Orienting Polygonal Parts without Sensors

1. Orienting Polygonal Parts without Sensors Kenneth Y. Goldberg Presented by Alan Chen

2. Outline • Background • Assumptions • Algorithm: Grasper • Push – Grasp • Proof • Discussion/Conclusion

3. Part Feeder • Hardware vs. Software • Redesigned vs. Reprogrammed • No sensors

4. Exact motion planning Compliant motion planning Open loop, sensorless Corner example Preimage backchaining Mechanical part feeders Vibrations Bowl feeder Bad for fragile parts Previous Work • Fence • Grasper/Pusher

5. Grasper • Squeeze action • No sensors

6. Assumptions • All motions in plane and inertial forces are negligible • Gripper consists of two parallel linear jaws • Gripper motion orthogonal to jaws • Convex hull treated as rigid planar polygon • Part is isolated • Part’s initial position is constrained within jaws • * Jaws make contact simultaneously • Once contact made, surfaces remain in contact • Zero friction between part and jaws

7. Q Definitions • Diameter function q s() • Squeeze function 

8. Symmetry • S(q+T) = s(q)+T; T period • T= 2p / r(1+(r mod 2)) • r - rotational symmetry • r = 1 T = p: no symmetry • r = 3 T = p/3: equilateral triangle • r = 4 T = p/2: square

9. Parts Feeding Problem • “Given a list of n vertices describing the convex hull of a polygonal part, find the shortest sequence of squeeze actions guaranteed to orient the part up to symmetry.” (Goldberg, 11)

10. Algorithm • Compute the squeeze function • Find widest step in the squeeze function and set Q1 = corresponding s-interval • While there exists |s(Q)| < |Qi| • Set Qi+1 = widest s-interval • Increment i • Return the list (Q1,Q2…) • Continue until |Qi| = T

11. a=atan2(3,2) a 2 3 s() = = =  0   +a +a 3/2 3/2 2-a 2-a 0 0 -a -a a a a a

12. Given  plans Recovering the Plan for j between 1 ~ A plan

13. s()  Q1 Q2

14. Push-Grasp vs Grasp • Jaws do not contact simultaneously • Radius function • Push function

16. Correctness • Plan will orient the part up to symmetry • r2 = (a1,a2) • No shorter plan orients up to symmetry Compare plans rj’ vs ri where j < i  |Qj’| > |Qj| Algorithm  |Q1’| < |Q1|  |Qk’| < |Qk| & |Qk+1’| > |Qk+1| Cannot happen so no shorter plan

17. Completeness Theorem – For any polygonal part, we can always find a plan to orient the part up to symmetry h – measure of some s-interval

18. Complexity • For a polygon of n sides, the algorithm runs in time O(n2 log n) and finds plans of length O(n2) • Compute squeeze function in O(n) • Step 2 takes O(n) time • Squeeze function defines O(n2) s-intervals • Traverse list once  I is O(n2) • Sorting  O(n2 log n)

19. Discussions • Plan verified experimentally • Carnegie Mellon University using a PUMA robot w/ electric LORD Co. gripper • USC using an IBM robot w/ pneumatic Robotics and Automation Corp. gripper • Occasionally failed: not enough pushing distance • Practicality • Feed rate: i actions with 1 gripper vs. i grippers performing 1 action • 1 D.O.F. not constrained • Limited to flat 2-D polygons • Working on curved edges • 3-D orient the part so its sitting on its most stable face before grasping

20. Conclusions • Algorithm that rapidly analyzes part geometry • Sensorless and easily reprogrammable • Complete and correct • Not practical for industry