e r a n g e f i n d e r a n d a n a n g l e 0 f r o m i t s a x is , it s p o ss i b i li t ie s t o b e

e m p t y o r o c c u p i e d a r e o b t a i n e d b y a d d i n g t h e p r e v i o u s v a lu e s . 1 F o r c o m p u t a t i o n a l r e a s o n s, t h e a lg e b r a i cp r o d u c t is us e d , h o w e v e r , i t c o u l d b e s u b s t it u t e d b y o t h e r o p e r a t o r s :

~ ' ( 0 , p ) = k E g b g c g , t , ( 12)o'(0, p) = k o , g . g c g d . ( 13)

T h e p r i m e d e n o t e s t h a t t h e se m e m b e r s h i p f u n c t i o n s a re r e fe r re d t o p o l a r c o o r d i n a t e s , t h e s a m e w a y t h e ya r e r e p r e s e n t e d ( n o r m a l i z e d ) i n F i g . 3 i n th e s a m p l e c a s e o f a m e a s u r e d d i s t a n c e e q u a l t o 2 .9 m .

T o b u i l d t h e m a p s e v er a l m e a s u r e s a r e t a k e n f r o m k n o w n p o i n t s in th e c o n s i d e r e d e n v i r o n m e n t . H e r e w ew i ll n e g l e c t t h e w a y t h e r o b o t r e a c h e s t h e s e p o i n t s . H o w e v e r , a s it w il l b e c o m e c l e a r i n t h e s e q u e l, t h e m a pc a n b e r e c o n s t r u c t e d s t e p b y s te p , a s th e r o b o t m o v e s a m o n g t h e f re e c el ls .F o r e a c h r e a d i n g k t h e t w o s u r f a c e s e ' ( O , p ) a n d o 9 ' ( O , p ) a r e p r o j e c t e d o n t h e c e ll s o f t h e C a r t e s i a n m a pu s i n g a r e c u r s i v e a l g o r i t h m w h o s e c o m p l e x i t y i s l i n e a r in t h e n u m b e r o f i n t e r e s t e d c e ll s. T h e o u t p u t s o f t h ep r o j e c t i o n s a r e t w o f u z z y s e ts e k ( i , j ) a n d o ( i , j ) , b e i n g i, j t h e i n d i c e s o f t h e c e l l m a t r i x .

1These functions are used to generate a fuzzy set from a measure. This fuzzification stage is quite different from that generally used infuzzy co ntrol.


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1' 1

2 I

2 1

F i g . 3. S e n s o r c e r t a i n t i e s f o r e m p t y l ef t) a n d o c c u p i e d c e l l s r i g h t) in a t y p i c a l c a s e .

T h e p i e c e s o f i n f o r m a t i o n c o n c e r n i n g t h e e m p t y a n d t h e o c c u p i e d c e l ls a r e s e p a r a t e l y c o l le c t e d d u r i n g t h em e a s u r i n g p r o c e s s u s i n g a u n i o n o p e r a t o r . T h e a s s o c i a t i v e p r o p e r t y o f th e s e o p e r a t o r s a l lo w s u s i n g t w o f u z z yse ts E i , j ) a n d O i , j ) a s a c c u m u l a t o r s . A s a c o n s e q u e n c e t h e p r o c e d u r e c a n b e u s e d a l s o w h i l e t h e r o b o t i sm o v i n g . A t t h e e n d o f t h e p r o c e s s t h e t w o s e t s a s s u m e t h e v a l u e s :

