1. Filtering Sampling Strategies: Gaussian Sampling and Bridge Test Valerie Boor, Mark H. Overmars and A. Frank van der Stappen Presented by Qi-xing Huang

2. Motivation 1. Connectivity checks between milestones are expensive.  Provide coverage with fewest possible milestones. 2. Collision checks to create milestones are cheap.  Take many samples, keep only the best.  Main idea: Sample many configurations, but retain only a small subset of promising ones

3. The Gaussian Sampling Strategy For Probabilistic Roadmap Planners Valerie Boor, Mark H. Overmars and A. Frank van der Stappen Presented by Qi-xing Huang

4. Motivation  Narrow passages are always close to the free space boundary  Goal: Identify and retain configurations sampled near the free space boundary Uniform Random SamplerThe Sampler we want

5. A Favorable Sample Distribution  The probability that a sample is added to the graph to depends on the amount of forbidden configurations nearby

6. Gaussian Sampler  1.Loop  2. a random configuration  3. a distance chosen according to a normal distribution  4. a random conf. at distance from  5. If and then  6. add to the graph  7.else if and then  8. add to the graph  9. else  10. discard both

7. Gaussian Sampler  1.Loop  2. a random configuration  3. a distance chosen according to a normal distribution  4. a random conf. at distance from  5. If and then  6. add to the graph  7.else if and then  8. add to the graph  9. else  10. discard both

8. Gaussian Sampler  1.Loop  2. a random configuration  3. a distance chosen according to a normal distribution  4. a random conf. at distance from  5. If and then  6. add to the graph  7.else if and then  8. add to the graph  9. else  10. discard both

9. Gaussian Sampler  1.Loop  2. a random configuration  3. a distance chosen according to a normal distribution  4. a random conf. at distance from  5. If and then  6. add to the graph  7.else if and then  8. add to the graph  9. else  10. discard both

10. Gaussian Sampler  1.Loop  2. a random configuration  3. a distance chosen according to a normal distribution  4. a random conf. at distance from  5. If and then  6. add to the graph  7.else if and then  8. add to the graph  9. else  10. discard both

11. Effect of the Parameter  If we choose a very small standard deviation  Require a lot of samples to generate sufficient number of surviving samples  If we choose a very large standard deviation  Almost uniformly distributed  A lot of surviving samples are redundant.  Tune the parameter based on the setting of the specific problem.

12. Experimental Results  A U-shaped robot has to twist to get through the narrow gap in the center.  The random sampler (which required10000 nodes) took about 13 times longer than the Gaussian sampler(which only required 750 nodes).

13. A more Complicated Example  5000(intersecting) obstacles.  Gaussian sampler needed 85 nodes to connect start and goal.  4 times as fast as the Random sampler, which required over 450 nodes.

14. The Bridge Test for Sampling Narrow Passages with Probabilistic Roadmap Planners David Hsu, Tingting Jiang, John Reif, and Zheng Sun Presented by Michael Graeb

15.  Review:  Provide coverage with fewest possible milestones.  Take many samples, keep only the best.  Gaussian Sampling: Best samples are near boundaries  Bridge Test: Milestones in narrow passages important.  Not all milestones near boundaries increase coverage. Motivations Gaussian Sampler Bridge Test Uniform Sampler

16. while( … ) pick a point x from configuration space at random if ( CLEARANCE( x ) == false ) pick a point x’ in the neighborhood of x if ( CLEARANCE( x’ ) == false ) point m is midpoint of x and x’ if ( CLEARANCE( m ) == true ) add m as new milestone The Bridge Test: Create samples in narrow passages

17. while( … ) pick a point x from configuration space at random if ( CLEARANCE( x ) == false ) pick a point x’ in the neighborhood of x if ( CLEARANCE( x’ ) == false ) point m is midpoint of x and x’ if ( CLEARANCE( m ) == true ) add m as new milestone The Bridge Test: Create samples in narrow passages

18. while( … ) pick a point x from configuration space at random if ( CLEARANCE( x ) == false ) pick a point x’ in the neighborhood of x if ( CLEARANCE( x’ ) == false ) point m is midpoint of x and x’ if ( CLEARANCE( m ) == true ) add m as new milestone The Bridge Test: Create samples in narrow passages

19. while( … ) pick a point x from configuration space at random if ( CLEARANCE( x ) == false ) pick a point x’ in the neighborhood of x if ( CLEARANCE( x’ ) == false ) point m is midpoint of x and x’ if ( CLEARANCE( m ) == true ) add m as new milestone The Bridge Test: Create samples in narrow passages

20. Bridge Test: Examples  4 loops of the algorithm, producing only 1 milestone

21.  x: uniformly random distribution  keep if CLEARANCE(x) == false Bridge Test: Distribution of Samples

22.  x: uniformly random distribution  keep if CLEARANCE(x) == false

23. Bridge Test: Distribution of Samples  x’: gaussian distribution in neighborhood of x  keep if CLEARANCE(x’) == false

24. Bridge Test: Distribution of Samples  x’: gaussian distribution in neighborhood of x  keep if CLEARANCE(x’) == false

25. Bridge Test: Distribution of Samples  m: midpoint of x and associated x’  keep if CLEARANCE(m) == true

26. Bridge Test: Distribution of Samples  m: midpoint of x and associated x’  keep if CLEARANCE(m) == true

27. Bridge Test: Examples 8 joint robot with mobile base ? → Point robot

28. Bridge Test: Complementary sampling  Bridge Test reliably provides milestones in narrow passages  Uniform sampling reliably provides milestones in open spaces.  A union of results from each algorithm provides good coverage with minimal milestones.  Issue: Number of milestones allotted to each algorithm. =U

29. Bridge Test: Complementary sampling  Bridge Test reliably provides milestones in narrow passages  Uniform sampling reliably provides milestones in open spaces.  A union of results from each algorithm provides good coverage with minimal milestones.  Issue: Number of milestones allotted to each algorithm. =U

30. Bridge Test: Complementary sampling Bridge TestUniform Sampling Union Solution

31. Bridge Test Hybrid: Bridge + Gaussian + Uniform 1)Sample two configurations q and q’ using Gaussian sampling technique 2)If both are in free space, then retain one as a milestone with low probability (e.g., 0.1) 3)Else if only one is in free space, then retain it as a node with intermediate probability (e.g., 0.5) 4)Else if qm = Midpoint(q, q’) is in free space, then retain it as a node with probability 1

32. Bridge Test: Results

33. Gaussian Sampler & Bridge Test: Conclusions  Connection tests are expensive, reducing them is key to faster runtimes.  We can exploit properties of the configuration space to provide better milestones.  Various algorithms have different strengths and weaknesses, don’t be afraid to mix and match.