industrial robots - polito.it€¦ · example 1.2 –vector representation if t t itd t th 11 11...
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![Page 1: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/1.jpg)
Industrial RobotsIndustrial Robots
A simple exampleA simple exampleBasilio Bona ROBOTICA 03CFIOR 1
A simple exampleA simple example
![Page 2: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/2.jpg)
Example 1.1 – vector representation
110
B
⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜vA very simple manipulator
v
P
0⎟⎜ ⎟⎜⎝ ⎠
BR
v
AR1( )q t
1 1 1 1
1 1 1 1
cos sin 0 c s 0
sin cos 0 s c 0AB
q q
q q
⎛ ⎞ ⎛ ⎞− −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= ≡ =⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟R
1 1 1 1
0 0 1 0 0 1B ⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
1
1
cos
sinAB
q
q
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟t1
0B
q⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠Basilio Bona ROBOTICA 03CFIOR 2
![Page 3: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/3.jpg)
Example 1.2 – vector representation
If t t i t d t th
1 11 1c sc s 0 1⎛ ⎞⎛ ⎞ ⎛ ⎞−− ⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜
If vector represents an oriented segment, thenBv
1 1
1 11 1c ss c 0 1
00 0 1 0A
⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟+= =⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜⎟ ⎟ ⎜ ⎟⎟⎜⎟ ⎟⎜ ⎜ ⎝ ⎠⎝ ⎠⎝ ⎠
v00 0 1 0 ⎜⎟ ⎟⎜ ⎜ ⎝ ⎠⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞If it represents a geometric point, then
1 1 1 1 1
1 1 1 1 1
c s 0 1 c s c
s c 0 1 c s sAA B
t
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟= + = + +⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟v
1 1 1 1 1
0 0 1 0 0 0
( )
A B⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞
1 1
1 1
(1 ) c s
(1 ) s c
⎛ ⎞+ − ⎟⎜ ⎟⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟
Basilio Bona 3ROBOTICA 03CFIOR
1 1( )
0⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
![Page 4: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/4.jpg)
Example 2.1 – rotations
Basilio Bona 4ROBOTICA 03CFIOR
![Page 5: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/5.jpg)
Example 2.2 – rotations
Basilio Bona 5ROBOTICA 03CFIOR
![Page 6: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/6.jpg)
Example 3.1 – DH parameters
3( )q t2( )q t
1 11 ( ) 0 90d a
q tθ α
− 1
2
2 2
3
2 ( ) 0 903 ( ) 0 0 0
q tq t−
3 ( )q
1( )q t
Basilio Bona 6ROBOTICA 03CFIOR
![Page 7: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/7.jpg)
2R
2q3
1R0R
1 2
1q Denavit – Hartenberg parametersi d θ
1 1 21 ( ) 2
i i i i
q t
i d aθ απ
1 1 2
2 30 ( ) 02 tq
Basilio Bona 7ROBOTICA 03CFIOR
![Page 8: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/8.jpg)
01R
12R
1 1 2 10 1 1 2 11
00
0 1 0
c s cs c s⎛ ⎞⎟⎜ ⎟⎜ − ⎟⎜= ⎟⎜ ⎟⎜
T2 2 3 2
1 2 2 3 22
00
0 0 1 0
c s cs c s⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜= ⎟⎜ ⎟⎜ ⎟
T1
10 1 00 0 0 1
⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠2 0 0 0
0 0 0 1⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
⎛ ⎞+02R
02t
1 2 1 2 1 3 1 2 2 1
1 2 1 2 1 3 1 2 2 10 11 2
2 2 3 2 10
c c c s s c c cs c s s c s c ss c s
⎛ ⎞− + ⎟⎜ ⎟⎜ − − + ⎟⎜ ⎟⎜= ⎟⎜ + ⎟⎜ ⎟T T
2 2 3 2 100 0 0 1s c s +⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠
Basilio Bona 8ROBOTICA 03CFIOR
![Page 9: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/9.jpg)
3 1 2 2 1( )
( )
x t c c c
t
= ++ E 1
3 1 2 2 1
3 2 1
( )
( )
y t s c s
z t s
= += +
Eqn. 1
c c c s s⎛ ⎞⎟⎜ ( ) ( )t tφ1 2 1 2 102 1 2 1 2 1
c c c s s
R s c s s c
⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟= − −⎜ ⎟⎜ ⎟⎜ ⎟
1( ) ( )
( ) 2
t q t
t
φθ π
==
2 20s c
⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
E l l
2( ) ( )t q tψ =
Euler angleseqn. (2.79) page 52
Basilio Bona 9ROBOTICA 03CFIOR
![Page 10: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/10.jpg)
Knowing the Euler angles, everything will be easy. Assume we do not know them.