E i , j ) = ~ ) e k i , j) , (14)O i , j ) = ~ J ~ O k i ,j ), (15)

b e i n g th e u n i o n p e r f o r m e d o v e r a ll th e a v a i l a b l e m e a s u r e s . F o r t h is o p e r a t i o n D o m b i ' s u n i o n (6 ) h a s b e e nu s e d . I n f a c t t h e m a x o p e r a t o r i s n o t s u i t a b l e i n t h i s c o n t e x t a s it d o e s n o t a l l o w t o t h e f i n al b e l ie f t o s u m u pa l l t h e a v a i l a b l e i n f o r m a t i o n . O t h e r o p e r a t o r s ( e.g . Y a g e r o n e s ( 7) ) s a t u r a t e , i.e . t h e y g i v e f o r m a l c e r t a i n t ya f te r a c e r t a in n u m b e r o f c o n c o r d a n t m e a s u r e s . T h e c h o s e n u n i o n f a m i ly o n l y a s y m p t o t i c a l l y r ea c h e sc e r t a i n t y ; in t h i s s en s e it i s m o r e a p p r o p r i a t e f o r o u r p u r p o s e s . F i g s . 4 a n d 5 r e f er a s i m p l e c o m p a r i s o n : t h e ys h o w h o w t h e d e g r e e o f b e l i e f i n c r e a s e s f o r a g i v e n c el l i n t h e c a s e t h a t t h e m e a s u r e s c o n t i n u o u s l y g i v ea c e r t a i n t y e q u a l t o 0 . 1. W i t h t h e c h o s e n v a l u e s f o r p a r a m e t e r s p a n d 2 t h e t w o f a m i l i es o f c u r v e s s h o w s i m i l a rv a l u es f o r a s m a l l n u m b e r o f m e a s u r e s .

I n t e r e s t in g c o m p a r i s o n s a m o n g t h e s e a n d o t h e r p a r a m e t r i c u n i o n o p e r a t o r s c a n b e f o u n d i n [1 9 , 2 2 ].T h e t w o o b t a i n e d s e ts ( m a p s ) o f t h e e m p t y c e l l s a n d o f t h e o c c u p i e d o n e s c a n b e c o m b i n e d i n s e v er a l w a y s

( f r o m n o w o n , t h e l in g u i s t ic a t t r i b u t e s w h i c h a s s u m e a f o r m a l m e a n i n g w i ll b e sl a n te d ) . T h e i r i n t e r s e c t i o n , f o re x a m p l e , r e p r e s e n t s t h e s e t o f c e ll s f o r w h i c h t h e m e a s u r e s a r e a m b i g u o u s , w i t h t h e m e m b e r s h i p v a l u e s g i v in ga d e g r e e o f c o n t r a d i c t i o n :

C i ,j ) = E i , j ) ~ O i, j ) . (16)T h e s e t o f t h e u n e x p l o r e d c e l l s c a n b e o b t a i n e d a s

Q i , j ) = f f , i , j) r n O i , j ) , (17)w h e r e t h e o v e r b a r i n d i c a t e s s e t c o m p l e m e n t a t i o n .

T h e k i n d o f e l a b o r a t i o n d e p e n d s o n t h e a p p l i c a t io n . I f t h e m a i n t a s k o f t h e r o b o t is to b u i l d a n a c c u r a t em a p , t h e n t h e c o n t r a d i c t o r y c e l l s s h o u l d b e e x p l o r e d f i r s t . F o r n a v i g a t i o n p u r p o s e s , w h a t i s n e e d e d i sa c o n s e r v a t iv e m a p o f t h e f r e e cells M i , j ) . A c c o r d i n g l y , t h is m a p c a n b e b u i lt b y s u b t r a c t i n g t h e a m b i g u o u sc e l l s f r o m th e e m p t y o n e s :

M i , j ) = E i , j ) ca C i ,j ) , (18)


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M . P o l o n i e t a l. / F u z z y S e t s a n d S y s t e m s 6 9 1 9 9 5 ) 1 5 - 2 7 2 1

0 9

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01i : i i i i " "

2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 l l 0 2 0 0

o . . . . . . . . . . . i . . . . . . . . . . . . ~ . . . . . . . . . . . .. : . . . . . . . . . . . ~ . . . . .. . . . .. . . .. ~ . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . i . . . . .

o . . . . . . . i i i i i

o , . . . . ~ . . . . . . i i . . .. . . . . . . . . . . . . . i0 . 6 . . . . . . . . . i . . . . . . . . .. . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . .. . . . . . . i . . . . . . . . .. . . . . . ~ . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . i . . . . . . . .. . . . . . . . i . . . . . . . . . . . . . .

