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Then for each arm the angle is chosen so it is inside the angle constraints for the robot. This
will convey to the first configuration, upper left corner in Figure 4.6.
4.5 VELOCITY KINEMATICS
The Jacobian matrix specifies a mapping from velocities in joint space to velocities in Cartesian
space. The Jacobian determines a linearized matrix of the first derivatives.
To calculate the Jacobian matrix for a Delta-3 robot one can use a set of constraint equations
linking the Cartesian space variables to the joint space variables (7). The three constraints
equations for the Delta-3 robot can be chosen as
2
Pc P i bi
2
B
2
− l = 0 i = 1,
2,
3
Eq. 4.3
assuming that the length of the forearms is constant.
Let si denote the vector PC P i b then can the Euclidean norm be written as s s
i
i
T
i . Consider Figure
4.2 for the following calculations. The vector si can be written as
⎡xn
⎤ ⎛⎡R⎤
⎡lA
cos( θi)
⎤⎞
⎜
⎟
R
= O ( )
⎢ ⎥
i R
⎢
⎜ 0
⎥ ⎢
0
⎥
iPb
− O P + P P =
⎢
y
⎥
−
⎢ ⎥
+
⎢ ⎥⎟
i = 1,
2,
3
i i Ai
Ai
C n
⎢ ⎥ ⎜
0 sin( )
⎟
⎣z
⎦ ⎝
⎢⎣
⎥⎦
⎢⎣
l ⎥
n
A θi
⎦⎠
s z
i i
Eq. 4.4
To introduce the first time derivative in joint space and in Cartesian space which the Jacobian
matrix maps between one can take the time derivative of
which gives
T 2
s s − = 0 i = 1,
2,
3
i
i
l B
Eq. 4.5
T T
s s&
+ s&
s = 0 i = 1,
2,
3
i
i
i
i
Eq. 4.6
With help of commutativity property of the product the Eq. 4.6 can be written as
s T
i i
s&
= 0 i = 1,
2,
3
Eq. 4.7
where the first time derivative out of si is given by
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Public Report ELAU GmbH, Marktheidenfeld
19