1. 1 Finding “Narrow Passages” with Probabilistic Roadmaps: The Small-Step Retraction Method Mitul Saha and Jean-Claude Latombe Research supported by NSF, ABB and GM Artificial Intelligence Lab Stanford University

2. 2 Probabilistic Roadmaps (PRM) [Kavraki, Svetska, Latombe, Overmars, 1996] start configuration goal configuration free-space c-obstacle Configuration-space components milestone local path Roadmap components

3. 3 PRM planners solve complicated problems Complex geometries: obstacles: 43530 polygons Robot: 4053 polygons High dimensional

4. 4 Main Issue: “Narrow Passages” free samples colliding samples colliding local path narrow passage low density of free samples high density of free samples The efficiency of PRM planners drops dramatically in spaces with narrow passages

5. 5 Problems with “narrow passages” are commonly encountered Main Issue: “Narrow Passages”

6. 6 ? Proposed strategies:  Filtering strategies, e.g., Gaussian sampling [Boor et al. ‘99] and bridge test [Hsu et al. ‘03]  rely heavily on rejection sampling  Retraction strategies, e.g., [Wilmart et al. ‘99][Lien et al. ‘03]  waste time moving many configurations out of collision

7. 7 Motivating Observation decreasing width of the narrow passage planning time easy narrow passages difficult narrow passages

8. 8 Roadmap construction and repair fattened free space widened passage Fattening free space c-obstacle start goal Small-Step Retraction Method 1.Slightly fatten the robot’s free space 2.Construct a roadmap in fattened free space 3.Repair the roadmap into original free space (1) (2 & 3)

9. 9 Small-Step Retraction Method Roadmap construction and repair fattened free space widened passage Fattening free space c-obstacle start goal -Free space can be “indirectly” fattened by reducing the scale of the geometries (usually of the robot) in the 3D workcell with respect to their medial axis -This can be pushed into the pre-processing phase

10. 10 Small-Step Retraction Method Roadmap construction and repair fattened free space widened passage Fattening free space c-obstacle start goal Repair during construction Repair after construction goal Pessimist Strategy Optimist Strategy fattened free space start

11. 11 Small-Step Retraction Method Roadmap construction and repair fattened free space widened passage Fattening free space c-obstacle start goal Repair during construction Repair after construction fattened free space goal Pessimist Strategy Optimist Strategy - Optimist may fail due to “false passages” but Pessimist is probabilistically complete - Hence Optimist is less reliable, but much faster due to its lazy strategy start

12. 12 Small-Step Retraction Method Roadmap construction and repair fattened free space widened passage Fattening free space c-obstacle start goal Repair during construction Repair after construction goal Pessimist Strategy Optimist Strategy  Integrated planner: 1. Try Optimist for N time. 2. If Optimist fails, then run Pessimist fattened free space start

13. 13 Quantitative Results Fattening “preserves” topology/ connectivity of the free space Fattening “alters” the topology/ connectivity of the free space Time SSRP (secs) Time SBL (secs) (a)9.412295 (b)325955 (c)2.141 (d)492863 (e)65631 (f)13588>100000 Time SSRP (secs) Time SBL (secs) (g)386572 (h)3365>100000 (a) (b) (c) (d) (e) (f) (g) (h) Alpha 1.0 Alpha 1.1 Upto 3 orders of magnitude improvement in the planning time was observed Our planner A recent PRM planner

14. 14 Quantitative Results Test environments “without” narrow passages –SSRP and SBL have similar performance Time SSRP Time SBL (i)1.681.60 (j)2.592.40 (i) (j)

15. 15 Conclusion SSRP is very efficient at finding narrow passages and still works well when there is none The main drawback is that there is an additional pre-computation step

16. 16 Finding “Narrow Passages” with Probabilistic Roadmaps: The Small-Step Retraction Method