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3RRR Kinematic Analysis of The 3RRR Parallel Robot

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Page 1: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

3RRR

Kinematic Analysis of The 3RRR Parallel

Robot

Page 2: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß
Page 3: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

ii

Page 4: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

iii

Page 5: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

iv

Abstract

Parallel robot is a usual important application in the industry. The advantage of

parallel robot is high hardness and parallel robot can absorb heavy load and it can

control accurately in some positions. We usual see parallel robot are 5RRR of 2

freedom, 3RRR of 3freedom, and unsymmetrical 2RRR1RRR.

The paper mainly describes kinematic analysis of the 3RRR parallel robot. The

structure of 3RRR parallel robot has an equilateral triangle and six poles. We put The

structure on the X-Y coordinate in order to analysis kinematics. The kinematic

analysis has two major parts: First part is the inverse kinematics; The second part is

the forward kinematics. First part inverse kinematics: Input coordinate of center of

equilateral triangle and rotative angles of equilateral triangle on the X-Y coordinate

work out rotative angles of poles, and simulates the inverse kinematics of the structure.

The second part forward kinematics: Input rotative angles of poles work out

coordinate of center of equilateral triangle and rotative angle of equilateral triangle,

and simulates the forward kinematics of the structure. These two parts of kinematic

analysis are to simulate by MATLAB software.

Page 6: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

v

Page 7: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

vi

7 8 9

4

1 2 3 4 5 6

1 2 3

Page 8: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

1

ISO/TC184/SC2/WG1(1984 )

Page 9: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

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Manipulator

Page 10: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

3

Page 11: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

4

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

c 1 0 0

b 0 1 0

a 0 0 1

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

c 1 0 0

b 0 1 0

a 0 0 1

⎥⎥⎥⎥

⎢⎢⎢⎢

0

z

y

x

Page 12: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

5

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 cos sin 0

0 sin- cos 0

0 0 0 1

θθ

θθ

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 cos 0 sin-

0 0 1 0

0 sin 0 cos

θθ

θθ

Page 13: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

6

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 1 0 0

0 0 cos sin

0 0 sin- cos

θθ

θθ

Page 14: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

7

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

c cos sin 0

b sin- cos 0

a 0 0 1

θθ

θθ

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

c cos 0 sin-

b 0 1 0

a sin 0 cos

θθ

θθ

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

c 1 0 0

b 0 cos sin

a 0 sin- cos

θθ

θθ

Page 15: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

8

Page 16: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

9

Page 17: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

10

Page 18: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

11

1 2 3 4 5

6 1 2 3 4 5 6 7 8 9

Page 19: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

12

1 2

3 1 2 3 1 2

3 4 4 4 5

5 5 6 6 6 7 7 7

8 8 8 9 9 9

Page 20: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

13

1 2 3 1 2 3

'7

'7

'7

'8

'8

'8

'9

'9

'9

'7

'7

'7

'8

'8

'8

'9

'9

'9

'7 2

1°×°×× 210cos30sec5.7

2

1°×°×× 210sin30sec5.7

'8 2

1°−×°×× 30cos30sec5.7

2

1°−×°×× 30sin30sec5.7

'9 3

2°×× 60sin5.7

7 8

9 7 8 9

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 1 0 0

Y 0 cos sin

X 0 sin- cos

ϕϕ

ϕϕ

7 8 9'

7'

8'

9

Page 21: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

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⎥⎥⎥⎥

⎢⎢⎢⎢

0

07

7

y

x

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 1 0 0

Y 0 cos sin

X 0 sin- cos

ϕϕ

ϕϕ

⎥⎥⎥⎥

⎢⎢⎢⎢

0

0

'

'

7

7

y

x

⎥⎥⎥⎥

⎢⎢⎢⎢

0

08

8

y

x

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 1 0 0

Y 0 cos sin

X 0 sin- cos

ϕϕ

ϕϕ

⎥⎥⎥⎥

⎢⎢⎢⎢

0

0

'

'

8

8

y

x

⎥⎥⎥⎥

⎢⎢⎢⎢

0

09

9

y

x

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 1 0 0

Y 0 cos sin

X 0 sin- cos

ϕϕ

ϕϕ

⎥⎥⎥⎥

⎢⎢⎢⎢

0

0

'

'

9

9

y

x

7

8 9

7 8 9

4 1 4

4 1

Page 22: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

15

7 7 7 1 7

2

)13( 7x+−

2

)0( 7y+

71aa

1 1

11−

)13(

0

7

7

−−

x

y

4

1

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 1 0 0

b 0 cos sin

a 0 sin- cos

11

11

TT

TT

1 4 41aa 47aa

1 4 7 4 71aa

Page 23: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

16

4 l 4'

4 l

l

l 2 2 2 2/1

4'

4

⎥⎥⎥⎥

⎢⎢⎢⎢

0

04

4

y

x

⎥⎥⎥⎥

⎢⎢⎢⎢

1 0 0 0

0 1 0 0

b 0 cos sin

a 0 sin- cos

11

11

TT

TT

⎥⎥⎥⎥

⎢⎢⎢⎢

0

0

'

'

4

4

y

x

5 6 5 5 6 6 1 2 3 4

5 6

1 2 3 1 2 3

1 2 3 1 2 3 1 2 3

11−

)13(

0

4

4

−−

x

y

21−

)13(

0

5

5

x

y

31−

0

23

6

6

x

y

1 2 3 1 2 3

Page 24: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

17

1 7

2 8 3 9

71aa 82aa 93aa

1 2 3

1 2 3

1 2 3 4 5 6

Page 25: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

18

1 2 3 1 2 3

4 5 6 1 2 3

4 4 4 5 5 5 6 6 6

4 1 1

5 2 2

6 3 3

1 2

3 1 2 3 4 5 6

1 2 3 1 2 3 4 5 6

7 7 7 8 8 8 9

9 9 47aa 58aa 69aa 87aa 98aa

79aa

Page 26: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

19

⎪⎪⎪⎪

⎪⎪⎪⎪

=−−+−

=−−+−

=−−+−

=−−+−

=−−+−

=−−+−

05.7)()(

05.7)()(

05.7)()(

05.7)()(

05.7)()(

05.7)()(

2279

279

2298

298

2287

287

2269

269

2258

258

2247

247

yyxx

yyxx

yyxx

yyxx

yyxx

yyxx

7 7 8 8 9

9 7 7 7 8 8 8 9 9 9

7 7 7 8 8 8 9 9 9

1 2 3

7

7 7 8 8 8 9 9 9

Page 27: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

20

3987 xxx ++

3987 yyy ++

1 2 3 1 2

3

1−

78

78

xx

yy

1 2 3 1 2 3

1 2 3 1 2

3 4 5 6

1 2 3

Page 28: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

21

1 2 3

Page 29: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

22

Page 30: ù µ â > O Ó À 3 â 3 Ù b U · n O é K ô ÿ V NBUMBC ì O ô j 3 p ù b n O ³ ö ç j n O ³ 3 û ô > Ã > Û ³ n O â ä ô n O â ä s æ ± 3 o ¤ æ ô U 9 ï ô n ß

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