1. Transformations and Euler Angles Sebastian van Delden USC Upstate svandelden@uscupstate.edu

2. Homogeneous Transformations • A 4x4 matrix that represents general transformations.

3. Transformation Equations • Consider the following transformations: • Notice that {D} can be expressed in two different ways.

4. Transformation Equations cont…

5. Transformation Equations cont… • A typical use of transformation equations:

6. 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.

7. More on Rotation Matrices • There are 6 dependencies in a rotation matrix.

8. More on Rotation Matrices • For example, given the following homogeneous transformation, find the missing values.

9. 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.

10. X – Y – Z Fixed Angles

11. X – Y – Z Fixed Angles cont…

12. X – Y – Z Fixed Angles cont…

13. X – Y – Z Fixed Angles cont… • The inverse problem:

14. Z – Y – X Euler Angles • Rotations are performed around the moving axes:

15. Z – Y – X Euler Angles cont…

16. 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)

17. Z – Y – Z Euler Angles

18. Z – Y – Z Euler Angles

19. 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

20. All combinations