2. Vision Based Motion Control CMPUT 610 2001 Martin Jagersand

3. How to go from Visual sensation to Motor action? Camera -> Robot coordRobot -> Object

4. Closed loop servoing We focus on the geometric transforms EE

5. Lots of possible coordinates Camera – Center of projection – Different models Robot – Base frame – End-effector frame – Object

6. Coordinate rotation Example: Around y-axis X’ Z’ X P

7. Euler angles Note: Successive rotations. Order matters.

8. Rotation and translation Translation t’ in new o’ coordinates X’ Z’ X P

9. Example Matlab robot Successive translation and rotation % robocop Simulates a 3 joint robot function Jpos = robocop(theta1,theta2,theta3,L1,L 2,L3,P0) Rxy1 = [cos(theta1) sin(theta1) 0 -sin(theta1) cos(theta1) 0 00 1]; Rxz2 = [cos(theta2) 0 sin(theta2) 0 1 0 -sin(theta2) 0 cos(theta2)]; Rxz3 = [cos(theta3) 0 sin(theta3) 0 1 0 -sin(theta3) 0 cos(theta3)]; P1 = P0 + Rxy1*[L1 0 0]'; P2 = P1 + Rxy1*Rxz2*[L2 0 0]'; P3 = P2 + Rxy1*Rxz2*Rxz3*[L3 0 0]'; Jpos = [P0 P1 P2 P3];

10. Homogeneous coordinates Write as matrix multiplications only 3-vectors -> 4 vectors Affine -> homogeneous

11. Denavit-Hartenberg Particular choice of homogeneous parameterization, see eq. 2.8 in Alexa’s thesis

12. Perspective Camera In homogeneous 4-vector Remove 3 rd row for standard camera plane proj

13. Hand-Eye system Motor-Visual function: y=f(x)