Modelling of mining manipulator in motors formalism

ZESZYTY NAUKOWE POLITECHNIKI 1989

S e r ia : GÓRNICTWO z . 180 Nr kol. 1043

INTERNATIONAL CONFERENCE: DYNAMICS OF ¡MINING MACHINES DYNAMACH '89

J ó z e f WOJNAROWSKI, A ndrzej NOWAK

I n s t y t u t M echaniki i Podstaw K o n str u k c ji Maszyn P o l i t e c h n i k i ś l . , G liw ic e

MODELLING OF MINING MANIPULATOR IN MOTORS FORMALISM

Summary. I n t h i s p a p e r th e way o f m o d e l lin g o f m in in g m a n ip u la to r b a s e d on t h e m o to r s f o r m a lis m , a s i t i s u n d e r s t o o d b y M is e a , h a s b e e n f o r m u la t e d . The r a n g e o f m o d e l li n g i n c l u d e s th e k in e m a tic ana­

l y s i s , i n t h e s e n s e o f s c r e w s v e l o c i t y and a c c e l e r a t i o n and th e dynam ic a n a l y s i s , from t h e d e s c r i p t i o n o f w ren ch p ro g ra m s. I t h a s b e e n p o in t e d o u t t h a t k i n e t o s t a t i c b a la n c e o f th e m a n ip u la to r can b e a c h ie v e d when t h e c e n t r a l l i n e o f w renoh f o r c e s i s c o l l i n e a r to t h e s c r e w a x i s o f w ren ch v e l o c i t y . G ra p h ic r e p r e s e n t a t i o n o f r e c u r ­ r e n t p r o c e d u r e o f c a l c u l a t i o n s h a s b e e n a c h ie v e d b y term o f w rench g r a p h s .

1 . INTRODUCTION

The i n c r e a s e o f m in in g o u tp u t from t h e d e e p e r and d e e p e r d e p o s i t s cau ­ s e s th e ch a n g e o f m eth od s t e c h n o l o g y and m eans o f r e a l i z a t i o n , b y th e u s e o f a u t o m a t i z a t io n and r o b o t s , i n th e m in in g p r o c e s s e s . M o d e llin g th e raa—

c h in e s - m a n i p u la t o r s i s on e o f th e ways o f f u l f i l l i n g c o m p lic a t e d m in in g w orks w i t h m in im a l p a r t o f mom—o p e r a t o r . Such t r i a l s h a v e b e e n d e s c r ib e d i n t h e work . I n t h a t p a p e r g r e a t a t t e n t i o n h a s b e e n d r iv e n t o th e k in e m a t ic and dynam ic a n a l y s i s o f m in in g m a n ip u la to r s a s on e s s e n t i a l s t e p i n th e m o d e l li n g o f t h e s e m a c h in e s .

The m ethod o f m a t r ix tr a n s f o r r a a t io n o f c o - o r d i n a t e s s o - c a l l e d m a t r ix o f k in e m a t ic p a i r s t r a n s i t i o n h a s b e e n a p p li e d i n ou r r e s e a r c h c o n s i d e ­ r a t i o n s . I n t h i s p a p e r th e way o f m o d e l lin g o f m in in g m a n ip u la to r w ith th e u s e o f m o to r s fo r m a lis m h a s b e e n fo r m u la te d £²] . Such an a p p ro a ch s i m p l i f e s t h e k in e m a t ic and dynam ic a n a l y s i s o f m a n ip u la to r on th e way o f g e n e r a t i n g a n a l y t i c a lg o r it h m s ^ 3 j. The u s e o f w ren ch r e d u c t i o n g ra p h s m akes e a s i e r th e o r g a n i z a t i o n s i n g r a p h ic r e p r e s e n t a t i o n s .

