1. Tao Zhang Gordon Smith Ken Goldberg ALPHA Lab, UC Berkeley The Toppling Graph: Designing Pin Sequence for Part Feeding Robert-Paul Berretty Mark Overmars CS Dept., Utrecht Univ.

2. The Problem:

3. Pin Sequence Design: An Example

4. The Solution: Designing Pin Sequence

5. Related Work Lozano-Perez [86]: part feeding as a dual of motion planning Erdmann and Mason [88]: sensorless manipulation Trinkle [92]: orienting parts in the vertical plane using gripper Goldberg [93]: orienting parts in the horizontal plane using gripper

6. Fences over conveyor belts: –Peshkin and Sanderson [88]: a numerical search algorithm –Akella et al. [97]: 1-JOC analysis –Berretty et al. [97]: a polynomial-time algorithm –Wiegley et al. [98]: a complete algorithm Toppling manipulation –Lynch [99]: toppling analysis –Zhang et al. [00]: compensatory grasping

7. Compute critical pin heights Approach Plan pin sequence

8. Critical Pin Heights Toppling Graph

9. Radius function R(  ): height of the COM as the part rotates Vertex height functions V i (  ): height of vertex i as the part rotates Functions 

10. Rolling Height Function X Z

11. 1 w il (  ) = (2  t z i cos  il –  cos(  il -  ) –  cos il - 2  t x i sin  il +  t  sin(  il -  ) +  t  sin il ) / (2  t sin(  il -  i )), H il (  ) = x i sin  + z i cos  + w il sin(  i +  ). Rolling Height Function: Computation

12. Rolling Height Function: Graph h =c=c cc h  =0

13. Jamming Height Functions X Z

14. Toppling Graph B BBB

15. Physical Experiment using an Adept Flex Feeder conveyor belt;  t = 53   2  and  p = 5   2 . Comparison of prediction with experiment.

16. Pin Planning

17. Pin Planning (Cont.) Total running time: O(n 3n ) in the worst case n n -1 1 O(n3)O(n3) O(n3)O(n3) O(n3)O(n3)

18. Conclusion Toppling Graph –Vertex height functions –Rolling height functions –Jamming height functions Pin sequence planning

19. Future Work Optimal gripper jaw design -- topple parts by a set of pins

20. Future Work(Cont.)