Recall our discussion of ZYX Euler angles

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Recall our discussion of Z-Y-X Euler angles
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Consider the A and B frames shown below.
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Inverse of a homogeneous transformation matrix
Review
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Inverse of a homogeneous transformation matrix
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Inverse of a homogeneous transformation matrix
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Inverse of a homogeneous transformation matrix
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Inverse of a homogeneous transformation matrix
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Inverse of a homogeneous transformation matrix
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Inverse
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Inverse
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Take transpose of rotation matrix.
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Take transpose of rotation matrix.
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Take transpose of rotation matrix.
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Reverse displacement vector.
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Reverse displacement vector.
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Reverse displacement vector.
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Refer reversed displacement vector to B frame
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Refer reversed displacement vector to B frame
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Refer reversed displacement vector to B frame
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Refer reversed displacement vector to B frame
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Returning for a moment to our Euler angles a, b,
g
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In the notation of Craig, this rotation matrix
(often called direction-cosine matrix) is
denoted by
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In the notation of Craig, this rotation matrix
(often called direction-cosine matrix) is
denoted by
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In the notation of Craig, this rotation matrix
(often called direction-cosine matrix) is
denoted by
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This direction-cosine matrix is an important
component of the homogeneous-transformation
matrix of our forward kinematics.
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Recall that we moved the stationary A frame over
to the moving E frame to illustrate the meaning
of the elements of this rotation matrix.
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Recall that we moved the stationary A frame over
to the moving E frame to illustrate the meaning
of the elements of this rotation matrix,
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Thus, 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 .
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Is there a systematic way, for a general
(holonomic) robot, to build in terms
of q1, q2 ?
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Is there a systematic way, for a general
(holonomic) robot, to build in terms
of q1, q2 ?
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Is there a systematic way, for a generic link
of a robot, to build in terms of
qi ?
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Call this link i-1.
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We apply to link i-1 the widely used
Denavit- Hartenberg convention.
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Denavit Hartenberg parameters
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Denavit Hartenberg parameters
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Denavit Hartenberg parameters
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Three constants
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and one variable.
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D-H Example
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D-H Example
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D-H Example Puma 560
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D-H Example Puma 560
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D-H Example Puma 560
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First three rotations of Puma
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First three rotations of Puma
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i-11
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i-11
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The first rotation q1 occurs about the Z1 axis.
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The second rotation q2 occurs about the Z2 axis.
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However, the Z2 axis and the Z1 axis intersect
one another.
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Therefore the X1 axis may be oriented arbitrarily.
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Therefore the X1 axis may be oriented arbitrarily.
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Since the two frames share their origin, a1d20
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Since the two frames share their origin, a1d20
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But what about a1?
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But what about a1?
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But what about a1?