3 1 2 2 1( )
( )
x t c c c
y t s c s
= += +
3 1 2 2 1
3 2 1
( )
( )
y t s c s
z t s
= += + 1
2
zs
−=
3
3 1 2 2 1( )
( )
x t c c c
y t s c s
= += +
Squaring and adding
( )23 1 2 2 1
3 2 1
( )
( )
y t s c s
z t s
+= + ( )22 2 2
2 3 2x y a c+ ≡ = +
2a
c−
=2
3
c
Basilio Bona 10ROBOTICA 03CFIOR
![Page 11: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/11.jpg)
12
3
zs
−=
2 1s z −
22
ac
−=
2 12
2 2
tan qc a
= =−
3
( )( )x t c c= + ( )1
( )x tc
c=
+( )( )3 2 2 1
3 2 2 1
( )
( )
x t c c
y t c s
= +
= +
( )
( )
3 2 2
1
( )
c
y ts
+
=( )1
3 2 2c +
1 3 2 2( ) ( )tan
s cy t y tq
+= = =
11 3 2 2
tan( ) ( )
qc c x t x t+
Basilio Bona 11ROBOTICA 03CFIOR
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Linear velocities
( ) ( ) ( )( )t t t+( ) ( ) ( )( ) ( ) ( )
( )
3 1 2 2 1 1 3 1 2 2
3 1 2 2 1 1 3 1 2 2
( )
( )
x t s c s q t c s q t
y t c c c q t s s q t
= − + −
= + −
( )3 2 2( )z t c q t=
Angular velocities: analytical approach
( )1( )
( ) 0
t q t
t
φθ
=
= W ll th “ l i l iti ”
( )1
0
q t⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ω( )2
( ) 0
( )
t
t q t
θ
ψ
=
=We call these “eulerian velocities”
( )2
0E
q t
⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
ω
( )⎝ ⎠
Basilio Bona ROBOTICA 03CFIOR 12
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Analytic Jacobian (by differentiation)
3 1 2 2 1 3 1 2s c s c s⎛ ⎞− − − ⎟⎜ ⎟⎜ ⎟⎜
3 1 2 2 1 3 1 2
3 20
Lc c c s s
c
⎟⎜ ⎟= + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
J Eqn. 2a
3 2⎝ ⎠
⎛ ⎞1 0
0 0
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜J Eqn. 2b0 0
0 1A ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
J q
Basilio Bona ROBOTICA 03CFIOR 13
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Transformation matrix (see textbook)2π
θ =
1 1
1 1
0 cos sin sin 00 sin cos sin 0E
c ss c
φ φ θφ φ θ
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜= − ≡ −⎟ ⎟⎜ ⎜M
2
1 10 sin cos sin 01 0 01 0 cos
E s cφ φ θθ
≡⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜⎜ ⎝ ⎠⎝ ⎠
M
( )0 c s s qq t⎛ ⎞⎛ ⎞ ⎛ ⎞⎟⎜⎟ ⎟⎜ ⎜( )
( )
1 1 1 21
1 1 1 2
0
0 0E E
c s s qq t
s c c q⎟⎜⎟ ⎟⎜ ⎜⎟⎟ ⎟⎜⎜ ⎜⎟⎟ ⎟⎜⎜ ⎜⎟⎟ ⎟⎜= = − = −⎜ ⎜⎟⎟ ⎟⎜⎜ ⎜⎟⎟ ⎟⎜ ⎟⎜ ⎜⎟ ⎟⎜
Mω ω
( ) 121 0 0 qq t
⎟⎜ ⎜⎟ ⎟⎜ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎟⎜⎝ ⎠ ⎝ ⎠⎝ ⎠
Basilio Bona ROBOTICA 03CFIOR 14
![Page 15: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/15.jpg)
From the previous slide we have the cartesian velocity in From the previous slide we have the cartesian velocity in base RF we can now compute the Jacobian matrix
1 2 1 10 0s q s s⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎟ ⎟ ⎟⎜ ⎜ ⎜⎛ ⎞⎟ ⎟ ⎟⎜ ⎜ ⎜1 2 1 1
11 2 1 1
2
0 0
1 0 1 0A
qc q c c
⎛ ⎞⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎟⎜⎜ ⎜ ⎜⎟ ⎟ ⎟⎟⎜= − = − ⇒ = −⎜ ⎜ ⎜⎟ ⎟ ⎟⎟⎜⎜ ⎜ ⎜⎟ ⎟ ⎟⎟⎟⎜⎜ ⎜ ⎜⎟ ⎟ ⎟⎝ ⎠⎜ ⎜ ⎜⎟ ⎟ ⎟⎟ ⎟ ⎟⎜ ⎜ ⎜
Jω Eq. 3
11 0 1 0q ⎝ ⎠⎜ ⎜ ⎜⎟ ⎟ ⎟⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠ ⎝ ⎠ ⎝ ⎠
This is the geometric angular JacobianN i i diff Now we compute it in a different way
Basilio Bona ROBOTICA 03CFIOR 15
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Both joints are rotoidal therefore considering results at pageBoth joints are rotoidal, therefore, considering results at page.