170 J. Wojnarowaki, A. Novak

2 . KINEMATIC ANALYSIS OF MANIPULATOR IN REPRESENTATION OF WRENCHES

The m odel o f m in in g m a n ip u la t o r h a s b e e n shown i n th e d iagram 1, The m echanism o f t h e m a n ip u la to r c o n s i s t s o f th e o p en c h a in o f f o u r l i n k s , t h e l a s t one b e i n g th e g r a b . L e a v in g o u t t h e l o c a l m o b i l i t y o f th e g r a b , th e t h i r d and t h e f o u r t h l i n k s w i l l be c o n s i d e r e d a s o n e . The l i n k 0 b u i l d s up t h e fram e a t t h e some tim e b e in g t h e b o d y o f t h e m a c h in e . T here a r e t h r e e s p h e r i c a l J o i n t s i n t h e m ech a n ism i th e J o i n t A h a s two d e­

g r e e s o f fr e e d o m , th e J o i n t B h a s t h r e e d e g r e e s o f freed o m and t h e J o i n t C a g a in h a s two d e g r e e s o f fr e e d o m . To e a o h o f t h e s e l i n k s a r e ­ l a t i v e s y s te m o f c o - o r d i n a t e s h a s b e e n c o n n e c t e d , t h e sy s te m ( x q , y o , z q) i s m o t i o n l e s s . The c h a in h a s m o b i l i t y s e v e n l e a v i n g o u t t h e l o c a l move­

m ents o f th e g r a b , b e c a u s e o f t h e s p h e r i c a l J o i n t s , t o k in e m a t ic and dy­

nam ic d e s c r i p t i o n o f m a n ip u la t o r we h a v e ta k e n E u l e r ' s a n g l e s a s g e n e r a l c o - o r d i n a t e s . T hey a r e show n i n d iagram 2 f o r e a c h J o i n t s e p a r a t e l y w ith th e l i n e s o f n o d e s . S e a r c h in g k in e m a t ic a n a l y s i s o f m a n ip u la t o r w i t h th e u s e o f m o to r s m eth o d , th e d i v i s i o n o f J o i n t movem ent h a s b e e n a c h ie v e d i n t o t h r e e em pty r o t a r y m o v em en ts, w h ic h d o e s n o t demand t h e c o m p lic a t e d from o f m a t r ix o f t r a n s i t i o n f o r J o i n t . I n t h e p r e s e n t e d m ethod we h a v e

ta k e n f o r e a c h J o i n t a s e p a r a t e ( r ^ - l ) l i n k s o f z e r o l e n g t h , w h ere r^

means a d e g r e e number o f fr e e d o m . Su ch an a p p r o a c h m akes p o s s i b l e r e s e a r ­ c h in g k in e m a t ic s o f m a n ip u la to r b y th e c o m p o s it i o n o f s im p le r o t a r y move­

m ents r l i n k s , w here r m eans m a n ip u la t o r m o b i l i t y .

I n th e d iagram 3 we h a v e shown t h e g r a p h o f t h e compound m o to r v e l o ­ c i t y o f th e m echanism f o r s e v e n r o t a r y m ovem ents a s t h e r e s u l t s o f th e d i v i s i o n s o f r o t a r y J o i n t m ovem ents i n t o t h e com ponent r o t a r y m ovem en ts.

Fig. 1.

Modelling of mining manipulator in... 171

Fig. 2.

CO. (O. 2

0 ^{« 2 1 CO22} CO23 CO31

Km -n°

Koi (

1

)

•V)2 io°21 ... __

if (2)

02

*°22

Kq

2 W2

Kf"

03

< 4 “>3°

0

_{b ab}

0 0

_{® BC}

0

1 1 1

i

VQ ° V ^{0 1} VB ¹ V VB V VB

F i g . 3 .

M a t r i c e s o f m ovem ents shown i n diagram 3 a r e :

V vc

of» -»1» O' "l o 0

mr( 1 ) _

*01 ~ 0 K 11 = O 0 .G* -SP2

0 0 1 _0 eV»2 ° r 2

K 03

B,

'CD

(1)

17 2

J , Wojwarowafct, A. ftowak

01

£(1)

1 2

*02

4 , ” K 11

of3 _-»v>3 0

s<P3 ⁰

0 0 1

:< o 0 2

k( 2 ) 12

= *0,

K — 3 )

0 2 " 0 2 12

(

²

)

(3)

(*)

c<p5 0 _{1 0} 0

k(2)

12 “ 0 1 0 ^k(3) _

12 ⁰ • n (5)

a <p5 0 ₀

•n, ° \

k o3} * K K (1)

02 23 SC - ^ ^ «[(2)

03 *-03 23 (6)

K ( 0 _ 23 “

> 7 -v*7 0 ₁ 0 0

0 r*f II 0

c?6 - n s (7)

