1. 1 Orienting Polygonal Parts without Sensors Author: Kenneth Goldberg Presented by Alan Schoen and Haomiao Huang

2. 2 Outline Problem statement Motivation Mechanical Design and Assumptions Algorithm Proofs Extension Conclusions

3. 3 Problem Statement Given a planer polygonal part with unknown initial orientation find the shortest sequence of mechanical gripper actions which will orient the part up to the symmetry of its convex hull

4. 4 Motivation Injection molding and other manufacturing processes produce parts in unknown orientations Purely mechanical systems are: –Difficult to Modify –Designed by trial and error A versatile mechanical design coupled with a good algorithm would help

5. 5 Mechanical Design Parallel sided gripper is proposed for versatility

6. 6 Assumptions Quasi-static Parallel gripper jaws Gripper motion orthogonal to jaws Convex hull of the part is a 2D polygon One part at a time Both jaws contact simultaneously Jaws close until doing so would deform part Zero friction between part and jaws

7. 7 Diameter Function Diameter function: d: S1  R

8. 8 Squeeze Function Squeeze function s: S1  S1 s-interval is a semi- closed interval [a,b) in the domain of s Θx is an s-interval. θx is the included bound s(Θ) = {s(θ)| θ in Θ} is the s-image

9. 9 Algorithm 1.Compute the Squeeze function 2.Find the widest step and call it Θ 1, Let i=1 3.While there is Θ such that |s(Θ)| < |Θ i | 1.Set Θ i+1 equal to the widest interval 2.Increment i 4.Return the list (Θ 1, Θ 2, …, Θ i )

10. 10 Algorithm 1.Compute the Squeeze function 2.Find the widest step and call it Θ 1, Let i=1 3.While there is Θ such that |s(Θ)| < |Θ i | 1.Set Θ i+1 equal to the widest interval 2.Increment i 4.Return the list (Θ 1, Θ 2, …, Θ i )

11. 11 Recovery of a Plan Call the i-step plan p i p i = (a i, a i-1, …, a 1 ) Set a i = 0 for j = i – 1 to 1 a j = s(θ j-1 ) – θ j – e j + a j+1 With e j = ½(|Θ j | - |s(Θ j+1 )|)

12. 12 Conceptual Strategy Create d and s Find organized list of Θ intervals Use the fact that Θ 1 is the largest interval to create a sequence of squeezes which collapses all orientations on Θ j to s(Θ 1 )

13. 13 Correctness Shortest plan: –Algorithm generates plan in i steps –Assume there exists a shorter plan in j steps –By definition, –j-plan: –Then at some point – But plan is monotonic in intervals 1 st Interval2 nd Interval

14. 14 Completeness & Complexity Completeness: For any polygonal part, we can always find a plan to orient up to symmetry –Can always find larger intervals until we reach a period of symmetry Complexity: –O(n^2 log n) algorithm –O(n^2) execution

15. 15 Push-Grasp Same idea as before, but including a single rotation from push action

16. 16 Push- Grasp

17. 17 Conclusions Advantages: –Adaptable to any polygonal part –No Sensing –Complete & Correct Disadvantages: –Only up to symmetry of convex hull –Fixed plan lengths & execution time –Polygonal parts only at this time –Slow mechanical implementation