ZYX Euler Angles

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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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By following in parallel with Craigs discussion
of ZYX Euler angles,determine the counterpart to
Eq. 2.72, i.e. the 3x3 overall rotation
matrix,but this time using Z-X-Z Euler angles.
 Keep the notationa b g for the first, second,
third rotations, respectively.
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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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Suppose the second rotation b had instead
occurred about the original YA axis?
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Suppose the second rotation b had instead
occurred about the original YA axis?
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Suppose the second rotation had instead occurred
about the original YA axis?
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Returning to the Z-Y-X Euler Angles
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take the last rotation g to be about the XB
axis.
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By following in parallel with Craigs discussion
of ZYX Euler angles,determine the counterpart to
Eq. 2.72, i.e. the 3x3 overall rotation
matrix,but this time using Z-X-Z Euler angles.
 Keep the notationa b g for the first, second,
third rotations, respectively.
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Beginning with the A frame, rotate a positive a
about the ZA axis.
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Note the rotation matrix between A and B
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As before, we consider the intermediate B
frame.
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This time, however, the second rotation b is not
about the intermediate Y axis, but rather about
the intermediate X axis.
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Craig, problem 2.38Imagine to unit vectors v1
and v2 embedded in a rigid body. Note that, no
matter how the body is rotated, the geometric
angle between these two vectors is preserved
(i.e. rigid-body rotation is an
angle-preserving operation). Use this fact to
give a concise (four or five line) proof that the
inverse of a rotation matrix must equal its
transpose and that a rotation matrix is
orthonormal.