0 0 1 0

»^6 °^6

R e d u c in g t h e g ra p h we d e te r m in e t h e m o to r s v e l o c i t y o f p o l e : 1) vA = { w ^ , 0}f

2 ) VB = i % ° - VB } '

3) v c = K ° - v c 3 }<

W1° = “ ,t * “ iS = * 0 l K , * < 2 >

VB = VA + EAB « , (8)

(9)

“ a° *

*02 M 22 * E 02 W 23 = W^{( 2.} I ¥■} _"2i^o i=l

VB + ®BC W2

«-° = «-° + _{3 2} T “ 0 3 <“ 3 1 ♦ k 03 * * 2 , * 2° + ^ + w °2

VD » K ° . v ^,

(

^{1 1}

)

(1 2) (13)

(1*)

Modelling of mining manipulator in.. 173 w h e r e : B n r , a r e a n t y s y m m e tr ie a l m a t r ic e s o f c o - o r d i n a t e s o f

A5 BC OJJ

p o i n t s B ,C ,D i n t h e l o c a l sy s te m o f l i n k s c o - o r d i n a t e s ,

CO* = [ o , 0 , c p j'1' < 2 = [ v 0 , o ]

( 15) C0* = [ ° , 0, ?3] < 2 = [°> °1

co*3 = [ ¥ V 0. °]T > “ 31 = [°’ °> ^ 7f . “ 32 = [V 6 ’ °’ °]

N e x t we d e te r m in e t h e m o to r s o f p o l e a c c e l e r a t i o n on t h e b a s e o f equa­

t i o n s

aA = ° ,

aB = aA + ®AB 61° + °AB “ i ° » CAB = ®AB “ l ’

aC = “b + ®BC £ 2 ° + CBC W2 ° * °BC X ®BC W2 *

®U = a C + ®CD ^ 3 ° + °CD “ 3° ’ CCD = ®CD W3 ’

fi,° « W 11 * “ t2

62° = " l l * W2° + “ 21 X " 2 2 + W21 X W23 + " 2 2 X " 2 3 ( 17)

6 3° = <o2° x o>3° + <0° , x w32

and t h e y a r e :

A- = { £ 1°» aAJ 9 AB = i £ 2 * “bJ*

c 0 3 A

Ac = ( e3°* ac) ' ad = ^ ^4 » ^d)‘

( 1 8 )

The ab ove d e p e n d e n c ie s and e q u a t i o n s a r e e a s y to b e programmed w i t ­ h o u t th e n e c e s s i t y o f m u l t i p l i c a t i o n o f m a t e r ic e s and w r i t i n g i n c o m p li­

c a t e d e x p r e s s i o n .

174 J. Vojnarowski, A. Novak

3. MOTORS IX KINETOSTATICS DESCRIPTION OF MANIPULATOR

The aim o f th e a n a l y s i s i s to d eterm in e th e d r iv in g moments f o r lin k s and th e r e a c t io n s in k in e m a tic p a i r s . Ve assume t h a t v e l o c i t i e s and a c c e l e r a t io n o f l i n k s , d e s c r ib in g th e movements o f th e grab i n th e sp a ce a re known. The k i n e t o s t a t i c a n a ly s is i s co n d u cted in th e c o n v e o t io n mo­

vem ent s e t t i n g o f th e m echanism .

In th e diagram ( 4 ) th e a c t i v e and p a s s i v e power sy stem s and th e r e s i s -

■ance to m otion th a t work on th e each l i n k have been shown.

sta n d f o r th e in s t a n ta n e o u s p o s i t i o n s o f a x is o f r o t a t i o n o f li n k s a lo n g w hich th e r e are s i t u a t e d th e v e c t o r s o f d r iv in g moments and moments o f f r i c t i o n i n j o i n t s .

F o rces o p e r a t in g on l i n k s a re d e s c r ib e d b y th e c o r r e sp o n d in g m otors o f f o r c e s . The a n a l y s i s begun w ith th e l a s t l i n k o f m a n ip u la to r and i s p roved i n th e l o c a l sy stem s o f c o o r d in a t e s o f l i n k s .

1 . Link 3

The main m otor o f a c t i v e and p a s s iv e f o r c e s i n th e p o le C i s d e fin e d

_Q___

F ig . 4 .

(19)

where s + IF.

