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where
&
⎡x&
n ⎤ ⎡− lA
sin( θi
) ⎤
R
=
⎢ ⎥
i R
⎢
0
⎥
⎢
y&
−
&
n ⎥ ⎢ ⎥
θ = X&
− b &
i n iθi
i = 1,
2,
3
⎢⎣
z&
⎥⎦
⎢⎣
l cos( ) ⎥
n
A θi
⎦
si z
For one robot arm the Eq. 4.7 can be written as
Eq. 4.8
⎡− lA
sin( θi
) ⎤
R
b R
⎢
i z 0
⎥
i =
⎢ ⎥
i = 1,
2,
3
⎢⎣
l cos( ) ⎥
A θi
⎦
s
T
i
Eq. 4.9
⎡xn
⎤ ⎡0⎤
⎢ ⎥ T
s
⎢
i 0
⎥
⎢
yn
⎥
− biθi
=
⎢ ⎥
⎢⎣
z ⎥⎦
⎢⎣
0⎥
n
⎦
&
&
&
&
Eq. 4.10
i = 1,
2,
3
which can be expressed in matrix form for all of the three robot arms as
T
T ⎡s
⎤ ⎡
1 s1
1
⎢ T⎥
⎢
⎢s2
⎥ − ⎢ 0
⎢ T⎥
⎢
⎣s3
⎦ ⎣ 0
&
b
X n
s
0 ⎤ ⎡0⎤
⎥
0
⎢ ⎥
⎥θ
=
⎢
0
⎥
T
s ⎥
⎦
⎢⎣
0⎥
3 3 ⎦
&
b
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Public Report ELAU GmbH, Marktheidenfeld
20
0
T
2
b
0
2
Eq. 4.11
From Eq. 4.11 the Jacobian matrix for a Delta-3 robot is obtained by considering
where
−1
θ& & X J
n =
Eq. 4.12
T T
⎡s
⎤ ⎡ 1 s1
b
⎢ T ⎥ ⎢
J = ⎢s
2 ⎥ ⎢ 0
⎢ T ⎥ ⎢
⎣
s3
⎦ ⎣
0
1
Eq. 4.13
s
0
T
2
b
0
2
0 ⎤
⎥
0 ⎥
T
s ⎥
3b3
⎦
The Jacobian matrix J is not only depending of θ, as is usually the case for serial robots, but
also a function of the TCP position Xn, which can be calculated with the forward kinematic
model of the Delta-3 robot.