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Transformations and Euler Angles

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:

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Transformations and Euler Angles

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  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

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