3 " 3 3 3 ’ 03 • (2 0)

(21)

Modelling of mining manipulator in,. 175

i s t h e moment o f i n e r t i a c o u n te d a c c o r d in g t o E u le r ’ s fo r m u la , i s t h e a b s o l u t e a n g le v e l o c i t y o f t h e l i n k d e te r m in e d i n th e sy s te m o f c o o r d i n a t e s o f th e l i n k 3 , i s a b s o l u t e a n g le a c c e l e r a t i o n o f th e l i n k .

The m o to r o f r e a c t i o n i n t h e p o l e C i s d e te r m in e d :

c • *3

and th e r e l a t i v e m otor o f v e l o c i t y :

( 2 2 )

v: = H » °)» ^{( 2 3 )}

where <*>3 i s th e r e l a t i v e a n g le v e l o c i t y o f th e l i n k .

The d r iv in g moment M o f th e l i n k we d e f in e from th e eq u a tio n o f po­

wer:

* 3 ° " 3 + (Mc * M cr) V c = 0 ( 2«0

from w hich we h ave:

«3* = M3* - ^ 3 + rcc3 * V 3> »3 * M 3° = M 3° C 3 ’ «3 = - 3 / I “ 3 I

(25) ( 26) The r e a c t i o n i n th e p o le C i s :

r£ 3) = - v 3 = - (b3 + c 3 + f3). ( 27)

The k in e m a t ic e n e r g y o f th e l i n k i n a b s o l u t e movement i s d e te r m in e d :

E3 = ° * 5 (Vc3 B3 V c 3 } = ° * 5 <»3Vc3 + W 3 J c 3 (2 8)

w h ere

Vc3 = i^3 » V c3) ^{( 2 9 )}

i s t h e a b s o l u t e m o to r o f v e l o c i t y i n th e c e n t e r s o f m a ss,

B 3 =

c3

(3 0 )

i s th e m a tr ix o f i n e r t i a o f th e l i n k .

176 J. Wojnarowski, A. Nowak On th e b a s i s (2 8) wo c o u n t th e r e d u c e d moment o f i n e r t i a o f t h e l i n k

J* - 2 \ - V°3 \

r3 ^k P K r

* 2 E_ vj B V

* 3 = _£3__ ? c3 (31)

2 . L in k 2

We make th e r e d u c t i o n o f m o to r (2 2 ) to th e c o o r d in a t e s sy ste m o f th e l i n k 2:

* i 2) = - K 2 ^ 3) = K 23W 3 = W 32) * (32)

m;(2) = - k2 3m ;<3>, k23 = 4 2)- (33)

I n th e p o l e B we d e f i n e t h e m ain m o to r o f l i n k 2:

«B = iW 2* *b) * ^(3*0

w here

V 2 = ®2 + C 2 + R i2) * ®2 * °2 + V 32)* (35)

« B + * b 2 + r BC2 1 W2 * r C2C 1 V 3 2 ) * (3 6 )

w here

* b2 = - [J c2 6 2 + ( J c2 (37)

I n th e p o l e B t h e m o to r o f r e a c t i o n i s d e f i n e d :

Mg = { 4 2) (38)

W© d e te r m in e th e d r i v i n g moment o f t h e l i n k u s i n g t h e e q u a t i o n o f pow er:

M 2e<°2 + <MB + MB,*> VB = ° • (39)

w here

< ={<• o). < ° < Z < °- (,t0)

Modelling of raining manipulator in.. 177

U sin g th a e q u a tio n ( 3 9 ) we c o u n t:

M e - (*L + r x V + r x V 12^)

2 2 t**b2 BCg 2 rc2C 3 '

< / | » * |

(

²

),

«I#2 =

The r e a c t io n i n t h e p o le B i s :

4 2> , - V 2 = - (B2 * o 2 4 V<2>) .

We cou n t th e k i n e t i c e n e r g y o f th e l i n k by:

E2 = ° * 5 (Vc I B2 VC2) = ° ’ 5 (» 2 VC2 + W2T J C 2 t° 2 ) * i

where

(M)

(*•2)

(**3)

(W.)

VC2 = { «2* VC2) » B2 =

0 Jt02

( 4 5 )

and th e red u ced moment o f i n e r t i a :

r2 ^{( M )}

3 . L in k 1

The k i n e t o s t a t i e a n a l y s i s o f th e l i n k 1 i s done a t th e l a s t s t e p . We make th e r e d u c t i o n o f m o to r (^{3 8}) t o th e sy s te m o f c o o r d i n a t e s o f th e

l i n k :

®( 1 ) _ Tic - k V - 1 )

*B = K 12 B “ K 12 * 2 ‘ * 2

- - SC M ^ 2 ^ K = ^ SC^2 ^ .