0 . 5 . . . . . . . . . . . . . . . . . . . . . . f . . . . . . . . . . . . . . : . . . . . . . . . . . . t . . . . . . . . . . . . . ~ . . . . . . . . . . . . ~ . . . ~

0 . 3 . . . . . . . . . i . . . .. . . .. . .. . . . i . . . .. . . .. . .. . . ~ . . . . . . . . . . . ~ . . . . . . . . . . . - . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . i . . . . . . . . . . . . ..

0 . 2 . . . . . i . . . . . . . . . i . . . . . . . . . . . i . . . . . . . . . i . . . .. . . .. . . .. i . . . . . . . . . i . . . . . . . . i . . . . . . . i . . . . . . . . . . i ............o.~ .............. .............. ................... .... .... ... ..~- ........~............ ............................................

Fig. 4. Certain ty accumulation according to D om bi union(certainty vs. iterations). Fig. 5. Certain ty accumulation according to Y ager union(certainty vs. iterations).

A m o r e c o n s e r v a t i v e m a p c a n b e o b t a i n e d g e t t i n g r i d a l s o o f t h e o c c u p i e d c e ll s . F o r m a l l y , th i s c a n b ee x p r e s s e d a s

M i , j ) = E i , j ) ~ O i , j ) c~ C i ,j ) . (19 )T h e r e s u l ts o b t a i n e d w i t h t h e s e t w o f o r m u l a t i o n s h a v e b e e n c o m p a r e d i n t h e n e x t se c t io n . A s f o r t h e s ett h e o r e t i c a l o p e r a t i o n s , b o t h t h e c o u p l e o f e q u a t i o n s ( 2 ), ( 4) a n d ( 3) , ( 5) h a v e b e e n u s e d .

4 . E x p e r i m e n t a l s e t - u p

T h e u s e d u l t r a s o n i c s e n s o r is a c o m m e r c i a l d e v i c e f r o m P o l a r o i d C o r p o r a t i o n [ 1 6, 1 7 ]. It i s c o m p o s e d o ft w o f u n d a m e n t a l e l e m e n t s : t h e a c o u s t i c t r a n s d u c e r w i t h t h e a s s o c i a t e d a n a l o g e l e c t ro n i c s a n d a r a n g i n gc i r c u i t b o a r d .

T h e d e v i c e e m i t s a c h i r p s i g n a l o f 1 m s d u r a t i o n , m a d e o f 56 p u l s es a t a f r e q u e n c y o f 49 .4 1 k H z . T h e m a i nr e d i a t i o n l o b e c o v e r s a s o l i d a n g l e o f a b o u t 3 0 a t - 3 8 d B .

T h e s t a n d a r d P o l a r o i d ' s r a n g i n g s y s t e m i s a b l e t o d e t e c t th e p r e s e n c e o f o b j e c t s i n a r a n g e f r o m 3 0 c m t o1 0 m .

T o o b t a i n b e t t e r p e r f o r m a n c e s a n e w r a n g i n g c i r c u i t h a s b e e n d e s i g n e d , w h i c h i s a b l e t o d r i v e e i g h t se n s o r s.N o t e , h o w e v e r , t h a t o n l y o n e s e n s o r a t a t i m e c a n b e f ir e d t o a v o i d c r o s s - i n t e r f e r e n c e p r o b l e m s .

