Transformations and Euler Angles

Sebastian van Delden

USC Upstate

svandelden@uscupstate.edu

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…

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…

X – Y – Z Fixed Angles cont…

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: RXYZ(90,90,90)

RZ’Y’X’(90,90,90)

Z – Y – Z Euler Angles

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