A systematic way to develop the 4x4 matrix Tq'

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Lecture 11
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By substituting back
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This product is the matrix required for problem 2
of HW4.
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This matrix represents the robots forward
kinematics
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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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Reverse displacement vector.
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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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Z-Y-X Euler Angles
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Z-Y-X Euler Angles

• - Just three numbers are needed to specify the
  orientation of one set of axes relative to
  another.

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Z-Y-X Euler Angles

• Just three numbers are needed to specify the
  orientation of one set of axes relative to
  another.
• One possible set of these numbers is the Z-Y-X
  Euler angles

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Consider the A and B frames shown below.
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How can we define just three quantities from
which we can express all nine elements of the
rotation matrix that defines the relative
orientations of these frames?
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Beginning with the A frame, rotate a positive a
about the ZA axis.
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Call this new frame B
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Note the rotation matrix between A and B
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Note the rotation matrix between A and B
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Note the rotation matrix between A and B
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Note the rotation matrix between A and B
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Note the rotation matrix between A and B
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Next consider just the intermediate B frame.
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Consider a positive rotation b about the YB axis.
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take the last rotation g to be about the XB
axis.
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Is there a systematic way to build ?
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Member i-1
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Denevit Hartenberg parameters
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Denevit Hartenberg parameters
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Denevit 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?