U s i n g t h e s u g g e s t i o n s g i v e n i n [ 1 6 ], t h e m i n i m u m r a n g e h a s b e e n l o w e r e d t o 1 0 c m a n d i t i s p o s s i b l e n o wt o s e l ec t t w o d i f f er e n t t y p e s o f r es o l u t i o n : h i g h ( ~ 1 c m ) a n d l o w ( 4 .5 c m ). A t h i g h r e s o l u t i o n , t h e m a x i m u md i s t a n c e t h a t c a n b e m e a s u r e d i s a b o u t 6 0 c m .

T h e b o a r d h a s b e en i n te r f ac e d t o a H e w l e t t P a c k a r d 1 6 M H z V E C T R A 3 8 6S X c o m p u t e r b y m e a n s o fa B u r r - B r o w n 3 2 b i t p a r a l l e l c o m m u n i c a t i o n c a r d . D a t a e x c h a n g e i s b a s e d o n a 1 6 b i t b u s : o n e b y t e is f o rb o a r d c o n t r o l , w h i l e t h e o t h e r c o n t a i n s t h e n u m e r i c a l d a t a p r o p o r t i o n a l t o t h e d i s t a n c e c o v e r e d b y th ea c o u s t i c w a v e . T h e e i g h t b i t s o f c o n t r o l a r e c o n f i g u r e d a s f o ll o w s : t h r e e a r e f o r t h e a d d r e s s e s o f th e u l t r a s o n i cs e n s o r s, t h r e e a r e f o r c o o r d i n a t i o n w i t h o t h e r s e n s o r y s y s t e m s o r f o r fu t u r e b o a r d c a p a b i l i t ie s , t w o g i ve s ,r e s p e c t i v e l y , t h e s t a r t s i g n a l f o r th e U S p u l s e a n d t h e d e s i r e d r e s o l u t i o n ( h i g h / l o w ) .

T h e P C s e le c ts a n d f ir e s t h e s e n s o r s , c o l l ec t s t h e r e s u l ts a n d e l a b o r a t e s t h e m a p s b y m e a n s o f t h ea l g o r i t h m s d e f i n e d i n t h e f o l l o w i n g . A s k e t c h o f t h e e x p e r i m e n t a l s e t - u p i s g i v e n i n F i g . 6.


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9O

O

Dig i t a l 6 0ln te f f a c e Ca r d 5 0C m m a n d s ( 8 b i t) J 1 D a t a (8 b i t ) 4 0

[ Cus tomlnterface oard I 3 0

. . . 1 0U l t r a s o n i c S e n s o r s r r a y

0 1 0 1 i

2 0

).1

4 0 6 0 8 0 1 0 0F i g . 6 . E x p e r i m e n t a l e t- u p . F i g . 7 . M a p o f th e o c c u p i e d ce l ls .

In the presented experiment s only one sensor is used, which is rotat ed 9 at each measure until a 360 scanis performed. Larger steps have been tried, but they gave worse results. The sensor is mounted on a trolley at1.3 m above the floor. Each measure takes about 0.25 s. Several independent readings from the same positionare averaged to reduce the effects of the tr ansduce r sensitivity fluctuations.

All the software is implemented in C language. The main functions include the low-level sensor interfaces,the pro jection of ~ (0, p) and d( 0, p) on the Cartesian frame, the accumul ation accordi ng to (14), (15), andfinally the ma p building. The execution of the whole pr ogr am for eight measuring points last a bout 1.25 s onthe aforesaid machine.

5 E x p e r i m e n t a l r e s u l tsThe described map building system has been tested in a corrido r of our departm ent. This is rather long and

narrow, with several book cabinets located by the walls, so that beams are easily reflected. Halfway thecorridor there is a widening where a slim irregular obstacle has been added. The ultrasonic sensor has beenmanual ly placed in eight known locations and 40 measures have been taken f rom each of them to cover thewhole circle. The chosen test-environment is coherent with our research goal, i.e. a mobile robot capable ofreconstr ucting a nd navigating in an u nkn own office-like environment. It emphasizes the issues of reflection,reconstruction of convex and concave corners and of the resolution of narrow walk-through passages thatcan be hidden from the ar c-shaped response of the sensor. In the following the maps are represented bycontour plots; the level lines are generally drawn at 0.3, 0.5 and 0.7. No scaling or normalization has beenapplied during data processing.

