1. On Delaying Collision Checking in PRM Planning G. Sánchez and J. Latombe presented by Niloy J. Mitra

2. Background PRM spends a large amount of time checking for collisions Most connection paths go unused Collision free paths take longer to verify Shorter connections have higher probability of being free paths

3. Improvement possible by Introducing better collision checking Designing smarter sampling strategies Avoiding testing all connections between milestones

4. SBL Planner Single-query Bi-directional Lazy collision-checking

5. Experimental Foundation Observations from Hsu’s planner led to SBL: Most local paths are not on the final path Collision-free tests are most expensive Short connections between two milestones have high prior probabilities of being free If a connection is colliding, it’s midpoint has high probability of being in collision

6. Short connections likely to be collision-free

7. If a connection is colliding, it’s midpoint likely to collide

8. “Fat Obstacles” A short colliding segment with collision free endpoints is necessarily almost tangential to an obstacle region in C, an event that has small probability of happening.

9. Description of the SBL Planner SBL Algorithm 1. Install q init and q goal as the roots of T init and T goal respectively 2. Repeat s times 1. EXPAND 2. τ ← CONNECT 3. If τ ≠ nil then return τ 3. Return failure

10. EXPAND EXPAND Algorithm 1. Pick T to be either T init or T goal, each with P=0.5 2. Pick a milestone m at random, with P π (m) ~ 1/ η (m) 3. For i = 1,2,… until a new q been generated 1. Pick a configuration q uniformly at random from B(m, ρ/i) 2. If q is collision-free, then install it as a child of m in T

11. Diffusion with a Grid Without diffusionWith diffusion

12. CONNECT CONNECT Algorithm 1. m ← most recently created milestone 2. m’ ← closest milestone to m in the other tree 3. If d(m,m’) < ρ then 1. Connect m and m’ by a bridge w 2. τ ← path connecting q init and q goal 3. Return TEST-PATH 4. Return nil

13. SBL Example q init q goal

14. N robot = 5,000; N obst = 83,000 T av = 4.42 s N robot = 3,000; N obst = 50,000 T av = 0.17 s Some Examples

15. Impact of Lazy Collision Checking Average performance with lazy collision checking Average performance without lazy collision checking  Speed-ups ranging from 4 to 40

16. Some More Examples

17. More Examples

18. Obstacle Jumping Example q init q goal

19. Comments Simple algorithm Big speed gain