Title: Recall our discussion of ZYX Euler angles
1Recall our discussion of Z-Y-X Euler angles
2Consider the A and B frames shown below.
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11In the notation of Craig, this rotation matrix
(often called direction-cosine matrix) is
denoted by
12In the notation of Craig, this rotation matrix
(often called direction-cosine matrix) is
denoted by
13In the notation of Craig, this rotation matrix
(often called direction-cosine matrix) is
denoted by
14In the notation of Craig, this rotation matrix
(often called direction-cosine matrix) is
denoted by
15This direction-cosine matrix is an important
component of the homogeneous-transformation
matrix of our forward kinematics.
16This direction-cosine matrix is an important
component of the homogeneous-transformation
matrix of our forward kinematics.
17Recall that we moved the stationary A frame over
to the moving E frame to illustrate the meaning
of the elements of this rotation matrix.
18Recall that we moved the stationary A frame over
to the moving E frame to illustrate the meaning
of the elements of this rotation matrix.
19Recall that we moved the stationary A frame over
to the moving E frame to illustrate the meaning
of the elements of this rotation matrix,
20Thus, this direction-cosine or rotation matrix
may be expressed either in terms of the min.
number of 3 coordinate, say a, b, g, or, as with
our forward kinematics, in terms of q1, q2 .
21Is there a more systematic way to get in terms
of q1, q2 for a given robot?
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23Consider a generic link i-1.
24Is there a systematic way to build ?
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44D-H Example Puma 560
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46D-H Example Puma 560
47D-H Example Puma 560
48First three rotations of Puma
49First three rotations of Puma
50i-11
51i-11
52The first rotation q1 occurs about the Z1 axis.
53The second rotation q2 occurs about the Z2 axis.
54However, the Z2 axis and the Z1 axis intersect
one another.
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56Therefore the X1 axis may be oriented arbitrarily.
57Therefore the X1 axis may be oriented arbitrarily.
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59Since the two frames share their origin, a1d20
60Since the two frames share their origin, a1d20
61But what about a1?
62But what about a1?
63But what about a1?