The overall maps of the o upied and of the empty cells are given, respectively, in the con tour plots of Figs.7 and 8. Using a value 0.6 of)+ in the Do mbi s union, the values of the member ship functions of the two m apsrange, respectively, between 0 and 0.85 and between 0 and 0.88. The actual map of the corridor and of theobstacle is superimposed to the plots and the locations of the points of measure are marked with asterisks.


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M . P o l o n i e t a l . / F u z z y S e t s a n d S y s t e m s 6 9 1 9 9 5 ) 1 5 2 7 23

90 9080 80

50 t~ P4030 3020 2010 1 0 ~

a20 40 60 80 1O0 20 40 60 80Fig . 8. M ap of the em pty ce l ls . Fig . 9 . M ap f rom Eq. (19) and BS opera tors .

100

[0

1020 40 60 80 1O0

Fig . 10. M ap f rom Eq. (19) and AP opera tors .

10090807 0 l ~ .605040302010

/9~3

20 40 60 80 100Fig . 11. M ap f rom Eq. (18) and BS opera tors .

T h e f ir s t o f t h e t w o m a p s i s c l u t t e r e d b y m a n y b e a m s p u r i o u s r e f l e c ti o n s , re v e a l e d b y m i s s e d w a l ls e .g .i n 4 5 - 6 5 , 2 0 - 3 0 ) . T h e e f f ec t o f t h e l o b e w i d t h i s e v i d e n t n e a r t h e c o r n e r s w h e r e a r c s h a p e d a r t i f a c t s a p p e a re.g. 5 5 - 8 0 , 4 0 - 5 0 ). T o m a k e t h e se p h e n o m e n a m o r e e v i d e n t a c o n t o u r l in e a t 0 .1 h a s b e e n a d d e d . T h e m a p o f

t h e e m p t y c el ls is c o m p a c t a n d s e e m s a g o o d r e p r e s e n t a t i o n o f t h e c o r ri d o r . H o w e v e r , n o t e t h a t e v e n t h ec o n t o u r l i n e 0 .7 , w h i c h e n c l o s e s a n a r e a a l m o s t ) c e r t a i n l y fr ee , i n 6 5 , 5 0 - 8 0 ) i s b e h i n d t h e a c t u a l w a l l.


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24 M . P o l o n i e t a l . / F u z z y S e t s a n d S y s t e m s 6 9 1 9 9 5 ) 1 5 2 7M o r e o v e r , t h e l a r g e d i s t a n c e a m o n g t h e c o n t o u r l i n es d e n o t e a la c k o f r e s o l u t i o n . T h i s i s o n e o f t h e m a i nq u a l i t y i n d i c e s c o n s i d e r e d i n th e f o ll o w i n g , t h e o t h e r o n e b e i n g o b v i o u s l y t h e r e li a b i l it y o f t h e n a v i g a t i o nm a p .

