1. EE 4315 / EE 5325 Robotics Lecture 11 February 25, 2015 Spring 2015 Indika Wijayasinghe & Dan Popa 1

2. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Analytical Jacobian 2 Indika Wijayasinghe & Dan Popa, EE4315/EE5325, Spring 2015

3. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Relationship b/w Geometric and Analytical Jacobian 3

4. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Relationship b/w Geometric and Analytical Jacobian 4

5. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Angular Velocity of Euler Angles ZYZ about Base Frame 55 By combining three components,

6. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Euler Angles 6 ZYZ Euler Angle Transformation

7. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Example – Elbow Manipulator 7 aiai 10d1d1 2a2a2 00 3a3a3 00

8. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Ex: Elbow Manipulator – Geometric Jacobian 8

9. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Ex: Elbow Manipulator – Analytical Jacobian 9

10. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Ex: Elbow Manipulator – Analytical Jacobian 10

11. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Singularities 11

12. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Singularities Singularities represent configurations from which certain directions of motion may be unattainable. At singularities, bounded end-effector velocities may correspond to unbounded joint velocities. At singularities, bounded end-effector forces and torques may correspond to unbounded joint torques. Singularities often correspond to points on the boundary of the manipulator workspace, that is, to points of maximum reach of the manipulator. Singularities correspond to points in the manipulator workspace that may be unreachable under small perturbations of the link parameters, such as length, offset, etc. 12

13. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Singularities 13

14. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Decoupling of Singularities 14 arm wrist Linear velocity Angular velocity armwrist

15. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Wrist Singularities 15

16. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Arm Singularities 16

17. Indika Wijayasinghe & Dan Popa, Robotics EE4315 / EE5325, Spring 2015 Arm Singularities 17