2 = -12 2 ’ 12 12 12 12

Ve d e f i n e t h e m ain m o to r i n th e p o l e A:

*A = ( v i*

w here

ri = ® 1 + ° 1 + V 2 ,(1)« B 1 --- 1 "C1

(2*8)

(<*9)

(5 0 )

(5 1 )

178 J. Vojnarowskl, A. Nowak

M A = *bt + r ACl * W 1 + * C $B f V i 0 (52)

“ bl * - [JC1 £ 1 * W 1 X (JC1 w i>] (53)

The m o to r o f r e a c t i o n i n th g e p o l e A i s d e te r m in e d :

MAr = <5<0

U s in g t h e e q u a t i o n o f pow er:

M18 < + (»A + M A r) Vt = °* (55)

w here

V* = {cn * , 0 }

i s t h e r e l a t i v e m o to r t h e n we d e f i n e t h e d r i v i n g momennt o f l i n k :

M1 = M1 “ (*S>1 + r AC1 x W 1 + ® * * ( 5 6 )

M l" = M l" ® * ’ ® T = W 1 / \ w * \ (57)

The r e a c t i o n i n th e n o d e A i s :

R i l) = - W 1 = " (®i + G 1 + (58)

The r e d u c e d moment o f i n e r t i a o f th e l i n k d e t e r m in e s d e p e n d e n c e :

v T D V

, * ... c i ^U1 C¹ . .

r = 7 - 7 - ( 5 9 )

1 («* )

w here

VC1 = ’VC l}

i s th e a b s o l u t e m o to r o f v e l o c i t y i n th e c e n t e r o f m a s s.

The re d u c e d moments o f i n e r t i a o f l i n k s r e f e r to t h e i r i n s t a n t a n e o u s a x i s o i r o t a t i o n . Now we ta k e u n d e r th e c o n s i d e r a t i o n t h e k i n e t o s t a t i c b a la n c e o f th e m a n ip u la to r w h ic h w i l l b e d e s c r i b e d on t h e b a s i s o f th e l i n k 1. The m ain m o to r i s r e d u c e d to t h e w ren ch o f f o r c e s :

Modelling of mining manipulator In.. 179

F 1 = i V 1 * «Si! <60>

w hore

* S1 = *, « V l » i l

Ve d e f i n e th e e q u a t i o n o f th e c e n t r a l l i n e o f th e w ren ch o f f o r c e s :

1, = x^Q1 + t ^ (^{6 2})

w here

x^{^ 1} m eans th e c o o r d i n a t e s o f th e a r b i t r a r y p o i n t o f s t r a i g h t l i n e . I n p a r t i c u l a r we d e f i n e th e p o i n t s i t u a t e d n e a r e s t t o t h e p o l e A:

rA Q

% x M

-1 (63)

lV ll

The r e d u c t i o n o f t h e v e l o c i t y m o to rs o f th e l i n k to th e k in e m a tic w rench i s d o n e , to o :

si = i“ r v sil » {6k)

where

u ) 1 f V .

V S 1 = ^ 1 “ V i * t = ■j j y • ( 6 5 )

B e c a u s e v e l o c i t y v = 0 w ren ch S i s r e d u c e d to m otor:

A 1

si = W 1 (66)

and t h e w ren ch l i n e ( t h e w ren ch a x i s ) o v e r la p s w it h th e i n s t a n t a x i s of r o t a t i o n P^.

The l i n k s i b a la n c e d k i n e t o s t a t i c a l l y when th e c e n t r a l l i n e i s p a r a l l e l o r o v e r l a p s w it h th e w ren ch l i n e , i n t h i s c a s e w it h t h e i n s t a n t a n e o u s a x i s o f r o t a t i o n o f th e l i n k . I t means t h a t th e d r i v i n g moment o f th e l i n k i s th e r e v e r s e v e c t o r t o th e moment o f r e s i s t a n c e t o m o tio n and th e r e a c t i o n i n th e n o d e A h a s t h e d i r e c t i o n o f th e i n s t a n t a n e o u s a x i s o f r o t a t i o n . The m a t t e r o f k i n e t o s t a t i c b a la n c e o f o p en c h a in h a s b een d e s c r i b e d i n d e t a i l s i n w orks F1*, 5] .

