1. Sampling Strategies for Narrow Passages Presented by Rahul Biswas April 21, 2003 CS326A: Motion Planning

2. Motivation Building probabilistic roadmaps is slow Two major costs: FREE - Check if points are in free space JOIN – Check if path between points in free space JOIN is 10 to 100 times slower than FREE Better points Fewer required edges Substantial speedups

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 What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

5. Overview Gaussian Strategy What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

6. What is Desired Goal:more samples in hard regions = more samples near obstacles Sampling Density of each point = Convolution(Gaussian, Obstacles) High DensityLow Density

7. Overview Gaussian Strategy What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

8. Proposed 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 saves time but not essential hence the name

9. Proposed Algorithm II 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 saves time but not essential

10. Overview Gaussian Strategy What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

11. Mixing and Parameterization Introduce some uniformly sampled points Sans mixing, inappropriate for simple regions Parameters Variance of normal (smaller = closer to obstacles) Mixing rate S G

12. Overview Gaussian Strategy What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

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

14. Narrow Passage uniform sampling took less time than algorithm 2

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

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

17. Overview Gaussian Strategy What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

18. 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

19. Overview Gaussian Strategy What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

20. 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, < 1) If points can be reused in a hybrid strategy Quality of sampled points Let p be prior probability of Free Assume I(p i,p j ) for i  j

21. Bridge vs. Gaussian StrategyCalls to Free Expected # Samples Reuse Points Point Quality Gaussian 12 2  p  (1-p) yes, tainted OK Gaussian 23 - p 2 3  p  (1-p) 2 yes, tainted Better Bridge1 + (1-p) + (1-p) 2 p  (1-p) 2 yesBest

22. Overview Gaussian Strategy What is Desired Two Proposed Algorithms Mixing and Parameterization Experimental Results Bridge Test Proposed Algorithm Comparison with Previous Paper Experimental Results

23. Clover

24. Two Squares

25. Depression

26. Zigzags

27. Bridge vs. Uniform RBB = Bridge

28. 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