Finding Narrow Passages with Probabilistic Roadmaps The SmallStep

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

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

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

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

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

Main Issue: “Narrow Passages” 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 6

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

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

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

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

start Small-Step Retraction Method Roadmap construction and repair Fattening goal c-obstacle free space start goal fattened free space Repair during construction Repair after construction fattened free space widened passage 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 11

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

Quantitative Results • Fattening “preserves” topology/ connectivity of the free space (a) (c) (b) Alpha 1. 0 (d) Our planner Alpha 1. 1 Time SBL (secs) (a) 9. 4 12295 (b) 32 5955 (c) 2. 1 41 (d) 492 863 (e) 65 631 (f) 13588 >100000 A recent PRM planner (h) (g) (f) (e) Time SSRP (secs) • Fattening “alters” the topology/ connectivity of the free space Time SSRP (secs) Time SBL (secs) (g) 386 572 (h) 3365 >100000 Upto 3 orders of magnitude improvement in the planning time was observed 13

Quantitative Results • Test environments “without” narrow passages – SSRP and SBL have similar performance (i) (j) Time SSRP Time SBL (i) 1. 68 1. 60 (j) 2. 59 2. 40 14

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 15

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