Transformations and Euler Angles Sebastian van Delden USC Upstate
Homogeneous Transformations A 4x4 matrix that represents general transformations.
Transformation Equations Consider the following transformations: Notice that {D} can be expressed in two different ways.
Transformation Equations cont…
A typical use of transformation equations:
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.
More on Rotation Matrices There are 6 dependencies in a rotation matrix.
More on Rotation Matrices For example, given the following homogeneous transformation, find the missing values.
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.
X – Y – Z Fixed Angles
X – Y – Z Fixed Angles cont…
The inverse problem:
Z – Y – X Euler Angles Rotations are performed around the moving axes:
Z – Y – X Euler Angles cont…
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: R XYZ (90,90,90) R Z’Y’X’ (90,90,90)
Z – Y – Z Euler Angles
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
All combinations