Download - Transformations and Euler Angles
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Homogeneous Transformations A 4x4 matrix that represents general
transformations.
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Transformation Equations Consider the following
transformations: Notice that {D} can be
expressed in two different ways.
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Transformation Equations cont…
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Transformation Equations cont… A typical use of transformation equations:
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More on Rotation Matrices
Rotation matrices can also be represented as follows. The significance is that only 3 parameters are
needed to specify an orientation … even though a rotation matrix has 9 values.
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More on Rotation Matrices
There are 6 dependencies in a rotation matrix.
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More on Rotation Matrices For example, given the following homogeneous
transformation, find the missing values.
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Representing A General Orientation A rotation matrix that represents any possible
orientation can be created by multiplying 3 pure axis rotation matrices together. Also, recall that matrix multiplication don’t usually
commute.
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X – Y – Z Fixed Angles
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X – Y – Z Fixed Angles cont…
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X – Y – Z Fixed Angles cont…
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X – Y – Z Fixed Angles cont… The inverse problem:
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Z – Y – X Euler Angles Rotations are performed around the moving axes:
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Z – Y – X Euler Angles cont…
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Z–Y–X Euler versus X-Y-Z Fixed They are the same… Conceptually the
difference is in the order that the matrix multiplications are made…
Do example: RXYZ(90,90,90)
RZ’Y’X’(90,90,90)
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Z – Y – Z Euler Angles
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Z – Y – Z Euler Angles
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Other Conventions 12 conventions for Fixed Angle, 12 for Euler
Angle. Because of duality, 12 unique conventions in
total. 3 x 2 x 2 combinations:
X Y Z X Y X X Z X …
Staubli’s V+ Machines use Z-Y-Z Euler Staubli’s VAL3 Machines use X-Y-Z Euler Fanuc TPP Machines use X-Y-Z Fixed
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All combinations