W e t e s t e d t h e fo u r c o m b i n a t i o n s o b t a i n e d i m p l e m e n t i n g E q s . (1 8) a n d ( 19 ) b y b o t h A P a n d B S o p e r a t o r s .A l l t h e c o m b i n a t i o n s g i v e g o o d r e s u l t s e s p e c ia l l y c o n s i d e r i n g t h e s m a l l n u m b e r o f m e a s u r e s . N o t e t h a t a l l th em a p s r a n g e b e t w e e n 0 a n d 0 . 8 8 .T h e b e s t r e s u l t s h a v e b e e n o b t a i n e d u s i n g E q . (1 9) a n d t h e B S c o u p l e o f o p e r a t o r s . T h e r e s u l t i n g m a p i sg i v e n i n F ig . 9. I t c a n b e c o m p a r e d w i t h F i g . 1 0, o b t a i n e d b y t h e s a m e e q u a t i o n b u t u s i n g t h e A P c o u p l e .N o t e i n t h e f o r m e r m a p t h e s h a r p n e s s o f t h e d e t e c t e d w a l l s in p a r t i c u l a r o n t h e r i g h t s i d e o f t h e c o r r i d o r , a l s ot h e d o o r s b e t w e e n t h e b o o k c a b i n e t s a r e w e l l d r a w n . T h e c o n t o u r l i n e a t 0 . 5 i s a l w a y s c o n t a i n e d i n t h e s a f ez o n e , e x c e p t f o r a sma l l a r e a n e a r t h e p o in t ( 65 , 5 2 ).

I n t h e l a t t e r m a p t h e l ev e l l i ne s a re g e n e r a l l y m o r e d i s t a n t a n d t h e o b s t a c l e r e s u l t s q u i t e l a r g e r t h a n i n t h ep r e v i o u s m a p a n d i s a l m o s t c o n n e c t e d t o t h e w a l l .

T h e ( s m a l l ) i m p r o v e m e n t s o f t h e B S c o u p l e o v e r t h e A P o n e , c a n b e a t t r i b u t e d t o t h e d if f e re n t s t r e n g t h o ft h e t w o i m p l e m e n t a t i o n s o f t h e i n t e rs e c ti o n :

iBs(a ,b)


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2 0

1 5

1 0

2 5 5 7 5 i0 12 5 15 17 5Fig. 13. En larged view from Eq. (19) and AP operators.

2 0

S i m i l a r c o n c lu s i o n s c a n b e d r a w n b y o b s e r v i n g t h e t h r e e e n l a r g e d v i e w s o f t h e r e g io n c o n t a i n i n g t h ec e n t r a l o b s t a c l e , sh o w n , r e sp e c t i v e ly , i n F ig s . 1 2 1 4 i n t h e s a me o r d e r a s b e f o r e . I n p a r t i c u l a r , i n t h e f ir s td e n s i t y p l o t t h e o b s t a c l e a p p e a r s v e r y w e l l o u t l i n e d a n d s u f f i c i e n t l y s e p a r a t e d f r o m t h e w a l l .

6 C o n c l u s i o n s

A n ew a p p r o a c h i s p r o p o s e d t o b u i l d m a p s f o r a u t o n o m o u s m o b i l e ro b o t s . F u z z y s et t h e o r y h a s b e e ns u c c e s sf u l ly u s e d t o m a n a g e t h e u n c e r t a i n t i e s r e s u l t i n g f r o m t h e c h a r a c t e r i s t i c s o f t y p i c a l s e n s o rs . M o r e o v e r ,i t a l l o w s t o d e s c r i b e , in a v e r y n a t u r a l w a y , t h e o p e r a t i o n s n e c e s s a r y t o e x t r a c t s p e c i fi c t y p e s o f i n f o r m a t i o nf r o m t h e m a p s .

