1. On Delaying Collision Checking in PRM Planning – Application to Multi-Robot Coordination By: Gildardo Sanchez and Jean-Claude Latombe Presented by: Michael Graeb and Samir Menon

2. Delayed Collision Checking Motivations Experimental Foundations: Collision checks removed from planner  Improved efficiency by 2-3 orders of magnitude  Most paths remained collision free Most time is spent checking connections Short connections likely to be collision-free anyway Most collision-free connections not part of final path Hence: Postpone testing a connection until it is absolutely needed

3. Single Query Motivations ◦ 90% - 99.9% of milestones in multi-query roadmap unused by final path. ◦ Most roadmaps with “good coverage” only used for a single task Hence: Single Query ◦ Build a roadmap with your specific tasks in mind ◦ Bi-Directional trees are an efficient query technique S G

4. SBL Algorithm – Overall 1. Start roadmap with two trees Rooted at start and goal, one node each 2. Try s times…

5. SBL Algorithm – Overall 2a) Grow a tree by one node

6. SBL Algorithm – Overall 2b) Find a path from start to goal ◦ We’re not certain all edges are valid Start Goal

7. SBL Algorithm – Overall 2c) Test unknown edges in path ◦ Stop once a collision is found ◦ Remember edges’ validity, for future use

8. SBL Algorithm – Overall 2c) Test unknown edges in path ◦ Stop once a collision is found ◦ Remember edges’ validity, for future use

9. SBL Algorithm – Overall 2d) If all edges valid, return path

10. SBL Algorithm - EXPAND 1. Pick which tree, T, will receive new milestone Uniformly random choice, 50/50 odds

11. SBL Algorithm - EXPAND 2. From T, pick an existing milestone, m  Random choice, with milestones in less-dense regions more likely to be picked

12. SBL Algorithm - EXPAND 3. Repeat until new milestone created: 3a) Take sample in neighborhood of m  Distance from m is initially p and shrinks with each successive attempt 3b) If sample collision free, add it as child of m

13. SBL Algorithm - EXPAND 3. Repeat until new milestone created: 3a) Take sample in neighborhood of m  Distance from m is initially p and shrinks with each successive attempt 3b) If sample collision free, add it as child of m

14. SBL Algorithm - EXPAND 3. Repeat until new milestone created: 3a) Take sample in neighborhood of m  Distance from m is initially p and shrinks with each successive attempt 3b) If sample collision free, add it as child of m

15. SBL Algorithm - EXPAND 3. Repeat until new milestone created: 3a) Take sample in neighborhood of m  Distance from m is initially p and shrinks with each successive attempt 3b) If sample collision free, add it as child of m

16. SBL Algorithm - EXPAND 3. Repeat until new milestone created: 3a) Take sample in neighborhood of m  Distance from m is initially p and shrinks with each successive attempt 3b) If sample collision free, add it as child of m

17. SBL Algorithm - EXPAND 3. Repeat until new milestone created: 3a) Take sample in neighborhood of m  Distance from m is initially p and shrinks with each successive attempt 3b) If sample collision free, add it as child of m

18. SBL Algorithm - EXPAND 3. Repeat until new milestone created: 3a) Take sample in neighborhood of m  Distance from m is initially p and shrinks with each successive attempt 3b) If sample collision free, add it as child of m

19. SBL Algorithm - CONNECT 1. m is new milestone 2. m’ is other tree’s nearest milestone to m

20. SBL Algorithm - CONNECT 3. If( distance(m, m’) < ρ ) 3.1 Connect m and m’ by bridge

21. SBL Algorithm - CONNECT 3. If( distance(m, m’) < ρ ) 3.1 Connect m and m’ by bridge

22. … 3.2 τ is path from start to goal 3.3 Return results of TEST_PATH( τ ) SBL Algorithm - CONNECT

23. SBL Algorithm – TEST_PATH(path) Continually test “most unsafe” segment until we we encounter a collision, or all segments are known to be safe. Save results for future use.

24. Results Results for configuration shown in the figure

25. Performance Evaluation & Convergence Rate

26. Comparative Performance Evaluation Experimental results for the full collision- check planner

27. Discussion Assumes two spatially close configurations in configuration space have low probability of collision Saves time by checking collision between milestones only when part of candidate path from start to goal. Valid assumption in practice & supported by experiments