180 J. Wojnarowski, A. Nowak

. CONCLUSIONS

The F orm alism o f th e m o to r s t h e o r y m akes p o s s i b l e f o r m u l a t i n g th e k in e m a t ic and dynam ic a n a l y s i s m eth od o f m in in g m a n i p u la t o r s . I t m akes e a s i e r th e i n t e r p r e t a t i o n s o f b a l a n c i n g o f t h e s e m a c h in e s a s t h e a r r a n g e ­ m ent o f c h a n g e a b le s t r u c t u r e . The p r e s e n t e d way o f m o d e l l i n g and d e s c r i ­ b ed a lg o r i t h m s c a n b® u s e i n t h e d e s i g n i n g p r o c e s s e s s p e c i a l l y i n t h e m in in g r o b o t i z a t i o n .

LITERATUR

[ 1] HepHoycfcKO FpaneiiKHÄ B .F ., $omhh J I .$ ., KmcaeBHi X)»H,, KoieHKo n .H ., KoB&ne-B A.M.., E£$hmo

6

B .C .: MexaHHKa nsaxiHKx poöoioiejCHH'iecKHx KOMaaeK- cob c Ga

3

o b u m uoöHJibHHM uaHHnyjiKTcpoti. łiHCtusyi IIpoÖJieM MeXaHHKH AjcaÄekHH EayK CCCP, MocKBa 1987,

[2J M is e s R .t M o to r r e n r e c h n u n g . Z e i t s c h r i f t f u e r A ngew andte M ath em atik und M ech a n ik , B e r l i n , 1 9 2 6,

P³j W ojn arow sk i J . , Nowak A .: Graph o f w r e n c h e s i n t h e k i n e m a t i o a l a n a ly ­ s i s o f t h e r o b o t i c s - r a a n l p u l a t o r s , Z b ió r r e f e r , XI K o n f e r e n c j i TMiM, Z .N , P o l i t e c h n i k i 3 l . s .M e c h a n ik a n r 8 6, G liw ic e - Z a k o p a n e 1 9 8 7 , ( i n P o l i s h ) .

Wojn a r o w s k i J . , Nowak A . : On th e b a la n c e o f m echanism o f m a n ip u la t o r i n t h e f o r m a l i z a t i o n o f s c r e w s . R e f e r . XXVII Symposium " M o d e llin g i n M e c h a n ic s " , G l i w i c e - W i s l a , 1988 ( i n P o l i s h ) .

£5] " I d e n t y f ik a c j a o b ie k t ó w b ad ań - m o d e lo w a n ie w y o d r ę b n io n y c h sy stem ó w i p o d sy stem ó w r o b o tó w i m a n ip u la to r ó w m etod ą g r a fó w " . P r a c a w ykonana pod k ie r u n k ie m J , Woj n a r o w s k ie g o w IMiPKM P o l i t e c h n i k i Ś l . w ram ach CPBR 7 ,1 "R oboty p r z e m y s ło w e " , G l i w i c e 1987 ( e t a p I i ) .

R e c e n z e n t : P r o f . d r h a b . i n ż . Bohdan M ochnacki

MODELOWANIE MANIPULATORA GÓRNICZEGO W FORMALIZMIE MOTORÓW

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s p i e s z e ń o r a z a n a l i z ę d y n a m iczn ą w o p i s i e s k r ę t n ik ó w s i ł a ż do g en ero w a ­ n i e a lg o r y tm ó w i program ów o b l i c z e n i o w y c h . W p r a c y p o n a d to w y k a za n o , ż e w yrów now ażenia k l n e t o s t a t y c z n e m a n ip u la to r a można u z y s k a ć , g d y l i n i a c e n ­ t r a l n a s k r ę t n i k a s i ł j e s t k o lin e a r n a z o s i ą śrubow ą s k r ę t n i k a p r ę d k o ś c i . D z i ę k i z a s t o s o w a n iu g r a fó w s k r ę t ó w u z y sk a n o g r a f i c z n ą r e p r e z e n t a c j ę r e k u r e n e y J n e j p r o c e d u r y o b l i c z e ń .

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