A p r o t o t y p e s y s t e m h a s b e e n r e a l iz e d in o u r l a b o r a t o r y , u s i n g t h e w e l l - k n o w n P o l a r o i d u l t r a s o n i c s e ns o r s.I t h a s b e e n s u c c e s sf u l ly te s t e d a n d h a s s h o w n t h e a d v a n t a g e s o f t h e d e s c r i b e d a p p r o a c h . T h e q u a l i t y o f t h eo b t a i n e d m a p s is n o t e w o r t h y , i n p a r t i c u l a r i f c o m p a r e d w i t h t h a t o f o t h e r m a p s p r e s e n t ed i n th e l i te r a tu r e .T h e p r o p o s e d s t r u c t u r e c a n b e a l s o u s e d t o in c l u d e i n f o r m a t i o n d e r i v i n g f r o m d i f fe r e n t k i n d s o f d e v i c es . A sa m a t t e r o f f a ct , o n l y t h e m o d e l i n g o f t h e u n c e r t a i n t i e s t h e y i n t r o d u c e i s r e q u i r e d . A l s o i n c r e m e n t a l m a pb u i ld i n g , t y p ic a l o f r o b o t s n a v i g a t i n g i n a n u n k n o w n e n v i r o n m e n t , c a n b e e as i ly a ll o c a t e d in t h e p r o p o s e df r a m e w o r k . T h e u s e o f a N O M A D 2 0 0 m o b i l e u n i t is p r o v i d i n g t h e w a y t o t e s t t h e a p p r o a c h i n a r e a l c as e .F u r t h e r r e s e a rc h e s in o u r l a b o r a t o r y i n c l u d e t h e s ea r c h o f a n o p t i m a l p a t h o n a f u z zy m a p . T h i s is a ni n t r i g u i n g p r o b l e m a s t h e u s u a l d e f i n i t i o n s o f o p t i m a l i t y , e .g . t h e s h o r t e s t p a t h , a r e d i ff i cu l t to a p p l y i np r e s e n c e o f u n c e r t a in t y , a s i n t h e ca s e o f s e n s o r - b a s e d r e c o n s t r u c t i o n o f t h e e n v i r o n m e n t . N e w m e t h o d s a r e


12/13

26 M . P o h m i e t a l . / F u z z ) S e t s a n d S y s t e m s 6 9 ( 1 9 9 5 ) 1 5 2 7

2 0

1 5

1 0

2 5 5 7 5 i0 12 5 15 17 5 20Fig. 14 . Enlarge d view f rom Eq. (18) and BS opera tors .

r e q u i r e d : a v i a b l e p o s s i b i l i t y is a t r a d e - o f f b e t w e e n l e n g t h a n d r is k ; t h i s a g a i n c a l ls fo r a f u z z y a p p r o a c h .A m o n g t h e a l g o r i t h m s u n d e r i n v e s t i g a t i o n s t h e r e a r e b o t h g r a p h - s e a r c h i n g (e .g . A * ) a n d p o t e n t i a l b a s e do n e s .

cknowledgmentsT h i s w o r k h a s b e e n p a r t i a l l y s u p p o r t e d b y M . U . R . S . T . 4 0 f u n ds .

References[ 1 ] H . Ba nde me r a nd W. Na t he r , F u z z y D a t a A n a l y s i s (Kluw er Academ ic Publ ishers , Dord recht , 1992) .[2] I .J. Co x and G .T. Wilfong, A u t o n o m o u s R o b o t V e h ic l es (Springer, Berlin, 1990).[3] J . Dom bi , A genera l c lass of fuzzy opera tors , the d e morg an c lass of fuzzy opera tors and fuzziness meas ure induced by fuzzyopera tors , F u z z y S e t s a n d S y s t e m s 8 [1982).[ 4 ] H . F . Dur r a n t - W hy t e , I n t e g r a t i o n C o o r d i n a t i o n a n d C o n t r o l c ~f M u l t i - s e n s o r R o b o t S y s t e m s , Vol. SECS 36, In t e rn a t i o n a l S e r i e s i n

E n g i n e e r i n g a n d C o m p u t e r S c i e n e e (Kluw er Academ ic Publ ishers , Norw el l , MA, 1988).[5] A. El fes and H.P. Moravec , High resolut ion maps f rom wide angle sonar , I E E E l n t e r n a t . C o n j . o n R o b o t i c s a n d A u t o m a t i o n ,St. Louis, M O, (1985) pp. 116 121.[6] G.D. Hager , T a s k - D i r e c t e d S e n s o r F u s i o n a n d P l a nn i n g : A C o m p u t a t i o n a l A p p r o a c h , Vol. SECS 99, In t e rn a t i o n a l S e r i e s i n

E n g i n e e r i n g a n d C o m p u t e r S c i e n c e (Kluw er Academ ic Publishers, Norw el l , MA, 1990).[ 7 ] J . Hwa n , L . Don g a nd W. Cho , Phys i c a ll y ba s e d s e ns o r mode l i ng f o r a s ona r ma p i n a s pe c u l a r e nv i r o nm e n t I E E E l n t e r n a t . C o n f

o n R o b o t i c s a n d A u t o m a t i o n , Nice, Fra nce (1992) pp. 1714~1719.


