1. Singularity-Robust Task Priority Redundancy Resolution for Real-time Kinematic Control of Robot Manipulators Stefano Chiaverini

2. Outline 1.Redundancy 2. Inverse differential kinematic control 3. Robust techniques for kinematic control 4. Task-priority redundancy resolution 5. Numerical analysis of existing solutions.

3. Redundancy N > M Degrees of Freedom Workspace

4. Redundancy Example: 7 DOF arm vs. 6 DOF Trajectory

5. Redundancy Many solutions per problem.

6. Redundancy Which one to pick?

7. Inverse Differential Kinematic Control: A redundant system!

8. Kinematic Control Gives an end-effector velocity which minimizes joint velocities. Jacobian Pseudo-Inverse :

9. Kinematic Control Generalized: Min NormNull space Arbitrary x Null Span

10. Kinematic Control Problem: singularites Min Norm Null space

11. Singularities Kinematic Singularities Occur when Jacobian has null singular values Inherent to all techniques. Algorithmic Singularities Occur when no solution exists which satisfies constraints and provides desired EE motion.

12. Singularities How to deal with kinematic singularities?

13. Robust Techniques First Technique: Damping Damping term From SVD

14. Robust Techniques *** First Technique: Damping Damped Norm Null Space

15. Robust Techniques Pros Good performance Robust to singularities Cons Accumulates error First Technique: Damping

16. Robust Techniques How to deal with error? Filter out singularities.

17. How to choose q0? Consequence of redundancy

18. Task Priority Redundancy Resolution How to choose q0? Minimizes an objective function H(q)

19. Task Priority Redundancy Resolution Or, define a task-space constraint: Complicates the update rule:

20. Task Priority Redundancy Resolution Task space constraint: Can satisfy (N – M) parameters

21. Task Priority Redundancy Resolution Problem: Leads to Algorithmic Singularities Might approach singular

22. Task Priority Redundancy Resolution Condition for algorithmic singularities Null spaces linearly dependant

23. Task Priority Redundancy Resolution  Algorithmic Singularities  Difficult to predict.  Arise because of competing demands.  Leads to extreme joint velocities. How do we get rid of them?

24. Task Priority Redundancy Resolution Solve for the constraint instead: A lot of math later…. Zeros out at singularities…

25. Numerical Analysis For a 7-dof manipulator: Exact solution (796 flops): Chiaverni’s robust simplification (632 flops): Naïve minimum norm (519 flops):

26. Numerical Analysis Experiments:

27. Numerical Analysis Constraint:

29. (damped)

30. (undamped)

31. (damped)

32. (undamped)

33. Conclusion  Need to avoid two kinds of singularities.  Presented approach which handles both.  New approach is more computationally efficient.

34. Limitations  Null space motions only  Constraints always prioritized lower