Representation of scene motion Stefano Soatto, UCLA

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

Inner and cross product ICRA 2003

Cross product, skew-symmetry and vee ICRA 2003

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

Representation of rigid motions ICRA 2003

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

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

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

Canonical exponential coordinates (cont.) ICRA 2003

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

General rigid motion (R, T) ICRA 2003

Homogeneous representation Points Vectors Transformation representation ICRA 2003

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

Exponential coordinates ICRA 2003

Rodrigues’ formula for rigid body motion ICRA 2003

Summary ICRA 2003