( )i ( ) ( ) ( ), ( ) ( ) ( )2

(1)

47

« –

» -

- -76,59.

1. ., ., . -

// -

. :

, 1980. . 5.

2. .

. . . , 2004.

2.02.07 .

. .

IV

- -

. -

IV -

-

1 T

3:

i

i ₁ t

1 (1) ,

,

, Y Y t Z Z t

t X

X_Ti _T _i _Ti _T _i _Ti _T _i i 1,4. ABCD

3 (BC), -

C l_CD=l₄

D 4 (CD)

1 1 1, _D , _D

D Y Z

X ,

[1]:

2 ,

4 2

4 l

l

q (2) 0

3 1

3

3Ci Ci Ci

i ax by cz

q , (3)

l

4 – C 3

D₁;

2 1 2 1 2 1

2

4 XD XCi YD YCi ZD ZCi

l ; (4)

a,b,c –

Z

O1

O 1

xD

1

A F

2

B

3

C

4

5

E Y

4

D

yD

X z1

x1

y₁

x₂ u2

2

w2

z2 v2 y2

y₅ u5

5

x5

w5

z5

v5 3

u3

w3

z₃ v3y3

x3 w4

y4

v4

x4

u4

z_D z4

T

(2)

48

2007. 2

; X_D₁, Y_D₁, Z_D₁, X_Ci, Y_Ci, Z_Ci – -

D₁ ( -

) C OXYZ

[2]. C -

3, -

T,

OXYZ -

- :

)) (

cos( ₁ ₂ ₃

3C

C x

X

, )) (

sin(

)) z₃_C ₁ ₂ ₃ X_C^'

)) (

sin( ₁ ₂ ₃

3C

C x

Y

, )) (

cos( ₁ ₂ ₃ ^'

3C YC

z

, cos ₃ ^'

3C C

C y Z

Z . (5)

) cos(

cos ₁ ₃_,₂ ₁ ₂

1 , 2

' a a

X_C ,

) sin(

sin ₁ ₃_,₂ ₁ ₂

1 , 2

' a a

Y_C ,

21 32 21 21

' c b b cos

Z_C .

- (2), (3)

( ):

. 1

, 1 , 1 , 1 , 1

, , ,

,

2 10 2 5

2 9 2 4

2 8 2 3

2 7 2 2

2 6 2 1

10 5 5 10 ' 5 8 3 7 2 2 7 ' 2 6 1

10 10 5 5 ' 5 3 3 7 7 2 2 ' 2 1 1

10 5 9 4 7 2 6 1

5 5 4 4 2 2 1 1

x x

b c x c x l x l c x c x l x l

a c x c x l x l c x c x l x l

b x l x l x l x l

a x l x l x l x l

(6) - x₁, x₂,x₃,x₄,x₅,x₆, x₇, x₈, x₉, x₁₀ – . - IV

(1). , -

- IV :

(7) - (6),

x₁,x₂, x₃, ..., x₂₁ -

IV . (7) -

-

.

0 ,...,

₂₁

2

1

x x

x

P

_n (8)

,

P

_n

x

₁

,..., x

_n -

. , (7), (8) -

.

: , -

(6) (7), (8) -

IV .

, -

-

[2]. ,

- ,

-

. , :

, 0 ,...,

...

, 0 ,...,

1 1 1

n m

n

x x P

(9)

m i x x

P

_i ₁

,...,

_n

, 1 ,

– n

.

11 1 12 2 14 4 15 5

11 6 12 7 14 9 15 10

11 1 16 2 18 7 19 13 3

17 5 20 10 21

11 6 16 7 18 2 19 13 8

17 10 20 5 21

2 2

1 6

2 2

2 7

, ,

,

, 1,

x x x x x x x x a x x x x x x x x b x x x x x x x x x

x x x x x a

x x x x x x x x x

x x x x x b

x x

2 2

3 8

2 2

4 9

2 2

5 10

2 2

18 19

2 2

20 21

1, 1, 1, 1,

1, 1.

x x

(7)

(3)

49

(9) I,

,

. ,..., ,...,

,..., ,

,...,

₂ ₁ ₁

1

x x

n

P x x

n

P

m

x x

n

P I

[3] ,

(9)

F x

₁

,..., x

_n -

,

n m

n

r x x

x x

r

₁ ₁

,... ,...,

₁

,...,

S,

m m

S

r P r P r P

F

₁ ₁ ₂ ₂

...

, . -

F = 0, -

, -

. [3],

: (9)

, 1 -

I, -

(9) F = 0,

m m

S

r P r P r P

F

₁ ₁ ₂ ₂

...

.

, : 1

m m

P r P

r P

r ...

1

₁ ₁ ₂ ₂ ?

[3].

- :

1 . -

P

m

P

₁

,

₂

,..,

(9).

mc c

c

P P

P

₁

,

₂

,...,

.

2 . P_i P_j, -

, .

wq

1

P

_ic

wq

2

P

_jc , w – .

3 .

1 2

, P P q

2

P q

P

S

_i _j _i .

4 .

S P

_i

, P

_j -

P

m

P

₁

,

₂

,...,

. , , . . -

(9) .

5 . 1-

, -

(9). , -

.

. - [3, 4],

. -

, (9) .

(9). -

-

IV .

[2] -

- IV

1 4

3 1 4 3 2

1 bx cx dx exx

x

a_i _i _i _i _i

. 4 , 3 , 2 , 1 ,

3

0

2

x g i

x

f

_i _i , (10)

i i i i i i

i

b c d e f g

a , , , , , ,

–

;x₁, x₂, x₃, x₄ – - .

x₄ >x₃>x₂ >x₁. - (10) P₁,P₂,P₃,P₄. - P₁ P₂

1 2 2 1 1 2 2 1 1 1 2 2 1 2

Pd P d a d a d x b d b d x

1 2 2 1 3 1 2 2 1 1 3

c d c d x e d e d x x

1 2 2 1 2 3 1 2 2 1 5

.

f d f d x x g d g d P

(11) (P₁ P₃), (P₁ P₄):

.

,

₁ ₄ ₄ ₁ ₇

6 1 3 3

1

d P d P P d P d P

P

(12)

(P₅ P₆), (P₅ P₇).

P₈ P₉, -

x₄,x₅ . P₈ P₉ x₂.

P₁₀ x₁. -

. , -

(10) ,

[3]: -

, I -

P₁, ..., P₁₀, -

x₁, ...,x_n

. -

4 1 3 2 2 3 4

1x , ax , bx , cx

d ,

.

(4)

50

2007. 2

1. ., ., -

. . .: , 1959.

1084 .

2. . -

IV

// . , 2003. . 2.

. 28-36.

3. . -

. , 2003. 68 .

4. ., . -

. .: , 1986. 544 .

IV i -

i i

, .

Summary

The application of the Grebner’s basis in solving tasks of kinematic synthesis of a spatial guide mechanism of IV class using multinomials in the form of a system of algebraic equations is proposed.

. . . 20.01.07 .

. .

, -

- -

. -

, ,

. -

.

- ,

, 949

20 2002 ,

- - - ,

, .

-

. . -

, « -

», 1968 ., :

, ,

, -

-

». .

(13 200 .) « -

»

-

351, « »

[1].

. 160

1977 , «

,

, -

-

». -

16

1978 . « , -

»,

, « -

( ) -

-

, ( -

) ,

- -

» [2].

.1 .3 .77

, -

«

,