1. Representation of scene motion
Stefano Soatto, UCLA

2. Notation Points and vectors are different! Source of confusion!
ICRA 2003

3. Inner and cross product
ICRA 2003

4. Cross product, skew-symmetry and vee
ICRA 2003

5. Rigid body motion, Euclidean Transform.
A transformation that preserves distances, angles and orientation
Preserves norm, inner product, cross product
Polarization identity
Preserve norm and cross product
ICRA 2003

6. Representation of rigid motions
ICRA 2003

7. Rotations Orthogonal change of coordinates Collect coordinates of one
reference frame relative to the other
into a matrix R
ICRA 2003

8. Rotations (contd.)
How does a rotation transform points?
ICRA 2003

9. Canonical exponential coordinates
SO(3) has 3 DOF, but is represented with 9 numbers?
i.e. it is skew-symmetric!
ICRA 2003

10. Canonical exponential coordinates (cont.)
ICRA 2003

11. Rodrigues’ formula Beautiful!
Toss Euler angles, Y-P-R, quaternions etc.
ICRA 2003

12. General rigid motion (R, T)
ICRA 2003

13. Homogeneous representation
Points
Vectors
Transformation
representation
ICRA 2003

14. What is the action of a rigid body motion on a vector?
Questions:
What is the action of a rigid body motion on a vector?
What is the representation of the inverse action?
ICRA 2003

15. Exponential coordinates
ICRA 2003

16. Rodrigues’ formula for rigid body motion
ICRA 2003

17. Summary
ICRA 2003