1. Sampling Strategies for Narrow Passages Presented by Irena Pashchenko CS326A, Winter 2004

2. Motivation Two major costs of PRMs: FREE - Check if sample point is in free space JOIN – Check if path between milestones is in free space JOIN is 10 to 100 times slower than FREE Idea: selectively pick milestones Try more samples ( n t ) Keep fewer samples ( n a ) by filtering out non-promising samples Filtering occurs before creating edges Running time: T = n t T t + n a T a n t – milestones tried T t << T a n a – milestones added to graph

3. Two Similar Approaches The Gaussian Sampling Strategy for PRMs Valerie Boor, Mark H. Overmars, A. Frank van der Stappen ICRA 1999 The Bridge Test for Sampling Narrow Passages with PRMs David Hsu, Tingting Jiang, John Reit, Zheng Sun ICRA 2003

4. Overview Gaussian Strategy Goals Two Proposed Algorithms Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

5. Goal More samples in hard regions Sampling density of each point = Conv ( N(c,  ), Obstacles) = more samples near obstacles

6. Parameters, Mixing Methods  indicates how close configurations are to obstacles  /2   s.t. most configs are at length of the robot Hybrid strategy: mix uniform sampling w/ Gaussian

7. Algorithm I loop c 1 = random config. d= distance sampled from Gaussian c 2 = random config. distance d from c 1 if Free(c 1 ) and !Free(c 2 ), add c 1 to graph if Free(c 2 ) and !Free(c 1 ), add c 2 to graph Intuition: Pick free points near blocked points Avoid adding configurations in large empty regions hence the name

8. Algorithm II Motivated by the case of convex obstacles: Difficult places lie close to at least two obstacles loop c 1 = random config. d 1,d 2 = distances sampled from Gaussian c 2,c 3 = random configs distance d 1,d 2 from c 1 if Free(c 1 ) and !Free(c 2 ) and !Free(c 3 ), add c 1 if !Free(c 1 ) and Free(c 2 ) and !Free(c 3 ), add c 2 if !Free(c 1 ) and !Free(c 2 ) and Free(c 3 ), add c 3

9. Narrow Passage uniform sampling took 60 times longer than algorithm 1

10. Narrow Passage uniform sampling took less time than algorithm II

11. Difficult Twist uniform sampling took 13 times longer than algorithm 1

12. Twisty Track uniform sampling took 4 times longer than algorithm 1

13. Is Gaussian Sampling Appropriate for Finding Narrow Passages? In n-dimensional configuration space, the obstacle boundaries still form an (n-1)-D subspace!

14. Overview Gaussian Strategy Goals Two Proposed Algorithms Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

15. Bridge Test loop c 1 = random config. if Free(c 1 ), continue (restart the loop) d= distance sampled from Gaussian c 2 = random config. distance d from c 1 if Free(c 2 ), continue (restart the loop) p = midpoint(c 1,c 2 ) if Free(p), add p c1c1 p c2c2

16. Bridge vs. Gaussian Paper mentions Gaussian but no comparison Want to compare: Expected # of calls to free (lower is better) Expected # points generated (higher better) If points can be reused in a hybrid strategy Quality of sampled points Let p be prior probability of Free

17. Bridge vs. Gaussian StrategyCalls to Free per iter Samples per iter Reuse Points Point Quality Gaussian 1 2 2  p  (1-p) yesOK Gaussian 2 3 - p 2 3  p  (1-p) 2 yesBetter Bridge 1 + (1-p) + (1-p) 2 p  (1-p) 2 yesBest Smallest

18. Clover

19. Two Squares RBB SamplerGaussian Sampler

20. Depression, Zigzags

21. Bridge vs. Uniform RBB = Bridge

22. Combined Algorithm (JCL) Loop until N milestones have been generated Pick a configuration q1 uniformly at random in C and a configuration q2 uniformly at random in a neighborhood of q1 If q1 and q2 are both in collision then If (q1+q2)/2 is collision-free, retain it as a milestone Else if q1 (or q2) is in collision Then retain q2 (or q1) as a milestone with high probability p’ Else retain q1 or q2 as a milestone with low probability p

23. Conclusion Better configurations = fewer configurations = less edge computations = faster running time Gaussian Points near obstacles Points near two obstacles Bridge Points between parts of obstacles Still un-tested in high-dimensional spaces