1Ai i−=J k
1 1,Li i i p− −= ×J k r
First we compute the angular Jacobian
0⎛ ⎞ 0⎛ ⎞ ⎛ ⎞
1 0
0
0A
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜J k
10
2 1 1 1
0
0A
s
c
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= = = −⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜J k R
1 0
1A ⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
2 1 1 1
1 0A ⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
These two columns are equal to those in Eqn. 3
Basilio Bona 16ROBOTICA 03CFIOR
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⎛ ⎞
0 p
x
y
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟r This relation is
b d f d0,py
z⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
⎛ ⎞⎛ ⎞
obtained from direct KF – Eqn. 1
( )1 0 0 0 0
0 1 0
1 0 0L p p
x
y
⎛ ⎞⎛ ⎞− ⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= × = = ⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟J k r S k r( )1 0 0, 0 0,
0 0 0L p p
y
z⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠⎝ ⎠
⎛ ⎞ ⎛ ⎞3 1 2 2 1
3 1 2 2 1
y s c s
x c c c
⎛ ⎞ ⎛ ⎞− − −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= = +⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟3 1 2 2 1
0 0⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
Basilio Bona ROBOTICA 03CFIOR 17
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Then we compute3 2c
s
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟r
1
1, 3 2
0p
R
s= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
r
1R
Instead of transforming it in RF 0 and after making the vector product,we make the vector product in RF 1 and then we transform the result
( )2 1 1 1 1L⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤= × =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦J k r S k r
to express it in RF 0
( )1 1 1 1
2 1 1, 1 1,
3 2 3 20 1 0
L p pR R R R
c s
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎛ ⎞⎛ ⎞ ⎛ ⎞− −⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜
3 2 3 21 0 0
0 0 0 0 0
s c⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟= =⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎟ ⎟ ⎟⎜ ⎜ ⎜
1
0 0 0 0 0R
⎟ ⎟ ⎟⎜ ⎜ ⎜⎝ ⎠⎝ ⎠ ⎝ ⎠
Basilio Bona ROBOTICA 03CFIOR 18
![Page 19: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/19.jpg)
Now we transform from RF 1 to RF 0
3 2 1 1 3 2 3 1 20s c s s c s⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞− − − −⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜
0
3 2 1 1 3 2 3 1 20
2 1 3 2 1 1 3 2 3 1 20
0 0 1 0 0L R
c s c c s s
c
⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎡ ⎤ ⎟ ⎟ ⎟ ⎟= = = −⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎢ ⎥ ⎜ ⎜ ⎜ ⎜⎣ ⎦ ⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎟ ⎟ ⎟ ⎟⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜
J R
In conclusions the two Jacobians are
1 1 03 2
0 0 1 0 0R R R
c⎟ ⎟ ⎟ ⎟⎜ ⎜ ⎜ ⎜⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
In conclusions, the two Jacobians are
3 1 2 2 1 3 1 2s c s c s⎛ ⎞− − − ⎟⎜ ⎟
( )3 1 2 2 1 3 1 2
1 2 3 1 2 2 1 3 1 2
0L L L
c c c s s⎜ ⎟⎜ ⎟⎜ ⎟= = + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
J J J
It coincides with the results of Eqn. 2a
3 20 c⎜ ⎟⎟⎜⎝ ⎠
q
Basilio Bona ROBOTICA 03CFIOR 19
![Page 20: Industrial Robots - polito.it€¦ · Example 1.2 –vector representation If t t itd t th 11 11 ⎛⎞⎛⎞⎜⎜cs01− ⎟⎟⎛⎞⎜cs− ⎟ ⎜⎜⎟⎟⎜ ⎟ If vector represents](https://reader034.vdocument.in/reader034/viewer/2022050204/5f57b31727ccbb6f704dda05/html5/thumbnails/20.jpg)
Linear Jacobians are independent from the methods used to compute them(since we use always Cartesian representation)
Instead, angular Jacobians, depends on the conventions used to expressthe TCP orientation
s c s c s⎛ ⎞⎟⎜In conclusions:
3 1 2 2 1 3 1 2
3 1 2 2 1 3 1 2L
s c s c s
c c c s s
− − − ⎟⎜ ⎟⎜ ⎟⎜ ⎟= + −⎜ ⎟⎜ ⎟⎜ ⎟J
3 20 c
⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
1 0
0 1
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟J1
0
0
s
c
⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⎜ ⎟J0 1
0 0A= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
J1
0
1 0A
c= −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠
J
Analytical Jacobian Geometric Jacobian
Basilio Bona 20ROBOTICA 03CFIOR