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F u z z y L o g i c A p p l i c a t i o n s i n I n t e l l i g e n t S y s t e m s Kluwer Academic Publishers, Dordrecht, 1992).[9] A. Kemurdjian, V. Granov and U. Mishkinyuk, Small marsokhod configuration, I E E E l n t e r n a t . C o n f o n R o b o t i c s a n d A u t o m a t i o nNice, France 1992) pp. 165-168.

[10] G.J. Klir and T.A. Folger, F u z z y S e t s , U n c e r t a i n t y a n d I n J b r m a t i o n Prentice-Hall, Englewood Cliffs, N J, 1988).[11] E. Krotkov and R. Simmons, Performance of a six-legged planetary rover: Power, posi tioning and autonomous walking, I E E E

l n t e r n a t . C o n f . o n R o b o t i c s a n d A u t o m a t i o n , Nice, France 1992) pp. 169 174.[12] R. Kuc and M.W. Siegel, Physically based simulation model for acoustic sensor robot navigation, I E E E T r a n s . P a t t e r n A n a l y s i sM a c h i n e I n t e l l i g e n c e PAMI-9 6) 1987) 766 778.

[13] Y. Maeda et al., Hierarchical control for autonomous mobile robo ts with behaviour decision fuzzy algorithm, IEEE In ter na t . C on f .o n R o b o t i c s a n d A u t o m a t i o n , Nice, France 1992).

[14] C. Pal and J. Bezdek, F u z z y M o d e l s f i~ r P a t t e r n R e c o g n i t i o n 1EEE Press, New York, 1991).[15] S.K. Pal, Fuzziness, image information and scene analysis, in: R. Yager and L.A. Zadeh, Eds., A n I n t r o d u c t i o n t o F u z z y L o g i c

Appl i ca t ions in In t e l l i gen t Sys t ems Kluwer Academic Publishers, Dordrecht, 1992).[16] Polaroid Corporation, P o l a r o i d U l t r a s o n i c R a n g i n g S y s t e m H a n d b o o k A p p l i c a t i o n N o t e s ~ T e c h n i c a l P a p e r s .[17] Polaroid Corporation, U l t r a s o n i c R a n g i n g S y s t e m .[18] F.G. Pin and H. Watanabe et al., Autonomous navigation of a mobile robot using custom designed qualitative reasoning vlsi chips,

I E E E I n t e r n a t . C o n f . o n R o b o t i c s a n d A u t o m a t i o n , Nice, France 1992).[19] H. Prade and D. Dubois, A review of fuzzy set aggregation connectives, Inform. Sci . 36 1985).[20] A. Pruski and C. Bourkis, The vahm project: a coopera tion between an autonomous mobile platform and a disabled person, I E E E

ln ter na t . C on f . on Robot i cs and Au tomat ion , Nice, France 1992).[21] B. Wilcox and L. Matthies et al., Robot ic vehicles for planeta ry exploration, I E E E l n t e r n a t . C o n j . o n R o b o t i e s a n d A u t o m a t i o n ,Nice, France 1992) pp. 175 180.

[22] R.R. Yager, On a general class of fuzzy connectives, I n t e r n a t . J . F u z z y S e t s a n d S y s t e m s 4 1980).[23] H.J. Zimmermann, F u z z y S e t T h e o r y a n d I t s A p p l i c a t i o n s Kluwer Academic Publishers, Dordrecht, 1991).

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