FORWARD KINEMATICS: THE DENAVIT-HARTENBERG ...

84CHAPTER 3. FORWARD KINEMATICS: THE DENAVIT-HARTENBERG CONVENTION
Table 3.1: Link parameters for 2-link planar manipulator.
determined from (3.10) as
The T-matrices are thus given by
Link ai αi di θi
1 a1 0 0 θ ∗ 1
2 a2 0 0 θ ∗ 2
∗ variable
A1 =
⎡
c1
⎢ s1
⎢
⎣ 0
−s1
c1
0
⎤
0 a1c1
0
⎥
a1s1 ⎥
⎥.
1 0 ⎦
(3.22)
A2 =
0
⎡
c2
⎢ s2
⎢
⎣ 0
0
−s2
c2
0
0 1
⎤
0 a2c2
0
⎥
a2s2 ⎥
1 0 ⎦
(3.23)
0 0 0 1
T 0
1 = A1. (3.24)
⎡
⎤
T 0
2
c12
⎢ s12
= A1A2 = ⎢
⎣ 0
−s12
c12
0
0 a1c1 + a2c12
0 a1s1 + a2s12
1 0
0 0 0 1
⎥
⎥.
(3.25)
⎦
Notice that the first two entries of the last column of T 0
2 are the x and y
components of the origin O 2 in the base frame; that is,
x = a1c1 + a2c12
y = a1s1 + a2s12
(3.26)
are the coordinates of the end-effector in the base frame. The rotational part
of T 0
2 gives the orientation of the frame o2x2y2z2 relative to the base frame.
⋄
Example 3.2 Three-Link Cylindrical Robot
Consider now the three-link cylindrical robot represented symbolically by
Figure 3.7. We establish O 0 as shown at joint 1. Note that the placement of