1. 1 Singularity Handling on PUMA in Operational Space Formulation Author: Denny Oetomo*, Marcelo Ang Jr*, Lim Ser Yong** * National University of Singapore, ** Gintic Institute of Manufacturing Technology ISER 2000, Honolulu, HI, Dec 12, 2000

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3. 3 Problem Statement Singularity Motion across singularities –increased usable workspace for task execution Traditional methods –motion-based –forcing Jacobian to be non-singular, etc Task (operational) space methods –Motion/Forces at End-Effector are directly controlled via Joint Torques

4. 4 Task-Based Control Operational space Force on hand/tool –virtual - to cause motion –real - actual forces exerted Singularity handling needs to be in both motion and force control Actuation signals to robot joints computed to effect forces

5. 5 Operational Space vs Inverse Kinematics Differential Motion Operational Space Formulation [Chang and Khatib, 1994]

6. 6 Work Done Analysis and resolution of singularities in operational space Remove degenerate directions –lower order (in terms of task space) non-singular but redundant mechanism Graceful escape algorithms using null space motion –stable and smooth motions from singular to non- singular regions Experimental verification

7. 7 ISER 2000: Singularity Handling on Puma in Operational Space Formulation Handling of Singularity Problem: existence of the inverse of the Jacobian matrix at singular configuration. Cause: Manipulator loses DOF(s). Jacobian is not full rank. There is a degenerate direction. Proposed Solution: Remove the degenerate component(s) from the Jacobian and collapse the Jacobian into a matrix of smaller dimension but with full rank.

8. 8 Singularities in PUMA 500 series: Wrist Elbow Head Z1Z1 X1X1 Y1Y1 Z2Z2 X2X2 Z3Z3 X3X3 Y3Y3 Z 4, Z 6 Y 4, Z 5 X 4, X 5 d2 d4 d3 a2 PUMA Singularities Det(J) = a 2 (d 4 C 3 - a 3 S 3 ) (d 4 S 23 + a 2 C 2 + a 3 C 23 ) S 5

9. 9 Jacobian in Frame 0, 0 J Z1Z1 X1X1 Head Singularity 1 J = 0 at head singularity Remove 2nd row

10. 10 Elbow Singularity Wrist point d2 d3 XBXB ZBZB Degenerate direction d4 a2 

11. 11 Wrist Singularity q4q4 q5q5 q6q6 Z4 Y4 X4

12. 12 ISER 2000: Singularity Handling on Puma in Operational Space Formulation The forces (and the Jacobian) is resolved into the frame which one of the axis represents the singular direction. The row of the Jacobian matrix that contains all zero is then removed. Collapsing the Jacobian q4q4 q5q5 q6q6 Z4 Y4 X4 Wrist point d2 d3 XBXB ZBZB Degenerate direction d4 a2  Zo Xo Wrist Elbow Head

13. 13 Four sets of result were collected, consisting of the position and orientation (tracking) error in: 1.PUMA tracing a non-singular trajectory 2.PUMA going through wrist singularity, not in the singular direction. 3.PUMA escaping from wrist singularity into a path in singular direction 4.PUMA escaping from elbow singularity into a path in singular direction Experimental Sets

14. 14 Escape into Singular Direction Utilising Null Space Motion: Type 1: Null Space Motion creates motion in singular direction Joint 3 Desired path Desired path and non-feasible path The initially non-feasible direction Type 2: Null Space Motion creates internal motion which shifts the singular direction.

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18. 18 Polishing Application

19. 19 Conclusions The singularity handling algorithm implemented. –By removing the singular component in operational space Graceful escape algorithms using null space motion –stable and smooth motions from singular to non- singular regions Experimental Verification

20. 20 Future Work This is one of the ‘infra structure’ of a larger project. (further work would be done on the larger project). Extension of the work into inherently redundant robots.

21. 21 ISER 2000: Singularity Handling on Puma in Operational Space Formulation Removing the Singular Component(s) Example on Head singularity: The resulting Jacobian: 1 J =